The Hipparcos Mission (1)

THE HIPPARCOS MISSION
FLOOR VAN LEEUWEN
Royal Greenwich Observatory, Madingley Road, Cambridge, UK
E-mail address: [email protected]
(Received 22 October, 1997)
Abstract. A review is presented of the European Space Agency’s astrometric satellite project Hipparcos, for which the final data catalogues were published in June 1997. The emphasis is on those
aspects that have or may have influenced the data as presented in the catalogues. It presents a brief
review of the satellite, the aims of the mission with their relation to ground-based astrometry, and
the mission history. This is followed by a description of the input data streams and a summary of
the data reductions. Data files presented in the catalogues are described in the context of the data
reductions, and explained in content and usage. The mission results comprise not only outstanding
astrometric data on both single and double stars, but also an unique all-sky photometric survey which
has been used for variability investigations. This has led to the discovery of thousands of variable
stars. The Hipparcos mission was complemented by the Tycho experiment, providing a complete
all-sky survey of astrometric and photometric parameters for one million stars down to magnitude
11, though with lower accuracies than obtained for the main mission. Astrometric and photometric
data for a selection of 48 minor planets, the Jovian moon Europa and the Saturnian moons Titan and
Iapetus were also obtained. The data quality verifications are reviewed and guidelines to the proper
use of the Hipparcos data are provided, followed by some of the first scientific results of the mission.
Experiences gained with this first ever space astrometry mission are considered in relation to a follow
up mission for Hipparcos.
Parallaxes measure stellar distances,
other methods rely on assumptions.
Key words: Data reduction techniques, Double Stars, Hipparcos, Satellite dynamics, Space Astrometry, Tycho, Variable Stars
Abbreviations: ABM – Apogee Boost Motor; CCCM – Coil Current Calibration Matrix; CERGA –
Centre d’Etudes et de Recherches Géodynamiques et Astronomiques; DMSA – Double and Multiple
Systems Annex; ESOC – European Space Operations Centre; FAST – Fundamental Astronomy by
Space Techniques; FFOV – Following Field of View; GCR – Great-Circle Reduction; IDT – Image
Dissector Tube; IFOV – Instantaneous Field of View; ITF – Intensity Transfer Function; INCA – Input
Catalogue Consortium; mas – milliarcsecond; NDAC – Northern Data Analysis Consortium; OTF –
Optical Transfer Function; PFOV – Preceding Field of View; SM – Star Mapper; RGC – Reference
Great Circle; RGO – Royal Greenwich Observatory; TDAC – Tycho Data Analysis Consortium.
1. Introduction
With the rapid advances of astrophysics over the past 100 years there has been an
increasing need for improvements in the quality and quantity of the most fundamental astronomical observations: the measurements of positions and trigonometric
distances, or parallaxes, of stars. The practical limits for such observations from the
Space Science Reviews 81: 201–409, 1997.
c 1997 Kluwer Academic Publishers. Printed in Belgium.
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F. VAN LEEUWEN
ground seemed to have been reached, however, and these limits were no longer providing adequate constraints for the researches in stellar structure, stellar evolution
and the Galactic distance scale, which form the basis for the extragalactic distance
scale and cosmology (see also Kovalevsky, 1984). Moreover, ground-based aquisition of astrometric data is a time consuming and laborious process, relying on
long-term investments. It is the kind of work for which there is, sadly, little space or
provision in today’s world. More recently, various techniques have been developed
that do allow very accurate relative astrometric measurements to be obtained from
the ground, but still only in very small quantities (Zacharias, 1997, Harris et al.,
1997).
1.1. PARALLAXES
The measurement of parallaxes from the ground has always been very difficult.
The traditional method made use of photographic plates, which, in order to get the
maximum parallactic displacements, had to be taken at far from ideal conditions,
away from the meridian and close to Sun-set or Sun-rise. In addition, the parallactic
displacements could only be determined relative to a selection of reference stars on
the same plate. The photographic plates used for this kind of work were exposed
with very long focal length instruments, covering only a very small area of the
sky on a plate. For example the refractors of Allegheny, McCormick, Sproul,
Yale and Yerkes, which were used for many parallax determinations, all had focal
lengths of 10 to 20 m. In a small area of the sky, all stars share the same parallactic
displacement direction, while the sizes of the displacements depend on the distances
of the individual stars. In the plate reductions, the reference stars were therefore also
subject to the parallactic displacements, and this led to the fact that ground based
parallax determinations always produced a measurement of the parallax relative
to the average parallax of the background stars. An estimate of this background
parallax gave an approximation for the correction to an absolute parallax. Such
estimates were, however, always difficult and uncertain. It was not uncommon to see
different parallax determinations for the same star with much larger differences than
the precisions indicated for these determinations (for further details and references
see Jenkins, 1952, Jenkins, 1963, Upgren, 1977, Gliese & Upgren, 1990, van Altena
et al., 1995).
Accuracies obtained for ground based parallaxes were at best of the order of
0.008 arcsec. More recently, using CCD technology, significantly higher reliability
(to a level of 0.002 arcsec) has been obtained for small numbers of ground based
parallax determinations for faint stars (Monet et al., 1992). The accuracies of the
parallax determinations can be translated into accuracies of distance moduli (but
note remarks made in Section 11) as well as in a distance range within which
distance moduli can effectively be determined. Relative errors on the distances of
less than 10 per cent were only reached for a very small number of objects, and
even such accuracies gave uncertainties in the distance moduli for those stars at a
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203
level of 0.1 to 0.2 mag, compared with accuracies of photometric measurements at
a level of 0.001 mag. The accuracies of the parallaxes limited the distance range for
useful measurements to a sphere with a radius of around 12 pc (0.2 mag accuracy
in distance moduli, some 250 to 300 stars) to 30 pc (0.5 mag accuracy in distance
modulus, some 4000 stars) from the Sun. The European Space Agency’s astrometric satellite Hipparcos has improved this situation dramatically. The parallax
accuracies were improved by a factor 6 to 8 with respect to average ground-based
accuracies, leading to a 100 to 1000 times larger volume within which significant
parallaxes could be measured, and to a very significant increase in the numbers of
stars available for distance determinations. Within its limiting magnitude, parallax
measurements with accuracies better than 10 per cent were obtained by Hipparcos
for around 20 000 stars.
1.2. ABSOLUTE POSITIONS
Absolute positions of stars used to depend completely on meridian observations
taken over long periods of time. This determined the stellar positions and hence a
reference frame for a given epoch as well as the stars’ proper motions, all linked
to the definitions of the Earth rotation (time) and precession of the equinoxes.
The most recent such reference frame is known as J2000, and its realization is
provided by a selection of 1535 bright stars for which the astrometric parameters
were presented in the FK5 Catalogue (Fricke et al., 1989). Its positions and proper
motions were derived from meridian observations that had been obtained with a
wide range of manually operated instruments and many different observers. Some
of the problems with the FK5 positions were exposed in the results of the first
automated meridian instruments (Morrison et al., 1990), and the overall systematic
errors in positions and proper motions became very clear after the comparison
between FK5 and Hipparcos positions (ESA, 1997). In particular, in the southern
hemisphere there were local discrepancies of several tenths of an arcsec. This is
an unacceptable level for comparisons between radio maps and optical images
obtained with the Hubble Space Telescope. Positional accuracies for radio maps
can reach levels of 1 to 10 mas, and optical images from the Hubble space telescope
have a resolution of around 0.1 arcsec. An error of a few 100 mas in positional
alignment can obscure the proper physical explanation of the observations.
1.3. THE EMERGING OF HIPPARCOS
In the 1960s P. Lacroute in France developed an alternative technology for measuring parallaxes, with particular emphasis on obtaining absolute rather than relative
values. The basic idea was to combine the images from two areas on the sky onto
one detector, and to do this at different times of the year. The parallactic displacements for the two fields would be uncorrelated and therefore allow the elimination
of the parallax zero point problem. It was soon realized that such technology could
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F. VAN LEEUWEN
not be employed successfully from the ground, as it required a very well determined and stable angle between the two fields of view, and viewing directions from
the ground are affected by a variable refraction. The first ideas of implementing
this technique in a space mission were submitted in France in 1966 and 1967. The
implementation appeared too complex at that time, in particular with respect to the
construction of the beam combiner, and in 1970 further studies were stopped (see
Lacroute, 1982).
The basic idea, however, survived and with the input of several new ideas
(many originating from E. Høg, Copenhagen Observatory), the concept of a space
astrometry mission became more and more realistic. A meeting on space astrometry
organized in 1974 in Frascati by the European Space Agency, ESA, marked the
start of what was to develop into the Hipparcos project. Its official status as a phase
A study by ESA was obtained in 1976, and it was approved as an ESA project in
1980.
Hipparcos is both an acronym and a dedication to the Greek astronomer Hipparchus. As an acronym it stands for “HIgh Precision PARallax COllecting Satellite”. Hipparchus, who lived in the 2nd century BC, is generally associated with
compiling the first catalogue of star positions from records of Babylonian observations as well as from some observations probably done under his auspices.
At a very early stage of the first studies the aims of the mission were set.
The astrometric data would be obtained with a scanning satellite over a nominal
2.5 year period, using an input catalogue of approximately 100 000 stars evenly
distributed over the sky (Høg, 1979). The 2.5 year period was required for a
reliable disentanglement of proper motions and parallaxes. The even distribution
and the density of stars were needed to provide distortion free rigidity to the
positional reference frame. The accuracy of the positions and parallaxes should
be 0.002 arcsec, and for the proper motions 0.002 arcsec yr,1 , in order to make
a sufficient impact with respect to the ground-based data. These clear aims were
a driving force behind many of the design features of the spacecraft and payload.
They required a mechanically (damping of vibrations) and thermally very stable
system, and optics polished to very high precision. The original idea of Lacroute
was incorporated in the two fields of view, projecting on the same detector. The
critical element in the optical system was the “beam combiner”, a polished mirror
which was cut in two halves and glued back at an angle, equivalent to half the angle
between the areas on the sky as seen through the two fields of view. This larger
angle, very close to 58 is referred to as the basic angle. The stability of the beam
combiner determined the stability of the basic angle, an important criterion in the
design of the payload.
This was not the first time an optical element cut in halves was used for triangularization of the sky: in the early 19th century Bessel (1841) used for obtaining
the most accurate relative positional measurements of those days a heliometer,
an instrument in which the objective was cut in two. The intersection of the two
halves of the objective was aligned with the great circle through a pair of stars. One
THE HIPPARCOS MISSION
205
element of the cut lens could be shifted such that the images of the two stars would
coincide. The measurement of the shift produced, like Hipparcos 150 years later, a
one dimensional positional measurement, the angular separation of the stars. The
measurements by Bessel of 61 Cyg provided the first reliable determination of the
parallax of a star. His measurements on stars in the Pleiades cluster formed the first
epoch of the 19th century astrometric studies of this cluster, which led to the distinction between field stars and cluster members (Lagrula, 1903) and the discovery of
the colour magnitude relation for stars by Hertzsprung (1907). Hipparcos returned
to the Pleiades cluster, and determined its distance (van Leeuwen & Hansen-Ruiz,
1997). A future astrometry mission, such as described in the GAIA concept (Lindegren & Perryman, 1995), will be able to determine the 3-dimensional structure
of the cluster (van Leeuwen, 1995).
The detector system for Hipparcos was in many ways similar to that which
is used in meridian instruments. It is based on photoelectric measurements in
conjunction with a slit system (Høg, 1960, Lindegren, 1979b). The purpose of both
types of instrument is the same, i.e. the determination of an accurate transit time. In
the case of the meridian telescope the images pass through a set of slits as a result
of the Earth’s rotation; for the Hipparcos satellite, the passages were the result of
the rotation of the satellite. The near synchronicity of measurements obtained in
the two fields of view together with the scanning of the satellite made it possible to
obtain a very accurate calibration of the basic angle, and thereby transforming very
precise relative separations in the combined fields of view to very precise large
angular separations on the sky. And this was the original aim of the proposals by
Lacroute, which made it possible to determine absolute parallaxes.
A requirement for the calibration of the basic angle was a smooth rotation of the
satellite. In order to simplify the effects of solar radiation torques, the outer structure
of the satellite was kept, in as far as possible, free from protruding structures.
The satellite’s attitude control, which monitored and corrected the rotations of the
satellite axes, was obtained through gyro and star mapper measurements for control
and by cold-gas thrusters, producing very short bursts that changed rotation rates,
for corrections.
The measurements of large angles over the sky provided the means to build a
very rigid positional frame. This frame was, however, “free floating”, not connected
to other independent reference points. This is different from ground-based measurements which are always linked to the Earth’s rotation parameters: the rotation
phase and the orientation in space of the rotation axis. The Hipparcos reference
frame had to be “attached” to another very accurate earthbound reference frame
in order to transform its precise positions into accurate positions. Several special
programs were set up for this purpose. Most of these programs aimed at linking
stellar positions to those of extragalactic sources.
Very accurate positions of radio sources, including a small number of radio
stars of which the optical counterparts were included in the Hipparcos catalogue,
were obtained by means of large interferometer arrays (Lestrade et al., 1992, 1995,
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F. VAN LEEUWEN
Morrison et al., 1997a). The International Celestial Reference System (ICRS) was
established on the basis of these radio positions and proper motions. A complete list
of the radio sources used to define the ICRS was presented by Kovalevsky (1996).
The ICRS is inertial through its linking to extragalactic radio sources and provides
a link to the Hipparcos catalogue through a small overlap in the objects. Other
methods, linking the Hipparcos proper motions and positions to the extragalactic
reference frame using photographic surveys (Brosche & Geffert, 1988) or through
the use of the Hubble Space Telescope (Hemenway et al., 1988) have also been
used. The Hipparcos reference frame, after careful transformations, is now the
optical realization of the ICRS (Lindegren & Kovalevsky, 1995, Kovalevsky et al.,
1997, Section 8.7.4).
1.4. DATA HANDLING
The handling of the data reductions was considered a major problem in the early
developments. All observations were interconnected, and it appeared that the whole
system needed to be solved in one go from the original phase estimates for the
stellar transits to the astrometric parameters. This was far beyond the possibilities
of the available computer technology. In 1976, L. Lindegren (Lund) designed a
reduction system in which the solution was split into three parts, each of which
was possible to handle. There was a theoretical small loss of accuracy associated
with some approximations that had to be made, but these methods made Hipparcos
possible in terms of data reductions. ESA, recognizing both the challenges in
the hardware design and the difficulties in the data reductions, decided when the
project was approved in March 1980, that all data reductions should be handled
by the astronomical community in Europe. It was also decided, that, as there was
no independent material available with which to compare Hipparcos results, it
would be beneficial to the project if more than one group carried out the full data
reductions. Two consortia applied for and were entrusted with the data-reductions
responsibility: the FAST consortium, based in France, Italy, Germany and the
Netherlands, and led by J. Kovalevsky, and the NDAC consortium, based in Sweden,
Denmark and the UK, led initially by E. Høg and later by L. Lindegren. A third
consortium, led by C. Turon, took the responsibility for the preparation of the Input
Catalogue.
1.5. PRE-LAUNCH DEVELOPMENTS
After the approval of the project, a second experiment was proposed by Høg et al.
(1982). Using the signal obtained from the star mapper detectors, whose primary
purpose was to help keep control of the satellite attitude, a complete survey of the
sky down to magnitude 11 could be obtained. By splitting the star mapper light into
two channels, roughly resembling the V and B bands from the Johnson system, the
survey would be supplemented with photometric information that would also be
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207
useful to the main mission. This parallel mission was named Tycho, after the Danish
16th century astronomer Tycho Brahe (1546-1601), who not only discovered the
nova in Cassiopeia in 1572, but also compiled the most accurate pre-telescope era
catalogue of positions for some 800 stars, using a very large quadrant. A fourth
consortium, TDAC, based in Denmark and Germany, took the responsibility for
the Tycho data reductions and was led by E. Høg. The Tycho astrometry was of
lower accuracy than that obtained with Hipparcos, but the completeness and the
density of stars in the survey (an average of 25 stars per square degree) offered
many possibilities for further research.
The years up to 1989 were spent by the data-reduction consortia on the development of the computer codes for the data reductions and the interfaces with the
data delivery. Extensive technological developments took place at the same time,
but these are not covered by the current review (see instead Volume I of ESA, 1989
and Volume 2 of ESA, 1997). Several experiments were done using simulated
satellite data created at ESTEC, RGO and CERGA. The data-reduction results of
these simulated data provided the input for the first comparison exercises, and the
first quality checks on the data reductions. These comparisons were going to be a
regular feature of the data reductions during the mission, and contributed much to
the outstanding quality of the final data.
1.6. POST-LAUNCH DEVELOPMENTS
On August 8, 1989 the satellite was launched successfully into a geostationary
transfer orbit by an Ariane 4 rocket (launch V33). Soon after, problems started to
develop when the apogee boost motor (ABM) failed to react on firing commands
and left the satellite stuck in this highly elliptical orbit. This caused many problems:
three ground stations were required to keep sufficient contact with the satellite
(see Figure 1); during the low perigee passages no contact was possible and no
observations could be made, often leading to loss of attitude control by the satellite;
the increased torques working on the satellite close to the perigee passage led to
short periods of relatively high usage of the cold-gas thrusters, shortening the lifetime of the satellite; in this orbit, the satellite crossed the Van Allen radiation belts
about once every five hours, leading to a gradual deterioration of solar panels and
electronic equipment. Despite pessimistic estimates in the autumn of 1989 (see for
example Waldrop, 1989, Neto, 1989) about the possible survival of the satellite
in this environment, it operated for 3.5 years and obtained results about 1.5 to 2
times better than the original aims (Perryman et al., 1997a). Radiation damage to
the on-board computer and the gyro electronics resulted in the termination of the
observations in March 1993 and of the mission in August 1993.
Behind the ultimate success of the mission was the enormous dedication from the
ground staff at ESOC (led by D. Heger) and the data-reduction consortia, the support
of NASA who made a receiver available at Goldstone, and last but not least the
commitment of the Hipparcos Science Team led by the Hipparcos project scientist
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Figure 1. The position of the satellite in the final orbit plotted relative to a stationary Earth. Points
in the orbit are drawn at equal time intervals. The just over 9 successive orbits shown are numbered.
Orbit 10 almost coincides with orbit 1. The ground stations are indicated by their initials: Odenwald,
Perth, Kourou and Goldstone. For each ground station the horizon is drawn as a light grey line,
showing the limited visibility of the satellite under various combinations of ground support.
M.A.C. Perryman (see Table I). The close collaboration between the data reduction
consortia and the ground staff at ESOC led to rapid recognition of any developing
problems (first look analysis, Schrijver, 1985) and to improved calibrations for
payload and spacecraft (gyro and torque calibrations done at RGO). The RGO was
also closely involved in developments to keep the satellite in operation despite a
break-down of all gyros. As a result of the additional ground stations and the ever
improving ability at ESOC to recover the satellite’s attitude, the final data return
was still approximately 65 to 70 per cent.
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Table I
The Hipparcos Science Team during the mission operations
phase.
U. Bastian
P.L. Bernacca
M. Crézé
F. Donati
M. Grenon
M. Grewing
E. Høg
J. Kovalevsky
F. van Leeuwen
L. Lindegren
H. van der Marel
F. Mignard
C.A. Murray
M.A.C. Perryman
R.S. Le Poole
H. Schrijver
C. Turon
Heidelberg
Asiago
Strasbourg
Torino
Genève
Tübingen
Copenhagen
Grasse
Cambridge
Lund
Delft
Grasse
Eastbourne
ESTEC
Leiden
Utrecht
Paris-Meudon
(D)
(I)
(F)
(I)
(CH)
(D)
(DK)
(F)
(UK)
(S)
(NL)
(F)
(UK)
(NL)
(NL)
(NL)
(F)
from 1993
from 1989
from 1983
from 1981
from 1982
from 1983
from 1981
from 1981
from 1986
from 1981
from 1987
from 1991
from 1981
from 1981
from 1981
from 1990
from 1981
Table II
Orbit characteristics.
Element
Mean value
Change/yr
Variation
Units
38340
24582
0.7196
6890
0.37
-20.4
-8.2
0.000
-3.0
4.5
2.5
0.005
140
0.07
s
km
km
/day
Period
Semi-major axis
Eccentricity
Perigee
Precession
1.7. THE OPERATIONAL ORBIT
Using the thrusters and fuel that was intended for moving the satellite to its correct
geostationary position, the perigee height was increased to move the satellite away
from the outer layers of the Earth’s atmosphere. The remaining fuel was drained
to avoid sloshing. An orbital period was obtained such that regular patterns would
occur in ground-station usage and occultations. The final orbital period was close
to 5 rotational periods of the satellite, and 9 orbital periods spanned almost exactly
4 days (see Figure 1). The average orbital parameters are summarized in Table II.
The data reduction requirements called for instantaneous knowledge of the satellite
position to 1.5 km and its velocity to 0.2 m s,1 (Perryman et al., 1992).
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Much more than would ever have been the case for the geostationary position,
the conditions of the satellite orbit determined the handling of the satellite and the
data reductions. With a guaranteed break in the observations of some 2 hours around
perigee, the “data set” units were effectively defined, giving the maximum length of
time for a wide range of calibration and data reduction processes. Given the orbital
period of just over 10 hours, the average data set covered some 6 to 8 hours of
observations. This was a shorter period than originally foreseen for the great-circle
reduction process, which was therefore slightly more vulnerable to instabilities
caused by the closeness of the basic angle of 58 to a 6th harmonic at 60 . Such
instabilities were controlled in the penultimate stage of the data reductions, the
sphere solution (see Section 8.6).
The orbits were numbered sequentially, starting at the first scientific data
obtained on November 5, 1989. The total mission covered a time-span of 2768
orbits, equivalent to 1230 days. Observations were obtained during 2341 orbits. In
a few cases successfully reduced data was obtained by only one of the reduction
consortia. This applied for 94 orbits, most of which concern short stretches of data.
An 11-week gap in the data occurred from orbit 2274 till orbit 2446, when the
satellite was being prepared for operations using two instead of three gyros (the
period indicted by ‘e’ in Table V and Figure 18).
Another consequence of the operational orbit were the much extended eclipse
periods and eclipse lengths. With the satellite moving to within 400 km from the
Earth’s surface, eclipses could last for almost 2 hours (Figure 2), compared to a
maximum length for the nominal mission of 72 min. This put severe demands on
the power supply, with the time between eclipses only marginally long enough to
reload the batteries. The satellite also cooled down very significantly during such
periods (see Section 3.1), which was observed in various instrument calibration
parameters despite the very strict thermal control of the payload. The eclipses also
caused problems with the reconstruction of the satellite attitude due to the changes
in the solar radiation torques working on the satellite (Figure 37). Two types of
torques, the gravity gradient and the interaction between the Earth’s magnetic field
and the magnetic moment of the satellite, increase by ,3 , where is the distance
between the satellite and the centre of the Earth. These torques would have been
insignificant in the geostationary orbit, but now played an important role (see
Section 8.2.2).
Occultation periods, when the image of the Earth or the Moon came too close to
one of the fields of view, were also on average longer than would have been the case
in the geostationary orbit. This led in some cases to extended interruptions around
perigee, when the perigee passage was preceeded or followed by a long occultation.
With the two fields of view, occultations usually came in pairs, separated by about
20 minutes. Further details on how the occultations affected the main-mission data
reductions can be found in Sections 8.2 and 8.5.
D
D
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211
Figure 2. Reflections of temperature changes in the spacecraft. Top graph: lengths of eclipses;
Middle graph: exposure factor (see text); Bottom graph: drift rate of the on-board time (OBT)
measured relative to the ground-station time (GST).
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1.8. DATA RELEASE
The results of the Hipparcos mission were made public in June 1997 in the form
of printed catalogues (for selected results only) and on ASCII CD-ROM for all
relevant data (ESA, 1997). Included in the printed catalogue is the documentation
of the data reduction processes and selected data files. A set of four letters to
Astronomy & Astrophysics described briefly the main-mission astrometric results
(Perryman et al., 1997a), the Tycho astrometric and photometric results (Høg et
al., 1997), the photometric data (van Leeuwen et al., 1997a) and the double star
investigations (Lindegren et al., 1997). Before this release date, selections of the
data had been made available on request to those involved in the preparation of
the Input Catalogue and the data reductions (September 1996) and to those who
made observing proposals in 1982 (release date January 1997). A symposium
in Venice in May 1997 rounded off the project and presented the first scientific
results (Perryman & Bernacca, 1997). In 1994, developments started on a possible
follow-up mission, building on the expertize obtained from Hipparcos and on much
improved technological possibilities. The ROEMER mission proposed to utilize
CCDs for drift scan in a Hipparcos-like satellite (Høg, 1993), (Høg & Lindegren,
1994), thus improving the light collecting efficiency of Hipparcos by a factor 105 .
This was further developed in the GAIA proposal, using interferometric techniques,
which should be capable of improving the Hipparcos results by about a factor 100
in accuracy, and 500 or more in numbers of stars (Lindegren & Perryman, 1995,
Perryman & van Leeuwen, 1995).
2. Possibilities of ground-based astrometry
In this section the Hipparcos aims and results are compared with the achievements and possibilities of ground-based astrometry. Although some aspects of
ground-based astrometry have become less meaningful with the availability of the
Hipparcos and Tycho data, in most cases, the Hipparcos and Tycho data will lead
to a significant enhancement of the possible results of ground-based astrometry
through providing a well defined high accuracy positional reference frame. At the
same time, the use of ground-based data can provide an extension to the Hipparcos
and Tycho catalogues to much fainter magnitudes (the determination of accurate
positions for large numbers of very faint objects in the International Celestial Reference System, ICRS) and an increase in the base line used for the proper motion
determinations (providing a better distinction for orbital disturbances to Hipparcos
proper motion and parallax determinations).
2.1. MERIDIAN ASTROMETRY
Traditional meridian astrometry, where the goal was to maintain the positional
reference system, is the only technique that has become largely redundant with
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213
the release of the Hipparcos data. The definition of the celestial reference system
using automatic meridian telescopes was accurate to a level of 0.04 to 0.1 arcsec,
compared to the Hipparcos reference frame (at epoch J1991.25) to a level of at least
0.001 arcsec. Due to its short time-basis, this accuracy will deteriorate relatively
quickly, but it will take some 50 to 100 years before the Hipparcos accuracy has
become comparable with the meridian-circle data. It is hoped that before that time
comes, at least one follow-up astrometric space mission will have substantially
extended this limit. In the case of the proposed GAIA mission it would take many
thousands of years for the absolute positions to deteriorate to the level of 0.1 arcsec.
The future for the meridian instruments, and in particular the automated ones, is
now in extending the positional reference frame from the Hipparcos and Tycho
limiting magnitude of 11, to magnitude 15 or 16, and thus provide a link between
the Hipparcos and Tycho reference frames and the positional measurements of
magnitude 13 to 21 objects on photographic plates or CCD frames. This can be
done by putting a CCD detector in the focal plane and taking images or making
drift scans (Stone, 1997).
2.2. WIDE FIELD PHOTOGRAPHIC ASTROMETRY
The main use of wide-field astrometry is as an intermediary in providing reference
positions for faint objects, and as an instrument for large-scale proper motion
studies (see for example Murray et al., 1989, Murray, 1989, Evans, 1992, Evans
& Irwin, 1995). The possibilities for providing reliable reference positions will be
very much improved with the release of primarily the Tycho data, as long as plates
are available sufficiently close in epoch to the mean Hipparcos and Tycho epoch,
J1991.25 (Robichon et al., 1995). This will then allow for a proper description of
the plate transformation and reliable positions for fainter objects. Even with plates
taken at larger epoch differences it will be at least partly possible to disentangle
the proper motion effects from the plate distortions. A major difficulty is, however,
caused by the magnitude range of Hipparcos and Tycho Catalogues in comparison
with the magnitude range of a typical Schmidt survey plate. All Tycho images
will normally be saturated on such a plate. This is where the newly equipped
meridian circles can step in: they can provide the link from the magnitude range
of Hipparcos and Tycho to the useful magnitude range of a survey plate. Very
significant contributions in this field are also expected from electronic digital sky
surveys, such as Sloan (Kron, 1995) and DENIS (deep near-infrared survey, Le
Betre et al., 1995).
In the use of survey plates for proper motion studies, the main contribution
from the Hipparcos data comes in the form of defining the reference system for
those proper motions, i.e. making them absolute rather than relative. Also, a better
understanding of the distortions to which those plates are subjected can improve
the comparisons between plates of different epochs and thus decrease the extent
to which these distortions enter the proper motion determinations. These improve-
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F. VAN LEEUWEN
ments are all important for the study of Galactic dynamics. Thus, it is expected
that Hipparcos and Tycho astrometry enhances the possibilities of ground-based
wide-field astronomy, and can help extending the Hipparcos reference frame to
fainter magnitudes.
2.3. SMALL-FIELD PHOTOGRAPHIC ASTROMETRY
Small-field astrometry explores the large numbers of photographic plates taken with
photographic refractors from as early on as the 1880s. Small-field photographic
astronomy started in the 1850s with wet-emulsion photographs of the moon, and
received a major boost with the work done at Paris Observatory by the brothers
Paul and Prosper Henry in the 1880s. Their success in producing photographic
doublets of flint and crown glass, demonstrated by many very good photographs
of e.g. the Moon and the Pleiades cluster, led, at the end of the 19th century, to
the Astrographic Catalogue initiative, which was later followed by the Carte du
Ciel project. Both projects used refractors with a fixed focal length of 3.4 m and an
objective diameter of 30 cm. It is interesting to notice that this first photographic
sky survey is now being re-examined for the use as a first epoch to the Tycho data
(Kuzmin et al., 1997).
Later developments led to refractors with longer focal lengths and larger objectives. The great astrometric value of the plate material accumulated with these
telescopes stems from the useful magnitude range (2 to 16), combined with a reasonable plate scale, field size and a large spread in epochs. All this is available
thanks to the long term views of astrometrists during the first half of the 20th century. This kind of material has provided the most precise relative proper motions
for stars in star open clusters, and more recently also in globular clusters, with
precisions as high as 0.1 to 0.2 mas yr,1 , ten times better than Hipparcos and
reaching to much fainter magnitudes (see for example Vasilevskis et al., 1979,
and references in van Leeuwen, 1985). The combination of this material with the
Hipparcos data is in principle possible, but hampered by distortions that occur
in proper-motion reference frames derived this way (Tian et al., 1996). In addition, there are problems with comparing short time-base proper motions such as
determined by Hipparcos with long time-base proper motions determined using
ground-based data (Wielen et al., 1997), problems which will become more severe
as improved positional accuracies are obtained over short epoch intervals only. In
some studies small-field astrometry will for a long time to come be the main or
only provider of high-precision astrometric data, most noticeably for measuring
internal motions in globular clusters, where star densities are generally too high to
handle for a scanning instrument.
Small-field astrometry for the use of parallax determinations can also be
improved through the much better possibilities for defining the parallax zero point.
Thus, ground-based parallaxes can be brought closer to absolute measurements.
THE HIPPARCOS MISSION
215
2.4. CCD ASTROMETRY
This is a recent development in astrometry and early results have shown very
good possibilities for determining parallaxes for fainter stars (Monet et al., 1992,
see also Connes, 1979 and Lindegren, 1980 on aspects of potential accuracies
of ground-based measurements). CCD frames are also being used to determine
accurate positions of faint objects, connected through many links of CCD frames
and photographic plates to the Hipparcos and Tycho catalogue positions. Examples
of such applications were presented by Reynolds et al. (1995) and Caraveo et al.
(1997).
3. The Hipparcos satellite
The satellite was designed and built by an industrial consortium led by Matra Marconi Space in France and Alenio Spazio in Italy under contract to ESA. The satellite
housed on its lower platform the spacecraft, which consisted of the hardware for
running and controlling the instrument, and on the upper platform the payload,
which contained the telescope and the detector chains (see Figure 3). The satellite
was controlled from the ground-segment at ESOC in Darmstadt through the ground
stations in Odenwald (Germany), Perth (Australia), Goldstone (USA) and Kourou
(French Guyana).
3.1. THE SPACECRAFT
The task of the spacecraft was to ensure the proper operation of the payload. This
ranged from getting it in the right orbit (in which it was not very successful)
through power supplies and handling all communications with the ground stations,
to controlling the scanning of the satellite through thruster firings. A major part of
the spacecraft platform was occupied by the apogee boost motor (MAGE 2 ABM)
tank (see Figure 3), which was still full with solid fuel. Due to this, the total
mass of the satellite during operations was considerably higher than was planned:
1100 kg instead of 640 kg. As a consequence, the satellite was less affected by
solar radiation torques (the same torques resulted in smaller accelerations around
the rotation axes), and the centre of mass for the satellite was shifted away from
its nominal position by 72 cm. The thrusters for the z axis of the satellite were
therefore no longer in a plane through the centre of mass of the satellite, and firings
of the z axis thrusters produced consequently residual torques on the x and y axes.
The latter effect allowed, using the reconstructed-attitude data, a determination of
the position of the centre of mass for the satellite to an accuracy of about 1 mm.
The spacecraft handled the control of the scan by means of 6 cold-gas thrusters,
supplied from one of two cold-gas tanks. There were two thrusters each for the x, y
and z axes, producing positive and negative changes to the rotation rates respectively. Thruster firings took place under control of the real time attitude determination
216
F. VAN LEEUWEN
Figure 3. Exploded view of the satellite, illustrating the position of the payload and below the payload
the spacecraft platform.
program running on the on-board computer. They lasted in general for less than half
a second (one quarter of an observational frame). The intervals between thruster
firings varied from a few minutes to more than half an hour. Near perigee passages
the thruster firings became more frequent and longer, compensating for the much
increased gravity gradient and magnetic field torques. Thruster firings affected
the data reductions by producing discrete changes in the satellite attitude, which
THE HIPPARCOS MISSION
217
had to be properly provided for in the attitude reconstruction and the great-circle
reduction.
The power supply on-board the satellite came from 3 solar panels, backed up
by batteries that were activated during solar eclipses. The total power consumption
during normal operations was around 270 W, with at the start of the mission a
power supply of 445 W by the solar panels, dropping to 360 W towards the end of
the mission as a result of radiation damage to the solar cells. During Sun-pointing
mode the power supply from the solar panels dropped due to the higher temperature,
and towards the end of the mission the satellite could no longer fully function in
Sun-pointing mode. The power subsystem provided a 50 V dc bus to the satellite
power users. A detailed report on the solar array performance and degradation as a
result of radiation can be found in Chapter 11 of Volume 2 of ESA (1997).
Some equipment on the spacecraft platform was thermally controlled, but overall
the conditions on the platform reflected the exposure of the satellite to sunlight.
This was most clearly demonstrated by measurements of drift rates in the on-board
time (OBT) that resulted from temperature changes. Times as provided by the OBT
were referred to the much more accurate time-tagging of the data at the ground
stations, and the evolution of the differences between the OBT and ground-station
times (GST, corrected for satellite to ground-station and receiver delays) provided
a measure of the OBT drift. The OBT was regulated by a quartz oscillator, situated
on the spacecraft deck close to the outside cover of the satellite, roughly halfway
between the apertures of the two fields of view. The long-term changes in the OBT
drift were directly correlated to the exposure factor, defined here as the fraction of
time of an orbit during which the satellite was exposed to the Sun, divided by the
square of the distance d between the Earth and the Sun:
x
E
=
Torb
,
Tecl
Torb
1
2
d
;
(1)
where Torb is the orbital period and Tecl the length of time of an orbit during which
the Sun was eclipsed by the Earth. Both the ellipticity of the Earth’s orbit and
the lengths of eclipses can easily be recognized in the drift of the on-board clock
(Figure 2). There were smaller effects relating to the height of perigee (warming
up of the spacecraft due to friction with the outer layers of the Earth’s atmosphere,
see also Section 8.2.6) and the rotation of the satellite (for more details see ESA,
1997, Volume 3, Chapter 8).
The communications from the spacecraft to the ground station were arranged
through a continuous telemetry stream of 23.04 kbits per second, containing all
science data, attitude-control-related data and housekeeping information such as
temperatures. Telecommands from the ground station to the spacecraft were used
to provide the instrument with a program star file, a listing of stars with predicted
positions and magnitudes, that were expected to become visible in either of the two
fields of view. This data was stored in a buffer on-board the satellite. The buffer
could store the star data for at least 5 minutes of observations.
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F. VAN LEEUWEN
Table III
Payload characteristics.
Optics
Telescope configuration
Field of view
Separation between fields
Diameter of primary mirror
Focal length
Scale at focal surface
Mirror surface accuracy
All-reflective Schmidt
0.9 0.9
58
290 mm
1400 mm
6.8m per arcsec
=60 rms (at 550 nm)
Main detector
Modulating grid
Slit period
Detector
Photocathode
Scale at photocathode
Sensitive field of view
Spectral range
Sampling frequency
2688 slits
1.2074 arcsec (8.2m)
Image dissector tube
S20
3.0m per arcsec
38 arcsec diameter
375-750 nm
1200 Hz
SM detector
Modulating grid
4 slits perpendicular to scan
4 slits at 45 inclination
Photomultiplier tubes
Bi-alkali
eff 430 nm, 90 nm
eff 530 nm, 100 nm
600 Hz
Detectors
Photocathode
Spectral range (BT )
Spectral range (VT )
Sampling frequency
=
=
=
=
=
3.2. THE PAYLOAD
The Hipparcos payload was built around an all-reflective Schmidt telescope, working in the visual part of the spectrum. It consisted of four main assemblies.
two external baffles to protect the telescope from stray light;
a carbon-fiber reinforced box containing the telescope assembly, consisting of
three mirrors, and mounted onto the spacecraft through three titanium bipods;
the focal plane assembly, containing all elements from the grid to the image
dissector tube and star mapper photomultipliers. This assembly could be
shifted to obtain an optimal focus position (see Figure 7);
the service electronics units.
The main characteristics of the payload are shown in Table III. The optical system
consisted of a beam combiner, which also took the role of Schmidt corrector
(providing the off-axis correction for spherical aberration), a flat folding mirror,
and a spherical primary mirror (see Figure 4). The mirrors were made from lightweighted Zerodur blank, polished to =60 rms at a reference wavelength of =
THE HIPPARCOS MISSION
219
Figure 4. Configuration of the Schmidt telescope. Light entered from the two baffle directions,
and was brought together at the beam combiner, which was configured as a Schmidt corrector. The
combined light was reflected from the flat folding and spherical primary mirrors onto the focal surface
where the modulating grids were located.
Figure 5. The grid assembly characteristics.
550 nm. The basic angle between the two fields of view had been chosen on the
basis of rigidity tests for the great-circle reductions and the provision of sufficiently
different parallax factors in the two fields of view.
In the focal plane of the telescope was situated the grid assembly. The characteristics of the detector grid are shown in Figure 5. The optical block on which the
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F. VAN LEEUWEN
grid was written consisted of a piece of Suprasil 1, a high quality silica. The convex
surface of the optical block matched the curved focal surface of the telescope. Its
rear side for the star mapper grids was flat, and for the main grid was ground with
a radius of curvature of 213.8 mm, which made it serve as a field lens. The surface
of the optical block had to be of extreme smoothness, as it was to support the
detector grids with sub-micron accuracy. After polishing and cleaning, a layer of
chromium was sputtered on the surface, followed by the deposition of a layer of
electron-sensitive resist.
The main grid and the star mapper grids were engraved in discrete “scan fields”,
with 46 168 of these scan fields covering the main grid field of view. The method
of engraving these grids meant that when projected orthogonally on a plane normal
to the optical axis of the optical block, these grids are strictly rectangular, but while
projected on the curved surface of the optical block, they are slightly distorted (see
Section 8.4). The spacing of the grid lines was 8.20 m, and their width 3.13 m.
The grid was made to very high specifications, with the rms variation in the main
grid parameters at a level of 0.010 m or less. A full specification of the main grid
and the star mapper grid is presented in Tables 2.5 and 2.6 of Volume 2 of (ESA,
1997). Figure 6 demonstrates the very high precision of the main grid as measured
on the ground.
The optimal focus of the telescope was determined by maximizing the modulation amplitude resulting from the modulating grid. It turned out, however, that
the optimal focus for the two fields of view were different, and a compromise
value halfway between the two best focus positions was adopted (see Figure 7).
As a result of the different focal positions, the effect of the telescope focus drifting
during the mission was noticeably different for the two fields of view, with the
preceding field of view (PFOV) moving towards a better focus, while the following
field of view (FFOV) was moving away from it (see for example Figure 20 or
Figure 47).
The payload units were kept under very strict thermal control in order to minimize changes in the optical system. The thermal stability of the payload can be
shown from the evolution of the basic angle as well as for various other calibration
parameters. Only periods of very long eclipses and excursions into Sun-pointing
mode can be recognized, but none of the other features shown in Figure 2 are
reflected in Figure 8. The range of short-term variations was of the order of 1 mas,
for long-term variations the range was about 20 mas. Only as a result of Sunpointing mode conditions and failures in the thermal control electronics did larger
variations occur.
