- Consider the secular evolution of two planets moving around a star in coplanar orbits of low
eccentricity. Let , , and be the orbital major radius, eccentricity, and longitude of the periastron (i.e., the point
of closest approach to the star)
of the first planet, respectively, and let , , and be the the corresponding parameters for the second planet. Suppose that . Let
,
,
,
and
. Consider normal mode solutions of the two planets' secular evolution equations of the form
,
,
, and
,
where
,
, , and are constants. Demonstrate that
- The gravitational potential of the Sun in the vicinity of the planet Mercury can be written
- The gravitational potential in the immediate vicinity of the Earth can be written
Demonstrate that, when averaged over an orbital period, the disturbing function due to the term takes the form

respectively. 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) raterespectively, where and are constants. (Modified from Murray and Dermott 1999.) - Demonstrate that, when averaged over an orbital period, the kinetic and potential energies of an object of mass
executing a Keplerian orbit of major radius about an object of mass (in a frame of reference in which the latter object is
stationary) are
- Using the notation of Section 10.6, show that when an artificial satellite interacts with the Earth's upper atmosphere, its
orbit-averaged perigee and apogee distances evolve in time as
respectively. Demonstrate that in the limit , in which the difference between the apogee and perigee distances is much greater than the scale height of the atmosphere, the previous expressions reduce to
- Consider an artificial satellite in a circular orbit of radius around the Earth. Assume that the altitude, , of the orbit is much less that the
terrestrial radius, . Using the notation of Section 10.6, show that the time, , for the orbit to decay from an initial
altitude to zero altitude is
- Consider a large dust grain orbiting the Sun. Making use of the notation of Section 10.7, demonstrate that
the time evolution of the grain's orbital major radius, , and eccentricity, , under the action of solar radiation pressure is such that