Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

TL;DR

This paper derives formulas for calculating the propagation time of signals between satellites on elliptical orbits using elliptic integrals, incorporating effects from General Relativity, and simplifies some integrals to elementary functions.

Contribution

It provides explicit expressions for signal propagation times involving elliptic integrals for satellites on elliptical orbits, including cases with space-distributed orbits, and simplifies certain integrals to elementary functions.

Findings

Propagation time expressed via elliptic integrals of various orders.

Second-order elliptic integrals can be reduced to elementary functions.

Higher-order elliptic integrals can be expressed recursively by lower-order ones.

Abstract

The propagation time of a signal, emitted by a moving along an elliptical orbit satellite from the GPS (or GLONASS) satellite confi gurations is a very important ingredient of the theory, based on the formalism of the null cone and accounting for the effects of the General Relativity Theory. For the case of satellites, orbiting along a plane elliptic orbit, it has been proved that the propagation time for the signal between the satellites is given by a combination of elliptic integrals of the first, second and third kind. For the more general case of satellites on a space-distributed elliptic orbit, the propagation time is expressed by higher (fourth) order elliptic integrals, which according to the standard theory can be expressed recurrently by means of lower-order elliptic integrals. In the concrete case, the elliptic integrals of the second and the fourth order are expressed by means of a combination of irrational functions and the zero-order elliptic integral in the Legendre form. It has been proved that for the investigated case, second-order elliptic integrals can be expressed by elementary functions.