4. The Input Catalogue
For a complete description of the Hipparcos Input Catalogue the reader is referred
to Volume II of ESA (1989) and Turon et al. (1992a). A summary of the properties
THE HIPPARCOS MISSION
221
Figure 6. The medium-scale irregularities of the main grid in the scan direction as determined through
ground-based measurements. The geometry of the grid corresponds to the configuration shown in
Figure 5. The scale in the upper figure is such that one horizontal scan-field unit corresponds to
40 nm. The lower figure shows the mean offset per slit line in the scan direction (solid line) and the
standard error of the offsets (dotted line).
of the Input Catalogue is provided by Turon et al. (1995). Here only some essential
points are reviewed. The aim of the Hipparcos Input Catalogue was to provide the
Hipparcos mission with a fixed selection of approximately 100 000 stars, evenly
distributed over the sky (in as far as possible), for which positions at the epoch
of observation are known to an a priori accuracy of better than 1.5 arcsec, and
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F. VAN LEEUWEN
Figure 7. Determination by ESOC of the optimal focus in the two fields of view for stars of different
colours, by maximizing the modulation amplitude. The optimal focus position for the two fields of
view differed, and a halfway position was adopted as the optimal focus.
Figure 8. The evolution of the basic angle as determined by NDAC. The feature near day 420 is
associated with the long eclipses around that time, and an overall cooling down of the satellite.
with sufficiently well known photometric properties. At the same time, the Input
Catalogue was supposed to incorporate stars requested in observing proposals made
in 1982. The preparation of the Input Catalogue was the responsibility of the INCA
consortium, led by C. Turon.
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THE HIPPARCOS MISSION
Table IV
The Scientific Proposals Selection Committee, 1982-1988.
A. Blaauw (chair)
J. Dommanget
W. Gliesey
M. Hack
E.P.J. van den Heuvel
C. Jaschek
J. Lequeux
P.O. Lindblad
A. Maeder
P.E. Nissen
B.E.J. Pagel
A. Renzini
C. de Vegt
P.A. Wayman
R.Wielen
4.1. OBSERVING PROPOSALS
Following an invitation by ESA at the end of 1981, some 200 proposals for objects
to be observed were submitted. They covered 214 000 different objects, and it was
clear that a fairly large number of these would have to be rejected. The proposals
covered a wide range of topics, from luminosity calibrations of specific types of
stars through distance calibrations for star clusters and associations to Galactic
dynamics. A selection from these proposals was made by the Scientific Proposals
Selection Committee (see Table IV).
A first selection was made by ranking those proposals which were possible to
be observed, and priorities were assigned to individual objects. It was realized that
faint objects required much more observing time than bright objects, and that an
increase in the number of faint objects included in the selection would lead to
an overall decrease in the total number of objects that could be included in the
catalogue. The distribution of stars over the sky resulting from this first selection
indicated that a large amount of work still had to be done to achieve a sky and
magnitude distribution better suited to the needs of the Hipparcos mission.
4.2. SELECTION MECHANISMS
As a first step, proposers were informed of, and asked to comment on, the recommendations by the selection committee, and to examine the consequences of
these recommendations. Based on the responses of the proposers, the selection
committee reviewed their recommendations. Also the overall contents of the Input
Catalogue were reviewed and the concept of a magnitude-limited complete survey
was introduced. Studies by the data reduction consortia provided details on the
feasibility of double star observations and disturbances by nearby bright stars.
Meanwhile, groups within INCA were improving the mission simulations,
which allowed for every selected object to obtain an estimate for the expected
accuracy of the astrometric parameters. Different algorithms, giving the observing
time as a function of stellar magnitude, were tested, allowing decisions to be made
about the total number of stars that could be retained in the catalogue. Other groups
in INCA were optimizing the observing lists for minor planets and variable stars,
and unifying proposals in dense regions of the sky. Final selections were often made
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F. VAN LEEUWEN
based on the results of the simulation software. The final results were presented to
the selection committee in the form of detailed statistics and performances for each
proposal, allowing the selection committee to verify that their recommendations
had been implemented satisfactorily.
4.3. PREPARATION OF THE CONTENTS OF THE INPUT CATALOGUE
For all stars appearing in the preliminary selections accurate coordinates and photometric properties had to be obtained. In many cases this meant new observations,
using plate measurements, meridian telescopes and photometers (Jahreiß et al.,
1992). In particular for many variable and large proper motion stars the coordinates provided by the proposers were often of very poor quality, and it is also mainly
among these stars that the final Input Catalogue contains a small number of misidentifications. The SIMBAD data base at the CDS in Strasbourg was extensively used
in these preparations. Photometric, spectroscopic and radial-velocity data were
collected from the literature for the selected objects. When no photometric data
was available, and the star was too faint to expect Tycho observations of sufficient
accuracy, stars were incorporated in ground-based observational programs of the
Danish, Dutch and Swiss telescopes on La Silla, ESO, providing Strömgren, Walraven and Geneva photometric measurements respectively (Grenon, 1988, Grenon
et al., 1992). A collaboration with the AAVSO was set up to assist with the observations of Miras and other large-amplitude variable stars (Mennessier & Baglin,
1988, Mattei, 1988). Special groups were set up for the aquisition of radial velocities for stars in the Input Catalogue (for further details see Burnage et al., 1988,
Udry et al., 1997, Fehrenbach et al., 1997).
4.4. PROPERTIES OF THE INPUT CATALOGUE
The Hipparcos Input Catalogue was published in printed form and on CD-ROM
(ESA, 1992). As was described in the preceding sections, the final Input Catalogue
was a compromise between observing proposals, mission requirements and limiting
magnitude. The contents are summarized in Figures 9 to 11.
4.4.1. Completeness
The Input Catalogue is nominally complete down to the following limits:
V
V
7 9 + 1 1 sin j j for spectral types earlier or equal to G5
7 3 + 1 1 sin j j for spectral types later than G5,
:
:
b
:
:
b
where b is the galactic latitude. Within these limits, 55 000 objects were selected
from the SIMBAD data base. Among the very few bright objects missing are components of very close systems of bright stars such as in the Orion Trapezium cluster.
For fainter magnitudes the selected objects had to fulfil one of two requirements:
THE HIPPARCOS MISSION
225
Figure 9. The cumulative numbers of stars in the Input Catalogue as a function of distance based
on the parallaxes in the final catalogue. The grey line represents an increase by r3 , showing the
differences between the numbers of stars included and the total number of stars expected to be
present within a given distance from the Sun (not corrected for density gradients).
Figure 10. The distribution of magnitudes in the Input Catalogue as based on the mission determinations of magnitudes. The grey line represents the completeness limit, showing the Input Catalogue
as essentially complete down to mag 7.
being a selected object from one of the approved observing proposals or assisting in
creating a homogeneous coverage of the sky. Thus, a substantial number of objects
in the Input Catalogue was finally selected mainly for reasons of the rigidity of the
final astrometric reference frame.
4.4.2. Double stars
The double stars included in the Input Catalogue were to some extent selected
on the basis of separation and magnitude difference, considering the possibilities
226
F. VAN LEEUWEN
Figure 11. The distribution over the colour index (V
, I )C .
as well as the limitations of the Hipparcos main detector (Dommanget, 1985,
Dommanget, 1988). In connection with the preparation of the Input Catalogue,
a new double star reference catalogue was created with specific emphasis on
astrometric properties: absolute positions and proper motions (Dommanget & Nys,
1994). Some 12 000 double stars were included in the Input Catalogue, and of
these 9500 were possible to resolve on the main grid, having separations of at least
0.1 arcsec and magnitude differences of 4 mag or less. The target positions for
these objects were the individual components (for separations beyond 10 arcsec),
or else the photocentre or the brightest component of the system.
4.4.3. Variable stars
Many pulsating variable stars, from Sct to Mira types, regular, semi regular and
irregular, were included in the Hipparcos Catalogue for the purpose of luminosity
calibrations, either directly through the measurement of parallaxes, or indirectly
through the use of proper motions (Mennessier & Baglin, 1988). A small number
of discoveries of pulsating variables were made among other stars in the catalogue,
using the Hipparcos main mission or the Tycho epoch photometry (Turon & van
Leeuwen, 1995, Bastian et al., 1996).
Several hundred eclipsing binaries were also included, and here the Hipparcos
discoveries almost doubled the number of cases in the Input Catalogue (see for
example van Leeuwen & van Genderen, 1997).
4.4.4. Identification problems
The main source of identification problems was the very inaccurate coordinates
provided in some observing proposals or available in the literature. This applied
primarily to two types of objects: faint variable stars and large-proper-motion stars.
Among those stars misidentified in the Input Catalogue and therefore not observed
THE HIPPARCOS MISSION
227
Figure 12. The heliotropic angles ( , , , and ) in which the satellite’s nominal attitude was
described, and their relation to the satellite axes (x, y, z). The plane of the ecliptic is indicated by
‘E’, the scanning circle by ‘G’. The two fields of view (PFOV, FFOV) and the basic angle are also
shown.
by the main mission are the variable stars GQ Ori (a Cepheid) and SS Cyg. One
of the consequences of misidentification was the use of the wrong colour index in
the data reductions, which affected primarily the main-mission photometric results
(see Section 9.3).
5. The scanning law
The scanning law (see Figure 12) was designed to describe a homogeneous coverage
of the sky. Restrictions on the scanning law came primarily from the brightness
of the Sun and from the requirement to provide approximate predictions of the
solar radiation torques, acting on the satellite, for the on-board attitude control.
This required a near constant angle between the direction to the Sun as seen from
the satellite, and the orientation of the satellite’s z axis. The solution that was
adopted described the scanning law in ecliptic coordinates, with the z axis of the
satellite precessing under an angle of = 43 around the direction to the Sun
( , see Figure 12). The precession rate for the z axis was 6.4 revolutions per
year (_ 6:4_ ). The satellite was spinning around the z axis at a rate of 11.25
revolutions per day ( _ = 168:75 arcsec s,1). The precession rate was adapted to the
ellipticity of the Earth’s orbit, to provide a smooth and homogeneous scan. Almost
the entire sky was scanned in at least two directions every half year. Figure 13
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F. VAN LEEUWEN
Figure 13. The scanning motion of the Hipparcos satellite on the celestial sphere, shown in ecliptic
coordinates. The top figure shows the path of the spin axis between 22 May 1990 and 23 September
1991. The scan direction is indicated by the arrows. The bottom figure shows one reference great
circle for each orbit (5 revolutions of the satellite) for the period between 22 May and 14 July 1990.
The actual scanning was 5 times denser.
shows the sky coverage for a 2-month period and the path of the z axis for a 16month period in the beginning of the mission. Figure 129 shows the scan-coverage
over a 6-month period.
On a few occasions, the satellite returned to the Sun-pointing mode (with the z
axis pointing towards the Sun), which was the fall back position for a serious failure
in the satellite attitude control. The thermal equilibrium status of the satellite was
disturbed at every such occasion, causing problems with some of the instrument
parameter calibrations (instrument parameters were often found to evolve rapidly
shortly after moving to Sun-pointing mode, while being assumed constant in the
calibration models). In the main-mission epoch photometry all data acquired during
THE HIPPARCOS MISSION
229
Sun-pointing mode observations is flagged as possibly suspicious. Main-mission
astrometric data obtained in Sun-pointing mode did not contain any information
on the stellar parallaxes, as in this configuration all parallax displacements were
perpendicular to the scan direction.
The peculiarities of the scanning law reflect strongly in the amounts and distribution of the data. For the astrometric data the distribution of scan directions
was quite important, in particular for double stars. The scan direction is defined
as the local angle between the reference great circle and the small circle of equal
declination or latitude. The distributions of scan-directions as a function of ecliptic
latitude are shown in Figure 14. The most important features are the missing scan
directions near the ecliptic plane, and the large concentration of observations near
ecliptic latitudes 47 . The asymmetry between scan directions near 0 and near
180 is the result of the motion of the z axis, which is moving along with the
changes in the direction to the Sun when at southern latitudes, and moving against
that direction when at northern latitudes (see also Figure 13). The peculiarities of
the scan directions for every individual star reflect in the covariance matrix of the
astrometric solution and have been preserved as such (see Section 8.7).
The scanning law affected the photometric observations in two ways: the total
number of available observations, and the statistics of the distribution of those
observations. These aspects are shown in Figures 15, 16 and 17. The numbers
of gaps and their lengths effectively limited the search for periodicity in variable
stars to periods less than about 5 days, except for very frequently observed stars
(see Section 9.4). The feature near the ecliptic plane in Figure 16 between 90 and
140 days results from the angular separation between stars near the ecliptic plane
and the Sun during three to five months of the year. An additional, much weaker
feature near 180 days is the result of a “closing error” in the scanning law that
affected observations for a small number of stars close to the ecliptic plane (see
also Figure 129).
The satellite was made to follow the scanning law through the use of coldgas thrusters, which primarily corrected for the effects of torques acting on the
satellite. The pointing was kept to within 10 arcmin from the nominal position,
and was controlled by the on-board computer, using star mapper transit times for
selected stars and gyro data. When the on-board attitude control was lost, which was
usually the case after a perigee passage, the attitude parameters were reinitialized
by the ground station. An elaborate software package was set up by A. McDonald
at the mission control, ESOC, for recovering the pointing of the satellite from the
continuous star-mapper data stream. This package operated during the last few
months before the onset of the final breakdown of the satellite with such a success,
that generally only very little time was lost after a perigee passage. When the
two-gyro mode came into operation, this recovery speed was the main reason that
some useful data were still obtained.
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F. VAN LEEUWEN
Figure 14. Distribution of the orientation angles of the scans for stars at different ecliptic latitudes.
Note the “missing” angles for latitudes between 47 and the large number of scans parallel to the
47 . The peaks at 90 and 270 are the result of Sun-pointing mode
ecliptic plane around
observations. The full lines represent positive and the dotted lines negative latitudes.
=
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231
Figure 15. The total number of photometric observations per star as a function of ecliptic latitude.
The bin size is 5 in latitude and 1 in number of field transits. The contour interval is 20 and the
highest contour represents 200 stars per bin.
Figure 16. Distribution of the lengths of gaps in the photometric data as a function of ecliptic latitude.
The bin size is 5 in latitude and 1 in gap length. The contours rise non-linearly from 200 points to
8900 points per bin.
6. Mission operations time line
The data quality and data return of the Hipparcos mission was affected by a wide
range of intended changes and unintentional circumstances. The intended changes
were generally aimed at improvements in quality and quantity of the data return:
improved calibrations and improvements to the Input Catalogue. Various failures
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F. VAN LEEUWEN
Figure 17. Distribution of the number of gaps in the photometric data as a function of ecliptic latitude.
Gaps longer than 1.5 days were counted. The bin size is 5 in latitude and 1 in number of gaps. The
contours follow the relative distribution of gaps. The contour interval is 2 per cent, and the highest
contours represent around 20 per cent of the data at the given latitude.
and a few communication problems led to loss of observations, while a range of
necessary calibrations and system tests also sometimes led to some temporary loss
of usually very small amounts of data (gyro de-storage, refocusing and various
routine instrument calibrations). Figure 18 shows schematically the main events
that affected the data quality and return. Table V lists the various features shown
in Figure 18.
6.1. ACTIVITIES LEADING TO IMPROVEMENTS OF THE DATA QUALITY
After the failure of the apogee boost motor, it was essential to incorporate additional
ground stations. Next to the main ground station in Odenwald (Germany), Perth
in Australia and Kourou in French Guyana were incorporated at an early stage.
Goldstone in the USA later took over from Kourou. Goldstone was positioned
much better than Kourou to fill in the gap between Perth and Odenwald (see
Figure 1). This brought the visibility of the satellite by ground stations to about 80
to 85 per cent. Odenwald alone could not have covered more than about 40 per cent.
The incorporation of additional ground stations also implied that the number of
satellite orbits without ground-station coverage at all was very small.
Improvements in the data quality were obtained from ground-based calibrations
of instrument parameters and Input Catalogue improvements. Most noticeable were
the determination of the orientation of the grid assembly with respect to the scanning
circle (by the First-Look processing in Utrecht) and the effect of improvements of
star positions on the attitude reconstruction. The latter were initially obtained from
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Table V
Summary of the main events. Start and end times are given in days from
1989 Jan 0. TCE: Thermal control electronics; MDE: Mechanism drive
electronics.
Start
328
382
389
612
632
592
755
819
900
940
1001
1013
1161
1285
1285
1315
1453
End
596
1004
1163
1288
1396
1466
Code
Event
I1
I2
I3
T1
T2
a
T3
V
S
H
b
I4
c
T4
d
e
f
Empirical correction of CCCM
Analogue mode anomaly detected
Grid rotation calibration implemented
Payload thermal control anomaly
Change from MDE1 to MDE2
Attitude lost, faulty RTAD initialization
Change from TCE1 to TCE2
Spurious undervoltage
Gas tanks swapped
Gyro-3 heater failure
Uplink command error
Anomalous IDT voltages
Data lost due to tape fault
TCE2 anomaly, TCE1 fully broken down
Non-z -gyro patch introduced
Suspension of data acquisitions
Gyro-2 anomaly, Sun-pointing for recovery
accumulated star mapper positions as obtained at the RGO (see van Leeuwen et
al., 1992a, Turon et al., 1992b and Section 8.3) and at later stages of the project
from the preliminary sphere reconstructions. The improved positions led to better
attitude control by the satellite and to improved pointing for individual objects.
Further improvements for the attitude control came from the gyro orientation and
drift calibrations at the RGO.
6.2. THE ROUTINE OPERATIONAL PHASE
A major influence on the data quality and return came from the varying background
levels. The main reason for this was the radiation background and the way this
affected the star mapper photomultipliers. The star mapper background would
normally increase during crossings of the Van Allen belts, but the amount of this
increase fluctuated strongly. The Hipparcos mission operated during a time of
high solar activity, and on several occasions the background levels for star mapper
were so high, that attitude control by the satellite became impossible. This led in
general to shorter data spans used in the great-circle reductions. Figure 19 shows
the evolution of the background in the star mapper V channel over the mission at
different orbital phases. The background levels are presented as the equivalent of
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F. VAN LEEUWEN
Figure 18. Summary of various events that affected the Hipparcos mission. Table V lists chronologically events that had direct bearing on the data reductions, and explains some of the symbols used in
the figure.
stellar signals of a given magnitude, to indicate to what magnitude stars could still
be observed.
The optimal focus of the telescope evolved slowly with time. In order to maintain
a close to optimal focus setting, the focal plane assembly had to be shifted slightly
every 2 to 3 weeks at the beginning of the mission, and at longer intervals later on.
These refocusing events can be recognized in several calibration parameters, but
should in most cases not affect the data, as they mostly took place outside the data
aquisition time. An example of the effects of the refocussing is shown in Figure 20.
The influence of some hardware failures on the focus of the telescope is clear from
the changes in the modulation coefficients that can be associated with these events.
7. The data streams
There were two primary and several secondary data streams. The primary data
streams provided the compressed photon counts for the main detector (the image
dissector tube or IDT) and the BT and VT star mapper detectors. These data formed
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235
Figure 19. Star mapper background levels in the V channel at different orbital phases over the
mission. The orbital phases range from -0.5 at perigee to 0.0 at apogee to 0.5 at perigee. The top
graph also indicates events of high solar activity (just below the mag 5 level).
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F. VAN LEEUWEN
Figure 20. The effect of refocussing on the modulation coefficient M1 of the first harmonic in the
I )C = 0:5 for the
main-mission signal. The top graph shows the coefficient for stars with (V
following field of view (FFOV) and the bottom graph for stars of the same colour index for the
preceding field of view (PFOV). The discontinuities are the result of refocussing of the telescope.
Various events that disturbed the thermal stability of the telescope are indicated in the top graph. The
codes refer to Figure 18 and Table V.
,
part of a 23 kilo bits per second down link from the satellite. The secondary data
streams provided information on gyro readings, temperatures, real time attitude
determination, thruster firings and various status bits. All data was received and
archived at ESOC in Darmstadt.
At ESOC, the satellite telemetry was split into data streams and put on magnetic tapes for dispatch to the data reduction consortia. A Data Delivery Interface
Document, defining the file structure, types and formats, had been established well
before the start of the mission. The simulated software as prepared at the RGO for
THE HIPPARCOS MISSION
237
the use in FAST, NDAC and TDAC had been written according to those specifications. When the real data came along, the reduction consortia were fully prepared
and ready to receive it. Very little interpretation and no data decompression was
applied to the data before being sent to the reduction consortia. The data collected
over 1 week generally took 6 to 7 9-track 6250 bpi tapes for transfer. A total of
around 1300 tapes each were sent over a period of almost 3 years to RGO (for
NDAC), CNES (for FAST) and Tübingen and Heidelberg (for TDAC). After a
period of testing of the interfaces and data reduction processes, which took place
between the start of the first observations in November 1989 and the end of 1990,
the production of all mission data on tape started in February 1991. The last tapes
were received in December 1993. The first task in each consortium was the archiving of the data tapes. In FAST and TDAC the tapes were copied and catalogued for
data contents, with the copies being used in the data reductions. In NDAC, the data
was transferred from the tapes into ready-to-reduce data sets, which were stored
on WORM-type optical discs, providing random access to all mission data.
Of the data streams present on these tapes, four are of particular interest in the
context of this review:
The main detector or IDT data stream, containing the compressed IDT counts
with some housekeeping giving information as to the association of counts
with stellar images and details allowing the reconstruction of the pointing of
the instantaneous field of view of the IDT (for NDAC and FAST);
The star-mapper data stream, containing sets of 250 samples of compressed
photon counts in BT and VT , centred on the expected transit times of reference
stars to be used in the reconstruction of the satellite attitude (for NDAC and
FAST);
The Tycho data stream, compressed counts in BT and VT from the continuous
stream of data obtained from the star mappers (for TDAC only);
The satellite attitude data stream, containing details on the real time attitude determination by the on-board computer (this allowed the possibility to
reconstruct the differences between the actual pointing of the satellite and
the pointing which the on-board computer considered correct). This file also
provided information on thruster firings and gyro readings.
In addition there were files giving details on the orbit of the satellite, the housekeeping data (for example temperature measurements) and timings of eclipses of
the Sun by the Earth or the Moon. The scientific data files are presented in more
detail below, preceded by a description of the decompression of the photon counts.
7.1. DECOMPRESSION OF THE PHOTON COUNTS
All photon counts were transmitted by the satellite to the ground station, and from
the ground station to the reduction consortia, compressed in 1 byte (8 bits) integers.
The compression law was semi-logarithmic, adding, when below saturation, a noise
well below the intrinsic Poisson photon count noise. In the decompression, correc-
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F. VAN LEEUWEN
Figure 21. Verification of the implemented intensity transfer function (ITF) for the IDT data stream,
using data for the brightest stars. The diagonal line is the expected relation (apart from an offset) for
properly decompressed counts. Counts for the brightest stars, and in particular for Sirius at 1:5 mag,
were still affected by uncorrected saturation.
,
tions for non-linearities of the photomultiplier responses were accommodated too,
as were, to some extent, the effects of near saturation. Decompression therefore
used tables rather than the mathematical form used for the compression of the
counts by the on-board computer. The transformation to decompressed responses
was described by the Intensity Transfer Functions (ITF). The low and intermediate
intensity range of each ITF was well determined. The high intensity range, where
the effects of saturation started to appear, was calibrated at the start of the mission,
using mission data for very bright stars (see Figure 21).
The IDT detector should have used for the 21 brightest stars (H p < 1:5) the
analogue observing mode. Due to a synchronization error between the observing
strategy and the switching between photon-counting and analogue mode, this mode
could not be used. As a result, the data for the brightest stars suffer from saturation,
affecting mainly their photometric results.
The star-mapper data was compressed in the same way as the IDT data. Here,
however, saturation affected the photon counts for stars of 2.5 mag and brighter. The
saturation also affected the astrometric results for the brightest stars (see Figure 34),
but in this case such results were all superseded by main-mission data.
7.2. THE IMAGE DISSECTOR TUBE DATA STREAM
The Image Dissector Tube (IDT) data stream consisted of compressed photon
counts obtained over 1/1200 s sampling periods. At nominal scanning speed
(168.75 arcsec s,1 ), 8 such sampling periods roughly covered the crossing of
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239
Figure 22. The average profile of the instantaneous field of view. Variations with star colour and with
position on the grid existed.
one slit (1.2074 arcsec) on the modulating grid. The sensitive area of the IDT,
referred to as the instantaneous field of view (IFOV), could be moved very quickly
across the grid, and had a diameter of about 30 arcsec (see Figure 22). It could
obviously not distinguish between data from the two fields of view. Due to its small
size, however, incidental images from the other field of view appearing within the
IFOV at the target position were relatively rare. Their influence depended very
much on the magnitude difference between the target object and the incidental
object. Significant disturbances were therefore more common for faint stars than
for bright ones.
The sampling periods were grouped in such a way that near simultaneous
observations were obtained for up to 10 stars over a period of an observational
frame, 32/15 s. The basic unit of these observations was the “slot”, a period of
1/150 s covering 8 samplings and being the minimum number of consecutive
samplings for the observation of a star. Sets of 20 slots, forming interlacing periods
of 2/15 s, were distributed among up to 10 programme stars visible in the field of
view. The number of slots assigned to each star depended on the magnitude and on
the competition from other stars (see also Figure 48). The observational frame was
made up of 16 interlacing periods, and was the unit time interval for the IDT data
reductions. Between the 16 interlacing periods, up to three different distributions
of observing time could be implemented. This way it was possible to accommodate
one star entering and one star leaving the field of view during the time-span of an
observational frame. Such observations were referred to as “partially observed”.
Only brighter stars were ever partially observed. Fainter stars were only “fully
observed”, i.e. when visible during the entire observational frame.
A reversal of priorities between stars from one frame to the next helped to
establish a proper overall distribution of observing time. This often meant that the
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F. VAN LEEUWEN
faintest stars, requiring many slots to get reliable observations, were observed in
alternating frames only.
The repositioning of the IFOV (moving from one stellar image to the next)
required a small but not negligible amount of time. The first sampling period after
a repositioning was therefore not used in the data reductions. The positioning of
the IFOV was controlled through coil currents, for which the values were included
in the data stream. The coil currents were calibrated at regular intervals; these
calibrations were provided in the form of coil current calibration matrices (CCCM)
on the data tapes. They allowed a reconstruction of the pointing to be made, and
a comparison with the a posteriori reconstruction of the best pointing. Figure 23
shows the evolution of the pointing accuracy over the mission.
The on-board computer determined the amounts of observing time to be spent
on each object, using a file with stellar positions and priorities as provided by the
ground station, and the on-board attitude reconstruction to see which objects would
be visible and where these would be situated in the field of view. The result of the
observing strategy was that a variable amount of observing time was assigned to
the observations of each star, which reflects in the accuracies of the reduced data
(see Figure 48).
The modulating grid produced a signal consisting of a zero level (signal plus
background) and a first and second harmonic sinusoidal modulation. The astrometric data were derived from the modulation phases; the mean intensity and
modulation amplitudes provided the photometric data. The relations between the
modulation amplitudes and between the phases of the first and second harmonics
for single stars were a function of stellar colour and position in the field of view.
Such relations, described by a calibrated optical transfer function (OTF, see Section 8.4.5), were used to recognize the superposition of more than one image (both
as a result of double stars and as a result of accidental superpositions), in which
situation these relations no longer applied (see Section 8.4).
The photometric passband for the IDT detector, referred to as Hp, was designed
for optimal return on astrometric data. It was therefore a broad band, which extended
towards the red, but with an effective wavelength close to the Johnson VJ band.
Figure 24 shows the Hp passband in comparison with passbands in the Johnson
and Cousins systems. The count-rate for the IDT detector depended on the stellar
magnitude, colour and time of the mission. The detector response decreased over
the mission as a result of radiation darkening of the relay optics in the IDT. The
effect was most pronounced for shorter wavelengths, and caused changes in the
passband (see Section 9.3). For an 8th magnitude star with B , V = 0, the mean
response was almost 7000 counts s,1 at the beginning of the mission, dropping to
4000 at the end; the relative drop for red stars was much less (see Figure 84).
THE HIPPARCOS MISSION
241
Figure 23. The observed pointing accuracy of the IFOV over the mission. The graphs show the
fraction of observations for which the reconstructed IFOV pointing was within the indicated radius
from the actual position of the stellar image. Only observations within 9.5 arcsec were used, those
outside were considered to be associated with failures of the on-board attitude control loop. The
overall improvements in the satellite attitude, the Input Catalogue and the grid geometry calibration
led to the increase in the pointing accuracy over the mission. The last part of the mission was hampered
by bad attitude due to gyro failures.
242
F. VAN LEEUWEN
Figure 24. Normalized response curves for the Hipparcos
on (from left top right) the BJ , VJ , RC and IC passbands.
Hp passband (solid line), superimposed
Figure 25. The BT and VT passbands (solid lines) compared with the BJ and VJ passbands (dotted
lines).
7.3. THE STAR-MAPPER DATA
The star-mapper data stream was obtained from the light detected through any
of the four slits of both slit groups, and from both fields of view. The slit-width
was 0.9 arcsec and the span of the slits perpendicular to the scan direction was
40 arcmin. The slit spacings were chosen such that at nominal scanning speed
(168.75 arcsec s,1 ) the slits were encountered at intervals of 40, 60 and 20 sampling periods of 1/600 s. At the nominal scanning speed one sampling period was
equivalent to a movement of 0.28 arcsec. The light behind the slits was split into
THE HIPPARCOS MISSION
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Figure 26. Photomultiplier tube signal (blue channel) for the transit of one star over vertical slit
group. The sampling period is 600 Hz. The four peaks, corresponding to the four aperiodically
spaced star mapper slits, yield the star intensity and transit time. The observation is for a transit of a
BT
6:62 mag star (HIP 35946).
=
two passbands, referred to as BT and VT , more or less resembling the B and V
passbands in the Johnson system (see Figure 25).
The star-mapper data were grouped in sets of 25 sampling periods, covering
1/24 s, equivalent to one telemetry frame. The data of 256 of such groups covered
one telemetry format (32/3 s), equal to 5 observational frames of 32/15 s. The Tycho
data reductions received all the star-mapper data in units of telemetry formats.
NDAC and FAST only received a selection of data, associated with stars bright
enough to be of use in the on-ground attitude reconstruction process. Centred on the
expected transit times of such stars (as based on the real time attitude determination,
see the next section), 10 groups of 25 samples were extracted. In cases when the
on-board attitude was incorrect and not converged, this meant that the extracted
samples would partly or completely miss the transit signal. Extracted signals were
identified in the data stream through the star number, the slit group and the field of
view for the transit. Figure 26 shows an example of a decompressed star mapper
transit signal for a relatively bright star. The average number of transits extracted
this way was 15 per minute, originating from both slit groups and both fields of
view.
In the Tycho data stream the record was identified with a format counter and
the ground-station time-tag, but identification of transits was left to the TDAC
consortium. The average number of transits present in one format of star mapper
data (32/3 s) down to the limiting magnitude 11, was 33, but there were considerable
variations.
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F. VAN LEEUWEN
7.4. THE ON-BOARD ATTITUDE DATA
The on-board attitude information was provided in the form of three streams: the
gyro measurements, the thruster firing information and the results of the on-board
real-time attitude determination. The knowledge of the thruster firing timings, and
to some extent their strengths, was crucial in the data reductions. Thruster firings
caused discontinuities in the rotation rates for the satellite axes which had to be
accounted for in the reconstruction of the satellite attitude. By calibrating the
changes in the inertial rates around the satellite axes with the lengths of the thruster
firings, constraints could be imposed on modelling of the attitude across firings.
In addition, the calibration results provided further insights in the working of the
spacecraft. Figure 27 shows the results of such calibrations over the entire mission.
Various features can be distinguished:
the discontinuity at day 900 is due to the change of cold-gas tank, when tank
1 was almost empty;
the slope in the relations is related to the depletion of the gas-tanks, with the
decrease in pressure being over-compensated;
some features are due to temperature variations in the spacecraft and can be
attributed to periods of very long eclipses. Here too, the decreased pressure
caused by the lower temperature was over-compensated and created a higher
thrust;
the variability of the thrusts was about twice as high for the second tank,
possibly related to its positioning on the spacecraft deck;
the results of thrusts could be predicted to within 0.02 arcsec s,1 .
Thrusters were fired only at the start of an observational frame. The main mission
data collected during such a frame was discarded.
Discontinuities in the rotation rates were also occasionally caused by the satellite
being hit by micrometeorites. Such discontinuities could only be recognized from
an examination of the gyro data (before the start of the data reductions, by NDAC)
or on the basis of the badly fitting attitude reconstruction results (by FAST and
NDAC). Further problems occurred during the monthly gyro de-storage exercises
(see Figure 28), which created sudden temporary torques within the satellite while
a redundant gyro was spinning up. Also these situations could only be recognized
from the gyro data, as they were not recorded in the telemetry.
The real time attitude determination data was a record of what the satellite
considered to be the pointing of its three principle axes and the inertial rotation
rates around these axes. These data were provided at intervals of 16/15 seconds
and were very useful for checking, as part of the data reductions, the convergence
and stability of the on-board attitude reconstruction, and thus of the quality and
reliability of the scientific data. The most direct comparison that could be made
was with the observed star mapper transit times (see Section 8.1 and Figure 29).
The real time attitude determination on-board the satellite was based on transit
times for selected stars and gyro data. The transit times were used to determine
THE HIPPARCOS MISSION
245
Figure 27. The evolution of the thrusts created by the cold-gas thrusters. Thrusters causing positive
velocity changes are indicated with filled symbols, those producing negative changes with open
symbols. The discontinuity is caused by a change of gas tank, the drift is caused by the emptying of
the gas tanks and a slight over-compensation for loss of pressure.
pointing directions: for transits through the vertical slits they provided the scan
phase estimate, and for the transits through the inclined slits of the two fields
of view the position of the satellite’s spin axis. Rotation rates were derived in
the same determination from the observed changes in positions. These rates were
246
F. VAN LEEUWEN
Figure 28. An example of the influence on the rate of change of the inertial rates caused by a gyro
de-storage exercise. The readings of gyro 4 are shown, calibrated to rates relative to the nominal
scan velocity. The redundant gyro 3 is spun up at position a. Its spin axis is aligned with the input
axis of gyro 4 (parallel to the satellite’s z axis), which is noted in the accelerations. At position b
the redundant gyro 3 is spun down, causing a deceleration. The discontinuities in the rates are the
result of thruster firings, which also compensated the changes in the satellite’s angular momentum
that resulted from the de-storage exercise.
used to calibrate on-board drifts for the gyros that were used to bridge the gaps
between accepted star mapper transits. The angles and rates of the real time attitude
determination could be overwritten through commands from the ground station.
The updating mechanism was regulated by a Kalman filter.
8. Overview of the main data reductions
Many features of the Hipparcos data reductions were first designed and proposed by
L. Lindegren, most noticeably, already in 1976, the division of the data reductions
from the phase estimates to the astrometric parameters into three discrete processes.
This so-called three-step method for the data reductions was used by both data
reduction consortia to extract from the individual phase estimates the final catalogue
of astrometric parameters and was crucial in the feasibility of the project. The
three-step method split the data reductions into the processes of the great-circle
reductions, the sphere reconstruction and the astrometric parameter determination.
As data reduction packages were developed and tested, the divisions shifted slightly,
and other aspects were also incorporated: the initial processes of the IDT and
star mapper photon count reductions, and the attitude reconstruction. At a later
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247
Figure 29. The differences expressed in 600 Hz sampling periods between observed transit times and
real time attitude determination (RTAD) based predictions. One field of view is converged from the
start, the second field of view converges around frame 2400, after which there is general agreement
between the observed transit times and the RTAD. Transits through the vertical slits are indicated
with squares, for the inclined slits with triangles for the upper branch and stars for the lower branch.
Filled symbols refer to the preceding field of view, open symbols to the following field of view. The
dotted vertical lines indicate thruster firings.
stage in the data reductions, when merging of the FAST and NDAC data was
investigated, it was realized that a determination of the final astrometric parameters
should be done on the combined results of the great-circle reductions and sphere
reconstruction from FAST and NDAC, rather than as weighted averages for the
independent determinations by the two groups. Thus, the determination of the
astrometric parameters is described in the context of the data merging, though
preliminary, consortia based, determinations took place too. Extensive descriptions
of the final data reductions can be found in Volumes 3 and 4 of ESA (1997) for
the Hipparcos and Tycho reductions respectively, and summaries by Kovalevsky
et al. (1992), Lindegren et al. (1992c) for the main-mission data, and by Høg et al.
(1992a) for the Tycho data.
This section describes the reduction steps more or less in the sequence in which
they took place. The following is a brief summary of these processes, the sections
in which they are described and the way they are interlinked.
A: Star mapper reductions. Described in Section 8.1. Used raw star mapper counts and scan velocity estimates from the first step in the attitude
reconstruction (B ). Provided the transit data for the final step in the attitude
reconstruction. The reduced data were also used in catalogue updating (C ) and
star mapper photometry (H).
B: Attitude reconstruction. Described in Section 8.2. Used star mapper transit
times (A) and auxiliary information from the satellite (gyro readings, thruster
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F. VAN LEEUWEN
firing information). The reduced data were used in the reduction of the image
dissector tube data (D ) and the great-circle reductions (E ).
C : Catalogue updating. Described in Section 8.3. Used star mapper transit
time residuals obtained from attitude reconstruction (B ), and at a later stage
early sphere solution results. Also collected photometric data from the star
mapper and the main detector. Used in star mapper reductions, attitude reconstruction, great-circle reductions and sphere solution, as well as in photometric
reductions and double star analysis.
D : Image dissector tube (IDT) data reductions. Described in Section 8.4.
Used raw photon counts from main-mission detector and attitude reconstruction results to relate measurements to proper positions on the modulating grid.
Derived the modulation parameters: mean intensity, phases and modulation
amplitudes. Modulation phase information was used in the great-circle reductions (E ) and mean level and amplitude information used in the main-mission
photometry. Full modulation information was also used in double star analysis.
E : Great-circle reduction. Described in Section 8.5. Used phase information
from IDT reductions (D ) and the star mapper based attitude reconstruction (B).
Derived abscissae for stars along reference great circles and an improvement
of the along-scan attitude. The abscissae were further processed in the sphere
solution (F ), the improved attitude was used in the Tycho reductions (H) and
in the double and multiple star analysis.
F : Sphere solution. Described in Section 8.6. Used data from great-circle
reductions (E ). Determined zero-points of reference great circles and preliminary astrometric parameters. Input for the catalogue merging process (G ).
G : Merging and astrometric parameters. Described in Section 8.7. Used data
from FAST and NDAC sphere solutions (F ), derived correlations and orientation corrections, and determined the final astrometric parameters. Includes
the transformation of the final catalogue to the ICRS.
H: Tycho reductions. Described in Section 8.8. Used the raw star mapper photon counts together with main-mission attitude reconstruction results, based
on the main mission astrometric catalogue. Produced a final catalogue aligned
with the ICRS, containing photometry and astrometry for just over one million
stars.
8.1. STAR MAPPER REDUCTIONS, INCLUDING TYCHO
In order to extract from a star mapper signal the transit time and intensity for a
star, several steps had to be taken, involving calibrations, recognition procedures
(including the recognition of any disturbing signals) and signal fitting. Many of
the calibrations were part of iterative processes, where preliminary calibrations
were used in the data processing first, followed by the first actual calibrations and
implementation of the first calibration results in a new reduction. These steps were
often repeated two to three times before calibrations were accepted as final. Thus,
THE HIPPARCOS MISSION
249
in the description of the calibrations, results from the transit time and intensity
determinations are used, and similarly in the description of the transit time and
intensity determinations the results of calibrations have been used. The calibrations
of the grid geometrical and response characteristics are presented first, followed
by aspects of the actual data reductions.
8.1.1. Single slit response functions
The single slit response functions described the normalized detector responses to
the crossing of a single slit by a point source, integrated over the 1/600 s periods, and
described as functions of the distance to the slit centre. These response functions
were obtained from accumulated transit data for stars brighter than 8th magnitude
and with a low background count. In the data reductions, transit times and signal
intensities had been determined for these stars, as well as background levels and
local scan velocities (see Section 8.1.6). Given this information, the observed
counts for all four slits could be related to a distance from each assumed slit
centre, and scaled (or weighted) according to the measured signal intensity. These
values were binned according to distance from the slit centre (at a time resolution
of 9600 Hz by NDAC and 5400 Hz by FAST), slit, slit group, within each slit
group the upper or lower branch, channel (BT or VT ), and field of view. Data for
20 000 to 30 000 transits, covering a time span of 2 to 3 months, were collected
for every single calibration. This allowed a check on any significant changes in
the response functions over the mission. Only small changes in the wings of the
response functions were noted.
The accumulated weighted data in each bin were divided by the accumulated
weights to provide response measurements. These response measurements were
fitted using cubic-spline functions Rq in least squares solutions. The index q in
Rq represented the 16 different combinations of slit group (the vertical slits and
the inclined slits), branch, field of view and passband. The centre t0 of each curve
Rq (t , t0 ) was defined by the point halfway responses of Rq = 0:5 on the rising
and descending branches. The curves were normalized to 1.0 at peak (NDAC) or
at t = t0 (FAST). The derivatives of the response curves were determined directly
from the cubic-spline fits. These derivatives were used in the determination of
transit times (Section 8.1.6). The widths of the response functions at a response
level of 0.5 ranged from 0.89 to 0.90 arcsec for the vertical slits, and from 1.00 to
1.07 arcsec for the inclined slits.
