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

Abstract

This is the second paper of a series of our works on the isotropic self-similar orbit-averaged Fokker-Planck (OAFP) equation and details physical properties of 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 thermodynamical aspects of the model based on the stellar DF obtained from the solution. Our calculation shows the following local properties (i) Equation of state in the core is local ideal gas p = 1.0ρ/χ esc where the p is the pressure, ρ density and χ esc the scaled escape energy, while it is polytropic p = 0.5ρ Γ /χ esc at large radii where Γ is the adiabatic index. (ii) If we consider the center is polytropic sphere, the polytropic index is 177. Also, as 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 negativity is directly related to the deep potential well or large scaled escape energy through the criterion condition φ = −6/χ esc where φ is the central mean field potential in a well-relaxed core. 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 stellarand heat-flow, rather than the isolation from surroundings due to self-gravity.

Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (iii) application of pre-collapse solution to Galactic globular clusters

“…The ss-OAFP model can model KM clusters since the model has flat (quasi-)isothermal core that can be directly inferred from our numerical result of (Ito, 2020b). There exists a similar structure in the cores between the ss-OAFP model and the isothermal sphere model as discussed in (Ito, 2020b) for the local properties. The ss-OAFP model has almost the same morphology in density profile as that of the isothermal sphere as shown in Figure 1.…”

“…Yet, it is not clear whether their DFs actually have reached a local Maxwellian DF considering that their core relaxation times t c.r. are long (1 Gyr) and even the self-similar solution to the OAFP equation can not strictly reach a Maxwellian DF as shown in (Ito, 2020b).…”

“…where the variables with subscript c corresponding to the time-dependent variables in self-similar analysis (Ito, 2020a) and the variables at a certain time t c of (Ito, 2020b). The boundary conditions for Poisson equation (2.4) are…”

“…This is the third paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation for isotropic dense star clusters in pre-collapse phase. The first paper (Ito, 2020a) showed an accurate Gauss-Chebyshev spectral solution of the equation and the second (Ito, 2020b) detailed the physical feature of the model focusing on the negative heat capacity of the core of the model. Based on their results, the present paper proposes a phenomenological model that reasonably fits to the projected structural profiles of Galactic globular clusters that include not only normal (King model) clusters but also core-collapsed (or core-collapsing) clusters with resolved cores.…”