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I am reading this paper from the NTRS, Passive stabilization of the Long Duration Exposure Facility (LDEF), about (not surprisingly) gravity-gradient stabilization of the LDEF satellite.

It looks like it says that gravity-gradient torque was used to stabilize the spacecraft in all three axes. It was my understanding that gravity-gradient torque tends to align the minor axis of inertia with the radius vector (perpendicular to the local horizon), giving only two axes of control. So how is 3-axis control possible?

The paper mentions that the third control torque is related to the difference between the two smaller axes of inertia, so it seems to be gravitational in nature. The stabilizing torque is also not magnetic or aerodynamic, since those torques are also discussed in the paper but only treated as perturbative. What causes the third component of torque? Is it related to the nonsphericity of the Earth?

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    $\begingroup$ I can see why you've misunderstood the paper - when I started reading it it looked like they where saying it's all gravity gradient based, however on pages 11, 12 and 13 they discuss how the magnetic damper acts to damp the initial motion and then causes an oscillation due to the alignment with the magnetic field. From these we can infer that the stabilisation is based on the magnetic damper not gravity gradient for the yaw axis. $\endgroup$ – ThePlanMan Apr 3 '16 at 13:00
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Reading the paper more carefully has provided the answer. The beginning of section 4.1 states:

Attitude stabilization is provided directly by gravity gradient on the pitch and roll axes, and by "dynamic torques" on the yaw axis. The "torques" result from the fact that the spacecraft prefers to rotate about its maximum moment of inertia, and will move toward that orientation when forced to rotate at orbital rate by gravity gradient torques.

(emphasis mine)

The analysis of gravity-gradient torque I saw in school was done in an LVLH (local vertical, local horizontal) reference frame in which the direction of the gravity gradient is constant. However, it was treated as an inertial frame, when it is actually a rotating frame (making one rotation per orbit), which is where the "dynamic torques" come from.

Put another way, the gravity-gradient torque spins the spacecraft, and then the third axis is spin-stabilized (at a very low spin rate). Quite a clever idea!

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