# Effect of viscosity and thermal conductivity on the radial oscillation and relaxation of relativistic stars

## TL;DR

This paper develops a linearized model for small radial oscillations in relativistic stars, incorporating viscosity and thermal conductivity effects, to analyze eigenfrequencies and relaxation times relevant for neutron star pulsations.

## Contribution

It introduces a generic formulation of the dynamical equations including viscosity and thermal conductivity effects, and applies a variational principle to determine eigenfunctions and eigenvalues for neutron star oscillations.

## Key findings

- Eigenvalues give squared frequencies and relaxation times.
- Framework allows integration with various nuclear matter models.
- Qualitative estimates of energy dissipation rates are provided.

## Abstract

In this paper we present a generic formulation of the linearized dynamical equations governing small adiabatic radial oscillations of relativistic stars. The dynamical equations are derived by taking into consideration those effects of viscosity and thermal conductivity of neutron-star matter which directly determine the minimum period of observable pulsars. A variational principle is applied to determine a discrete set of eigenfunctions with complex eigenvalues. The real and imaginary parts of eigenvalues represent the squared natural frequencies and relaxation time of radial oscillations of non-rotating neutron stars, respectively. We provide a suitable framework which may be supplemented with various potential species of cold-nuclear-matter models to compute the spectra of the normalized eigenfrequencies with a certain numerical precision. In the last section, we provide a qualitative estimation of the rate at which viscosity and thermal conductivity drain the kinetic energy of radial oscillation mode in reasonably uniform neutron stars, without relying on explicit numerical computations.