Late-time Cooling of Neutron Star Transients and the Physics of the Inner Crust

We follow the thermal evolution of the neutron star crust using the thermal evolution code dStar  (Brown 2015) that solves the fully general relativistic heat diffusion equation using a method of lines algorithm in the MESA numerical library (Paxton et al. 2011, 2013, 2015). The microphysics of the crust follows Brown & Cumming (2009). The results are verified with the code crustcool ⁶ that solves the heat diffusion equation assuming constant gravity through the crust.

We model the $\approx 2.5\,\mathrm{year}$ outburst in MXB 1659-29  (Wijnands et al. 2003, 2004) using a local mass accretion rate $\dot{m}=0.1\,{\dot{m}}_{\mathrm{Edd}}$, where ${\dot{m}}_{\mathrm{Edd}}=8.8\times {10}^{4}\,{\rm{g}}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ is the local Eddington mass accretion rate. The model uses a neutron star mass of $M=1.6\,{M}_{\odot }$ and radius of $R=11.2\,\mathrm{km}$ that are consistent with the MXB 1659-29  quiescent light curve fits from Brown & Cumming (2009). The model includes a ${Q}_{s}=1\,\mathrm{MeV}$ per accreted nucleon shallow heat source spread between $y=2\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ and $y=2\times {10}^{14}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$, in addition to deep crustal heating from electron capture and pycnonuclear reactions (Haensel & Zdunik 1990, 2003, 2008). For the crust composition we use the accreted composition from Haensel & Zdunik (2008) that assumes an initial composition of pure ⁵⁶Fe (see their Table A3).

The thermal conductivity in the inner crust is largely set by impurity scattering. The impurity parameter of the crust is given by

$$\begin{eqnarray}&&{Q}_{\mathrm{imp}}\equiv \displaystyle \frac{1}{{n}_{\mathrm{ion}}}\displaystyle \sum _{j}\,{n}_{j}{({Z}_{j}-\langle Z\rangle )}^{2},\end{eqnarray} \tag{ 1 }$$

where ${n}_{\mathrm{ion}}$ is the number density of ions, n_j is the number density of the nuclear species with Z_j number of protons, and $\langle Z\rangle$ is the average proton number of the crust composition. The impurity parameter in the neutron star crust was constrained to ${Q}_{\mathrm{imp}}\lt 10$ in MXB 1659-29  (Brown & Cumming 2009) assuming a constant impurity parameter throughout the entire crust. We show a model of crust cooling in MXB 1659-29 with ${Q}_{\mathrm{imp}}=2.5$ and ${T}_{\mathrm{core}}=4\times {10}^{7}\,{\rm{K}}$, consistent with the fit from Brown & Cumming (2009), in Figure 1. In this model, the crust reaches thermal equilibrium with the core by $\approx 1000$ days into quiescence, and so predicts a constant temperature at later times.

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We also run two models with a disordered inner crust with ${Q}_{\mathrm{imp}}=20$ for $\rho \gt 8\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (and ${Q}_{\mathrm{imp}}=1$ for $\rho \lt 8\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$) to represent the low conductivity expected for nuclear pasta, as done in Horowitz et al. (2015); both models have a neutron star mass $M=1.6\,{M}_{\odot }$, radius $R=11.2\,\mathrm{km}$, and ${T}_{\mathrm{core}}=3\times {10}^{7}\,{\rm{K}}$. The two models use different choices of the neutron superfluid critical temperature profile ${T}_{c}(\rho )$. The first uses a ¹S₀ gap that closes in the inner crust (Gandolfi et al. 2008, hereafter G08), and the second uses a gap that closes in the core (Schwenk et al. 2003, hereafter S03). The difference in the ${T}_{c}(\rho )$ profiles for each pairing gap model are shown in Figure 2.