The response curves showed significant differences between various combinations of slit group, branch, passband and field of view. Figure 30 shows two
examples of reconstructed single slit response functions and the corresponding
derivatives. No significant differences in response functions were detected as a
function of colour index, but no sufficient accumulation of data was possible for
the extreme red stars (V , I > 2).
The measurements for stars brighter than mag 2.5 were affected by saturation of
the highest counts in the signal. Here the knowledge of the wings of the response
250
F. VAN LEEUWEN
Figure 30. Reconstructed single slit response functions (left) and their derivatives (right) for the
inclined slits (solid line) and vertical slits (dotted line), upper branch, in the preceding field of view,
recorded in the BT channel. The derivative is given as the change of response per 600 Hz sampling
period at nominal scan-velocity, when one sampling period corresponds to 0.28 arcsec.
Table VI
Calibrated distances between the slit positions dj
in the star mapper vertical slit group.
Interval
Arcsec
mm
ground (mm)
d1 to d2
d2 to d3
d3 to d4
11.247
16.867
5.623
0.07638
0.11455
0.03819
0.07640
0.11460
0.03820
functions became very important. In TDAC special calibrations of these wings
were carried out, also in order to be able to determine proper background levels in
the presence of very bright stars.
8.1.2. Grid geometry calibrations
There were four aspects of the grid geometry that required calibrating: the spacings
between the individual slits; the position of the inclined slit group with respect
to the vertical slit group; the orientation of the slit groups; and the medium-scale
distortions for each group.
The spacings of the slits were very accurately calibrated as part of the above
described single slit response function calibrations. By determining the responses
for each of the four slits individually, the corrections to the assumed spacings were
determined by cross-correlating neighbouring responses. The values that were
THE HIPPARCOS MISSION
251
obtained were very close to the ground-based grid measurements, as can be seen
from Table VI.
The orientation of the slit groups and their relative positions were calibrated
as part of the attitude reconstruction, either direct as instrument parameters or a
posteriori from the residuals of the star mapper transit times after fitting of the
satellite attitude. These calibrations were also carried out as part of the Tycho data
processing, from which a record of star mapper related instrument parameters as a
function of time over the length of the mission was obtained (Høg et al., 1995).
The medium-scale distortions described small deviations of the grids from
linearity. These distortions had been determined from measurements of the grid in
the laboratory before the start of the mission. They ranged from 0.01 to 0.05 arcsec
when projected on the sky. Their proper implementation was checked using the
star mapper transit time residuals obtained from the attitude reconstruction process.
Empirical calibrations of the medium-scale distortions were obtained in the Tycho
astrometric reduction.
8.1.3. The star mapper transit signal
The star mapper photon-counts Nk;c , collected at times tk by either channel
c = B; V during the passage of a star, were described as a sequence of statistically independent Poisson-distributed counts, having the following time-varying
average:
E[Nk;c ] Ik;c
XR t ,
4
= Ib;c + Is;c
j
=1
q
(
k
j
);
(2)
where Ib;c and Is;c are the intensities of the background and star respectively. Rq
is the single slit response function as defined above and j are the transit times
for the four slits. The transit time for the slit group was defined as the unweighted
mean = 0:25 4j =1 j . The transit times j were linked to the individual slit
positions dj through the instantaneous local scan velocity vq :
P
j = + dj =vq :
(3)
In the NDAC applications the scan velocity vq was obtained from the gyro data,
and a single mean transit time estimate was obtained. In the FAST determinations
the transit time per slit was obtained, and from this the mean transit time .
8.1.4. Transit detection
The transit detection processes that were developed in the different consortia all
had one aim: to reduce the signal of four peaks to a signal of one peak, in order to
be able to distinguish more easily the often partly overlapping signals of different
transits. This involved various filters and algorithms to examine the filtered results.
The task of a filter was to recognize the characteristic pattern of the four slits. Filters
were generally applied to the combined BT and VT photon counts for increased
252
F. VAN LEEUWEN
Figure 31. NDAC processing of star-mapper data for a transit of the double star HIP 51560/51561
through the vertical slits. The first two bars show the original counts, the third bar the cumulated
signal J , and the fourth bar the filtered signal F . Between the V and B signals is indicated which
samples were excluded from the calculation of the background. Below F are indicated the recognized
transits: here the program star transit is fainter than the companion. The arrows below indicate the
predicted position, as based on the on-board attitude. The lower half of the graph shows the original
B counts folded with the final signal fit, which excluded the first 26 and last 35 samples, within which
it was not possible to identify transits unambiguously.
signal sizes. The filter as implemented in the reductions for NDAC at the RGO is
given here as an example. The FAST filter can be found in Section 6.4 of Volume 3
and is also partly described in the context of the FAST background determination
in Section 8.1.5. The TDAC non-linear 4-peak filter is described in Chapter 2 of
Volume 4 of ESA (1997).
A multiplicative filter was implemented by NDAC. It was applied to the the
following reduced resolution signal:
X
(Ik,1;c + Ik;c , 2I~b;c );
J (k=2) =
(4)
k = 2; 4; 6; : : : 250:
=
c
V ;B
All negative values of J (k=2) (counts below the estimated background level I~b;c )
were set to zero. This signal was subjected to the multiplicative filter, which
consisted of four Dirac functions, spaced like the expected slit responses: 20, 30
and 10 intervals at 300 Hz:
1=4
Y
J (l + n)
F (l ) =
; n = ,30; ,10; 20; 30:
(5)
n
THE HIPPARCOS MISSION
253
Special provisions were made for signals with only two or three out of four peaks
visible within the 250 counts provided. A transit signal was recognized from a cluster of F (l) values different from zero. More than one cluster could be recognized,
and up to four could be accounted for in the further processing. The peak values
of the clusters provided the first estimates of the transit times and indicated to
the further processing which samples were associated with star transits, and which
samples could be used for background determination. With more than one transit
present, the one closest to the expected brightness and transit time was selected as
the intended transit (the transit of the star for which the counts were extracted). For
known double stars the expected transit time difference between the components
was incorporated in the recognition mechanism. Figure 31 shows these steps in an
application to the star mapper transit signal for a known double star.
In the Tycho data stream the identification of transits with detections was a
different problem. A special process, referred to as ‘prediction’ produced initially
predicted transit times based on the Tycho Input Catalogue (Egret et al., 1992)
and the real time attitude determination. At later stages the NDAC attitude reconstruction results were used in combination with the revised Tycho Input Catalogue
(Bässgen et al., 1992, Halbwachs et al., 1992, see also Section 8.8).
8.1.5. Background determination
The background determination was important for the fitting of the single slit
response functions, on which both the transit time and the signal intensity determinations depended. Two examples of the background determinations are presented
here, the implementations by FAST and TDAC.
In the FAST processing a set of five variables Tj (m); j = 0; 1; : : : 4 was obtained
from the convolution of the cumulated counts in the B and V channels with the
single slit response function:
Tj (m) =
X X Nk;cRq tk , tm , dj =vq ;
(
c=B;V k
)
(6)
where dj = 0 for j = 0, and defined in Table VI for j 6= 0. This signal, shown in
Figure 32, was transformed in a set of Boolean test variables Lj (m), which gave a
value of one when Tj (m) was above a suitably chosen test value s, indicating the
presence of a star transit signal, and zero otherwise.
A generic sample m was accepted as a background sample if L0 (m) = 0 and it
was separated by at least n0 samples from a position for which L0 (n) = 1 applied.
The last condition avoided the inclusion of samples in the background calculation
that were affected by the wing responses of a single slit transit. The background
samples detected in a record were averaged to produce the (local) background
estimates for the transit:
I^b;VT
=
1
X Nk;V ;
n k2A
T
I^b;BT
=
1
X Nk;B
n k2A
T
(7)
254
F. VAN LEEUWEN
Figure 32. Top: a photon count record for a 6.6 mag star (HIP 47115) through the vertical slit group,
with the estimated average signal superimposed. The abscissae are in sample units (600 Hz), the
ordinates are in counts per sample. A spike, which occurred in the V channel, is indicated by an
arrow. Bottom: the corresponding convolution T0 (m) as provided by the FAST processing.
where n is the number of accepted background samples and A is their sequence in
the record.
A running estimate of the background, Ib , was maintained during a data set,
based on the estimates obtained from successive semi-contiguous records. The
running estimates were updated at each transit i using the local estimates I^b;B (i)
and I^b;V (i). A first order Kalman filter was used for each channel:
Ib;c (i) = [1
, K(i)]I
b;c
(i
, 1) + K(i)I^
b;c
(i)
(8)
where the time varying gain was set to K(i) = n(i)=(n(i) + 250), depending
on the number n(i) of detected background samples, but restricted to the range
THE HIPPARCOS MISSION
255
Figure 33. The estimated background intensity over an orbit on 10 June 1991, coinciding with a
period of very high solar activity. The background levels detected at apogee (centre of the graph) were
more than 10 times higher than under normal conditions. The top curve represents the V channel, the
lower curve the B channel. The Van Allen Belt crossings are indicated by VA, resets of the Kalman
filter by arrows.
0:05 K (i) 0:5. This range was designed to trade off between two contrasting
goals: attenuating the photon noise error (typically when the background was low
and few samples could be detected), and tracking the real background variations.
The Kalman filter was reset when there was a significant time gap in the data. An
example of the running background estimate for a time interval with particularly
strong background variations is shown in Figure 33.
In the Tycho processing two kinds of background estimates were obtained,
a global background based on the median of all 6400 samples contained in a
telemetry format, and a local background, based on 192 samples around every
estimated transit position. The latter was used when background was disturbed due
to very bright stars, dense star clusters, Galactic plane, zodiacal light, radiation or
stray light (Wicenec & Bässgen, 1992).
With its sensitivity to the radiation from outside the satellite, the star mapper
background provided a kind of record of the variations in the radiation environment
to which the satellite was subjected, both within each orbit and over the mission
(see also Figure 19). This allowed an investigation into the evolution of the activity
of the radiation-belts over the lifetime of the mission (Daly et al., 1994). The star
mapper background records also provided information on the zodiacal light and
the glow of the Galaxy (Wicenec & van Leeuwen, 1995). A complete record of the
star mapper background over the mission is kept at the RGO.
256
F. VAN LEEUWEN
8.1.6. Transit time and signal intensity determination
All methods used to determine transit times aimed at a way to optimize the fitting of
the single slit response functions to the data. Here the implementation by NDAC is
shown as an example because of its capability of handling more than one detected
transit at a time, and the way this assisted some of the main-mission double star
processing.
The estimates of the transit times and signal intensities were done as two independent solutions in an iterative loop. All estimates were obtained using linear least
squares. A first solution was obtained using unit weights for all observations, subsequent solutions used weights according to the estimated signal plus background
for each sample. This was effectively equivalent to a joint maximum likelihood
estimation for the transit times and the intensities. The basic equation for the intensity determinations is Equation 2, which, rewritten in the form of single transit
times for m transits, gives:
E[Nk;c ] Ik;c
=
Ib;c +
m
X
s=1
Is;c
4
X
j =1
Rq (tk , s , dj =vq );
(9)
The equation for the transit time corrections is obtained by differentiating Equation 9 with respect to the transit times s . Information from the BT and VT channels
was used together to determine single transit time corrections ds :
m
X X
c=B;V s=1
dIk;c
=
m
X X
c=B;V s=1
q
Is;c @R
@ ds:
(10)
s
This is where the derivatives of the single slit response functions, introduced in
Section 8.1.1, are used. These derivatives are defined as ,Rq0 = @Rq =@ . The
values of dIk;c are reflected in the differences between the observed counts and
the predicted counts using the previous estimates of the transit times ~s and signal
intensities I~s;c . Thus, we can write:
m
X X
c=B;V s=1
X c=B;V
ds
, I~s;c
Nk;c , Ib;c ,
m
X
s=1
4
X
j =1
Is;c
~
Rq (tk , ~s , dj =vq )
0
4
X
j =1
=
Rq (tk , ~s , dj =vq ) :
(11)
A practical limit of m 4 signals was applied, beyond which the resolution of
individual signals became too bad. In order to provide the proper Rq and vq for all
identified transits, each would have had to be identified with a field of view and
a slit group. This was considered too time-consuming when the data reductions
were performed and of little effect to the quality of the parameter estimates of the
target transit. Thus, all parasitic transits were treated with the scan velocity and slit
responses of the target transit.
THE HIPPARCOS MISSION
257
Figure 34. The formal standard errors ( , in arcsec) of the transit time determinations as a function
of the total signal intensity, transformed to an approximate magnitude scale. The graph shows results
for 9400 reduced star mapper transits obtained over a period of 9 hours in January 1990. The line
represents the development of accuracies when only affected by the photon noise of the signal.
An example of formal standard errors of the transit time determinations is shown
in Figure 34. There were on average some 900 star mapper transits of Hipparcos
stars available for every hour of observations, varying from 600 to 1200 depending
on the inclination of the scanning circle with respect to the Galactic plane. Of
these transits, half contributed to the determination of the scan phase, and a quarter
each to the determination of the two coordinates of the spin axis of the satellite (see
Section 8.2). The average standard error on a transit was approximately 0.03 arcsec,
with a range from 0.1 to 0.003 arcsec. Transits with transit-time accuracies worse
than 0.1 arcsec were rejected, as well as transits failing basic statistical tests for the
model fits.
In the Tycho astrometric processing (Section 8.8), an average of about 10000
transit times were derived from every hour of observations. These transit times
were transformed into one-dimensional positions through the reconstructed satellite
attitude.
8.1.7. Duplicity effects
Double stars with separations above 1.5 to 2 arcsec could be resolved as separate
transits in the star-mapper data processing. Below these separations a merged signal
was observed and processed. The reconstructed intensity of such a signal was a
function of not only the intensities of the component stars, but also of their projected
separation. Thus, unresolved double stars can show up in the Tycho photometry
file with magnitudes that are a function of the scan-position angle. The results of a
search for such cases among 480 000 Tycho stars with VT < 10:5 are indicated in
the Tycho Catalogue (field T49). For 8441 systems clear indications of duplicity
were found (indicated by ‘D’ in field T49 in the Tycho Catalogue, ESA, 1997).
258
F. VAN LEEUWEN
Figure 35. Distribution of the ordinates for all measurements used in the great-circle reductions,
relative to their reference great circles. The solid curve represents the NDAC data, the dotted curve
the FAST data.
The Tycho Epoch Photometry Annex file (tyc ep.dat on disk 4) provides magnitudes
and scan-position angles (field TT9) for a selection of 34 446 stars from the Tycho
catalogue.
8.2. ATTITUDE RECONSTRUCTION
This section describes the satellite attitude (the rotations of the satellite around
its major axes and the resulting pointing of the satellite as a function of time)
and the processes to reconstruct it using star mapper transit times, star positions,
gyro data and torque modelling. The emphasis here is on the modelling of the
motions and the information that was extracted from it. The Hipparcos data have
provided an unique insight in the torques working on a satellite, with many of the
details still to be resolved. Much more detailed accounts of the methods used to
reconstruct the attitude are presented by Donati et al. (1992), van Leeuwen et al.
(1992b) and Chapter 7 of Volume 3 of ESA (1997). This section is divided into four
parts, describing the aims and target accuracy, the dynamical model, the FAST and
NDAC approaches to the attitude reconstruction and the final accuracy assessment.
The dynamical model and the NDAC processing were closely interlinked: the
modelling used the reconstruction results, and the reconstruction used the model
as a first approximation. The FAST processing was, to some extent, independent
of this model.
THE HIPPARCOS MISSION
259
8.2.1. The aims and target accuracy
The attitude reconstruction for the Hipparcos satellite required an accuracy well
beyond any preceding, and so far also following, satellite. The accuracy with which
the pointings of the three main axes of the telescope reference frame (x, y, z) were
to be determined was mainly set by the requirements of the great-circle reduction
process: here transit positions were projected onto a reference great circle, to be
related to other transits. The maximum projection distance (the ordinate of a star
in the great-circle reference frame) was 1:5 . Figure 35 shows the histograms of
all observed ordinates over the mission by FAST and NDAC. The small difference
between the two curves resulted from the slightly higher tolerances of the NDAC
attitude reconstruction to the more extreme observing conditions close to perigee,
thus giving generally longer stretches of data per reference great circle (see also
Figure 49). For 99.9 per cent of the data the ordinates were less than 1.278 .
The target accuracy for the attitude reconstruction set before the start of the
mission was for the root mean square of the errors to be less than 0.1 arcsec. For
an ordinate at 1 this would add a noise of 1.6 mas in the form of small projection
errors to the IDT transit time estimates, which is well below the noise on the phase
estimates for most individual IDT transits (see Figure 48).
8.2.2. The satellite dynamical model, I: The equations of motion
A first attempt to reconstruct and model the torques working on the Hipparcos
satellite was presented by van Leeuwen et al. (1992b) on the basis of a small
amount of mission data used for testing data interface and reduction software.
The dynamical model of the motions of the satellite presented here describes the
motions around its major axes, and not the orbital disturbances. The model is based
on a mixture of observational evidence and theoretical considerations, and will
be illustrated by the observations of various torques that affected the satellite. A
reconstruction of torques was possible for the Hipparcos satellite due to the very
high requirements put on the attitude reconstruction, and the fact that even these
high requirements were most of the time surpassed by a significant margin. This
allowed the resulting attitude to be examined in the form of rates and accelerations
to a very high precision.
The attitude of the satellite is described with respect to an orthogonal set of three
axes, (x, y, z), fixed with respect to the satellite and defined by the geometry of the
star mapper grid: the xy-plane was defined by the great circle through the corner
points of the inclined slits as seen through the preceding and following fields of
view. The x axis was defined halfway these two points, the z axis perpendicular
to the plane, and the y axis completed the triad. This is schematically shown in
Figure 36. As a result of a small rotation (5 arcmin) of the grid assembly, the centre
of the main grid did not coincide with the xy plane.
The satellite is assumed to be a rigid body, which, considering the very small
torques working on it and the rigidity of the satellite that was required for the proper
260
F. VAN LEEUWEN
Figure 36. Definition of the major axes with respect to the telescope and the grid geometry. The
xy-plane is defined by the great circle through the corner points of the inclined star mapper slits as
seen through the preceding and following fields of view.
operation of the instrument, seems a reasonable assumption. Thus, its motions could
be described by the Euler equation:
I
d!
dt
=N
, ! I!;
(12)
where I is the inertia tensor of the satellite, N the external torques, and ! the inertial
rotation rates around the axes of the satellite reference frame. An approximate
analytical solution for the Euler equations was obtained by Bois (1986) in an
application to a model of Hipparcos affected by solar radiation torques.
In the analysis of the satellite attitude both the rotation rates ! and the accelerations dd!t were measured. Following the approximations made in Chapter 8 of
Volume 3 of ESA (1997), Equation 12 can be written as:
0 N =I 1 0 ! ! (I , I )=I , ! ! I =I 1
yy xx
x z xy xx
d! @ x xx A @ y z zz
Ny =Iyy + !z !x(Ixx , Izz )=Iyy + !y !z Ixy =Iyy A :
dt
Nz =Izz
!x!z Iyz =Izz + !y !z Ixz =Izz
(13)
This equation, expressed in the observable inertial rates and accelerations, allowed
for the calibrations of the inertia tensor element ratios as a function of time over the
mission. Changes in the inertia tensor took place primarily as a result of the depletion of the cold-gas tanks used for thruster firings. Table VII gives the calibrated
inertia tensor elements scaled to a fixed, ground-based value of Ixx . This calibrated
inertia tensor was subsequently used for all other torque calibrations, realizing,
however, the arbitrary scaling to the Ixx element, which may be responsible for
some small drifts in the observed torques. In most cases, however, observed torques
were a function of the same inertia tensor element ratios as those calibrated above.
THE HIPPARCOS MISSION
261
Figure 37. Removal of the Euler-equation cross-terms and the gravity gradient contributions from
the observed accelerations. Top graph: the observed accelerations around the x axis (black) and the
predicted gravity gradient contribution (grey). Bottom graph: the remaining accelerations. The data
cover one orbit (orbit number 307, March 19, 1990), with apogee at phase 0.5. The effect of a solar
eclipse is clearly visible between phases 0.17 and 0.32. Discontinuities in the top graph relate to
thruster firings, causing discrete changes in the rotation rates.
262
F. VAN LEEUWEN
Table VII
The inertia tensor elements in kg m2 for the Hipparcos satellite. The
reference epoch is day 70 in 1991 and t is measured in years from
that epoch. The inertia tensor has been scaled to a fixed value of Ixx .
x
x
y
z
534:220
3:459
2:031
y
+ 0 000
, 0 469
+ 0 082
:
t
:
t
:
t
3:459
590:680
0:806
,
z
, 0 469
+ 0 209
+ 0 005
:
t
:
t
:
t
2 031 + 0 082
,0 806 + 0 005
459 420 , 1 390
:
:
t
:
:
t
:
:
t
8.2.3. The satellite dynamical model, II: The gravity gradient
After the determination of the inertia tensor of the satellite, the torques due to the
gradient of the Earth’s gravitational field across the satellite were fully described.
Following Spence (1978), the gravity gradient induced torque is written as:
NG
^r I^r;
= 3GE
r3
(14)
where ^r = hri is the unit length vector indicating the geocentric direction to the
satellite’s centre of mass, and r = jrj is the geocentric distance of the satellite. GE
is the Earth’s gravitational constant. The remaining environmental torques were
derived from the following equations:
0 (! I !) =I 1 0 N
0 N =I 1
G
d
!
@ N =I A = + @ (! I !) =I A , @ NG
dt
(! I !) =I
NG
N =I
x
xx
x
xx
y
yy
y
yy
z
zz
z
zz
=Ixx
;y =Iyy
;z =Izz
;x
1
A;
(15)
where all terms on the right-hand side have now been determined. Figure 37
illustrates these corrections for one orbit.
8.2.4. The satellite dynamical model, III: The solar radiation torques
The solar radiation reflected and absorbed by the outer surface of the satellite caused
a force working on the satellite which was not balanced with respect to its centre
of mass, and therefore resulted in torques. The size of these torques depended on
the orientation of the satellite with respect to the direction to the Sun, which was
determined by the nearly fixed angle between the satellite’s z axis and the Sun
( = 43 ), and the scan-phase , which described the angle between the x axis of
the satellite and the great circle through the direction to the Sun and the satellite’s
z axis (see Figure 12). Because the satellite had a basic three-fold symmetry (see
Figure 3), the torques caused by the solar radiation were predominantly a function
of 3 .
The periodicity at multiples of 3 applied when the torques were seen in a
y z ] be such a
reference frame fixed with respect to the solar direction. Let [ x
on the great circle
reference frame with z along the satellite nominal spin axis, x
= z x
through z and the direction to the Sun as seen from the satellite, and y
263
THE HIPPARCOS MISSION
Table VIII
The solar radiation related torque components as observed in the accelerations. The observed values, after having been corrected for Sun–Earth
distance variations, were fitted to a linear function of time with day 800
as reference point. The coefficients (at day 800) are measured in mas s,2
and the derivatives in mas s,2 year,1 .
Axis
Term
x
sin
sin 2
sin 4
sin 5
sin 7
x
x
x
x
Coefficient
,2 28005 0 00031
0 59429 0 00012
,0 20310 0 00008
0 09826 0 00008
,0 01112 0 00009
,2 16908 0 00024
,0 64572 0 00011
,0 17513 0 00008
,0 09407 0 00010
0 00948 0 00011
,0 77013 0 00022
,0 12250 0 00008
,0 02593 0 00006
,0 01035 0 00006
cos
cos 2
cos 4
cos 5
cos 7
y
y
y
y
y
sin 3
sin 6
sin 9
sin 12
z
z
z
z
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
Derivative
,0 00980
,0 00895
,0 00028
,0 00230
:
:
:
:
0:00039
0:01662
,0 00968
:
0:00319
0:00099
0:00152
,
0:00261
0:00337
0:00049
0:00106
0 00042
0 00016
0 00010
0 00011
0 00012
0 00032
0 00015
0 00011
0 00013
0 00014
0 00029
0 00011
0 00008
0 00007
:
:
:
:
:
:
:
:
:
:
:
:
:
:
completing the triad. The solar radiation torques were approximated in this frame
by:
0 0 1 m 0 a sin 3n 1
n
N R @ b0 A + X @ bn cos 3n A
n=1 cn sin 3n
0
(16)
N
The torques R as experienced in the satellite reference frame were obtained
through a rotation by around the z axis:
0 cos
NR = @ , sin
sin
cos
0
0
1
0
0A
1
N R
As a result, the solar radiation torques around the satellite
approximated by:
NRx b0 sin + 12 Pmn=1(an , bn) sin(3n , 1)
+ 1 Pm (a + b ) sin(3n + 1)
2
n=1 n
n
(17)
x and y axes were
(18)
and:
NRy b0 cos , 12 Pmn=1(an , bn) cos(3n , 1)
+ 12 Pmn=1(an + bn ) cos(3n + 1)
(19)
264
F. VAN LEEUWEN
Figure 38. The primary components of the solar radiation torques as observed in the accelerations
around the three main satellite axes. The modulation has a period of a year and results from the
varying distance between the Earth and the Sun, due to the ellipticity of the Earth’s orbit.
This defined the coefficients characteristic for torques caused by the solar radiation.
A further peculiarity of these torques was the fact that their strength varied
inversely proportional to the square of the distance between the Earth and the Sun,
i.e. by about 3:5 per cent. This can be seen in Figures 38 and 39, which show
the main torque components (expressed in the measured accelerations) as observed
over the mission before and after correction for the varying distance to the Sun.
THE HIPPARCOS MISSION
265
Figure 39. The primary components of the solar radiation torques, corrected for the varying distance
between the Earth and the Sun.
The solar radiation torques around the x and y axes were of the order of
2 Nm, which translated into accelerations around these satellite axes at a level
of 2.3 mas s,2 . Around the z axis the torque (or acceleration) was about a factor
3 smaller. Table VIII shows the amplitudes of the main solar radiation related
torque components. The observed changes in the solar radiation torque coefficients
probably relate to changes in the light reflection and absorption by the satellite and
its solar panels due to the exposure to the space environment.
266
F. VAN LEEUWEN
The coefficients in Table VIII can be transformed back into the coefficients an ,
bn and cn as given in Equation 16, using the inertia tensor values given in Table VII.
From a comparison of the results obtained for the x and y axis for the an and bn
coefficients it is clear that although the results are in general agreement, additional
contributions not described by Equation 16 also play a role. This was also noted
from the presence of various other harmonics, most noticeably a 1st harmonic ( )
around the z axis, for which the amplitude increased much towards perigee. These
coefficients revealed the magnetic moment of the satellite and its interaction with
the Earth’s magnetic field.
8.2.5. The satellite dynamical model, IV: The magnetic torques
The torque caused by the interaction of the magnetic moment vector
Earth’s geocentric magnetic flux density vector is given by:
B
N M = m B
m and the
(20)
As seen from the satellite reference frame, is rotating. In the satellite reference
frame the flux density vector is expressed as:
0 cos
B = @ , sin
0
B
1
m sin m 0
m cos m 0 A B
0
(21)
1
B
where m describes the instantaneous orientation of with respect to the satellite
x axis. Within the satellite reference frame was assumed constant. This gave the
following relations:
m
0m B ,m B 1
y z
z y
NM = @ mz Bx , mxBz A
mxBy , my Bx
(22)
where at apogee the torques around all three axes were approximately modulated
by . Away from apogee this modulation became disturbed due to the changes in
, and both the amplitude and the period were varying strongly.
The description for used the 1985 coefficients of the International Geomagnetic Reference Field (IGRF) (Barraclough, 1985), describing the magnetic field
as a series of spherical harmonics. The most important (dipole) coefficient is proportional to r ,3 . The magnetic torques were smaller on average than the gravity
gradient torques, but increased in a similar fashion with decreasing distance to the
Earth.
Two magnetic moment components were observed: along the y and z axes of the
satellite. Their behaviour over the mission is illustrated in Figure 40. The magnetic
moment along the z axis changed significantly during eclipse periods, when the
power supply switched from solar panels to batteries. Studies of the wiring of the
spacecraft could not reveal the likely causes for these effects. The magnetic torques
would have been insignificant for the nominal mission.
B
B
THE HIPPARCOS MISSION
267
Figure 40. Measurements of the magnetic moments of the satellite. The y component was most
clearly present. There was probably also a significant z component during normal operations (middle
graph) and there appeared to be a significant z component during eclipse conditions (bottom graph),
which could, however, only be solved for under good attitude convergence conditions, and had to be
fixed otherwise.
No corrections were applied for compression or stretching of the Earth’s magnetic field due to interaction with the solar wind. The analysis of the data indicated,
however, that such effects were probably present. The analysis results for torque
components that are likely to be related to the magnetic moment of the satellite
show a modulation with the precession phase of the satellite’s z axis as well as
268
F. VAN LEEUWEN
Figure 41. Three coefficients, showing through their variability a possible link with the magnetic
moment of the satellite and a distortion of the Earth’s magnetic field at high altitudes. The short
period variation is probably related to the spin period of the satellite’s z axis (57 days). The long
period modulations could be related to the time interval between two conjunctions of the Sun and the
satellite at apogee (588.7 days).
with the orientation of the satellite’s orbit with respect to the direction of the Sun
as seen from the Earth, that may well reflect these compressions (Figure 41).
8.2.6. The satellite dynamical model, V: Miscellaneous effects
Among the various miscellaneous effects noted in the torque analysis was a local
warming-up during perigee passages, followed by local radiation and a resulting
THE HIPPARCOS MISSION
269
Figure 42. Short term variations for the constant for the y axis torques at a time of very low perigee
passages. The orbits with even numbers are connected to show the effect of earth longitude on this
coefficient. The period of the modulation is 8 days, equivalent to 18 orbits.
windmill effect. This effect was very sensitive to the density of the outer layers
of the atmosphere at perigee, and occurred only during very low perigee passages
(400-500 km altitude). The warming-up of the spacecraft during these passages
was clearly noticed from the drift of the on-board time (OBT, see Section 3.1).
Considerable variations appeared to exist in the atmospheric densities as a function
of Earth longitude: clear correlations were observed for the constant coefficient in
the y axis torques between perigee passages at nearly the same longitude. As can
be seen from Figure 1, perigee longitudes for alternating orbits were much closer
related than for successive orbits. Figure 42 shows the evolution of the constant
coefficient for the y axis solutions over a period of low perigee height, illustrating
this effect.
Some gyro de-storage exercises produced a considerable additional torque on
the satellite, depending on the orientation of the angular momentum vector of the
gyro with respect to the spin axis of the satellite. This allowed a calibration of the
angular momentum for gyros 4 and 5 to be made. The observed offsets showed
that orientation parameters were as expected, while the gyro angular momentum
appeared to be 4 percent higher than specified.
Finally, impulse like torques were experienced occasionally as result of accidental hits of the satellite by micrometeorites. Most hits were very minor, but two
reached a level comparable to a thruster firing, producing a change in rotation rate
of a few arcsec s,1 . Such occasions were recognized from the routine inspections
of the gyro data by NDAC as a discontinuity in the measured rotation rates that
was not associated with a thruster firing.
8.2.7. The attitude reconstruction, NDAC approach
The attitude reconstruction as performed by NDAC was based on the torque calibrations and modelling described in the preceding sections. The descriptions of
the torques were used to model the accelerations from which, through a numerical
integration of the Euler equations, estimates for the inertial rates were obtained.
270
F. VAN LEEUWEN
These integrations took place over intervals between thruster firings. Integrations
across thruster firings used the calibrated thruster responses (see Figure 27) and
provided starting points for the inertial rates for the integration over the next thruster
firing interval. The zero-point offsets of the inertial rates were calibrated using the
gyro data. The corrected rates were integrated to provide a first approximation
of the attitude angles. These approximate angles (given at three points for every
observational frame) were used to predict star mapper transit times. The differences
with the observed transit times were expressed as low order polynomials, which,
together with the model that served as a first approximation, provided the actual
attitude reconstruction.
The main vulnerability of this method appeared to be around thruster firings
where sometimes, due to a lack of star mapper observations, insufficient constraints
for the solutions were available. A detailed description of the NDAC attitude
reconstruction can be found in Chapter 7 of Volume 3 of (ESA, 1997).
8.2.8. The attitude reconstruction, FAST approach. I: the model
The FAST approach was based on a simplified integration of the Euler equations
while the satellite was near apogee, and a polynomial approach closer to perigee.
The simplified model shows the harmonics of the solar radiation torques to appear
in the modelling of the attitude angles, accompanied by discrete functions over
thruster firing intervals, which described the effects of the thruster firings on the
attitude. Pre-launch research aimed at reducing the modelling error under normal
geostationary conditions, while using a minimum number of degrees of freedom.
This model worked well in apogee conditions, but was no longer adequate near
perigee.
The use of the rigid body equations to completely describe the satellite motion
was excluded by FAST as it was considered too difficult to calibrate with the
required accuracy the telescope reference frame position with respect to the inertial
axes of the satellite and to maintain such an accuracy during all the satellite life.
8.2.9. The attitude reconstruction, FAST approach. II: the attitude angles
The FAST attitude representation was described relative to the reference great-circle
frame (Section 8.5), defined by the position of its pole in ecliptic coordinates, R
and R . This pole was selected as the position of the z axis in the nominal scanning
law for a reference time halfway between the start and the end of the data-set
concerned. Relative to this reference system, three attitude angles were defined: ,
and . Particular attention was devoted to the choice of the attitude angles. In
fact, the various sets of three angles are related among themselves by non-linear
transformations so that their choice influences the simplicity of the attitude model,
the estimation method, and accuracy. The chosen angles are related to the classical
3-2-1 Euler angles e , e and e by:
= e ;
= e ;
=
sin sin e , arctan
cos + cos (23)
THE HIPPARCOS MISSION
271
The chosen representation of the angle was motivated by the fact that, as its
rate of change is much closer to the inertial rotation rate around the satellite
z axis, it improved the modelling of the scanning motion. Experiments showed that
estimation results obtained with these angles were a significant improvement upon
those obtained by using e .
8.2.10. The attitude reconstruction, FAST approach, III: Approximate motion
equations
The , , attitude angles were modelled as the outputs of the following two independent systems of differential equations, which were obtained as an approximation
of the satellite dynamic equations in the case of small perturbations.
!
_ z = z (t)
_ = !
z
(24)
and:
!
_ x = x (t)
!_ y = y (t)
_ =
_ =
,!0 + !
x
!0 + !y
(25)
where !0 is a constant parameter assumed known and equal to the satellite’s
nominal spinning rate.
The functions x (t), y (t), z (t) are the derivatives of the rate components !x ,
!y , !z . In a first rough approximation they model the effect of the torques acting
on the satellite, but more precisely they have been introduced not only to model
the torque effect but also to compensate the approximation introduced by using
the above simplified equations to model the telescope motion. Considering that
any given set of three continuous time functions (t), (t), (t) can be obtained
as the output of the above dynamic system by the application of a suitable set of
input functions x (t), y (t), z (t) and by initial conditions, the model error could
be reduced as much as required by attributing a sufficient number of degrees of
freedom to the input functions.
8.2.11. The attitude reconstruction, FAST approach, IV: System Input Functions
The system inputs x;y;z (t) were decomposed into the sum of two terms, the control
inputs ux;y;z (t) and the disturbances dx;y;z (t) respectively. The control inputs were
introduced to describe the effects of thruster firings on the on-board attitude control:
they were assumed to be a sequence of ideal pulses (Dirac functions) of unknown
amplitude applied at the central time of the thruster firings. The disturbances
were introduced to describe the effects produced by the perturbing torques and to
keep into account, at the same time, the difference between the above simplified
dynamic system and the reality. The disturbances were expressed by different
272
F. VAN LEEUWEN
Figure 43. Flow charts for the FAST attitude reconstruction models.
mathematical models (Fourier and polynomial models) depending on the length of
the time interval considered and the satellite perturbing conditions. In the case of
Fourier models the disturbances were described by a Fourier series with unknown
coefficients and the response of the system was computed on the basis of the above
differential equations. In the case of polynomial models, the system response to the
disturbances was directly described by polynomial series. The resulting attitude
model is presented for the two cases in Figure 43. The following different models
have been used:
long-term Fourier model: this was the standard model, used in normal operating conditions over a time interval corresponding to a full rotation around
the satellite z axis, 128 min. Independently from the actual duration of the
considered time interval, the disturbances were described by a Fourier series
whose fundamental harmonics had the frequency !0 , corresponding to the
satellite’s nominal spinning rate used in the approximate equation of motion
(Equation 25). In this particular case the forced solution of the considered
differential equations had the following expression:
d (t)
= bs0 t sin !0 t + bc0 t cos !0 t +
Pn k=2 bsk sin(k!0 t) + bck cos(k!0 t)
d (t)
= cs0 t sin !0 t + cc0 t cos !0 t +
Pn 2 csk sin(k!0 t) + cck cos(k!0 t)
k
=
Pm (26)
d (t) = k=1 ask sin(k!0 t) + ack cos(k!0 t)
THE HIPPARCOS MISSION
273
A variable number of harmonics, in the range 12 to 36, was used in modelling , (m), and in modelling and , (n). The number of harmonics was
determined by an adaptive method based on the computation of the FisherSnedecor test (Cramér, 1946). The adaptation was made by increasing the
number of harmonics by 3 at a time, starting from 12. The optimal number
was the maximum number for which the Fisher-Snedecor test value was still
above a given threshold;
intermediate term Fourier model: this model was used over intervals of 30 min
to 115 min. The perturbing torques were modelled like above by a finite Fourier
series of unknown amplitude harmonics (Equation 26), but without the mixed
terms for and (describing the torque caused on through the change of and vice versa). Here the fundamental period was chosen equal to the length
of the time interval, and the minimum number of harmonics at the start of the
computation was determined by the interval length. They were increased by
one at a time, followed by the tests described above, until a test value within
a given threshold was obtained;
short-term polynomial model: this model was used for the shortest considered
time intervals, viz. a period between two thruster firings (varying from a few
minutes to more than half an hour, but on average about 12 to 15 minutes). The
perturbing torque effect was modelled by Legendre polynomials, the degree
of which was adapted to the considered case, independently for and for
and . The polynomial degrees were increased by one at a time until the
Fisher-Snedecor test value obtained was within a given threshold;
short-term polynomial model in penumbra: while the satellite was in penumbra, before and after an eclipse, the solar radiation torque followed a transient
from high values to very low or vice versa. Conditions during different transients were similar but never the same. The transient was modelled as a
stochastic process for which a Karhunen-Loève expansion (Papoulis, 1991)
was computed. This provided an ordered set of orthonormal functions, of
which each one was approximated by a Legendre polynomial representation.
The first 10 ( ) and 15 ( and ) functions were stored. The order of the
Karhunen-Loève expansion used was adapted to the particular case considered, starting from a minimum value determined by the number of star mapper
transits available, up to the maximum number of stored functions, applying at
each step the Fisher-Snedecor test.
The numerical methods used for the FAST attitude reconstruction are detailed
in Chapter 7 of Volume 3 of ESA (1997).
8.2.12. Accuracy assessment
The accuracy assessment was primarily based on comparisons between FAST and
NDAC reduction results, as carried out at the Centre for System Studies (CSS)
in Torino by F. Donati (Donati & Sechi, 1992). The comparisons also involved
274
F. VAN LEEUWEN
Table IX
Estimated standard deviations of the random terms in the differential rotation in
body axes, expressed in milliarcsec. Column (a): all frames included; column (b)
excluding outliers.
Component
Estimates (mas)
(a)
(b)
129
81
12
90
47
10
82
54
24
14
3 3
improvements in the along-scan attitude as obtained with the great-circle reductions
(see Section 8.5).
Before comparisons could be made, the data obtained from NDAC and FAST had
to be synchronized and rotated to coincide. The rotation matrix reflected differences
in the Input Catalogue positions (due to independent updating mechanisms, see
Section 8.3) and differences in the star mapper geometry calibrations.
The remaining standard deviations between the results of the two consortia were
examined for three coordinates:
0 1
= @ A ;
(27)
where (, , ) describe the rotations around the x, y and z axes respectively.
Denoting by 1q and 2q the standard deviations of the two attitude reconstructions
to be compared, in any of the angles (q = , , ), the following holds:
q2
= 12q + 22q , 21q 2q ;
(28)
where is the correlation coefficient between the two realizations of the attitude.
The correlation was close to one, as both solutions used the same star-mapper transit
data. Table IX shows the results of the last attitude comparison, where it should
be noted that the data for the coordinate had been improved by the great-circle
reductions. As is very clear from these results and the discussion in Section 8.2.1,
the projection errors in the great-circle reduction that resulted from the errors on
the reconstruction of the z axis position (angles and ), were for most of the time
negligible. There were, however, occasions when a very small amount of data had
much larger uncertainties in the attitude determinations. These occurred very close
to thruster firings, where control over the attitude reconstruction was partly lost,
and at times of very high background signals in the star mapper detectors, when
the numbers of available transits were much reduced. Most of these problems were
detected and taken care of by the great-circle reduction process.
