Quadrupole potential of a circular binary
= Quadrupole potential of a circular binary
A <binary star> of total mass $M$, <reduced mass> $\mu$ and separation vector $\mathbf a$ has far-field <potential function>
$$
\Phi=-\frac{GM}{x}-\frac{G\mu a^2}{2x^3}(3\cos^2\theta-1)+O(GMa^3/x^4),
$$
where $\theta$ is the angle between $\mathbf x$ and $\mathbf a$. The dipole term vanishes in <centre of mass> coordinates. Averaging a circular coplanar binary over its fast phase gives $\overline\Phi=-GM/x-G\mu a^2/(4x^3)$.