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Jupiter is approximately one thousandth the mass of the sun. small stars that we see are about one tenth the mass of the sun and the smallest stars that glow are probably about one hundredth the mass of the sun so Jupiter needs about ten times as much material to start to glow from mass alone. However in the story of 2010 the monoliths didn't just add mass to Jupiter they added energy by all flying together to meet at the centre of the planet so if you arranged for this to happen you could probably get away with a lot less mass.The Chandrasekhar limit for the collapse of stars to neutron stars is about 1.4 solar masses and the limit for the collapse of neutron stars to black holes is about 2-3 solar masses but this does not mean that all stars more than three times the mass of the sun are in imminent danger of collapse into a black hole because this is in fact very difficult for several reasons. notably the gravitational and nuclear fusion energy and residual angular momentum so you need to start with a star more than ten times the mass of the sun before it has a good chance of collapsing int a black hole.
It has been recognised recently that there is a continuum between planets and stars with a class of dwarf star that heats up due to gravitational commpression but does not become sufficiently hot in its centre for nuclear fusion to take place.Over many millions of years these cool down and become planet like objects
Yes it would in theory be possible for a rocky body to become a star if it was big enough.