Note on the construction of a Long-Term Solar Almanac

Due to the relative smooth and regular motion of the Earth around the Sun, compact long-term ephemeris data for the Sun can be compiled. This section describes how such tables can be constructed and also how they are used. These compact Sun Ephemeris tables may be used as a backup alternative if a regular Nautical Almanac is not available.

The Motion of the Earth

The apparent motion of the Sun originates from two independent motions of the Earth in space: the spinning of the Earth on it's axis of rotation and the orbiting of the Earth around the Sun. Roughly speaking, the spinning of the Earth is related to time through the day, whereas the date through the year is related to the orbiting around the Sun. Since ancient times it is known that the periodicity of these two basic motions have no common factor and the quest to keep a certain synchronism between (solar) time and date led to the science of calender making. Today this synchronism is arranged through the concept of leap days (rough adjustment) and leap seconds (fine adjustment).

Let's observe the Earth on it's orbit around the Sun, starting the journey on the first day of January at the moment the Geographical Position of the Sun is on the Greenwich Meridian. Each day around 12 UTC, the Geographical Position of the Sun is on the Greenwich Meridian again, but the Earth will have proceeded a little bit along it's orbit around the Sun. After 365 days, while the Greenwich Meridian is again oriented towards the Sun, the Earth will not be exactly on the same orbit position as it was at the start of the journey. The Earth still has about six hours to go along it's orbit in order to reach that previous starting position.
After four years (4 times 365 days), this time offset has accumulated to about 24 hours and now about a full day is missing to reach the original orbit starting point. In modern calenders a leap day at the end of February is inserted every four years in order to realign these time- and date scales.

Because the yearly misalignment between time and date scales is not exactly 6 hour (but slightly less), a more complex leap day scheme has been adopted. If no leap days were inserted and each year would have the same number of days, the seasons would be shifting slowly through the calender year by about 6 hours per year (the effect of axial precession is considerably smaller - less than 0.34 hours per year).

If the Almanac for a specific year is available, and the exact time offset for the current year with respect to the validity year of the Almanac is known, the Ephemeris Data for the Sun can be used but the UT has to be adapted according to the time offset between the years that are involved. Almanac Data can be re-used in the original form after roughly four years. Based on this, a scheme for a compact, and long-term Almanac can be constructed. Notice, that this is only valid for the Ephemerides of the Sun.

Determining GHA of the Sun from UTC

The true Solar Time is directly related to the GHA of the Sun. Civil time scales such as UTC or GMT however are based on mean Solar Time. The difference between true- and mean Solar Time is called "Equation of Time" (EoT) and it is required to calculate the solar GHA from UTC:

 GHA = 180°  +  15°/h x (UTC + EoT)                 range: 0° through 360°

Time is expressed in hours and GHA in degrees. The factor 15 is related to the Earth rotation: 360° in 24 hours (15°/h). The value of EoT is in the range ±16min. Expressed in degrees of GHA, this corresponds to a range of ±4°.

So, if UTC is known, the corresponding EoT can be determined from the Nautical Almanac and from these data the correct GHA for the Sun can be calculated.

Long-Term Solar Almanac

A long-term Solar Almanac has the EoT and Declination data for a certain reference year. The Almanac Data for the other years is determined by correcting the applied UTC time scale by an appropriate time offset (which will be approximately a multiple of 6 hours). To account for this, an adapted time scale called "Orbit Time" (OT) is used to enter the Solar Almanac.
Orbit Time is the UTC time corrected by a time offset value, which is obtained from an extra table in the Long-Term Almanac. Other tables in the Almanac allow for interpolation of EoT- and Declination values and for calculating GHA from UTC as indicated above.

In order to simplify the calculations, EoT is tabulated in degrees (15°/h * EoT) and with an offset of 5° to avoid negative values. The resulting values are in a range of [1.5° - 9.1°]. The compensation of the 5°-offset in the "E" value, is taken into account in the Tables for the Diurnal Arc from which the final value for the GHA is obtained.

An example of a Long-Term Solar Almanac is available. An analysis of the accuracy over the valid time frame of 20 years, shows an error in GHA of less than 2'. For the Declination, the values are accurate within 3'. So the expected accuracy of these long-term almanacs is about twenty times less than the accuracy of a standard Nautical Almanac in which GHA and Declination values are given with 0.1' accuracy. However, for a backup method, this should be tolerable.

Table Structure

The main table gives "E" (5° + Equation of Time) and Declination of the Sun for the argument "Orbit Time" OT, which is formed by applying the "h" correction from the "Corrections for OT" to the nearest integral hour of UTC. In leap years, the upper value of the correction is to be used for January and February and the lower value for the rest of the year. Thus, OT’s corresponding to 2016 February 29 - 16h 31m UTC and 2016 March 01 - 05h 29m UTC, are February 29 - 11h 00m and March 01 - 23h 00m respectively.

Corrections to E and Declination for OT are determined by entering the Table "Interpolation for Hours of OT" with the differences between consecutive values of E and of Declination respectively as the horizontal argument ("Diff."), and with the number of hours of OT ("h") as the vertical argument. The Declination differences are given in the main table.
The GHA is obtained by adding to the corrected E, the value of the Diurnal Arc obtained from the Tables "Diurnal Arc". The latter two tables must be entered with argument UTC.

Example of how to use the Tables

Given the long-term Almanac 2017-2036, find the GHA and Declination of the Sun on 2020 January 18 at 03h 30m 35s UTC.

In a first step, the correct date and time (OT) to be used in the Tables for EoT and Declination is determined. Then the appropriate values for "E" and "Dec" are taken from the main Table as well as the hourly increments for both values. Subsequently the increments are interpolated for the previously determined OT integral-hour value. Remember to apply the correct signs of each of the increment values.
Adding the interpolated increment values to the corresponding main values gives the final values for "E" and "Dec". No further interpolation for minutes and seconds of UTC are performed.
Finally, for obtaining the GHA, the obtained value for E must be added to the Diurnal Arc for the given UTC time (03h 30m 35s). The Diurnal Arc is simply the translation of UTC-time into degrees of Longitude using the 15°/h factor. This is obtained in two steps: the angle for hours and tens of minutes is added to the angle for minutes and seconds.

OT = UTC (nearest integral hour) + Corr. (Table "Correction for OT")
   = Jan. 18 / 04h - 6h 
   = Jan. 17 / 22h.
   
 Main Table, Jan. 17 OT,   E   2° 32’ (-4’)     Dec S 20° 50’ (-12’)
 Interp. Table,  22h OT,          -4’                    -11’
                             --------                 -------
 Corrected Jan. 17 22h OT, E   2° 28’           Dec S 20° 39’
 Diur. Arc 03h 30m UTC       227° 30’
 Diur. Arc 00m 35s UTC         0° 09’
                             --------
 GHA                         230° 07'