Renewable Energy 184 (2022) 176e181 Contents lists available at ScienceDirect Renewable Energy journal homepage: www.elsevier.com/locate/renene The position of the sun based on a simplified model Finley R. Shapiro Department of Engineering, Leadership, and Society, Drexel University, Philadelphia, PA, 19104, USA a r t i c l e i n f o a b s t r a c t Article history: Received 20 May 2021 Received in revised form 6 October 2021 Accepted 23 November 2021 Available online 26 November 2021 A simplified model for the position of the sun at any location on earth is shown to produce fairly accurate results when compared to values calculated in a spreadsheet from the United States National Oceanic and Atmospheric Administration (NOAA). A straightforward derivation of the model is presented, in which all terms and parameters have clear physical meaning. In the first version of the model, the only parameters are the tilt of the earth's axis and the date and time of the nearest northern hemisphere winter solstice. The differences between the elevation and azimuth calculated using the model and the NOAA results are typically a few degrees or less, and it is seen that the primary cause of these differences is the eccentricity of the earth's orbit. A simple correction is then added to the model, in which the only parameters are the earth's orbital eccentricity and the date and time of the perihelion nearest the solstice already in the model. This removes most of the differences between the model and the NOAA results. The model is shown to work in both the northern and southern hemispheres. © 2021 Elsevier Ltd. All rights reserved. Keywords: Sun position Solar position Analemma Eccentricity Azimuth Elevation 1. Introduction The position of the sun in the sky as a function of time is important for predicting the performance of solar energy systems [1e7], determining the sunlight striking building walls and passing through windows [8], and other applications [9]. A review is found in Ref. [10]. Many papers on calculations of the position of the sun, such as [11e19], show equations for the elevation and azimuth of the sun's position as a function of time and date. No derivation is given for many of the equations in these papers, and some parameters have seven or more digits but are not given any physical meaning. Complicated derivations of equations for the position of the sun are found in Refs. [20,21]. This paper provides a very simplified and easily understood astronomical model to show the origins of equations for the elevation and azimuth of the sun. In the initial model, the earth's orbit is a circle and the earth is a sphere. The only parameters are the date and time of the nearest winter solstice, the tilt of the earth's axis, and the length of a year by the Gregorian calendar. The results are compared to the output of a spreadsheet from the Earth System Research Laboratories' Global Monitoring Laboratory of the United States National Oceanic and Atmospheric Administration (NOAA) [22]. The comparison shows how close the calculated position in the basic model is to the actual solar position. The E-mail address: [email protected]. https://doi.org/10.1016/j.renene.2021.11.084 0960-1481/© 2021 Elsevier Ltd. All rights reserved. differences due to other phenomena are relatively small, but still significant, and analysis shows that the primary cause of the differences is the eccentricity of the earth's orbit. A simplified correction for the eccentricity of the earth's orbit is then added to the model, using only the date and time of the perihelion nearest to the winter solstice already used in the calculation, the eccentricity of the earth's orbit, and Kepler's second law. This correction removes most of the differences between the original model and the NOAA calculation. There are many applications in which the errors in this simple model are small compared to other uncertainties. These include applications in which the weather has a major influence on the overall sunlight that is received by a photovoltaic or solar thermal collector, or when the exact orientation of a collector varies over its surface or over the time of the experiment. 