

5.5 Circular orbit with a small inclination from
the equatorial plane around a Kerr black hole
Next, we consider a particle in a circular orbit with small
inclination from the equatorial plane around a Kerr black
hole [52]. In this case,
apart from the energy
and
-component of the angular momentum
, the particle motion
has another constant of motion, the Carter constant
. The orbital plane of
the particle precesses around the symmetry axis of the black hole,
and the degree of precession is determined by the value of the
Carter constant. We introduce a dimensionless parameter
defined by
Given the Carter constant and the orbital radius
, the energy and
angular momentum is uniquely determined by
, and
. By
solving the geodesic equation with the assumption
, we find that
is equal to
the inclination angle from the equatorial plane. The angular
frequency
is
determined to
and
as
We now present the energy and the
-component angular
flux to
:
Using Equation (188), we can express
in terms of
as
We then express Equations (189) and (190) in terms of
as