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As can be seen in Figure 1, the solid blue curve corresponding to the G08 gap shows a long decline in temperature even at late times near $\approx 4000\,\mathrm{days}$, and thermal equilibrium is reached near $\approx 5000\,\mathrm{days}$ into quiescence. Note, however, that is difficult to reproduce the observations near $\approx 1500\,\mathrm{day}{\rm{s}}$ and $\approx 2500\,\mathrm{day}{\rm{s}}$ simultaneously. By contrast, the red dashed curve using the S03 gap does not show a decline at late times, but instead levels off to a constant temperature after $\approx 2000\,\mathrm{days}$. This difference arises because significant late-time cooling only occurs if there is a normal layer of neutrons at the base of the crust, giving a large heat capacity there (Figure 2). Furthermore, as demonstrated by the blue dotted curve in Figure 1, crust cooling with normal neutrons, but without a low thermal conductivity pasta layer, does not exhibit late-time cooling.

As we show in the following section, a low conductivity pasta layer maintains a temperature difference between the inner crust and core during quiescence, and a layer of normal neutrons survives at the base of the crust that has a long thermal time.

Some analytic estimates are useful to understand why the late-time cooling occurs and the crucial role of the normal neutron layer. First, we consider the temperature contrast ${\rm{\Delta }}T$ between the inner crust and the core that develops during the accretion outburst. This is set by the value at which the heat flux through the pasta layer balances the nuclear heating in the crust (mostly located at shallower densities near the neutron drip region). The heating rate is ${\epsilon }_{\mathrm{nuc}}=\dot{m}{E}_{\mathrm{nuc}}$, where ${E}_{\mathrm{nuc}}\approx 2\,\mathrm{MeV}$ per accreted nucleon comes from deep crustal heating. The equivalent heat flux is ${F}_{\mathrm{in}}\approx 2\times {10}^{22}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ for an accretion rate of $\dot{m}=0.1\,{\dot{m}}_{\mathrm{Edd}}$.

The heat flux through the pasta layer is $F\approx K{\rm{\Delta }}T/H$, where K is the thermal conductivity and H the pressure scale height. Neutrons set the pressure in the inner crust, so that⁷ $H=P/\rho g\approx 7\times {10}^{4}\,\mathrm{cm}\ ({\rho }_{14}^{2/3}{Y}_{n}^{5/3}/{g}_{14})$, where Y_n is the neutron fraction, ${\rho }_{14}$ is the mass density in units of ${10}^{14}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, and the surface gravity of the neutron star is $g={({GM}/{R}^{2})(1-2{GM}/{{Rc}}^{2})}^{-1/2}$ in units of ${10}^{14}\,\mathrm{cm}\,{{\rm{s}}}^{-2}$. The thermal conductivity is primarily set by electron–impurity scattering, with scattering frequency (Itoh & Kohyama 1993; Potekhin et al. 1999)

$$\begin{eqnarray}&&{\nu }_{{eQ}}=\displaystyle \frac{4\pi {e}^{4}{n}_{e}}{{p}_{{\rm{F}},e}^{2}{v}_{{\rm{F}},e}}\displaystyle \frac{{Q}_{\mathrm{imp}}}{\langle Z\rangle }{{\rm{\Lambda }}}_{{eQ}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\approx \,3\times {10}^{18}\,{{\rm{s}}}^{-1}\left[{\left(\displaystyle \frac{{\rho }_{14}{Y}_{e}}{0.05}\right)}^{1/3}\left(\displaystyle \frac{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}{\langle Z\rangle }\right)\right].\end{eqnarray} \tag{ 3 }$$

Here, ${p}_{{\rm{F}},e}$ and ${v}_{{\rm{F}},e}$ and the Fermi momentum and velocity of the electrons, ${{\rm{\Lambda }}}_{{eQ}}$ is the Coulomb logarithm, and Y_e is the electron fraction. The quantity ${Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}/\langle Z\rangle$ is of order unity in the inner crust. The resulting thermal conductivity is

$$\begin{eqnarray}{K}_{e} & = & \displaystyle \frac{\pi }{12}\displaystyle \frac{{E}_{{\rm{F}},e}\,{k}_{{\rm{B}}}^{2}\,{Tc}}{{e}^{4}}\displaystyle \frac{\langle Z\rangle }{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}\\ & \approx & 4\times {10}^{19}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-1}\,{{\rm{K}}}^{-1}\ \left[{T}_{8}{\left(\displaystyle \frac{{\rho }_{14}{Y}_{e}}{0.05}\right)}^{1/3}\,\displaystyle \frac{\langle Z\rangle }{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}\right],\end{eqnarray} \tag{ 4 }$$

