We develop a simple analytic two-component model that links the thermal evolution of self-gravitating systems to black-hole thermodynamics. The model consists of a monatomic nonrelativistic ideal gas and an effective barotropic support calibrated by the specific heat $\hat c_V^{\rm J}(\Theta)$ of a J"uttner gas. From the partition function we obtain an exact modified Bessel expression for $\hat c_V^{\rm J}(\Theta)$ and supply compact Pad'e approximants with good accuracy across all temperatures. These inputs yield a closed virial-capacity formula, which we use to determine where the heat capacity changes sign as the support pressure becomes competitive with the thermal pressure. We then contrast this behavior with Schwarzschild anti-de Sitter black holes. Despite the disparate microphysics, both systems exhibit a switch in heat-capacity behavior governed by competing pressure contributions. The surrogate model provides compact analytic insight into stability thresholds in self-gravitating systems and their black-hole analogues.