Neutron Stars—Cooling and Transport

Abstract

Observations of thermal radiation from neutron stars can potentially provide information about the states of supranuclear matter in the interiors of these stars with the aid of the theory of neutron-star thermal evolution. We review the basics of this theory for isolated neutron stars with strong magnetic fields, including most relevant thermodynamic and kinetic properties in the stellar core, crust, and blanketing envelopes.

Appendices

Appendix A: Electron Thermal Conductivities

In this Appendix, we briefly overview the physics of electron heat conduction in the neutron-star envelopes, which is the most important heat conduction mechanism as regards the neutron-star thermal evolution, in the case of $B=0$. The magnetic field effects on the heat conduction are considered in Sect. 4.4.

1.1 A.1 Weakly Degenerate Electron Gas

In the case of non-degenerate and non-relativistic electrons (Spitzer and Härm 1953; Braginskiĭ 1958; Spitzer 1962), the effective energy-averaged electron-ion collision frequency is

$$ \nu_{e\mathrm{i}} = \frac{4}{3} \sqrt{\frac{2\pi}{ m_{e}}} \frac{Z^{2} e^{4}}{T^{3/2}} n_{\mathrm{i}}\varLambda_{e\mathrm{i}}, $$

(A.1)

where $\varLambda_{e\mathrm{i}}$ is the Coulomb logarithm. In the considered case $\varLambda_{e\mathrm{i}}$ is a slowly varying function of density and temperature. Its precise value depends on the approximations used to solve the Boltzmann equation, but its order of magnitude is given by the elementary theory, where the Coulomb collision integral is truncated at small and large impact parameters of the electrons. Then $\varLambda_{e\mathrm{i}}\sim\ln(r_{\mathrm{max}}/r_{\mathrm{min}})$, where $r_{\mathrm{max}}$ and $r_{\mathrm{min}}$ are the maximum and minimum electron impact parameters. The parameter $r_{\mathrm{max}}$ can be set equal to the Debye screening length, $r_{\mathrm{max}}^{-2}=4\pi(n_{e}+Z^{2} n_{\mathrm{i}}) e^{2}/T$. The second parameter can be estimated as $r_{\mathrm{min}} = \max(\lambda _{e},\,r_{\mathrm{cl}})$, where $\lambda _{e}$ (defined in Sect. 2.3) limits $r_{\mathrm{min}}$ in the high-temperature regime (where the Born approximation holds), and $r_{\mathrm{cl}} = Ze^{2}/T$ is the classical closest-approach distance of a thermal electron, which limits $r_{\mathrm{min}}$ in the low-temperature, quasiclassical regime.

A similar effective frequency

$$ \nu_{ee} = \frac{8}{3} \sqrt{\frac{\pi}{ m_{e}}} \frac{e^{4}}{T^{3/2}} n_{e}\varLambda_{ee} $$

(A.2)

characterizes the efficiency of the $ee$ collisions. If $\varLambda_{ee}\sim\varLambda_{e\mathrm{i}}$, then $\nu_{e\mathrm {i}}/\nu_{ee}\sim Z$, therefore for large $Z$ the $e\mathrm{i}$ collisions are much more efficient than the $ee$ collisions.

1.2 A.2 Strongly Degenerate Electron Gas

1.2.1 A.2.1 Electron-Ion Scattering

The thermal conductivity of strongly degenerate electrons in a fully ionized plasma is given by Eq. (12) with $a=\pi^{2}/3$. In order to determine the effective collision frequency that enters this equation, we use the Matthiessen rule $\nu=\nu_{e\mathrm{i}}+\nu_{ee}$.