where ${T}_{8}\equiv T/({10}^{8}\,{\rm{K}})$. Therefore, the temperature difference between inner crust and core is

$$\begin{eqnarray}&&{\rm{\Delta }}T\approx 3\times {10}^{7}\,{\rm{K}}\\ &&\,\times \left[\displaystyle \frac{{\rho }_{14}^{1/3}}{{g}_{14}{T}_{8}}\displaystyle \frac{{Y}_{n}^{5/3}}{{({Y}_{e}/0.05)}^{1/3}}\left(\displaystyle \frac{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}{\langle Z\rangle }\right)\left(\displaystyle \frac{\dot{m}}{0.1{\dot{m}}_{\mathrm{Edd}}}\right)\right],\end{eqnarray} \tag{ 5 }$$

which is in reasonable agreement with the temperature jumps seen in Figure 2(a) between $\rho \approx 8\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and $\rho \,\approx 1.5\times {10}^{14}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

We can understand the cooling timescale of the normal neutron layer as follows. The specific heat capacity of the normal neutrons is

$$\begin{eqnarray}{C}_{V} & = & \displaystyle \frac{{\pi }^{2}}{\rho }\displaystyle \frac{{n}_{n}{k}_{{\rm{B}}}^{2}T}{{p}_{{\rm{F}},n}{v}_{{\rm{F}},n}}\\ & \approx & 3\times {10}^{4}\ \mathrm{erg}\,{{\rm{g}}}^{-1}\,{{\rm{K}}}^{-1}\ \left[{Y}_{n}^{1/3}{\rho }_{14}^{-2/3}\left(\displaystyle \frac{{T}_{7}}{3}\right)\right],\end{eqnarray} \tag{ 6 }$$

where n_n is the number density of free neutrons, ${p}_{{\rm{F}},n}$ is the neutron Fermi momentum, and ${v}_{{\rm{F}},n}$ is the neutron Fermi velocity. The thermal diffusivity is

$$\begin{eqnarray}D & = & \displaystyle \frac{K}{\rho {C}_{V}}\\ & \approx & 4\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}\ {\left(\displaystyle \frac{{Y}_{e}}{0.05{Y}_{n}}\right)}^{1/3}\left(\displaystyle \frac{\langle Z\rangle }{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}\right);\end{eqnarray} \tag{ 7 }$$

D is independent of temperature and depends only weakly on density. The thermal timescale is then

$$\begin{eqnarray}{t}_{\mathrm{therm}} & \approx & \displaystyle \frac{{H}^{2}}{D}\\ & \approx & 4000\,\mathrm{days}\left[{\rho }_{14}^{4/3}{Y}_{n}^{11/3}{\left(\displaystyle \frac{{g}_{14}}{2}\right)}^{-2}\left(\displaystyle \frac{{Q}_{\mathrm{imp}}{{\rm{\Lambda }}}_{{eQ}}}{\langle Z\rangle }\right){\left(\displaystyle \frac{{Y}_{e}}{0.05}\right)}^{-1/3}\right].\end{eqnarray} \tag{ 8 }$$

Again, this is in good agreement with the cooling timescale we see in the numerical models. It also highlights the role of the large heat capacity from the normal neutrons. Without the normal neutrons, the electrons would set the heat capacity in the inner crust (see, e.g., Figure 6 of Brown & Cumming 2009); in this case, the thermal conductivity is $K=\rho {C}_{V}^{e}{v}_{{\rm{F}},e}^{2}/3{\nu }_{{eQ}}$, where C_V^e is the heat capacity of electrons, and we see that the thermal diffusivity is then $D\approx {c}^{2}/3{\nu }_{{eQ}}$, which is about two orders of magnitude larger than that given by Equation (7). Without normal neutrons, the inner crust cools in months, so that no late-time cooling signature of the pasta region is seen.

