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 functionwhere $\theta$ 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 $\overline{\Phi }=-GM/x-G\mu a^2/(4x^3)$.

For a distant coplanar circular orbit about a binary star, the averaged quadrupole potential of a circular binary givesAt fixed outer radius $x$, the quadrupole shortens the period relative to the monopole result. If the inner binary evolves slowly, the outer specific angular momentum stays constant in the axisymmetric averaged potential, but the outer energy can change. The outer radius must then be allowed to adjust when computing a period derivative.