telescopes - How do we stabilise satellites so precisely?

Look at the Hubble Ultra Deep Field photo. The stars in it are on the order of 1 arcsecond across. To an order of magnitude, this is $10^{-6}$ radians in a $10\text m$ telescope which was held steady for $10^6$ seconds.

In other words, the velocity of the aperture of the telescope around the light sensors had to be on the order of one angstrom per second.

Perhaps my maths is wrong, but this seems like an extraordinary feat of control. I can't quite believe it. The computer programmer in me suspects that, since the image was captured across a number of occasions, each occasion would be smeared somewhat less than a single long exposure and some kind of correspondence-finding algorithm could alight the images (and infer the drift of the telescope). In any case, even if that is what they did, the satellite is held amazingly steady.

How do we achieve this?

Answer

Actually reaction wheels or control moment gyros are only part of the answer. To maintain the the accuracy and precision on the order of what Hubble demands requires a fully integrated Feedback Control System of actuators and sensors. For microradian pointing, reaction wheels provide only the first stage of isolating disturbances in a multi-stage pointing control system.

Disturbances that can interfere with attitude stabilization include those from outside the spacecraft, such magnetic anomalies and atmospheric drag from planetary orbits, or solar winds from spacecraft further away from a planet - as examples. Or disturbance can come from the spacecraft itself such as vibrational modes excited by solar array stepping.

Reaction wheels or CMG's can be used to change the attitude of the spacecraft, and together with feedback from gyros or inertial measurement units (IMU's) closed loop control systems maintain the attitude to perhaps 10's of microradians in the face of the disturbances.

But to get down to microradian or submicroradian stability usually requires optical components in the line of sight that compensate for the residual higher frequency jitter that the reaction wheel control system is unable to remove. A fast steering mirror for example can be tipped or tilted to re-align the optical path according to what the imaging sensor reads from the target star or galaxy.