Any dynamical interaction other than gravity present in an eccentric system perturbs the binary orbit forcing it to evolve in time.
Lagrange formalism allows us to compute time-evolution equations for the orbital elements of an eccentric binary once we know the perturbing force form as a function of the radius and velocity along the orbit.When the timescale over which the perturbing dynamical force acts is much longer than the orbital timescale, we can orbit-average the time-evolution equations over many single orbits to derive secular equations for the orbital elements of the binary. These equations are useful to combine different dynamical processes and compare our analytical predictions with the results of numerical simulations.Many different processes can act as perturbations of an eccentric binary. The effect of all these dynamical interactions can be described using the Lagrange formalism for the perturbed two-body problem described above. Below is what we find applying this formalism to Bondi-Hoyle accretion in X-ray binaries.Bondi-Hoyle accretion in X-ray binaries. Part of the spherical stellar wind is accreted by the compact object (neutron star or black hole) in the binary through gravitational focusing. Our eccentricity evolution equation is different from previous studies (Eggleton 2006) indicating that Bondi-Hoyle accretion
can increase eccentricity
for a mass ratio q < 0.78 contrary to the often assumed Bondi-Hoyle accretion circularization.
