Autoguiding accuracy and pixel scale
Your sensor is made up of a dense matrix of pixels. For a small star to elongate, the ray of light coming from that star has to move from one pixel to its neighbour. You start to have a problem if your average guiding error is larger than the angular size of your pixels. If your average error is smaller than this, the light ray won’t hit the next pixel.
So, what does one pixel translate to in terms of arcseconds in the sky? That depends on your pixel size (which can be between around 2µm and 5µm) and your focal length. The formula is 206.265 x pixel size / focal length. Astronomy tools tells me that the “pixel scale” for my refractor and camera works out at 1.4 arcseconds per pixel. My Labradors can whack my tripod with their big waggy tails and shake the scope up to 1.4 arcseconds without damaging my image – whatsoever.
So, if your average guiding accuracy is less than your pixel scale, quit worrying. Your guiding is fine.
Exposure time and seeing
One of the first things you learn when getting into autoguiding is that the atmosphere itself is very dynamic. Look carefully, and it’s like a drunken duck. Pockets of warmer and colder air are swirling and mixing above us all the time. This causes lensing effects that change constantly. In theory, you could have your autoguide system follow all these little wobbles, but it’s a pointless waste of time. The saying in autoguiding is “don’t try to chase the seeing”.
The way to avoid this rabbit hole is to take longer autoguide exposures. This way, your computer will see the average position of your guide star, rather than a swirling, wandering target. Exposures of 2-3 seconds are normally sufficient to achieve this.
Seeing and autoguiding accuracy
While we’re on the subject of seeing, it’s pretty obvious that poor atmospheric conditions are going to muck up your image. But image degradation resulting from poor seeing is far more significant than image degradation from poor guiding.
I’ve used criteria from https://astronomy.tools/calculators/ccd_suitability. On a night with “OK” seeing conditions, the “full-width half-maximum” size of a star (that is, the amount the point source light gets smeared into a blob) can be between 2 and 4 arcseconds. So even if I have perfect guiding, my rig, with its 1.4 arcseconds per pixel, just can’t get a star onto a single pixel. I’d need a night with “exceptional seeing” (between 0.5 and 1 arcseconds FWHM) to be able to achieve that.
Of course, this seeing error will be equally spread between RA and Dec axes. So those stars will still look round, just larger. Add a little poor guiding to the mix and they’ll go a bit egg-shaped, but the worse the seeing the less obvious it will be.