Effective-one-body binary dynamics

Use total mass M, reduced mass mu and symmetric mass ratio nu as before. In an illustrative nonspinning EOB representation define an effective energy divided by mu c squared. The real Hamiltonian is related to this dimensionless energy by a nonlinear square-root map. Subtracting total rest energy gives the binding energy. Confusing these normalizations can produce a result off by the mass ratio even when the potential itself has been evaluated correctly.

The effective potentials are chosen to reproduce known post-Newtonian information under the appropriate canonical mapping. In the small-mass-ratio limit they reduce to Schwarzschild test-particle dynamics in the nonspinning example. Finite-mass-ratio corrections deform that limit. Higher-order potentials, spins and nongeodesic momentum terms enrich practical models; the simplest displayed potential is a controlled illustrative truncation, not a complete universal EOB prescription.

In the Schwarzschild test-mass limit, set effective radial momentum to zero and study the squared effective energy at fixed angular momentum. A circular orbit is a stationary point of this radial potential. A stable circular orbit requires a local minimum. The innermost stable circular orbit occurs when that minimum becomes marginal, so one must examine derivatives rather than looking only at the numerical size of a correction to the potential.

Writing the test-mass potential in terms of inverse radius makes its derivatives elementary. The circular angular momentum is one divided by u times one minus three u, under a square root. Its radial curvature vanishes at u equal to one sixth. This result checks the representation against a known general-relativistic limit. At comparable masses, the location of an effective stability transition and its coordinate radius depend on the specified model and its higher-order information.

A finite PN series can be reorganized through resummed potentials and fluxes to improve its behavior in a wider regime. Different resummation choices that agree through known orders can disagree in uncomputed terms. Their usefulness is tested against additional analytic limits, self-force information and numerical-relativity calculations. A compact square root or an effective horizon does not by itself make an approximation exact or provide a certified uncertainty.

A coalescence model also evolves dissipative orbital motion, constructs radiative multipoles and connects the late signal to a perturbed remnant. Calibration parameters and matching choices can matter for inference, especially outside the domain where a model was checked. Eccentricity, precession and extreme mass ratios pose distinct demands. EOB is consequently best understood as a structured framework combining physical inputs, whose accuracy belongs to a specified implementation and parameter region rather than to the name alone.