Heat capacity of the neutron star inner crust within an extended nuclear statistical equilibrium model

Background: Superfluidity in the crust is a key ingredient for the cooling properties of proto-neutron stars. Present theoretical calculations employ the quasiparticle mean-field Hartree-Fock-Bogoliubov theory with temperature-dependent occupation numbers for the quasiparticle states.

Purpose: Finite temperature stellar matter is characterized by a whole distribution of different nuclear species. We want to assess the importance of this distribution on the calculation of heat capacity in the inner crust.

Method: Following a recent work, the Wigner-Seitz cell is mapped into a model with cluster degrees of freedom. The finite temperature distribution is then given by a statistical collection of Wigner-Seitz cells. We additionally introduce pairing correlations in the local density BCS approximation both in the homogeneous unbound neutron component, and in the interface region between clusters and neutrons.

Results: The heat capacity is calculated in the different baryonic density conditions corresponding to the inner crust, and in a temperature range varying from 100 KeV to 2 MeV. We show that accounting for the cluster distribution has a small effect at intermediate densities, but it considerably affects the heat capacity both close to the outer crust and close to the core. We additionally show that it is very important to consider the temperature evolution of the proton fraction for a quantitatively reliable estimation of the heat capacity.

Conclusions: We present the first modelization of stellar matter containing at the same time a statistical distribution of clusters at finite temperature, and pairing correlations in the unbound neutron component. The effect of the nuclear distribution on the superfluid properties can be easily added in future calculations of the neutron star cooling curves. A strong influence of resonance population on the heat capacity at high temperature is observed, which deserves to be further studied within more microscopic calculations.

Figure 1

${}^1{S}_0$ pairing gap as a function of density for homogeneous neutron matter at zero temperature as obtained from Brueckner-Hartree Fock calculations (full line from [28]). The figure also shows the energy gap deduced solving the BCS gap equations at finite temperature (symbols from [26]).

Figure 2

Temperature evolution of the global proton fraction obtained by imposing the neutrinoless $\beta$-equilibrium condition (43) for four representative cells.

Figure 3

Temperature evolution of the unbound gas component in the same representative cells as in Fig. 2. (Full line) Complete NSE calculation. (Dashed line) The value of the global proton fraction is assumed equal to the one calculated from $\beta$ equilibrium at the lowest temperature, $y_p\left(T\right)=y_p\left(0.1\mathrm{MeV}\right)$. (Dotted line) As the full line, but neglecting the pairing interaction.

Figure 4

Normalized cluster size distribution at different temperatures in the same four representative cells as in Fig. 2 and at $\beta$ equilibrium.

Figure 5

Normalized isotopic cluster distribution in the same four representative cells as in Fig. 2, as obtained at the highest temperature considered, $T=2$ MeV. The different symbols indicate results at $\beta$ equilibrium (triangle) or assuming a global proton fraction equal to that one corresponding to $\beta$ equilibrium but at the lowest temperature, $y_p\left(T\right)=y_p\left(0.1\mathrm{MeV}\right)$.

Figure 6

Temperature evolution of the baryonic energy density for the same representative cells as in Fig. 2. (Full line) Complete NSE calculation. (Dashed line) As the full line, but the value of the global proton fraction is assumed equal to that one calculated from $\beta$ equilibrium at the lowest temperature, $y_p\left(T\right)=y_p\left(0.1\mathrm{MeV}\right)$. (Lines with symbols) As the full line, but the NSE distribution is replaced with the most probable Wigner-Seitz cell.

Figure 7

Temperature evolution of the heat capacity for the same representative cells as in Fig. 2. (Full line) Complete NSE calculation. (Dashed line) As the full line, but the value of the global proton fraction is assumed equal to that one calculated from $\beta$ equilibrium at the lowest temperature, $y_p\left(T\right)=y_p\left(0.1\mathrm{MeV}\right)$.

Figure 8

Temperature evolution of heat capacity for Cell 6, obtained considering a pairing interaction with the same parameters of Ref. [5]. (Full line) Complete NSE calculation. (Dashed line) As the full line, but the value of the global proton fraction is assumed equal to that one given in Ref. [5] (see also Table 1). The inset shows a zoom at low temperature to facilitate the comparison with the results of Ref. [5].

Figure 9

Temperature evolution of heat capacity for four intermediate cells. (Full lines) Complete NSE calculation making use of the analytical expressions, Eq. (26), for binding energies (labeled as LDM in the figure). (Line with symbols) As the full line, but experimental binding energies, from [38], are used whenever available.

Figure 10

Normalized cluster size distribution at different temperatures, as obtained from complete NSE calculation using experimental binding energies, from [38], whenever available. Results are shown for two representative cells where the transition to a dominance of light clusters is observed in the range of temperatures considered.

Figure 11

Temperature evolution of heat capacity for the same two cells as in Fig. 10. (Full line with symbols) Complete NSE calculation using experimental binding energies, from [38], whenever available. (Dashed line) As the full line but neglecting in-medium effects.