= Gravitational-wave energy and angular-momentum balance
{title2=$\dot E=\Omega\dot J$}
For a slowly evolving circular <binary star> with separation $a$, <reduced mass> $\mu$ and orbital angular frequency $\Omega$, the leading <gravitational wave> losses satisfy $\dot E=\Omega\dot J$. One direct check uses the standard quadrupole angular-momentum loss law
$$
\dot J_i=-\frac{2G}{5c^5}\epsilon_{ijk}\left\langle\ddot Q_{j\ell}\dddot Q_{k\ell}\right\rangle,
$$
where $Q$ is the <mass quadrupole moment> and $\epsilon_{ijk}$ is the <Levi-Civita symbol>. For $\mathbf r=a(\cos\Omega t,\sin\Omega t,0)$, put $A=\mu a^2$; then $Q_{xx}=A(1/6+\cos(2\Omega t)/2)$, $Q_{yy}=A(1/6-\cos(2\Omega t)/2)$, $Q_{xy}=A\sin(2\Omega t)/2$, and $Q_{zz}=-A/3$. Substitution gives $\dot J_z=-32GA^2\Omega^5/(5c^5)$ and the <quadrupole formula> gives $\dot E=-32GA^2\Omega^6/(5c^5)$. Their ratio proves the balance relation. The same relation follows from the rotating wave's angular harmonic number two and frequency $2\Omega$. The additional angular-momentum flux law, not the energy flux alone, is needed to justify preservation of secular circularity.
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