= Circumbinary orbital-period correction
For a distant coplanar <circular orbit> about a <binary star>, the averaged <quadrupole potential of a circular binary> gives
$$
P=2\pi\sqrt{\frac{x^3}{GM}}\left(1+\frac{3\mu a^2}{4Mx^2}\right)^{-1/2}.
$$
At 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.
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