Asymptotic model for a bi-component system with a very large mass ratio mh/ml of their constituting particles

Abstract

We address the asymptotic limit of the γ-exponential model for a bi-component system with a very large ratio mh/ml between the masses of their constituting particles. The simplest model that captures this limit situation is an isothermal gas of non-relativistic point particles under an external harmonic potential $V\left(\mathbf {r}\right)=m\, \omega ^{2}_{0}\, \mathbf {r}^{2}/2$ that mimics the gravitational influence of the lightest particles in the central regions. We revisits the study of this self-gravitating model by including the anisotropy due to the system rotation. The thermodynamics of this situation is hallmarked by the gravitational radius$r_{c}=\left(G\, M/\omega ^{2}_{0}\right)^{1/3}$ and finite values of the rotation frequency ω < ω0, where M is the system total mass. The axially symmetric solutions of this model predicts spheroidal profiles with any value of the angular momentum, whose anisotropy is driven via the dimensionless parameter $\varepsilon =\omega ^{2}/\omega ^{2}_{0}$. In general, the thermodynamics is significantly affected by the system rotation. Considering the limit ϵ → 1, in particular, our calculations predict that the temperature at gravothermal collapse goes to zero and the main components of the inertia tensor diverge. Finally, we apply the present model to perform an analysis of stars distributions under the gravitational influence of dark matter haloes in some dwarf spheroidal galaxies. Our interest is focused on the applicability conditions of the present model to these stellar systems.