Magnetic topology in coupled binaries, spin-orbital resonances, and flares

We consider topological configurations of the magnetically coupled spinning stellar binaries (e.g., merging neutron stars or interacting star-planet systems). We discuss conditions when the stellar spins and the orbital motion nearly `compensate' each other, leading to very {\it slow} overall winding of the coupled magnetic fields; slowly winding configurations allow gradual accumulation of magnetic energy, that is eventually released in a flare when the instability threshold is reached. We find that this slow winding can be global and/or local. We describe the topology of the relevant space $\mathbb{F}=T^1S^2$ as the unit tangent bundle of the two-sphere and find conditions for slowly winding configurations in terms of magnetic moments, spins and orbital momentum. These conditions become ambiguous near the topological bifurcation points; in certain cases they also depend on the relative phases of the spin and orbital motions. In the case of merging magnetized neutron stars, if one of the stars is a millisecond pulsar, spinning at $\sim$ 10 msec, the global resonance $\omega_1+\omega_2= 2 \Omega$ (spin-plus beat is two times the orbital period) occurs approximately a second before the merger; the total energy of the flare can be as large as $10\%$ of the total magnetic energy, producing bursts of luminosity $\sim 10^{44}$ erg s$^{-1}$. Higher order local resonances may have similar powers, since the amount of involved magnetic flux tubes may be comparable to the total connected flux.