Nonchaotic evolution of triangular configuration due to gravitational radiation reaction in the three-body problem

Continuing work initiated in an earlier publication [H. Asada, Phys. Rev. D 80, 064021 (2009)], the gravitational radiation reaction to Lagrange’s equilateral triangular solution of the three-body problem is investigated in an analytic method. The previous work is based on the energy balance argument, which is sufficient for a two-body system because the number of degrees of freedom (the semimajor axis and the eccentricity in quasi-Keplerian cases, for instance) equals that of the constants of motion such as the total energy and the orbital angular momentum. In a system with three (or more) bodies, however, the number of degrees of freedom is more than that of the constants of motion. Therefore, the present paper discusses the evolution of the triangular system by directly treating the gravitational radiation reaction force to each body. The perturbed equations of motion are solved by using the Laplace transform technique. It is found that the triangular configuration is adiabatically shrinking and is kept in equilibrium by increasing the orbital frequency due to the radiation reaction if the mass ratios satisfy the Newtonian stability condition. Long-term stability involving the first post-Newtonian corrections is also discussed.

Figure 1

Perturbations in Lagrange’s orbit. Here, ${\chi }_{12}$ and $\sigma$ correspond to the scale transformation of the triangle and the change of the angle of the system to a reference direction, respectively. On the other hand, $X$ and $\psi$ are the degrees of freedom of a shape change from the equilateral triangle.

Figure 2

A contour map of the critical values of $\lambda$, which are marginal points of Eq. (2), as a function of ${\nu }_2$ and ${\nu }_3$. Note that since Eq. (2) is valid only for small values of $\lambda$, the lower-left region may not be accurate. Indeed, one can see $V<\lambda$ in this region; thus, the critical values of $\lambda$ are very sensitive to higher PN corrections.