MONDian three-body predictions for LISA Pathfinder

Abstract

In previous work it was shown that MOND theories predict anomalously strong tidal stresses near the saddle points of the Newtonian gravitational potential. An analytical examination of the saddle between two bodies revealed a linear and a non-linear solution, valid for the outer and inner regions. Here we present a numerical algorithm for solving the MOND equations. We check the code against the two-body analytical solutions and explore the region transitioning between them. We then develop a a realistic model for the MONDian effects on the saddles of the Sun-Earth-Moon system (including further sources is straightforward). For the Sun-Earth saddle we find that the two-body results are almost unchanged, with corrections increasing from full to new Moon. In contrast, the Moon saddle is an intrinsically three-body problem, but we numerically find a recipe for adapting the two-body solution to this case, by means of a suitable re-scaling and axis re-orientation. We explore possible experimental scenarios for LISA Pathfinder, and the prospect of a visit to the saddle(s) at the end of the mission. Given the chaotic nature of the orbits, awareness of the full range of the possibilities is crucial for a realistic prediction. We conclude that even with very conservative assumptions on the impact parameter, the accelerometers are abundantly sensitive to vindicate or rule out the theory.