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In the weak-field, slow-source approximation, the leading luminosity is the quadrupole formulain units $G=c=1$, whereOnly the $\ell =2$ symmetric trace-free coefficient contributes. Indeed,and contraction with traceless $a_{kl}$ removes the first term. ThereforeThe power crossing the large sphere at time $t$ is consequentlyThe monopole is conserved total mass, the dipole is center-of-mass motion, and every $\ell \ne 2$ term is orthogonal to the trace-free quadrupole at this leading order.

The quadrupole formula in units $G=c=1$ isHere $I_{kk}=m{a}^2/3$ is constant, so its trace subtraction has no third derivative. If $B=4m{a}^2{\omega }^3/3$, thenThus ${\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}_{ij}=2B^2$ andRestoring units multiplies this by $G/c^5$.