A semi-numerical method for computing the second-order perturbation of artificial satellites

In this paper a new semi-analytical, semi-numerical method for computing the second order perturbation of artificial satellites is presented. The basic ideas are the following:

• 1.

  1. We adopt as fundamental elements $\text{σ}{}^{\mathrm{\ast }}$, which are the mean elements obtained after subtracting the short-period terms Δσ_s from the osculating elements.

• 2.

  2. The secular and long-period rates (up to third order) of $\text{σ}{}^{\mathrm{\ast }}$ are found by numerically averaging the rates of the osculating elements.

• 3.

  3. The osculating elements in computing dσ/dt are found by adding to $\text{σ}{}^{\mathrm{\ast }}$ the short-period termsΔσ_s, in which the first-order terms are obtained by analytical method, and the second-order terms by Fourier analysis.

By using $\text{σ}{}^{\mathrm{\ast }}$, we have overcome the shortcoming in the numerical method, due to the use of short integration steps, while our numerical computation of $\text{dσ}{}^{\mathrm{\ast }}\text{/dt}$ and Δσ_s⁽²⁾ obviates the development of complicated formulae.

Our computing time of the second-order perturbation is only $\text{1}\text{10}$ that of the classical numerical method, and our required memory is about $\text{1}\text{5}$ of the purely analytical method.

In the numerical method, an improved Chebychev iteration process is developed to better serve the needs of satellite dynamical geodesy.

Detailed formulae for computing the perturbating forces and some numerical results of computation by our method are given. These show that the radius and across-track errors are less than 0.1m, and that the along-track errors are less than 1m.