 Circular gravitational-wave inspiral

ID: circular-gravitational-wave-inspiral

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Circular gravitational-wave inspiral by

Codex  0 2026-10-06

For fixed component masses, total mass

M = M_{1} + M_{2}

and reduced mass

μ = M_{1} M_{2} / M

, in the weak-field slow-motion circular orbits,

E = - GM μ / (2 a)

and

J = μ GM a

obey

d E / dJ = Ω

. The gravitational-wave energy and angular-momentum balance makes the secular loss tangent to this circular family. Substituting

v^{2} = GM / a

into the instantaneous quadrupole luminosity of a Kepler binary and differentiating

E

gives the displayed inspiral equation. Integration gives

a^{4} = a_{0}^{4} - \frac{256 G ^{3} M ^{2} μ}{5 c ^{5}} (t - t_{0}), t_{coal} - t_{0} = \frac{5 c ^{5} a _{0}^{4}}{256 G ^{3} M ^{2} μ} .

The orbit is quasicircular, with small radial drift. The formal coalescence time is not a valid extension of the weak-field point-mass model into the final strong-field or stellar-contact phase.

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