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The short answer is that Jupiter isn't big enough. Hydrogen on the Sun isn't "burning" as we know it -- that would combine with oxygen and release water. There's no oxygen at the Sun. Its mass and resulting gravity cause the hydrogen to begin fusing into helium, which is an exothermic reaction.
So for Jupiter to ignite we'd need to increase its mass by about three orders of magnitude (Jupiter is about 1.9*10^27 kg, the Sun is about 1.9*10^30 kg). This sounds simple enough, we just need to find some more Jupiter-stuff and throw it on the pile, right?
Are there any physicists around who can check my work here? ;)
If you'd settle for something smaller, it turns out that the requirement for Jupiter to "become" a star (self sustaining hydrogen fusion reaction in its core) is lots less. 8 Jovian masses does the trick.
Ok, so Jupiter * 8 == star. How long would the reaction last, though, if we only gave it the bare minimum mass?
In any case, I'm glad we have a physicist here in the pub with us, and not a cosmologist. Cosmologists are morons who make stuff up. A particularly stupid one named Stephen Hawking just said this week that you can pass through a black hole and come out in another universe. Yeah, I saw that movie 36 years ago. He then went on to say that if something goes into a black hole, it turns into a hologram. You cannot make this crap up! I wonder if he's trying to out-moron Dawkins and Tyson.
Black holes shenanigans: You go over the rim and come out a Rimmer? Scary!!
https://en.wikipedia.org/wiki/Arnold_Rimmer
The smaller the initial mass of any star, the longer it spends before becoming a dwarf. In this case, it would 'almost' be a dwarf at birth.
The dwarf lifetime of a small mass star is hugely long. Again, if I remember this 40-plus years ago exposure to the thermonuclear "chain of events" in a small mass decaying star. There is no nova event; and certainly no supernova event - these are what speeds up the end-stage of most stars. It won't even do the "red giant" thing - not enough mass. It will die a very very slow and very very boring death. Which, strange as it may seem, is ideal for the concept of Jovian moons developing life. They'd have, as we would say in Philly, "ooh-gobs" of time in which to do that!!
But Jupiter itself is nowhere near massive enough for that stuff. Any life developing on Io or Europa will have to do that on its own (or have help from The Monolith!!)<evil grin>.
Europa my old friend... We'll be the Pirate Twins again. Bartender, I seem to have dropped my glass.
But you'll need to supercool it to get it back into a liquid state.
(the *fun* version where you throw them at each other instead of at the dart board)
I am not talking about the new style of Jartz, I am talking about the old kind, a large metal spike with fins. A large dart.