Thermal conductivity due to phonons in the core of superfluid neutron stars

We compute the contribution of phonons to the thermal conductivity in the core of superfluid neutron stars. We use effective field theory techniques to extract the phonon scattering rates, written as a function of the equation of state of the system. We also calculate the phonon dispersion law beyond linear order, which depends on the gap of superfluid neutron matter. With all these ingredients, we solve the Boltzmann equation numerically using a variational approach. We find that the thermal conductivity $\kappa$ is dominated by combined small- and large-angle binary collisions. As in the color-flavor-locked superfluid, we find that our result can be well approximated by $\kappa \propto 1/{\Delta }^6$ at low temperature, where $\Delta$ is the neutron gap, the constant of proportionality depending on the density. We further comment on the possible relevance of electron and superfluid phonon collisions in obtaining the total contribution to the thermal conductivity in the core of superfluid neutron stars.

Figure 1

One-loop Feynman diagrams needed for the computation of the phonon dispersion law, corresponding to (a) diagonal and (b) off-diagonal terms in the Nambu-Gorkov basis.

Figure 2

Feynman diagrams contributing to $2\leftrightarrow 2$ phonon scattering processes.

Figure 3

The ${}^1{S}_0$ and averaged ${}^3{P}_2$ neutron gaps as a function of the nucleon particle density in units of saturation density, $n_0=0.16{\mathrm{fm}}^{-3}$. We use two very different gap models as a function of the density to illustrate the gap dependence of our results. In the first scheme we consider the ${}^1{S}_0\left(A\right)+{}^3{P}_2\left(i\right)$ model, where the ${}^1{S}_0$ neutron gap is calculated in the BCS approach using different bare nucleon–nucleon interactions that converge toward a maximum neutron gap of about 3 MeV at $p_F\approx 0.85{\mathrm{fm}}^{-1}$ (parametrization $A$ of Table I in Ref. [14]) while for ${}^3{P}_2$ we have taken the parametrization $i$ (strong neutron superfluidity in the core). The second model is the ${}^1{S}_0\left(c\right)+{}^3{P}_2\left(k\right)$ model, where for the ${}^1{S}_0$ channel the calculation of the gap goes beyond the BCS theory and in the ${}^3{P}_2$ channel we have taken into account a parametrization assuming weak neutron superfluidity in the core with maximum values for the gap of the order of 0.1 MeV [14].

Figure 4

Variational calculation for the thermal conductivity due to superfluid phonons up to order $N=6$ for nuclear matter saturation density, $n_0$, as a function of temperature. The end temperature is the critical temperature, $T_c=0.57\Delta \left(n_0\right)=3.4\times {10}^9$ K. The calculation is performed for the ${}^1{S}_0\left(A\right)+{}^3{P}_2\left(i\right)$ model for the gap.

Figure 5

Phonon contribution to the thermal conductivity using the ${}^1{S}_0\left(A\right)+{}^3{P}_2\left(i\right)$ model for the gap as a function of temperature for different particle densities in units of normal saturation density $n_0$: (a) for $0.5n_0,T_{c,0.5n_0}=5.8\times {10}^9$ K, (b) for $n_0,T_{c,n_0}=3.4\times {10}^9$ K, and (c) for $2n_0,T_{c,2n_0}=6.3\times {10}^9$ K. The end temperature for each density is the critical temperature, $T_c=0.57\Delta$.

Figure 6

The same as in Fig. 5, but using the ${}^1{S}_0\left(c\right)+{}^3{P}_2\left(k\right)$ model for the gap. In this case, the different particle densities in units of normal saturation density $n_0$ and critical temperatures are (a) $0.5n_0$ with $T_{c,0.5n_0}=1.6\times {10}^8$ K, (b) $n_0$ with $T_{c,n_0}=5.8\times {10}^8$ K, and (c) $2n_0$ with $T_{c,2n_0}=5.8\times {10}^8$ K.

Figure 7

Mean-free path due to the phonon thermal conductivity using the ${}^1{S}_0\left(A\right)+{}^3{P}_2\left(i\right)$ model for the gap as a function of temperature for three different densities in units of $n_0$: (a) for $0.5n_0,T_{c,0.5n_0}=5.8\times {10}^9$ K, (b) for $n_0,T_{c,n_0}=3.4\times {10}^9$ K, and (c) for $2n_0,T_{c,2n_0}=6.3\times {10}^9$ K. The end temperature for each density is the critical temperature, $T_c=0.57\Delta$. The mean-free path has to be compared with the radius of the star, which we take as $R=10$ km.