Magnetar superconductivity versus magnetism: Neutrino cooling processes

We describe the microphysics, phenomenology, and astrophysical implication of a $B$-field induced unpairing effect that may occur in magnetars, if the local $B$ field in the core of a magnetar exceeds a critical value $H_{c2}$. Using the Ginzburg-Landau theory of superconductivity, we derive the $H_{c2}$ field for proton condensate taking into the correction $\left(\le 30\%\right)$ which arises from its coupling to the background neutron condensate. The density dependence of pairing of proton condensate implies that $H_{c2}$ is maximal at the crust-core interface and decreases towards the center of the star. As a consequence, magnetar cores with homogenous constant fields will be partially superconducting for “medium-field” magnetars $({10}^{15}\le B\le 5\times {10}^{16}\text{G})$ whereas “strong-field” magnetars $(B>5\times {10}^{16}\text{G})$ will be void of superconductivity. The neutrino emissivity of a magnetar's core changes in a twofold manner: (i) the $B$-field assisted direct Urca process is enhanced by orders of magnitude, because of the unpairing effect in regions where $B\ge H_{c2}$; (ii) the Cooper-pair breaking processes on protons vanish in these regions and the overall emissivity by the pair-breaking processes is reduced by a factor of only a few.

Figure 1

Zero-temperature equation of state of dense matter composed of neutrons, protons, electrons, and muons in $\beta$ equilibrium derived from the relativistic DF with the parametrization of Ref. [4].

Figure 2

Dependence of particle abundances $n_i/n_b,i\in n,p,e,\mu$ on the net baryon density $n_b$ in units of saturation density $n_0=0.152{\mathrm{fm}}^{-3}$. The vertical line shows the approximate Urca threshold $Y_{\mathrm{Urca}}=0.11$ for proton fraction.

Figure 3

Dependence of $S$- and $P$-wave pairing gaps of neutrons (dash-dotted and solid lines) and of $S$-wave protons (dashed line) on their respective Fermi momenta.

Figure 4

Upper panel: Dependence of pairing gaps for neutrons $({}^1{S}_0$ and ${}^3{P}_2$ channels) and for protons $({}^1{S}_0$ channel) on baryonic density in units of nuclear saturation density. Lower panel: Dependence of the critical unpairing field $H_{c2}$ on baryonic density with account for the coupling between the neutron and proton condensates (full line) and without (dashed line).

Figure 5

The emissivity of the Urca process in the forbidden region $(x>0)$ in units of the zero-field emissivity ${\epsilon }_0$ at fixed density $n=n_0$ for temperatures $T=0.01$ and $0.1$ MeV. The Urca emissivity is shown for the cases of (a) normal matter (dotted line), (b) paired neutrons and normal protons (dashed lines), and (c) paired neutrons and protons (dashed-dotted lines). However the case (c) cannot be realized because of the unpairing effect for all the plotted values of $B>H_{c2}=16.57$. Note that for fixed density the scaling of the kinematical factor $x$ along the $B$ axis is given through its dependence on the number of Landau levels, i.e., $x\propto N_{Fp}^{2/3}\propto B^{-2/3}$.

Figure 6

The emissivity of the Urca process in the allowed region $(x<0)$ in units of the zero-field emissivity ${\epsilon }_0$ at fixed density $n=1.5n_0$ for temperature $T=0.1$ MeV. The magnitude of proton and neutron gaps are ${\Delta }_p=0.73$ MeV and ${\Delta }_n=0.28$ MeV. The emissivity is shown in the cases of (a) unpaired matter, (b) paired neutrons and for $B>H_{c2}$ (unpaired protons), and (c) paired neutrons and for $B<H_{c2}$ (paired protons).

Figure 7

Neutrino emissivity via pair-breaking processes as a function of baryon density in units of $n_0$ for $B_{16}=0$ (a), $B_{16}=0.5$ (b), and $B_{16}=1$ (c). The full pair-breaking emissivity is shown by solid lines and consists of ${}^1{S}_0$ neutron pair emission for $n_b/n_0>0.5$ and of the sum of ${}^1{S}_0$ proton and ${}^3{P}_2$ neutron pair emission for larger densities. The separate contributions of ${}^1{S}_0$ proton and ${}^3{P}_2$ neutron pairs are shown by dash-dotted and dashed lines, respectively.