In 2010: Odyssey Two Jupiter is turned into a star, but what kind of star is it turning into?
While it's common, especially in sci-fi and popular science, to call Jupiter a "failed star", it's rather a misleading description. Let's look at some stars...
The smallest objects science considers "stars" are brown dwarfs. These are objects that are not massive enough to fuse ordinary hydrogen in their cores, but can fuse deuterium, an isotope of hydrogen that has a single neutron in its nucleus (ordinary hydrogen doesn't have any neutrons). Of all stellar bodies, these are the closest to Jupiter's mass - but even the smallest brown dwarf has AT LEAST 13 times the mass of Jupiter.
Next up in the scale, and arguably the smallest 'true' stars, are red dwarfs (no, not the mining ship). These range from about 0.075 solar masses to around 0.5 solar masses, and they're the smallest stars that exist on the Main Sequence.
For comparison, Jupiter's mass is 0.001 solar masses.
Mass is the critical component here. It's the gravitational energy from the enormous mass of even the smallest stars that compresses them enough to sustain nuclear fusion. Jupiter's mass isn't even close to high enough for that
If we can't use mass, how about temperature?
Well, there's bad news there, too...
Europa's average orbit around Jupiter is around 670,000 km. That would change a bit with the reduction in Jupiter's own diameter when it became Lucifer, so let's round up to something like 700,000 km. How hot would Lucifer have to be for Europa to be in its habitable zone?
Here is a habitable zone calculator. It returns the size in AU of the habitable zone for a star with a given temperature and luminosity. That unit, AU, gives us the first clue of how much trouble we're in. 1 AU is equal to the distance from the Earth to the sun, around 149,600,000 km. Europa's orbit equals roughly 0.004 AU.
I played with the numbers in the calculator a little bit. To get a habitable zone of 0.003 AU, I reduced the star's temperature to 1,000 kelvin, and 0.00001 times the luminosity of the sun.
These numbers are just in the range for a very cool, very dim brown dwarf (though most measures of luminosity don't go anywhere near that low). That's probably the closest to an answer we'll see.
The trouble is, of course, that Lucifer was supposed to light the night side of Earth; a very dim brown dwarf would be little brighter than the planet Jupiter, and wouldn't have anything like that great an effect.
It seems that there is no good answer - a star that could light the night side of Earth from Jupiter's orbit could not also sustain life on a moon as close as Europa. While Arthur C. Clarke's writing was generally very scientifically literate, on this one he just didn't get the numbers right.