Step-size effect in the time-transformed leapfrog integrator on elliptic and hyperbolic orbits

TL;DR

This paper investigates the step-size effects of the time-transformed leapfrog integrator on elliptic and hyperbolic orbits, revealing phase transitions and energy conservation properties crucial for accurate hyperbolic encounter simulations.

Contribution

It provides a mathematical explanation for the step-size dependent behavior of the TSI method on hyperbolic orbits, highlighting constraints for its proper use.

Findings

Exact orbit conservation for elliptic orbits regardless of step size

Phase transition occurs for hyperbolic orbits when step size exceeds a threshold

Energy conservation shifts from total energy to kinetic minus potential energy in hyperbolic cases

Abstract

A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a time-transformed separable Hamiltonian (or time-transformed symplectic integrator; TSI) has been known to conserve the Kepler orbit exactly. We find that for an elliptic orbit, such feature appears for an arbitrary step size. But it is not the case for a hyperbolic orbit: when the half step size is larger than the conjugate momenta of the mean anomaly, a phase transition happens and the new position jumps to the nonphysical counterpart of the hyperbolic trajectory. Once it happens, the energy conservation is broken. Instead, the kinetic energy minus the potential energy becomes a new conserved quantity. We provide a mathematical explanation for such phenomenon. Our result provides a deeper understanding of the TSI method, and a useful constraint of the step size when the TSI method is used to solve the hyperbolic encounters. This is particular important when an (Bulirsch-Stoer) extrapolation integrator is used together, which requires the convergence of integration errors.