Determination of Ecliptic Longitude

The sun's *ecliptic
longitude* is defined as the angle subtended at the earth between the vernal equinox and the sun.
Hence, from Fig. 21,

(96) |

(97) |

where

(99) |

The mean longitude increases
uniformly with time (since both and increase uniformly with time) as

where

(102) |

the

Our procedure for determining the ecliptic longitude of the sun is described
below.
The requisite orbital elements (*i.e.*, , , ,
, and ) for the J2000 epoch (*i.e.*, 12:00 UT on January 1, 2000 CE, which corresponds to
JD) are listed
in Table 30. These elements are calculated
on the assumption that the vernal equinox *precesses* at the uniform
rate of
.
The ecliptic longitude of the sun is specified by the
following formulae:

These formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE (see

The ecliptic longitude of the sun can be calculated with the aid of Tables 32 and 33. Table 32 allows the mean longitude, , and mean anomaly, , of the sun to be determined as functions of time. Table 33 specifies the equation of center, , as a function of the mean anomaly.

The procedure for using the tables is as follows:

- Determine the fractional Julian day number, , corresponding to the date and time at which the sun's ecliptic longitude is to be calculated with the aid of Tables 27-29. Form , where is the epoch.
- Enter Table 32 with the digit for each power of 10 in and take out the corresponding values of and . If is negative then the corresponding values are also negative. The value of the mean longitude, , is the sum of all the values plus the value of at the epoch. Likewise, the value of the mean anomaly, , is the sum of all the values plus the value of at the epoch. Add as many multiples of to and as is required to make them both fall in the range to . Round to the nearest degree.
- Enter Table 33 with the value of and take out the corresponding value of the equation of center, , and the radial anomaly, . (The latter step is only necessary if the ecliptic longitude of the sun is to be used to determine that of a planet.) It is necessary to interpolate if is odd.
- The ecliptic longitude, , is the sum of the mean longitude, , and the equation of center, . If necessary, convert into an angle in the range to . The decimal fraction can be converted into arc minutes using Table 31. Round to the nearest arc minute.