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.