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 } ] ^ { - \alpha } , where r is the distance from the Sun .
Using a probabilistic inference technique , we infer 1.989 < \alpha < 2.052 ( 95 percent 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 .