THE HIPPARCOS MISSION
275
8.3. EARLY IMPROVEMENTS OF THE INPUT CATALOGUE
The processes described in this section made use of the star mapper and attitudereconstruction data, and had only indirect influences on the mission results. The
techniques used, however, may have much wider applications. The aim of the
improvements to the catalogue was to obtain at an early stage improved star positions for the epoch of observation, leading to better results for the on-board as well
as the on-ground attitude reconstruction (for which the accuracies as provided in the
Input Catalogue were inadequate), to obtain improved photometric data required
for various calibrations, to detect misidentifications early in the mission, and to
provide much improved input positions of double stars to the main-mission double star processing (where absolute positions were often very poorly known). The
processes as described here were implemented by NDAC. Positional improvements
were also implemented by FAST, in particular as obtained from provisional sphere
solutions and as part of the final iteration between the attitude reconstruction and the
great-circle reduction, sphere solution and astrometric parameter determinations.
The accuracy of the attitude reconstruction and the star mapper transit times was
about 5 to 10 times higher than the average accuracy of the positional information
in the Input Catalogue. As a result, the residuals obtained during the first runs of the
attitude reconstruction processes reflected mainly the errors in the star positions and
not the noise on the transit time determinations. By collecting sufficient numbers
of these residuals, positions could be adjusted.
Accurate photometric data were available for only a small fraction of the Input
Catalogue stars. The intensities obtained from the star mapper processing were
used to improve this situation as early in the mission as possible, and to provide
accurate colour information for a wide range of calibration processes.
8.3.1. The updating mechanism
The updating mechanism used a running least squares solution, based on Householder transformations as described by Bierman (1977). The advantage of this
method is the way the normal equations are represented: as an upper triangular
matrix with dimensionally the same form as the observation equations. Thus, an
old solution can be combined with new observations simply in the form of a set
of additional observation equations. The positional and photometric updates for
each star were given by these normal equations stored in a working catalogue.
For example, the normal equations for the positional updates described the differential corrections to the current catalogue positions. The updates for the two
positional coordinates were linked, as all information came from one-dimensional
observations, giving correlated corrections to right ascension and declination. The
provision of vertical and inclined slits effectively produced a two-dimensional position for a transit through both slit groups, which was the way positional updates
were implemented by FAST. For the magnitudes the information consisted of sums
of weights and weighted residuals. The accumulation of the information on the
276
F. VAN LEEUWEN
magnitudes was done in a pseudo intensity scale to avoid a bias in the accumulated
results.
In order to avoid faulty data to enter these updates, a minimum of 8 to 10
observations was required before a first solution was made. Before this minimum
number of observations was reached, observations were put into a storage file, from
where they could be retrieved when more data for the same star became available.
Data for single stars was processed without supervision, while data for double
stars (for which component confusion was not uncommon) were processed in an
interactive mode first.
8.3.2. The use of the early improvements
The use of the positional improvements was described by van Leeuwen et al.
(1992a) and Turon et al. (1992b). Positional improvements for over 15 000 stars
were already obtained before the start of the routine data reductions, using the
1300 hours of data sent from ESOC to the consortia for testing interface and
reduction software (Lindegren et al., 1992a). Much improved positions for some
60 000 stars had been obtained before the first sphere reconstruction results became
available. A small number of these involved corrections of over 2 arcsec, at which
level the pointing of the instantaneous field of view for the main-mission data
became affected. These updates were incorporated in the star catalogue used for
the mission operations, and led to a significant improvement in the on-board attitude
control, with much less influence from position errors on the attitude results. For a
small number of stars the positional corrections were so large that only observations
obtained after the corrections to the Input Catalogue had been applied could be
used in the preparation of the final catalogue results.
The star mapper processing for double stars by NDAC resulted in very much
improved positions and in some cases improved separations and orientations for
1643 systems unresolved in the Input Catalogue and with a minimum separation
of 1.5 arcsec. For all unresolved double stars with separations above 1.5 arcsec the
components were entered in the working catalogue: the brighter component under
the original HIC (Hipparcos Input Catalogue) number of the system, and the fainter
component with a new number. The initial positions were set by the separation and
orientation parameters and the original position for the HIC number. The two
components were treated as two separate stars in the star mapper processing, but
with the knowledge of the proximity of the other component.
The star mapper double star results were of considerable help in the processing
of the main-mission double star information, which, due to slit ambiguities, had
often problems with setting proper starting parameters for solutions (Söderhjelm
et al., 1992). Figure 44 compares separations and orientations as given in the Input
Catalogue with the data derived from the star mapper processing.
The catalogue updating process accumulated photometric data for over 90 000
stars, which were used to improve the colour index information required in several
calibration processes. Figure 45 shows the standard errors of the individual photo-
THE HIPPARCOS MISSION
277
Figure 44. Star mapper results for double stars, compared with the Input Catalogue values. A total
of 1643 stars that were entered as a single component in the Input Catalogue were resolved using
star-mapper data.
278
F. VAN LEEUWEN
Figure 45. The standard errors for the accumulated BT and VT magnitudes for constant stars as a
function of magnitude. The solid diagonal lines show the expected relation for Poisson noise. The
increase in noise towards fainter magnitudes is due to background contributions, towards brighter
magnitudes to modelling inaccuracies and saturation.
metric star mapper observations. Errors on the mean values for constant stars were
about 10 times smaller. The photometric processing of the star-mapper data was
very similar to the Tycho photometric reductions, described in Section 8.8.3.
279
THE HIPPARCOS MISSION
8.4. IMAGE DISSECTOR TUBE DATA REDUCTION
The task of the image-dissector-tube-data reductions was to determine the modulation parameters for the transits of program stars passing over the main grid. Observations were collected over time spans of an observational frame, 32/15 s, containing 2560 sampling periods of 1/1200 s. The modulation parameters described
the mean intensity of the signal and the amplitudes and phases of the first and second harmonics of the modulation. In this section the methods used to derive these
modulation parameters are described, as well as statistical tests and calibrations
performed on the results. The reduced data formed the input for the great-circle
reductions, the double star analysis and the main-mission photometric reductions.
8.4.1. The signal model
The estimation procedures used a reference phase pk for each sample k . This reference phase measured the local modulation phase relative to a phase equal to zero
at mid-frame time at the fiducial reference line on the main grid, halfway between
slits 1344 and 1345 (see Figure 5). Methods used to assign precise reference phases
pk to each sampling have been described in detail in Chapter 5 of Volume 3 of ESA
(1997). Using the attitude reconstruction results, the local scan velocities for each
field of view were calculated, and together with the projection-related distortions of
the grid (which was accounted for by describing the stellar positions on the grid in
direction cosines) and the catalogue position of the star (transformed to the epoch
of observation and corrected for aberration), the relative phase for each sampling
was calculated. The grid positions obtained this way were compared with the grid
positions for the positioning of the instantaneous field of view (IFOV), as derived
from the coil current values for the observation and the coil current calibration
matrix. If the discrepancy was more than 10 arcsec, the observation was rejected. A
record was kept of the observed differences as an overall quality control parameter
(see Figure 23). With the precise relative phase estimates for each sample available,
the processing could start.
The signal model described the sequence of statistically independent and
Poisson-distributed counts Nk as having ideally a time-periodic behaviour with
an expected value of E (Nk ) Ik . Various representations of the modulated signal
have been used, but only one of these was implemented in the actual fitting of the
data:
b) = b1 + b2 cos pk + b3 sin pk + b4 cos 2pk + b5 sin 2pk :
(29)
All other representations were derived from this “b-parameters” fit through transIk (
formations. The expression in parameters with direct physical interpretation was
given by:
Ik
= Ib + Is
1 + M1 cos(pk + g1 ) + M2 cos 2(pk + g1 + g2 )
;
(30)
where M1 and M2 are the modulation amplitudes and g1 and g1 + g2 the modulation
phases for the first and second harmonic. The signal and background intensities
280
F. VAN LEEUWEN
were given by Is and Ib respectively. The model as used by FAST was closely
related to this representation:
I (a) = a1 + a2 cos(p + a3 ) + a4 cos 2(p + a5 ):
k
k
(31)
k
The reference phase used by FAST was based on a weighted mean of the first and
second harmonic phases:
g0 = 0:75a3 + 0:25a5:
(32)
For single stars the amplitude ratio of the two harmonics was given by = a4 =a2
and their phase difference by = a5 , a3 , where (, ) are instrument parameters
that are functions of star colour, field of view and position on the grid, as well as
time.
In NDAC a model was used which represented the second harmonic relative to
the first harmonic parameters:
I ( )=
k
1
+
2
cos(pk +
3)
+
4 cos 2(pk
+
3)
+
5 sin 2(pk
+
3)
: (33)
In this representation, 4 and 5 are the instrument parameters that describe the
amplitude ratio and phase difference of the first and second harmonic in the modulation (see Section 8.4.5: The OTF calibration). In the NDAC reductions, g0 = 3
was used as reference phase.
Next to these five-parameter models, three parameter models were used. These
three-parameter models were obtained through the implementation of the instrument parameter calibration values. In the NDAC representation the three parameter
model was simply given by ( 1 , 2 , 3 ), as the remaining two parameters were not
used in defining the signal phase and amplitude. In the FAST representation the
three parameter model was given by:
I (r) = r1 + r2 cos(p + r3 ) + r2 cos 2(p + r3 + ):
k
k
k
(34)
The three parameter models were important in the recognition of double stars. In a
signal composed of the contributions of more than one star, the calibration values
for (, ) or ( 4 , 5 ) no longer applied (see Section 8.4.6).
8.4.2. Relations for double and multiple images
The most important parameter relations were those between the coefficients in
Equations 29 and 30, as they allowed the interpretation of signals obtained from
single and multiple images. Assuming average modulation coefficients for all
1 and M
2 , and the same reference phases pk
components in a multiple image as M
2
for all components, the following relations were derived (where the parameters M
and g2 were eliminated using the OTF calibration):
b1 = I +
X
I
s;j
b
j
1
b2 = M
X
281
THE HIPPARCOS MISSION
Is;j cos g1;j
j
1
b3 = M
X
Is;j sin g1;j
j
1
b4 = M
X
Is;j cos 2(g1;j + )
j
1
b 5 = M
X
Is;j sin 2(g1;j + );
(35)
j
This was used (in a somewhat different form) as a test for single stars (see Section 8.4.6) and as the basic equations for the double star investigations (see Section 9.1).
8.4.3. Data binning
At the time of the development and early implementation of the data reduction software, it was essential to look for ways of limiting the time spent on the processing of
the 1200 IDT samples received every second. Both data reduction consortia opted
for a binning strategy, reducing the main processing tasks by a very considerable
factor in both computing time and complexity.
The binning strategy assigns a reference phase pi to all samples with phases pk
falling in a phase interval pi , dp to pi + dp. With pk = pi + pkji , and ni samples
falling in bin i, the expected count in bin i can be expressed as a modification of
Equation 29:
1 X
ni
Nkji = b1 + b2 n1
i
cos pi
+b3 n1i sin pi
P
P
P
+b4 n1i cos 2pi
+b5 n1i sin 2pi
P
cos pkji , sin pi
cos pkji + cos pi
P
P
cos 2pkji , sin 2pi
cos 2pkji + cos 2pi
sin pkji
sin pkji
P
P
sin 2pkji
sin 2pkji
(36)
The advantage of the binning resulted from the pk being small and the values of
pi being fixed, thus avoiding repeated trigonometric calculations.
Two different implementations were used: the FAST consortium applied a strategy using 64 bins, and neglected the effect of pk , in which case binning corresponded to a quantization of the relative phase. It was estimated that the error on
the relative phase pk resulting from the binning should be less than 10 mas, in
which case the quantization effects on the grid phase estimate would be negligible
with respect to the photon noise. The quantization noise q for l bins was given by:
q =
ps
2 3l
;
(37)
282
F. VAN LEEUWEN
where s is the grid-period of 1.2074 arcsec. For l > 35 it follows that q < 10
mas. l was rounded up to the nearest power of 2, i.e. l = 64, giving q = 5:5 mas
and a noise contribution to the grid phase estimates less than 0.5 mas.
NDAC performed a substantial number of simulations and tests with various
binning strategies, using both noise-free and Poisson-noise simulated data. The
results of these tests showed that a substantial reduction in computing time without
any significant loss of accuracy could be obtained using l = 12 bins and carrying
along up to second order corrections for pk . The noise contribution to the phase
estimates resulting from the binning this way, was less than 0.2 mas.
In the FAST approximation Equation
P 36 reduces to Equation 29, with pk being
replaced by the nearest pi and Ni = Nkji being the total count per bin. In the
NDAC approximation the equations obtained were:
1 X
ni
Nkji = b1 + b2 n1 cos pi
i
P
(1
i
k i
P
pkji
, 12 p2 j ) + cos p P p j
P
P
1
+b4
cos 2p (1 , 2p2 j ) , sin 2p 2 p j
P
P
1
sin 2p (1 , 2p2 j ) + cos 2p 2 p j :
+b5
1
+b3 n
i
sin pi
i
ni
ni
i
P
, 12 p2 j ) , sin p
(1
k i
k i
k i
i
k i
i
k i
i
k i
(38)
The relevant trigonometric terms were calculated once only and stored in memory.
For certain values of the scan-velocity the relative grid phases pk would assume
only a limited number of different values. At nominal scan velocity (168.75 arcsec s,1 ), the phase shift from one sample to the next was 41.92 . At a scanning
speed of 170.456 arcsec s,1 , the phase shift to the next sample was 42.35 , which
meant that the phase shifts of 17 samplings fitted in exactly 2 complete modulation
cycles. Similar situations happened for scan velocities of 173.866 arcsec s,1 (25
samplings in 3 cycles) and 167.178 arcsec s,1 (26 samplings in 3 cycles). In the
case of 64 bins without phase correction, this left a large number of bins unoccupied, and could cause systematic differences between assumed bin-phases and
actual mean bin phases. The latter effect was, however, small with respect to other
sources of noise, most notably the Poisson noise on the photon counts.
The scan velocity was changing continuously, and a resonance situation like
that described above would normally not persist over more than a few frames. No
measures were taken to remedy these phenomena by FAST, while in NDAC they
were implicitly taken care off in the standard reductions. In FAST, the statistical
index T (Section 8.4.4) was able to detect non-negligible biases in the grid phase
such as due to resonances.
A full description of the solution techniques used for the binned equations can be
found in Chapter 5 of Volume 3 of ESA (1997). Solutions were based on weighted
least squares applied iteratively, using in the final solution the previous intensity
estimates as an indicator for the Poisson noise on the photon count.
THE HIPPARCOS MISSION
283
8.4.4. Statistical tests
Two test parameters were derived at this stage: F and T . The first of these, F ,
tested the null hypothesis that no modulation was present in the signal, in other
words, that the observed photon counts collected for a transit were a stationary
white Poisson noise; hence F appeared significant when the photon-counts were
sufficiently modulated. The second of these, T , tested the null hypothesis that
the residuals left after application of the five-parameter model represented a zeromean stationary white noise. Hence, T appeared significant when the residuals
were modulated (failure of the 5-parameter model) or the original photon counts
were not Poisson distributed. The tests were carried out on the binned counts.
In NDAC, a 2 value for the T statistic was derived for the binned mean counts,
^ ):
assuming them to be represented by Ik (b
i2
h1 P
^)
N
,
I
(
b
k
k
i
n
2 =
1 P
Nk i
n
i=1
l
X
j
i
i
(39)
j
where l = 12 is the number of bins and where the denominator represents the
expected variance of the residuals. The 2 value was transformed into a pseudoGaussian variable, which under the null hypothesis should have a unit-normal
distribution:
3
s 2 2 !1=3
T = 92d 4 d
, 1 + 92d 5
(40)
where d is the degrees of freedom, normally equal to l , 5 = 7 for the T statistics.
Transits for which jT j > 4:0 were flagged in the Epoch Photometry Annex file
(hip ep.dat) as suspicious.
In FAST, a binned mean square error was computed as:
h
i2
^)
l
N
,
I
(b
X
i
i
s2 =
(41)
~
i=1 (l=m)Ii (b)
where l = 64 is the number of bins and m the total number of samples used. The
statistic T was then computed as:
s
"
#
l
,
5 s2
T = 2 l,5 ,1
(42)
Under the null hypothesis, s2 follows the 2 distribution with l , 5 = 59 degrees
of freedom and T approaches asymptotically the unit-normal distribution. The null
hypothesis was rejected if jT j > 3:5, which happened in less than 0.5 per cent of
all transits.
284
F. VAN LEEUWEN
In NDAC, a 2 value for the F statistic was derived for the binned mean counts,
assuming them to be stationary:
2 =
h
i
X P NP , b
N
1
l
1
=1
1
2
(43)
k ji
ni
i
^
k ji
ni
As for the T statistic, the 2 value was transformed into a pseudo-Gaussian variable,
except that the degrees of freedom was d = 11.
In FAST the binned mean square of the estimated modulation components was
computed as:
p2 =
X
i
h
i
I (b^ ) , ^b1
=1 (l=m)I (b~ )
l
i
2
(44)
i
having four degrees of freedom. The statistical index F was then defined as a Fisher
ratio between the statistics in Equations 44 and 41:
2
F = s2=p(l=,4 5)
(45)
Since (l , 5) was rather large, 4F almost followed a 2 distribution with four
degrees of freedom. Accordingly, the null hypothesis was accepted (no modulation
found) if F < 4:5. This happened in less than 1.5 per cent of all transits.
8.4.5. Calibration of the Optical Transfer Function
The Optical Transfer Function (OTF) described the characteristics of the modulated
signal over both fields of view. The calibration of the optical transfer function used
the values for (; ) or ( 4 , 5 ) collected over one orbital period of the satellite. The
calibrated values were used in the 3-parameter model described in Section 8.4.6.
The modulation parameters were associated with a position on the grid (G; H )
at mid-frame time, and with the colour index for the star observed. The instrument
parameters ( 4 , 5 ), obtained for stars not known or found to be double, were
modelled with a two-dimensional third order polynomial in position, first and
second order in colour index, and cross-terms between colour and position on the
grid, 14 parameters in total. Every calibration covered the data collected during
one orbit. The calibrations were done independently for the two fields of view. Of
the parameters, the positional dependence was the most important. An example
of this dependence is shown in Figure 46. The calibration was time dependent, as
is shown in Figure 47, and affected by refocussing of the instrument. The colour
dependence in the preceding field of view for 4 increased during the mission by
a factor 2, while for 4 in the following field of view and for 5 this dependence
was much less a function of time. The amplitude ratio 4 decreased towards redder
stars, but could not be calibrated for the very red stars, as too few measurements
THE HIPPARCOS MISSION
285
Figure 46. The dependence of 4 and 5 on the grid position at the beginning of the mission. The
scan direction is indicated by the arrow, the two graphs for 4 and 5 refer to the two fields of view.
Scales for the two fields of view are identical.
were available. At V , I = 2 the decrease was approximately 10 per cent in the
following field of view, less in the preceding field of view. For extreme-red stars
these calibrations were not possible, while an extrapolation of the calibrations was
very unreliable. Duplicity indicators for such stars should therefore be treated with
caution.
In FAST, the calibration of the optical transfer function was, as were most
calibrations, performed twice. An OTF calibration was done in Utrecht by the first
look task once a week. The mean for a calibration period was made and used
by the IDT data processing as described in this chapter. In a second run, the five
286
F. VAN LEEUWEN
Figure 47. Evolution with time for the phase difference and the amplitude ratio . Data for the
preceding field of view are shown by filled symbols (points in the lower parts of the graphs), for the
following field of view by open symbols (points in the upper parts of the graphs). Features in the top
graph around days 450 and 1020 are related to variations in the satellite’s exposure factor, shown in
Figure 2.
parameter solutions obtained in this processing were used in a more refined analysis
intended to be used by the multiple star and photometry tasks. About twenty orbits
contributed to each calibration period, chosen among those giving the best results
in the great-circle reduction.
The procedure was as follows. All the results of the five parameter solution were
examined and several tests performed in order to exclude known double stars, faint
287
THE HIPPARCOS MISSION
stars (magnitude > 11), stars with an unknown colour, and all stars for which either
the ratio of the ‘ac’ to ‘dc’ photometry or the difference a3 , a5 suggested that the
star was double. Additional rejections were made from the analysis of histograms
of these quantities. The ‘ac’ intensity was computed and used to determine M1 and
M2 from a1 , a2 and a4 .
Each quantity M1 ; M2 and a3 , a5 for each field of view was expressed as a
third order polynomial in the grid coordinates G and H plus a similar polynomial
multiplied by C , 0:5, where C is the colour index; this gave 20 parameters in total.
The equations were solved by least squares giving for each a weight proportional
to the intensity of the star (a1 ).
In addition, the reference intensity response was calibrated for a mesh of 19 19
points on the grid for three classes of colours and also by a polynomial of the third
order in G and H , second order in colour and some mixed terms. This formed part
of the FAST photometric calibrations described in Section 9.3.
8.4.6. The Three Parameter Solution
In NDAC, the optical transfer function was applied to the (G; H ) coordinates and
star colour of each frame transit to provide predicted values for 4 and 5 . These
provided, together with the relevant elements of the covariance matrix ,1 , a 2
estimate for the likelihood of the signal being the result of one point-source only:
B
2 = B,1 ]44 + 2
2
4[
B,1 ]45 + 52 [B,1 ]55
4 5[
(46)
where 4 is the difference between the observed and predicted value of 4 , and
similarly for 5 . The 2 values were collected per star for the purpose of double
star recognition. This was done using a transformation of 2 into a variable that has
a flat distribution for single point-source objects, but becomes increasingly skew
for disturbed objects:
c = 8 exp(,0:52 )
(47)
where the factor 8 allowed the accumulation into eight discrete bins by taking
the integer part of c as the index of the bin (from 0 to 7) (see Section 9.1 and
Figure 68). A second criterion for detecting similar problem cases was derived
from the photometric reductions and is described in Sections 9.1 and 9.3.
In FAST the optical transfer function calibration allowed for the representation
of the signal through the three-parameter model, Equation 34. For a true single star
transit, the residuals Ik ( ^ ) , Ik (^) should be a zero-mean white noise, with the
variance estimated by s2 as in Equation 41. The statistical test used the Fisher ratio
F35 :
b
F35 =
p235 =2
s2 =(l , 5)
r
(48)
288
where:
p235 =
F. VAN LEEUWEN
Xm hIk b , Ik r i
(^ )
k=1
(^)
2
Ik (b^ )
(49)
was estimated from the residuals of Equation 34. Also, in this case, the Fisher
variable 2F35 could be assumed to be asymptotically distributed as a 2 distribution
having two degrees of freedom. The null hypothesis (single star) was rejected when
F35 > 3:5. The F35 statistic became unreliable for stars of extreme colour, as these
could not be covered by the calibration model as described above.
8.4.7. Accuracy assessment
At various stages during the software development and testing, first on simulated
data and later on actual mission data, comparisons were made between the FAST
and NDAC results. These comparisons, led by M.A.C. Perryman, showed initially
some problems with the processing of the partially observed stars (see Section 7.2),
but after this was solved, any remaining differences were well below the intrinsic
noise levels.
Both consortia had performed intensive testing before launch using simulated
data (see Chapter 2 of Volume III of ESA, 1989). These tests verified the correct
implementation of the mathematical model used for the IDT data processing to a
high degree of accuracy (0.001 mas for the NDAC tests, using noise free data).
Figure 48 shows an example of formal standard errors obtained for the phase and
mean intensity estimates for frame transits over part of one orbit. Clearly visible is
the effect of variable amounts of observing time assigned to different observations.
These data and their error estimates formed the input for the great-circle reduction,
the main-mission photometric analysis and the double star analysis. In the greatcircle reductions the data for on average 35 frame transits were combined (all frame
transits for a star as obtained during one orbit), while in the photometric reductions
on average 9 frame transits were combined into a field of view transit. From these
graphs it can be seen that for the majority of frame transits the errors on the phase
estimates are between 10 and 60 mas. As attitude errors were highly correlated
over short time intervals, the more relevant errors were those averaged over field
transits. These were about 3 times smaller: 3 to 20 mas, which, compared with
the projection error resulting from errors on the attitude reconstruction, was the
dominant noise contributor for all but the brightest stars (see also Section 8.2.1).
8.5. GREAT-CIRCLE REDUCTIONS
The scanning of the sky by Hipparcos was such that the z axis moved by about
24 arcmin over one revolution of the satellite. There was, as a result, a considerable
amount of overlap between successive full circle scans of the sky. This characteristic
led to the concept of the reference great circle and the great-circle reductions. The
THE HIPPARCOS MISSION
289
Figure 48. Formal errors on the phase estimate ( 3 , expressed in mas) and the mean intensity
( 1 = 1 ) as a function of mean intensity. The errors are clearly dominated by Poisson noise, shown
by the linear relations in the log-log diagram. The discrete line-like features are the result of different
amounts of observing time assigned to the measurements at different times, depending on competition.
290
F. VAN LEEUWEN
Figure 49. Histogram of the total number of single stars per reference great circle. The full line
represents the NDAC data, the dotted line the FAST data.
data was accumulated in batches, referred to as data sets. In the original mission
concept a data set would have spanned 11 to 12 hours, but due to the orbital
problems of the satellite this time limit was effectively set at the total observing
time accomplished during an orbital period. Great-circle data sets can therefore be
recognized in the final mission products through the orbit number. A reference great
circle was associated with each data set. This circle was defined by means of the
position of its pole, chosen either as the mean pole position for the data set or as the
nominal position of the pole for a time halfway the start and end times of the data
set. One data set contained up to 70 000 individual phase estimates, obtained over
4 000 to 16 000 observational frames for between 800 and 2100 different single
stars (see Figure 49).
The task of the great-circle reductions was to relate the phase estimates obtained
from the frame transits for star i in frame k to an actual position on the modulating
grid, Gik (using the best a priori positional information for the observed stars as
available at the time of the reductions), and to relate this position to an abscissae
and ordinate (vi , ri ) with respect to a reference great circle, using the results of
the attitude reconstruction. The relation between Gik and (vi , ri ) was described by
the instrument parameters, improvements to the star positions and improvements
to the along-scan satellite attitude.
The satellite’s attitude was provided in three coordinates: two describing the
instantaneous position of the z axis, and one describing the scan phase. The first
two provided the information on the instantaneous orientation of the modulating
grid with respect to the reference great circle, which defined the projection of the
observed grid position onto the reference great circle. The scan phase provided the
291
THE HIPPARCOS MISSION
provisional linking between the observed transit times and angular displacements
along the reference great circle.
The phase estimates obtained from the reduction of the IDT data defined a position relative to the modulating grid pattern, without determining where on the grid
the image was situated at the reference time. This information was obtained from the
a priori positions given in the Input Catalogue and its updates (see Section 8.3). By
providing positions with accuracies well below 0.1 arcsec there was generally little
or no ambiguity in the identification of the reference slit belonging to a star transit
phase estimate. For some 20 000 faint stars these positional improvements were
only obtained as a result of the first preliminary sphere solutions. At early stages
of the reductions it was not uncommon for the first estimates of the grid position of
a star to be wrong by one or more grid steps. These grid-step errors (inconsistent
choice of the reference grid line over one great circle) and grid-step ambiguities
(consistent but wrong choice of the reference grid line over one great circle) were
resolved in the great-circle reductions and sphere solutions respectively.
A comprehensive description of the great-circle reduction process was presented
by van der Marel (1988) (pre-launch status) and for the post-launch status by van
der Marel & Petersen (1992).
8.5.1. The reduction mechanism
Using a provisional instrument calibration, the star mapper attitude and the best
known catalogue position for an observed star (corrected for proper motion, aberration and relativistic light bending by the Sun and the Earth), predicted values
were obtained. The differences between the predictfor the grid positions, Gcalc
ik
ed and observed grid positions were modelled as a function of corrections to the
along-scan attitude (one parameter for every observational frame), the instrument
calibration parameters (34 parameters, not all of which were always estimated),
and the corrections to the star abscissae (about 1500 parameters):
@G
@G
G = @G
v +
+
d + @v
@
@d
ik
ik
ik
i
i
ik
k
k
ik ;
(50)
where ik is a general noise term including:
the photon noise error 0ik ;
the projection error on the reference great circle:
00
ik
@G
@G
+
;
r +
= @G
@r
@
@
ik
i
i
ik
k
k
ik
k
k
(51)
modelling errors in the instrument parameters, the attitude modelling and the
star position;
the linearization errors
The values of Gik were occasionally affected by grid-step errors due to poor
quality a priori star positions or due to problems with the attitude reconstruction.
These problems largely disappeared with the introduction of better star catalogues.
Thus, on average some 50 000 observations equations with 12 000 unknowns
were to be solved. This is the kind of problem that is relatively rare in astronomy but
292
F. VAN LEEUWEN
fairly common in geodetics. In both consortia, the great-circle reduction software
was prepared at or in collaboration with geodetic institutes: for FAST by D. van
Daalen and H. van der Marel of the Geodetic Institute of the Technical University
in Delft, and for NDAC by C.S. Petersen Copenhagen University Observatory in
collaboration with K. Poder of the Geodetic Institute in Copenhagen.
The observation equations were written such that the parameters in each of the
three groups (stars, attitude and instrument) appeared together: first the very sparse
attitude parameters, then the sparse stellar parameters, and last the well covered
instrument parameters. Giving the observed grid-coordinate residuals Gik and
the unknown corrections to the parameters by the m-vector and the n-vector ,
the observation equations were written as:
y
y = Ax + e = Aa xa + Asxs + Aixi + e;
x
(52)
where the subscripts a, s and i refer to attitude, stellar and instrument parameters
respectively. The underscores are used to emphasize the stochastic nature of the
observations and the residual noise . Given the (diagonal) covariance matrix y
of the observations , the least squares solution was computed by solving the
normal equations:
y
e
Q
x^
(AT Qy,1A)^x = AT Q,y 1y;
(53)
using Cholesky factorization. After solving and backsubstituting the instrument
parameters (see Section 8.5.3), the remaining Cholesky factors were rewritten
as two triangular systems, which were solved by simple forward and backward
substitution.
Even after backsubstitution of the attitude parameters, the stellar part of the
normal equations was still very sparse. By storing only non-zero elements and
reorganizing the normal equations considerable savings in computing time were
obtained. A subset of the non-zero elements of the covariance matrix for the starsolutions was obtained as part of these calculations. This type of solution, where
one attitude parameter was solved for every observational frame, was referred to
as the geometric solution. The smoothness of the attitude of the satellite allowed,
however, the attitude corrections in consecutive observational frames to be linked.
8.5.2. Attitude smoothing and accuracy assessment
In the smoothed attitude solution the individual frame parameters a were replaced
by a functional representation, using cubic splines, thus reducing considerably the
number of unknowns. At times of thruster firings, a knot was introduced with
a discontinuity in the first derivative. Other knots were distributed over a thruster
firing interval according to the requirements of the data. The smoothed solution was
computed as an update to the geometric solution, using the already well determined
instrument parameters. The smoothed solution produced much reduced noise levels
on the stellar abscissae (see Figure 50).
x
293
st.dev. g/c abscissa (smoothing) [mas]
THE HIPPARCOS MISSION
st.dev. g/c abscissa (geometric) [mas]
Figure 50. Star-abscissa improvements by attitude smoothing (data from 21 May 1990, day 506).
Figure 51. The abscissa residuals correlation coefficient as a function of the separation on the
reference great circle for all data sets. The NDAC curve can be distinguished from the FAST curve by
its lower minima and higher maxima, differences that become more pronounced as data sets become
shorter.
The attitude updates, linked to the positional updates, introduced correlations
between the abscissae for neighbouring stars and for stars separated by integer
multiples of 58 , the basic angle between the two fields of view. These correlations,
294
F. VAN LEEUWEN
Figure 52. Standard deviations ( in mas) for the abscissa residuals as a function of the Hp magnitude.
Open symbols refer to FAST data, filled symbols to NDAC data. The diagonal line represents the
expected relation for Poisson noise on the photon counts only, given one and the same integration
time for all intensities. The flattening of the relation towards brighter magnitudes shows the 2 mas
noise contribution from the instrument parameters and attitude modelling contributions.
which were estimated before the start of the mission by Lindegren (1988), are shown
averaged over all mission results in Figure 51. They tended to be stronger for data
sets containing smaller amounts of data (see van Leeuwen & Evans, 1997 and
Section 11, where it is also described how to account for such correlations when
combining proper motion or parallax data for stars in a small field on the sky).
Various comparisons were made between NDAC and FAST results for the greatcircle reductions. One of the prominent features was some instability with respect
to a 6th harmonic, causing a modulation of the abscissa results. Such modulations,
which could not be suppressed in the great-circle reduction process, were removed
in the sphere solution (see Section 8.6). The sensitivity to these harmonics increased
with decreasing length of the data sets and with the presence of gaps in the data
(resulting from repeated occultations), leading to not fully closed great circles.
Occultations were most frequent for great circles having a small inclination with
respect to the ecliptic plane. The position of the pole of every great circle is stored
in hip rgc.dat on disk 5 of the ASCII CD-ROM set.
The final standard errors of the abscissa residuals were the result of two main
error contributors: the Poisson noise of the original photon counts and errors in
the instrument and attitude modelling. Figure 52 shows the observed standard
errors for FAST and NDAC. The better performance of the FAST results for the
intermediate brightness range is due to the inclusion of the second harmonic phase
information. This gain is becomes less for the fainter stars and causes additional
modelling problems for the brighter stars. Towards the brighter stars the errors are
dominated by contributions from the instrument and attitude modelling, giving a
standard error of close to 2 mas.
295
THE HIPPARCOS MISSION
The diagonal line in Figure 52 indicates the expected relation for Poisson
noise only. The actual relation deviates from this due to the increasing amounts
of observing time assigned to fainter stars. As a result, the signal intensity is not
directly proportional to the total number of photons received for an observation,
and the relation between standard errors and magnitudes for fainter stars reflects
both observing time and Poisson noise.
8.5.3. Instrument parameters
The instrument parameters described the relations between an ideal grid and the
actual grid for both fields of view, as well as the basic angle between the two
fields of view. The instrument parameters evolved over the mission as a result of
the satellite and payload settling down to the conditions in space. This was most
clearly seen in the settings for the optimal focus.
The instrument model consisted of a 2-dimensional third-order polynomial in
positional coordinates, the basic angle and chromaticity components for the basic
angle, scale and grid orientation. Fourth order terms were in some cases also
included. Thus, there was a basic set of 25 instrument parameters, extended to 34
when fourth order terms were included. In the NDAC representation the parameters
for the two fields of view were represented by the mean values (g -parameters) and
half the differences (h-parameters) between the fields of view (thus, gij hij gives
the relevant parameter ij for the preceding field of view, and gij , hij similarly
for the following field of view). The parameters were given a two-digit index,
indicating the polynomial degree for the x and y coordinate on the grid. Thus, h00
represents half the basic angle (shown in Figure 8), g10 the mean scale along the
scan-direction and h10 is half the difference in scale for the two fields of view. g01
represents the mean orientation of the grid and h01 half the difference in orientation
for the two fields of view.
In the FAST representation the instrument parameters were given per field
of view, with the same two-digit index as for the NDAC parameters. The FAST
parameters for the preceding field of view are given as apij and for the following
+
field of view as afij , supplemented by
0 for the basic angle correction at the
centre of the main grid.
Figures 53 and 54 show the evolution of the first order parameters over the
Hipparcos mission. The scale parameter was clearly affected by refocussing (discontinuities), while the orientation was affected by problems with the thermal
control electronics. Both parameters show the results of the thermal changes of
Sun-pointing mode observations. Also visible, and indicated in Figure 53, are the
results of a number of anomalies, such as failures of the thermal control electronics
(see also Section 6 and Figure 18). These show as a sudden drift in parameters,
followed by a restoration of the normal proceedings at the time the problem was
detected and cured.
Differences between the FAST and NDAC instrument parameters occurred
because of the differences in the definition of the reference phase, with the NDAC
296
F. VAN LEEUWEN
Figure 53. Variations in the first order parameters for the FAST instrument model. Discontinuities
are in general the result of refocusing, but could also be caused by thermal control problems in the
payload. Known problems are indicated in the second graph. Table V provides further information
on these anomalies.
THE HIPPARCOS MISSION
Figure 54. As the previous Figure, but for the NDAC data.
297
298
F. VAN LEEUWEN
Figure 55. Definition of the angles and directions used to calculate the relative abscissae on a
reference great circle. s is the direction to a star, p the position of the pole of the reference great circle,
r the direction of the crossing of the reference great circle with the equator, and v the abscissa value
gives the ordinate for this star.
for the star at s. The angle 0:5
,
reference phase depending only on the first harmonic, while FAST used both the
first and the second harmonics (see Section 8.4.1).
The instrument parameters were also examined in terms of physical parameters
of the optical system by Lindegren et al. (1992b) for the first 12 months of the
mission. An examination of all mission results is presented in Section 10.5 in
Volume 2 of ESA (1997). This model showed the first order positional changes
for the optical elements of the Hipparcos telescope that could be derived from the
instrument parameters.
8.6. SPHERE SOLUTION
The product obtained from the great-circle reduction was a set of 2341 reference
great circles with very precise relative abscissae for on average 1500 stars each.
These abscissae had been corrected for aberration and relativistic light bending
effects, so the only remaining information consisted of the astrometric parameters:
the position as a function of time and the parallax. The sphere solution process
was designed to determine consistent zero-points for all reference great circles, so
that the very precise relative abscissae could be compared and used to calculate
the astrometric parameters. The zero-points were defined by the crossing of the
reference great circle and either the ecliptic plane (FAST) or the equatorial plane
(NDAC) (see Figure 55). The sphere reconstruction process also took care of
the removal of instrument parameters that were out of reach of the great-circle
reduction process (chromaticity) and parameters related to instabilities in the great-
THE HIPPARCOS MISSION
299
circle reductions (the 6th harmonic). Reports on the pre-launch status of the sphere
solution were presented by Lindegren & Söderhjelm (1985), Tommasini Montanari
et al. (1985), and in Chapters 7, 8 and 9 in Volume III of ESA (1989). A full report
on the implementation of the sphere solution in the data reductions is provided in
Chapter 11 of Volume 3 of ESA (1997). Intermediate results on the sphere solutions
were presented by Kovalevsky et al. (1995) and in Chapter 16 of Volume 3 of ESA
(1997).
In the sphere solution three sets of parameters were to be estimated:
Corrections to the astrometric parameters of the stars used as reference points;
Corrections to the assumed zero-points of the reference great circles;
Global parameters.
Using a catalogue of a priori astrometric parameters, predicted abscissae were
calculated for a selection of primary reference stars, all of which had positions
reliable enough to avoid any slit errors and ambiguities to disturb the processing.
The differences between the predicted and measured abscissae were collected
in a vector . The sphere solution modelled those differences as functions of
corrections to the assumed astrometric parameters, and the global parameters
, giving the corrections to the zero-points, the 6th harmonic for each reference
great circle and the chromaticity in each field of view:
v
a
b
Aa + Bb = v + ;
(54)
A
where the vector represents the normalised measurement errors. The matrix
contains the derivatives of the abscissae with respect to the astrometric parameters
and are described in more detail in Section 8.7. The full set of normal equations is
given by:
0
AA
0 B a A0 v A
B0A B0 B b = B0 v :
(55)
a
Through elimination of the stellar unknowns , two sets of equations were
obtained:
i
h 0
B B , B0 A(A0 A),1A0 B b = B0v , B0 A(A0A),1 A0v
(56)
and
A0Aa = A0 v , A0 Bb:
AA
(57)
Given Np primary reference stars, the matrix 0 becomes a 5Np 5Np blockdiagonal matrix, i.e. zero everywhere except for Np blocks of size 5 5 along the
diagonal. Thus, it becomes a simple process to compute the following two vectors:
a A0 A,1 A0 v ;
(58)
a provided the first estimate of the astrometric parameter corrections, and:
where f
f
v = v , A0 fa:
(59)
f =
300
F. VAN LEEUWEN
Substituting Equations 58 and 59 into Equation 56, a set of equations was obtained
for the first approximation of the parameters :
h
i
B0 B , B0 A(A0 A),1A0 B b = B0fv;
b
(60)
where the left-hand side contains a practically filled symmetric matrix of size
Nb Nb .
After sorting the observation equations on star number, Equation 58 could be
solved on a star-by-star basis, giving the provisionally corrected abscissa results
of Equation 59. In the process of calculating these provisional results, the information for Equation 60 was accumulated, providing as soon as all stars had
been processed. These equations were solved by means of Cholesky factorization.
Using the first estimates of the abscissae zero-points and global parameters ,
the abscissa residuals were recalculated and the process was repeated. The iteration continued until corrections to the parameters became negligible, typically
contributing less than 0.001 mas. With this convergence, also the astrometric parameters had reached their final values. This was the NDAC approach. The FAST
approach differed in a number of details, in particular with respect to the selection
of reference stars, and can be found in Chapter 11 of Volume 3 of ESA (1997).
