Astrophysical unipolar inductors powered by GW emission

We consider the Unipolar Inductor Model (Goldreich & Lynden-Bell 1969, ApJ, 156, 59) applied to Double Degenerate Binaries (DDBs) with ultrashort periods (Wu et al. 2002, MNRAS, 331, 221). In this model a magnetized primary white dwarf has a slight asynchronism between its spin and orbital motion, so that the (non-magnetic) secondary experiences a motional electric field when moving through the primary field lines. This induces a current flow between the two stars and provides an electric spin-orbit coupling mechanism for the primary. We study the combined effect of Gravitational Wave emission and electric spin-orbit coupling on the evolution of the primary degree of asynchronism and the associated rate of electric current dissipation in such systems, assuming that the primary's spin is not affected by other mechanisms such as tidal interactions with the companion. In particular, we show that in ultrashort period binaries the emission of GW pumps energy in the electric circuit as to keep it steadily active. This happens despite the fact that spin-orbit coupling can rapidly synchronize the primary, because GW represent a slow desynchronizing mechanism that steadily substracts orbital angular momentum to the system. A slightly asynchronous steady-state is thus achieved, determined by the balance between these two competing effects. This can be shown to correspond to a condition where the total available electric energy is conserved, because of GW emission, while dissipation, synchronization and orbital shrinking continue.