5 billion years from now the sun would be dead and the solar system would be in darkness. :( But what if in 5 billion years humans would be advanced enough to move the dead iron(EDIT: I meant carbon) core of our sun, and place it at the center of Jupiter. Would the mass and gravitational effects of the dense core of our sun would be enough to start fusion of Hydrogen on Jupiter, and thus we have another sun? If not, what type of comparable extremely dense object would be required to be placed in the center of Jupiter, to cause it to start nuclear fusion due to temperature increase due to gravity?
The sun will lose about half its mass on the way to becoming a white dwarf, which will leave it with about 500 times the mass of Jupiter. It doesn't make sense to speak of "placing it at the center of Jupiter".
If not, what type of comparable extremely dense object would be required to be placed in the center of Jupiter, to cause it to start nuclear fusion due to temperature increase due to gravity?
Jupiter just doesn't have enough material to contribute meaningfully to even the smallest of hydrogen-burning stars, except perhaps in the form of something like a nova produced by a planet merging into a degenerate star. It would be hundreds of times easier to just move Jupiter and what's left of the rest of the solar system to another star. It's not as if the solar system would survive having that amount of mass replacing Jupiter anyway.