In the NDAC approach, the ‘pre-adjustment’ of the astrometric parameters, by
means of the provisional updates , had some advantages. It allowed any well
determined astrometric solution to contribute to the sphere solution and it allowed
the early detection of outliers and grid-step errors.
One problem in the determination of the great-circle zero-points was that the
solution had a small rank-deficiency, which did not allow for independent determination of all zero-points as well as the astrometric parameters of the stars. This
problem was partly solved by adopting a minimum-norm solution, which effectively defined the average correction for all the zero-points to be zero. The errors
on the zero-points were of the order of 0.1 mas were defined as the standard errors
on the ‘centred’ variables: identical to the minimum norm values (see Chapter 11
in Volume 3 of ESA, 1997). The rigid but free-floating reference frame in positions
and proper motions, as based on the Hipparcos measurements only, was later fixed
to a ground-based reference frame (see Section 8.7.4).
b
b
b
a
8.7. MERGING AND ASTROMETRIC PARAMETER DETERMINATION
The comparisons between sphere solutions and astrometric parameter determinations as obtained by the two consortia showed that there would be a substantial
gain to be made in merging the two solutions rather than choosing a single one.
The results obtained in the two data reduction chains showed an average correlation coefficient of 0.7 between the remaining abscissa residuals, and merging the
two solutions would reduce the uncorrelated errors that resulted from differences
in instrument and attitude modelling. In addition, the combined consortia results
provided more independent abscissa data than each consortium alone.
301
THE HIPPARCOS MISSION
1.1
Unit-weight error of abscissae
FAST
NDAC
1.0
0.9
200
400
600
800
1000
1200
1400
1600
Days from 1989 January 0
Figure 56. Evolution of the unit weight errors of the abscissa residuals, based on the formal abscissa
errors supplied by the reduction consortia. Each curve is a running median over 50 orbits ( 22 days).
8.7.1. Determination of the variances and correlations
In order to preserve the proper statistical information on the final astrometric
solutions (the covariance matrices) it was necessary to merge the data at the level
of abscissa results, i.e. the great-circle reduction results corrected for zero-points,
chromaticity and other global parameters. The basic ideas for the merging of the
abscissa results were proposed by C.A. Murray and implemented by F. Arenou at
Meudon.
During the mission a number of sphere solutions were carried out, including
or followed by the determination of the astrometric parameters. The final sphere
solutions and astrometric parameter determinations from both consortia were used
in the merging process. It was essential to first reduce the two sets to one common
reference frame through purely orthogonal rotations. These rotations were at a
level of 30 to 50 mas, and do not have any physical meaning. They only reflect
the convergence of the working catalogue used in each consortium. The rotations
that were applied to the final NDAC and FAST catalogues (referred to as N37.5
and F37.3 respectively) aligned these catalogues with an earlier merged catalogue,
referred to as H30. The abscissa records for the N37.5 catalogue did not include
data for many of the double and multiple stars, but this type of data (for which
the abscissa data are disturbed) was included in the abscissa records for the F37.3
catalogue.
302
F. VAN LEEUWEN
1.2
Unit-weight error of abscissae
FAST
NDAC
1.1
1.0
0.9
0.8
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Standard errors of abscissae (mas)
Figure 57. Unit weight errors of abscissa residuals as functions of the formal standard errors.
After applying the rotations, averaged (preliminary) astrometric parameters
were determined for stars in common to the two solutions. Abscissa residuals
for both catalogues were determined relative to these averaged solutions. New
standard error estimates for the astrometric solutions were obtained for both sets.
These errors were compared with the standard errors of the abscissae and examined
as a function of time over the mission (see Figure 56) and as a function of the
abscissa error estimates provided by the consortia (Figure 57). A scaling of the
error estimates was introduced that reproduced the proper unit weight errors. The
total number of abscissae thus obtained for single stars is shown in Figure 58. The
two peaks and the wing towards large numbers of observations are the same features
as observed for the photometric data in Figure 15. A spatially resolved graph of
these data can be found on page 329 of Volume 1 of ESA (1997), Figure 3.2.3.
The correlations between abscissa residuals for the two solutions were modelled
as a function of the stellar magnitude, the quality of the great-circle reduction results
and the time in the mission. They were obtained by accumulating abscissa residuals
for single stars, in common to both solutions. The correlations between the FAST
and NDAC residuals were derived from the observed variance of the differences
THE HIPPARCOS MISSION
303
Figure 58. Histogram of the number of abscissa data per star for all single stars. As both FAST
and NDAC abscissae were counted, even numbers of abscissae are more common than odd numbers,
which explains the saw-tooth shaped distribution.
between the FAST and NDAC abscissa residuals,
deviations F and N :
= F + N , 2Var(vF , vN ) :
2
vF , vN , and the standard
2
F N
(61)
8.7.2. Fitting models to the abscissa residuals
The determination of the astrometric parameters was split into two different problems: the stars with undisturbed abscissa measurements (single stars and double
stars with separations less than 0.1 arcsec or a magnitude difference of more than
4 mag), and stars with disturbed abscissa measurements (double and multiple stars
and stars in planetary nebulae). The abscissa results could only be used for the
first group. Stars in the second group were treated in the double stars analysis (see
Section 9.1).
The abscissae for the first group, which will be referred to as single stars even
though in many cases the stars are not really single, can be fitted with a range of
models. Each model is described by a set of parameters a and the changes to the
abscissa residuals v resulting from changes in these parameters:
v + =
X
i
@v
@ai
ai;
(62)
where the vector represents the standard errors on the abscissae. The vector v
contains observations from both reduction consortia. Equation 62 is solved by
304
F. VAN LEEUWEN
minimizing the 2 value, taking into account the correlations that exist between
the consortia data:
2
v , @@va a 0V,1 v , @@va a ;
(63)
where V is a block-diagonal matrix, i.e. zero everywhere except for blocks of
size 2 2 along the diagonal wherever there are FAST and NDAC measurements
for the same abscissa value. These data, which are referred to as the Intermediate
Astrometric Data, can be accessed on the ASCII CD-ROM, disk 5, and are described
as the hip i.dat file in Section 2.8 of Volume 1 of ESA (1997). Their use is also
described by van Leeuwen & Evans (1997), from which the following simplification
for solving these equations is obtained.
The matrix V,1 can be written as the product of a lower triangular matrix and
its transverse:
)
= (
(
V,1 = T0T:
)
(64)
The lower triangular matrix T is referred to as the Cholesky square root of the
symmetric matrix V,1 . After substituting Equation 64 into Equation 63 and some
reorganization, the following equation is obtained:
0 @v
@v
2
T v , @a a T v , @a a :
(65)
= (
) (
)
The matrix T removes the correlation between NDAC and FAST determinations
of the same abscissa residual. Measurements can be recognized as such by means
of the orbit number (field IA1 in file hip i.dat). This involves never more than two
observations at one time:
!
1
0
F
;
T
(66)
p,NF
p1
=
F
,
1 2NF
N
,
1 2NF
where NF is the correlation coefficient between the FAST and NDAC determinations for the abscissa residual. This correlation coefficient, as well as the standard
errors F and N are given in the Intermediate Astrometric Data file in fields
IA9 and IA10, with the abscissa residual given in field IA8. Thus, a pair of uncorrelated and properly weighted observation equations is obtained by first writing
out the full observation equations for the FAST and the NDAC determination, and
then multiplying these two equations by T as defined in Equation 66. When only
a FAST or a NDAC observation is available, matrix T reduces to a single element,
giving the inverse of the standard error on that observation. This also applies to the
remaining observation if either the FAST or NDAC observation was rejected in the
solution.
8.7.3. Astrometric parameter modelling
The basic astrometric parameter model for fitting the abscissa residuals provided
by the Intermediate Astrometric Data file assumes a linear proper motion in right
305
THE HIPPARCOS MISSION
Positional Offset
J1992.55
J1992.15
30
J1991.25
J1989.95
40
J1990.35
50
cubic
quadratic
20
standard
10
0
-10
G
-20
200
400
600
800
1000
1200
1400
1600
Days from 1989 January 0
Figure 59. Schematic representation of the baricentric position of the photocentre, as a function of
time, in a -type (acceleration) solution. The three curves show an example of cubic solution (full
line) and the related quadratic and standard solutions. At the indicated epochs the quadratic or the
cubic term vanishes from the solution. The heavy dot corresponds to the position at the reference
epoch J1991.25.
G
ascension and declination and a parallax, giving a total of 5 parameters: ( cos ,
, , cos , ). In the Hipparcos and Tycho Catalogues cos is given as :
(t)
=
0 + t
p (t)
0 + t + p (t);
+
(67)
(t) =
where t is measured in years from the reference epoch J1991.25, and p (t) and
p (t) are the parallax factors in right ascension and declination. For the majority of
stars the 5 parameter model was all that was required to fit the abscissa residuals.
They are the 100 038 stars for which field H59 in the Hipparcos Catalogue was left
blank (not counting the 263 stars for which no solution was obtained). For these
stars all astrometric data (the five parameters given above, their standard errors and
their covariances) are given in the Hipparcos Catalogue, fields H8 till H28. The
coefficients @ v=@ a for the five-parameter solution are given for every observation
in the Intermediate Astrometric Data file, as fields IA3 to IA7. The five-parameter
solution relative to which the abscissa residuals were determined are given for each
star in the file header records, fields IH3 to IH7, which also identifies the type of
solution accepted for the final catalogue (field IH8).
In the process of fitting the astrometric data, measurements could be rejected.
The percentage of rejected data is given in field H29. If too many measurements
were rejected or the goodness of fit of the solution was unsatisfactory, a solution
within which the proper motion was a linear or second order function of time was
306
F. VAN LEEUWEN
attempted. These solutions are referred to as ‘G’-type solutions. They added two
or four coefficients to Equation 67:
(t) = : : : + 12 (t2 , 0:81)g
(t)
=
::: +
+
1 2
(t , 1:69)tg_
6
1 2
1 2
(t , 0:81)g + (t , 1:69)tg_ ;
2
6
(68)
where g d =dt, g d =dt, g_ d2 =dt2 and g_ d2 =dt2 . The
time t is again measured in years from the reference epoch J1991.25. The reference
times for the first and second order time dependence coefficients were chosen such
that their influence on the values derived for the linear proper motion was minimal
(see Figure 59).
The significance of the g and g_ terms was determined by means of the statistics
Fg and Fg_ , derived from covariance matrix of the solution:
2
g ;
Fg = g g g g2 g
(69)
g
g g g
and similarly for Fg_ . By requiring Fg > 3:44, the same level of significance
(0.27 per cent) was obtained as for two-sided 3 criterion for a normal distribution.
All the information for the G-type solutions is contained in the DMSA/G file:
2
,
hip dm g.dat on disk 1 of the ASCII CD-ROM set. There were in total 2163 7parameter solutions accepted, and 459 9-parameter solutions. Most of these cases
are likely to be associated with orbital motions in astrometric binaries with periods
of 10 years and longer. Some cases will turn out to be spurious, which is most
probably the case for a 7-parameter solution for one of the Large Magellanic Cloud
stars.
With the provision of the Intermediate Astrometric Data it always remains
possible to redo a 7- or 9-parameter solution with different constraints, or as part
of an orbital motion solution. When applying 7- or 9-parameter solutions to the
Intermediate Astrometric Data, one or two pairs of additional derivatives have to
added:
@v
@ai+5
=
1 2
@v
;
(t , 0:81)
2
@ai
=
1 2
@v
(t , 1:69)
;
2
@ai+3
i = 1; 2;
(70)
and
@v
@ai+7
i = 1; 2:
(71)
The abscissa residuals for 235 stars were solved in orbital motion solutions, often
incorporating ground-based data. For these stars the 5 basic astrometric parameters
were supplemented by 7 orbital elements. In most cases only some of these were
estimated based on Hipparcos data alone, and often estimates were obtained for
orbital parameters from a combination of Hipparcos and ground-based data. Some
THE HIPPARCOS MISSION
307
examples of these solutions were presented by Söderhjelm et al. (1997) and Martin
& Mignard (1997). The DMSA/O file (hip dm o.dat on disk 1 of the ASCII CDROM set) provides all information on the systems dealt with before the finalization
of the data reductions. However, in particular for orbital binaries there is still scope
for improvements and some new detections, in particular for short period (less
than 3 years) systems, as was shown by Bernstein & Bastian (1995) and Bernstein
(1997). The systems that qualify for further investigations are those for which a
so-called stochastic solution was obtained for the astrometric parameters.
The stochastic solution, indicated by ‘X’ in field H59, was applied to 1561 stars
for which there was a clear indication of unmodelled disturbances to the abscissa data. These disturbances, which could be signs of orbital motion, unresolved
duplicity or multiplicity, but may also be an indication of disturbing nebulae very
close to the star, were ‘modelled’ in the form of an additional ‘cosmic noise’,
added in quadrature to the standard errors of the observations. The level of cosmic
noise added was such that the unit weight variance of the solution became equal
to one. If the amount of ‘cosmic noise’ required was more than 0.1 arcsec, the
solution was rejected, and no astrometric parameter solution has been provided in
the Hipparcos Catalogue. The original abscissa data for all these stars are available
in the Intermediate Astrometric Data file, and can be examined further. The details
of the stochastic solutions, i.e. the amount of cosmic noise added, is provided in
the hip dm x.dat file on disk 1 of the ASCII CD-ROM set. It should be realized
that the goodness-of fit parameter in field H30 has no meaning for such solutions.
Finally there was a small group of stars (288) for which the abscissa residuals
appeared to be magnitude dependent. This group contains mainly large-amplitude
red variable stars. There are two possibilities for stars in this group. A companion
may become visible when the main component of the system is at a photometric
minimum, or, while the main component is at a minimum, its signal becomes
much more vulnerable to incidental superpositions. It was not always possible to
distinguish these two types of disturbances. In addition, there were problems with
properly recognizing duplicity for very red stars.
The variability induced movements of the photocentre were modelled by adding
the following parameters to Equation 67:
(t) = : : : +
(t)
=
::: +
100:4(Hptot (t),Hpref ) , 1
100:4(Hptot (t),Hpref ) , 1
D
D ;
(72)
where D = 10,0:4(HpC ,Hpref ) d and D = 10,0:4(HpC ,Hpref ) d . The differences in magnitude and in position could not be separated. An example of a type ‘V’
solution is shown in Figure 60. A statistical test very similar to the one described
for the g -parameters in the 7- and 9-parameter solutions was applied to (D , D ).
However, the criterion for acceptance was set lower, allowing a larger fraction of
spurious systems to enter the catalogue. Thus, any variable star that, on the basis
of these measurements, could be suspected to be a double star, has been marked as
308
F. VAN LEEUWEN
50
J1991.25
40
Positional Offset
30
photocentre
20
standard
10
0
-10
V
-20
200
400
600
800
1000
1200
1400
1600
Days from 1989 January 0
Figure 60. Schematic representation of the baricentric position of the photocentre, as a function of
time, in a ‘V’ type solution (variability induced mover). The photocentre describes an oscillating
motion between the binary components, in phase with the variations of total luminosity of the system.
The dashed line gives the hypothetical motion of the photocentre had the total magnitude been
constant, corresponding to the astrometric parameters given in the main catalogue.
such in the final catalogue. All information for the ‘V’ solutions can be found in
the hip dm v.dat file on disk 1 of the ASCII CD-ROM set.
8.7.4. Transformation to the ICRS
The Hipparcos reference frame as obtained from the sphere solutions, the merging
and the determination of the astrometric parameters, was rigid but not attached to
inertial reference points. An inertial reference frame at mas level accuracy had been
established already in radio astronomy, based on radio positions of extragalactic
point sources: the ICRF. The final realisation of the Hipparcos Catalogue within the
extragalactic reference system was described by Kovalevsky et al. (1997), and the
reader is referred to that paper for details. To summarize, the following individual
link solutions were available:
VLBI observations, multi-epoch observations of 12 radio-emitting stars
obtained between 1984 and 1994 (Lestrade et al., 1995);
Observations with Merlin, astrometric observations of 9 radio-emitting stars
(Morrison et al., 1997a);
Observations with VLA of radio-emitting stars between 1982 and 1995
(Florkowski et al., 1985);
Optical positions of compact sources, a Hamburg/USNO program to determine
precise optical positions of about 400 to 500 selected compact radio sources
THE HIPPARCOS MISSION
309
which display optical counterparts (Ma et al., 1990, Johnston et al., 1995,
Zacharias et al., 1995);
Observations with the Hubble Space telescope, measurements of angular separations between Hipparcos stars and extragalactic sources known from VLBI
observations (Benedict et al., 1992);
The Lick proper motion program, proper motions for 149 000 stars from 899
fields of the Northern Proper Motion program, determined relative to galaxies
(Klemola et al., 1994);
Catalogue of faint stars (KSZ), containing 977 Hipparcos stars (Rybka &
Ytsenko, 1997);
The Yale/San Juan Southern Proper Motion program, containing 4100 Hipparcos stars in 63 fields (Platais et al., 1995);
The Bonn link solution, using large epoch differences to obtain very accurate
proper motions (70 to 100 years); Hipparcos stars were measured relative to
extragalactic radio sources (Tucholke et al., 1997, Geffert et al., 1997);
The Potsdam link solution, based on plates taken with the Tautenburg Schmidt
telescope; proper motions for 360 Hipparcos stars in 24 fields (Dick et al.,
1987);
Comparison of Earth Orientation parameters as determined by optical astrometry and VLBI, providing an indirect link of the Hipparcos coordinate system
to the extragalactic coordinate system (Vondrák, 1996, Vondrák et al., 1997).
The results of all these methods are summarized in Tables X and XI as residuals relative to the adopted solution for the spin components and the orientation
components respectively.
In the general synthesis of the accumulated data two independent approaches were used, referred to as Method A (by L. Lindegren) and Method B (by
J. Kovalevsky). The main problem was posed by the claimed accuracies of some
of the determinations, and the main difference between the two methods used to
combine the data was in determining appropriate weights for the different determinations. In Method A a first solution was made using the a priori weights. This
showed the presence of some clearly unrealistic accuracy estimates. The weight
for the most discrepant solution was halved, and the overall solution was repeated.
This process was iterated until the overall goodness-of-fit agreed with the expected
value. The most severely down-weighted solutions were the SPM program and the
Earth orientation parameters.
In Method B the covariance matrix for the solution, using the provided accuracies
of the various methods, was obtained. From this covariance matrix were derived
(under the assumption that all error distributions were Gaussian) the standard
error corrections for each contribution. This was done for the spin and orientation
parameters separately, as well as for a combined solution. In the final solution the
mean of both methods was retained, with differences between the results for the
two methods at a level of 0.15 mas/yr for the spin parameters and at 0.10 mas for
the orientation components. The final standard deviations for the spin components
310
F. VAN LEEUWEN
Table X
Summary of the spin components results as residuals with respect to the
adopted solution. Accuracies are given in brackets.
Method
!x
VLBI
HST/FGS
NPM (Heidelberg)
NPM (Yale)
KSZ Kiev
SPM (blue, )
SPM (blue, )
SPM (visual, )
SPM (visual, )
Bonn plates
Potsdam plates
EOP
Spin components (mas/yr)
!y
!z
,0.16 (0.30) ,0.17 (0.26) ,0.33 (0.30)
,1.60 (2.87) ,1.92 (1.54) +2.26 (3.42)
,0.77 (0.40) +0.15 (0.40)
+0.09 (0.18) ,0.20 (0.18)
,0.27 (0.80) +0.15 (0.60) ,1.07 (0.80)
+0.23 (0.13)
+0.07 (0.15)
+0.44 (0.12)
+0.30 (0.12)
+0.93 (0.34)
+0.22 (0.52)
0.93 (0.28)
,
+0.50 (0.20)
+0.58 (0.08)
+0.71 (0.18)
+0.76 (0.06)
0.32 (0.25)
+0.43 (0.50)
0.32 (0.28)
,
,
0.00 (0.08)
,0.30 (0.07)
+0.17 (0.33)
+0.13 (0.48)
Table XI
Summary of the orientation components results as residuals with respect to the
adopted solution. Accuracies are given in brackets.
Method
VLBI
MERLIN
VLA
Hamburg/USNO
HST/FGS
EOP
x
Orientation components (mas)
,0.10 (0.47)
+1.41 (2.60)
+4.27 (4.70)
+3.38 (5.00)
6.10 (2.16)
+2.33 (0.88)
,
y
z
+0.08 (0.49)
+0.16 (0.50)
,0.64 (2.20) +0.51 (2.40)
,3.75 (5.30) ,5.76 (5.20)
,0.06 (4.90) ,9.20 (4.70)
,3.25 (1.49) +5.42 (2.14)
+7.80 (0.90)
epoch
1900+
91.3
94.0
86.3
88.5
94.3
85.0
was estimated at 0.25 mas/yr and for the orientation components at 0.6 mas. Further
improvements of the extragalactic link are still in progress, using new observations
from which more accurate radio proper motions and reference positions are derived
(Morrison et al., 1997a).
8.7.5. Accuracy assessment of the astrometric parameters
As was explained in Section 5, the amount and distribution of the observations
obtained with the Hipparcos satellite was very much determined by the characteristics of the scanning law, and as a result the accuracies and covariance matrices
as obtained for the astrometric parameters show a clear dependence on ecliptic
311
THE HIPPARCOS MISSION
Table XII
Median standard errors in parallax, , for the entries in the Hipparcos Catalogue with
associated astrometry, divided according to ecliptic latitude and median magnitude
(field H44), expressed in milliarcsec (mas).
jj
Ecliptic latitude ( , deg)
20–30 30–40 40–50 50–60
Hp
0–10
10–20
<2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
10–11
11–12
12
0.88
0.88
0.91
0.88
0.88
0.92
1.02
1.21
1.49
2.01
2.98
4.35
0.98
0.81
0.82
0.86
0.85
0.90
1.01
1.19
1.47
1.97
2.96
4.19
0.90
0.77
0.79
0.81
0.82
0.86
0.97
1.16
1.43
1.93
2.81
4.11
0.78
0.76
0.74
0.76
0.75
0.81
0.92
1.12
1.37
1.82
2.74
4.10
0.64
0.68
0.65
0.66
0.67
0.73
0.86
1.07
1.35
1.84
2.73
4.00
all
1.23
1.24
1.20
1.15
1.11
60–70
70–90
0.60
0.56
0.55
0.55
0.56
0.61
0.72
0.89
1.11
1.54
2.26
3.67
0.53
0.49
0.52
0.51
0.52
0.55
0.63
0.77
0.94
1.24
1.88
3.49
0.49
0.48
0.49
0.48
0.49
0.52
0.59
0.71
0.86
1.13
1.73
2.51
0.89
0.77
0.71
latitude. This is shown extensively in Section 3 of Volume 1 of ESA (1997). The
accuracies are, naturally, also a function of stellar magnitude. Table XII summarizes the accuracies for the parallax determinations. Similar tables for the positional
accuracies at the catalogue reference epoch J1991.25 and for the proper motions
are presented in Section 3.2 of Volume 1 of ESA (1997). In general, the accuracies
surpassed the aims set for the mission by almost a factor of two. In the case of
the parallax measurements this meant that the relative parallax accuracies aimed
for before launch were achieved for a 5 to 10 times larger volume of space, thus
significantly enhancing the scientific impact of the mission. The fact that this was
achieved despite the orbital problems has to be attributed to all parties involved in
the manufacturing of the satellite, the operations and the data reductions.
The parallax accuracies were verified, using the methods described by Lindegren
(1995), from the distribution of small and negative parallaxes, through a Bayesian
deconvolution in which the distribution functions for the parallaxes and the errors on
the parallaxes are estimated simultaneously under the assumption that no negative
parallaxes can exist. The method dates back to Hertzsprung (1952), and has been
refined since then by several authors. An application of this method, using the
preliminary 30-months sphere solution results, showed the predicted external errors
to be within 1 per cent of the observed internal errors, well within the accuracy
of the determination, and confirming the very high accuracy (Figure 61) as well
as the reliability of the formal errors on the Hipparcos parallax determinations.
The distribution of parallaxes for 100 019 solutions of the simple 5-parameter type
312
F. VAN LEEUWEN
Figure 61. Histogram of the parallax accuracies for 100 019 stars with 5-parameter solutions.
Figure 62. Histogram of the distribution of parallaxes for 100 019 stars with 5-parameter solutions.
is shown in Figure 62. The small tail of negative parallaxes seen in this figure
confirmed the claimed accuracies (see also Section 10.2).
The positional accuracies are very much a function of time, and will deteriorate
quite rapidly away from the reference epoch J1991.25. Approximate ‘deterioration
factors’ can be calculated from the provided accuracies and covariances in the main
catalogue. This factor represents the average increase per year within a magnitude
interval of the formal standard errors on the positional coordinates and is closely
linked to the average proper motion accuracies. Table XIII and Figure 63 show these
313
THE HIPPARCOS MISSION
Figure 63. Average positional deterioration factors for (filled circles) and
function of Hp magnitude.
(open circles) as a
Table XIII
Average positional accuracies at reference epoch
J1991.25 (in mas) and their deterioration factors (in
mas/yr) as function of Hp magnitude.
Hp
mag
<2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
10–11
11–12
> 12
< >
< >
mas
0.60
0.58
0.59
0.59
0.58
0.62
0.70
0.84
1.04
1.44
2.12
3.46
D
D
mas/yr
0.48
0.48
0.48
0.48
0.48
0.51
0.58
0.70
0.87
1.17
1.74
2.83
0.67
0.69
0.68
0.69
0.68
0.73
0.83
1.00
1.26
1.76
2.61
4.34
0.53
0.56
0.57
0.56
0.56
0.60
0.69
0.83
1.04
1.41
2.10
3.45
values as a function of magnitude. For an individual star the error propagation is
fully described in Sections 1.5.4 and 1.5.5 of Volume 1 of (ESA, 1997), and
algorithms with the implementation of the error propagation are provided on the
ASCII CD-ROM as ‘C’ and as ‘Fortran’ programs (pos prop.c and pos prop.f).
314
F. VAN LEEUWEN
8.8. TYCHO ASTROMETRY AND PHOTOMETRY
The reduction of the Tycho data (the continuous stream of star mapper photon
counts) was at various stages interlinked with the Hipparcos data reductions. The
task of the Tycho data reductions was to reduce the star mapper photon counts
to a catalogue of astrometric and photometric parameters for all stars that could
be detected in these signals, approximately one million. This involved as basic
steps the prediction of transits (based on preliminary catalogues), the detection of
transits (described in the context of the main-mission reductions in Section 8.1), the
recognition of transits and the reduction of the intensity information to magnitudes.
The relation between transit times and positions on the sky was finally described
by the main-mission attitude reconstruction results, including the improvements
obtained with the great-circle reductions (Section 8.5), the calibration of the grid
geometry as a function of time, which was obtained from the astrometric parameter
determinations, and the great-circle zero-points as provided by the sphere solution.
8.8.1. The prediction process
The prediction process aimed at deriving from the available information on:
the satellite attitude description,
the satellite’s and Earth’s orbit parameters,
a catalogue of astrometric parameters for a selection of stars,
and instrument parameters describing the star mapper grid geometry,
transit times for all objects that could possibly be visible in the star mapper data
stream. Initially, the attitude used was the on-board real time attitude determination
(see Section 7.4), which was not very accurate, in particular early in the mission.
The star catalogue used initially was the Tycho Input Catalogue (TIC), based on
a magnitude limited selection of stars contained in the Hubble Space Telescope
Guide Star Catalogue, upgraded with any information available from the INCA
consortium for the preparation of the Hipparcos catalogue and from SIMBAD
for general information (photometric and spectroscopic data). The relatively large
uncertainties of the star positions in the TIC and the on-board attitude meant that
the errors in the predicted transit times were often fairly large, requiring large
search windows, and resulting often in misidentifications.
Using the NDAC attitude reconstruction for the first few months of the mission,
much improved transit time predictions were obtained, leading to more secure
identifications. The results of these runs were accumulated in Strasbourg where
they were used to create the Tycho Input Catalogue Revision (TICR) based on
transits identified during the first part of the mission. In comparison with TIC, TICR
contained far fewer stars (1.26 million instead of 3.15 million), but the positional
information for these stars was in most cases very much better than in TIC. This
catalogue was used in the final processing of the data, together with the on-ground
attitude reconstruction by NDAC, improving considerably the prediction accuracy
THE HIPPARCOS MISSION
315
and thereby the detection chances, which was very important for accumulating data
on the faintest stars visible with the star mapper detectors.
8.8.2. Identification of transits
The process of identification of transits had as task to connect predicted transit times
with observed transit times, and to predict and identify any possible disturbances
through overlapping transits. It was important in this process to clearly distinguish
the origins of an observed transit: the field of view, slit group and upper or lower
branch for the inclined slits, and to avoid double identifications of observed transits.
Any parasitic transits (transits of other catalogued stars that were close enough to the
target transit to have caused a significant astrometric and photometric disturbance)
were recorded with the transit information, providing the angular separation and
the brightness of the disturbing object.
After identification of the predicted transits, there remained a number of unidentified observed transits. Information for such transits was only preserved if they
were close to identified transits, and could possibly indicate a companion star for
the identified object. A total of 160 serendipity stars and 6600 new components
were derived from these data.
8.8.3. Tycho photometry
In the Tycho photometric reductions the observed intensities were calibrated as
functions of slit group, position along the slits, star colour, and field of view for
BT and VT . Only 10 000 stars between mag 4.5 and 9 could be used as calibration
standards (see Scales et al., 1992, Großmann et al., 1995). Brighter stars could
be affected by saturation, and fainter stars were affected by a bias. This bias was
due to the detection statistics for the fainter stars. Given a normal distribution of
errors around a mean value that is close to or even below the detection limit, the
non-detection of all observations below the detection limit will give the impression
that the object is much brighter than it really is. A process called de-censoring,
described in Chapter 9 of Volume 4 of ESA (1997), was developed to correct
photometric data for faint stars for the non-detections. The de-censoring process
is illustrated in Figure 64. The comparison between the reference magnitudes and
the de-censored magnitudes is shown in Figure 65.
The star mapper detector sensitivity was modelled as a function of the ordinate
of the star (the position along the slit group) and the colour, both up to second
order, and a cross-term of the ordinate and the colour, a total of 6 coefficients. The
zero-point defined the response half-way the slits for a colour index B , V = 0:7.
Separate calibrations were made for the two slit groups, two fields of view and the
two pass-bands (Großmann et al., 1995). The star mapper photometric reductions
in NDAC were done with a very similar model and comparisons were carried
out between reduced data at various stages of the data reductions by D.W. Evans
(RGO).
316
F. VAN LEEUWEN
800.0
Number of measurements
600.0
400.0
200.0
0.0
-1.0
0.0
log (signal)
1.0
Figure 64. The distribution of the observed transit intensities for standard stars within a narrow range
of magnitudes close to the detection limit. The thin drawn line is the expected actual distribution, the
dotted line the observed distribution, showing the bias due to the detection cut-off. The thick drawn
line is the derived corrected distribution.
de-censored VT - on-ground VT
0.5
0.0
-0.5
8.0
9.0
10.0
on-ground VT
11.0
12.0
Figure 65. The comparison between the final de-censored VT magnitudes and the true magnitudes
for standard stars, showing that de-censoring has largely removed the bias from the final magnitudes.
A small remaining bias was removed before the data were entered in the catalogue.
The star mapper detector response changed much less over the mission than the
main detector response, as the number of optical elements in the detector chain
that could get affected by radiation was much smaller. Typical response levels at
the start, middle and end of the mission are summarized in Table XIV. An example
of the evolution of the star mapper detector response over the mission is shown
317
THE HIPPARCOS MISSION
Figure 66. The star mapper response for the
over the mission for three colour indices.
V
channel, vertical slits and preceding field of view
Table XIV
Star mapper responses in counts/sec for a magnitude 10 star as function of channel, slit group and
field of view, over the mission. The reference colour
index is B V
0:7.
, =
Channel
Slits
Field
B
Incl
Vert
Incl
Vert
Incl
Vert
Incl
Vert
P
P
F
F
P
P
F
F
B
B
B
V
V
V
V
400
900 1500
days since 1989.0
2200
2970
2350
3060
1640
2325
1550
2180
2100
2930
2250
2920
1560
2270
1470
2070
2050
2840
2200
2870
1530
2200
1440
2040
in Figure 66, a similar figure for a different combination of channel, slit group
and field of view can be seen in Figure 10.2 in Volume 2 of ESA (1997). These
figures were obtained from the NDAC reductions, and were confirmed by the Tycho
reductions.
The passbands for the Tycho photometry are shown in Figure 25. The relation
between the Tycho and Johnson photometry is shown in detail in Section 1.3,
Appendix 4 of Volume 1 of ESA (1997). These relations are a function of luminosity
318
F. VAN LEEUWEN
,
,
Figure 67. The relationship between (B V ) (Johnson) and (BT VT ) for luminosity class V stars
with accurate photometry. The solid line shows the mean reddening direction for O-B2 stars.
class and are affected by interstellar reddening. Figure 67 shows the relation for
luminosity class V stars and the displacement direction resulting from reddening.
The Tycho photometric data appear at various places in the Hipparcos and Tycho
Catalogues. The Tycho Catalogue gives the complete information on the final BT
and VT magnitudes and estimates of their standard errors in fields T32 to T36,
including a flag indicating how these data were obtained. For Hipparcos stars, any
available Tycho photometry can be found in fields H32 to H36. Epoch photometry
is in addition available for a selection of 34 446 brighter stars as file tyc ep.dat on
disk 4 of the ASCII CD-ROM set, and for a much larger selection (481 553 stars),
through the CDS in Strasbourg. General characteristics of the Tycho astrometric
and photometric data are summarized in Table XV.
8.8.4. Tycho astrometry
The Tycho astrometry process derived the final positions, proper motions and
parallaxes for some 1.05 million stars, using the transit time information, the
reconstructed satellite attitude and the astrometric parameters for 100 000 single
stars derived in the main Hipparcos data reduction. In the process, the large- and
medium-scale distortion characteristics of the star mapper detectors were calibrated
by means of the 1000́00 Hipparcos stars. This is analogous to the calibration of the
instrument parameters for the main detector in the great-circle reductions (see Høg
et al., 1992b, Høg et al., 1995, and Chapter 7 in Volume 4 of ESA, 1997).
319
THE HIPPARCOS MISSION
Table XV
Summary of the data contained in the Tycho Catalogue. Errors on positions (epoch
J1991.25), parallaxes and proper motions are given in mas or mas/yr.
V
Interval of T
Median T , mag
V
N (Tycho)
N (not in HIP)
<6.0
>11.0
5.38
6–7.0
6.63
7–8.0
7.62
8–9.0
8.62
9–10.0
9.61
10–11.0
10.58
4553
4
9550
55
27750
3485
78029
36511
211107
182773
515029
506720
205934
205275
4.0
5.3
5.0
6.7
8.6
8.3
12.9
16.4
16.0
27.2
34.3
33.5
39.2
49.6
48.6
0.018
0.014
0.024
0.036
0.027
0.049
0.084
0.064
0.117
0.128
0.122
0.200
Median standard errors in astrometry:
Position
1.8
2.6
Parallax
2.5
3.6
Proper motion
2.3
3.3
Median standard errors in photometry (mag):
0.003 0.006
0.010
T
0.003 0.005
0.008
T
0.005
0.008
0.014
T
T
B
V
B ,V
11.19
All Tycho astrometric data is presented in the Tycho Catalogue, available on
disk 1 of the ASCII CD-ROM set as file tyc main.dat. When considering Tycho
astrometric data, and in particular parallaxes and proper motions, it should be
realized that intrinsic accuracies are low, especially so for fainter stars. In the case
of parallaxes this means the occurrence of many (large) negative values. It is not
advised to use Tycho parallaxes on an individual basis, and when using Tycho
parallaxes on a statistical basis to always include those negative parallax values,
as is also explained in Section 11. The proper motions as provided in the Tycho
Catalogue are in the process of being upgraded through the use of the positional
information collected in the Astrographic Catalogue (Kuzmin et al., 1997).
8.8.5. Quality assessment
An astrometric quality flag for the Tycho data was derived from a combination of
various criteria:
The maximum standard error on any of the 5 astrometric parameters, max ;
The signal-to-noise ratio for the detection of the stars, defined on the basis
of the distribution of residuals for the astrometric solution. This distribution
consisted in general of a sharp peak, flanked by wings. The ratio between the
number of points in the peak (n1 ) and the number of points in the wings (n2 )
was given by:
p
Fs = (n1 , n2 )= n1 + n2:
(73)
pNast , where
In the limit of a sharp image on a low background, Fs
Nast n1 is the number of accepted astrometric observations;
320
F. VAN LEEUWEN
Table XVI
The astrometric quality Q and associated quantities. N is the number of stars in the
Tycho Catalogue of each quality class. med gives the median standard error of a position
coordinate at the catalogue epoch J1991.25 (the errors at the mean epoch of observation
of any given star are typically 5 per cent smaller).
Q
max
Fs
(mas)
1
2
3
4
5
6
7
8
9
<5
, 10
, 25
, 50
, 150
5
10
25
50
< 150
< 150
< 150
200
–
obs
N
(mas)
>5
>5
>5
>5
>5
>5
3
3
< 300
< 300
< 300
< 300
< 300
300
, 5 < 300
, 5 300
–
–
–
–
23147
70945
259695
430182
146520
41695
37821
28949
13077
6301
med
(mas)
Astrometric
quality
2.6
5.5
13
26
39
44
45
54
–
–
very high
very high
high
high
medium
perhaps non-single
low
perhaps non-stellar
low, ‘R’ in Field T42
unassigned, ‘H’ in Field T42
The formal standard error of a single observation, obs , which is a measure of
the observed half-width of the image:
2
2
2
obs
= 0:25(x + y )(Nast , 5);
(74)
a half-width of more than 300 mas was generally associated with duplicity;
Table XVI shows the relation between the quality parameter Q, given in field T40
of the Tycho Catalogue, and these criteria.
9. Parallel processes in the data reductions
The data reduction processes described in this section were handled (semi)independently from the main stream processing described in the preceding section.
The results of two of these processes were, however, incorporated directly or indirectly in the final data reductions for the main catalogue: the double star reduction,
with its capabilities to recognize likely double stars, also indicated which stars
could safely be regarded as single stars. The results of the photometric processing
were used in the double-star data reductions, and in case of the star mapper photometry, as colour information in a wide range of instrument calibrations. Other
processes described in this section are the treatment of data for solar system objects
and the variability analysis of the main-mission photometric data.
9.1. DOUBLE AND MULTIPLE STARS
The possibilities to detect and resolve double stars with an astrometric satellite
were first put forward by Lindegren (1979a). This was followed by several papers
THE HIPPARCOS MISSION
321
Table XVII
The double star working group
Name
Consortium
F. Mignard (chair)
S. Söderhjelm
L. Lindegren
M. Badiali
H.-H. Bernstein
J. Dommanget
P. Lampens
R. Pannunzio
FAST
NDAC
NDAC
FAST
FAST
INCA
INCA
FAST
on detection possibilities by Vaccari & Delaney (1983), Boriello & Delaney (1985),
Söderhjelm (1987), and on techniques for resolving the double star information
by Delaney (1983), Delaney (1985), Farilla (1985), Lindegren (1987b), Delaney
(1987), Söderhjelm et al. (1992), Mignard et al. (1992a), Pannunzio et al. (1992),
Mignard et al. (1995), Gazengel et al. (1995). A detailed account of the final
procedures used in the double star analysis was presented in Chapter 13 of Volume 3
of ESA (1997). The double star processing and the preparation of the merged results
for the final catalogue were supervised by the Double Star Working Group, of which
the members are given in Table XVII.
The analysis of the double, and in particular the multiple star information was
among the most difficult tasks in the Hipparcos reductions. Although the problem
was well defined for double systems with separations above 0.1 arcsec and with
magnitude differences below 3 to 4 mag, near to these limits more than one solution
for the data was often possible.
The information available for the double star processing consisted of three parts:
A priori information as collected in the CCDM (Dommanget & Nys, 1994),
further enhanced by a data base on speckle interferometric measurements set
up in Torino (see Figure 74) and by CCD observations for selected objects;
statistical indicators for duplicity, as obtained from the main mission modulated signals;
the measured modulation parameters, calibrated for response variations over
the grid.
In the following sections the detection mechanisms, signal processing, quality
assessment and merging are briefly described. For detailed descriptions the reader
is referred to Chapter 13 of Volume 3 of ESA (1997).
9.1.1. Detection of double and multiple systems
There were three independent indicators for duplicity: two associated with the
phase difference and amplitude ratios between the first and second harmonics (in
322
F. VAN LEEUWEN
Figure 68. Histograms of the accumulated 2 values for the double star HIP 55 (solid line) and the
single star HIP 48 (dotted line), showing the effect of duplicity as an asymmetry in the distribution
of 2 values for the three-parameter solutions (see Equations 46 and 47).
NDAC combined to a single 2 estimate and accumulated in a histogram, see
Section 8.4.6), and one relating the amplitude of the first harmonic to the mean
intensity of the signal, reflected in the difference between the dc and ac magnitudes
(Section 9.3). These data were available for all 118 000 stars that were observed.
