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Stars are powered by nuclear fusion and the energy released radiates through their surface as heat. But all objects radiate away some form of electromagnetic radiation depending on their temperature.

Is it possible for a planet to be large enough (i.e. Have a large surface area to radiate away heat) and be hot enough (to increase the total energy emitted but NOT enough for it to visibly glow in the visible-light spectrum) to heat up a moon orbiting it to habitable temperatures?

And if so, at what distance would the moon have to orbit?

PS: You can assume the moon has an Earth-like atmosphere to moderate the temperature extremes and keep heat from radiating away like it does in our Moon.

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    $\begingroup$ An atmosphere won't "keep heat from radiating away". It might make the moon cooler than it would be without the atmosphere if the atmosphere were partly reflective (e.g., because it had clouds, like Earth or Venus), because then some of the light from the star (and the planet) would be reflected away. $\endgroup$
    – Peter Erwin
    Commented Sep 13, 2021 at 8:30
  • $\begingroup$ Are you thinking of a rogue planet? Because if it is orbiting a star, the moon will be heated up anyway. $\endgroup$
    – Alchimista
    Commented Sep 13, 2021 at 9:02
  • $\begingroup$ On a semi related note, tidal forces can be a huge factor in giving moons geothermal heat, which then can be transported through various means to the surface; take Io for example. If a moon has an atmosphere to keep heat from escaping, that would definitely heat it up, but perhaps a bit more than at the inhabitable level, and potentially with side effects that would be undesirable for life. $\endgroup$
    – Justin T
    Commented Sep 13, 2021 at 15:58

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Consider a planet with effective temperature $T$ radiating as a blackbody, emitting a total luminosity $L=4 \pi \sigma R^2 T^4$. At distance $a$ the power per square meter will be $$P = \frac{L}{4\pi a^2} = \sigma \left(\frac{R}{a}\right)^2 T^4.$$ If we demand a sun-like energy flow of $P_{required}$, we get a necessary temperature $$T=\left[\frac{P_{required} }{\sigma }\right]^{1/4}\sqrt{\frac{a}{R}}.$$

For $P_{required}=1000$ W/m$^2$, a Jupiter-like $R=70,000$ km and $a=4R$ (safely outside the Roche limit) gives $T=730$ K - hot (Jupiter is 88 K), but not extreme by any means.

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    $\begingroup$ Could you please add relevant parentheses so that the denominator in $P=L/4\pi a^2$ is easily identified? $\endgroup$
    – Stef
    Commented Sep 13, 2021 at 16:31
  • $\begingroup$ @Stef - Switched to display mode instead. Generally parentheses are not used to make denominators explicit in the works I read. $\endgroup$
    – Anders Sandberg
    Commented Sep 14, 2021 at 18:22
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Partial problem with Anders' answer: 730K is a "hot Jupiter", which I think are usually kept hot by stellar insolation, which would also heat the proposed moon. Removing that heat-source: Shortly after Jupiter formed, assuming it orbited at its present distance from the Sun, at some point it might have had a surface radiation temperature of 730K, by gravity/compression heating. Then it would gradually cool to a sunlight-controlled temperature more like 88K. So the situation in the question could exist, but only for some brief time. Any estimates how long? And perhaps it could last longer if the core had some internal heat source other than stellar fusion. Such as a large concentration of actinides, whose decay heat would not let it get cold, but also let it be a "planet" rather than a "star".

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    $\begingroup$ Yes, hot Jupiters are externally powered. But you can also get deuterium fusion (for a while) in heavier gas giants, and brown dwarfs certainly maintain their heat longer (and are roughly Jupiter sized; if they counts as planets here is mostly a matter of taste). $\endgroup$
    – Anders Sandberg
    Commented Sep 13, 2021 at 12:10
  • $\begingroup$ You could estimate the time the temperature on the Moon is reasonable by using the Luminosity calculations in Mordasiniet al .2012 aanda.org/articles/aa/pdf/2012/11/aa18457-11.pdf, particularly fig. 10 there $\endgroup$
    – AtmosphericPrisonEscape
    Commented Sep 13, 2021 at 13:34

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