In our local astronomy club we are debating the effectiveness of available amateur tip-tilt adaptive optics at removing the effects of seeing. The promotional literature shows tighter stars with higher peaks using the AO. Could it be that the AO is only compensating for errors in the telescope mount? Or is it possible that AO can partially compensate for bad seeing? They advertise 5 to 10 corrections per second.
Adaptive Optics (AO) systems measure and correct any distortions coming into your telescope/detector. It does this by measuring in real-time the distortion on the incoming 'wavefront' and applying a correction to it, flattening out the distortions so you obtain a sharper image on your detector. Typically AO is used to measure and correct the atmospheric effects (i.e. bad seeing) on a wavefront coming into a telescope system but it can be used to correct systematic problems. As such, you could use it to correct for errors in the mount providing they are not too severe although as these effects are most likely repeatable, mounts can either be trained or the use of an auto-guider may be as effective and cheaper.
If you are trying to remove the majority of the atmospheric distortions you would need a system running at many hundred times a second and this is what many of the professional large telescopes around the world do today. It is however possible to get a correction with a much simpler system by thinking about turbulence being made up of different scales of distortion. You have larger scales i.e. tip-tilt, defocus etc. which will affect the whole of your image while much smaller scales will only effect a smaller part. This means you can think of these large scale distortions causing most of the spread of the PSF and if you can remove these effects from the system, you can improve your image. For tip-tilt, typically you will get a factor of two improvement in image quality but this improvement will only be as good as the measurements and corrections of the AO system - think of it more as a 'theoretical' limit. If you can also remove the defocus term, by that point you'll have image quality by about a factor of 5. The more corrections you do, the greater this factor becomes but to do these extra corrections adds complexity, cost etc.
Regarding timescales and seeing - the timescale of correction is set by how fast the atmospheric turbulence changes. This is principally set by the 'wind crossing time' of a telescope (i.e. how long it takes cell of air (think of it as a cubic metre) to move from one side of the telescope entrance aperture to the other) and how many 'turbulent cells' you have across an aperture. Turbulent cells of air are typically 10-20cm across in size (this is seeing dependent - on a clear night it could be bigger) and so across the diameter of a 4 metre telescope we have approximately 20. As such, you need to correct at least 20 times a second on a 4m telescope and for smaller telescopes, you can work slower than that. The slower you can go, the easier the system is as the correcting mirror can move more slowing and you can collect more photons from your reference star before needing to make a measurement i.e. improving sensitivity. There is however a limit to the level of corrections you can make - the correcting mirror will have a limited 'stroke' on each element which is the maximum movement possible. As such, if the tip-tilt value is greater than this (at the edge of the mirror) you will be unable to correct the distortion. This is why typically AO systems on large telescopes do not work well with seeing greater than 1-1.5" (there is additional complexity) although on smaller telescopes some level of correction should still be possible.