The gravitational potential in the immediate vicinity of the Earth can be written
where , is the terrestrial mass, , , are spherical coordinates that are centered on the Earth, and
aligned with its axis of rotation, is the Earth's equatorial radius, and
(Yoder 1995). In the preceding expression, the term involving is caused by
the Earth's small oblateness, and the term
involving is caused by the Earth's slightly asymmetric mass distribution between its northern and southern hemispheres. Consider an artificial satellite in orbit around the Earth.
Let , , , and be the orbital major radius, eccentricity, inclination (to the Earth's equatorial plane),
and argument of the perigee, respectively. Furthermore, let
be the unperturbed mean orbital angular velocity.
Demonstrate that, when averaged over an orbital period, the disturbing function due to the term
takes the form
Hence, deduce that the term causes the eccentricity and inclination of the satellite orbit to evolve in time
Given that the (much larger) term causes the argument of the perigee to precess at the approximately constant (assuming that the variations in and are small) rate
deduce that the variations in the orbital eccentricity and inclination induced by the term can be written
where and are constants. (Modified from Murray and Dermott 1999.)