The effective electron-ion collision frequency can be written in the form (Lee 1950; Yakovlev and Urpin 1980)

$$ \nu_{e\mathrm{i}} = \frac{ 4 Z m_{e}^{\ast}e^{4} \varLambda_{e\mathrm{i}}}{ 3 \pi\hbar^{3} } =\frac{Z\varLambda_{e\mathrm{i}} \sqrt{1+x_{\mathrm{r}}^{2}}}{5.7\times10^{-17}~\mbox{s}}. $$

(A.3)

Lee (1950) gave an estimate of the Coulomb logarithm $\varLambda_{e\mathrm{i}} = \ln (r_{\mathrm{max}}/r_{\mathrm{min}})$, with the minimum impact parameter $r_{\mathrm{min}} = \hbar/2 p_{\mathrm{F}}$ and the maximum impact parameter $r_{\mathrm{max}} = a_{\mathrm{i}}$. Yakovlev and Urpin (1980) calculated the conductivities for relativistic degenerate electrons, neglecting electron screening, and obtained a more accurate estimate $r_{\mathrm{max}} \approx0.4 a_{\mathrm{i}}$ in the liquid regime. In the solid regime, where the electrons scatter on phonons (collective ion excitations), Yakovlev and Urpin (1980) obtained different approximations for the two distinct cases, $\varTheta_{\mathrm{D}} < T < T_{\mathrm{m}}$ and $T< \varTheta_{\mathrm{D}}$.

Potekhin et al. (1999) derived a unified treatment of the electron conductivities in the Coulomb liquid and solid and described both regimes by Appendix A.3. Then qualitatively, by order of magnitude, $\varLambda_{e\mathrm{i}}\sim1$ in the ion liquid, and $\varLambda_{e\mathrm{i}}\sim T/T_{\mathrm{m}}$ in the Coulomb solid with a melting temperature $T_{\mathrm{m}}$. The effects of multiphonon scattering, electron screening, and non-Born corrections, have been taken into account, and the Coulomb logarithms in both liquid and solid phases have been fitted by a single analytical formula. A Fortran code and a table of thermal conductivities, based on this formalism, are available online.⁴

At the conditions typical for the envelopes of neutron stars, the electron-phonon scattering proceeds mainly via the Umklapp processes, where the wave vector corresponding to the change of electron momentum lies outside the first Brillouin zone. Raikh and Yakovlev (1982) noticed that if $T\lesssim T_{\mathrm{U}} = T_{\mathrm{p}}Z^{1/3} \alpha _{\mathrm{f}}\sqrt{1+x_{\mathrm{r}}^{2}}/3x_{\mathrm{r}}$, then the Umklapp processes occur less often (“freeze out”). Then the scattering rate decreases. Raikh and Yakovlev (1982) assumed an extremely strong (exponential) decrease. This implied that at $T < T_{\mathrm{U}}$ the conductivity would be in practice determined by impurities and structure defects of the lattice, rather than by the electron-phonon scattering (Gnedin et al. 2001). However, Chugunov (2012) showed that distortion of electron wave functions due to interaction with the Coulomb lattice destroys this picture and strongly slows down the increase of the conductivity. As a result, the conductivities in neutron star envelopes can be treated neglecting the “freezing-out” of the Umklapp processes.

1.2.2 A.2.2 Electron-Electron Scattering

Although the electron-ion scattering is usually most important for degenerate plasmas, the electron-electron scattering still can be non-negligible for relatively light elements ($Z\lesssim10$) (Lampe 1968). The expression of $\nu_{ee}$ for the relativistic degenerate electrons at $T\ll T_{\mathrm{p}}$ was obtained by Flowers and Itoh (1976). Urpin and Yakovlev (1980) extended it to higher temperatures, where $T_{\mathrm{p}}\lesssim T\ll \varepsilon _{\mathrm{F}}$.

Shternin and Yakovlev (2006) reconsidered the problem including the Landau damping of transverse plasmons, neglected by the previous authors. This effect is due to the difference of the components of the polarizability tensor, responsible for screening the charge-charge and current-current interactions: the transverse current-current interactions undergo “dynamical screening.” Shternin and Yakovlev (2006) showed that the Landau damping of transverse plasmons strongly increases $\nu_{ee}$ in the domain of $x_{\mathrm{r}}\gtrsim1$ and $T\ll T_{\mathrm{p}}$ and presented a new fit to $\nu_{ee}$ (also implemented in the code referenced in footnote 4).

