### 5.6 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 [94]. 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
Update 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 (198), we can express in terms of as
We then express Equations (199) and (200) in terms of as