Efficient Gravity Field Computation for Trajectory Propagation near Small Bodies

We present an approximation scheme for gravitational force near arbitrarily shaped small bodies. The approximation uses polynomial interpolation with an adaptive spatial data structure. This data structure allows us to drive errors in the model to within user defined thresholds. The result is a method that is expected to reduce the run time of trajectory integrations around small bodies without compromising accuracy. Our method is likely to provide a substantial improvement in the speed of Monte Carlo analysis of spacecraft motion in proximity to small bodies. INTRODUCTION Gravitational force computation is the bottleneck of Monte Carlo simulation used in design and analysis of small body missions. This expense has lead researchers to investigate ways to reduce cost. 1 The current approach for this force evaluation, developed by Werner and Scheeres, 2, 3 requires a summation of contributions from every face and edge over a polyhedral model representing the surface of the attracting body. While tractable over a single evaluation, this process becomes prohibitive when one considers the fact that numerically integrated trajectories consist of hundreds of elementary time steps, each of which rely on multiple force evaluations. Furthermore, Monte Carlo simulations represent another summation over this already expensive process. While the improvements proposed in Ref. 1 result in significantly faster running times, ultimately their complexity remains the same O(F + E), where F is the number of faces and E is the number of edges in the model of the attracting body. We propose to alleviate this computational burden by pre-computing gravitation throughout the domain, and reducing force computations to a fast, sub-linear (i.e. O(log N)), interpolation operation. Indeed, given the availability of cheap, fast memory storage, trading memory against on-line computation seems advantageous. The idea of representing the gravitational potential using interpolation schemes has already been investigated by Junkins and Engels (Ref. 4 and 5) for the case of nearly spherical bodies. The extension of such methods to the case of arbitrarily shaped bodies presents several challenges: the irregularity of the domain and the taxing error bounds required for trajectory integration. This paper presents a solution to these challenges. In the next section we review two computational models relevant to the formulation of our method. Then, the section 'Approximating an Asteroid's Gravitational Force' presents our approach to the computation of gravitational forces. The final two sections give a discussion of our results and present some challenges that remain. We …