Dynamical inference from a kinematic snapshot: The force law in the solar system

If a dynamical system is long-lived and non-resonant (that is, if there is a set of tracers that have evolved independently through many orbital times), and if the system is observed at any non-special time, it is possible to infer the dynamical properties of the system (such as the gravitational force or acceleration law) from a snapshot of the positions and velocities of the tracer population at a single moment in time. In this paper, we describe a general inference technique that solves this problem while allowing (1) the unknown distribution function of the tracer population to be simultaneously inferred and marginalized over, and (2) prior information about the gravitational field and distribution function to be taken into account. As an example, we consider the simplest problem of this kind: we infer the force law in the solar system using only an instantaneous kinematic snapshot (valid at 2009 April 1.0) for the eight major planets. We consider purely radial acceleration laws of the form a _r = -A [r/r ₀]^-α, where r is the distance from the Sun. Using a probabilistic inference technique, we infer 1.989 < α < 2.052 (95% interval), largely independent of any assumptions about the distribution of energies and eccentricities in the system beyond the assumption that the system is phase-mixed. Generalizations of the methods used here will permit, among other things, inference of Milky Way dynamics from Gaia-like observations.