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I remember being taught about Jupiter being believed to be a failed Sun. I also am aware that Jupiter releases energy like the Sun. I am sure some of this is in relation to the sun as well. But due to the distance Jupiter is, I imagine any planet that far is beyond the Goldilock zone from a star, if the same distance as Jupiter is from ours. Could Jupiter-like planets radiate enough energy to create another Goldilock zone?

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Talking about Jupiter like a failed star can be somewhat misleading. However, it is true that Jupiter radiates more heat than what it receives from the Sun.

Based on this paper Jupiter emits around 7.5 $W/m^2$ of internal heat while it reflects 0.5 $W/m^2$ of solar energy.

That energy is way way less than what the Sun gives us to sustain life, which is in excess of 1 $kW/m2$ outside the atmosphere. Probably when the planet was newly formed and hotter it could have radiated more, but I doubt that phase would have lasted enough to allow life to evolve.

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    $\begingroup$ The comparison is very much more flattering to Jupiter than it should be. The solar constant is 1.3 kW/m² at Earth's orbit, whereas those 8 W/m² emitted by Jupiter are at Jupiter's 1 bar pressure boundary, and definitely not at the orbit of Europa or Ganymede. The sun emits about 60 MW/m². $\endgroup$
    – AlexP
    Commented May 19 at 16:43
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Geothermal energy might do it from tidal forces. This can heat up a moon a lot, probably enough to sustain life(just look at the volcanos on Io). Possibly any life might also take advantage of the radiation, but that would be much less than the energy from the sun on Earth. But it is possible for life to take advantage of geothermal energy.

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Planets always form white-hot because of gravitational energy of the protoplanetary matter falling together. Then they slowly cool down by radiaton, big planets like Jupiter can have red hot outside even after a billion years.

Still, any "Goldilocks zone" there is much shorter lived than around a star. Likely not long enough for life to evolve. Any moon that is close to planet from the start will be baked and stripped of all water and other volatiles before it cools down. Plus, if it orbits too close, the tidal heating will be too strong and it will end up volcanic wasteland hostile to life like Io.

Only possibility is that a icy moon gets captured into the right orbit after the planet partially cools and its temperature decays only slowly. Requires the right combination of tidal heating and light from the planet. Very unlikely but space is also very big. Any life sustained there would have to use infrared light for photosynthesis.

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