The large heat capacity of the normal neutrons suggests they may have a large thermal conductivity that could contribute significantly to the thermal conductivity near the base of the crust. Heat conduction by normal neutrons has been considered in the neutron star core (e.g., Baiko et al. 2001), but not in the crust. We calculate the scattering frequency for neutrons scattering from nuclei in the inner crust, either from thermal vibrations (phonons) or irregularities in the structure (impurities). The details are given in the Appendix; the total scattering frequency is given in Equation (24). The neutron thermal conductivity is then $K={\pi }^{2}{n}_{n}{k}_{{\rm{B}}}^{2}T/(3{m}_{n}^{\star }\nu )$, or

$$\begin{eqnarray}{K}_{n} & = & \displaystyle \frac{9{\pi }^{3}}{4}\displaystyle \frac{{n}_{n}{k}_{{\rm{B}}}^{2}T}{{m}_{n}^{\star }}\displaystyle \frac{{\hslash }}{{m}_{n}^{\star }{c}^{2}}\displaystyle \frac{{n}_{n}}{{n}_{\mathrm{ion}}}{\left(\displaystyle \frac{{\hslash }c}{{V}_{0}{R}_{A}}\right)}^{2}{\left[{{\rm{\Lambda }}}_{n,\mathrm{phn}}+\displaystyle \frac{{Q}_{\mathrm{imp}}}{\langle Z{\rangle }^{2}}{{\rm{\Lambda }}}_{{nQ}}\right]}^{-1}\\ & \approx & 3\times {10}^{17}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-1}\,{{\rm{K}}}^{-1}\ {T}_{8}{Y}_{n}{\rho }_{14}{\left(\displaystyle \frac{{m}_{n}^{\star }}{{m}_{n}}\right)}^{-2}\\ & & \times \left(\displaystyle \frac{{n}_{n}/{n}_{\mathrm{ion}}}{100\,}\right){\left(\displaystyle \frac{{\hslash }c}{{V}_{0}{R}_{A}}\right)}^{2}{\left[{{\rm{\Lambda }}}_{n,\mathrm{phn}}+\displaystyle \frac{{Q}_{\mathrm{imp}}}{\langle Z{\rangle }^{2}}{{\rm{\Lambda }}}_{{nQ}}\right]}^{-1}.\end{eqnarray} \tag{ 9 }$$

In this expression, ${m}_{n}^{\star }={p}_{{\rm{F}},n}\,{[\partial \varepsilon (p)/\partial p]}_{p={p}_{{\rm{F}},n}}^{-1}$ is the Landau effective mass and $\varepsilon (p)$ is the neutron single particle energy including the rest mass (see, e.g., Baym & Chin 1976). The dimensionless quantity ${V}_{0}{R}_{A}/{\hslash }c$ is of order unity and measures the strength of the neutron–nucleus interaction; R_A is the typical size of the scattering structure and energy V₀ is the magnitude of the scattering potential. Note that the neutron scattering frequency can be approximately reproduced by making the substitutions ${e}^{2}\to {V}_{0}{R}_{A}$ and ${p}_{{\rm{F}},e}^{2}{v}_{{\rm{F}},e}\to {p}_{{\rm{F}},n}^{2}{v}_{{\rm{F}},n}$ in Equation (2) for the electron–impurity scattering rate. The quantities ${{\rm{\Lambda }}}_{n,\mathrm{phn}}$ and ${{\rm{\Lambda }}}_{{nQ}}$ are the Coulomb logarithms for phonon and impurity scattering, respectively. As we discuss in the Appendix, we are able to write the impurity scattering for neutrons in terms of the impurity parameter for electron–impurity scattering ${Q}_{\mathrm{imp}}$.

The thermal conductivity of electrons and neutrons is compared in Figure 3 for the crust temperature profile in Figure 2 where we show the separate contributions from phonon and impurity scattering as a function of density. As has been discussed previously, impurity scattering dominates phonon scattering for electrons in the inner crust when ${Q}_{\mathrm{imp}}\gtrsim 1$ (e.g., Brown & Cumming 2009). We find for neutron scattering that the phonon contribution is larger where ${Q}_{\mathrm{imp}}=1$, and the impurity contribution is larger in the pasta layer where ${Q}_{\mathrm{imp}}=20$, as can be seen in Figure 3 (see also Equation (25)).

