Relationship between temperature of nebula and size of star

Question

I was wondering, in the process of star formation, does the temperature of the nebula that produces a star play a role in the size of that star? I mean, it's only logical that the size would depend on the amount of matter around.But then, if the temperature is lower, that means that the net density is lower, so the star would 'absorb' less matter. Or is that not a factor, and the growing star would gobble up anything on its vicinity? I've encountered some contradicting sources and I'm trying to double check.

Answer 1

This is such a complex issue that I'm not going to try to be comprehensive. One major thing you are missing is that stars tend not to form in isolation, especially massive stars, but in groups and clusters.

The basic unit in star formation is the Jeans mass. $$ M_J \propto T^{3/2} \rho^{-1/2}$$ where $T$ is the temperature, $\rho$ is the density and $M_J$ is the minimum mass for a cloud to gravitationally collapse.

So from this point of view, a higher temperature cloud must be more massive in order to collapse in the first place.

But as the cloud collapses, the density goes up and the Jeans mass decreases - and the cloud fragments into smaller sub-units and ultimately into a cluster of stars.

Exactly how this pans out in practice is basically the whole research field of star formation. It depends not only on relatively simple things like the temperature of the gas, its density, the equation of state; but also on much more complicated ideas like magnetic field support, competitive accretion between stars, feedback from newly-formed stars and protostellar outflows and the rate at which turbulence is generated and then dissipated.

There almost certainly is no simple relationship as you propose. In fact the outcome of star formation seems to produce a mass function (number of star per unit mass) that is only weakly dependent on the initial conditions, but may differ a bit in low density and high density environments.