Distant Retrograde Orbits

Abstract

Distant Retrograde Orbits (DROs) are special orbits for third bodies in two-body systems. The third body revolves around the secondary – the smaller of the two primaries – in a retrograde way, meaning the direction is opposite to the direction that the primaries revolve around each other. DROs are not close to either of the primaries, making it difficult to model them as perturbed two-body orbits.

There is no analytical solution for the initial conditions of DROs. This thesis presents a novel method of calculating an initial velocity guess which is then fed into a differential corrector that is able to calculate the initial conditions. In contrast to the state-of-the-art, this happens without the method of incremental steps in the initial position, which requires to go through all possible DROs for a specific two-body system first.

For the calculation of DROs, numerical integration is done. Optimal integrator settings are determined, which is in this case an eighth-order Runge-Kutta method (RK8). By setting the tolerance to the lowest possible value, the accuracy requirements are satisfied.

Furthermore, this thesis explores a different method of modeling DROs that makes use of Fourier series and polynomials, which had already been proposed by Hirani in 2006 for a different set of parameters. By exploiting explicit knowledge about the shape of DROs, this approach is made more efficient in terms of accuracy per Fourier/polynomial parameters needed and thus the computation time is enhanced.

The second part of this study addresses the stability of DROs. This is analyzed in order to get an idea of what DROs would be suitable for future missions. For mass ratios of primary and secondary that realistically occur in the Solar System, all DROs that are closer to the secondary than the primary turn out to be stable when disregarding perturbations. Perturbations are modeled as a constant external acceleration with a constant direction, which is only a first step towards modeling the Sun's and other planet's point mass gravity (p.m.g.), the solar radiation pressure (s.r.p.), and other perturbations, as they are usually depending on time and position. With this rough estimate, only the Sun's p.m.g. is identified as a possible source of instability for DROs in the Earth-Moon system, as all other perturbations are too small.