Electrical conductivity of a warm neutron star crust in magnetic fields

We study the electrical conductivity of finite-temperature crust of a warm compact star which may be formed in the aftermath of a supernova explosion or a binary neutron star merger as well as when a cold neutron star is heated by accretion of material from a companion. We focus on the temperature-density regime where plasma is in the liquid state and, therefore, the conductivity is dominated by the electron scattering off correlated nuclei. The dynamical screening of this interaction is implemented in terms of the polarization tensor computed in the hard-thermal-loop effective field theory of QED plasma. The correlations of the background ionic component are accounted for via a structure factor derived from Monte Carlo simulations of one-component plasma. With this input we solve the Boltzmann kinetic equation in relaxation time approximation taking into account the anisotropy of transport due to the magnetic field. The electrical conductivity tensor is studied numerically as a function of temperature and density for carbon and iron nuclei as well as density-dependent composition of zero-temperature dense matter in weak equilibrium with electrons. We also provide accurate fit formulas to our numerical results as well as supplemental tables which can be used in dissipative magneto-hydrodynamics simulations of warm compact stars.

Figure 1

The temperature-density phase diagram of dense plasma composed of iron ${}^{56}\mathrm{Fe}$ (a) and carbon ${}^{12}\mathrm{C}$ (b). The electron gas degeneracy sets in below the Fermi temperature $T_F$ (short dashed lines). The ionic component solidifies below the melting temperature $T_m$ (solid lines), while quantum effects become important below the plasma temperature (dash-dotted lines). For temperatures above $T_{\mathrm{C}}$ (long dashed lines) the ionic component forms a Boltzmann gas. Note that for ${}^{12}\mathrm{C}$ the quantum effects become important in the portion of the phase diagram lying between the lines $T_p\left(\rho \right)$ and $T_m\left(\rho \right)$. The present study does not cover the shaded portion of the phase diagram.

Figure 2

The temperature-density phase diagram of dense stellar matter in the crust of a neutron star. Various phases are labeled in the figure. The vertical arrows show the density at which the indicated element first appears. The crust composition is taken from Ref. [32]. The shaded portion of the phase diagram indicates the regime which is not covered by our study.

Figure 3

Feynman diagram describing the scattering of electron of charge $e^{*}=\sqrt{4\pi }e$ off a nucleus of charge $e^{*}Z$ (left and right straight arrows, respectively) via exchange of photon (wavy line). The photon self-energy is given by the polarization tensor ${\mathrm{\Pi }}_{\mu \nu }$ shown by the closed loop. Dots stand for QED vertices.

Figure 4

Dependence of the structure factor of one-component plasma on the magnitude of momentum transfer $q$ in units of inverse $a_i$. For $\mathrm{\Gamma }\ge 2$ the structure factor is taken from Monte Carlo calculations of Galam and Hansen [35]. For $\mathrm{\Gamma }<2$ we obtain the structure factor from the analytical expressions provided by Tamashiro et al. [36].

Figure 5

Dependence of three components of the electrical conductivity tensor on density for various values of temperature indicated in the plot and $B_{12}=1$ ($B_{12}\equiv B/{10}^{12}$ G). (a)–(c) show the conductivities for ${}^{56}\mathrm{Fe}$, (d)–(f) the same for ${}^{12}\mathrm{C}$.

Figure 6

The temperature dependence of three components of the electrical conductivity tensor for $B_{12}=1$ and three values of density—${log}_{10}\rho =10$ (solid lines), ${log}_{10}\rho =8$ (dashed lines), ${log}_{10}\rho =6$ (dash-dotted lines) in units [g ${\mathrm{cm}}^{-3}]$. (a)–(c) show the conductivities for ${}^{56}\mathrm{Fe}$, (d)–(f) the same for ${}^{12}\mathrm{C}$. The dots represent the results obtained with Eqs. (34) and (39).

Figure 7

Dependence of conductivity on the scaled temperature for various densities. The minimum of the conductivity occurs roughly at $T\simeq T^{*}$.

Figure 8

Dependence of ${\sigma }_0$ [(a) and (c)] and ${\sigma }_1$ [(b) and (d)] components of the conductivity tensor on density for various values of the $B$-field. (a) and (b) show these components for ${}^{56}\mathrm{Fe}$, (c) and (d) the same for ${}^{12}\mathrm{C}$. The temperature is fixed at $T=1$ MeV.

Figure 9

Dependence of ${\sigma }_0$ (upper panels) and ${\sigma }_1$ (lower panels) components of the conductivity tensor on the magnetic field for ${}^{56}\mathrm{Fe}$ (left panels) and ${}^{12}\mathrm{C}$ (right panels) at six values of density—${log}_{10}\rho =11, {log}_{10}\rho =10, {log}_{10}\rho =9, {log}_{10}\rho =8, {log}_{10}\rho =7, {log}_{10}\rho =6$ (from top to bottom). The temperature is fixed at $T=1$ MeV.

Figure 10

The ratio ${\sigma }_0/\sigma$ as a function of scaled temperature $T/T^{*}$ at various densities ${\log }_{10}\rho =9$ (solid lines), ${\log }_{10}\rho =8$ (dashed lines), ${\log }_{10}\rho =7$ (dash-dotted lines), and ${\log }_{10}\rho =6$ (double-dash-dotted lines) and for three values of magnetic field $B_{12}=1$ [(a) and (d)], $B_{12}=10$ [(b) and (e)], and $B_{12}=100$ [(c) and (f)]. (a)—(c) show the ratio for ${}^{56}\mathrm{Fe}$, (d)–(f) the same for ${}^{12}\mathrm{C}$.

Figure 11

The function $f(\rho ,T)$ for ${}^{56}\mathrm{Fe}$ at magnetic field $B_{12}=1$.

Figure 12

Dependence of three components of the electrical conductivity tensor on density and temperature for $B_{12}=1$ and composition taken from Ref. [32].

Figure 13

The integration region in Eq. (30) in the plane $(\omega ,q)$ (light shaded area) bounded by the functions $q_{\pm }\left(\varepsilon \right)$. The major contribution to the integral comes from the triangle bound by the lines $\omega =\pm \theta q$ with the narrow opening angle $2\theta$ (dark shaded area).

Figure 14

The relaxation time [(a) and (c)] and dimensionless product ${\omega }_c\tau$ [(b) and (d)] as functions of density. The temperature is fixed at $T=0.1$ MeV in (a) and (b) (degenerate regime) and at $T=10$ MeV in (c) and (d) (nondegenerate regime). The magnetic field is fixed at $B_{12}=1$.

Figure 15

The relaxation time as a function on dimensionless ratio $T/T_F$ for three values of density: ${log}_{10}\rho =10$ (solid lines), ${log}_{10}\rho =8$ (dashed lines), ${log}_{10}\rho =6$ (dash-dotted lines) for ${}^{56}\mathrm{Fe}$ [(a) and (c)] and ${}^{12}\mathrm{C}$ [(b) and (d)]. (a) and (b) correspond to the degenerate and (c) and (d) to the nondegenerate regime. We show the effect of setting $S\left(q\right)=1$ for ${log}_{10}\rho =10$ by short-dashed lines. The open circles reproduce the results of Ref. [15].