Computation of elliptical functions in problems of celestial mechanics

Elliptical functions are widely used in celestial mechanics in the theory of motion of artificial Earth satellites and resonance asteroids, in qualitative study of the restricted three-body problem and in simple computation of Laplace coefficients. However, advances in computer technology dictate a reexamination of existing algorithms for computing elliptical functions for increasing their accuracy. Having the k modulus, it is usually necessary to find the value of the elliptical function for a stipulated argument u. The easiest way to do this is a changeover to theta functions, which are power series relative to the q parameter with coefficients being trigonometric functions of the argument u. Using expressions of the elliptical functions through the theta function, it is relatively easy to compute the required function. The fundamental problem is therefore a determination of the q parameter on the basis of the known k value. Formulas are derived for computing the theta function with high precision. A numerical example of the computations is given.