Caloric curve of star clusters

Self-gravitating systems, such as globular clusters or elliptical galaxies, are the prototypes of many-body systems with long-range interactions, and should be the natural arena in which to test theoretical predictions on the statistical behavior of long-range-interacting systems. Systems of classical self-gravitating particles can be studied with the standard tools of equilibrium statistical mechanics, provided the potential is regularized at small length scales and the system is confined in a box. The confinement condition looks rather unphysical in general, so that it is natural to ask whether what we learn with these studies is relevant to real self-gravitating systems. In order to provide an answer to this question, we consider a basic, simple, yet effective model of globular clusters: the King model. This model describes a self-consistently confined system, without the need of any external box, but the stationary state is a nonthermal one. In particular, we consider the King model with a short-distance cutoff on the interactions, and we discuss how such a cutoff affects the caloric curve, i.e., the relation between temperature and energy. We find that the cutoff stabilizes a low-energy phase, which is absent in the King model without cutoff; the caloric curve of the model with cutoff turns out to be very similar to that of previously studied confined and regularized models, but for the absence of a high-energy gaslike phase. We briefly discuss the possible phenomenological as well as theoretical implications of these results.

Figure 1

Caloric curve of the isothermal sphere. The part of the curve drawn as a red dashed line corresponds to unstable states.

Figure 2

Sketch of a typical caloric curve of a model of a regularized and confined self-gravitating system.

Figure 3

Caloric curve of the King model. The solid line is the $\vartheta =-\frac{2}{3}\varepsilon$ virial law and blue circles are the values of $\vartheta$ and $\varepsilon$ calculated using King distribution functions.

Figure 4

Central dimensionless temperature ${\vartheta }_0$ of the King model as a function of the dimensionless energy $\varepsilon$ (blue solid line), compared to the caloric curve $\vartheta \left(\varepsilon \right)$ (black dashed-dotted line) already reported in Fig. 3.

Figure 5

Caloric curve of the King model with a short-distance cutoff. The dimensionless squared cutoff length is ${\alpha }^2={10}^{-3}$.

Figure 6

As in Fig. 5, with ${\alpha }^2={10}^{-5}$.

Figure 7

Caloric curves of the King model with a short-distance cutoff (blue dashed line: ${\alpha }^2={10}^{-3}$; red dotted-dashed line: ${\alpha }^2={10}^{-5}$) compared with the caloric curve of the King model without cutoff (black solid line). The dotted line is the continuation of the virial law $\vartheta =-\frac{2}{3}\varepsilon$ for energies below ${\varepsilon }_{\text{min}}^{\text{King}}\simeq -2.13$.

Figure 8

Dimensionless density profile $\psi \left(x\right)$ as a function of the dimensionless radius $x$ for the King model with short-distance cutoff, with ${\alpha }^2={10}^{-3}$. Black solid curve: $\varepsilon =-1.27$, belonging to the energy range in common with the King model without cutoff; blue dashed curve: $\varepsilon =-3.91$, belonging to the intermediate energy region with negative specific heat; red dotted-dashed curve: $\varepsilon =-9.02$, belonging to the lowest energy region with positive specific heat. The dotted vertical line marks the cutoff length scale $\alpha$.

Figure 9

As in Fig. 8, with ${\alpha }^2={10}^{-5}$ and energies $\varepsilon =-1.37$ (black solid curve); $\varepsilon =-7.19$ (blue dashed curve); $\varepsilon =-96.4$ (red dotted-dashed curve).

Figure 10

Comparison between the caloric curve of the King model with a short-distance cutoff with ${\alpha }^2=7.5\times {10}^{-6}$ (blue dotted-dashed line) and the caloric curve of the King model without cutoff (black solid line). The dotted line is the continuation of the virial law $\vartheta =-\frac{2}{3}\varepsilon$ for energies below ${\varepsilon }_{\text{min}}^{\text{King}}\simeq -2.13$. A change in the slope of the caloric curve with cutoff is apparent for energies between -3 and -2.

Figure 11

Dimensionless energy $\varepsilon$ as a function of the control parameter $W_0$ (dimensionless central potential) for the King model with a short-distance cutoff. Blue dotted-dashed curve: ${\alpha }^2={10}^{-5}$; red solid curve: ${\alpha }^2=5\times {10}^{-6}$, values of $\varepsilon$ corresponding to converged (nonoscillating) solutions; red dashed and dotted curves: ${\alpha }^2=5\times {10}^{-6}$, values of $\varepsilon$ corresponding to upper and lower bounds of oscillating solutions.

Figure 12

Caloric curve of the King model with a short-distance cutoff, in an energy range similar to that of Fig. 10. The dimensionless squared cutoff length is ${\alpha }^2=5\times {10}^{-6}$. The solid lines are converged values, while the dotted lines refer to the lower and upper bounds of oscillating solutions. The upper (lower) part of the dotted line appears as the continuation of the rightmost (leftmost) solid line representing the high-energy (low-energy) converged values and is plotted in the same red (blue) color in the online version.

Figure 13

As in Fig. 8, with ${\alpha }^2=5\times {10}^{-6}$ and energies $\varepsilon =-1.37$ (black solid curve); $\varepsilon =-17.65$ (blue dashed curve); $\varepsilon =-136$ (red dotted-dashed curve).