Constraints on the equation of state from the stability condition of neutron stars

Abstract

The stellar equilibrium and collapse, including mainly white dwarfs, neutron stars and super massive stars, is an interplay between general relativistic effects and the equation of state of nuclear matter. In the present work, we use the Chandrasekhar criterion of stellar instability by employing a large number of realistic equations of state (EoSs) of neutron star matter. We mainly focus on the critical point of transition from stable to unstable configuration. This point corresponds to the maximum neutron star mass configuration. We calculate, in each case, the resulting compactness parameter, $\beta =GM/c^{2}R$, and the corresponding effective adiabatic index, $\gamma _{\mathrm{cr}}$. We find that there is a model-independent relation between $\gamma _{ \mathrm{cr}}$ and $\beta$. This statement is strongly supported by the large number of EoSs, and it is also corroborated by using analytical solutions of the Einstein field equations. In addition, we present and discuss the relation between the maximum rotation rate and the adiabatic index close to the instability limit. Accurate observational measurements of the upper bound of the neutron star mass and the corresponding radius, in correlation with present predictions, may help to impose constraints on the high density part of the neutron star equation of state.

Appendix

The numerical integration of the integrals related with the definition of $\langle \gamma \rangle$ and $\gamma _{\mathrm{cr}}$ can easily be performed. However, following this procedure it is difficult to perceive the final results. Actually, this is easy only in some approximated cases, e.g. in the Newtonian and post-Newtonian limit. In the following, we try to generalize the finding of Chandrasekhar (1964a) to even higher values of the compactness where the relativistic effects become important. The expression of the critical adiabatic index, with the help of the TOV equations (2), (3), using the trial function $\xi (r)=r e^{\nu /2}$ (in order to be consistent with the pioneering work of Chandrasekhar (1964a)), and, performing a Taylor expansion inside the integrals in each case, we found for the uniform and the Tolman VII solution that (see also Merafina and Ruffini 1989)

$$ \gamma _{\mathrm{cr}}(\beta )=\frac{4}{3}+\frac{38}{42}\beta { \mathcal{P}}( \beta ) $$

(21)

where, for the uniform solution, $\mathcal{P}(\beta )$ takes the form

$$ {\mathcal{P}}_{\mathrm{uniform}}(\beta )=1+2.13\beta +4.65\beta ^{2}+10.22 \beta ^{3}+\mathcal{O}\bigl(\beta ^{4}\bigr) $$

(22)

and for the Tolman VII solution:

$$ {\mathcal{P}}_{\mathrm{Tolman}}(\beta )=1.19+2.93\beta +7.34\beta ^{2}+19.36 \beta ^{3}+\mathcal{O}\bigl(\beta ^{4}\bigr) . $$

(23)

Obviously, the approximation (21), using Eq. (22), to a linear term, confirms the Chandrasekhar expression (12). The above expressions are good approximations for $\beta <0.2$. However, they fail for higher values of $\beta$ and consequently additional terms must be included. In particular, $\gamma _{\mathrm{cr}}$ increases very fast for $\beta >0.25$ due to the strong effects of general relativity.