2. Derivation of the model We begin with the earth at its position at the moment of the northern hemisphere winter solstice, as shown in Fig. 1. The z direction is perpendicular to the plane of the earth's orbit, and the y direction is from the sun to the earth's position at this moment. At the winter solstice it is solar midnight at some longitude that is at a position directly opposite the sun. The winter solstice in December 2020 was on 21 December at 10:03 universal time, so the longitude of solar midnight was 4s ¼ -150.75 , where negative longitude is to the west and positive longitude is to the east of the F.R. Shapiro Renewable Energy 184 (2022) 176e181 Fig. 1. The northern end of earth's axis points approximately toward Polaris, the North Star. At the winter solstice, the earth's orbital position is also in the direction of Polaris. The z and y axes in the coordinate system are shown. Greenwich meridian. For comparison, Anchorage Alaska is at 149.9 and Honolulu Hawaii is at 157.9 . If the earth had no tilt with respect to the orbital plane, then an observer at longitude 4s and latitude L would have a zenith vector ð0; cos L; sin LÞ at the winter solstice. An observer at a different longitude 4 would have a zenith vector cos qZ ¼ cos L sinðst þ 4 4s Þsin ut cos a cos L cosðst þ 4 4s Þcos ut sin a sin L cos ut The same form is found in other models. The unit vector pointing from the center of the earth to the South Pole is 0 1 0 @ U ¼ sin a A cos a 0 1 cos L sinð4 4s Þ @ Z ¼ cos L cosð4 4s Þ A sin L 0 (1) At a time t after the winter solstice, the direction of this zenith vector would be 1 cos L sinðst þ 4 4s Þ Z ðtÞ ¼ @ cos L cosðst þ 4 4s Þ A sin L H ¼ U ðU , ZÞZ (2) 1 ZðtÞ ¼ @ 0 0 0 0 cos a sin a R ¼ S ðS , ZÞZ (9) R and H do not have unit length. The sun's azimuth angle relative to due south as seen by the observer is the angle between these two vectors. These equations were programmed into a spreadsheet in Microsoft Excel. The sun's azimuth angle as measured by the observer clockwise from north is calculated as 1 0 0 sin a AZ ðtÞ cos a 1 cos L sinðst þ 4 4s Þ ¼ @ cos a cos L cosðst þ 4 4s Þ þ sin a sin L A sin a cos L cosðst þ 4 4s Þ þ cos a sin L (8) The projection on to the same plane of the vector pointing to the sun is where s is the rotation speed of the earth. However, the earth is actually tilted at an angle a ¼ 23:44 toward Polaris, the North Star. So the actual zenith vector of the observer is 0 (7) which is independent of time in this model. The observer will see due south as the direction of the projection of this vector on to the plane perpendicular to the unit zenith vector Z. This vector is 0 0 (6) (3) g ¼ DEGREESðATAN2ðR , H; Z , ðR HÞÞÞ þ 180 (10) The time of the earth's rotation is found from the length the which has unit length. At a time t, the unit vector 0 1 sin ut SðtÞ ¼ @ cos ut A 0 (4) points from the earth to the sun, assuming a circular orbit where u is the earth's angular velocity around the sun. Therefore, the observer will see an angle between zenith and the sun as qZ ¼ cos1 ðZ , SÞ (5) The elevation of the sun above the horizon is 90 Combining equations (3)e(5), we get qZ . Fig. 2. Elevation and azimuth of the sun at Latitude 39.956 and Longitude 75.1878 as calculated by the model, for 7 October 2020, 21 December 2020, 20 April 2021 and 20 June 2021. 177 F.R. Shapiro Renewable Energy 184 (2022) 176e181 Gregorian year, 365.2425 days. The period of a rotation, the sidereal period, is calculated as 24 , 365:2425 ¼ 23 hours 56 minutes 4:09 seconds 366:2425 (11) The rotation rate is s¼ 2p 366:2425 radians hour , 24 365:2425 (12) The times in the calculation shown here are referenced to the winter solstice on December 21, 2020 at 10:03 UT. Fig. 2 shows the position of the sun as calculated by the model for 4 dates at Latitude þ39.956 and Longitude 75.1878 , the approximate location of the author's office. Fig. 4. The difference between the azimuth calculated by the model and the values from the NOAA spreadsheet. 