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Figure 3 shows that the conductivity due to neutrons can be comparable to the electron conductivity near the base of the crust, but is otherwise not important. To see this in more detail, it is useful to calculate the ratio ${K}_{n}/{K}_{e}$. The electron thermal conductivity is given by Equation (4) and taking the impurity contribution to the neutron conductivity only (since ${\nu }_{n,\mathrm{phn}}\lesssim {\nu }_{{nQ}}$ at the base of the crust), we find

$$\begin{eqnarray}&&\displaystyle \frac{{K}_{n}}{{K}_{e}}=9{\alpha }^{2}\langle Z{\rangle }^{2/3}{\left(\displaystyle \frac{{\hslash }c}{{V}_{0}{R}_{A}}\right)}^{2}\displaystyle \frac{{{\rm{\Lambda }}}_{{eQ}}}{{{\rm{\Lambda }}}_{{nQ}}}{\left(\displaystyle \frac{{n}_{n}}{{n}_{\mathrm{ion}}}\right)}^{4/3}{\left(\displaystyle \frac{{v}_{{\rm{F}},n}}{c}\right)}^{2},\end{eqnarray} \tag{ 10 }$$

where ${v}_{{\rm{F}},n}={p}_{{\rm{F}},n}/{m}_{n}^{\star }$ is the Fermi velocity of the neutrons and $\alpha ={e}^{2}/{\hslash }c\approx 1/137$ is the fine structure constant. All the factors in Equation (10) are $\lt 1$, except for ${n}_{n}/{n}_{\mathrm{ion}}$, which is typically ∼100. Therefore, we expect ${K}_{n}\lesssim {K}_{e}$, consistent with our numerical evaluation shown in Figure 3. Also, since ${v}_{{\rm{F}},n}={p}_{{\rm{F}},n}/{m}_{n}^{\star }$, we see that ${K}_{n}/{K}_{e}$ increases with density approximately as ${\rho }^{2/3}$, so that neutron thermal conductivity is most important at higher densities. Therefore, we do not expect that neutron thermal conductivity will remove the late-time cooling effect of a disordered pasta layer.

We note that the calculation of the neutron scattering rate could be improved, especially at low temperature and small impurity concentrations. Under these conditions, we found that the scattering rate was large because the neutron–phonon Umklapp process played an important role. In this context, it is known that band gaps in the neutron spectrum can suppress these processes and their effects need to be accounted for before firm conclusions can be drawn, and our estimate should be considered as an upper bound. This uncertainty is unlikely to change our main conclusions, however, since even a modest impurity concentration is sufficient to ensure ${K}_{n}\lt {K}_{e}$, except perhaps in the densest layers. In the highest-density regions where pasta phases are likely, the neutron density contrast and therefore V₀ is reduced, which makes neutron scattering less efficient. Here, the details of the neutron density distribution and the appearance of non-spherical rod- and slab-like structures in the pasta region can be important and warrants further study.

An additional piece of physics that could affect late-time cooling is neutrino emission from the pasta layer (Leinson 1993; Gusakov et al. 2004; Newton et al. 2013). To compete with the inward flux of ${F}_{\mathrm{in}}\approx 2\times {10}^{22}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ (see Section 2.2), the neutrino emissivity would have to be ${\epsilon }_{\nu }\approx \rho {F}_{\mathrm{in}}/y\sim {10}^{18}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}$ at $T\sim (3\mbox{--}6)\times {10}^{7}\,{\rm{K}}$. Neutrino cooling in the pasta layer can be enhanced when the neutrons are in the normal phase. Two mechanisms for such enhancement have been considered earlier. In one scenario, enhanced neutrino pair emission arises from spin flip transitions of neutrons due to their spin–orbit interactions with the density gradients in the pasta phase (Leinson 1993). In this case, estimates indicate that the neutrino emissivity ${\epsilon }_{\nu \bar{\nu }}^{\mathrm{pasta}}\approx 4\times {10}^{23}{T}_{9}^{6}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}$. In the second scenario, the direct Urca processes ${e}^{-}+p\to n+{\nu }_{e}$ and $n\to {e}^{-}+p+{\bar{\nu }}_{e}$ are kinematically allowed due to coherent Bragg scattering of nucleons from the the pasta (Gusakov et al. 2004); the resulting neutrino emissivity is ${\epsilon }_{\mathrm{Urca}}^{\mathrm{pasta}}\approx 4\,\times {Y}_{e}^{1/3}\times {10}^{21}{T}_{9}^{6}$ erg cm⁻³ s⁻¹.

