Transport coefficients of leptons in superconducting neutron star cores

I consider the thermal conductivity and shear viscosity of leptons (electrons and muons) in the nucleon neutron star cores where protons are in the superconducting state. I restrict the consideration to the case of not too high temperatures $T\lesssim 0.35T_{\mathrm{c}p}$, where $T_{\mathrm{c}p}$ is the critical temperature of the proton pairing. In this case, lepton collisions with protons can be neglected. Charged lepton collision frequencies are mainly determined by the transverse plasmon exchange and are mediated by the character of the transverse plasma screening. In our previous works [Shternin and Yakovlev, Phys. Rev. D 75, 103004 (2007); Phys. Rev. D 78, 063006 (2008)] the superconducting proton contribution to the transverse screening was considered in the Pippard limit $\mathrm{\Delta }\ll \hslash q{v}_{\mathrm{F}p}$, where $\mathrm{\Delta }$ is the proton pairing gap, $v_{\mathrm{F}p}$ is the proton Fermi velocity, and $\hslash q$ is the typical transferred momentum in collisions. However, for large critical temperatures (large $\mathrm{\Delta }$) and relatively small densities (small $q$) the Pippard limit may become invalid. In the present study I show that this is indeed the case and that the older calculations severely underestimated the screening in a certain range of the parameters appropriate to the neutron star cores. As a consequence, the kinetic coefficients at $T\ll T_{\mathrm{c}p}$ are found to be smaller than in previous calculations.

Figure 1

Proton contribution to the zero-temperature transverse polarization function in the static limit. The polarization function is normalized to $q_M^2$. Dashed line shows the asymptote in the Pippard limit.

Figure 2

Ratios of characteristic screening momenta to $2p_{\mathrm{F}e}$ versus baryon density (in units of the nuclear saturation density $n_0=0.16{\mathrm{fm}}^{-3}$) for two EOSs desribed in the text. Thick lines correspond to the HHJ EOS, while thin lines show the results for the BSk21 EOS. Solid lines show the longitudinal (Thomas-Fermi) screening momentum $q_{\mathrm{TF}}$, dashed lines give the Meissner momentum $q_M$. Dash-dotted and double-dot-dashed lines show the characteristic screening momenta in Pippard limit for $T_{\mathrm{c}p}={10}^9\mathrm{K}$ and $10^{10}\mathrm{K}$, respectively.

Figure 3

The coherence scale parameter $A$ as a function of the total baryon density for two selected EOSs. Effective masses are set to $m_p^{*}=0.8m_u$ and $\mathrm{\Delta }=1\mathrm{MeV}$.

Figure 4

The ratio $R(A)$ for $n=0$ (solid lines) and $n=2$ (dashed lines). Thin lines represent the low-$A$ approximation (25).

Figure 5

The ratios $R$ for (a) thermal conductivity ($n=0$) and (b) shear viscosity ($n=2$) as function of $n_B$ for the HHJ equation of state and three values of $T_{\mathrm{c}p}={10}^{10}\mathrm{K}$ (solid lines), $3\times {10}^9\mathrm{K}$ (dashed lines), and $10^9\mathrm{K}$ (dash-dotted lines). Thin lines give a low-$A$ approximation, where the interaction is screened by the pure Meissner mass. Double-dot-dashed lines show with right vertical scales the combination $A{T}_{\mathrm{c}p,9}$ which is independent of $T_{\mathrm{c}p}$. Here $T_{\mathrm{c}p,9}\equiv T_{\mathrm{c}p}/({10}^9\mathrm{K})$.

Figure 6

Comparison of the thermal conductivity and shear viscosity in the leading approximation, Eqs. (23) and (24), to the results of complete calculations. Upper pair of curves shows the ratio ${\eta }_{e\mu }^{t,\text{Lead}}/{\eta }_{e\mu }$, while lower pair of curves marked $\kappa$ shows ${\kappa }_{e\mu }^{t,\text{Lead}}/{\kappa }_{e\mu }$. Results for the HHJ EOS are shown and for two values of the proton critical temperature. $T_{\mathrm{c}p}={10}^{10}\mathrm{K}$ (solid lines) and $10^9\mathrm{K}$ (dashed lines). Variational solutions are employed. See text for details.

Figure 7

Lepton thermal conductivity ${\kappa }_{e\mu }$ as a function of $n_B$ (a) for the HHJ EOS and (b) for the BSk21 EOS, $T={10}^8\mathrm{K}$, and for three values of $T_{\mathrm{c}p}={10}^{10}$, $3\times {10}^9$, and $10^9\mathrm{K}$ as indicated in the plot. Solid lines show the results of the present paper, dash-dotted lines correspond to the Pippard limit, and dashed lines to the London limit of transverse screening. Dotted lines show the calculations for normal (not superconducting) matter.

Figure 8

Lepton shear viscosity (a) for the HHJ EOS and (b) for the BSk21 EOS as a function of $n_B$. The parameters of calculations and notations are the same as in Fig. 7. Notice, that the linear scale is used in this plot. Insets enlarge the region of $n_B$ up to $2n_0$, where the logarithmic scale for ${\eta }_{e\mu }$ is used.

Figure 9

Lepton thermal conductivity (a) and shear viscosity (b) calculated for $T={10}^8\mathrm{K}$, $n_B=4n_0$, and proton fraction $x_p=0.15$ as functions of the proton effective mass $m_p^{*}$. The notations are the same as in Fig. 7.

Figure 10

Shear viscosity (a)–(b) and thermal conductivity (c)–(d) in the HHJ NS core for weak proton superconductivity model [panels (a) and (c)] and strong superconductivity model [panels (b) and (d)] discussed in the text. The temperature is set to $T={10}^7\mathrm{K}$. As in previous figures, solid lines give full results of the present work, dash-dotted lines correspond to calculations in the Pippard limit, and dashed lines—to calculations in the London limit. Dotted lines show the transports coefficients in normal matter. Lower hatched strips marked “n, pnorm” and upper filled strips marked “n, pSC” give the uncertainty bands for neutron transport coefficients in normal and superconducting matter, respectively. Thin solid lines and the right scales in each panel show the values of $T_{\mathrm{c}p,9}=T_{\mathrm{c}p}/{10}^9\mathrm{K}$ and $A$ for each model. Vertical dashed lines ($A=\sqrt{2}\pi$) divide type-II and type-I superconductivity regions. See text for details.

Figure 11

Same as Fig. 10 but for the BSk21 EOS. Notations are the same as in Fig. 10.