3. Comparison with a more accurate model Figs. 3 and 4 show the difference between the elevation and azimuth from the model and the values from the NOAA spreadsheet on three dates, 30 December 2020, 16 February 2021, and 6 April 2021. The first is a local minimum in the daily root mean square difference (discussed below), the third is a local maximum, and the second is a date about half way between the first and the third. While these differences, up to 2.15 in elevation and up to 3.1 in azimuth, are too big for navigation applications, they may be acceptable for calculating the absorption by sunlight collectors. It is important to note that the errors in elevation are mainly early and late in the day. Stationary solar collectors are often pointed due south and at a tilt approximately equal to the latitude, so absorption early and late in the day is a small part of the total. An analemma showing the position of the sun as calculated by the model at noon Eastern Standard Time (UT -5) from 20 June 2020 to 20 June 2021 is shown in Fig. 5, compared the position of the sun from the NOAA spreadsheet at the same times. The figure shows that the characteristic figure-8 of an analemma is due only to the tilt of the earth's axis. Other phenomena alter the details of the plot but are not the cause of the shape. While differences between the azimuths calculated by the model and the NOAA spreadsheet are very visible on this scale, especially around 27 July 2020 and 15 May 2021, they are only rarely more than 4 . Even on these dates the differences in calculated elevation at noon are less than 0.5 . Fig. 6 shows the root mean square difference of the elevation for each day as calculated by the model relative to the NOAA spreadsheet, from 20 June 2020 to 20 June 2021. Fig. 7 shows the root mean square difference of the azimuth over the same period. The minima occur in Fig. 6 on 5 July and 30 December 2020, and in Fig. 7 occur on 5 July 2020 and 2 January 2021. The aphelion in the earth's orbit was on 4 July 2020 and the perihelion on January 2, 2021. On these days the earth's motion is parallel to that of the circular model, so the effect of the eccentricity of the earth's orbit is at its minimum. The proximity of the minima in the differences to the dates of the aphelion and perihelion suggest that the primary cause of the difference in the model is that it does not include the earth's orbital eccentricity. 4. A simple correction for eccentricity To correct for the eccentricity of the earth's orbit in a manner that is consistent with the simplicity of this model, we can replace ut in equation (4) with ut þ f ðtÞ. Kepler's second law, which is also the conservation of angular momentum, requires that uþ df A ¼ dt r 2 (13) for some constant A, where r is the distance from the sun to the earth. A simple estimate for f ðtÞ is f ðtÞ ¼ C sin u t tp (14) where tp is the time of perihelion. Combining equations (13) and (14), 2 C 1 1 r2 ra2 ¼ 1p þ r12 rp2 a 1 ¼ rp ra 2 1þ (15) rp ra where rp is the distance to the sun at perihelion and ra is the distance to the sun at aphelion. The ratio of rp to ra can be written in terms of the earth's orbital eccentricity as rp 1 ε ¼ ra 1 þ ε (16) So C¼ Fig. 3. The differences between the elevation calculated by the model and the values from the NOAA spreadsheet. 2ε 1 þ ε2 (17) The eccentricity of the earth's orbit is 0.0167. Graphs of the daily 178 F.R. Shapiro Renewable Energy 184 (2022) 176e181 Fig. 5. Analemma showing the position of the sun at noon Eastern Standard Time (UT -5) 20 June 2020 to 20 June 2021, at the same location as Figs. 2 and 3, as calculated by the model. The dates of the points are A: 27 July 2020, B: 18 December 2020, C: 15 May 2021, and D: 19 June 2021. The dashed line shows the same values from the NOAA spreadsheet. Fig. 8. Root mean square error of the elevation of the sun as calculated by the model as in Fig. 6, but with an approximate correction for the eccentricity of the earth's orbit. Fig. 6. Root mean square error of the elevation of the sun as calculated by the model relative to the results in the spreadsheet from the United States National Oceanic and Atmospheric Administration, for each day from 20 June 2020 to 20 June 2021. Fig. 7. Root mean square error of the azimuth of the sun as calculated by the model relative to the results in the spreadsheet from NOAA, for each day from 20 June 2020 to 20 June 2021. Fig. 9. Root mean square error of the azimuth of the sun as calculated by the model in Fig. 7, but with an approximate correction for the eccentricity of the earth's orbit. 