1.3 A.3 The Case of Intermediate Degeneracy

In the case where the electron gas is partially degenerate, that is $T\sim \varepsilon _{\mathrm{F}}$, the thermal and electrical conductivities determined by the electron-ion scattering are satisfactorily evaluated by the thermal averaging procedure [Eq. (33) in Sect. 4.4.2]. For conductivities determined by the electron-electron collisions, there is no such averaging procedure, but we can use an interpolation between the two limiting cases,

$$ \nu_{ee} = \nu_{ee}^{\mathrm{deg}} \frac{1+625 (T/ \varepsilon _{\mathrm{F}})^{2}}{1+25 T/ \varepsilon _{\mathrm{F}}+ 271 (T/ \varepsilon _{\mathrm{F}})^{5/2}}. $$

(A.4)

A satisfactory accuracy of this interpolation has been verified by Cassisi et al. (2007).

1.4 A.4 Impurities and Mixtures

If the plasma in an envelope is not a pure substance of a single chemical element, then the effective collision frequency $\nu_{e\mathrm{i}}$ should be modified. The required modification can be different, depending on the state of the plasma and on the amount of impurities. For example, Flowers and Itoh (1976), Yakovlev and Urpin (1980), and Itoh and Kohyama (1993) considered electron scattering by charged impurities in a Coulomb crystal. If the fraction of impurities is small and they are randomly distributed, then electron-impurity scattering can be treated as scattering by charge fluctuations, controlled by the impurity parameter $Q = \langle(Z-\langle Z\rangle)^{2}\rangle$, where $\langle Z\rangle\equiv\sum_{j} Y_{j} Z_{j}$, $Y_{j}=n_{j}/\sum_{j} n_{j}$ is the number fraction of ions of the $j$th kind, and $Z_{j}$ is their charge number. Then, using the Matthiessen rule, one can obtain $\nu_{e\mathrm{i}}$ as a sum of the terms corresponding to the electron-phonon scattering in a homogeneous lattice and to the electron scattering by charge fluctuations. The effective relaxation time for the latter term is given by Appendix A.3 with $Z\varLambda_{e\mathrm{i}}$ replaced by $\sum_{j} Y_{j}(Z_{j}-\langle Z\rangle)^{2} \varLambda_{j}/\langle Z\rangle$, where the Coulomb logarithm $\varLambda_{j}$ depends generally on $j$. Neglecting the differences between the Coulomb logarithms, one can thus simply replace $Z$ by $Q/\langle Z\rangle$ in Appendix A.3 to estimate the conductivity due to electron scattering by charged impurities.

An alternative approach is relevant when there is no dominant ion species which forms a crystal (e.g., in a liquid, a gas, or a glassy alloy). In this case, one can use Appendix A.3 with $Z^{2} n_{\mathrm{i}}\varLambda_{e\mathrm{i}}$ replaced by $\sum_{j} Z_{j}^{2} n_{j} \varLambda_{j}$. An approximation to $\varLambda_{j}$ based on the plasma “additivity rule” has been suggested by Potekhin et al. (1999). Neglecting the differences between the Coulomb logarithms, one arrives at Appendix A.3 with $Z$ replaced by $\sqrt{\langle Z^{2}\rangle}$. If tabulated conductivities $\kappa_{j}$ for pure substances are used, then the best agreement with calculations based on the “additivity rule” is usually given by the estimate

$$ \kappa\approx\frac{\sum_{j} Y_{j} Z_{j} \kappa_{j} }{ \sum_{j} Y_{j} Z_{j}} \equiv\frac{\langle\kappa Z \rangle}{ \langle Z \rangle}. $$

(A.5)

Appendix B: Temperature Relations for Envelopes of Neutron Stars with Magnetic Fields

