Let me only comment about seeing, primarily responding to the very first post from Tony.
To quantitatively understand seeing, there are at least three fundamental parameters:
1. Height (H) of atmosphere where most of the turbulence occurs. In clear, calm nights, that's about 10 km or so, plus a ground layer.
2. Freid parameter (r0), where within this size, the air can be treated as uniform and therefore little deformation of incoming wavefront. (This is more like a diameter, and I don't really know why it's written as r rather than d.). In other words, the resolution you get is the diffraction limit associated with r0, not your telescope aperture. At top-notch astronomical sites like Maunakea, this can be about 30 cm, corresponding to a seeing of about 0.4". At sites accessible to most amateurs, let's say r0 is probably between 5 cm to 10 cm.
3. Wind speed (v) in the layer that produces turbulence. This can be highly variable. r0/v is basically the time scale of seeing. For example, if r0=5cm, v=1m/sec, seeing time scale would be 0.05 sec, i.e., your image will change at a speed of 20 Hz.
So Tony is correct that when you use a small guide scope whose aperture is smaller than r0, essentially you are seeing through a more or less uniform air. The incoming wavefront is still more or less planar. As a result, you can still see the diffraction pattern of the telescope. However, although the wavefront is more or less flat, it can be tilted. A tilted wavefront appears as light coming from a slightly different direction. So this effect will make a star "dance around" at a high frequency (v/r0) in your image. This effect is called tip-tilt effect.
When you use a much larger telescope, whose aperture is a few times larger than r0, it's looking through air that contains several different zones (N = (D/r0)^2). Air in a zone can be treated as uniform, but each zone is independent. Light from these zones form diffraction limited images that dance around independently. However, their motion do not cancel each other out perfectly. The center of these speckles would still move around in the image. In many cases, even on very large telescope (D > 1m), the tip-tilt mode still contains most of the power, compared to the high order mode (the mode associated with the uncorrelated motion of the many diffraction limited speckles). So even if you use a large telescope with OAG, you can still chase after seeing.
Another important thing is the field of view r0/H. Because H is usually so large, only objects in a very small field move or scintillate together. For example, take H = 10 km and r0 = 5 cm. The associated FoV would be 1 arcsec. (5cm/10km = 0.000005 radian = 1 arcsec) So essentially every object on your focal plane dances differently, if they are more than 1 arcsec away from each other. In other words, no matter it's OAG or guide scope, you are chasing a different seeing than your main target. Well, even for your main target (usually larger than 1 arcsec), different parts of the target suffer from different seeing effects. So from this point of view, an OAG and a guide scope are equally bad.
(1 arcsec sounds bad, and this is a bit too pessimistic. The reality can be slightly better, since only part of the turbulence is from the 10km layer. The ground layer usually has some contribution and its H is much smaller. But this won't change the overall picture.)
To avoid chasing seeing, what can really make a difference is the use of multi-star guiding. The multiple objects in the guider FoV (no matter guide scope or OAG) essentially move independently from each other. By averaging their motion if there are enough of them, you can average a good fraction (if not all) of their tip-tilt mode motion so you are no longer chasing the seeing. Immediately after PHD2 implemented the multi-star guiding function, I gave it a try. There is often a noticeable difference is guiding rms before and after turning on multi-star guiding. So I now always use it as default. In some sense, this slightly favors a guide scope, since a guide scope can usually (not always) offer many more guide stars than an OAG.
So theoretically speaking, from seeing's point of view, there shouldn't be a great difference between guide scope and OAG. But there can be other factors, such as flexure. This favors OAG. So I can still believe the claim by many people that OAG improves their guiding. However, I don't think too many of them reached such a conclusion by doing carefully controlled tests, such as quickly switching back and forth between an OAG and a guide scope on the same imaging scope within a short time when conditions are similar. So although I can believe the conclusion, the exact reason (seeing? flexure? anything else?) is not too clear to me.
My 2 cents.
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