Abstract
Equations of secular dynamics for stellar quadruple systems in 2+2 hierarchy are formulated. Non-singular, angular momentum and Laplace vector variables are used to describe orbital evolution of both inner and outer orbits.
Given a typical wide separation of the binaries in these systems, gravitational interactions are truncated at the octupole approximation. Secular equations are propagated numerically and the results compared to the complete numerical integration on a long time-scale.
Our basic formulation uses a point-mass model, but we also extend it by including the simplest description of the quadrupole interaction among the components of close (inner) binaries. Evolution of orbital planes of the binaries is discussed analytically in a simplified model and numerically using a more complete model.
Maximum angular separation of the two orbital planes reaches only 20-40 per cent of the simple geometric maximum value for low-eccentricity cases with small inclination with respect to the orbital plane of the relative motion. This may be a pre-requisite for occurrence of quadruple systems with both binaries showing eclipses.
However, statistical occurrence of eclipses at any time for a synthetic population of quadruples with initially isotropic distribution of orbital planes is about equal to the model where the orbits do not evolve due to gravitational interactions. We also show that the model is potentially suitable for long-term studies of the initial evolutionary tracks of the 2 + 2 quadruple systems.