179 F.R. Shapiro Renewable Energy 184 (2022) 176e181 6. Conclusions This work shows the derivation of equations for the elevation and azimuth angles of the sun in a simple model. The only parameters are readily available values with clear physical meaning: the date and time of the nearest northern hemisphere winter solstice, the tilt of the earth's axis, and the length of a year in the Gregorian calendar. In the model, the earth's orbit is assumed to be circular, the earth is assumed to be a perfect sphere, and many other phenomena are excluded. However, results are generally within a few degrees of a more accurate calculation in a spreadsheet by NOAA throughout the year centered at the solstice. The characteristic figure-8 of the analemma is found even with this simple model. Most of the difference between the simple model and the NOAA calculation is due to the eccentricity of the earth's orbit. An approximate correction for the eccentricity, based on Kepler's second law, substantially reduces the differences between the simple model and the NOAA calculation. The only parameters in the correction are the eccentricity of the earth's orbit and the date and time of the perihelion nearest to the solstice already used in the model. The non-spherical shape of the earth, the influence of the moon, diffraction by the earth's atmosphere, and other phenomena cause only small variations in sun's elevation and azimuth. The model is shown to work for trial locations in both the northern and southern hemispheres. Fig. 10. Analemma showing the position of the sun at noon Eastern Standard Time (UT -5) 20 June 2020 to 20 June 2021, as calculated by the model. The dashed line shows the same values from the NOAA spreadsheet. root mean square difference from 20 June 2020 to 20 June 2021 with this correction are shown in Figs. 8 and 9, using the perihelion on 2 January 2021 at 13:50 UT. The reductions in the difference from Figs. 6e8 and from Figs. 7e9 are substantial, although there is still some structure. The root mean square differences in Figs. 8 and 9 are less than the sun's diameter as viewed from earth, which is approximately 0.5 . Fig. 10 shows the analemma using the model with the eccentricity correction, and the values for the same times from the NOAA spreadsheet. The results of the model and the spreadsheet are almost indistinguishable. CRediT authorship contribution statement Finley R. Shapiro: Writing e original draft, This work was written exclusively by the author, who also did all of the work described. The author benefitted from discussions and comments from colleagues and the referees, as mentioned in the acknowledgements. 5. Southern hemisphere Calculations of the solar position according to the model and the NOAA spreadsheet were also done for the position Latitude 12 Longitude 77, the approximate location of Lima, Peru. Lima is south of the equator and north of the Tropic of Capricorn, so at noon the sun can be north, south, east or west of zenith. In order to see a Fig. 8 shape similar to those in Figs. 5 and 10, the graph in Fig. 11 shows qZ sin g on the x-axis and qZ cos g on the y-axis. The origin of the graph represents zenith. The skew to the right is because local solar noon at Longitude 77 is approximately 8 minutes after noon for the time zone (UT -5). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author is grateful for helpful suggestions from Gerard F. Jones, Michael E. Evans, and the anonymous reviewers. This paper did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] J.A. Duffie, W.A. Beckman, Solar Energy Thermal Processes, 5th ed., John Wiley & Sons, also, 1974, 2020. [2] P.I. Cooper, The absorption of radiation in solar stills, Sol. Energy 12 (1969) 333e346. [3] A. Goetzberger, A. Zastrow, On the coexistence of solar-energy conversion and plant cultivation, Int. J. Sol. Energy 1 (1982) 55e69. [4] C. Dupraz, H. Marrou, G. Talbot, L. Dufour, A. Nogier, Y. Ferard, Combining solar photovoltaic panels and food crops for optimizing land use: towards new agrivoltaic schemes, Renew. 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