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?

  • $\begingroup$ Angular velocity can be expressed in any reference frame, expressing $\omega$ in a reference frame fixed to the body ($\omega=\Omega$) has the advantage of obtaining a constant inertia matrix (for a rigid body) with respect to time, so @uhoh I do not understand the Hubble issue that you are pointing out. I can only think of situations where you have attached moving parts to your satellite, so "no matter" what reference frame you choose you cannot have a constant inertia matrix. $\endgroup$ – Julio Nov 17 '17 at 12:58
  • $\begingroup$ @Julio oh, I think I've completely misread the paragraph. Never mind. $\endgroup$ – uhoh Nov 17 '17 at 13:02

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: Molinya 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".

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