Figure 68 shows an example of accumulated 2 data for the double star HIP 55
(separation 3.7 arcsec, magnitude difference 1.7 mag), in comparison with a single
star of similar magnitude. Figure 69 shows the ac- and dc-photometric data for the
same stars. The ac photometry for a double or multiple star shows systematically
fainter magnitudes and has a larger to much larger scatter than the ac magnitudes
for a constant star of the same brightness.
The effect of duplicity on the modulated signal was a function of the orientation
angle of the double star system with respect to the grid lines and the difference between the magnitudes of the components. This showed from a comparison
between the observed parameters for the second harmonic in the modulated signal,
4 and 5 in Equation 33, and the values expected on the basis of the OTF calibration (see Section 8.4.5). Two functions, Fc (for the relative amplitude of the second
harmonic) and Fs (for the relative phase of the second harmonic) were defined as
the ratios between the expected values for single stars and the observed values for
double stars of various separations and magnitude differences. Assuming the same
colour for the two double star components, Fc and Fs become functions of only
the orientation of the pair with respect to the modulating grid, and the magnitude
difference, as is shown in Figure 70.
The effect of the third criterion, the comparison between the dc and ac magnitudes, is shown in Figure 71. In case all stars were single, the histograms would only
show a narrow peak around Hpac , Hpdc = 0. Instead, a surplus of positive residuals is clearly visible in the distributions. These positive residuals indicate weaker
signals for the ac than for the dc magnitudes, which is as expected for double and
multiple stars for which the modulation amplitude is disturbed. The occurrence of
many stars with positive magnitude differences showed that at least approximately
323
THE HIPPARCOS MISSION
Figure 69. Epoch-photometry data for the same two stars as used in the previous figure. The top
diagram shows the single star HIP 48, the bottom diagram the double star HIP 55. The open circles
represent the dc magnitudes, the dots the ac magnitudes, which coincide for single stars, but show
large discrepancies for double stars.
F5c
F4s
∆ m = 0.5
4
3
∆ m = 0.5
2
3
1
1.5
1
1
2
2
0
1.5
2
-1
1
-2
0
-3
-1
-4
0
60
120
180
240
300
360
0
60
120
180
240
300
360
Figure 70. Variation of the functions Fc and Fs with the projected separation and magnitude
1 2
difference of a double star. The separation is given here as the phase difference between the signals of the individual stellar components; runs from 0 to 360 over the grid period
of 1.2074 arcsec. The single star limits for the two functions are one and zero respectively.
=
,
10 per cent of all stars in the Hipparcos Catalogue could be resolved as double or
multiple stars by the main-mission detector. A summary of the detection results
in Table XVIII shows, by means of the differences between the FAST and NDAC
detections, the difficulties associated with detection close to the separation and
324
F. VAN LEEUWEN
Hp
Hp
Figure 71. Differences between median values for the
ac and
dc magnitudes. The occurrence
of many positive differences is due to the presence of double stars. Detection possibilities vary with
magnitude, due to both the accuracy of the photometry and the sensitivity to fainter companions.
magnitude difference limits, with in both solutions the occurrence of solved binary
systems that were not even detected by the other consortium.
325
THE HIPPARCOS MISSION
Table XVIII
Number of entries solved or detected only as double stars by each consortium, divided into solved, detected or undetected entries for the other
consortium.
NDAC
Solved
FAST
Detected
Undetected
Total
Solved
Detected
Undetected
12710
500
2310
1030
210
160
2250
2190
-
15990
2900
2470
Total
15520
1400
4440
21360
Q
✰ Secondary
Celestial Pole
Scanning circle
ρ
θ
γ
Primary ✰
Gs
Gp
ky
he s
Grid
nt
ed o
ject
pro
Figure 72. Geometry of the observation of a binary star with separation and position angle . The
angular coordinate along the scanning circle is denoted G. The signal observed depends only on the
projected phase difference between the phase of the secondary (at GS ) and that of the primary
(at GP . The orientation of the scanning circle is indicated by .
)
9.1.2. Resolving double stars
The fundamental data from which the double star parameters were derived are the
modulation parameters obtained in the IDT data processing as given by Equation 29.
For a double or multiple star these parameters describe the combined effect of the
instantaneous transit times and magnitudes of the various components. Unlike
for a single star, the resolution of the data in transit times and magnitudes for
individual stellar components from a single transit is ambiguous. Only the combined
information of many transits at different scanning angles can provide this resolution.
Two different approaches to extract the component data were used by FAST and
NDAC. In FAST, the relative astrometry (separation, orientation and any changes
over the mission in these parameters) was solved first, followed at a later stage by
the absolute astrometry (the absolute position, system proper motion and parallax).
Figure 72 shows the basic double star geometry for the relative astrometry, as used
by FAST. In NDAC all phases were first related to absolute measurements on the
326
F. VAN LEEUWEN
sky, and then solved for the relative and absolute astrometry together. The data were
collected for this purpose in case-history files, which have been made available in a
slightly modified form on on disk 6 of the ASCII CD-ROM set for 38 535 Hipparcos
Catalogue entries, including all those that are known or suspected to be double or
multiple stars as the Transit Data file (hip j.dat). Considering that this file provides
the only data available for further analysis, and has been directly derived from the
NDAC processing, the main emphasis on the double star analysis here will focus on
the NDAC approach. A complete description of the FAST approach can be found
in Chapter 13 of Volume 3 of ESA (1997).
The first step in the NDAC processing was to relate all modulation information
to a calibrated and fixed reference system. This meant application of:
the photometric calibrations (to make intensities over the grid and over the
mission compatible);
the instrument parameters obtained in the great-circle reductions (ensuring
that phases could be properly related);
the along-scan attitude improvements obtained in the great-circle reductions
(to obtain accurate relative phases);
the zero-points of the great circles as obtained in the sphere solution (creating
absolute phase estimates).
The phase difference between the first and second harmonic as determined for single
stars in the OTF calibration (Section 8.4.5), were eliminated, and the amplitudes
= 0:7100
of the first and second harmonics were transformed to fixed values, M
= 0:2485, giving for a unit-intensity point source at the reference position
and N
the following modulation signal:
Ak = 1 + M cos pk + N cos 2pk
(75)
The reference position was defined in barycentric coordinates in the reference frame
that resulted from the final sphere solution. A component of a double star has an
offset, as measured in baricentric coordinates, of ( , ) with respect to this reference
position. The observed instantaneous offset, which is a function of time, is given
by this offset plus the offset caused by the parallax of the star. These offsets cause a
phase shift in the modulated signal by , which can be obtained from the observed
modulation parameters. Defining three derivatives: fx = d=d , fy = d=d and
fp = d=d, all dependencies can be taken into account rigorously:
= fx + fy + fp (76)
The factors fx , fy and fp are given in the Transit Data file in fields JT3 to JT5.
The target position is given in fields JP2 and JP3, and in case of two-or more
pointing systems in further fields of the Pointing Record.
The observed signal for an object consisting of m components, after applying
all calibrations as described above, is given by:
Ak =
Xm Ii
i=1
cos(pk + i ) + N
cos 2(pk + i )];
[1 + M
(77)
THE HIPPARCOS MISSION
327
where Ii = 6200 10,0:4H p are the intensities of the components, expressed in
counts per sample. This relates to the (calibrated) observed modulation coefficients
through the following relations:
b1 =
X
X
X
,
X
X
,
Ii
i
b2 = M
Ii cos i
i
b3 =
M
Ii sin i
(78)
i
b4 = N
Ii cos 2i
i
b5 =
N
Ii sin 2i
i
The observed parameters b1 to b5 and their accuracies are given in fields JT6 to
JT15 in the Transit Data file. As these parameters were dependent on the assumed
colour for the system, and this colour may be wrong, colour correction factors are
also provided (see Section 2.9.3 of Volume 1 of ESA, 1997). By writing the phases
i as functions of the component positions (i ,i ) and parallaxes (i ), a set of 5
non-linear observations was obtained for each field transit, giving an average of
some 500 observations per system.
In the analysis of double stars a set of 12 parameters was solved for (see
Table XIX). This type of solution was necessary for optical double stars, identified
by ‘I’ in field DC3 of the Double and Multiple Systems Annex, file hip dm c.dat on
disk 1 of the ASCII CD-ROM set. However, for physical binaries it was possible
to reduce these 12 parameters to a smaller subset. In solutions of type ‘L’ the two
components were assumed to have the same parallax, and in solutions of type ‘F’
the relative geometry of the system was assumed constant.
In order for the least squares solution to converge, a priori coordinates were
required with errors much smaller than the grid-period of 1.2074 arcsec. This
information was not always available, and in many cases a double iteration was
required: one iteration providing a range of a priori positions, and the other iteration solving the non-linear equations. It was not uncommon to find a number of
indistinguishable solutions, with possible positions for the secondary component
found at intervals equal to the grid period. In many cases, however, the solutions
were well defined, in particular when magnitude differences were less than 2.5 mag
and separations more than 0.3 arcsec. For less well defined solution, a single star
solution was obtained as alternative, and accepted if the 2 for the double star
solution was not significantly better.
A further complication came from double star systems containing at least one
variable star. A number of double star systems are in fact triple star systems,
with one of the components being a tight eclipsing binary (van Leeuwen & van
Genderen, 1997). The modelling as described above no longer strictly applies
328
F. VAN LEEUWEN
Table XIX
Astrometric and photometric parameters solved for in the NDAC doublestar analysis. Three different physical models used for double stars resulted
from the use of various constraints: without any constraints the solution
0 a solution of type ‘L’ results, and with
is of type ‘I’, with u10
u10 u11 u12 0 the solution is of type ‘F’. The parameter u13
applies only to double-pointing systems, independent of solution type.
=
=
=
=
Parameter
Description
u1
u2
u3
u4
u5
u6
u7
u8
u9
u10
u11
u12
[u13
magnitude of primary (Hp1 )
offset (1 ) in of primary from reference point
offset (1 ) in of primary from reference point
offset ( 1 ) in parallax of primary from reference point
offset (1 ) in of primary from reference point
offset (1 ) in of primary from reference point
magnitude difference ( m Hp2 Hp1 )
relative position in of secondary (2 1 X )
relative position in of secondary (2 1 Y )
relative parallax of secondary (2 1 )
relative proper motion in of secondary (2 1 X )
relative proper motion in of secondary (2 1 Y )
correction factor for the attenuation profile ( 1)]
_
_
=
,
,
, =
, =
_,_ = _
_ ,_ = _
'
to these systems, and improvements of the astrometric parameters may still be
possible for such systems.
Systems containing more than 2 components depended even more on good a
priori coordinates. Without, the search mesh became unmanageable. The basic
equations remained, however, the same. Another approach for multiple star recognition, through imaging, was presented by Boriello (1987). Similar ideas, applied
to double stars, were presented by Quist et al. (1997), from which was obtained Figure 73, showing an image of a binary discovered by Hipparcos (see Section 12.2).
9.1.3. Quality assessment and merging
The NDAC processing produced a total of 16 016 resolved double-star solutions
(this included some of the systems detected initially only by FAST), as well as
solutions for 180 triple or quadruple systems. A quality parameter Q was assigned
to each solution, based on the difference in magnitude and the separation of the
components, as given by:
8
< m , 20 log10(%=0:138) % < 0:10 arcsec
D = : m , 5:5 log10 (%=0:32) 0:10 < % < 0:32
(79)
m
otherwise
The discovery limit for a double star was around D = 3:5 to 4. On the basis of D ,
the quality parameter Q was set to range from 3 (reliable solution) to 0 (marginal
THE HIPPARCOS MISSION
329
Figure 73. An image of a binary discovered by Hipparcos. The two objects (HIP 15719) have a
separation on the sky of 0.32 arcsec and a magnitude difference of 2.8. The restoring beam is shown
in the lower left-hand corner. (from Quist et al., 1997)
solution). This quality flag, together with the quality assessment by FAST, was
used to define the overall solution quality as given in field DC5.
When a priori information was available, the solutions had little chance to
converge wrongly. The speckle interferometry information as collected in Torino
helped to resolve and verify the solutions for almost 700 systems (see Figure 74).
Comparison exercises took place at various times during the mission, mostly
associated with one of the intermediate sphere solutions. These comparisons had
shown what problems could be expected in the final merging of the consortia
results. In the final merging, some 1800 entries for which the solutions as obtained
by FAST and NDAC differed by multiples of the grid period, still had to be resolved.
Some 700 of these were in the end reduced to single star solutions, and for the
remaining 1100 various criteria were applied to decide which should be regarded
as the most reliable. The merging of the remaining solutions, for which general
agreement had been reached, proceeded following a simple recipe: a straight mean,
giving equal weight to the two solutions, was obtained for both the absolute and
the relative astrometry.
The different approaches in the double star analysis by FAST and NDAC caused
another problem: the covariance matrix for the solutions as given by NDAC included correlations between the relative and absolute astrometry, while in the FAST
processing these correlations were by default assumed to be non-existing. In the
merging this was accommodated through the introduction of neutral points: posi-
330
F. VAN LEEUWEN
δ(ρ cos θ) mas
40
20
0
-20
-40
-40
-20
0
20
40
δ(ρ sin θ) mas
Figure 74. Relative positions on the tangent plane on the sky between the position of the secondary
with respect to the primary obtained by Hipparcos and speckle interferometry. The difference is
defined in the sense Hipparcos minus speckle.
tions in the system where there was minimal coupling between the absolute and
relative astrometry. These positions could be derived from the covariance matrices
as provided by NDAC, and were a function of the separation of the two components and their magnitude difference. Within the double star system, it ranged from
the position of the primary for separations above 0.7 arcsec to the position of the
photocentre for separations below 0.25 arcsec. The merging was thus done in two
steps: the relative astrometric parameters from the two solutions were averaged and
the absolute astrometry for the neutral points in the two solutions were averaged.
The final solution was the combination of these two averaged solutions.
Finally, there were some 1400 cases with discrepant FAST and NDAC solutions
to be resolved. 60 of these were resolved by means of dedicated CCD observations
by Le Poole and others (see Figure 75). 600 other cases showed much better fits
in the FAST solution and were kept as such. Remaining systems were investigated
using the NDAC imaging approach, which resolved a further 100 cases. All cases
retained as valid double-star solutions, but with an alternative solution from the
other consortium, were graded as ‘uncertain’ in fields H61 (Hipparcos Catalogue)
and DC5 (Double Star Annex C). Some key parameters from the alternative solution
are given in the Notes to the Double and Multiple Systems Annex, file hd notes.dat
on disk 1 of the ASCII CD-ROM set. There remains the possibility that for newly
discovered systems (field H56=‘H’) with relatively poor solutions (field H61=‘C’)
both solutions are wrong by one or more grid steps. In some cases improved solutions were obtained after the official completion of the catalogues. Such solutions
are indicated by notes to the main catalogue and double star catalogues.
331
THE HIPPARCOS MISSION
15
12
9
%
6
3
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Reduced δ(∆m)
Figure 75. Histogram of the reduced (normalized) differences of the relative photometry between
Hipparcos and the CCD observations obtained with the JKT 1-meter telescope on La Palma. The
dotted line is a normal distribution of zero mean and standard deviation 1.4.
Table XX
Quality assessment of double star systems in the
main catalogue. A: good; B: fair; C: poor; D: unreliable.
Known
New
Total
A
B
C
D
Total
7360
1621
8981
904
590
1494
399
428
827
536
356
892
9199
2995
12194
The impact of the Hipparcos double star discoveries is shown in Figure 76. As
could be expected, the largest impact is for the smaller separations, although there
are also discoveries among the larger separations, even at not very large magnitude
differences. Table XX summarizes the statistics for known and newly discovered
systems with respect to the quality parameter as given in field H61 in the Hipparcos
Catalogue. The relative number of good quality solutions among known systems is
obviously much larger than for newly discovered systems due to the advantages of a
priori knowledge and generally more favourable parameters for system resolution.
9.2. SOLAR SYSTEM OBJECTS
A selection of solar system objects was measured by the main-mission detector
and/or by the Tycho detector. 48 Minor planets and three planetary satellites (J II
Europa, S VI Titan and S VIII Iapetus) were observed by the main mission. The
selected objects had to be bright enough and small enough (diameter < 1 arcsec)
to be observed (Lindegren, 1987a). The data for these objects was prepared for
332
F. VAN LEEUWEN
Figure 76. The resolved double star systems in the Hipparcos Catalogue. Top diagram: newly
discovered systems, bottom diagram: previously known systems. The dotted line in both diagrams
indicates the detection limit D=4 as used by NDAC.
publication by D. Hestroffer at ESTEC, in collaboration with F. Mignard (see also
Hestroffer et al., 1995).
Two problems were known to occur when observing these objects: they can
not be regarded as point sources in most cases, which affects the phase estimates
and their relation to the stellar positions (see Figure 77); their motion on the sky
could be so fast, that individual observational frame estimates had to be preserved
333
THE HIPPARCOS MISSION
Figure 77. Theoretical differences between the observed position and the photocentre versus apparent
20 . Lower
diameter for a spherical object of uniform brightness viewed with a solar aspect angle
curve: positions derived using only the first harmonic (NDAC); upper curve: positions derived using
first and second harmonics (FAST).
=
500
400
300
200
100
0
10
15
20
25
30
35
Figure 78. Distribution of the solar phase angle of minor planets during the Hipparcos mission. Due
to the characteristics of the scanning law, observations only occurred in the vicinity of the quadratures.
rather than the frame transits combined into single field transits. The first problem
resulted from the effect of the solar aspect angle on the phase estimates. The solar
aspect angle varied between 13 and 33 : due to the restrictions of the scanning
law, observations only occurred in the vicinity of the quadratures (see Figure 78).
Reference phases do not refer to the centre of the object, but the differences are
generally too small to be of much concern.
9.2.1. Presentation of the data on solar system objects
Each observation presented for a solar system object relates to a transit of that
object over the main grid, and is based on suitably averaged data obtained from the
frame transits, i.e. taking into account the predicted drift of the object during the
334
F. VAN LEEUWEN
Figure 79. Definition of the abscissa correction v for a predicted position (
the catalogue reference position ( 0 , 0 ) and the orientation angle .
m,
m ), with respect to
transit. All Hipparcos observations are one-dimensional transit times. This applies
to the solar system objects too. However, the Hipparcos Solar System Objects
Astrometric Catalogue, file solar ha.dat, gives for each transit both coordinates,
a reference 0 and 0 , fields SHA2 and SHA3. This is no more than a reference
position, derived using a recent ephemerides. The actual positional information is
in this position, combined with the position angle (field SHA6), which define
a great-circle arc corresponding to the actual transit position as observed (see
Figure 79). The accuracy of the transit position is given in field SHA7. A similar
situation exists for the solar system observations for Tycho (file solar t.dat). All
observations are expressed in ICRS coordinates, and to a position as seen from
Earth.
In addition to the astrometric data, also photometric data are provided, but only
for the minor planets. The Jovian and Saturnian moons were too much disturbed by
veiling glare from the planets. Photometric values for minor planets were obtained
by FAST only, and were an implementation of the photometric reductions for stellar
images using a fixed colour index of B , V = 0:5. The ac magnitudes are affected
by the non-negligible sizes of these objects. All magnitudes have been presented
in the solar hp.dat file, and are complemented by information on aspects that
affected the observed magnitudes: the distance Sun-asteroid, the distance Earthasteroid, and the solar phase angle. Figure 80 shows an example of the photometry
for the minor planet Papagena (471), corrected for distances and aspect angle, and
folded with the rotation period of 7.112 d.
335
THE HIPPARCOS MISSION
7.4
7.2
7
0
2
4
6
Figure 80. Folded light curve for (471) Papagena obtained at epoch
correspond to observations made about 3 days later.
t
1140. The circled points
9.2.2. Using the solar-system-object data
The use of the solar-system-object data is very similar to the methods described for
single stars in Section 8.7.3 and was described by van Leeuwen & Evans (1997)
and used as such by Morrison et al. (1997b) for the observations of Europa and
Titan. It is based on providing estimated positions ( m , m ) based on a reasonably
accurate ephemerides. Comparing these positions with the observations (the great
circles through the reference positions at angles ) gives the abscissa residuals v
as:
v = ( m , 0) cos sin + (m , 0 ) cos :
(80)
These abscissa residuals can be modelled in various ways by expressing corrections
to the residuals as corrections in right ascension and declination:
(v) = d cos sin + d cos + ;
(81)
with < 2 > = v . By representing d and d as functions of time or as functions
d
1
2
of orbital parameters, systematic corrections to the predicted ephemerides positions
2
P,
can be obtained through minimizing i vi,vdvi . Observations from differently
i
oriented reference great circles (angles ) are needed to solve these equations.
9.3. PHOTOMETRY
The photometric reductions aimed at the calibration of the intensity measurements
for both main-mission (IDT) and star-mapper data as functions of the field of view,
the position in the field of view and the star colour. Like for the main-mission
astrometric data, the reduction procedures for the photometric data were developed
independently at RGO for NDAC (see Evans et al., 1992) and at CERGA for FAST
336
F. VAN LEEUWEN
Table XXI
The photometry working group
Name
Consortium
D.W. Evans (chair)
F. Mignard
V. Großmann
M. Grenon
F. van Leeuwen
M.A.C. Perryman
NDAC
FAST
TDAC
INCA
NDAC
ESTEC
by Mignard et al. (1992b). All photometric reductions, comparisons and merging
was supervised by the photometry working group, with members from NDAC,
FAST, TDAC and INCA (see Table XXI).
The photometric reductions were made with respect to a set of some 20 000
standard stars distributed all over the sky. The selection of standard stars was partly
based on pre-mission ground-based photometry, and partly on photometry accumulated over the first 18 months of the mission. Examination of this photometry
also led to the elimination of a number of variable stars among the initial selection
of photometric standards. The calibration values for these stars were representing
a photometric reference passband close to the actual passband of the system at the
beginning of January 1992, as calibrated by M. Grenon at Geneva Observatory
from the preliminary photometric calibration parameters obtained by FAST for the
period December 1989 till October 1991.
The photometric reductions considered the following aspects:
the calibration of the background signal.
the calibration of the sensitivity of the detectors over the field of view;
the calibration of the colour dependence of the detector chain, representing
the differences between the actual passband for the data set and the reference
passband;
the adjustments of the error estimates;
the final corrections for positional and time-dependent systematic discrepancies.
Each of these aspects was treated independently by FAST and NDAC. The colour
dependence was, however, calibrated once again a posteriori by M. Grenon for
both consortia, to take properly into account the effects of ageing of the optics on
very red stars.
There were two photometric indicators: the mean intensity of the signal and
the modulation amplitude (see Section 8.4.1). The first is referred to as the dc
photometry, the latter as the ac photometry. The dc photometry was about two
times more accurate than the ac photometry, but was affected by background
THE HIPPARCOS MISSION
337
contributions for stars fainter than 9th magnitude. The ac photometry was sensitive
to duplicity and parasitic transits. Most of the calibration procedures, the exception
being the background determination, applied to both the dc and the ac photometry.
All published data were calibrated with respect to the final set of standards.
There were some essential differences between the FAST and the NDAC calibration mechanisms, described in Chapter 14 of Volume 3 of ESA (1997), but
various detailed comparisons during and after the data reductions showed no reasons for concern with respect to the overall results of either reduction scheme (see
Section 9.3.4).
9.3.1. Background determinations
There were at least four contributors to the background in the IDT signal:
Galactic glow (the accumulation of faint stars in the Galactic plane), with peak
levels of around 20 counts s,1 ;
Zodiacal light (reflection of sunlight by dust in the plane of the ecliptic), with
peak levels around 30 counts s,1 ;
Various detector related background signals, not much more than a few
counts s,1 ;
Radiation related background (primarily during Van Allen Belt crossings),
very variable, peak levels above 100 counts s,1 .
The first two of these could be modelled to an acceptable accuracy, while the
third could be solved from the available data. The last contribution could only be
incorporated by linking it with the measurements of the background in the star
mapper detectors. This required a calibration between background responses in
the star mapper and the IDT. A few (very rare) orbits within which the attitude
control on-board the satellite did not reach convergence (either for the entire orbit
or for a large part of it) were used for this purpose. Attitude convergence problems
did not hamper the star mapper background determinations, which were obtained
independent of whether a star signal was detected. IDT data was still provided for
such orbits, but the measurements contained only occasionally, and by accident, a
star signal. In most cases all that was measured was a background signal. Figure 81
shows an example of the measured and modelled background for one such orbit.
The estimated background corrections were subtracted in intensity scale from
the actual observations, leaving only a constant offset to be estimated by the calibrations. This residual background component was generally very small and had
little influence on the final results. More complex models could not be fitted as only
the intensities of the faintest stars were affected significantly by background contributions, and very few of these faint stars were measured. A more complex model
would have had too many degrees of freedom and could have fitted measurement
errors rather than real background contributions.
Different models were used for the contributions of the zodiacal and the Galactic
light by FAST and NDAC. The contribution from the radiation was only accounted for by NDAC. Because of these differences in approach, and the influence of
338
F. VAN LEEUWEN
Figure 81. An example of the IDT measurements for a data set without attitude convergence. As a
result, the data points show only the background in the IDT detector. The solid line is the modelling
of this background by NDAC. Contributions related to zodiacal light are indicated by below the
graph, Galactic plane contributions are indicated by . The reaction on the Van Allen belt radiation
is much less for the IDT than for the star mapper, as can be seen from the inset, which shows the star
mapper background over the same interval.
G
Z
the background contribution on the magnitudes of the faintest program stars, the
individual background determinations from both consortia, given in Hz, were preserved in the Epoch Photometry Annex Extension file (hip ep e.dat) as field HTE3
for FAST and field HTE4 for NDAC. The equivalent stellar intensity in Hz at any
time during the mission and for any colour is approximately given by:
,8
10,0 4 0 4658 ,
(82)
where = ,9 143 for data obtained before the interruption around day 1350 (see
Table V and Figure 18), and = ,9 20 for data obtained after the interruption.
:
I
a
(a+
:
(t
t0 )+F (t)
C +H p
)
;
:
a
:
The function F (t), the reference times t and t0 , and the pseudo colour index C are
defined in Section 9.3.3 and Tables XXII and XXIII.
9.3.2. Positional dependence of the detector sensitivity
Two approaches were used in calibrating the sensitivity of the detector chain as a
function of position in the field of view. In both groups residuals obtained from
preliminary calibrations were accumulated in grids for the two fields of view. In the
FAST approach, these residuals were stored as a maps of corrections to be applied
339
THE HIPPARCOS MISSION
Figure 82. The observed distribution of residuals in the preceding field of view for the dc photometry.
The contour intervals are at 0.005 mag, with the dotted contour representing the zero-level. The range
and
represents the entire 0.9 by 0.9 field. The graph is based on data
in the coordinates
obtained between February 5 and March 14, 1990.
G
H
G
H
Figure 83. As the previous figure, but after removal of the gradients along the and coordinates,
showing clearly the circular symmetry of the detector sensitivity over the field of view. Very similar
graphs were obtained for the following field of view and for the ac photometry.
to the data before the actual reductions. Maps were created for three colour indices,
the two fields of view and the ac and dc components of the photometry. In NDAC
the general features of the residual distribution were modelled and entered as
parameters in the photometric calibration (see Figures 82 and 83). After removing
the gradients in G and H , the remaining general feature was a circular symmetry
340
F. VAN LEEUWEN
Figure 84. Time variation of the response in the centre of the field for 8th mag stars of two different
colour indexes, showing the effect of ageing of the detector chain through the decrease in response
over the mission and the difference in this decrease for stars of different colour.
around a fixed position and a variable slope. This characteristic was entered as a
set of 6 parameters in the NDAC reductions. After all data had been reduced, final
position dependent corrections were determined and applied (see Section 9.3.4).
9.3.3. Colour corrections and ageing
The passband for the main mission photometry was changing over the mission due
to radiation darkening of the IDT relay optics. The loss of transmission was more
severe towards the blue than towards the red (see Figure 84). As a result, the effective
passband changed over the mission. This change in passband reflected in the colour
coefficients derived in the data reductions. A major problem with these colour
coefficients was their behaviour towards the extreme red stars ((V , I )C > 2). Most
of these stars are large-amplitude variables and highly unsuitable as calibration
standards. A special program was set up in collaboration with the AAVSO and
observers at ESO, La Silla, for the observations of a large number of such variables.
This allowed the determination of a number of reference points for the detector
response at extreme red colours over the mission. It was also found that the (V , I )C
index was better suited to represent the passband changes than the (B , V )J index.
For further simplification of the the relations, a pseudo colour index C (based
on the (V , I )C index) was created, for which the ageing effects were practically
linear (see Table XXII and Figure 85). The introduction of the C -index allowed the
photometry of stars with V , I > 2:0 to be reliably calibrated within the Hipparcos
H p system. The evolution of the colour correction factor in the final reductions
341
THE HIPPARCOS MISSION
Figure 85. The pseudo-colour index C as a function of (V
Table XXII
Definition of the pseudo-colour index C = a + b(V
3
I.
d(V
I ) for different intervals of V
,
,
Interval
V
0:85
, 0 85
,0 48729
, 2 00 ,1 03936
, 20
0 5152
I
< V
V
a
:
I
I >
:
:
:
:
:
,
I)
,
+ c(V
C.
I)
,
2
I)
+
b
c
d
0.98554
2.14720
0.592
,0 31968
,0 416
,0 0256
0.592
0.0
0.0
:
:
:
(as determined by M. Grenon) is shown in Figure 86. The differences between the
colour coefficients for the two fields of view and for the dc and ac photometry
that can be seen in Figure 86 reflect the small differences that existed between the
passbands of the two fields of view and the different types of photometry. Any such
differences were removed in the data reductions. The functional representation
given in Table XXIII allows an accurate reproduction of the colour corrections
applied to the data. This is important in two cases: variable stars with significant
colour variations and stars reduced with the wrong , colour index. For variable
stars an average colour was used (sometimes only based on an estimate), and no
account was taken of possible colour variations, as such information was simply
not available at the time of the data reductions. The result for a variable star
light curve can be an increased noise towards the beginning of the mission. For a
star reduced with the wrong colour index, the result is a general drift in the
magnitudes over the mission. A number of these cases were detected already before
the publication of the data, and improved , indices were entered in the main
catalogue in field H42, while the value used in the data reductions was preserved in
field H75 (not contained in the printed catalogue, but only on the ASCII CD-ROM
V I
Hp
V I
342
F. VAN LEEUWEN
Figure 86. The colour correction factor as derived for the dc (top) and ac (bottom) magnitudes. The
preceding field of view is indicated with open squares, the following field of view with crosses. The
curves show the polynomial fits given in Table XXIII, columns 3 to 6.
file hip main.dat). The correction for an H p magnitude measured at epoch t for
the difference between the assumed colour index Cold and the newly determined
index Cnew is given by:
H pnew
=
H pold
+ F (t)
(
Cold
,
Cnew );
where the function F (t) is defined in Table XXIII.
(83)
343
THE HIPPARCOS MISSION
Table XXIII
Definitions of the colour correction factors F (t) expressed as polynomial in
t
t0 , for dc and ac magnitudes. t and t0 are measured in units of 1000 JD,
t0 = 2448:6225 = 1 January 1992.
,
dc
approx
Stand.dev.
const
(t
(t
(t
(t
(t
(t
(t
(t
,
,
,
,
,
,
,
,
t0 )
2
t0 )
3
t0 )
4
t0 )
5
t0 )
6
t0 )
7
t0 )
8
t0 )
dc
ac
preceding
following
preceding
following
0.0031
0.0012
0.0012
0.0017
0.0017
0.0084
0.00303
0.00873
0.0416
,0.0389
0.09000
0.07720
0.1317
0.340
1.989
0.553
,0.0537 ,0.04022 ,0.04098 ,0.0431
,0.18863 ,0.13772 ,0.2384
,0.6746 ,0.6735 ,1.9387
0.598
0.1322
,2.9443
2.018
2.052
,1.3327
,0.742
0.0478
0.0338
,0.8277
,2.9383
,3.6662
,1.5601
9.3.4. Final quality checks and merging
All data reductions were done at observational frame level. In the published epoch
photometry, however, these were merged into field transit data. A total of about
13 million field transits for 118 000 stars were obtained by RGO and CERGA. A
field transit contained up to 10 frame transits, with an average close to 8. Some
outlying frame transit data were rejected in this process. Standard deviations were
determined on the basis of the spread in the observations, but as the number of
observations was generally low, these estimates were affected by a Student’s-t
distribution, which in particular created some over estimates of accuracies.
The accuracy estimates were corrected through an examination of the unitweight residuals for constant stars as a function of magnitude at a range of estimated
accuracies. The observed variances provided the correction factors necessary to
adjust the estimated standard errors. These adjustments were essential for the
variability investigations, described in Section 9.4. They were determined and
applied for the FAST and NDAC data, and later for the merged data. Figure 87
shows the final distribution of the unit weight errors for standard stars in the merged
epoch photometry.
All NDAC epoch photometry was checked for the possibility of accidental
superpositions from the other field of view, using the Tycho TICR Catalogue
(Section 8.8). A total of about 100 000 such transits were found with a minimum
perturbation of 0.01 mag. All these transits are flagged in the Hipparcos Epoch
Photometry Annex Extension file (hip ep e.dat) in fields HTE8 and HHE10, which
together provide an entry point to the Coincidence file (hip ep c.dat), in which
details of the perturbing object(s) are provided.
344
F. VAN LEEUWEN
Figure 87. The observed distribution of unit weight residuals for standard stars, compared with the
expected probabilities for a unit variance Gaussian distribution.
Figure 88. The correlation coefficients between the FAST and NDAC photometric results as a
function of magnitude.
In the examination of standard star residuals it was noted that during some data
sets discrete changes in the detector sensitivity had taken place. These changes, of
the order of a few millimagnitudes, remain unexplained, and have been corrected
in the FAST and NDAC data a posteriori.
After all corrections had been applied, the data sets were ready for comparisons.
Given the relatively straightforward reductions of the photometric data, no large
discrepancies were expected. The main emphasis was on the correlations between
the residuals for standard stars as obtained by the two groups. These correlations
were an indication of the success of the instrument modelling in the calibrations, and
of the gain there would be in merging the two data sets. Similar to the astrometric
THE HIPPARCOS MISSION
345
data, the errors on the photometric data had two contributors: the photon noise on
the original data and an instrument modelling error. The first error was the same
for both groups, the second error was largely uncorrelated due to the different
reduction methods used. The correlation coefficients were generally high, with the
exception of very bright and very faint stars (see Figure 88). The very bright stars
are most sensitive to the instrument modelling, and the faintest stars were affected
by differences in the background modelling.
The high degree of correlation meant that there was little to be gained by complicated weighting systems, and instead the photometric data were merged by taking
the mean value of the two determinations available. In case only one determination
was available, this determination was entered in the catalogue, suitably flagged. The
data were split in two files: the Hipparcos Epoch Photometry Annex (hip ep.dat)
and the Hipparcos Epoch Photometry Annex Extension (hip ep e.dat). The first of
these contains the more accurate dc photometry plus accuracies, epoch information
and a quality flag. The second file adds to this the ac photometry plus accuracy,
the FAST and NDAC background determinations, the field of view, the position on
the sky of the other field of view and the coincidence index, thus allowing more
detailed investigations.
9.3.5. Properties of the photometric data
Various aspects of the distribution of observations and the total number of observations were described in connection with the scanning law (Section 5, Figures 15
to 17). The general conclusion about the distribution is, that although the average
number of observations is relatively high (110), the distribution of those observations is far from optimal for periodicity research on variable stars.
In any investigation involving the Hipparcos epoch photometry it is important
to consider the standard errors on the individual data points. Due to characteristics
of the observing strategy different amounts of observing time were assigned to
observations of the same star at different times of the mission. In addition, different
numbers of frame transits were used in creating the field transits. This was due to
a variety of reasons, such as interruptions in the data stream, loss of data due to
telemetry errors and competition from other stars. There are therefore significant
real variations in the accuracies of the final data.
The distributions of errors for median magnitudes are shown in Figure 89. The
wings towards larger errors on the median magnitudes are the result of variability
for the dc components, and variability and duplicity for the ac components. The
median magnitudes, the associated standard errors and the standard deviations of
the epoch photometry are all available in the Hipparcos Catalogue in fields H44
to H47, also giving the number of photometric measurements available in the
Hipparcos Epoch Photometry Annex.
346
F. VAN LEEUWEN
Figure 89. The distribution of errors on the medians in three magnitude intervals. The distribution for
the dc magnitudes is shown by solid lines, for the ac magnitudes by dotted lines. The accumulation to
the right of each histogram represents the variable stars, and for ac magnitudes also the double stars.
9.4. VARIABLE STARS
All main-mission photometry was subjected to variability investigations. In the
majority of cases a statistical test on the photometric data showed no significant
signs of variability, but for those stars that did show such signs various tests for
periodicity were made. The variability investigations were done in parallel by
THE HIPPARCOS MISSION
347
Figure 90. The observed distribution of 2 values (full line) compared with the distribution expected
under the assumption that all stars are constant (dotted line). The lowest probability bin accumulated
all observations with probabilities below 10,15 .
M. Grenon and L. Eyer at Geneva Observatory, and by M.J. Penston and the author
at RGO, with assistance from various other people. Reports on aspects of the
treatment of the variable stars were presented by van Leeuwen (1997a), Eyer &
Grenon (1997), van Leeuwen et al. (1997a) and van Leeuwen et al. (1997b).
9.4.1. Statistical indicators
The Hipparcos photometry was reasonably well suited to derive statistical indicators
for variability. Care had to be taken, however, of the varying accuracies of the
individual observations of a star. A general variability statistic was entered in the
Hipparcos catalogue, field H52, as a one-letter code. This statistic was derived
from 2 tests on the histogram of residuals as obtained for every star with respect
to the median magnitude. The residuals were binned over a range of 5 in unitweight standard error. The number of bins was set according to the number of
observations available, ranging from 9 bins for less than 93 observations to 33 bins
for more than 200 observations. For every binned distribution there was an equally
binned Gaussian distribution with variance equal to 1. The differences between the
observed (ho ) and expected (hc ) distributions, for bins with at least 3 observations,
were accumulated in a 2 value (Pearson’s test, Papoulis, 1991):
2 =
Xm
ho (i) , n hc (i))2
;
n hc (i)
i=,m
(
(84)
where n is the total number of observations for a star. The 2 value was transformed
in a probability using the NAG-library function G01ECF (returns the lower or upper
348
F. VAN LEEUWEN
Table XXIV
The minimum peak-to-peak amplitude for stars to be confirmed
as variables. Stars having variability at these levels have an
average probability of being constant of less than 0.001. The
reference points are given for the average of 110 observations.
Hp
T
Hp
T
4.5
5.0
5.5
6.0
6.5
7.0
0.010
0.011
0.012
0.013
0.015
0.018
7.5
8.0
8.5
9.0
9.5
10.0
0.022
0.026
0.031
0.037
0.044
0.052
110
110
Hp
T
10.5
11.0
11.5
12.0
12.5
13.0
0.063
0.076
0.098
0.124
0.18
0.36
110
Figure 91. The distribution of amplitudes and estimated errors on amplitudes for the 5542 stars in
the unsolved variables file.
tail probability for a 2 distribution with real degrees of freedom). Figure 90 shows
the observed distribution of probabilities compared with the distribution that would
have been obtained if no Hipparcos stars were variable. The difference between
these two distributions indicate the presence of some 20 000 variables among the
Hipparcos stars, but for many of those the variability was too small to allow further
investigations. The probabilities are reflected in the main catalogue through a ‘C’
in field H52 for stars found to have a more than 50 per cent probability of being
constant (but note that this probability is very much magnitude dependent due to
the increasing noise levels towards fainter stars, see Table XXIV), and by ‘ ’ for
stars with a probability of being constant between 0.5 and 10,4 .
349
THE HIPPARCOS MISSION
It was also possible to estimate statistically the amplitude of variations for stars
that were apparently variable, but for which the amplitude and/or the distribution
of the observations did not allow a successful period search. For simplicity, a
sinusoidal variation was assumed:
mi = m0 + m1 cos i ;
(85)
where m1 is the amplitude that has to be estimated. The variance of the observations
mi is then approximated by:
R 2 2
2
2 0 cos d
< (mi , m0) >= m1 R 2
(86)
= 0:5m21:
d
0
The observed variance in the observations also contained the noise from the observational errors. Thus, taking into account the standard errors on the individual
measurements i through weights wi i,2 , and defining mi mi , m0 , the
amplitudes of variations were estimated as follows:
=
s P
mf = 2 ( i Pmiwwi=)(=n(n,,1)1) , 1 ;
i
=
f
2
1
i
(87)
where n is the number of observations. The amplitudes of the variations equaled
2m1 and were checked against the amplitudes obtained from the fitting of sinusoidal
functions for a large number of small-amplitude periodic stars. No systematic
discrepancies were detected. These amplitudes can be found in field U11 of the
Variability Annex for Unsolved Variables, file hip va 2.dat on disk 1 of the ASCII
CD-ROM set. Figure 91 shows the distribution of amplitudes and their estimated
errors as obtained for the 5542 stars in this file.
