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When fitting N-body models to astronomical data - including transit times,radial velocity, and astrometric positions at observed times - the derivativesof the model outputs with respect to the initial conditions can help with modeloptimization and posterior sampling. Here we describe a general-purposesymplectic integrator for arbitrary orbital architectures, including those withclose encounters, which we have recast to maintain numerical stability andprecision for small step sizes. We compute the derivatives of the N-bodycoordinates and velocities as a function of time with respect to the initialconditions and masses by propagating the Jacobian along with the N-bodyintegration. For the first time we obtain the derivatives of the transit timeswith respect to the initial conditions and masses using the chain rule, whichis quicker and more accurate than using finite differences or automaticdifferentiation. We implement this algorithm in an open source package,NbodyGradient.jl, written in the Julia language, which has been used in theoptimization and error analysis of transit-timing variations in the TRAPPIST-1system. We present tests of the accuracy and precision of the code, and showthat it compares favorably in speed to other integrators which are written inC.

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