This famous problem is discussed in most books on satellite dynamics (see, e.g., [22, 133]). The 24 solutions discussed subsequently date to Lagrange [154, 156] in the late eighteenth century. Among the remarkable aspects about Lagrange’s extraordinary work on this topic in [156] is the (early) use of his celebrated equations of motion in the context of a rigid body and his clear discussion of a set of (what are now known as) 3–1–3 Euler angles. As shown in Figure 9.17, consider a rigid body B of mass m that is in motion in a central gravitational force field about a massive fixed body of mass M. The center of this force field is assumed to be located at a fixed point O. The force, moment, and potential energy of the field are given by approximations (8.18).
(a) Verify that
What is the physical relevance of this result?
(b) Why are the angular momentum HO and the total energy E of the satellite conserved?
(c) Using the balance of linear momentum, show that it is possible for the body to move in a circular orbit x¯ = r0er about O with a constant orbital angular velocity θ˙ 0 , which is known as the modified Kepler frequency, ωKm:
where the Kepler frequency was defined previously (see (2.13)):
In (9.39),
and this vector is an eigenvector of J. That is, er is parallel to one of the principal axes of the body.
(d) Using the results of (c), show that a steady motion of the rigid body, that is, one in which ω˙ = 0, is governed by the equation
(e) Suppose that the body is asymmetric. That is, the principal values of J0 are distinct. We seek solutions of (9.40) such that ω · er = 0. Show that there are six possible solutions for ω that satisfy (9.40) and four possible solutions for er. Here, you should assume that J is known and as a result Q is known. Hence, there are 6 × 4 possible solutions of (9.40).
(f) Suppose that the body is such that J = µI, where µ is a constant. Show that any constant ω satisfies (9.40) and consequently any orientation of the rigid body is possible in this case.
(g) Using the results of (e), explain why it is possible for an Earth-based observer to see the same side of a satellite in a circular orbit above the Earth.