Electromagnetic power of merging and collapsing compact objects

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Understanding possible electromagnetic signatures of merging and collapsing compact objects is important for identifying possible sources of the LIGO signal. Electromagnetic emission can be produced as a precursor to the merger, as a prompt emission during the collapse of a neutron star and at the spin-down stage of the resulting Kerr-Newman black hole. For the neutron star--neutron star mergers, the precursor power scales as $L\ensuremath{\approx}{B}_{\mathrm{NS}}^{2}G{M}_{\mathrm{NS}}{R}_{\mathrm{NS}}^{8}/({R}_{\mathrm{orb}}^{7}c)$, while for the neutron star--black hole mergers, it is $(GM/({c}^{2}{R}_{\mathrm{NS}}){)}^{2}$ times smaller. We demonstrate that the time evolution of the axisymmetric force-free magnetic fields can be expressed in terms of the hyperbolic Grad-Shafranov equation, and we formulate the generalization of Ferraro's law of isorotation to time-dependent angular velocity. We find an exact nonlinear time-dependent Michel-type (split-monopole) structure of magnetospheres driven by spinning and collapsing neutron stars in Schwarzschild geometry. Based on this solution, we argue that the collapse of a neutron star into a black hole happens smoothly, without the natural formation of current sheets or other dissipative structures on the open field lines; thus, it does not allow the magnetic field to become disconnected from the star and escape to infinity. Therefore, as long as an isolated Kerr black hole can produce plasma and currents, it does not lose its open magnetic field lines. Its magnetospheric structure evolves towards a split monopole, and the black hole spins down electromagnetically (the closed field lines get absorbed by the hole). The ``no-hair theorem,'' which assumes that the outside medium is a vacuum, is not applicable in this case: highly conducting plasma introduces a topological constraint forbidding the disconnection of the magnetic field lines from the black hole. Eventually, a single random large scale spontaneous reconnection event will lead to magnetic field release, shutting down the electromagnetic black hole engine forever. Overall, the electromagnetic power in all the above cases is expected to be relatively small. We also discuss the nature of short gamma-ray bursts and suggest that if the magnetic field is amplified to $\ensuremath{\sim}{10}^{14}\text{ }\text{ }\mathrm{G}$ during the merger or the core collapse, the similarity of the early afterglow properties of long and short gamma-ray bursts can be related to the fact that in both cases a spinning black hole can retain a magnetic field for a sufficiently long time to extract a large fraction of its rotational energy and produce high energy emission via the internal dissipation in the wind.