In cases where a significant variability was detected with an amplitude below
0.03 mag, the star was indicated as a possible micro variable, ‘M’, in field H52 of
the Hipparcos catalogue. A more empirical approach, ignoring the variation of the
individual errors on the measurements and fitting the observed estimated intrinsic
dispersion to the expected amplitudes as observed for periodic stars, was used in
Geneva (Eyer & Grenon, 1997).
As was indicated in Section 9.3.3, errors in the V , I colour index used in
the data reductions could cause a drift in the final photometry. All data showing
significant drifts were checked by M. Grenon to see whether such an error could
have been the cause of it. Stars for which this was likely to be the case are indicated
by ‘R’ in field H52. Improved V , I values were introduced for the final catalogue,
but the epoch photometry was not changed.
One source of apparent variability was in fact caused by the observation technique. For double stars with separations of between 5 and 35 arcsec (approximately)
and small magnitude differences, the position of one or both stars with respect to the
instantaneous field of view (IFOV) during the observations gave rice to variations
in the observed intensities. These variations were caused by the exact positioning
e
350
F. VAN LEEUWEN
Hp
Figure 92. Hipparcos
dc photometry for a newly-discovered Algol type eclipsing binary, HIP 270,
showing through the unequal spacings of the minima a highly elliptical orbit. This star received the
name V397 Cep.
of the IFOV with respect to the images. With at least one of the stars situated there
where the detector sensitivity changed rapidly, small positional changes led to large
intensity variations. The measurements for these stars are completely unsuited for
variability investigations. They are indicated by ‘D’ in field H52 in the Hipparcos
catalogue.
Incidental (bright) outliers could occur as a result of the superposition of a stellar
image from the other field of view. These were identified in as far as possible and
given in the Hipparcos Epoch Photometry Annex Extension and the corresponding
Coincidence file, see also Section 9.3.4. The remaining 10 000 variable stars were
further investigated for periodicity in the variations.
9.4.2. Periodicity investigations
The Hipparcos photometry was rather poorly suited for periodicity investigations,
in particular for periods longer than a few days. The distribution of data gaps
and their lengths (see Section 5) caused a duty cycle value as defined by Wilcox
& Wilcox (1995) of 0.01 to 0.003, well below what these authors considered
acceptable. The difficulties with detecting periodicity in the photometric data for
variable stars affected in particular the semi-regular variables, for which all too
often (probably) spurious periods were found.
A range of different methods was used in the period searches. At the RGO, an
extensive literature search led to two methods more suitable to be applied to sparse
data: the discrete Fourier analysis as described by Scargle (1982) and Scargle (1989)
and the analysis of variances by Schwarzenberg-Czerny (1989). The two methods
were used in parallel. The folded light curves resulting from periods detected
by both methods were visually inspected before being accepted. In the case of
eclipsing binaries, a manual intervention was often required, although measures
had been taken to recognize probable eclipsing binaries from the distribution of the
351
THE HIPPARCOS MISSION
Table XXV
Summary of the variable stars investigations.
Number of entries variable or possibly variable
Periodic Variables
Cepheids
Sct & SX Phe
Eclipsing binaries
Other types
Non-periodic and unresolved
Not investigated
11597
2712
273
186
917
1238
5542
3343
(8237 new)
(970 new)
(2 new)
(9 new)
(343 new)
(576 new)
(4145 new)
(3122 new)
magnitude residuals. In such cases, periods as found with the methods mentioned
above were doubled and checked for performance too. This was mainly relevant to
the analysis of variance method, which did not depend on any assumptions about
the shape of the light curve. Periods for some Algol type eclipsing binaries were
best detected “manually”. An example is the star HIP 270, which due to its elliptical
orbit had rather unevenly spaced minima, as can be seen in Figure 92.
Alongside the visual checking of the detected periods, a reference data base for
papers on photometric measurements of stars contained in the Hipparcos Catalogue
was used. This data base had been developed over a period of three years at the
RGO, and was also used to enter notes and remarks from the inspections of the light
curves. The literature and the more relevant notes have been made available both
in printed form and on the ASCII CD-ROM, disk 1, files hp refs.doc, hp auth.doc
and hp notes.doc and in Volume 11 of ESA (1997). The backbone of the data base
was formed by the General Catalogue for Variable Stars (Kholopov, 1985) and its
updates as provided by the IAU Information Bulletin for Variable Stars.
At Geneva Observatory different period finding routines were applied to three
groups of variable stars:
Fourier analysis for small-amplitude variables (Deeming, 1975), (FerrazMello, 1981);
Dispersion minimization for large-amplitude variables (Stellingwerf, 1978);
Renson’s method for eclipsing binaries (Lafter & Kinman, 1965), (Renson,
1980).
Here too, all results were examined manually before acceptance, but no references
to literature were used.
The final list of periodic variables, as presented in Volume 11 of ESA (1997)
and on ASCII CD-ROM disk 1 as file hip va 1.dat, is based on a wide ranging
verification of any discrepant periods obtained with the RGO and the Geneva
analysis. In those cases where the final choice was not clear, a possible alternative
period is given in the notes. Stars for which the period appeared possible but
uncertain have in most cases been entered in the table for unsolved variables,
352
F. VAN LEEUWEN
Figure 93. The distribution of periods given in the Hipparcos Catalogue as Hipparcos determinations
(solid line) and as available in the literature (dotted line).
with the possible period given in the notes. Some of the discovery statistics are
summarized in Table XXV.
The distribution of periods as obtained from the Hipparcos data is shown in
comparison with the periods that were known already in Figure 93. This figure
shows the problems for ground-based measurements with detecting periods around
1 day, a problem not existing for Hipparcos and consequently the range of periods
where most discoveries were made. It also shows that Hipparcos did not contribute
much to periods longer than about 8 days. This was partly due to the distribution
of the data, but possibly also to the larger amplitudes of variations that are generally associated with long-period variables. These large-amplitude variables were
generally well known already. The concentration of new periods between 0.1 and
0.6 days is due to Scuti stars, small-amplitude red variables and Cep stars.
For all periodic variables an optimization of the period was obtained through
curve fitting. At the RGO a semi-regular spline fit of every light curve was obtained
by repeating the sequences of observations and knots 3 times, and using the central
section of the spline fit. This did not always provide a perfectly closed light curve,
but provided a curve that was well suited for a period optimization. The optimal
period was obtained through minimizing the sum of the squared residuals of the
observations Oi with respect to the fitted curve f , as a function of a change to
the period and the derivative of the fitted curve. As was shown by (van Leeuwen et
al., 1997b), these residuals can be given by:
()
t , t p @f ()
;
Oi , f (i ) = , i 0
p~ p~
@ =i
(88)
353
THE HIPPARCOS MISSION
Figure 94. The accuracies for periods of Cepheid variables as determined from the Hipparcos data
alone. The dotted line represents the relation p 10,5 p2 .
=
~
where ti is the epoch of observation, t0 an arbitrary reference epoch, p the last
estimate of the period and p the correction to this period, such that the new
p
p. This solution was iterated a few times between the curve
period p0
fitting and the period optimization.
Along with the determination of the period correction was obtained an estimate
of the accuracy of the period. These accuracies are approximately proportional to
the squared periods, with the proportionality factor depending on the amplitude
and coverage of the light curve (see Figure 94). A different method was used in
Geneva, where the curve fitting was done by means of Fourier components.
A complete set of fitted light curves for 2712 periodic variables is presented in
Volume 12 of ESA (1997). A selection of unsolved light curves is shown in the
same volume. All light curves can also be found as encapsulated postscript files on
disk 1 of the ASCII CD-ROM set, directory curves/.
The assignment of variability types to newly discovered variables was often
difficult and dubious due to the lack of complementary information. This led, in
particular for various (semi-)regular pulsating stars, to types that may very well
turn out to be wrong. A special effort was made by C. Waelkens to classify many of
the low amplitude variables. Names were assigned to newly discovered variables
whenever the discovery appeared reliable. This was done, as is done for all newly
discovered variable stars, at the Sternberg Institute in Moscow, where over 3000
discoveries from Hipparcos were given names.
~ = ~+ 354
F. VAN LEEUWEN
10. Verifications of the Hipparcos and Tycho data
The verification of the Hipparcos astrometric and photometric data was in many
ways provided through the parallel reductions by the two consortia. This ensured
an optimal reduction of the data, but could not eliminate any problems that may be
inherent to the data itself. There is currently very little material with which Hipparcos astrometric results can be compared. A number of studies were, nevertheless,
carried out.
10.1. POSITIONS AND PROPER MOTIONS
The only verification of positional precisions possible was with the milliarcsecaccuracy positions obtained in the radio reference frame, ICRF (see Section 8.7.4).
This comparison showed that, after transformations had been applied, the positions
in the Hipparcos reference system were fully consistent with those obtained with
large interferometric arrays in radio astronomy. Differences between the Hipparcos
optical reference frame and its predecessor, the FK5, are, on the other hand, locally
as much as 0.1 arcsec. Such differences had been noted before from data obtained
with the automated meridian circles (Morrison et al., 1990).
The verification of proper motions was more complicated because of the differences in epoch coverage between the Hipparcos observations (3.5 yr) and groundbased observations (> 50 yr). As was pointed out by Wielen (1996) and Wielen
et al. (1997), orbital motions on a time-scale of 5 to 25 years can disturb the comparison between the Hipparcos and ground-based determinations. This may be the
reason why the differences between the proper motions as given in the FK5 Catalogue and proper motions for the same stars given in the Hipparcos Catalogue show
a larger than expected dispersion. Only 1233 FK5 stars could be compared, 302
others showed various signs of duplicity (actual double stars, non linear motions,
orbital motions etc). The proper motion comparison clearly shows the effect of the
meridian-telescope data on which the FK5 is based: as a function of right ascension
there are no systematic differences larger than 1 mas/yr, but as a function of declination (where the input from different instruments becomes relevant) differences
reach a level of 3 to 4 mas/yr. Thus, instead of being due to orbital motions, the
differences could still be at least partly due to problems in the FK5 proper motions.
The use of Hipparcos proper motions in the analysis of the Hyades cluster by
Dravins et al. (1997) provided an indirect test of their quality. In the analysis,
all cluster star are assumed to share the same space motion. Without making
assumptions concerning the radial velocities of the cluster stars, the projection
effect of the proper motion and the parallax alone are sufficient to reconstruct the
spatial structure of the cluster (see Section 11.3). The constraint of the shared space
motion leads to improvements by a factor two of the individual parallax estimates.
This shows in a cluster HR-diagram through much reduced noise (see Figure 95).
355
THE HIPPARCOS MISSION
(a)
0
MHp [mag]
2
4
6
8
All (195 stars)
Distances from parallax
10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(b)
0
MHp [mag]
2
4
6
8
All (195 stars)
Distances from model fit
10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(c)
0
MHp [mag]
2
4
6
8
Sample AB (95 stars)
Distances from model fit
10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
B − V [mag]
Figure 95. The HR-diagram for the Hyades, using data for all presumed member stars (a and b), and
for a subset of 98 stars with a good fit to a model assuming a constant space velocity for all stars,
and excluding spectroscopic binaries (c). In (a) the observed parallaxes were used to calculate the
absolute magnitudes, in (b) and (c) the estimated parallaxes from the model fit. (from Dravins et al.,
1997)
356
F. VAN LEEUWEN
Any inhomogeneity in the proper motions would have shown in a disturbed HRdiagram. Instead, the HR-diagram turns out to be of very small intrinsic width.
A comparison between differential high precision proper motions in the Pleiades
cluster and the Hipparcos proper motions show a general agreement, but possibly
also some problems of the kind described by Wielen et al. (1997).
10.2. PARALLAXES
The parallax determinations as provided by the Hipparcos mission are in principle
absolute, with no systematic errors expected above 0.1 to 0.2 mas. There are few
possibilities to check parallaxes for a reliability at such a high level, most of which
were presented by Arenou et al. (1995). One of the few fully reliable methods
examines the parallaxes for objects that can safely be assumed to be too far away
to give a measurable parallax in the Hipparcos data. Using the methods described
by van Leeuwen & Evans (1997), a common parallax was solved for the stars in
the Magellanic Clouds, giving the following results:
LMC
SMC
= 0:27 0:30 mas
= ,0:44 0:65 mas
31 stars
8 stars
The results are consistent with the assumed parallax values for the Magellanic
Clouds of around 0.002 mas, but do not constrain systematic errors to a very high
level. A comparison with dynamical parallaxes for 369 orbital systems (Dommanget & Nys, 1982) showed a consistency with the Hipparcos determinations
of 0.25 mas. Accurate parallaxes for binary stars in the Hyades by Torres et al
(1997a), Torres et al (1997b) and Torres et al (1997c), using orbital parameters,
show excellent agreement with the Hipparcos results, as was shown by Perryman
et al. (1997b).
There are two other sources of parallax measurements with similar accuracies
as provided by Hipparcos: 88 stars in common with the program of the USNO
(Monet et al., 1992), (Harris et al., 1997) and 12 parallaxes obtained with VLBI
observations of radio stars, 5 of which concern double stars of various types. The
comparisons between these data and the Hipparcos determinations are shown in
Figure 96 and 97.
In Chapter 20 of Volume 3 of ESA (1997) two more comparisons were made,
with cluster star parallaxes and with photometric parallaxes using the Strömgren
photometric system. Both indicate a reliability of the Hipparcos parallaxes to better
than 0.1 mas, but these methods cannot be considered as entirely independent
verifications, and some doubt about the use of cluster stars and their ground-based
distance determinations appears to be justified after the discovery of a 0.3 mag
discrepancy between the photometric and the Hipparcos parallax for the Pleiades
cluster by van Leeuwen & Hansen-Ruiz (1997) and for the Pleiades and other open
clusters by Mermilliod et al. (1997).
357
THE HIPPARCOS MISSION
200
180
160
USNO parallaxes (mas)
140
120
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
200
Hipparcos parallaxes (mas)
Figure 96. Comparison between Hipparcos and USNO parallaxes.
50
45
40
VLBI parallaxes (mas)
35
30
25
20
15
10
5
0
-5
down-weighted stars
in DMSA annex
-5
0
5
10
15
20
25
30
35
40
45
50
Hipparcos parallaxes (mas)
Figure 97. Comparison between Hipparcos and VLBI parallaxes. Two stars were down-weighted
and five stars are members of resolved double or multiple stars.
358
F. VAN LEEUWEN
V
Hp
Figure 98. Comparison of Walraven -band measurements with Hipparcos median
magnitudes.
Above: the direct comparison of the data (the W band is expressed in log ), and the fitted curve.
Below: the residuals left after subtracting the fitted curve.
V
I
10.3. PHOTOMETRIC DATA
Hp
Comparisons between the Hipparcos
photometry and photometric observations
obtained in the Johnson, Geneva and Walraven (Lub & Pel, 1977) systems were
made. Unlike the ground-based data, the Hipparcos data is not affected by seasonal
effects or limited to one hemisphere. A comparison between ground-based systems
was made by Pel (1991) which showed systematic discrepancies for the Geneva
photometric system. These discrepancies showed again in the comparison between
the Hipparcos and the Geneva photometry.
passband of the Hipparcos main mission photometry,
Due to the wide
relations with the intermediate bands of the Walraven, Strömgren, Geneva and
similar systems are complicated when examined in detail, and have to be resolved
as functions of luminosity class. Examining only constant stars of luminosity
and the Walraven
band shows
classes III to V, the comparison between
systematic differences to be well below a level of 0.001 mag (see Figure 98). This
Hp
Hp
V
THE HIPPARCOS MISSION
359
,
Figure 99. The residuals VGen VHip for stars with declinations below 10 (bottom) and above 10
(top) as a function of right ascension.
is in practice only possible if both the Walraven and the Hipparcos photometric
systems are free from systematic errors to at least 0.001 mag.
The comparison with the Geneva system showed systematic differences as a
function of right ascension, as can be seen in Figure 99. In this comparison the
magnitudes were transformed to the Gen passband and are referred
Hipparcos
to as Hip . The observed differences as a function of right ascension are due to
errors at a level of approximately 0.005 mag in the Geneva system, as nothing
similar is noted in comparisons of the Hipparcos photometry with the Walraven or
with the Johnson UBV system, and similar discrepancies had already been noticed
in comparisons between the Geneva and the Strömgren, Walraven and Johnson
systems (Pel, 1991).
V
Hp
V
360
F. VAN LEEUWEN
,
Figure 100. The mean differences Vground VHip as function of the apparent V magnitude from the
Geneva photometry (filled symbols) and from the UBV photometry (open symbols). The observed
differences for the bright stars were the result of saturation effects.
For the handful of brightest stars there is likely to be a problem with the
Hipparcos photometry due to the failure of the analogue mode (see Section 7.1).
Stars that were supposed to be measured in analogue mode had to be measured
in photon counting mode, knowing that the signals would be partly saturated.
A comparison between Hipparcos, Johnson and Geneva magnitudes is shown in
Figure 100. For the brightest star, Sirius (HIP 32349), the situation is obviously the
worst, and the underestimate of the brightness by Hipparcos reaches 44 per cent.
11. On the proper use of the Hipparcos data
Although the application of parallax data as obtained by Hipparcos may seem
simple and straightforward, this only applies when relative errors are very small
(well below 10 per cent, Lutz & Kelker, 1973). The user has to remain constantly
aware of the way the parallax data are implemented and how accuracies of astrometric parameters are accounted for. Failing to do so can lead to unreliable results
and wrong conclusions. In positional comparisons, such as for the use of plate
transformations, the Hipparcos positions and their estimated accuracies have to
be properly transferred to the epoch of the plate, considering the full covariance
matrix in recalculating the standard errors of the transformed positions. This is
in particular the case for nearby stars where also the radial velocity can play a
361
P(πH)
THE HIPPARCOS MISSION
π
0.0000
0.0025
0.0050
πH
0.0075
0.0100
Figure 101. The distribution of the observational error in the observed parallax for a star at 200 pc.
(from Luri, 1997)
role in the propagation of the astrometric parameters and their estimated errors. In
addition, proper motions for these stars are sometimes non-linear.
11.1. CONSIDERING SINGLE STAR OBSERVATIONS
Hipparcos measured parallaxes, from which distances can be derived. To obtain
these distances is, however, not as straightforward as it may seem, in particular
when the relative error on the parallax is larger than about 5 per cent. The error
on the Hipparcos parallax is well determined, but this is not the case for derived
quantities (Lutz & Kelker, 1973, Luri & Arenou, 1997, Brown et al., 1997a). The
probability density function (pdf) for the parallax measurements can be assumed
Gaussian. The pdf for derived quantities is given by the pdf for the parallaxes times
the derivative of the parallax with respect to the derived quantity, the Jacobian.
In the case of distances, this means multiplication of the pdf for parallaxes by
(1=R)2 , where R = 1= . Figures 101 and 102 demonstrate this for the parallax
measurement of a star at 200 pc. The first of these figures shows the probability
distribution for measuring a parallax H for a star with parallax = 0:005 arcsec.
The second plot shows the probability distribution for the distances given a parallax
of 0.005 arcsec as measured by Hipparcos. Though the direct conversion would give
a distance of 200 pc, the most likely distance associated with this parallax is higher.
This effect is also known in the literature as the Lutz-Kelker effect (Lutz & Kelker,
1973) and demonstrates the need to interpret the Hipparcos parallax data in terms of
parallaxes and not in terms of distances or distance moduli. It was shown by Brown
et al. (1997a) how the estimated value for a distance modulus differs from the
calculated value on the basis of the observed parallax (see Figure 103), depending
362
P(R)
F. VAN LEEUWEN
R0
0
200
400
600
R
Figure 102. The probability density function for the distribution of distances given a measured
parallax of 0.005 arcsec with a formal standard error of 0.001 arcsec. (from Luri, 1997)
1.5
∆M
1.0
0.5
0.0
-0.5
π=10 mas
π=1 mas
π=0.1 mas
0
1
2
3
σπ /π
Η
Figure 103. The bias in absolute magnitudes as a function of the relative accuracies of the parallaxes.
(from Brown et al., 1997a)
on accuracy of the measured parallax relative to the actual stellar parallax. It is
not possible, however, to correct on an individual basis for a bias, as the bias can
only be calculated when the actual parallax of the star is known, in which case the
correction is of no interest. Correcting on an individual basis, using the measured
parallax, will essentially make results only more unreliable.
When relative errors on parallaxes are large, the data should only be used in testing models in the sense that the observed parallax is or is not in agreement with the
parallax predicted by the model. Low accuracy individual parallax determinations
should not be transformed into measurements of distance or distance modulus.
363
THE HIPPARCOS MISSION
2.5
2.0
σM
1.5
1.0
π=10 mas
π=1 mas
π=0.1 mas
0.5
0.0
0
1
2
3
σπ /π
Η
Figure 104. The precision on the computed absolute magnitude as a function of H = . (from Brown
et al., 1997a)
Error estimates for these derived parameters can become completely meaningless
(see Figure 104).
An example of the strange results authors can obtain when examining Hipparcos
data in terms of distances was encountered in a paper submitted to a refereed journal,
in which the Hipparcos parallax measurement of 0:05 0:82 mas for the Cepheid
V Cen (HIP 71116, for which the expected distance is approximately 721 pc) was
interpreted as a distance of 20 000 pc, which was subsequently used to derive a
value for the distance modulus. However, the estimated distance predicts a parallax
of 1.4 mas, which is within 2 from the observed parallax value of 0.05 mas.
The occurrence of negative parallax values can always be accounted for, as
long as expected distances are translated into expected parallax values before
comparisons are made. The occurrence of negative parallaxes is a simple statistical
effect, and ignoring these or trying to explain them as the result of some other
astrophysical phenomenon is in fact a misinterpretation of the data.
11.2. CONSIDERING GROUPS OF STARS
There are better possibilities in dealing with lower accuracy parallaxes when considering groups of stars. There are, however, also more pitfalls. The best known
of these is the Malmquist bias: the error on the estimation of the mean parallax
for a group of stars when their apparent magnitudes are affected by absorption
(Malmquist, 1922, Malmquist, 1936). This has its equivalent in an easily made
selection bias which will distort a data sample. An obvious example is ‘ignoring
negative parallaxes’. By doing so, the distribution of parallax values is truncated
and as a result shifted to a higher average value. When examining data purely in
terms of parallaxes such a cutoff is unnecessary, unlike when an examination of the
data is (wrongly) made in terms of distances or distance moduli, and negative par-
364
F. VAN LEEUWEN
Figure 105. The relation between photometric parallaxes, calibrated using Hipparcos parallax data,
and the observed parallaxes with their estimated formal errors for RRab and RRc stars. The solid line
represents the one-to-one relation.
allaxes can no longer be properly accounted for. Only final results, when significant
in terms of a parallax measurement, should be translated into related parameters,
still noting the often asymmetric error distributions for these derived parameters
and the fact that the most probable value for a derived parameter is usually not
identical to what is found from a direct mathematical derivation of the derived
parameter. As was shown by Lutz & Kelker (1973), no attempts should be made to
obtain derived parameters when relative accuracies are worse than 17 per cent.
An example of how to express a luminosity calibration in terms of the parallax
measurements was presented by van Leeuwen & Evans (1997), using the calibration
of the absolute magnitude of RR Lyrae stars as an illustration, and applied by (van
Leeuwen et al., 1997c) for the calibration of the Mira PL relation.
The basic equation for a luminosity calibration describes the absolute magnitudes for a group of stars as a function of a set of observable parameters:
M^ = M (ai )
(89)
The relation between the observed magnitude m (corrected for interstellar reddening and measured in the same photometric passband as M ), the calibrated
magnitude and the parallax (here expressed in mas) is then given by:
^ = 100: e,0:4605(m,M (a )) :
i
(90)
^
Thus, a correction to the estimated parallax is in first approximation given by:
= ,0:4605^
X @M ai:
n
i=1 @ai
(91)
As these relations are non-linear, they have to be iterated through applying and
obtaining smaller and smaller corrections for the assumed parameters ai . The
THE HIPPARCOS MISSION
365
proper criterion for the solution of the parameters ai is the 2 value of the residuals
of the parallax measurements. This 2 value is also an indicator for the quality of
the model used for the luminosity calibration. In cases when observable parameters
in the model are having relatively large errors, methods beyond simple least squares
will have to be used to minimize the variances of not only the parallaxes, but also
for these specific parameters. Figure 105 shows graphically the implementation of
such a solution for RR Lyrae stars.
The bias in estimates as described by Lutz & Kelker (1973) does not enter
the definition of the luminosity calibration parameters, as the entire description
of the problem is done in parallax space. Only when applying the calibration in
order to obtain an absolute magnitude estimate, and comparing this estimate with
an individual, measured apparent magnitude, do the bias problems reoccur. The
application of the calibration provides an estimate of the parallax with a formal
standard error, which translates non-linearly into uncertainties in estimated distance
and distance modulus. At this stage, however, the uncertainty will in general be
reduced in comparison with the parallax uncertainties for the original single-star
data. Only if the relative formal accuracies for the parallaxes obtained from the
luminosity calibration are sufficiently small (less than 5 to 10 per cent), can the
luminosity calibration be applied directly in magnitude space.
A somewhat different method was presented by Luri et al. (1996) and Luri
& Arenou (1997). Examples of the implementation of their “LM method” were
presented by Arenou & Gómez (1997), Gómez et al. (1997a), Mennessier et al.
(1997) and Alvarez et al. (1997). The “LM method” uses all available information
for a given star: the parallax, proper motion, radial velocity and apparent magnitude
as well as any other available physical parameters. The distributions of these
parameters are represented by a probability function, from which the expected
value of the distance R for a star is calculated, which is then used to obtain an
estimate of the true distance RLM . Monte Carlo simulations on a sample of A0
stars showed a very good performance of this method in comparison with various
other methods. It produces for every individual star an unbiased estimate of the
distance. The “LM method” segregates stars according to kinematical groups, based
on proper motions, radial velocities (when available) and distribution in space.
11.3. POSSIBILITIES WITH THE INTERMEDIATE ASTROMETRIC DATA
The Intermediate Astrometric Data file (hip i.dat) contains the data used to derive
the astrometric parameters for undisturbed single stellar images. These data allow
for a re-examination of the original astrometric measurements in the context of
alternative astrometric models, which can involve just one star or groups of stars
(van Leeuwen & Evans, 1997). When considering single stars, an alternative solution may include a new orbital solution or other additional parameters in the
astrometric solution.
366
F. VAN LEEUWEN
The main application for the intermediate astrometric data is for determining
common parameters (parallax, space motion) of groups of stars (star clusters,
associations, Magellanic Clouds) and in luminosity calibrations. The luminosity
calibrations proceed along the same lines as described in the previous section, with
the exception that the parallax values for all stars are determined from the original
abscissa data rather than taken from the catalogue. The advantage of this is a better
recognition of bad data points, and the possibility to eliminate weak and potentially
damaging data, like the astrometric data obtained for Miras near minimum light,
which can be disturbed much more easily than the other measurements by duplicity.
In clusters and associations the problem is to derive the best possible reference
parallax and space motion for the system. However, as was shown in Section 8.5,
the original abscissa data from which the astrometric parameters for stars in a small
area of the sky were formed are correlated. Thus, by going back to the abscissae
data, these correlations can be accounted for in the determination of the parameters
of the system. The methods and the data needed to remove these correlations were
described by van Leeuwen & Evans (1997).
The problem in all solutions for groups of stars is to express the abscissa data as
functions of the model parameters or corrections thereof. This was already shown
for luminosity calibrations before. For a star cluster the relevant parameters are the
space velocity and distance for a fiducial reference point in the center of the group.
The observed proper motions and parallax (and if relevant and available also the
radial velocity), are related to expected proper motions (which will be a function
of parallax and position on the sky), and the differences are expressed as functions
of corrections to the assumed space velocity and parallax of the reference point. As
the centre of an open cluster is in general rather poorly defined, the exact position
of the reference point should be of little importance.
The position for the cluster centre is given by the vector u0 in pc:
0 cos
u0 = 0,1 @ sin
1
A;
0 cos 0
sin 0
0 cos 0
(92)
and its derivative by
0 cos
u_ 0 = 0,1 @ sin
0 cos 0
sin 0
0 cos 0
, sin
cos
0
0
0
, cos
, sin
10
1
V 0 0
A @ 0 A ;
0 sin 0
0
cos 0
0 sin 0
(93)
where the vector u_ 0 is expressed in arcsec yr,1 , and the radial velocity V0 in
parsec yr,1 (obtained from dividing V0 , when expressed in km s,1 , by 4.74074).
The vector on the right-hand side will be referred to as the velocity vector v:
0V 1
i i
vi = @ A ;
i
i
(94)
367
THE HIPPARCOS MISSION
=
where the index i refers to a star if i 6 0 and to the centre of the cluster if i
The inverse of Equation 93 can easily be derived:
= 0.
0 cos cos sin cos sin 1
0
0
0
0
0
cos 0
0 A u_ 0 :
v0 = 0 @ , sin 0
(95)
, cos 0 sin 0 , sin 0 sin 0 cos 0
The space velocity u_ 0 is assumed constant over the cluster : u_ = u_ 0 for all values
i
of i. Only in the centre of the cluster is this condition not completely true due to
internal motions. These are, however, in general quite small (of the order of 500
to 1000 m s,1 ). The astrometric parameters for a star at position ( i , i ) can be
related to the astrometric parameters for the centre of the cluster. The relations for
cluster star i are given by Equation 95, through replacing the index 0 by i. But,
as i
0 , Equation 93 can be back substituted to obtain relations between the
observed velocity vector i and the reference velocity vector 0 :
u_ = u_
v = 0 Av0;
i
v
v
i
where the matrix
for columns):
(96)
A has the following elements (first index for rows, second index
A11 = cos cos cos 0 + sin sin 0 1
A21 =
sin cos A31 = , cos cos sin 0 + sin cos 0 sin , sin cos 0
A12 =
A22 =
cos 1
A32 =
sin cos 0
A13 = , cos sin cos 0 + cos 0 sin , sin A23 =
, sin sin (97)
A33 = cos sin sin 0 + cos cos 0 1
where = , 0 and = , 0 . Equations 96 and 97 contain the classic
i
i
i
i
i
i
i
i
i
i
i
i
convergent point method for cluster distance determination. The convergent point
is found at the ( i , i ) value for which the matrix aligns the velocity vector 0
with the radial velocity direction. As the convergent point has to lie on the great
circle through the proper motion of the cluster centre, the variables ( i , i ) are
not independent. Thus, a set of two equations with two unknowns can be obtained
for the observed proper motions, solving for the angular distance between the
direction of the cluster centre and the direction of the convergent point, and V0 0 .
Combined with the measured radial velocity V0 , the distance 0 of the cluster centre
is obtained.
The same equations are at the basis of the study by Dravins et al. (1997), who
used them to derive for individual stars in the Hyades cluster the parallaxes and
A
v
368
F. VAN LEEUWEN
radial velocities, based on the assumption that all stars in the cluster share the
same space velocity. After having derived the space velocity using the parallax,
radial-velocity and proper-motion data for individual members, Equation 95 can be
used to derive two equations for the observed proper motion components of a star,
with two unknowns, the radial velocity and the parallax of the star (see Figure 95).
For a more distant cluster the spread in positional coordinates is small and
the approximations made in Equation 97 can be used. In most cases it will also
be correct to ignore the projection effects of the radial velocities and the relative
distance variations. What is left then is a set of equations that describe the proper
motion of star i with respect to the proper motion of the cluster centre:
i
i
+ sin sin 0 , sin sin i + 0
0
0
0
(98)
Using the Intermediate Astrometric Data it is now possible to solve for the cluster
proper motion using all cluster members. The procedure, described in detail by van
Leeuwen & Evans (1997), is as follows:
define an estimated cluster proper motion as a reference value;
correct the abscissa residuals for all member stars for the difference between
the measured proper motion and the reference proper motion;
describe the corrections to the abscissae residuals for all members as corrections to the proper motion of the cluster centre using Equations 98;
correct these observation equations for the correlations between abscissae
with small separations contained on the same reference great circle;
apply a least squares solution to solve for corrections to the assumed cluster
proper motion.
In this procedure faulty data points can be eliminated if required. Experience shows
that some data points rejected in the single star solutions can be reinstated. This type
of solution effectively treats the cluster star observations as if they are observations
of a single star, situated in the centre of the cluster. The same procedure can also
include an estimate of the parallax of the cluster. This is done in the same way as
was done for the common proper motion.
The least squares solution provides a 2 value, showing the correctness of the
model that was applied (in this case the constant space velocity and common
parallax). It is expected and observed that this model is not entirely correct due to
the internal motions in the centre of a cluster. It may be possible to combine the
Hipparcos solution with the information from the much more precise differential
proper motion studies, but this appears to be a far from trivial exercise, with
difficulties caused by the much longer time-span used for ground-based proper
motions, and distortions in ground-based proper motion reference frames (Tian et
al., 1996).
To summarize, the Intermediate Astrometric Data offer a range of possibilities,
from a re-examination of the data for an individual star, to a proper treatment for
THE HIPPARCOS MISSION
369
Figure 106. An example of uv coverage for a Hipparcos object. This object (HIP 15719) was scanned
from virtually all angles. The middle point gives the total flux from all scans, while the two circles
represent the two spatial frequencies for the Hipparcos sampling. (from Quist et al., 1997)
stars in open clusters, associations and the Magellanic Clouds (Kroupa & Bastian,
1997, van Leeuwen & Evans, 1997).
11.4. POSSIBILITIES FOR THE TRANSIT DATA
The Transit Data in file hip j.dat make it possible to re-examine signal-modulation
data for signs of duplicity. The main areas of application are double stars of relatively small separation or large magnitude difference, double stars containing one
variable component, and multiple stars. Most other situations have already satisfactorily been resolved in the standard reductions. This type of investigation requires
input parameters: an estimate of the separation(s) and magnitude difference(s) in
the system, to be obtained from independent measurements.
As was shown by Quist et al. (1997), the Hipparcos main-mission data has
features very similar to data obtained with radio interferometers, and similar techniques can be used in examining those data. The two harmonics in the main detector
signal produced 3 unique points in the uv plane, equivalent to baselines of 0, 10,
and 20 cm, with the centre of the uv plane being equivalent to the zero-baseline,
and giving the total flux of the observed object. Figure 106 shows an example of a
well scanned star (HIP 15719).
370
F. VAN LEEUWEN
Figure 107. The raw image (dirty map) of Hipparcos object HIP 15719. This map is the Fourier
transform of the previous figure. (from Quist et al., 1997)
An image obtained by the Hipparcos satellite is derived by taking the Fourier
transform of the uv plane along with the phase and amplitude information. This
produces what is called a dirty map, requiring further image processing. Figure 107
shows this for the same star as used in Figure 106. Using the CLEAN algorithm
as contained in the Caltech Difmap package, the original image is restored (see
Figure 73). A users package for examination of the Transit Data, describing these
data in UV FITS files (Greisen & Harten, 1981) as used in radio-interferometry, is
in preparation by C.F. Quist and L. Lindegren (Lund Observatory).
12. First scientific results
This section can by no means be a comprehensive review of the scientific results
of the Hipparcos mission, the range of results and details is simply too wide and
too many studies have only just started. In preparing it, much use was made of the
proceedings of the Venice’97 symposium, supplemented by a few papers that have
already appeared in refereed journals, or for which preprints were available.
12.1. NEARBY STARS, LOCAL GALACTIC KINEMATICS
Using nearby stars the Hipparcos data clarified to some extent the reliability of
photometric/spectroscopic distance determinations for stars of different luminosity
THE HIPPARCOS MISSION
371
Figure 108. The distribution with respect to distance of 5610 objects selected on spectral type and
apparent luminosity to lie within 80 pc. 2384 of these were found to have larger distances. (from
Binney et al., 1997)
classes, and the kinematics and population of the solar neighbourhood. Studies of
this type are providing a better basis for distance calibrations of objects too distant
to have their parallax measured, and of the history of the local group in our Galaxy.
A first examination of stars contained in the Catalogue of Nearby Stars (Gliese,
1969, Gliese & Jahreiß, 1979, CNS3 as available at the CDS in Strasbourg) confirmed aspects of distance determinations based on photometric and spectroscopic
data that had been suspected before by Grenon (1984) on the basis of a Geneva photometric survey of these stars. The Hipparcos parallax data established that some
40 per cent of stars in the CNS3 lie beyond 25 pc, the limit set for this catalogue
(Perryman et al., 1995). A significant number was found to have parallaxes of less
than 20 mas. This was confirmed as a more general trend by an examination of
a selection of stars for which the Michigan Spectral Catalogue (Houk & Cowley,
1975, Houk, 1978, Houk, 1982) indicated a luminosity class V, which, given their
apparent brightness, would put these stars within 80 pc (Binney et al., 1997, Murray et al., 1997). A histogram of the Hipparcos distances derived for these stars
is shown in Figure 108. A diagram like this should, however, be presented as a
histogram giving the numbers of stars as a function of measured parallax rather
than derived distance, as was done by Perryman et al. (1995) for the stars within
25 pc. This allows it to be compared with the formal errors on the parallaxes. A
careful examination will now be required to find out how the discrepancies between
the expected and observed distances for these stars came about, and what relevance
they may have with respect to other studies.
One field which has depended for a long time on simple associations between
spectral features and luminosities is the study of the kinematics and distributions
372
F. VAN LEEUWEN
Figure 109. The variation with colour B
dispersion S . (from Binney et al., 1997)
,V
of the components of solar motion U , V , W and the
Figure 110. Velocity dispersion as a function of age. Open symbols: CNS2-values; filled symbols:
new results. Circles: main sequence stars; triangles: K and M dwarfs. (from Jahreiß & Wielen, 1997)
of stars within 100 to 200 pc from the Sun. New relations between spectral classifications and absolute magnitudes and their dispersions were determined using
selections of nearby stars (Houk et al., 1997). This study, however, seems to show
signs of using a sample truncated at a fixed parallax value similar to those indicated
by (Luri & Arenou, 1997). The overall results confirm to within a few tenths of
a magnitude the classical values tabulated by Schmidt-Kaler (1982), but provide
in addition estimates of the intrinsic dispersions in absolute magnitudes for different spectral types and luminosity classes. These dispersions were found to be
considerable larger than than they were expected to be.
Preliminary results of various kinematic studies using Hipparcos data are now
available. The mean velocities and velocity dispersions for groups of stars within 80 pc was investigated by Binney et al. (1997) as a function of colour (see
THE HIPPARCOS MISSION
373
Figure 111. The dependence of U , V and W on S 2 . The dotted lines correspond to the linear relation
fitted (V ) or the mean values (U and W ). (from Binney et al., 1997)
Figure 109). The velocity dispersions were represented by a combined value S 2 ,
which shows a clear relation with stellar colour: from increasing over the blue stars
to constant over the red stars. This effect, first noted by Parenago (1950), reflects
the varying mean ages of stars of different colours, and the increase in velocity
dispersion towards higher ages. Similar effects were derived from the Hipparcos
data by Gómez et al. (1997b) and Jahreiß & Wielen (1997), who split up stars
according to age-groups (see Figure 110). Gómez et al. (1997b) also claimed that
the velocity ellipsoid gets rounder with increasing age.
Strömberg’s asymmetric drift equation (Binney & Tremaine, 1987) predicts
a linear dependence of the V -component in the velocities (the direction of the
rotation in the Galaxy) and S 2 : when the velocity dispersion in a group increases,
this group will rotate more slowly around the Galactic centre. This relation is shown
clearly by the Hipparcos data (see Figure 111), and can be extrapolated to a zero
dispersion, which represents the velocity of the Sun (U0 , V0 , W0 ) with respect to
the local standard of rest: the velocity in a closed orbit in the Galactic plane that
passes through the location of the Sun. The following values were found for the
local standard of rest:
U0
V0
W0
= 11 0 0 6km s,1
= 5 3 1 7km s,1
= 7 0 0 6km s,1
:
:
:
:
:
:
(99)
:
These values are significantly different from the classical values of Delhaye (1965)
in U0 and V0 . This may be related to the difficulties in associating the correct
luminosity class to some types of stars (see Figure 108).
The distribution of nearby stars was used by Crézé et al. (1997) to derive
the potential well and local dynamical mass. The availability of distances and
proper motions from the Hipparcos Catalogue has now made it possible to trace
374
F. VAN LEEUWEN
Figure 112. The stellar luminosity function derived from the stars within 25 pc. The units are the
number of stars within 20 pc per one magnitude bin. The dashed histogram shows the LMF from
Wielen et al. (1983). (from Jahreiß & Wielen, 1997)
the potential well to 125 pc from the Sun, using 2977 A-type stars. The local
dynamical density was found to be 0 = 0:076 0:015M pc,3 , close to the figure
for known matter of 0:085M pc,3 . Using the observed value, a dynamical model
of the Galactic disk was built, showing that there may be just room for a spherical
halo of dark matter (0.008 M pc,3 ), but there is no room for a significant amount
of dark matter to be concentrated in the disk.
