Coalescing binary systems of compact objects to ${\left(\mathrm{post}\right)}^{\frac{5}{2}}$-Newtonian order. III. Transition from inspiral to plunge

Late in its evolution, a binary system of compact objects will undergo a transition from an adiabatic inspiral induced by gravitational radiation damping to an unstable plunge, induced by strong spacetime curvature. This transition from inspiral to plunge is studied in detail using higher-order post-Newtonian methods. First, we study the innermost stable circular orbits of binary systems of nonrotating, compact objects of arbitrary mass ratio in the absence of gravitational radiation reaction. The method uses "hybrid" two-body equations of motion that are valid through ${\mathrm{}\left(\mathrm{post}\right)\mathrm{}}^2$-Newtonian order [order ${\left(\frac{\mathrm{Gm}}{r{c}^2}\right)}^2$], but that also include the test-body limit in the Schwarzschild geometry exactly. Using a critical-point analysis, we show that circular orbits inside this innermost orbit are unstable to plunge. The separation of the innermost stable orbit (in harmonic, or de Donder coordinates) is found to vary with mass ratio, from the test-body value of $5m$ to about $6m$ for equal masses, where $m$ is the total mass of the system. The orbital energy, angular momentum, and frequency of the innermost stable orbit are also determined as a function of the ratio of the two masses. We study the sensitivity of these values to higher-order post-Newtonian corrections. Incorporating gravitational radiation reaction in the hybrid equations of motion, we evaluate such variables as radial velocity, angular velocity, energy, and angular momentum for a coalescing binary at the corresponding innermost stable orbit, as a function of mass ratio. These variables could be used as initial conditions for numerical calculations of the ensuing coalescence.