Neutrino emission from electron bremsstrahlung reactions can be enhanced due to impurities since the rate of this reaction is roughly proportional to the electron scattering rate. In earlier work (Ofengeim et al. 2014) it has been shown that in the absence of impurities, when electron–phonon processes dominate, the neutrino energy loss rate at $T={10}^{8}\,{\rm{K}}$ is about ${\epsilon }_{\nu }\sim {10}^{13}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}$ in the pasta region and roughly scales as T⁶. This is significantly smaller than the emission expected from normal neutrons, and even a very large enhancement due to impurity scattering is unlikely to be relevant. From these estimates and the preceding discussion we conclude that neutrino cooling in the pasta layers, even with normal neutrons, is unlikely to be relevant at the temperatures encountered during thermal relaxation of the neutron stars and magnetars studied here.

The accreting neutron star KS 1731-260 also has quiescent cooling measurements at late times. Merritt et al. (2016) recently reported a new temperature measurement for KS 1731-260 taken $\approx 5300\,\mathrm{days}$ into quiescence. They found that the temperature was consistent with the previous value measured $\approx 3000\,\mathrm{days}$ into quiescence, implying that the neutron star crust has now reached thermal equilibrium with the neutron star core near ${T}_{\mathrm{core}}\approx 9.3\times {10}^{7}\,{\rm{K}}$. Furthermore, the cooling curve could be fit equally well with or without a disordered pasta layer at the base of the crust. This is a similar result to that found by Horowitz et al. (2015), finding that they could fit KS 1731-260 equally well with or without a disordered pasta layer. The role of the normal neutrons, however, has not been examined in the late-time cooling in this source.

We now model the quiescent cooling of KS 1731-260 following its $\approx 12.5\,\mathrm{year}$ outburst including a disordered pasta layer and the G08 gap that closes in the crust. The model uses a neutron star mass $M=1.4\,{M}_{\odot }$ and radius $R=10\,\mathrm{km}$, consistent with the spectral fits and crust models from Merritt et al. (2016). The model uses an iron envelope and an accretion rate $\dot{m}=0.1\,{\dot{m}}_{\mathrm{Edd}}$ as done in Merritt et al. (2016) and Cumming et al. (2017). The model fits to the quiescent cooling of KS 1731-260 can be seen in Figure 4.

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Although the cooling of KS 1731-260 can be fit well without a low conductivity pasta layer (Merritt et al. 2016), the cooling model with a ${Q}_{\mathrm{imp}}=20$ pasta layer and the G08 gap provides a better fit to the data. The fit requires a lower core temperature near ${T}_{\mathrm{core}}\approx 9.1\times {10}^{7}\,{\rm{K}}$. The high core temperature means that there is not a large temperature difference ${\rm{\Delta }}T$ (see Equation (5)) between the inner crust and the core during quiescence. The normal neutrons, however, still release heat at late times and the crust reaches thermal equilibrium with the core near $\approx 5000\,\mathrm{days}$ into quiescence.

The magnetar SGR 1627-41 has had two outbursts, one in 1998 (Woods et al. 1999) and one in 2008 (An et al. 2012). The measured luminosities are shown in Figure 5. The flux decay following the 1998 outburst was fit with a crust cooling model by Kouveliotou et al. (2003). The energy source for magnetar outbursts is thought to be the decaying magnetic field of the star, but the mechanism driving the outbursts and the distribution of the dissipated energy is not understood at present. Therefore the approach of these cooling models is to attempt to reproduce the observed flux decay by varying the amount of energy deposited and its location within the crust. An et al. (2012) showed that the flux decay after both the 1998 and 2008 outbursts could be reproduced by depositing energy in the outer crust, although with about an order of magnitude difference in the depth and magnitude of the energy deposited ($\sim {10}^{44}\,\mathrm{erg}$ at $\rho \lesssim 2\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ for 1998; $\sim {10}^{43}\,\mathrm{erg}$ at $\rho \lesssim 3\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ for 2008).

