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I've done some Google research to find out.

All found sources approached the question from a mass viewpoint. That is, Jupiter has to be about 70 - 80 times more massive to be a red dwarf (through hydrogen fusion) or 13 times more massive to be a brown dwarf (through deuterium fusion).

If Jupiter was somehow given the critical mass it needs to initiate thermonuclear fusion, how hot would it be and how much pressure would it have right before fusion starts?

According to this, the minimum temperature required for hydrogen fusion is 100 million kelvin.

If Jupiter's critical mass to fuse hydrogen was met, does this automatically mean its temperature would be 100 million Kelvin? And what about its pressure?

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    $\begingroup$ "If Jupiter was somehow given the critical mass it needs to initiate thermonuclear fusion" That's a bit vague. Star formation theories assume that a roughly uniform cloud of gas and dust collapses due to gravity. As the cloud collapses, gravitational potential energy is converted to kinetic energy, which heats up the cloud and opposes the collapse process, and the cloud must shed heat for collapse to proceed. $\endgroup$
    – PM 2Ring
    Commented Oct 18, 2022 at 15:31
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    $\begingroup$ Are you just asking what is the pressure and temperature at the centre of a $0.08$ solar mass object just capable of hydrogen fusion? $\endgroup$
    – ProfRob
    Commented Oct 19, 2022 at 7:03
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    $\begingroup$ @ProfRob Yes. The reason I'm asking is that I've seen a popular science video on YouTube claiming Jupiter is "trying to fuse the first two H atoms and is very close to that". I know this to be untrue. The video bases this claim on a a recent Nature research paper trying to explain Jupiter's energy crisis. The paper claims no such thing. The video has gained a lot of popularity in my country Egypt and I'm concerned about the spread of misinformation, but I wanted to collect my facts before my next move. $\endgroup$
    – William
    Commented Oct 19, 2022 at 8:47
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    $\begingroup$ @William I fear the facts may be against you. Nuclear fusion will always proceed at some rate. There will be some finite probability of two H nuclei fusing inside Jupiter over some time interval. What matters as far as a star is concerned is whether that fusion rate is enough to supply the energy leaving its surface. $\endgroup$
    – ProfRob
    Commented Oct 19, 2022 at 8:49
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    $\begingroup$ @ProfRob I didn't know that. But the video is full of lots of other unfounded and sensational claims. For example, it tells us that "it's possible that oneday we might wake up to find two stars in our sky; the Sun and Jupiter." I'm sure you don't agree with that, do you? $\endgroup$
    – William
    Commented Oct 19, 2022 at 8:56

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The answer you link to contains this self-contradictory statement.

The minimum temperature required to fuse hydrogen is about 100 million Kelvin, which is about six times the temperature in the core of our Sun.

I cannot guess what the author means, but the statement is incorrect. The fusion of protons (the nuclei of hydrogen) occurs at about 15 million Kelvin in the Sun and at slightly lower temperatures in lower mass stars.

The energy generation rate by nuclear fusion isn't an on/off process, it occurs at a rate determined by the temperature and density of the reactants. However, the temperature dependence is strong - about $T^4$ in a star like the Sun - which means the fusion rate ramps up quickly with increasing temperature. Because of that, it is often said that there is a temperature threshold for [significant] fusion.

In the case of Jupiter, what is being discussed is not the fusion of protons but the fusion of deuterium nuclei (deuterons - consisting of a proton and a neutron). It turns out that this is far easier to accomplish and the "threshold temperature" for significant fusion is about 500 thousand Kelvin.

How close is Jupiter to that? Not very close. To a reasonable approximation, the central temperature of a ball of gas is proportional to $M/R$. The threshold temperature for deuterium burning is reached in objects that have about the same radius as Jupiter but about 13 times the mass. Using the proportionality above we might guess that the central temperature of Jupiter is at least an order of magnitude too low to initiate significant fusion of deuterium.

If you add mass to Jupiter and allow it to settle, then the core will become hotter and eventually, if enough mass is added, it would begin deuterium fusion, with almost exactly (chemical composition plays a minor role) the same temperature and interior pressure as a low-mass brown dwarf.

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