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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 ($\le 30\%$) 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 ($10^{15}\le B\le 5 \times 10^{16}$ G) whereas "strong-field" magnetars ($B>5\times 10^{16}$ 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.