Cored DARKexp systems with finite size: numerical results

In the DARKexp framework for collisionless isotropic relaxation of self–gravitating matter, the central object is the differential energy distribution $n(E)$, which takes a maximum–entropy form proportional to $\exp [-\beta (E-\mathrm{\Phi }(0))]-1$, $\mathrm{\Phi }(0)$ being the depth of the potential well and $\beta$ the standard Lagrange multiplier. Then the first and quite non–trivial problem consists in the determination of an ergodic phase–space distribution which reproduces this $n(E)$. In this work we present a very extensive and accurate numerical solution of such DARKexp problem for systems with cored mass density and finite size. This solution holds throughout the energy interval $\mathrm{\Phi }(0)\le E\le 0$ and is double–valued for a certain interval of $\beta$. The size of the system represents a unique identifier for each member of this solution family and diverges as $\beta$ approaches a specific value. In this limit, the tail of the mass density $\rho (r)$ dies off as $r^{-4}$, while at small radii it always starts off linearly in $r$, that is $\rho (r)-\rho (0)\propto r$.