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).