4. Planetary Ephemerides for Navigators

  The practical application of the previous results has been the implementation of the ANDI, a computer application providing astronomical data for navigators. For the selection of the ephemerides representation the following considerations are in order: on one hand truncated series for planetary motion comprise large spans, on the other hand polynomial approximations have the advantage of its simple and fast evaluation, what is rather convenient due to the iterative procedures carried out during the program execution.

A precision better than 0.1 arc minutes is required for navigational purposes. Despite of this low value the precision requirement could be unfulfilled using truncated series for planetary motion; thus, in "Planetary Programs and Tables'' (Bretagnon & Simon 1956) the accuracy is bounded to . That circumstance and the fact that the computation of certain critical phenomena could spent an appreciable time, convinced us to use polynomial approximations; and because of the results of Sect. 3 (click here) we decided to use the best approximation method.

Again, we based our computations on DE200/LE200 basic ephemerides. By properly corrections of precession, nutation and aberration, apparent coordinates are obtained. Then the coefficients of all necessary polynomials are computed using the Schmitt algorithm. For a desired error the polynomial degree and the amplitude of the interval cannot be selected in an arbitrary manner. From practical considerations we decided to use medium or low degree polynomials and, except for the moon, we selected an interval of a year when possible; otherwise the interval was reduced to half year. Table 3 (click here) presents results corresponding to 1997. In order to use the computed polynomial approximations like interpolating polynomials it was enough to work with data bases tabulated every 4 hours except for the moon; in this last case we used a much more dense data base tabulated every 20 minutes.

 

Body days degree

Sun Right Ascension 183 19

Declination 365 16

Moon Right Ascension 031 19

Declination 031 18

Venus Right Ascension 183 18

Declination 365 21

Mars Right Ascension 183 22

Declination 365 24

Jupiter Right Ascension 365 15

Declination 365 14

Saturn Right Ascension 365 12

Declination 365 10

Table 3: Degrees of approximating polynomials. Fixing an error of 0.05 arc minutes we use different time spans depending on the body. Polynomials corresponding to 1997

 

  
Figure 2: Error functions for the right ascension (RA) and declination () of the Sun (above) and Mars (below). Time intervals and polynomial degrees correspond to the values presented in Table 3 (click here). Ordinates are arc seconds. Black dots correspond to errors in critical points

Figure 2 (click here) shows the error functions for the right ascension and declination of the Sun and Mars. For the right ascensions the time span is 183 days and for the declinations 365 days. Polynomial degrees are detailed in Table 3 (click here). Notice the alternance in sign of the extreme values predicted by Remez (1957). Though the error functions for declinations are not as smooth as for right ascensions, notice again the alternance of sign in critical points (black dots). Uniform approximation is ensured in all cases. From the smoothness of the error functions we conclude that the error has also the same bound for interpolated points of the data base and consequently the approximating polynomials can also be used for fast evaluation at any point.