Exactly solved model of a one dimensional self gravitating system

TL;DR

This paper presents an exactly solvable one-dimensional self-gravitating system with particles of equal energy, providing explicit analytic expressions for particle orbits, potential, and density, which could inform dark matter and stellar dynamics studies.

Contribution

It introduces a novel exactly solvable model of a self-gravitating system with explicit solutions for particle trajectories, potential, and density.

Findings

Particle orbits are given by truncated sine and cosine functions.

Potential and density have simple analytic expressions in terms of time.

The model offers insights into caustic behavior in dark matter simulations.

Abstract

A model one-dimensional self consistent steady state collisionless self-gravitating system in which all the particles have the same energy is presented. This has the remarkable property that the position and velocity of the particles orbiting in their own self consistent potential are given exactly, in terms of time, by the truncations of sine and cosine functions to the first two terms in their respective Taylor series. The potential and density also have simple analytic expressions in terms of time as parameter. It is not being claimed that this system has any direct astronomical application. However, it does motivate a conjecture about the behaviour of the density, potential, and orbits near caustics in simulations of cold collisionless dark matter. It is a rather surprising result which might interest practitioners of stellar dynamics and serve as an elementary example in teaching the subject.