Dynamics of quadruple systems composed of two binaries: stars, white dwarfs, and implications for Ia supernovae

We investigate the long-term secular dynamics and Lidov-Kozai (LK) eccentricity oscillations of quadruple systems composed of two binaries at quadrupole and octupole orders in the perturbing Hamiltonian. We show that the fraction of systems reaching high eccentricities is enhanced relative to triple systems, over a broader range of parameter space. We show that this fraction grows with time, unlike triple systems evolved at quadrupole order. This is fundamentally because with their additional degrees of freedom, quadruple systems do not have a maximal set of commuting constants of the motion, even in secular theory at quadrupole order. We discuss these results in the context of star-star and white dwarf-white dwarf (WD) binaries, with emphasis on WD-WD mergers and collisions relevant to the Type Ia supernova problem. For star-star systems, we find that more than 30 per cent of systems reach high eccentricity within a Hubble time, potentially forming triple systems via stellar mergers or close binaries. For WD-WD systems, taking into account general relativistic and tidal precession and dissipation, we show that the merger rate is enhanced in quadruple systems relative to triple systems by a factor of 3.5-10, and that the long-term evolution of quadruple systems leads to a delay-time distribution similar to 1/t for mergers and collisions. In gravitational wave-driven mergers of compact objects, we classify the mergers by their evolutionary patterns in phase space and identify a regime in about 8 per cent of orbital shrinking mergers, where eccentricity oscillations occur on the general relativistic precession time-scale, rather than the much longer LK time-scale. Finally, we generalize previous treatments of oscillations in the inner binary eccentricity (evection) to eccentric mutual orbits. We assess the merger rate in quadruple and triple systems and the implications for their viability as progenitors of stellar mergers and Type Ia supernovae.