Stability bounds on compact astrophysical objects from information-entropic measure

We obtain bounds on the stability of various self-gravitating astrophysical objects using a new measure of shape complexity known as configurational entropy. We apply the method to Newtonian polytropes, neutron stars with an Oppenheimer-Volkoff equation of state, and to self-gravitating configurations of complex scalar field (boson stars) with different self couplings, showing that the critical stability region of these stellar configurations obtained from traditional perturbation methods correlates well with critical points of the configurational entropy with accuracy of a few percent or better.

Figure 1

Normalized modal fraction $\tilde{f}(|\mathbf{k}|)$ for sample values of the polytropic index $\gamma$. From left to right, $\gamma =1.2$, 1.4, and 1.7.

Figure 2

Configurational entropy times ${\rho }_0^{-1}$ (continuous line) and mass (dotted line) versus polytropic index $\gamma$ for ${\rho }_0={\rho }_c$.

Figure 3

Contour plot for the mass of polytropes as a function of the central density ${\rho }_0/{\rho }_c$ and polytropic index $\gamma$. There is an instability ridge—a saddle line—for $\gamma =4/3$. The vertical bar specifies the values in units given at its bottom.

Figure 4

Contour plot for the configurational entropy of polytropes as a function of the central density ${\rho }_0/{\rho }_c$ and polytropic index $\gamma$. There is an instability ridge—a saddle line—for $\gamma =4/3$. The vertical bar specifies the values in units given at its bottom.

Figure 5

Configurational entropy versus polytropic index $\gamma$ for polytropes. We display results for several choices of cutoff for $k_{\min }$.

Figure 6

OV neutron star mass vs central density ${\rho }_0$. Stars with ${\rho }_0>{\rho }_c=0.588b{a}^{-3}$ are known to be unstable to gravitational collapse.

Figure 7

Configurational entropy times ${\rho }_0^{-1}$ for the OV neutron star vs central density ${\rho }_0$. Stars with ${\rho }_0>{\rho }_c=0.588b{a}^{-3}$ are perturbatively unstable to gravitational collapse. The inset shows the result near the CE minimum.

Figure 8

Behavior of the quantity $S{\rho }_0^{-1}M$ as a function of central density ${\rho }_0$ for OV neutron stars. The approximate linear scaling persists for a wide range of central densities. The inset shows the result near ${\rho }_c=0.588b{a}^{-3}$, the critical value for stability.

Figure 9

Boson star mass (continuous line) and conserved charge $Q$ (dot-dashed line) vs central value of the scalar field ${\sigma }_0$. We also show the binding energy $E_b$ (dotted line). Stars with ${\sigma }_0>{\sigma }_c=0.271$ are known to be unstable to gravitational collapse.

Figure 10

The configurational entropy for boson stars multiplied by inverse central density $S{\rho }_0^{-1}$ as a function of the scalar field’s central value ${\sigma }_0$ for different values of the scalar field coupling $\mathrm{\Lambda }$. The dashed vertical line denotes ${\sigma }_c$, the instability boundary for the star under radial perturbations. As in the case with neutron stars, the CE provides a reliable estimate for the critical mass, with precision better than $\sim 1\%$. Values of maximum stellar masses for different values of $\mathrm{\Lambda }$ are listed in Table 1.

Figure 11

The inset shows the quantity $S{\varphi }_0^{-4}M$ as a function of the scalar field’s central value ${\sigma }_0$ for $\mathrm{\Lambda }=0$. The main plot shows the second derivative of $S{\varphi }_0^{-4}M$ with respect to ${\sigma }_0$, displaying the same saddle ridge behavior found for polytropes.

Figure 12

The inset shows the quantity $S{\varphi }_0^{-4}M$ as a function of the scalar field’s central value ${\sigma }_0$ for $\mathrm{\Lambda }=100$. The main plot shows the second derivative of $S{\varphi }_0^{-4}M$ with respect to ${\sigma }_0$, displaying the same saddle ridge behavior found for polytropes.