### 9.4 Equations of motion and energy for circular orbits

Most inspiralling compact binaries will have been circularized by the time they become visible by the
detectors LIGO and VIRGO. In the case of orbits that are circular - apart from the gradual 2.5PN
radiation-reaction inspiral - the complicated equations of motion simplify drastically, since we have
, and the remainder can always be neglected at the 3PN level. In the case of circular
orbits, up to the 2.5PN order, the relation between center-of-mass variables and the relative ones
reads [16]
To display conveniently the successive post-Newtonian corrections, we employ the post-Newtonian
parameter
Notice that there are no corrections of order 1PN in Equations (187) for circular orbits; the dominant term
is of order 2PN, i.e. proportional to .
The relative acceleration of two bodies moving on a circular orbit at the 3PN order is
then given by

where is the relative separation (in harmonic coordinates) and denotes the angular
frequency of the circular motion. The second term in Equation (189), opposite to the velocity
, is the 2.5PN radiation reaction force (we neglect here its 3.5PN extension),
which comes from the reduction of the coefficient of in Equations (182, 183). The main
content of the 3PN equations (189) is the relation between the frequency and the orbital
separation , that we find to be given by the generalized version of Kepler’s third law [37, 38]:
The length scale is given in terms of the two gauge-constants and by Equation (184). As for
the energy, it is immediately obtained from the circular-orbit reduction of the general result (170). We have
This expression is that of a physical observable ; however, it depends on the choice of a coordinate
system, as it involves the post-Newtonian parameter defined from the harmonic-coordinate separation
. But the numerical value of should not depend on the choice of a coordinate system, so must
admit a frame-invariant expression, the same in all coordinate systems. To find it we re-express with
the help of a frequency-related parameter instead of the post-Newtonian parameter . Posing
we readily obtain from Equation (190) the expression of in terms of at 3PN order,
that we substitute back into Equation (191), making all appropriate post-Newtonian re-expansions. As a
result, we gladly discover that the logarithms together with their associated gauge constant have
cancelled out. Therefore, our result is
For circular orbits one can check that there are no terms of order in Equation (194), so our result for
is actually valid up to the 3.5PN order.