Dynamical onset of superconductivity and retention of magnetic fields in cooling neutron stars

A superconductor of paired protons is thought to form in the core of neutron stars soon after their birth. Minimum energy conditions suggest magnetic flux is expelled from the superconducting region due to the Meissner effect, such that the neutron star core is largely devoid of magnetic fields for some nuclear equation of state and proton pairing models. We show via neutron star cooling simulations that the superconducting region expands faster than flux is expected to be expelled because cooling timescales are much shorter than timescales of magnetic field diffusion. Thus magnetic fields remain in the bulk of the neutron star core for at least ${10}^6\text{–}{10}^7\text{yr}$. We estimate the size of flux free regions at ${10}^7\text{yr}$ to be $\lesssim 100\text{m}$ for a magnetic field of ${10}^{11}\text{G}$ and possibly smaller for stronger field strengths. For proton pairing models that are narrow, magnetic flux may be completely expelled from a thin shell of approximately the above size after ${10}^5\text{yr}$. This shell may insulate lower conductivity outer layers, where magnetic fields can diffuse and decay faster, from fields maintained in the highly conducting deep core.

Figure 1

Proton superconductor states. Top: CCDK model of singlet pairing energy gap $\mathrm{\Delta }$ (left axis) and critical temperature $T_{\mathrm{cp}}$ (right axis) as a function of baryon number density $n_{\mathrm{b}}$, calculated using the APR EOS model. Middle: Ratio $\kappa$ between magnetic field penetration length scale $\lambda$ and fluxtube size $\xi$. The two shaded regions separated at $n_{\mathrm{b}}=0.48{\mathrm{fm}}^{-3}$ denote the regime where only the Meissner (flux expulsion) state is allowed (when $\kappa <1/\sqrt{2}$) and the regime where the superconductor can be in Meissner state or in fluxtubes (when $\kappa >1/\sqrt{2}$). Bottom: Dependence of superconductor state on $n_{\mathrm{b}}$ and magnetic field $H$. Superconductivity is destroyed when $H>H_{\mathrm{c}2}$. The fluxtube state exists when $\kappa >1/\sqrt{2}$ and $H_{\mathrm{c}1}\lesssim H\lesssim H_{\mathrm{c}2}$, and the Meissner state exists otherwise. The behavior of $H_{\mathrm{c}1}$ for $\kappa <2$ is approximated using results from [9].

Figure 2

Top: Temperature $T$ as a function of radius $r$. The crust-core boundary is at $r\approx 10.7\text{km}$, and total radius is $11.2\text{km}$ for this $1.9M_{\mathrm{Sun}}$ NS built using the APR EOS. $T_{\mathrm{cp}}$ denotes the density-dependent critical temperature for onset of proton superconductivity (when $T<T_{\mathrm{cp}}$) using the CCDK gap model. The separation between the two shaded regions is defined by $\kappa =1/\sqrt{2}$ (see Fig. 1). Nearly horizontal curves show temperature profiles at various ages. Bottom: Radial profile of electrical conductivity ${\sigma }_{\mathrm{c}}$ at ages corresponding to temperature profiles shown in top panel.

Figure 3

Top: Density $n_{\mathrm{b}}\left(T_{\mathrm{cp}}\right)$ at which onset of proton superconductivity occurs as a function of time $t$ [since $T_{\mathrm{cp}}=T\left(t\right)$]. Horizontal dotted line (at $n_{\mathrm{b}}=0.48{\mathrm{fm}}^{-3}$, where $\kappa =1/\sqrt{2}$) delineates regimes of Meissner and Meissner/fluxtube states (see Fig. 1). Middle: Crosses are radial distance over which the superconducting region grows at each cooling epoch and we define as cooling length scale $l_{\mathrm{cool}}$. Dashed line is the length scale $l_{\mathrm{mag}}$ obtained using Eq. (6) and setting ${\tau }_{\mathrm{OhmH}}=t$, while the solid line corresponds to using Eq. (7) and setting ${\tau }_{\mathrm{sc}}=t$. Bottom: Flux diffusion timescales ${\tau }_{\mathrm{OhmH}}$ and ${\tau }_{\mathrm{sc}}$ with $l_{\mathrm{mag}}=l_{\mathrm{cool}}$, where $l_{\mathrm{cool}}$ is from the middle panel. Dotted line is $\tau =t$. In middle and bottom panels, $H/2H_{\mathrm{c}2}={10}^{-5}$ is assumed.

Figure 4

Top: Temperature $T$ as a function of radius $r$. The crust-core boundary is at $r\approx 10.6\text{km}$, and total radius is $11.6\text{km}$ for this $1.4M_{\mathrm{Sun}}$ NS built using the APR EOS. The separation between the two shaded regions is defined by $\kappa =1/\sqrt{2}$ (see Fig. 1). Nearly horizontal curves show temperature profiles at various ages. Middle: Crosses are radial distance over which the superconducting region grows at each cooling epoch and we define as cooling length scale $l_{\mathrm{cool}}$. Bottom: Flux diffusion timescales ${\tau }_{\mathrm{OhmH}}$ and ${\tau }_{\mathrm{sc}}$ with $l_{\mathrm{mag}}=l_{\mathrm{cool}}$, where $l_{\mathrm{cool}}$ is from the middle panel. In middle and bottom panels, $H/2H_{\mathrm{c}2}={10}^{-5}$ is assumed.

Figure 5

Top: Temperature $T$ as a function of radius $r$. The crust-core boundary is at $r\approx 11.5\text{km}$, and total radius is $12.6\text{km}$ for this $1.4M_{\mathrm{Sun}}$ NS built using the BSk21 EOS. The separation between the two shaded regions is defined by $\kappa =1/\sqrt{2}$ (see Fig. 1). Nearly horizontal curves show temperature profiles at various ages. Middle: Crosses are radial distance over which the superconducting region grows at each cooling epoch and we define as cooling length scale $l_{\mathrm{cool}}$. Dashed line is the length scale $l_{\mathrm{mag}}$ obtained using Eq. (6) and setting ${\tau }_{\mathrm{OhmH}}=t$, while the solid line corresponds to using Eq. (7) and setting ${\tau }_{\mathrm{sc}}=t$. Bottom: Flux diffusion timescales ${\tau }_{\mathrm{OhmH}}$ and ${\tau }_{\mathrm{sc}}$ with $l_{\mathrm{mag}}=l_{\mathrm{cool}}$, where $l_{\mathrm{cool}}$ is from the middle panel. Dotted line is $\tau =t$. In middle and bottom panels, $H/2H_{\mathrm{c}2}={10}^{-5}$ is assumed.