The local luminosity function has been investigated by Murray et al. (1997) and
Jahreiß & Wielen (1997). The first of these studies looked at stars within 80 pc,
the second within 25 pc. Hipparcos discovered 119 new stars within 25 pc and 37
previously unknown companions of nearby stars. The nearest of these is HIP 103039
at 5.5 pc. The completeness of the sample is very different for declinations north of
= ,30 than for southern declinations as a result of the Luyten surveys for high
proper motion stars, and the fact that high proper motion stars were selected for the
Hipparcos Input Catalogue. Figure 112 shows the derived luminosity function. The
main differences with earlier determinations are that the density of giants appears
to be lower, while the density of the faintest main sequence stars is higher. The
estimate of the total density of stars has decreased from 0.045 pc,3 to 0.039 pc,3 .
375
THE HIPPARCOS MISSION
3.2
M/Msun
1.6
0.8
0.4
0.2
0.1
0
2
4
6
8
10
Hp(abs)
12
14
16
Figure 113. The mass-luminosity plot for all masses with an uncertainty below 15 per cent. (from
Söderhjelm et al., 1997)
12.2. STELLAR MASSES
Further details on the properties of the nearby stars were obtained from mass
determinations derived from orbital parameters of binary systems for which the
Hipparcos data also provided an accurate parallax measurement (Söderhjelm et al.,
1997). A re-examination of the orbital-system data improved in a few cases the
parallax determination with respect to the value published in ESA (1997). The sums
of the masses were determined for 45 systems, and accurate dynamical component
masses for 15 of these. For the remainder the mass ratio was estimated from the
magnitude difference. The well determined masses range from 0:36 0:02M to
3:12 0:37M . Only main sequence stars were investigated. Figure 113 shows the
relation for these stars between their H p magnitudes and masses.
A similar study, but covering astrometric binaries in which the motion of the
photocentre for 43 systems were investigated, was presented by Martin & Mignard
(1997). Here masses range from 0.3 to 5 M . Many stars contained in Söderhjelm
et al. (1997) were also contained in this study, and differences between masses
found in the studies are often larger than expected from the formal standard errors.
This may at least partly be due to a reexamination of the orbital and parallax data
for these stars in both studies, and shows some of the difficulties and uncertainties
encountered in interpreting such data.
A preliminary study of the mass-luminosity relation by Lampens et al. (1997)
indicated a wide range in slopes and an average value somewhat higher (K = 3:9)
than it was previously assumed to be. The high level of noise on the observed
relations and the differences in results noted above seems to indicate that further
improvements to the analysis of these data can still be made.
Some extreme cases of astrometric binaries were examined by Bernstein (1997).
Examining those stars for which the single star solution showed the presence of
376
F. VAN LEEUWEN
He
.014
40 Eri B
.012
EG 50
.01
Procyon B
Fe
GD 140
.008
Mg
C
.006
.004
.3
.4
.5
.6
.7
.8
.9
1
1.1
Figure 114. Observations of white dwarfs compared with various mass-radius relations. (from
Provencal et al., 1997)
disturbances on the astrometric data, a number of low mass companions were
discovered. In one case, for the 0.03 M companion of Gliese 433 (HIP 56528), the
companion was “seen” in the results of 2D IR speckle interferometer observations
at ESO SHARP on the NTT by Leinert and Woitas (see Bernstein, 1997). The
observed separation and position angle were in very good agreement with the
values predicted from the Hipparcos data analysis.
Using a combination of theoretical relations and Hipparcos distance determinations, mass estimates were obtained for horizontal-branch stars by de Boer et
al. (1997a, 1997b). Using spectroscopic determinations of Teff and the surface
gravity, log g , combined with the apparent luminosity transformed to an absolute
luminosity by means of the Hipparcos parallax determinations, the masses of
horizontal-branch stars were found to be MHB = 0:38 0:07M , considerably
below what was expected, although a similar result had been obtained for HB-stars
in globular clusters (see de Boer et al., 1995 and references therein). The findings
of these studies show that the HB-stars in the field and in globular clusters are very
similar, but results would become very different (by 20 per cent) if the change in
the galactic distance scale as proposed by Feast & Catchpole (1997) is adopted.
The difference between the expected masses and the calculated masses is most
likely due to the estimates of the surface gravity values, and an increase in these
values by 0.2 would bring the calculated masses more in agreement with predicted
masses.
377
THE HIPPARCOS MISSION
Figure 115. Cumulative distributions for stars in the Hyades cluster versus distance from the centre
1 M (solid line, 68 objects); 1
2 M (dotted line, 97 objects); and
2M
for
(dashed line, 26 objects). (from Brown et al., 1997b)
M
<M M>
Improvements have also been obtained in the determination of masses and radii
for white dwarfs (Vauclair et al., 1997, Provencal et al., 1997). In addition to 3
white dwarfs in binary systems, there are now also data for 7 white dwarfs in
common proper motion pairs, for each of which masses and radii were measured.
The masses are all in the range 0.45 to 0.75 M , the radii are between 0.011 and
0.015 R . There is some marginal evidence of a confirmation of the mass-radius
relation expected for white dwarfs (see Figure 114). Radii were also estimated for
11 field white dwarfs.
12.3. STAR CLUSTERS AND ASSOCIATIONS
Members of some 200 open clusters had been selected for the Hipparcos Input
Catalogue by J.-C. Mermilliod with the aim of determining cluster distances. The
biggest impact of the results of the mission is for the nearest clusters: the Hyades,
Pleiades, Praesepe and Per, all of which have now accurately determined distances
and well defined main sequence tracks in the HR-diagram.
The parallax and proper-motion data for Hyades-cluster stars for the first time
allowed a spatial resolution of the cluster (see Perryman et al., 1997b, Brown
et al., 1997b, Dravins et al., 1997) and a reconciliation of the various distance
determinations made in the past. A total of 218 probable members were selected,
of which 39 had not been considered before as members. From this sample, the
378
F. VAN LEEUWEN
Figure 116. Distribution of individual parallaxes and their accuracies for the 54 single Hipparcos
Pleiades members. (from van Leeuwen & Hansen-Ruiz, 1997)
position and space velocity of the cluster centre were calculated. This was done
for two selections: considering stars within the tidal radius of 10 pc, and for all
stars within 20 pc. The differences between the results are small. They show a
distance for the cluster centre of 46:34 0:27 pc, equivalent to a distance modulus
of 3:33 0:01 mag, and a velocity of 45:93 0:23 km s,1 . The available proper
motion, radial velocity and parallax data show a full consistency for stars within
10 pc from the cluster centre to have a uniform space motion, with an additional
velocity dispersion in the cluster centre of 0.3 km s,1 . Beyond the radius of 10 pc
there appears to exist a halo of stars no longer bound to the cluster, as was expected
for an older open cluster from numerical N-body simulations (Terlevich, 1987).
The consistency of the space motion in the cluster was shown by a very narrow
main sequence for the Hyades by Dravins et al. (1997) (see Figure 95), who used
it to derive more accurate distances for individual cluster members. The spatial
resolution of the cluster also made it possible to distinguish very clearly the mass
segregation in the cluster (see Figure 115). A subsequent spectral analysis of 20
single stars in the Hyades cluster by Lebreton et al. (1997) gave the [Fe=H] value for
the cluster and from this a value of X = 0:024 0:003. ZAMS modelling was used
to derive the cluster age (625 50 Myr) and helium abundance of Y = 0:26 0:02
(Cayrel de Strobel et al., 1997). The helium abundance is very close to the solar
value, despite the higher than solar metallicity of the cluster.
The much larger distance of the Pleiades cluster implied that there was no
possibility to do as detailed an investigation as was done for the Hyades. The
improvement in the accuracies of the astrometric parameters from the original aims
of 2 mas by almost a factor two had a very significant impact on the possibility
THE HIPPARCOS MISSION
379
Figure 117. Colour-magnitude diagram for the Hipparcos Pleiades members. Data for 60 stars are
shown, 54 as used in the combined astrometric solution and 6 stars with non-single solutions. (from
van Leeuwen & Hansen-Ruiz, 1997
to determine a reliable distance for this cluster. Figure 116 shows the distribution
of parallaxes and their accuracies for the selected stars, Figure 117 shows the HRdiagram for the selected stars as well as 6 other members that were known to be
non-single. In the distance determination the methods as outlined by van Leeuwen
& Evans (1997) were used (see also Section 11.3). Combining the intermediate
astrometric data for 54 single stars, van Leeuwen & Hansen-Ruiz (1997), and
independently Mermilliod et al. (1997), derived a parallax of 8:61 0:23 mas,
which, at an accuracy of better than 3 per cent, can be translated into a distance of
116 pc and a distance modulus of 5:32 0:05 mag. In the same process the cluster
proper motion was determined to an accuracy of about 0.2 mas/yr, thus enabling
the very precise studies of internal motions to be calibrated to absolute motions.
The value found for the cluster distance is substantially different from any
determination made on the ground using mostly photometric properties of cluster
members (130 to 135 pc), and puts the entire main sequence of the Pleiades at
the blue envelope of the Hipparcos HR-diagram (based on stars with parallax
determinations better than 10 per cent). Although it is in principle possible to fit the
380
F. VAN LEEUWEN
Table XXVI
Parallax determinations for open clusters by Mermilliod et
al, 1997 (left) and improved values by Robichon (Turon,
priv.comm.) (right)
Cluster
N
mas
mas
N
mas
mas
Coma Ber
Pleiades
IC2602
IC2391
Praesepe
Per
Blanco 1
26
51
13
7
24
32
11
11.34
8.60
6.81
6.78
5.65
5.43
3.96
0.22
0.24
0.22
0.25
0.31
0.22
0.48
31
61
13
12
24
49
11
11.49
8.54
6.81
6.79
5.57
5.49
3.96
0.21
0.22
0.22
0.22
0.29
0.19
0.48
Pleiades main sequence with a synthetic isochrone, the helium abundance value
required seems far too high to be realistic: Y 0:35. No signs of such high helium
abundances are observed in other young open clusters (Brown et al., 1986). It
appears close to impossible to measure the helium abundance in the Pleiades, as
its most massive stars are of type B6III and are known to rotate very rapidly. The
metallicity of the cluster stars is supposed to be close to solar, with [Fe=H] = 0:026
(Mermilliod et al., 1997).
Without some observational parameter able to explain the observed difference
of 0.3 between the expected and the observed distance modulus, further detailed
interpretation of the Hipparcos data becomes rather uncertain. It seems very unlikely that the parallax determination is wrong. If this were the case, then all Hipparcos
parallaxes could be assumed wrong, including those that confirm earlier estimates.
There seems to be no reason why the Pleiades parallax values should be singled out
as wrong. Accepting that the parallax is well determined, and that the metallicity of
the cluster is indeed close to solar (but new determinations are required, in particular concerning abundances of oxygen and other elements), there is an uncertainty
at a level of about 0.3 mag in photometric distance determinations, which may
well interfere with photometric determinations of abundances. Until the position
of the Pleiades main sequence is properly explained as a function of observational
parameters, determinations of the age of the cluster and of the masses of its stars
will remain uncertain.
Like the Hyades cluster, the HR-diagram for Pleiades cluster is intrinsically
of very narrow width. The observed width is, however, affected by differential
reddening in the cluster, duplicity and the spread in distance moduli (Hansen-Ruiz
& van Leeuwen, 1997). This narrowness of the main sequence must mean that
these clusters were formed from very homogeneous, well mixed clouds.
A difference exists between the distance determinations for the Praesepe cluster
by van Leeuwen & Evans (1997) and Mermilliod et al. (1997), citing values for
THE HIPPARCOS MISSION
381
Figure 118. Distances derived from the mean parallaxes of the Hipparcos members of OB associations
versus previous best estimates of distances to these associations. (from de Zeeuw et al., 1997)
the mean parallax of 5:04 0:36 mas (based on 22 members) and 5:65 0:31 mas
(based on 24 members) respectively. This leads to a difference in distance modulus
of 0.26 mag, and possibly shows the vulnerability of cluster parallax determinations
to the selection of probable members. Further cluster parallax determinations by
Mermilliod et al. (1997) are summarized in Table XXVI, together with a more
recent update by Robichon (Turon, private comm.).
Stars in OB-associations formed a significant part of the Hipparcos Input Catalogue, primarily as a result of the program proposed by the SPECTER group of Leiden Observatory (de Zeeuw et al., 1997). The aims for the Hipparcos data requested
by this project were much improved membership determinations, providing in particular information on probable fainter members, and improved information on the
kinematics of these groups. The data was supplemented by a wide range of other
observations: Walraven photometry, mm and radio observations of the interstellar
medium and precise radial and rotational velocities. The method used to select
members is described by de Bruijne et al. (1997). It uses the stellar position, proper
motion and radial velocity to select a group of stars all moving towards the same
converging point. The dispersion of the proper motion components perpendicular
to the direction of the convergent point is used to determine a 2 value for the solution, and to detect and reject non-members as outliers. Examples of the application
of this method are shown by de Bruijne et al. (1997), Hoogerwerf et al. (1997) and
de Zeeuw et al. (1997). Figure 118 shows a comparison between Hipparcos and
pre-Hipparcos distance determinations for 11 associations. The general trend is for
Hipparcos distances to be smaller than previous ground-based estimates.
382
F. VAN LEEUWEN
12.4. LUMINOSITY CALIBRATIONS
As a result of the higher than expected parallax accuracies obtained by Hipparcos,
it became possible to obtain the first direct, parallax based calibrations of the
period-luminosity (PL) relations of Scuti stars, Cepheids and Miras.
The best possibilities for a PL relation calibration were provided by the Sct
stars, and in particular for a small selection with high amplitudes such as SX Phe,
AI Vel and V474 Mon (Høg & Petersen, 1997). It was shown that these stars are
most likely main sequence stars, probably related to a similar type of variable
occurring in globular clusters among the so-called blue straglers. For these stars it
is possible to distinguish fundamental mode and overtone pulsation periods (and
many show both), and to relate these periods to masses and a period-luminosity
relation, for which (Høg & Petersen, 1997) give:
MV =
,3:74 log P0 , 1:91;
(100)
where P0 is the period for the fundamental mode. This relation was, however,
derived in magnitude space (using 6 stars), where it is susceptible to bias due to nonlinear transformations (see Section 11.2). A reevaluation by means of Equation 91,
using the accuracies on the parallax measurements as weights, gave the following
result:
,
MV = ( 3:98
0:52) log(P0 =0:1) + (1:80 0:09);
(101)
with a unit weight standard error of 1.16, and using 5 stars with fundamental mode
periods: V474 Mon (HIP 28321), AI Vel (HIP 40330), V703 Sco (HIP 86650),
RS Gru (HIP 107231) and SX Phe (HIP 117254), and VZ Cen (HIP 42594), for
which a fundamental mode period of 0.2263 d was used (the observed first overtone
period divided by 0.77). At the reference period of 0.1 d the covariance between
the slope and zero-point solutions is only 0.14. An apparently similar star, AD CMi
(HIP 38473), lies very much outside the PL relation, as was noted by Høg &
Petersen (1997). The comparison between the predicted and measured parallaxes
using this relation is shown in Figure 119. A similar (pre-Hipparcos) relation for
SX Phe stars in fundamental mode pulsation given by Nemec & Lutz (1993) is
very similar but has larger uncertainties:
,(4:19 0:70) log P0 , 1:71 + 0:32[Fe=H]:
(102)
With [Fe=H] = ,1:70 for SX Phe, this gives a zero-point equivalent to 1:94 in
MV =
the Hipparcos relation (at P0 =0.1 d). The selection of stars used in the calibration
appears from the proper motions to be a mixture of Population I and II stars,
with SX Phe and possibly also RS Gru being Population II, and the remainder
Population I. There were 21 stars classified as SX Phe in the Hipparcos Catalogue,
primarily on the basis of the amplitude and period of their variations, offering still
further scope for checking and improving their PL relation.
THE HIPPARCOS MISSION
383
Figure 119. Comparison between calibrated (C) and Hipparcos (H) parallaxes for SX Phe stars.
Figure 120. HR diagram for a sample of 75 Scuti stars. The continuous lines are the ZAMS and
the upper limit of the main sequence, while the dotted lines are evolutionary tracks for the indicated
masses and metallicity of Z 0:020. (from North et al., 1997)
=
384
F. VAN LEEUWEN
Figure 121. The positions in the period versus amplitude diagram for three groups of stars following
the classification in the Hipparcos Catalogue. Open circles: Sct; filled circles: SX Phe; open stars:
RRc; filled stars: RRab and RR. Also shown, with filled triangles, are periodic variables of spectral
types A and F in the same range of periods, but without types assigned in the Hipparcos Catalogue.
Most of the stars with low amplitude and periods longer than 0.5 d are Ap stars (see Figure 124).
The PL relation for low-amplitude Sct stars is much less well defined due to
the mixture of pulsation modes for these stars, as was shown by North et al. (1997).
The luminosities derived from the Hipparcos parallaxes and Strömgren photometry
show a wide spread, with some 15 per cent lying above the theoretical upper limit
of the ZAMS (see Figure 120). The available theoretical models and Hipparcos
luminosities indicate a mixture of pulsations in fundamental mode, first and third
overtone. The luminosities obtained from photometric calibrations for these stars
are, according to Liu et al. (1997), now found to be on average too low, although
a considerable scatter is shown to exist, with discrepancies of over 0.3 mag. This
may well be related to a similar discrepancy observed for the Pleiades cluster by
van Leeuwen & Hansen-Ruiz (1997).
The RR Lyrae stars did not provide serious possibilities for a parallax based
calibration of their absolute magnitude, while a kinematical calibration did not
change much with respect to similar ground based calibrations (Fernley et al., 1997).
Only for RR Lyrae itself was a marginally useful parallax obtained: 4:38 0:59 mas,
which, following Brown et al. (1997a), gives an uncertainty of about 0.26 mag on
the absolute magnitude, and a possible bias at a level of about 0.01 mag. The
relation given by Fernley et al. (1997) for the RR Lyrae magnitudes is:
MV = (0:18
0:03)([Fe=H] + 1:52) + (0:72 0:04):
(103)
THE HIPPARCOS MISSION
385
It is interesting to notice that the relation for the SX Phe stars, when extrapolated
to a period of 0.4 d, gives an absolute magnitude within a few hundredth of the
absolute magnitude of the RR Lyrae stars.
In the Hipparcos variability annexes, the distinction for A and F type stars
with periods in the range 0.05 to 0.8 d, between Sct, SX Phe and RRc stars
was not always clear from the Hipparcos photometry only, as is demonstrated by
Figure 121. Classifications for all these stars need to be reinvestigated with the aid
of additional ground-based information, and supplemented with the kinematical
evidence provided by Hipparcos. A group of small-amplitude periodic variables
with periods from 0.5 to 25.0 d also seen in diagram primarily consists of peculiar
A stars with almost one quarter having Ap Si as spectral classification (see also
next section).
The Cepheids, being intrinsically much brighter than the RR Lyrae stars, contained a larger number of significant or marginally significant parallax determinations which were used by Feast & Catchpole (1997) for a calibration of the zero
point in the PL relation:
hMV i = log P + ;
(104)
where P is the photometric period. The parallaxes for a selection of 23 stars were
used in a solution of the type described by Equation 91, in which the zero point was
described as a function of the observed parallax, photometric period and reddening
corrected V magnitude:
100:2
= 0:01100:2 hV i, log P :
(
0
)
(105)
The Hipparcos data does not contain sufficient information to calibrate the slope
of the PL relation, which was fixed at = ,2:81, a value derived for 88 Cepheids
in the LMC by Caldwell & Laney (1991). Two types of solution were made, one
assuming an intrinsic scatter in the Hipparcos solution of 0.1, and one assuming
no such intrinsic scatter. The results from the two solutions are very similar, and
a final value of = ,1:43 0:10 was adopted. This is equivalent to a value of
100:2 = 0:517 0:025, giving an uncertainty in parallax values derived from
this relation of approximately 5 per cent. At this accuracy level the bias from the
non-linear transformations to distances and distance moduli is much smaller than
the uncertainties in the calibration itself. Thus, the Hipparcos data were interpreted
as parallax data, at which stage there was no bias entering the descriptions. Only at
the last stage, when a derived parallax value was transformed to a distance modulus
value could a bias as described by Lutz & Kelker (1973) become relevant, but the
relative accuracy of the derived parallax was such that this bias is insignificant.
When applied to the LMC Cepheids, the calibration of gives a distance
modulus for the LMC of 18:70 0:10, rather different from what is derived using
RR Lyrae stars or the ring around SN1987A. A similar calibration of the PL relation
for Miras showed results compatible with the Cepheids (van Leeuwen et al., 1997c,
Whitelock et al., 1997). A calibration of oxygen-rich Miras using the “LM method”
386
F. VAN LEEUWEN
6.0
7.5
-5
8.0
Mv
8.5
0
9.0
9.5
5
Supergiants-Bright giants
Giants 1
Giants 2
Clump giants
Old giants
Main sequence
Subgiants
-1.0
-0.5
0.0
0.5
(B-V)0
1.0
1.5
2.0
Figure 122. Groups resulting from the “LM-Method” calibration for G and K stars. Isochrones with
log(age) are also shown. (from Gómez et al., 1997a)
by Alvarez et al. (1997) showed the need of considering different populations when
examining the Miras.
A number of more general luminosity calibrations have already been started,
most noticeably by Gómez et al. (1997a). Bright stars with luminosity classifica-
387
THE HIPPARCOS MISSION
8.0
Mv (Phot)
7.0
a)
6.0
5.0
4.0
3.0
2.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Mv (Hip)
Mv (Phot) − Mv(Hip)
1.0
b)
0.5
0.0
−0.5
−1.0
−3.0
−2.0
−1.0
0.0
1.0
[Fe/H]
Figure 123. The new photometric calibration of absolute magnitudes by Jordi et al. (1997). Full
circles are stars with [Fe=H] < 0:5, and open circles stars with [Fe=H] > 0:5. (from Jordi et al.,
1997)
,
,
tions were subjected to the “LM method” described by Luri et al. (1996), Arenou
& Gómez (1997) and Luri & Arenou (1997) (see also Section 11.2). The “LM
method” split the stars in different groups according to kinematic properties and
spatial distribution and showed clearly a much larger range in absolute magnitudes
than was assumed before, with a range as large as 2 mag for especially earlier type
stars. It also showed the difficulties in obtaining fully reliable luminosity classifications from spectral properties, with some stars classified as luminosity class V
found near the horizontal giant branch and vice versa. This is the same problem
that was encountered in the selection of some nearby stars on spectral and luminosity characteristics (see Section 12.1). Figure 122 shows how the LM calibration
created different groups of G and K stars within the HR diagram.
A comparison between various photometrically calibrated luminosities and Hipparcos luminosities was presented by Jordi et al. (1997) for FGK dwarfs and
subdwarfs. A preliminary new calibration was presented, removing the metallicity
related drifts from existing calibrations. The remaining rms of the residuals between
388
F. VAN LEEUWEN
the new photometrically calibrated luminosities and those obtained with Hipparcos
is of the order of 0.3 mag, a factor two reduction with respect to pre-Hipparcos
calibrations (see Figure 123). A similar study for B-type stars, using the Geneva
photometric system, has been presented by Cramer (1997).
All of these detailed luminosity calibrations provided by Hipparcos are expected
to contribute significantly to the study of stellar structure and evolution, providing
a much more detailed observational grid to compare theoretical isochrones with.
12.5. VARIABLE STARS
The greatest impact of the Hipparcos results for variable stars is probably among
the eclipsing binaries, where the known number of such stars as contained in the
Hipparcos Catalogue was almost doubled (from 574 to 917), and among short
period and small-amplitude variables of various types. The statistical aspects of the
Hipparcos variable star results will be of great importance, in particular in better
defining the regions of instability in the HR-diagram (Eyer & Grenon, 1997).
An examination of Figure 93 shows clearly the main contribution of Hipparcos
for periodic stars as a function of the photometric periods: between 0.1 and 8 days,
with a peak around 1 day. This peak is not only the result of the difficulty for
ground-based observations to detect regular variability on a time scale of 1 day, but
also of observing strategy used by Hipparcos, which made detection of periods of
about 1 day relatively easy.
An example of a major contributor to this group can be seen in Figures 121 and
124, showing the Hipparcos amplitudes and periods for low amplitude periodic
variables among A and F type stars. Although classifications were not clear at
the time of the preparation for the Hipparcos Catalogue (and given the very large
number of variable stars that had to be examined, there was not really time for
detailed classifications) it is now clear that the concentration of stars with periods
less than 0.2 d and amplitudes below 0.1 mag are probably all Scuti stars, while
those with periods longer than 0.4 d and amplitudes less than 0.1 mag are most
likely variable Ap stars and related types. Noteworthy seems to be a group of
spectral type F0 stars with periods between 0.6 and 2 days, many of which are of
luminosity class III or IV. Among the A and F stars in these groups are likely to
be a small number of low amplitude close binary systems, which have not yet been
properly recognized. An example is the early A-type star HIP 85501, which could,
judging on the shape of the minimum, be an eclipsing binary with a 2.215 d period.
It is clear that there is still a lot of research to be done on these stars to explain their
various types of variability, and the way this may be connected to their structural
parameters and evolutionary status.
Various studies on semi-regular, small- and large-amplitude variables have started already (for Cyg stars by van Leeuwen et al., 1997d, van Leeuwen et al., 1997d,
for B and Be stars by Hubert et al., 1997, for Herbig Ae/Be stars by van den Ancker
et al., 1997). A possible detection of rapid variations in Mira stars by de Laverny
389
THE HIPPARCOS MISSION
Figure 124. Periods, types and amplitudes for low amplitude intrinsic variables of spectral types
A and F, many of which were discovered by Hipparcos. The meaning of the symbols is as follows
(bearing in mind however the uncertainties in Hipparcos classifications): open circles: Scuti; filled
circles: SX Phe; open stars: RRc; filled stars: RRab; open triangles: Ap stars with no variability type;
filled triangles: remaining A stars; : remaining F stars.
+
8000
8500
9000
BJD - 2 440 000
Figure 125. The Hipparcos observations of the Mira T Cas (HIP 1834), shown with respect to the
fitted and transformed AAVSO observations (dotted line), in the way these stars are presented in
Volume 12 of the Hipparcos Catalogue.
et al. (1997) was deduced from the Hipparcos data, but has to be treated with some
care, in particular as these variations take place near minimum light, which makes
the Hipparcos photometric data much more vulnerable to incidental superpositions.
This can show up in particular through a large difference between the magnitudes
as measured in the preceding and following fields of view.
The scanning mode and observing frequency for Hipparcos was not well suited
for the slowly varying red variables (Miras, semi-regulars and slow irregulars).
This was compensated by a collaboration between the Hipparcos project and the
390
F. VAN LEEUWEN
Figure 126. Period versus amplitude for M-Miras, showing the upper limit of the amplitudes as a
function of period. (from Mattei et al., 1997)
AAVSO, which led to an unique data base for astrometric and photometric information on these stars (Mattei et al., 1997, Mennessier et al., 1997). During the
Hipparcos mission, this data base was used in planning the observations of slowly
varying large-amplitude variables (Mennessier et al., 1992). The combination of
the AAVSO and Hipparcos observations is shown in Volume 12 of ESA (1997),
in Section B of the folded light curves. There, the Hipparcos data are shown with
respect to the fitted light curves of the AAVSO data, carefully transformed to the
same system by M. Grenon. An example is shown in Figure 125.
The AAVSO data base contains extensive visual data for 255 Mira and semiregular variables that were observed by the Hipparcos satellite, with for most at least
1000 observations, and for many in excess of 10 000 observation, covering a time
span from October 1961 till August 1996. Most of these stars, 170, are classified
as Mira-type, the remainder are types of SR variables. Various relations between
periods, amplitudes and variation of amplitudes were used by Mattei et al. (1997)
to re-examine the classification of these stars, leading to 7 revised classifications: 5
SR-types that are in fact Miras, and two Miras that are SRa-types. Also recognized
in the data is an upper limit to the amplitudes of Miras as a function of their periods,
expressed as:
A 5:72 log P , 7:85 mag:
(106)
THE HIPPARCOS MISSION
391
Figure 127. Periods, types and amplitudes for eclipsing variables. Upper diagram: new discoveries;
lower diagram: previously known variables. Open circles: EW types; filled circles: EB types; open
stars: EA types; crosses: ELL types; filled stars: probably eclipsing, but insufficient coverage for type
determination.
This relation is shown in Figure 126. The analysis also revealed a group of 30 stars
having two periods with a ratio between 1.7 and 1.95, below the expected ratio.
Most of these are SRb types, and 3 are Miras.
A different grouping of Miras was done by Mennessier et al. (1997). Using the
“LM method”, the stars were divided into kinematical groups, for which the mean
luminosity, distribution and velocity characteristics were calibrated. It was shown
how the different types of red variables may be related to these kinematical groups,
which indirectly relate types of long period red variables to stars of different age
and mass.
A very rich source of material may become available through the discovery of
over 300 eclipsing binaries. Most of the new discoveries are in the lower amplitude
range and for periods between 0.8 and 3.0 d, a range which is difficult to cover with
ground-based measurements. The first paper on one of the newly discovered EB
type binaries ( CMa) was presented by van Leeuwen & van Genderen (1997). The
notes accompanying the variable star annexes in the Hipparcos Catalogue show
392
F. VAN LEEUWEN
many cases where existing ephemerides have been compared with the timing of
minimum light as measured by Hipparcos, in most cases leading to a more accurate
period determination. In one case at least, notably for Lyr, the Hipparcos light
curve confirmed the changing period as given by Sahade (1980).
12.6. REFERENCE FRAME IMPROVEMENTS
The first proof of the impact of the improvement in the accuracy of the stellar
reference frame was obtained through a comparison between radio and optical
images for SN1987A (Reynolds et al., 1995). The alignment of the Hubble Space
Telescope image with the radio map to an accuracy of at least 0.1 arcsec made it
possible to understand much better and with greater confidence the relation between
the radio and optical signals.
An extreme example of the effect of a positional improvement was presented by
Caraveo et al. (1997) for the mV = 25:5 optical counterpart of the Geminga radioquiet pulsar. Connecting through 5 steps the Hipparcos and Tycho positions to the
HST image of the pulsar, a positional accuracy of 0.04 arcsec was obtained. This
accuracy was required to properly relate the observations to the barycentre of the
solar system, and thus combine the observations from SAS-2, COS-B and EGRET,
so that the second time derivative in the pulsar frequency could be determined.
The procedures described above rely on the possibility to obtain much improved
absolute positions from photographic plates, using the Hipparcos and Tycho reference frames (Robichon et al., 1995, Platais et al., 1995). It was shown for plates
taken at an epoch close to the Hipparcos reference epoch of J1991.25, that the
rms of the residuals could be significantly reduced with respect to pre-Hipparcos
reference frames (from 0.3 arcsec down to 0.07 arcsec accuracy for positions on
a Schmidt plate). High order polynomials were, however, needed to represent the
distortions of positions on a Schmidt plate.
The possibilities for smaller astrograph plates are less favourable due to the
smaller numbers of reference stars available. Some of these problems can be
overcome when many plates of different parts of the sky are available, and an
overall field distortion pattern can be determined (Zacharias et al., 1997). Using the
Hipparcos and Tycho Catalogues for a new reduction of the photographic plates
used for the CPC2 catalogue, the accuracies of the derived positional coordinates
was in all cases significantly improved. The same conclusion was reached from
the implementation of the Hipparcos positions as reference points in meridian
circle observations by Réquième et al. (1997). In comparison with using PPM
reference stars, the mean standard errors dropped from 0.3 arcsec to 0.06 arcsec,
with accuracies for the measured positions better than 0.1 arcsec for stars brighter
than mV = 14:5.
The much improved star positions and magnitudes are also already assisting
satellites currently in operation, such as ISO, to improve the mission operations
393
THE HIPPARCOS MISSION
Table XXVII
DE200: FAST solution for the rotation and spin parameters.
Values of are given in mas, and values of ! in mas/yr.
1
2
3
!1
!2
!3
,11.90 ,12.64 4.32 ,9.51 14.91
2.62 4.12 1.18 1.47 3.68
3.71
2.00
(Batten et al., 1997) and are already incorporated in the planning of future satellites
such as ENVISAT (Ratier et al., 1997) and AXAF (Green et al., 1997).
12.7. SOLAR SYSTEM RELATED STUDIES
The duration of the Hipparcos mission was rather short in comparison with orbital
periods in the solar system. The impact of the solar system observations will therefore be rather limited, providing in most cases a few high accuracy reference points.
However, improvements for orbital parameters have been obtained by Viateau &
Rapaport (1997) through incorporating Hipparcos data in ground-based data for
two asteroids, Pallas and Bamberga. The improvements were most significant for
the inclinations and nodes of the orbits. A determination of the mass of Ceres,
using disturbances to these orbits and incorporating Hipparcos data for improved
orbit descriptions, showed a mass some 5 per cent lower than the value adopted by
the IAU. The new value for the mass is:
MCeres
= (4:785
0 039)10,10M
:
:
(107)
In a different approach, examining the instantaneous effects of close encounters
between asteroids, Bange & Bec-Borsenberger (1997) derived masses for a small
number of asteroids, though with relatively low accuracy. In the same paper, the
link between the dynamical reference frame given by the DE200 ephemerides and
the ICRS was determined, using the minor-planet data, to an accuracy of a few
mas. The rotation and spin values are given in Table XXVII. Further details on the
methods used for solving these parameters are given by Bec-Borsenberger et al.
(1995) and Bougeard et al. (1997).
Indirect improvements of information on orbits in the solar system can be
obtained through the use of the Hipparcos reference positions in the reduction of
existing plate material (Fienga et al., 1997). Improvements by a factor 5 to 10 were
obtained from a number of experiments, using only small numbers of Hipparcos
stars per plate.
An alternative way of interpreting the Hipparcos data on solar system objects
was presented by van Leeuwen & Evans (1997) and Morrison et al. (1997b). In
an application to the ephemerides of Jupiter and Saturn (measured through the
positions of the moons Europa, Titan and Callisto), a small number of reference
394
F. VAN LEEUWEN
Figure 128. Comparison of the observation positions of Jupiter with DE200 and DExxx ephemerides
in right ascension (upper graph) and declination (lower graph). The Hipparcos normal points derived
from observations of Europa are shown as filled squares. Individual Tycho observations derived from
observations of Calisto are plotted as light error bars. The opposition means derived from Carlsberg
Meridian Telescope observations are plotted as heavy error bars. (from Morrison et al., 1997)
points was constructed from the available data. These reference points are twodimensional rather than the one-dimensional observations, and can be used as
(very accurate) positional determinations in combination with ground-based measurements in new ephemerides calculations. The data confirmed the discrepancies
observed with the Carlsberg Meridian telescope between the DE200 ephemerides
and the measured positions. An example of these ephemerides comparisons, including a preliminary improvement of DE200 (DExxx), is shown in Figure 128.
13. Future projects
The scientific basis for further astrometric missions, obtaining higher accuracies
and measuring fainter stars, is very strong. A wide range of fundamental topics
in astronomy can be covered, ranging from stellar structure and stellar evolution through kinematics, masses and cluster dynamics to understanding of our
Galaxy, extragalactic astrophysics and general relativity, while planetary searches,
supernovae detections and a host of other scientific objectives become feasible
too (Perryman et al., 1997c, Peterson & Shao, 1997). The almost automatic “byproduct” of a photometric survey adds to this the discovery and surveying of
any detectable variable stars, among which there will be thousands of Cepheid,
RR Lyrae, Mira and other pulsating stars, for all of which very well determined
distances can be obtained. The most substantial new astrometric survey is planned
by the GAIA mission, for which ESA has recently started the first studies. GAIA
THE HIPPARCOS MISSION
395
aims at obtaining astrometric parameters to 1 to 10 arcsec accuracy for more than
100 million objects, and can be complemented with a survey down to 18-20 mag,
with astrometric parameters to accuracies 10 times better than Hipparcos.
Hipparcos has led the way and has shown that obtaining high quality astrometric
data from space is not only possible, but reaches in general far higher accuracies.
It is also much more efficient than obtaining similar data from the ground. Hipparcos has, however, also shown some weaknesses: the relatively short length of
the mission introduced an unwanted correlation between orbital motions and proper motions, an effect that will only get worse when accuracies get higher. More
accurate missions should therefore be planned to last longer. The first research
on the Hipparcos results exposed the need for supporting data, primarily detailed
photometric measurements, radial velocities and spectral classifications. This was
recognized well before the launch of the mission, but only limited support was
received (from the ESO key programmes) for a large-scale radial velocity survey.
Again, obtaining much larger amounts of data than Hipparcos, such deficiencies
become only more severe, as much more automated processing and interpretation
of the data is required. Improved photometric observations can be incorporated in a
future mission (Straižys & Høg, 1995, Straižys et al., 1997), and there may be possibilities for spectral classifications and radial velocities too (Favata & Perryman,
1995, Favata & Perryman, 1997, Bastian, 1995, LaSala & Kurtz, 1995).
Although there are some small advantages in a pointing satellite, in particular through the accumulation of observing time for specific objects, the scanning
satellite, using two or more apertures, remains by far the most effective way of accumulating absolute parallaxes for large numbers of stars. Significant improvements
in positional accuracy can be obtained incorporating interferometric techniques,
such as planned for the GAIA design (Lindegren & Perryman, 1995) and in DIVA
(Röser et al., 1997), a smaller, German version of the GAIA proposal, or LIGHT,
a Japanese version of GAIA (Yoshizawa et al., 1997). A summary of the kind of
problems faced in designing an astrometric interferometry space mission were outlined by Gai et al. (1997): detector efficiency, resolution and dynamic range, optical
design, stability of the system, attitude control, are only a few of the problems that
have to be considered from the point of view of the much higher requirements set
by such missions in comparison with the Hipparcos satellite.
As an example of the complexity of the data analysis, the investigations on variable stars are considered here. Given the very large numbers of stars to be observed
by GAIA, and the two or possibly three interferometers, an estimated 1011 observations would be obtained for 108 stars, in 7 photometric channels, accompanied
by parallax information for most of these objects, determining distances to a very
high accuracy. Information on colours and distances (absolute magnitudes) can
assist in variability classifications, which will have to be made unsupervised by
computer software, possibly incorporating Bayesian methods. Due to the scanning
law, however, the window functions for the period searches on those data remain
very bad, in particular for time scales of 5 to 100 d. Small changes in the scanning
396
F. VAN LEEUWEN
Figure 129. The coverage of the sky by Hipparcos over a period of 6 months. Two small areas near
the ecliptic plane were not covered, which is reflected in the occurrences of gaps in the photometric
data as shown in Figure 16.
law can make large differences on the coverage. In particular, a larger angle in
the scanning law will lead to a decrease in the lengths of the largest gaps in the data
for stars near the ecliptic plane, and at the same time will produce a better overall
coverage of scanning directions, which is of benefit to the double star analysis.
A closing gap (a small part of the sky centered on the ecliptic plane that was not
covered at all during a half-year scan, see Figure 129) existed for Hipparcos and
introduced unnecessarily very large gaps in the photometric data for a small number
of stars near to the ecliptic (see also Figure 16). Such gaps, which do much harm to
automated period searches, should be avoided for any further astrometry mission.
Contrary to the astrometric data, which are all interconnected and can therefore
only be obtained at the end of a mission, the photometric data stand on their own.
It would be advisable to consider early publication of details on variable stars
discovered during a new mission, so that ground-based observations of the kind
not provided by a mission can be obtained simultaneously with the further mission
observations.
Finally, with several missions being planned for more or less the same epoch,
the combined interpretation of data should be considered, in particular for the
astrometric data. Ultimately, one should be able to combine abscissa residuals from
Hipparcos and future missions to obtain the best possible astrometric parameters.
This applies in particular to proper motions and their possible disturbance by orbital
motions. Thus, the publication of abscissa residuals as was done for Hipparcos
should be considered very seriously for any further astrometric mission too.
THE HIPPARCOS MISSION
397
Acknowledgements
A large number of people worked together to make Hipparcos a success, and this
paper tried to do some justice to their work. Above all should be mentioned the
project scientist Michael Perryman and the consortia leaders Jean Kovalevsky, Erik
Høg and Lennart Lindegren, and the remaining members of the Hipparcos science
team as well as the various working groups. The team at ESOC, led by Dietmar
Heger and Alastair McDonald, rescued the mission from what appeared to be
an early exit. The three data reduction teams then succeeded to turn the mission
into a success even well beyond what was expected for the nominal mission, not
least thanks to an element of competition and the regular comparison exercises.
Much use was made of the more detailed descriptions of the data presentation
and reduction procedures in Volumes 1 to 4 of the Hipparcos Catalogue, of which
Volume 1 contains a list of the many people that contributed to the mission from
within and outside the Input Catalogue and Data Reduction Consortia. All of these
people helped to make the Hipparcos mission a total success. Finally I like to
thank Andrew Murray who introduced me into project 16 years ago, the members
past and present of the NDAC consortium, and my wife Małgorzata, who at many
occasions contributed, behind the scenes, to the success of the NDAC team at the
RGO, and Michael Perryman, Hans Schrijver, Dafydd Evans, Margaret Penston,
Rudolf Le Poole and Erik Høg who read (parts of) earlier versions of the text and
provided numerous corrections.
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