You know, despite your username, I never realized you were THE Bad Astonomer responsible for badastronomy.com, which is one of the all-time coolest places on the Internet, in the finest tradition of the Straight Dope, Snopes, etc. I bow in humble worship of your awesome splendor.
If fusion didn’t make the sun so hot, it too would have a rocky core. As this article on sungrazing comets describes, the sun continues to accrete heavy elements even today.
Certainly Jupiter doesn’t have sufficient mass to ignite in its current form, otherwise we would have a second star in the system.
But what if Jupiter weren’t so “fluffy”. What if Jupiter (with the same mass) were not a gas giant but made of rock instead?
Not that I expect a difinitive answer, but I’m just pointing out that the “star threshhold” can’t possibly be based on mass alone. (can it?)
The Sun’s core would be hot anyway, even without fusion. That much pressure would heat things up considerably.
Jupiter, last I heard, probably does not have a rocky core so much as one made of liquid metallic hydrogen. That exhausts my knowledge of it. 
And yes, I’m the one at badastronomy. I can’t accept any supplication because I think I have totally botched my bulletin board upgrade. 
If I fix it, then I will happily take whatever praise and glory I can reap.
What bulletin board are you using, BA? I’ve done coding work on several using tForum, and I’m conversant with PHP and Perl programming.
Well, you’d be hard-pressed to find that much rock and bring it together to one place. Pretty much everything in the Universe of that size is mostly hydrogen, so of the things which realistically can change, mass is the deciding factor.
But even if you did somehow manage to get a lump of rock that size, I’m not sure what would happen. The central pressure would, indeed, be higher, but then again, it’s a lot harder to fuse silicon and the other heavy elements (in astronomy, anything heavier than helium is “heavy”) than it is to fuse hydrogen, so I don’t think you’d have fusion. I suppose you’d end up with some sort of degenerate matter in the center, perhaps similar to a white dwarf.
The model we have been using for Jupiter has a rocky core about 10 Earth masses in the middle. I suppose if Jupiter were hot enough to be a star, this rock, what ever it is made of, would show up as ‘metallic’ vapour in the star’s spectrograph - metallic in this context means anything non volatile, I believe.
http://www.solarviews.com/eng/jupiter.htm (not an OA link, btw)
After several thousand years of gas mining, you see, you could end up with a rocky planet to live on- easier than trying to live in a gas giant’s atmosphere.
But if you did use the Clarke method of igniting Jupiter, I don’t think you would get a very bright sun- it is still very small, and far away, compared to Sol.