# How gravitational softening affects galaxy stability I. Linear mode analysis of disc galaxies

## TL;DR

This study examines how different gravitational softening kernels influence the accuracy of linear mode analysis in disc galaxy simulations, highlighting the importance of minimizing gravity bias for precise eigenmode frequency estimation.

## Contribution

It introduces and tests softening kernels with reduced gravity bias, demonstrating improved accuracy in galaxy stability mode analysis compared to traditional kernels like Plummer.

## Key findings

- Softening kernels with minimal gravity bias yield more accurate mode frequencies.
- Plummer softening performs poorly in reproducing unsoftened mode properties.
- Super-Newtonian forces near the softening length improve mode analysis accuracy.

## Abstract

Linear perturbation is used to investigate the effect of gravitational softening on the retrieved two-armed spiral eigenmodes of razor-thin stellar discs. We explore four softening kernels with different degrees of gravity bias, and with/without compact support (compact in the sense that they yield exactly Newtonian forces outside the softening kernel). These kernels are applied to two disc galaxy models with well-known unsoftened unstable modes. We illustrate quantitatively the importance of a vanishing linear gravity bias to yield accurate frequency estimates of the unstable modes. As such, Plummer softening, while very popular amongst simulators, performs poorly in our tests. The best results, with excellent agreement between the softened and unsoftened mode properties, are obtained with softening kernels that have a reduced gravity bias, obtained by compensating for the sub-Newtonian forces at small interparticle distances with slightly super-Newtonian forces at radii near the softening length. We present examples of such kernels that, moreover, are analytically simple and computationally cheap. Finally, these results light the way to the construction of softening methods with even smaller gravity bias, although at the price of increasingly complex kernels.