Your "circles" problem is usually called "banding". It is a form of "posterization" when the gradient involved is geometrically simple (eg, part of a circle). It is most often seen in very bright sections of featureless skies.
There has been a lot written about this on the web. Google {sky banding 8 bpc}. Read the 1st hit and the links therein such as,
http://www.cambridgeincolour.com/tutorials/posterization.htm (IMHO, contains a nice example, but is written at an introductory level).
The fundamental reason for this problem is that you are trying to represent a smooth transition between two slightly different brightness levels / colors in a file format (ie, JPG) that inherently has only a limited number of of different RGB levels in that range, specifically, one which only has 256 levels available in each of the color channels (ie, an 8 bit per channel or 8 bpc image).
To verify that this is what is occurring in your 2nd image, move an eyedropper (set for single pixel resolution) over the image and carefully watch the RGB values as you move radially out from the subject. You will see that you can move the eyedropper for a considerable distance before one of the R, G or B values changes, and when you see a change in one of the numbers, you will see that your eyedropper just passed over one of the boundaries.
A similar but fundamentally unrelated problem can sometimes occur even on 16 and 32 bpc images if you have the misfortune of using a monitor (and software drivers) that actually have lower bit depth than you might be led to believe from its advertising / specs.
Obviously, the question is how to eliminate the problem. The real answer is to do all of your processing at 16 bits per channel, not 8 bpc. This should eliminate the problem when viewed or printed from PS (unless you have the problem described in the preceding paragraph). If the image is meant to be printed on a different machine, save it in a file format (eg, 16 bit per channel, flattened TIF) that supports the larger bit depth. Of course, you obviously need to be printing it on a device which also has a bit depth of more than 8 bpc.
OTOH, if your image needs to be a JPG meant for general web viewing (ie, using viewing software and computer monitors that you can't control), you've got a problem. In this case, about the best you can do is add some noise or texture to the background to break up the regularity (and obviousness) of the banding. Some folks recommend simply adding some Gaussian noise, but I think this looks artificial. I recommend overlaying that part of the image (soft blend mode, low opacity) with an image of something mid-gray and organic / realistic such as clouds, a studio backdrop, a scanned frame of film grain, etc.
HTH,
Tom M
PS - BTW, in 99% of the cases, the problem has absolutely nothing to do with screen or print resolution.
