# What's lowest mass object that can form from collapsing gas?

Is there a minimum mass for an object that can form on its own from collapsing interstellar gas without being bound to a larger star's protoplanetary disk first?

There are red dwarfs, brown dwarfs, sub-brown dwards and there may be even lower mass objects we just can't detect because they are to faint. What I read about rogue planets and rogue comets it seems they can only be ejected from a star system and can't form on their own. Why?

The Jeans instability occurs when there is a cloud with a mass larger than the Jeans length, or $$M_J > \frac{\pi}{6}\frac{c_s^3}{G^{3/2}\rho^{1/2}}\sim (2M_\odot)\left(\frac{c_s}{0.2 \text{km s}^{-1}}\right)^3\left(\frac{n}{1000 \text{cm}^{-3}}\right)^{-1/2}$$ where $$c_s$$ is the speed of sound in the gas and $$n$$ is the number density. The above values make sense for normal astrophysics.

However, by making $$n$$ larger we can clearly make $$M_J$$ smaller. In an ideal gas the speed of sound only depends on temperature, not the pressure or density since they increase in lockstep ($$c_s=\sqrt{\gamma (p/\rho)}=\sqrt{\gamma k_B T/m}$$ where $$m$$ is the molecular mass). In practice this will fail when intermolecular forces become significant. So somewhere between air density ($$n\approx 2.53\times 10^{17}$$ per cubic centimeter) and water density (about 1000 times more) this formula will really break down, mostly because the cloud is no longer a gas but a liquid. That implies a Jeans mass somewhere close to $$4.24\times 10^{-7} M_\odot$$. That is 14% of an Earth mass.

But even smaller gas clouds could of course collapse if we make more extreme assumptions, like assuming a very low temperature. One could even have a virialized tiny cloud of gas in an otherwise empty universe, and this would slowly lose energy from gravitational radiation until it coalesced. In this case even two atoms orbiting each other would collapse (but it would take long time; for two hydrogen atoms starting 1 meter apart the coalescence timescale is around $$10^{143}$$ years).

In normal astrophysical environments tiny objects do not condense because temperatures are too high and densities too low. Even the cool molecular clouds allowing star formation are some tens of kelvin and have at most a thousand atoms per cubic cm; the super-dense clouds envisioned above will not occur unless you are inside a collapsing cloud. There, of course, you do get planet and comet formation as a side effect of the star formation.

• “tiny clouds of gas” would not be losing energy through gravitational radiation but instead would be “evaporating” as some molecules would gain velocities large enough to escape. The remaining cloud would become smaller and hotter. Nov 4, 2019 at 13:45
• @A.V.S. Doesn't evaporation cool the remaining gas? Nov 4, 2019 at 14:19
• Gas cloud has (being gravitationally bound object) negative heat capacity, so as the fastest molecules escape, specific energy (with gravitational potential energy) decreases, but the temperature rises. Nov 4, 2019 at 15:08
• "In normal astrophysical environments tiny objects do not condense because temperatures are too high and densities too low." So does this mean that all free floating sub-brown dwarves are actually ejected gas giants? Nov 4, 2019 at 15:13
• @Calmarius - No. Check what I wrote in the last paragraph: stellar nurseries have local fragmentation, producing denser sub-clouds. That is where you get the brown dwarves. If I remember right there is a discussion about whether there is a "desert" of sub-brown dwarf before one reaches the ejected rogue planet population. Nov 4, 2019 at 23:27

There is a well-known minimum mass to objects that can form from the direct collapse of a gas cloud and it is known variously as the fragmentation limit or the opacity limit (Low & Lynden-Bell 1976). As the name suggests, it depends on the opacity of the gas, and is usually estimated to be in the range 3-10 Jupiter masses and perhaps lower in low metallicity gas (e.g., Boyd & Whitworth 2005).

The theory of the fragmentation limit is discussed extensively by Whitworth (2018) in the context of a possible formation mechanism for low-mass brown dwarfs or "rogue planets". The basic idea is that if a collapsing core is able to cool effectively (by emitting infrared radiation), then it's temperature can stay roughly constant whilst it collapses. The increased density results in a decreasing Jeans mass (the mass at which a cloud becomes unstable to collapse, which is proportional to $$T^{3/2}\rho^{-1/2}$$) and this allows small perturbations in the collapsing cloud to grow and fragment into separately collapsing cores of lower mass.

However, at very low masses there comes a limit where the heat cannot be shifted out of the core quickly enough - the cooling time become longer than the dynamical collapse time. These objects are stable to further fragmentation and this defines the lower limit I described above.

There are various ways around the opacity limit (there must be, since Jupiter has formed!), which usually involve formation in a pre-existing protostellar disc and/or accretion onto a solid core (i.e. not in a collapsing gas cloud), and that is why it is claimed that "rogue planets" (with masses below the fragmentation limit) must have formed around another star rather than as isolated objects.

The above answer by Anders Sandberg led me to find a 2001 paper by Alan P. Boss where he proposes mathematically that an object as low as 13 jupiter masses could form directly via the collapse of an interstellar gas cloud. He proposes that such objects by called sub-brown dwarf stars. A link to the article is below.

The real difficulty would be finding such an object, no fusion of any kind would be happening in a sub-brown dwarf so its surface will be cold and dark, if it has objects orbiting it it may produce a fair amount of radio. Gravitational lensing does provide the possibility to detect planet sized objects that are floating free from any stars. But how might a sub-brown dwarf be differentiated from a rouge planet? The only suggestion I have is to have a look at metallicity as it would be lower in a sub-brown dwarf.

https://iopscience.iop.org/article/10.1086/320033/meta

• Young, low-mass objects are luminous. They can and have been found. There would be no metallicity difference between a low-mass (sub) brown dwarf and a "rogue planet" of similar mass. Jun 2, 2023 at 11:40