Meissner effect and vortex dynamics in quark stars

We present a new model for soft gamma-ray repeaters based on a quark star born with temperatures above the critical value (Tc) for the onset of the colour-flavor locked superconductivity. The quark star then quickly cools below Tc, expelling a fraction of the surface magnetic field via the Meissner effect. We show that if a small fraction (≤$10\%$) of the surface magnetic field ($10^{14}{-}10^{15}\, {\rm G}$) is expelled, it quickly decays via magnetic reconnection and heats up the quark star surface to temperatures >$10^9\,{\rm K}$. Created $({\rm e}^{+},{\rm e}^{-})$ pairs annihilate into gamma rays emitted in a giant burst (the first burst in our model), with a luminosity of ~$10^{45}\, {\rm erg}\, {\rm s}^{-1}$. Subsequent bursts result from the restructuring of the surface magnetic field following the formation and relaxation of a vortex lattice which confines the internal magnetic field. During this phase, energy is sporadically released as a consequence of magnetic reconnection events in the entangled surface magnetic field as it evolves into a smooth, more stable, configuration. The star eventually enters a quiescent phase in which energy is continuously supplied by vortex annihilation at the surface. As the star spins down, the outermost vortex lines will be pushed to the surface where they annihilate and release their confined magnetic field. We show that the corresponding luminosity is $L_v \sim 10^{36}\, {\rm erg}\, {\rm s}^{-1}$ for a typical soft gamma-ray repeater spinning with a period of $8\, {\rm s}$ and a surface magnetic field not exceeding $10^{15}\, {\rm G}$. Our model can be applied to any situation where a $T>T_{\rm c}$ quark star is generated. We discuss the connection between anomalous X-ray pulsars and soft gamma-ray repeaters in the context of our model.