Evolutionary dynamics while trapped in resonance: a Keplerian binary system perturbed by gravitational radiation

Abstract

The method of averaging is used to investigate the phenomenon of capture into resonance for a model that describes a Keplerian binary system influenced by radiation damping and external normally incident periodic gravitational radiation. The dynamical evolution of the binary orbit while trapped in resonance is elucidated using the second order partially averaged system. This method provides a theoretical framework that can be used to explain the main evolutionary dynamics of a physical system that has been trapped in resonance.

Delay equations and radiation damping

“…But for Ω = 0 (that is when all forces are retarded), it is easy to prove that no periodic orbits exist and most solutions are unbounded. A typical plot of x versus t for system (16) for Ω 2 − α > 0 is given in figure 1, where the delay increases from its initial value w(0) = 0 to ν = 10.5. The initial response of the system (where the delay is small) is characterized by an oscillation as expected from equation (17), which follows from the expansion of equation (16) to first order in w ≪ 1.…”

“…A typical plot of x versus t for system (16) for Ω 2 − α > 0 is given in figure 1, where the delay increases from its initial value w(0) = 0 to ν = 10.5. The initial response of the system (where the delay is small) is characterized by an oscillation as expected from equation (17), which follows from the expansion of equation (16) to first order in w ≪ 1. But as w increases, the qualitative behavior of the system is affected by three additional bifurcations not accounted for by equation (17).…”

“…for the linearization of the delay equation (16). The bifurcation points (corresponding to the existence of centers) are given by…”

“…In the quadrupole approximation under consideration here, the gravitational waves carry away energy and angular momentum, but not linear momentum [9,10,11,12,13,14,15,16,17,18,19]; therefore, the total momentum of the binary system is conserved and this fact is responsible for the absence of a force term proportional to c −3 in (1). Moreover, all tidal, spin-orbit, and spin-spin interactions are neglected in (1); the only parameters in equation (1) are the masses and the separation between the centers of mass of the members of the binary system.…”

Dynamical friction and resonance trapping in planetary systems

“…and F 12 = ∂ F 11 /∂ L are evaluated at ( L 0 , G , L 0 −3 t + ϕ , g ). In labelling these terms, I have followed the convention of Chicone et al (1997b).…”