Magnetar superconductivity versus magnetism: Neutrino cooling processes

Abstract

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 $(\ensuremath{\le}30%)$ 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 $({10}^{15}\ensuremath{\le}B\ensuremath{\le}5\ifmmode\times\else\texttimes\fi{}{10}^{16}\text{G})$ whereas ``strong-field'' magnetars $(B>5\ifmmode\times\else\texttimes\fi{}{10}^{16}\text{G})$ 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 $B\ensuremath{\ge}{H}_{c2}$; (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.