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The late-time observations of SGR 1627-41 after its 1998 outburst are reminiscent of the drop in flux seen in MXB 1659-29. The luminosity appeared to have leveled off after $\approx 1000\,\mathrm{days}$, but then showed a drop by a factor of two in an observation at $\approx 3500\,\mathrm{days}$. Moreover, An et al. (2012) showed that the flux $\approx 1000\,\mathrm{days}$ after the 2008 outburst was similar to the flux at the same time after the 1998 outburst. This is unexpected because the energy deposition in the 2008 outburst is much shallower, and by $\approx 1000\,\mathrm{days}$ the crust should have relaxed to the core temperature and the luminosity should be at its minimum value. An et al. (2012) suggested that perhaps the last flux measurement after the 1998 outburst was a statistical deviation (it is within 2σ of the previous flux value), and that the luminosity seen at $\approx 1000\,\mathrm{days}$ in both outbursts reflects the core temperature. This would mean SGR 1627-41 then has a hot core with a temperature near ${T}_{\mathrm{core}}\approx {10}^{8}\,{\rm{K}}$.

Here, we pursue the possibility of a colder core (as originally envisioned by Kouveliotou et al. 2003) and investigate whether the drop at $\approx 3000$ days is due to inner crust physics. We model the flux decay of SGR 1627-41 using crustcool, which includes envelope models and thermal conductivities that take into account the strong magnetic field (averaged over angle around the star; Scholz et al. 2014). The blue solid curves in Figure 5 show the best-fitting model with a constant energy density deposited in the outer crust. The amount of energy deposited is similar to An et al. (2012). In units of ${10}^{25}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$, the 1998 outburst model has an energy density ${E}_{25}=13$ deposited in the density range $7\times {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}\lt \rho \,\lt 3\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, and the 2008 outburst model has ${E}_{25}=1.0$ in the density range ${10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}\lt \rho \lt 3\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. We set the impurity parameter ${Q}_{\mathrm{imp}}=3$, constant throughout the crust, and core temperature ${T}_{\mathrm{core}}=7\times {10}^{7}\,{\rm{K}}$ (as measured at the crust–core boundary) so that the flux in the 1998 outburst continues to decline at $\approx 3000\,\mathrm{days}$. As An et al. (2012) pointed out, the 2008 outburst then cools much too quickly to agree with the flux measured at $\approx 1000\,\mathrm{days}$.

We introduce a pasta layer with ${Q}_{\mathrm{imp}}=25$ at $\rho \gt 8\,\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ to try to increase the luminosity at $\approx 1000\,\mathrm{days}$ after the 2008 outburst. The light curve is shown as the blue dashed line in Figure 5. Unlike accreting neutron stars, we find that introducing a low conductivity layer in the pasta region is not enough to delay the cooling. The difference is that in the accreting case the heating is over a long timescale so that the temperature of the inner crust is increased substantially. In the magnetar case, the energy is deposited at the beginning of the outburst and the high temperature of the inner crust must be established rapidly.

We find that with our colder core temperature, the only way to get agreement with the $\approx 1000\,\mathrm{day}$ 2008 outburst measurement is to deposit energy directly into the inner crust during the 2008 outburst. The black curves in Figure 5 show models in which we extend the heating into the inner crust, depositing ${E}_{25}=13$ in the inner crust but leaving the outer crust heating at ${E}_{25}=1.0$ as before. Note that in this model the total energy deposited during the 2008 outburst is then comparable to the energy deposited in 1998, $\sim {10}^{44}\,\mathrm{erg}$. The dramatic difference between the outburst light curves is due to the different radial distribution of the heating, rather than total energy. Although in the model shown we deposit energy throughout the inner crust, we find that we can match the observations as long as the crust is heated up to a density $\rho \lesssim 3\times {10}^{13}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, so that the energetic requirements can be reduced by a factor of 2–3 compared to the model shown. These models predict that if the core temperature is low ${T}_{\mathrm{core}}\lesssim 7\times {10}^{7}\,{\rm{K}}$ in SGR 1627-41, future observations should show a further decline in flux. We find that the future evolution is sensitive to the choice of ${Q}_{\mathrm{imp}}$ in the pasta layer and the choice of ${T}_{c}(\rho )$. This is shown by the black dashed, solid, and dotted–dashed curves in Figure 5 that have different choices for those parameters (these models are all consistent with the 1998 outburst, left panel of Figure 5).