Here we present an analytical fit to the temperature distribution over a surface of a neutron star with a non-accreted envelope and a dipole magnetic field. We have chosen $\rho _{\mathrm{b}}=10^{10}~\mbox{g}\,\mbox{cm}^{-3}$ and used the BSk21 EoS (Pearson et al. 2012) in the parametrized form (Potekhin et al. 2013). The numerical data have been produced with the 2D code of Viganò et al. (2013) for 5 values of internal temperature $T_{\mathrm{b}}$ from $10^{7}~\mbox{K}$ to $10^{9}~\mbox{K}$, 5 values of the magnetic field at the pole $B_{\mathrm{p}}$ from $10^{11}~\mbox{G}$ to $10^{15}~\mbox{G}$, and 20 values of magnetic colatitude $\theta$ at the surface of the neutron star from 0 to $\pi/2$. The use of the 2D code corrects the temperature distribution near the magnetic equator, because the non-radial heat flow increases the equatorial $T_{\mathrm{s}}$ as compared to the 1D model that was employed previously (see Fig. 11 in Sect. 5.2). These data have been supplemented with more detailed calculations at the magnetic pole ($\theta=0$) using the 1D code of Potekhin et al. (2007) for 36 values of $T_{\mathrm{b}}$ from $10^{6.5}~\mbox{K}$ to $10^{10}~\mbox{K}$ and 9 values of $B_{\mathrm{p}}$ from $10^{11}~\mbox{G}$ to $10^{15}~\mbox{G}$. An important difference from the old results is the inclusion of the neutrino emission from the crust, which is especially important for the magnetars (see Sect. 5). Because of the 2D treatment and the allowance for neutrino emission, the new fit supersedes the previous one (Potekhin et al. 2003), whenever $B>10^{12}~\mbox{G}$ or $T_{\mathrm{b}}\gtrsim10^{8}~\mbox{K}$. We stress that its use is restricted by non-accreted (i.e., composed of heavy chemical elements) envelopes in the range of $10^{6.5}~\mbox{K}\lesssim T_{\mathrm{b}}\lesssim10^{10}~\mbox{K}$ and $B_{\mathrm{p}}\lesssim10^{15}~\mbox{G}$, which is covered by the underlying numerical data. For envelopes with $B\lesssim10^{12}~\mbox{G}$ (either non-accreted or accreted), the previous fit can be used, however the surface temperature $T_{\mathrm{s}}$ (but not the flux at the inner boundary, $F_{\mathrm{b}}$—see item 4 below) should be limited for hot stars according to Eq. (B.4) below.

The fit consists of 3 stages: (1) an expression for the surface temperature at the magnetic pole, $T_{\mathrm{p}}$, as function of $T_{\mathrm{b}}$, $g$, and $B_{\mathrm{p}}$; (2) an expression for the ratio of the polar to the equatorial surface temperatures, $T_{\mathrm{p}}/T_{\mathrm{eq}}$; (3) an expression for the dependence of $T_{\mathrm{s}}$ on the magnetic colatitude $\theta$. Since the thermal conductivities for quantizing magnetic fields (Sect. 4.4.2) are known for the electron-ion but not electron-electron collision mechanism, we multiplied $T_{\mathrm{s}}$ by a correction factor, obtained numerically from a comparison of the results of thermal-structure calculations with and without the $ee$ collisions at $B=0$. At the end of this Appendix we suggest a recipe for relating the flux $F_{\mathrm{b}}$ at the bottom of the heat-blanketing envelope to temperature $T_{\mathrm{s}}$ and thereby to $T_{\mathrm{b}}$.

• 1. At the magnetic pole, the effective surface temperature, neglecting neutrino emission from the crust, is approximately given by the expression

  $$ T_{\mathrm{p}}^{(0)}= \bigl[g_{14}\bigl(T_{1}^{4}+ (1+0.15\sqrt{B_{12}})T_{0}^{4}\bigr) \bigr]^{1/4} \times10^{6}~\mbox{K}, $$

  (B.1)

  where

  $$\begin{aligned} \textstyle\begin{array}{l} T_{0} = \bigl(15.7 T_{9}^{3/2}+1.36 T_{9}\bigr)^{0.3796}, \qquad T_{1} = 1.13 B_{12}^{0.119}T_{9}^{a}, \\ a=0.337/(1+0.02\sqrt{B_{12}}\,), \end{array}\displaystyle \end{aligned}$$

  (B.2)

