Long-Term Ephemerides for the Sun

Scope and Purpose

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 accuracy of the obtained GHA and Declination data is better than 3'.
The effort to obtain the ephemeris data with these long-term tables is higher compared to using a regular Almanac in which the data is recorded at integral hour intervals with an accuracy of 0.1' for both GHA and Declination. But as a backup solution, these compact tables may be a feasible alternative.

Background

Time is experienced in the first place by the rotation of the Earth resulting in the day-night succession and in the second place by the orbiting of the Earth around the Sun manifesting itself primary in the succession of the different seasons over the year but also in the changing star constellations at night. However, there is no direct relationship between the time periods of a day and a year. For the Earth, it takes about 365.25 rotations to arrive at the same position on it's orbit around the Sun.

To make things more complicated, the duration of a true solar day varies in length over the course of a year. This became already obvious in ancient times as people tried to keep track of time with mechanical instruments such as wax candles, water clocks or pendulum clockworks. Mechanical timekeeping presumes a constant time flow and thus a constant length of a day. This resulted in the development of the concept of a Mean Solar Day in contrast to the True Solar Day. The difference between Mean Solar day and True solar day is the Equation-of-Time (EoT). The value for the EoT varies over the year in a range or ±16 min or in a range of ±4° expressed in degrees (of Longitude). Expressed in time, the EoT gives the time difference between UTC-Noon based on Mean Solar time and True-Solar-Noon as shown by a sun dial.

Some more background information is also available in the Note on the construction of a Long-Term Solar Almanac.

Construction of the Tables

The ephemeris data for the Sun is directly related to the orbit position of the Earth relative to the Sun. The basic idea behind the long-term Sun ephemeris tables, is to construct the data for a reference year and then to find the orbit position in the reference year corresponding to the situation at the time-of-observation for which GHA and Declination are to be evaluated. This "adjusted" orbit position will be expressed in units of time and is called Orbit Time.

Table Layout

The main table, supplies the basic ephemeris information. It uses Orbit Time as argument and gives the values of the Equation-of-Time (E) and Declination (Dec) of the Sun for the start of each day (00:00 UTC). These values are recorded for a one year time period. This main table is constructed for a reference year, which may be chosen at about half of the valid time range. The data for the other years is obtained by entering the main table with the correctly adjusted time for the Orbit Time.

This time adjustment or hour offset (in integral hours) is obtained from table (a) using the year of interest as argument. From one year to the next year, the time adjustment decrements by approximately 6 hours to flip forward by about 24 hours in the course of a leap year. Hence, for a leap year two time adjustment values are in place. The first value is valid for the first two months including the 29th of February. The second value is valid from March till the end of the year.

Note, that in order to obtain positive values for E, the value of the Equation-of-time is increased by 5°: E = Equation-of-time + 5°. This offset is taken into account in the tables for determining the Diurnal Arc, which start with a value of 175°, instead of 180° for 00h UTC.

Table (b) is used for interpolation of the data obtained from the main table.

Adjusting the orbit position to the reference year, is sufficient to obtain the Declination value, but for obtaining the GHA information, also the rotation of the Earth must be considered. This requires the correct value for the Equation-of-Time (E) and the value of Diurnal Arc obtained from the table (c) and table (d) using the time-of-observation (in UTC) as argument.

Explanation and Usage of the Tables

The first step to obtain the Sun Ephemerides for a UTC-time of observation from the long-term table, is to determine the "Correction for OT" (h) from table (a). This time correction is the time difference (in integral hours) between the year of interest and the reference year for which the main table was compiled.
The next step is to calculate the Orbit Time (OT) from the UTC time of the Sun sight observation: OT = UTC + h.
With the value of OT, the E and Dec values and their increments can be extracted from the main Table. Next, the integral UTC hour value is used to interpolate both the E and the Dec value.

Finally, the Diurnal Arc obtained from tables (c) and (d), is added to the interpolated E value to determine the GHA for the given UTC time.

Example

Get the Sun Ephemerides for 2024, August 08, 17h 23m 44s UTC:

OT = UTC (nearest integral hour) + Correction (tab.a)
   = Aug. 08, 17h + 18 
   = Aug. 09, 11h
   
Main table,    Aug. 09, 00 OT,   E  3° 35'  (+3')    Dec N 15° 58'  (-18')
Interp. table (tab.b)   11 OT,         +1'                     -8'

Corrected:     Aug. 09, 11 OT,   E  3° 36'           Dec N 15° 50'
Diur. Arc (tab.c) 17h 20m UTC,     75°
Diur. Arc (tab.d) 03m 44s UTC,      0° 56'
                              GHA 079° 32'

Usage

The long-term Ephemerides for the Sun are primarily desiged to be used as backup solution for the more precise Nautical Almanac. Combining these long-term Sun Ephemerides, with the simplified Sight-Reduction Method as proposed by Robert Doniol, allows for a complete electronic-free method for Sun-based Celestial Navigation, which consists of only four pages of tables. These will fit in any sextant box and with the knowledge of how to use both the instrument and the required calculation schemes, a complete emergency navigation kit can be assembled.

Precession

The procedure used here to construct the described long-term Ephemerides Tables for the Sun, does not account for the effect of precession, so that in the very long term some adaption to this procedure may become necessary. The reason is that the reference system slowly drifts due to precession. However, the basic principle to construct the Sun Ephemeris data as described above remains valid. In the rest of this section, a short introduction to the precession mechanism is given.

The axis of the Earth is undergoing a precessional motion similar to that of a top spinning with its axis tilted. In about 25800 years the axis completes a cycle and returns to the position from which it started. Since the celestial equator is 90° from the celestial poles, it too is moving. The result is a slow westward movement of the equinoxes and solstices, which has already carried them about 30°, or one constellation, along the ecliptic from the positions they occupied when named more than 2000 years ago.

Since sidereal hour angle is measured from the vernal equinox, and declination from the celestial equator, the coordinates of celestial bodies would be changing even if the bodies themselves were stationary. This westward motion of the equinoxes along the ecliptic is called precession of the equinoxes. The total amount, called general precession, is about 50 seconds of arc per year. It may be considered divided into two components: precession in right ascension (about 46.10 seconds per year) measured along the celestial equator, and precession in declination (about 20.04" per year) measured perpendicular to the celestial equator. The annual change in the coordinates of any given star, due to precession alone, depends upon its position on the celestial sphere, since these coordinates are measured relative to the polar axis while the precessional motion is relative to the ecliptic axis.

Due to precession of the equinoxes, the celestial poles are slowly describing circles in the sky. The north celestial pole is moving closer to Polaris, which it will pass at a distance of approximately 28 minutes about the year 2102. Following this, the polar distance will increase, and eventually other stars, in their turn, will become the Pole Star. The precession of the Earth’s axis is the result of gravitational forces exerted principally by the Sun and Moon on the Earth’s equatorial bulge. The spinning Earth responds to these forces in the manner of a gyroscope. Regression of the nodes introduces certain irregularities known as nutation in the precessional motion.