Effects of the equation of state on the core-crust interface of slowly rotating neutron stars

We systematically study the symmetry energy effects of the transition density $n_{\mathrm{t}}$ and the transition pressure $P_{\mathrm{t}}$ around the crust-core interface of a neutron star in the framework of the dynamical and the thermodynamical methods respectively. We employ both the parabolic approximation and the full expansion, for the definition of the symmetry energy. We use various theoretical nuclear models, which are suitable for reproducing the bulk properties of nuclear matter at low densities, close to saturation density as well as the maximum observational neutron star mass. First we derive and present an approximation for the transition pressure $P_{\mathrm{t}}$ and crustal mass $M_{\mathrm{crust}}$. Moreover, we derive a model-independent correlation between $P_{\mathrm{t}}$ and the slope parameter $L$ for a fixed value of the symmetry energy at the saturation density. Second, we explore the effects of the equation of state on a few astrophysical applications which are sensitive to the values of $n_{\mathrm{t}}$ and $P_{\mathrm{t}}$ including neutron star oscillation frequencies, thermal relaxation of the crust, crustal fraction of the moment of inertia, and the $r$-mode instability window of a rotating neutron star. In particular, we employ the Tolman VII solution of the TOV equations to derive analytical expressions for the critical frequencies and the relative time scales, for the $r$-mode instability, in comparison with the numerical predictions. In the majority of the applications, we found that the above quantities are sensitive mainly to the applied approximation for the symmetry energy (confirming previous results). There is also a dependence on the used method (dynamical or thermodynamical). The above findings lead us to claim that the determination of $n_{\mathrm{t}}$ and $P_{\mathrm{t}}$ must be reliable and accurate before they are used to constrain relevant neutron star properties.

Figure 1

The thickness parameter $t_{90-10}$ (in fm) and the surface tension ${\sigma }_{\mathrm{snm}}$ (in $\mathrm{MeV}/{\mathrm{fm}}^2$) as a function of the parameter $D$ for the employed models.

Figure 2

The mass as a function of the radii for the MDI model (for $L=80$ MeV) including constraints from neutron star seismology according to Eqs. (63) and (64) and for the four selected cases. The straight lines correspond to the equation $\beta =\beta \left(h_{\mathrm{t}}\right)$, which emerges from the elimination of $R$ in Eqs. (63) and (64), for each of the four cases.

Figure 3

(a) The thermal relaxation time $t_w$ as a function of the total mass for the four selected cases. (b) Constraints on the $M\text{−}R$ diagram form the thermal relaxation time $t_w$ for the four selected cases. The $M\text{−}R$ dependence for the MDI model ($L=80$ MeV) has been included also for comparison. For more details see text.

Figure 4

(a) The crustal fraction of the moment of the inertia as a function of mass presented for the four selected cases. For comparison we include the horizontal lines, each one representing a possible $I_{\mathrm{crust}}/I$ constraint, deduced for the Vela pulsar (assuming a mass $M=1.4M_{\odot }$). (b) The mass-radius diagram for various nuclear EoS and the constraints $I_{\mathrm{crust}}/I=0.014$ are derived from the four selected cases.

Figure 5

The critical frequency-temperature dependence for a neutron star with mass $M=1.4M_{\odot }$ (a) and $M=1.8M_{\odot }$ (b) constructed for the selected EoSs. The observed cases of LMXBs and MSRPs from Haskell et al. [88] are also included for a comparison. The cases IGR J00291+5934, XTE J1751-305, and SAX J1808-3658 with well-known observation spin-down rate are also indicated.