Dynamical Stability of Cusps and Cores: Implication to the Centers of Galaxies and Cluster of Galaxies

We study the stability of a spherically symmetric density profile. We analyze the case of a collisionless system with a power-law profile given by rho propto r^-alpha, in the Newtonian regime using the Jeans equation. The Jeans equation is mathematically identical to the equation of hydrostatic equilibrium as long as p_J = rho sigma^2 is regarded as the pressure counterpart for ordinary gas, where sigma is the local velocity dispersion. Since a self-gravitating collisionless system is at best metastable, if it is treated statistical mechanically, the dynamical equilibrium described by the Jeans equation will have a shorter life time than the thermodynamically stable equilibrium for ordinary gas. Here we take a heuristic approach, using the pressure counterpart, p_J. From the requirement that p_J > 0 or the normalcy of the formal EOS, p_J = p_J(rho) obtained by eliminating r, we find the following heuristic stability criterion. For 0 < alpha < 1, p_J < 0, the density profile is unstable. We interpret this profile to be a transient one evolving toward a flat-top profile, which seems to be the final product with absolute stability. For 1<alpha <3, the profile is dynamically stable. It is an interesting coincidence that dark matter halo profiles obtained by cosmological N-body simulations, fall into this category. For 3<alpha, p_J<0. We suspect that such a steep profile is in the process of collapsing toward r=0.