### 5.4 Slightly eccentric orbit around a Schwarzschild black hole

Next, we consider a particle in slightly eccentric orbit on the equatorial plane around a
Schwarzschild black hole (see [71], Section 7). We define as the minimum of the radial
potential . We also define an eccentricity parameter from the maximum radius of the
orbit , which is given by . These conditions are explicitly given by
We assume . In this case, is given to by
where is the orbital angular frequency in the circular orbit case. We now present the
energy and angular momentum luminosity, accurate to and to beyond Newtonian order.
They are given by
and
where is the Newtonian angular momentum flux expressed in terms of ,
and the -independent terms in both and are the same and are given by the terms
in the case of circular orbit, Equation (178).