Pete, 0.55" arc is basically Raleigh for a 10" aperture. That would be located about 0.82 spacial frequency well beyond the 'hump' formed by the perfect obstructed aperture. I suppose 0.55 is near Dawes for an 8" aperture (the red lines), so the spacial frequency would be 0.9 or so. Both are well above where the 5" (green line) craps out.
Now, I am sure a 5" can easily observe the polar cap that small just as it could a star, but it will be more of a point source at about half the Raleigh limit for that aperture. Being at half the Raleigh limit, one might imagine the polar cap might exhibit some elongation in the 5" not unlike observing a tight double at half the Raleigh limit (you and I have done that.) But, the greater resolution of the 8 and 10" apertures have more resolution space to work with being the polar cap in withing Raleigh and Dawes, respectively.
For the 5" aperture, 0.55" arc is well below the sparrow limit of 107/127mm = 0.84" arc. So, at that very high spacial frequency there is not enough contrast to see much. Contrast fell to zero in the 5" at about 0.62 spacial frequency which correlates to the maximum spacial frequency of the 5" scope which (correct me if I am wrong) would be near the Sparrow limit of 0.84" arc. This is what the green line falling to zero is telling us....the aperture cannot put any contrast into the image for something that small - that high a spacial frequency.
Now, on an object that small, but not quite a point source (at 1/4th Raleigh), it may be possible to observe some brighter, high contrast albedo not unlike observing Osiris on Ganymede (which isn't really resolved, technically, as much as it is just seen.) But, such features that small will just not have the contrast of a Dawes split at around 5% contrast (because contrast transferred to the image is zero) for actual resolution of two separate points within the tiny polar cap. The 8 and 10", however, being the polar cap is at Raleigh and Dawes, respectively, should be able to separate two distinct points on a polar cap 0.55" arc across. The 5" cannot, contrast fell to zero closer to 0.84" arc.
That is what the 8" red lines are telling us as they fall to zero well above the 5" green line. Notice at Raleigh and Dawes for the 8", way out at 0.82 and about 0.9 spacial frequency, contrast (left hand scale) is not at zero. Contrast transferred to the focal plane does not get to zero until the red lines hit the maximum spacial frequency at 1. And actually, the obstructed scope's contrast falls to zero out to about 1.1 because in the graph above that "hump" formed by the obstruction is made to - it's normalized - to 1 for the unobstructed aperture 8" aperture. That hump actually extends out to about 1.1, give or take. Still, with low actual object contrast on a bright polar cap, it's possible neither the 8" or 10" could resolve two separate bright points either even though they are separated by enough. But, they stand a better chance of it.
That's kind of my take on it.
One caveat, though. It's always dangerous to talk about extended object resolution in terms of Raleigh and Dawes because both apply to point sources separated by some angular dimension with black space between them. They are two bright stars sitting in black object space with 100% contrast between them. As the aperture forms an image of these two stars very close together, the image contrast falls off as shown by the line. At Dawes (o.55" arc), or about 0.9 spacial frequency (for the 8" scope - red line) the image looses about 90% of the image's 100% actual contrast. In the case of double stars, there remains enough contrast to see a dark space between two points.
In the case of two bright spots on on a very small bright polar cap, the initial contrast is low to begin with. After being imaged and loosing 90% of it's already low contrast, the actual contrast transferred to the image might be too low to be perceived. But, imagine if the polar cap were black, those two points could be resolved in the 8" (Dawes), more so in the 10" (Raleigh), but not at all in the 5" (well below sparrow)...all with a polar cap of 0.55" arc.
Edited by Asbytec, 13 October 2014 - 05:13 AM.