### 9.3 Equations of motion 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 quite complicated acceleration (131) simplifies drastically, since all
the scalar products between and the velocities are of small 2.5PN order: For instance,
, and the remainder can always be neglected here. Let us translate the origin
of coordinates to the binary’s center-of-mass by imposing that the binary’s dipole
(notation of Part A). Up to the 2.5PN order, and in the case of circular orbits, this condition
implies [7]
Mass parameters are the total mass ( in the notation of Part A), the mass
difference , the reduced mass , and the very useful symmetric mass ratio
The usefulness of this ratio lies in its interesting range of variation: , with in the
case of equal masses, and in the “test-mass” limit for one of the bodies. To display
conveniently the successive post-Newtonian corrections, we employ the post-Newtonian parameter
Notice that there are no corrections of order 1PN in Eqs. (143) 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 Eq. (146), opposite to the velocity , is
the 2.5PN radiation reaction force, which comes from the reduction of the coefficient of in the
expression (131). The main content of the 3PN equations (146) 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 [21, 22]:
The length scale is given in terms of the two gauge-constants and by
As for the energy, it is immediately obtained from the circular-orbit reduction of the general result (133).
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 Eq. (147) the expression of in terms of at 3PN order,
that we substitute back into Eq. (149), 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
The constant is the one introduced in Eq. (128). For circular orbits one can check that there are no
terms of order in Eq. (152), so our result for is actually valid up to the 3.5PN order. In the
test-mass limit , we recover the energy of a particle with mass in a Schwarzschild
background of mass , i.e. , when developed to 3.5PN
order.