Floating Saturn calculations

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] You’ve probably seen somewhere that the density of Saturn is less than that of water. If there were a bathtub big enough to hold it, Saturn would float. I think I first read this in one of Isaac Asimov’s collections of science essays. If you do an image search, you can easily find many illustrations of Saturn floating in water. Most of these show no more than half of Saturn under the water. Could that be right? Because there were several things I should have been doing this afternoon, I decided to work out how much of Saturn should be underwater in these images. Even better, I’d do the more general problem: how much of any sphere would be submerged in a liquid if the density of the sphere is less than that of the liquid? Here’s a cross-section of the problem: We’ll say the sphere has a radius r, a diameter d=2r, and a uniform density of ρs. The…