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Substituting Eqs. (17-28) and (17-29) into Eq. (17-27) and performing some algebraic manipulation, the gravity-gradient torque may be rewritten as

where Jr;dm(=0 by definition of the center of mass and M is the total mass of the satellite. Note that the first term is zero when the geometric center is chosen to be the center of mass. The integral in the second term may be rewritten in terms of the moments of inertia. Defining the vectors r, and R, along the body reference axes (X,Y,Z), the gravity-gradient torque (assuming p=0) can be expressed as

where / is the moment-of-inertia tensor. From Eq. (17-31), several general characteristics of the gravity-gradient torque may be deduced: (1) the torque is normal to the local vertical; (2) the torque is inversely proportional to the cube of the geocentric distance; and (3) within the approximation of Eq. (17-29), the torque vanishes for a spherically symmetric spacecraft.

Many spacecraft rotate about one of the principal axes. Because the transverse axes (the principal axes normal to the axis of rotation) are continuously changing their inertial position, it is convenient to replace Eq. (17-31) with the average torque over one spacecraft rotation period. Let the spacecraft spin about the Z axis with spin rate a. The body coordinate system at time I can be expressed in terms of an inertially fixed reference frame Xg, Yq, and Z0 at f=0 as

X=cos0Xo+sin0Yo

Y=-sin0Xo+cos0Yo (17-32) Z=Zo where B=ut. The unit vector R4 can also be written as

where R°„ R°2, and R?3 are components of R, along Xq, Y„, and Z0 at /=0. The instantaneous gravity-gradient torque from Eq. (17-31) is averaged over one spin period to obtain

Substituting Eqs. (17-31) and (17-33) into Eq. (17-34), the spin-averaged gravity-gradient torque becomes

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