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Self-gravitating clusters of Fermi-Dirac gas with planar, cylindrical, or spherical symmetry: evolution of density profiles with temperature

We calculate density profiles for self-gravitating clusters of an ideal Fermi-Dirac gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions $\mathcal{D}=1,2,3$, clusters with cylindrical symmetry in $\mathcal{D}=2,3$, and clusters with spherical symmetry in $\mathcal{D}=3$. Wall confinement is imposed where needed for stability against escape. The length scale and energy scale in use render all results independent of total mass and prove adequate at all temperatures. We present exact analytic expressions for (fully degenerate) $T=0$ density profiles in four of the six combinations of symmetry and dimensionality. Our numerical results for $T>0$ describe the emergence, upon quasistatic cooling, of a core with incipient degeneracy surrounded by a more dilute halo. The equilibrium macrostates are found to depend more strongly on the cluster symmetry than on the space dimensionality. We demonstrate the mechanical and thermal stability of spherical clusters with coexisting phases.