First, even as is, it's better than any image of Saturn I've managed (though I admit I haven't tried in years and I was using horrible webcams).
The trouble is that there colour noise in the data. So when you bump up the saturation it comes out as looking grainy which to the eye does not look sharp. If you split the image into red, green and blue channels, you'll see blue is very blotchy, and red a little blotchy, but green is good. The more you sharpen the more this blotchiness is standing out and the less sharp it looks.
You've got a few options here. (I'm using GIMP and old version of Paintshop Pro terminology. The ideas are the same though).
1) Greyscale the image and recolourise. The trouble is the easiest way to colourise - basically change it to yellow - doesn't look at all natural and you lose detail. I wouldn't bother with this one.
2) Decompose into channels. Copy and individually blur the blue, and perhaps the red channels AND BLUR THEM. Copy them back in and recompose them. The trouble is if you blur 2 out of 3 channels you're going to end up with a less sharp image.
3) A better way is to posterize or decrease the colour depth. It looks better to me if I decrease the colour depth to 256 colours (using whatever options best smooth the changes between colours do it doesn't look like a mess) then apply moderate saturation. This worked well for me on your image...but you can do better...
4) Finally consider how you're sharpening. You don't want to sharpen the colour noise. There are techniques for sharpening just the edges. If you use a mask to just sharpen the edges you want to, your image will look sharp but noise in the less defined parts won't be affected.
Here's one of the most comprehensive reviews of different sharpening techniques I've seen:
http://ronbigelow.com/articles/sharpen1/sharpen1.htm
through
http://ronbigelow.com/articles/sharpen1/sharpen6.htm
Here's a simpler one
http://bythom.com/sharpening.htm
