Colloquia

Summary

This report aims to solve the satellite-to-site visibility problem through the use of a so-called controlling equation, an analytically developed transcendental equation formulated by Escobal. Once or twice per orbital revolution, this equation is solved numerically for a rise and set eccentric anomaly. From this, the visibility window, where communication between ground station and satellite is possible, can be determined. This method is more efficient in comparison with the traditional brute force method, where for each point of a satellite's ephemeris, a constraint equation is solved.
The two-body problem forms the foundation from which the orbit of a satellite can be determined. Once established, perturbations due to a nonspherical celestial body and third body attraction are introduced in the form of zonal harmonics. Expressions are obtained through the gravitational potential and series expansion. These perturbations are incorporated through the Lagrange equations, an analytical solution method that describes the rate of change of the orbital elements, a set of variables that define an orbit.
To verify the implementation of first the Lagrange equations, and later on the controlling equation, Cowell's method is applied, a method that numerically integrates the equation of motion of the satellite in time. The brute force method is subsequently applied to yield a visibility window.
At first, this is applied to a satellite in orbit around the Earth, for which the behaviour is well documented in literature. Afterwards, a lunar orbiter is analysed. The Moon, in contrast to the Earth, has a less dominant single zonal harmonic, meaning it is imperative to extend the series and include more degrees. In this report, degrees up till the tenth are considered.