Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (ii) physical properties and negative heat capacity of pre-collapse core

TL;DR

This paper analyzes the physical properties and thermodynamics of pre-collapse stellar clusters using a self-similar orbit-averaged Fokker-Planck model, revealing negative heat capacity driven by collisionless stellar dynamics.

Contribution

It provides a detailed thermodynamic analysis of the pre-collapse core using the self-similar Fokker-Planck equation, highlighting the origin of negative heat capacity.

Findings

Core equation of state: p=ρ/χ_esc

Polytropic index of core: 177

Negative heat capacity caused by collisionless stellar dynamics

Abstract

This is the second paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation and details physical properties of isotropic pre-collapse solution. The fundamental core collapse process at the late stage of relaxation evolution of spherical star clusters can be described by the self-similar OAFP equation. The accurate spectral solution was found recently in the first paper. The present work details the thermodynamic aspects of the model based on the stellar DF obtained from the solution. Our calculation shows the following local properties (i) Equation of state is $p=1.0\rho/\chi_\text{esc}$ in the core where $p$ is the pressure, $\rho$ the density and $\chi_\text{esc}$ the scaled escape energy, while it is $p=0.5\rho^{1.1}/\chi_\text{esc}$ at large radii. (ii) If we consider the center of the core is polytropic, the polytropic index is 177. Also, as a global property we construct caloric curves of the model to discuss the heat capacity together with Virial. Special focus is the cause of negative heat capacity of the core; the well-relaxed core can be directly compared to the isothermal sphere with positive heat capacity. Comparing our results to the previous works, we conclude, in the self-similar evolution, the negative heat capacity in the core holds due to collisionless and high-temperature stars that experience a rapid change in mean field potential through stellar- and heat- flows, rather than due to the (quasi-)isolation of the core from surroundings.