Astron. Astrophys. 329, 329-338 (1998)

4. Results

Our analytical solution is close to the results of Bretagnon et al. (1997). The differences arise from the use of the complete solution ELP 2000 for the motion of the Moon and of a better accuracy in the computation of the right-hand sides of the equations. So, our analytical solution is more accurate than the former one.

4.1. Comparison to numerical integration using analytical theories

To test the precision of our solution we performed numerical integration using VSOP87A and ELP 2000. Instead of using directly the Fourier and Poisson series of the theories, that would be very time-consuming, we have first at all built a representation with Chebychev polynomials. The variables used are the rectangular coordinates of the bodies in the reference frame defined by the dynamical equinox and ecliptic J2000. The parameters of the representation are given in Table 1. In this table, the accuracy column contains the largest discrepancies between the theories and the Chebychev polynomials over the time span 1800-2100.

Table 1. Parameters of Chebychev polynomials used for the representation of ELP 2000 (Moon) and VSOP87A (Sun and planets).

We have compared our analytical solution with a numerical integration run over 50 days from J2000 to test the accuracy of the diurnal terms (terms of period 24, 12 and 8 hours) and with a numerical integration run over 150 yr (1900-2050). Over 50 days, the differences , , are about 0.02 as. Over 150 yr they are smaller than 1.5 as. Those results are given in Table 2 and illustrated by the Fig. 1 and the Fig. 2.

Table 2. Differences between the analytical solution and the numerical solution calculated from ELP2000 and VSOP87A over 50 days and 150 years (1900-2050). Unit is as (microarcsecond).

Fig. 1. Theory - numerical integration using ELP2000 and VSOP87A over 50 days

Fig. 2. Theory - numerical integration using ELP2000 and VSOP87A over 1900-2050

Figs. 1 and 2 illustrate the periodic differences. For this purpose they have been corrected by linear terms. So, the Fig. 2 corresponds to

where t is measured in tjy from J2000.

Those corrections give an estimate of the accuracy of the computation of the secular terms. For instance for given by (1), we have . The relative precision is and the digits of are given only to obtain the value for the constant of the general precession. The accuracy of given by (3) is . The accuracy of the Earth rotation is

and the relative precision is .

4.2. Comparison to numerical integration using DE403/ LE403

The differences , , between our analytical solution and a numerical integration computed using DE403/LE403 are greater than the differences given in Table 2. They come from the differences between the models and from the reference frames used in the solutions. We have compared our analytical solution with numerical integrations run over 50 days, 150 yr and also 55 yr (1968-2023) to test the terms with period less than 50 years. The differences are given in Table 3 and they are illustrated by Figs 3, 4 and 5. For the diurnal terms the differences are equivalent in Table 3 and in Table 2. Over 150 yr, the differences are about 12 as for . They correspond to long period and polynomial terms. Over 50 yr the differences are smaller 2.20 as for , 0.65 as for and as for .

Table 3. Differences between the analytical solution and the numerical solution calculated from DE403/LE403 over 50 days, 150 years (1900-2050), and 55 years (1968-2023). Unit is as (microarcsecond).

Fig. 3. Theory - numerical integration using DE403/LE403 over 50 days

Fig. 4. Theory - numerical integration using DE403/LE403 over 1900-2050

Fig. 5. Theory - numerical integration using DE403/LE403 over 1968-2023

© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997
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