Stellar Stability by Thermodynamic Instability

For gravity-dominated systems the three features $\mathrm{\text{shrinking}}\iff \mathrm{\text{energy decrease}}\iff \mathrm{\text{temperature increase}}$ are dynamically linked together. So are their inverses: $\mathrm{\text{expansion}}\iff \mathrm{\text{energy increase}}\iff$temperature decrease. We exhibit these features by one classical particle in a suitable environment, and by many particles with purely attractive interactions. We then show how the ensuing negative heat capacity tames an explosive energy input.

Figure 1

Geometry and short trajectory for the jumping-board model. The gravitational force points into the negative $y$ direction, and the particle is elastically reflected at the bottom, which is given by $V(y_{\max })$ as specified in the main text. Since the particle is also reflected at the line $x=0$, the spatial domain may be restricted to positive $x$. The energy, $E=1$ for this figure, corresponds to the maximum height, $y_{\max }$, the jumping particle may reach, if both momentum components $p_x$ and $p_y$ vanish.

Figure 2

Dependence of the temperature on the total energy for the mechanical jumping-board model. Simulation results (points) are compared to the theoretical expression (solid line). The number of computed jumps is gradually increased from $4\times {10}^6$, for $E=0.8$, to $40\times {10}^6$, for $E=2.8$.

Figure 3

Negative Gauss potential: typical particle configuration at a time $t_0=3000$. The particles are represented by circles with a diameter equal to unity.

Figure 4

Time evolution for the temperature $T_c$ of the largest cluster. Line $A$ belongs to the equilibration step, line $B$ to the heating step, and line $C$ to the final heating-cooling step.

Figure 5

Time evolution for the energy $E$ of the system. The lines $A$, $B$, and $C$ refer to the respective equilibration, heating, and heating-cooling steps mentioned in the main text.