where R1 = d2R1/dt2 and t represents time. It can readily be shown that the equations of relative motion are r = "7r (2.2.3)

where ¡i = G{M1 + M2). In the case of an arti ci al satellite of the Earth, let M1 represent the Earth and M2 the satellite. Since the Earth is so massive, even a large satellite like a fully loaded space station results in

to very high accuracy (better than one part in 1015).

The general characteristics of the motion follow from manipulation of Eqs. (2.2.1) and (2.2.2):

1. Motion of Two-Body Center of Mass. The de nition of the position of the center of mass of the two spheres, Rcm, is

where C1 and C2 are constants of the motion. Equation (2.2.5) states that the center of mass of the two spheres moves in a straight line with constant velocity. Note that the straight line motion of the center of mass is de ned in an inertial coordinate system.

2. Angular Momentum. Taking the vector cross product of r with both sides of Eq. (2.2.3) gives d rxr = 0 or — (r x r) = 0 .

Defiling h = r x r, the angular momentum per unit mass (or specific angular momentum), it follows that h is a vector constant:

An important interpretation of h = constant is that the orbital motion is planar, since r and r must always be perpendicular to h. The plane of the motion is referred to as the orbit plane, which is a plane that is perpendicular to the angular momentum vector. Note that the angular momentum in Eq. (2.2.6) is defiled for the motion of M2 with respect to Mi.

3. Energy. Taking the vector dot product of r with Eq. (2.2.3), it can be shown that

2 r where £ is the scalar energy per unit mass (also known as specific energy or vis viva) and r is the velocity of M2 with respect to M1. Thus, the energy per unit mass is constant; that is, the sum of the kinetic and potential energies is constant. Note that this expression describes the energy in terms of the motion of M2 with respect to M1.

In summary, there are ten constants of the motion: C1, C2, h, £. The following general properties of the orbital motion can be stated:

a. The motion of M2 with respect to M1 takes place in a plane, the orbit plane, defined as the plane that is perpendicular to the constant angular momentum vector (h = r x r);

b. The center of mass of the two bodies moves in a straight line with constant velocity with respect to an inertial coordinate system;

c. The scalar energy per unit mass is constant, as given by Eq. (2.2.7).

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