Theorie litterale du neuvieme satellite de Saturne, Phoebe

Abstract

We study a theory for the ninth satellite of Saturn, Phoebe, based on the literal solution we have obtained in the main problem of the lunar theory.

These series were computed by solving, by successive approximations, the Lagrange's equations expressed in variables, functions of the elliptic elements.

We may consider the case of Phoebe simpler than a lunar case because we seek less precision (1/10″ geocentric) than in the Lunar case, although the eccentricity of Phoebe is stronger.

Main problem: our series are computed to the complete seventh order and a great part of the perturbations of the eighth and ninth order, where we have attributed to the small lunar parameters the order 1 tom ₀=n′/n ₀,e ₀,′e, sin (i ₀/2), the order 2 to α₀=(a ₀/a′)((M ₁−)/(M ₁+M)) and the order 4 toµ ₀(a ₀/a′)M ₁ M/M ²₁ −M ².

In the case of Phoebe,µ ₀ equal zero and ±₀ is the ratioa ₀/a′.

We study the further development of these series by using, instead of parameterm ₀, the quantity Δm ₀=n′/n ₀−m ₁ wherem ₁ is an approached value ofm ₀, in order to accelerate the convergence of the series with respect tom ₀.

Comparison with a numerical integration we are adjusting a numerical integration to the observations. We have already more than 100 observations, for the period 1900–1957.

At first, we compare the series of the main problem to a numerical integration of the Keplerian problem.