Does a cloud (essentially a nebula) have to turn into a star or can it directly become a neutron star/black hole? I might've read somewhere that some primordial black holes might have formed this way, currently am looking for a link.
Does the radiation pressure always have to be strong enough to stop the gravitational collapse?
If a cloud hypothetically could collapse into one, could someone point me to the right direction in calculating the mass range of this cloud?
I've tried looking into simulations but I think that is completely out of question because of the complexity involved (in terms of the density and number of particles).
Answer
In the present day universe gas clouds cannot collapse directly to black holes. The main reason for this is that gas enriched by metals from previous generations of star can cool effectively and this leads to fragmentation of a collapsing gas cloud.
Let's back up a step and follow the collapse. Instability is governed by the Jeans mass, the smallest mass that is likely to collapse, scales as $T^{3/2}/\rho^{1/2}$, where $T$ is the temperature and $\rho$ the density. If the gas can effectively cool as it collapses, then the temperature remains roughly constant, the Jeans mass falls and the cloud breaks up into smaller cores. These cores usually end up being of stellar size.
The fragmentation ceases because at some point in the collapse, the gas becomes opaque to infrared radiation and the cloud achieves a rough hydrostatic equilibrium. Thermal energy that is lost results in contraction and the centre of the protostar heats up. Your question is essentially asking whether it is possible to get the cloud inside it's Schwarzschild radius before it ignites nuclear fusion? The answer is no.
The Schwarzschild radius is $R_s = 2 GM/c^2$; we can use a form of the virial theorem to work out how hot the centre of the gas cloud would be at this point. $$ \Omega = -3 \int P\ dV,$$ where $\Omega$ is the gravitational potential energy, $P$ is the pressure and the integral is over the volume of the gas cloud. Making the crude assumption (which will only make a difference of a small numerical factor) that the pressure in the cloud is constant, then we rewrite this as $$-\frac{3GM^2}{5R} = -3\frac{P}{\rho}\int dM = -3\frac{PM}{\rho}.$$ Now if we assume an ideal gas with a mean particle mass of $m$, then $$ \frac{GM}{5R} = \frac{\rho k_B T}{\rho m}$$ $$ T = \frac{Gm}{5k_B}\left(\frac{M}{R}\right)$$
Now we can substitute $R=R_s$ and find $$T = \frac{mc^2}{10k_B}$$ In other words, the temperature attained is independent of the mass of the gas cloud and, assuming $m \sim 1.67\times 10^{-27}/2$ kg (for ionised hydrogen atoms), it is $5\times 10^{11}$ K. This is far above the temperature required for the initiation of nuclear fusion, so the collapse can never get to the point of producing a black hole before forming a star.
However, in the early universe, it might be possible for a gas cloud to collapse directly to a supermassive black holes and this may be why quasars exist only a few hundred million years after the big bang.
Primordial gas made of just hydrogen and helium atoms cannot cool very efficiently however, hydrogen molecules can radiate efficiently. The key to direct collapse to a black hole is to prevent the cooling and fragmentation of the gas. This can be achieved if an external source of UV radiation, provided by the first stars, is able to dissociate the the hydrogen molecules. The primordial clouds are then less susceptible to fragmentation because they heat up as they get more dense and the Jeans mass cannot become small. These large clouds are not as dense as a smaller mass cloud as they approach their Schwarzschild radii, so do not become opaque to the radiation they produce and they may be able to collapse directly to large black holes ($10^4$ to $10^5$ solar masses).
See this press release for an alternative summary of this idea and links to recent academic papers on the topic (e.g. Agarawal et al. 2015; Regan et al. 2017; Smith, Bromm & Loeb 2017).
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