5.9 Adiabatic evolution of constants of motion for orbits with generic inclination angle and with small
eccentricity around a Kerr black holeUpdate
The calculation in Section 5.8 was extended to orbits with generic inclination angle by Ganz et al. .
We specify the geodesics by the semi-latus rectum and the eccentricity and a dimensionless
inclination parameter . The outer and inner turning point of the radial motion is here define as
The inclination parameter is defined by
which is the same as Equation (210). We define . By solving these equations with respect to
and , we obtain
The average rate of change of , and become up to ,
Here, a term is factored out. We can express in terms of by using
Equation (3.15) in  as
The average rate of change of , and are rewritten as
If we assume that the inclination angle is small and , we find that
Equations (229) – (231) reduce respectively to (220) – (222) in Section 5.8. As discussed in , in the case
of largely inclined orbits, the fundamental frequency of gravitational waves is expressed not only with
but also the frequency of -ocillation, .