A differentiable N-body code for transit timing and dynamical modelling – I. Algorithm and derivatives

Abstract

ABSTRACT When fitting N-body models to astronomical data – such as transit times, radial velocity, and astrometric positions at observed times – the derivatives of the model outputs with respect to the initial conditions can help with model optimization and posterior sampling. Here, we describe a general purpose symplectic integrator for arbitrary orbital architectures, including those with close encounters, which we have recast to maintain numerical stability and precision for small step sizes. We compute the derivatives of the N-body coordinates and velocities as a function of time with respect to the initial conditions and masses by propagating the Jacobian along with the N-body integration. For the first time, we obtain the derivatives of the transit times with respect to the initial conditions and masses using the chain rule, which is quicker and more accurate than using finite differences or automatic differentiation. We implement this algorithm in an open source package, NbodyGradient.jl, written in the Julia language, which has been used in the optimization and error analysis of transit-timing variations in the TRAPPIST-1 system. We present tests of the accuracy and precision of the code, and show that it compares favourably in speed to other integrators that are written in C.