  $T_{9} = T_{\mathrm{b}}/10^{9}~\mbox{K}$, and $B_{12}=B_{\mathrm{p}}/10^{12}~\mbox{G}$. The limiting temperature, at which $T_{\mathrm{p}}(T_{\mathrm{b}})$ levels off due to the neutrino emission from the crust is approximately given by

  $$ T_{\mathrm{p}}^{(\mathrm{max})}= \bigl(5.2 g_{14}^{0.65}+ 0.093\sqrt{g_{14} B_{12}}\,\bigr) \times10^{6}~\mbox{K}. $$

  (B.3)

  The corrected surface temperature at the pole, which takes this limit into account, is reproduced by the expression

  $$ T_{\mathrm{p}} = T_{\mathrm{p}}^{(0)} \bigl[ 1+ \bigl(T_{\mathrm{p}}^{(0)}/T_{\mathrm{p}}^{(\mathrm{max})} \bigr)^{4} \bigr]^{-1/4} $$

  (B.4)

• 2. The ratio of the polar to equatorial surface temperatures can be roughly evaluated as

  $$ \frac{T_{\mathrm{p}}}{T_{\mathrm{eq}}} = 1+ \frac{(1230 T_{9})^{3.35}B_{12} \sqrt{1+2B_{12}^{ 2}} }{ (B_{12}+450 T_{9}+ 119 B_{12} T_{9})^{4}} + \frac{0.0066 B_{12}^{5/2} }{ T_{9}^{1/2}+0.00258 B_{12}^{5/2}}. $$

  (B.5)

  The numerically calculated $T_{\mathrm{p}}/T_{\mathrm{eq}}$ ratio has a complex dependence on $T_{\mathrm{b}}$ and $B$ at $B > 10^{13}~\mbox{G}$. In order to keep our fitting formulae relatively simple, we do not reproduce these oscillations, but instead force the ratio (B.5) to converge to some average value at $B \gg10^{13}~\mbox{G}$. The numerical data oscillate in a complicated manner around this average, with deviations reaching up to 35 %. For smaller fields, $B\lesssim3\times10^{12}~\mbox{G}$, Eq. (B.5) reproduces the numerical data with typical errors of several percent (up to 10 %). Note that these significant deviations affect only nearly tangential field case, viz. the equatorial region, which is substantially colder than the rest of the surface. Therefore its contribution to the observed flux is usually not very important.

• 3. Finally, the dependence of the surface temperature on the magnetic colatitude $\theta$ is approximately described by the expression

  $$\begin{aligned} &\frac{T_{\mathrm{s}}(\theta)-T_{\mathrm{eq}}}{T_{\mathrm{p}}-T_{\mathrm{eq}}} = \frac{(1+a_{1}+a_{2}) \cos^{2}\theta}{ 1+a_{1}\cos\theta+a_{2}\cos^{2}\theta}, \\ & \quad\mbox{where}\ a_{1}= \frac{a_{2} T_{9}^{1/2}}{3}, \ a_{2} = \frac{10 B_{12}}{ T_{9}^{1/2} + 0.1 B_{12} T_{9}^{-1/4}}. \end{aligned}$$

  (B.6)

• 4. Note that the outer boundary condition to the thermal evolution equations (4) involves the relation between the heat flux density $F_{\mathrm{b}}$ through the boundary at $\rho=\rho _{\mathrm{b}}$ and the temperature $T_{\mathrm{b}}$ at this boundary. In the absence of the neutrino emission from the crust, this boundary condition is directly provided by the $T_{\mathrm{b}}$–$T_{\mathrm{s}}$ relation, because in this case (in the plane-parallel approximation) $F_{\mathrm{b}}=\sigma _{\mathrm{SB}}T_{\mathrm{s}}^{4}$. It is not so if a significant part of the energy is carried from the outer crust by neutrinos. In this case we suggest to evaluate the flux through the boundary by the relation $F_{\mathrm{b}}=\sigma _{\mathrm{SB}}T_{\ast}^{4}$, where $T_{\ast}$ is given by the above approximations for $T_{\mathrm{s}}$, but without the correction (B.4).