Exploring the dynamics of collisionless spherical stellar systems using the matrix method: Insights from the dilation mode

¹ Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr 12-14, 69120 Heidelberg, Germany
e-mail: epolyachenko@uni-heidelberg.de
² Institute of Astronomy Russian Academy of Sciences, 48 Pyatnitskaya st., 119017 Moscow, Russia
³ Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Branch, PO Box 291 Irkutsk 664033, Russia
e-mail: shukhman@iszf.irk.ru

Received: 10 November 2023
Accepted: 18 December 2023

Abstract

Context. Analytical solutions to the perturbed equations that govern self-gravitating collisionless stellar systems are crucial for both code testing and theoretical insights. For spheres, a solution has been known for years that corresponds to the entire object’s shift from the origin. We recently introduced a new exact stationary solution, relevant for models with a single length parameter. This solution, referred to as the scale-invariant or dilation mode, has led to insights regarding the concept of perturbation energy within the linear theory framework.

Aims. Our aim is to use Hénon’s isochrone model as an example to verify the ability of the standard matrix method to successfully predict the existence of a dilation mode, and to explore its potential application as a test disturbance.

Methods. We used the standard matrix method for radial perturbations and applied Clutton-Brock potential-density pairs to determine the properties of the perturbations.

Results. In this particular case of stationary radial perturbations, the typical relationship between the perturbations of the distribution function and the potential fails. This discrepancy poses a challenge when attempting to use the dilation mode as a test. When using Clutton-Brock pairs with the matrix method, a mass conservation equation as an additional equation to the ordinary set of linear equations is required. With this added equation, it’s possible to obtain the needed test: identical vanishing of the determinant of this modified set of equations with an increasing number of included basis functions.