Superfluid density in disordered pasta phases in neutron star crusts

In the inner crust of neutron stars one expects phases in which nuclei adopt rodlike and platelike forms, so-called pasta phases. For ordered phases, the superfluid density of nucleons is anisotropic and in this paper we calculate the effective superfluid density of disordered pasta phases. We use an effective medium approach which parallels that previously used for calculating the electrical conductivity of terrestrial matter. We allow for the effect of entrainment, the fact that the current density of one species of nucleon depends on the gradient of the phase of the condensate pair wave function not only of the same species but also of the other species. We find that for protons, the results of the effective medium formalism can be quite different from those of simple approximations.

Figure 1

Effective superfluid density component $n_{\alpha \alpha }^{\mathrm{s},\mathrm{e}}$ in units of $n_{\alpha \alpha }^{\mathrm{s}\perp }$ as a function of $n_{\alpha \alpha }^{\mathrm{s}\parallel }/n_{\alpha \alpha }^{\mathrm{s}\perp }$ for the lasagna phase in the absence of entrainment. The full line is the result of the effective medium theory, the dashed line is the Voigt approximation, and the dotted line is the Reuss approximation.

Figure 2

Effective superfluid density component $n_{\alpha \alpha }^{\mathrm{s},\mathrm{e}}$ in units of $n_{\alpha \alpha }^{\mathrm{s}\parallel }$ as a function of $n_{\alpha \alpha }^{\mathrm{s}\perp }/n_{\alpha \alpha }^{\mathrm{s}\parallel }$ for the spaghetti phase in the absence of entrainment. As in Fig. 1, the full line shows the result of the effective medium theory, the dashed line the Voigt approximation, and the dotted line the Reuss approximation.