In what circumstance would it be useful for a satellite’s angular velocity to differ from its reference frame’s angular velocity?

Question

Casually reading “orbital mechanics for engineering students” on rigid body attitude dynamics, i see the following passage:

$$M_{net}=\dot{H}_{rel} + \Omega\times H$$ Keep in mind that, whereas $\Omega$ (the angular velocity of the moving xyz coordinate system) and $\omega$ (the angular velocity of the rigid body itself) are both absolute kinematic quantities, ... If the comoving frame is rigidly attached to the body frame, then ... $\Omega=\omega$.

I can’t fathom any plausible reason why one would have $\Omega\ne\omega$ at all, much less for a satellite. Is there a situation that comes up where it would be useful to describe a rigid body satellite with a rotating coordinate axis that differs from the rotation of the body itself? It seems like the authors are extremely careful to avoid saying that $\Omega=\omega$ universally, which makes me suspect that there may be cases that come up where it is best to keep them separate. What are those situations in the context of a satellite’s attitude dynamics?

Answer 1

Obviously this depends on the purpose of the satellite.

If the satellite is a space telescope, it needs to focus on distant stars; $\omega = 0$.

If it's a spy satellite focusing on a specific point on the surface, it will turn to aim at that point.

In case of satellites in highly eccentric orbit, angular velocity is often controlled on ad-hoc basis - since $\Omega$ is non-constant, the satellite traveling much slower near apoapsis than near periapsis; meanwhile $\omega$ "left to its own devices" would be constant; if the satellite is to focus at Earth, it needs to be rotated actively (example: Molniya orbit satellites).

There are many other applications where specific spin, or lack of spin is desirable - orienting panels towards the Sun, scanning surface in narrow strips through narrow beam, inertial stabilization, artificial gravity - in all these cases $\omega$ will be "custom".