From what I'm looking up, 13x Jupiter mass to become a brown dwarf star and around 80x Jupiter mass to become a red dwarf star.
Yep those are the typical values however an additional nuance is that mass is only one factor, what also plays an important role in determining the minimum amount of mass needed to be a star is the rate of heat loss/retention. The critical threshold for fusion is directly controlled by temperature in the core. Thus other critical factors are metallicity, which controls properties like the thermal conductivity and emissivity of a substance, and the timescale of formation. The key determining factor is whether the core of a body is able to reach the temperature threshold for nuclear fusion before it can cool off enough to reach electron degeneracy pressure.
Electron degeneracy is a quantum mechanical property that arises as electrons are a type of particle known as fermions which are forbidden from sharing the same quantum state. When there are more available electrons than a substance's temperature and pressure can thermodynamically support it becomes electron degenerate.
Incidentally electron degeneracy is indeed a very exotic and truly quantum mechanical state of matter, but in my graduate level statistical mechanics class I learned that it is also one shockingly familiar one that answered a fair number of questions about chemistry and why various elements act the way they do.
It turns out that while electron degenerate matter can arise under very high pressures it also naturally arises when an element has too many electrons relative to what their temperature can support. In particular have less excited electron states when they have a filled valence shell configuration. This coupled with the sheer number of electrons around heavier atoms and certain atoms with just a few unpaired electrons in a higher energy shells can result in these elements having too many electrons compared to what temperatures on Earth thermodynamically supports, and no way to get rid of them. This is to say the outer electron orbitals are in higher states of energy than their surroundings but is unable to fall into a lower energy state) beyond their filled valence electron shell configuration becomes electron degenerate. These materials have high thermal and electrical conductivities are largely incompressible regardless of other physical properties due to quantum mechanics since electron degeneracy only fails at around 1.44 solar masses or if a sudden injection of heat is sufficient to heat it over a critical degeneracy pressure
Once electron degeneracy pressure is reached, i.e. the gas becomes metallic gravitational collapse stops and thus is cut off as a heat source from that point on it would take a considerable effort. Higher metallicities for instance can mean more silicate and or metal oxide clouds which trap heat etc. since the upper atmospheres of these kinds of objects are cool enough for refractory elements/compounds to condense out. As usual the more closely you look at something the more complicated things get.
Because of these properties there is a fuzzy bit of wiggle room between star and not star so you can have brown dwarfs above 80 Jupiter masses (
https://iopscience.iop.org/article/10.3847/1538-4357/aafac8) and stars down to perhaps as low as 75 Jupiter masses with enough of a heat injection. As a consequence there is a fuzzy region where the characteristic neutrino detection of proton proton chain fusion would be the only way to tell for sure you are dealing with a star and not a very heavy brown dwarf at least without a reliable independent age estimate.
Either way Jupiter is very far away from that limit and Saturn is "barely" a gas giant
Which is somewhat surprising to me. As a kid, I had always heard that Jupiter was a "failed star" which meant to me that it was on the brink of becoming a star; just on the threshold and anything large enough crashing into it could cause ignition. In my mind, I thought that if Saturn were ever captured by Jupiter and sucked in, we'd have a second star.
Technically the notion of a "failed star" isn't entirely false its just a gross underestimate. Back in 2018 there was a story about the very long baseline interferometry of a extremely massive protostar (about 40 solar masses and still actively accreting) revealing that within its accretion disk was a planetary like system that features a 0.5 solar mass object with its own accretion disk probably not unlike how Jupiter once did. With the extreme mass ratio models indicate the smaller star almost certainly formed like a planet.
So if there is enough material available during star formation you can get a planet accreting enough material to become a star.
Technically thanks to more comprehensive study of the galactic center resolving the anomalously young stars to a planar disk around Sagittarius A* this process of planet formation seems to be general to accretion with the accumulated mass being limited to that active accretion window and the amount of mass an object was able to accrete in that interval allowing young massive stars to form in orbit of Sagittarius A* and models suggest it is very likely that its more massive brethren across the universe form similar systems during their active accretion intervals.
So by the same standard that we can arguably call all gas giants failed brown dwarfs and all brown dwarfs failed stars we can also call ice giants failed gas giants and so on.
If you want to get into the technical details evidence is building up based on isotope fractions which suggest that most of Earth and other planets volatiles are native to the original world likely having been accreted in complex interstellar dust grains rich in volatiles with the limiting factor on volatile retention being the amount of material able to be accreted before differentiation starts to dominate. (That is to say the isotopic ratios of Earth's Nitrogen appears to match that of undifferentiated asteroids of the inner solar system and not the icy bodies of the outer solar system rich in ammonia and molecular nitrogen ices. If this holds up then planets largely might start out as the result of either very rapid and or direct collapse of interstellar dust and other molecules within giant molecular clouds or similar environments with the composition largely governed by what a world has the mass to hold onto.
In this extended case then one could argue that any accumulated body is a failed star and every star is a failed black hole. Its a useless distinction but it isn't necessarily false just vastly underscores the complexity.
But yeah Even if Saturn and Jupiter collided it wouldn't be anywhere near enough to make a star.
Something about 18x the mass of Jupiter, IIRC, is needed to get it to "stardom".
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