where the products of inertia average to zero over the spin period.

Some spacecraft consist of both an inertially fixed component and a spinning component. For example, the lower portion of the OSO-8 spacecraft spins to provide gyroscopic stability while the upper portion, which consists of solar and instrument panels, is servo controlled to keep the panels pointing toward the Sun in azimuth. For such a composite spacecraft, Hooper [1977] has shown that Eq. (17-31) can be used to calculate die gravity gradient torque along the principal body axes frame by defining an effective moment of inertia. For a composite satellite, with both spinning and inertially fixed components, the effective moments of inertia applicable to gravity-gradient torques are defined as

where the subscripts 5 and / refer to the spinning and inertially fixed components, respectively. The moments of inertia on the right-hand side of Eq. (17-36) are defined about their respective component's center of mass. The symbols M and p are the total component mass and the distance of the component's center of mass from the center of mass of the composite structure.

For some spacecraft, it is convenient to average the gravitational torque over an orbit to obtain the net angular momentum impulse imparted to the spacecraft. The magnitude of the time-averaged or secular torque is often needed for the design of attitude control systems [Hultquist, 1961; Nidey, 1961 J. The time averaged value of the gravity-gradient torque (Ncc)0 for an inertially fixed satellite is defined by integrating Eq. (17-33) over one orbit,

where ju is the mean anomaly which is proportional to the elapsed time. The integration can best be carried out by changing the variable of integration from the mean anomaly to the true anomaly (see Section 3.1):

where e, a, and v are the orbital eccentricity, semimajor axis, and true anomaly, respectively. Because the spacecraft is inertially fixed, the body reference axes, X, Y, Z are constant and only R5 is a function of v. This relation is a( l-e2)

Choosing a coordinate system (h,p,q) such that h is the direction of the_ orbit normal, p is in the direction of perigee, and q = hXp, the components of Rs are given by

Substituting Eq. (17-40) into Eq. (17-38) and performing the integration, the average torque can be written as

<NCC>0=-?^=x[(/^-/IJ(Zi.)(Yh) + /v(X-ii)(Z.i.)

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