An analytical solution for Kepler's problem

TL;DR

This paper introduces an analytical, smooth transformation framework for solving the gravitational two-body problem, enabling precise orbital analysis and improved observational strategies for exoplanet systems.

Contribution

It presents a novel, differentiable formalism that avoids singularities and simplifies calculations of orbital parameters and their derivatives.

Findings

Provides a smooth, analytical transformation between coordinates and orbital elements.

Enables optimal timing of radial velocity measurements for exoplanets.

Facilitates detection of secular variations in orbital parameters.

Abstract

In this paper we present a framework which provides an analytical (i.e., infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits all singular variables which otherwise would yield discontinuities. This method is based on two simple real functions for which the derivative rules are only required to be known, all other applications -- e.g., calculating the orbital velocities, obtaining the partial derivatives of radial velocity curves with respect to the orbital elements -- are thereafter straightforward. As it is shown, the presented formalism can be applied to find optimal instants for radial velocity measurements in transiting exoplanetary systems to constrain the orbital eccentricity as well as to detect secular variations in the eccentricity or in the longitude of periastron.