The Accelerated Kepler Problem

TL;DR

This paper extends the classical Kepler problem by incorporating constant acceleration, modeling astrophysical phenomena like accretion disks with jets, and analyzes the resulting dynamics and stability boundaries.

Contribution

It introduces the accelerated Kepler problem, providing analytical insights into its Hamiltonian structure, resonance origins, and stability limits relevant to astrophysical disk dynamics.

Findings

Identifies the origin of secular resonance in the accelerated Kepler problem.

Derives the stability boundary for circular orbits under acceleration.

Provides analytical tools for understanding radial migration and disk truncation.

Abstract

The accelerated Kepler problem is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem's Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the accelerated Kepler problem and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.