

Subject:
Equation to know where the Sun is at a given place at a given datetime
Category: Science > Math Asked by: herixga List Price: $10.00 
Posted:
15 Nov 2006 03:07 PST
Expires: 15 Dec 2006 03:07 PST Question ID: 782886 
The goal is to predict the apparent position of the Sun in the Earth sky at a given date, a given hour and a given latitude. Let alpha be the angle between North and the direction of the Sun, and beta then angle of the Sun above the horizon, L the latitude, l the longitude, and t the datetime. I'm looking for the function F : ( albha, beta ) = F( L , t ) For exemple, right now, I am at 48°49'N 02°17'23 E, and we are the 15th of november 2006, time is 1035 GMT. **Roughly** observed, the Sun is at 165° (North being zero) and 22° (horizon being zero) I want F that could have told me : F( 48°49' , 2°17', 2006/11/15, 1035) = (165, 22) 

Subject:
Re: Equation to know where the Sun is at a given place at a given datetime
Answered By: liviofloresga on 16 Nov 2006 09:24 PST Rated: 
Hi!! First of all, a direct function will give a very long and difficult to compute equation. There are some intermediate steps that must be done, so an algorithm (using several different formulas), instead of a single formula, is the way to predict theposition of the Sun in the Earth sky at a given date, a given hour and a given latitude. Here is the result of my research: Nice graph of what we will calculate: http://www.srrb.noaa.gov/highlights/sunrise/azelzen.gif First thing you need to compute is the fractional year g in degrees: g = (360/365.25)*(N + hour/24) where: N = day number > January 1 = day 1, January 2 = day 2, etc. Note about hour: during the period when Daylight Saving Time (DST) is in force one hour should be added to obtain clock time. Note that hour is the local hour and it is expressed in fractions of hours, that is 10h 35m = 10 + 35/60 = 10.58333 November 15, 2006 is the day number 319, then: g = (360/365.25)*(319 + 10.58333/24) = 314.849 Follows the calculation of the declination of the sun, again you an use a table or a formula: "Table of the Declination of the Sun": November 15 declination = 18° 20' (= 18.33333°) http://www.wsanford.com/~wsanford/exo/sundials/DEC_Sun.html Or use the following formula: D = 0.39637222.91327*cos(g)+4.02543*sin(g)0.387205*cos(2*g)+ +0.051967*sin(2*g)0.154527*cos(3*g) + 0.084798*sin(3*g) In this case: D = 18.6162 Now calculate the time correction for solar angle: TC = 0.004297+0.107029*cos(g)1.837877*sin(g)0.837378*cos(2*g) 2.340475*sin(2*g) In your case: TC = 3.72763 Now we can calculate the Solar Hour Angle (SHA) SHA = (hour12)*15 + Longitude + TC Longitude in degrees (this figure should be negative Weast of Greenwich and positive East of Greenwich). Your longitude in degrees is: 02°17'23 E = 2+17/60+23/3600 = 2.28972 Then: SHA = (10.5833312)*15 + 2.28972 + 3.72763 = 15.2327 Note that if SHA is greater than 180, then you must add (360) to the result and if SHA is lower than 180, then you must add 360 to the result. Now we can calculate the Sun Zenith Angle (SZA): cos(SZA) = sin(Latitude)*sin(D)+cos(Latitude)*cos(D)*cos(SHA) Your Latitude is 48°49'N = 48 + 49/60 = 48.81667 cos(SZA) = sin(48.81667)*sin(18.6162)+ +cos(48.81667)*cos(18.6162)*cos(15.2327) = = 0.36184 SZA = arccos(0.36184) = 68.78676 Well SZA is the complementary angle of the Sun Elevation Angle or Altitude(SEA), therefore SEA = 9068.78676 = 21.213° (21°12'48'') NOTE: if cos(SZA) formula gives a figure greater than 1 use 1 and if it gives you a figure lower than 1 use 1. To finish we will calculate the Azimuth Angle (AZ): cos(AZ) = (sin(D)sin(Latitude)*cos(SZA))/(cos(Latitude)*sin(SZA)) = = 0.96367 AZ = arccos(0.96367) = 164.50848 (164°30'30.5'') As you can see this method gives you from the inputs ( 48°49'; 2°17'23''; 2006/11/15; 10:35) the results (164°30'30.5''; 21°12'48''); your rough figures gave a good aproximation. You can use the above formulae to make an Excel spreadsheet. Sources: NOAA (National Oceanic and Atmospheric Administration) Surface Radiation Research Branch: "NOAA GVI GUIDE APPENDIX L: Software to Calculate Relative Azimuth from Third Generation Weekly Composite GVI Date": http://www2.ncdc.noaa.gov/docs/gviug/html/l/appl.htm "Horizontal coordinate system  Wikipedia, the free encyclopedia": http://en.wikipedia.org/wiki/Horizontal_coordinate_system "Positional Astronomy": http://starwww.stand.ac.uk/~fv/webnotes/ "Basics of Positional Astronomy": http://www.geoastro.de/elevaz/basics/index.htm You can check other results using a Java applet here: "Sun Calculator" http://users.zoominternet.net/~matto/Java/Solar%20Calculator.htm Search strategy: solar time calculation solar azimuth calculation solar azimuth formula I hope this helps you. Feel free to request for a clarification if you need it. Best regards, liviofloresga  
 

herixga
rated this answer:
This is really the best answer I could have expected for my question. Many thanks + congratulations. 

Subject:
Re: Equation to know where the Sun is at a given place at a given datetime
From: liviofloresga on 23 Nov 2006 06:00 PST 
Thank you so much for trusting on us by asking questions. I hope you continue using our service in the future. I am also thankful for the good rating and nice comments on my answer. Best regards, liviofloresga 
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