Calculating the Sun's Altitude and Azimuth

To determine whether a transit is visible from a specific geographic location, it is simply a matter of calculating the Sun's altitude and azimuth during each phase of the transit. The calculations can be performed on any pocket calculator having trig functions (SIN, COS, TAN). Armed with the latitude and longitude of the location, the transit catalogs provide all the additional information needed to make the calculations.

The altitude 'a' and azimuth 'A' of the Sun during any phase of a transit depends on the time and the observer's geographic coordinates. Neglecting the effects of atmospheric refraction and planetary parallax, 'a' and 'A' are calculated as follows:

             h  = 15 * (GST + t - ra ) + l
             a  =  ArcSin  [ Sin d Sin f + Cos d Cos h Cos f ]
             A  =  ArcTan  [ - (Cos d Sin h) / (Sin d Cos f -  Cos d Cos h Sin f) ]

     where:
             h  =  Hour Angle of the Sun (in degrees)
             a  =  Altitude (in degrees)
             A  =  Azimuth (in degrees)
           GST  =  Greenwich Sidereal Time at 00:00 UT
             t  =  Universal Time
            ra  =  Right Ascension of the Sun (in hours)
             d  =  Declination of the Sun (in degrees)
             l  =  Observer's Longitude (East +, West -)
             f  =  Observer's Latitude (North +, South -)

For example, the Venus catalog lists the following record for the transit of 2004 June 08:

                      Transit Contact Times (UT)
                -------------------------------------  Minimum   Sun      Sun           Transit
     Date         I      II   Greatest   III     IV      Sep.     RA      Dec     GST   Series
                 h:m     h:m     h:m     h:m     h:m      "       h        °       h

  2004 Jun 08   05:13   05:33   08:20   11:07   11:26   626.9    5.121   22.89   17.137    3 

Using the preceeding formulas and the data above, we will now determine whether the Sun will be above the horizon at greatest transit as seen from Washington DC. The geographic coordinates of Washington DC are:

                  Latitude:   f =  38°53´N  =   +38.9°
                 Longitude:   l = 077°02´W  =  -077.0°

From the transit catalog record, we have:

             Time of Greatest Transit:     t =  08:20 = 8.333
  Greenwich Sidereal Time at 00:00 UT:   GST =   17.137
           Right Ascension of the Sun:    ra =   5.121 
               Declination of the Sun:     d =  22.89 

Usung this data to solve for Hour Angle, Altitude and Azimuth, we have:

     Hour Angle of the Sun:  h  = 15 * (GST + t - ra ) + l
                                = 15 * (17.137 + 8.333 - 5.121) + -077.0
                                = 15 * (20.346) + -077.0
                             h  = 228.2°

     Altitude of Sun:   a = ArcSin [Sin d Sin f + Cos d Cos h Cos f]
                          = ArcSin [Sin(22.89) Sin(38.9) + Cos(22.89) Cos(228.2) Cos(38.9)]
                          = ArcSin [0.386 * 0.628 + 0.921 * -0.666 * 0.778]

                          = ArcSin [0.242 + -0.477]
                          = ArcSin [-0.235]
                          = -13.6°

With an altitude of -13°, the Sun will be below the horizon and will not be visible at greatest transit as seen from Washington DC. Repeating the calculation using the time of contact III (at 11:07 UT), reveals that the Sun's altitude is then 14°. Thus Washington DC will see the end of the transit. The Sun's azimuth can be calculated similarly using the above formula.

Geocentric contact times have been used in the above calculations in order to greatly simplify the problem. This approximation may introduce of an error in the actual contact times of up to 3 minutes for Mercury and up to 10 minutes for Venus. This translates into a possible error in the calculated altitudes of a up to 1° for Mercury and 2.5° for Venus. Thus, the proceedure has more that enough accuracy for most historical inquiries.

As an aid to historical research, four Excel 97 spreadsheet files have been prepared which perform the above calculations automatically. You simply enter the location name, latitude and longitude. Each of the tables then calculates the altitude of the Sun at that location for every contact and for every transit in the table. The four tables are similar but cover different time periods for Mercury and Venus: