The role of dark matter in black holes and star formation

Question

In my understanding, there exists a critical mass for which a star needs to be in order for it to collapse into a black hole. This also applies to a certain critical density of gas in order for stars to form in the first place. However, given a massive star or a massive amount of gas, it must have a relatively large gravitational field and therefore will interact with dark matter, attracting dark matter towards itself. This makes me question, doesn't the critical mass or critical density that we've figured out before our knowledge of dark matter now need to take that into account? OR Shouldn't the critical mass and critical density resemble the following pseudo equations?

Critical mass (normal and dark) for a star to turn to black hole = A*normal matter mass + B*dark matter mass, where A and B are fractional amounts

Critical density for a cloud of gas (normal and dark) for star formation = C*normal matter mass + D*dark matter mass, where C and D are fractional amounts

So if the answer is yes, then what are the fractions, A, B, C, and D? Moreover, What kind of experiment or calculation would allow us to figure out these fractions?

Answer 1

Yes, you do need to add in the mass of dark matter if it's present, however on small scales the dark matter is almost uniformly distributed.

To see this, consider formation of the Solar System from the original dust cloud. If you take some test particle far from the Sun and let it fall towards the Sun it will accelerate towards the Sun, then pass it and head on out again. If every particle in the original dust cloud behaved this way the Solar System could never have formed since the dust cloud would simply oscillate about its centre of mass and stay the same overall size. The reason the Sun formed is that electrostatic interactions between the dust particles allowed the cloud to dissipate energy as heat and settle towards the centre.

Now you see why the Sun isn't full of dark matter. Dark matter only interacts by the weak and gravitational forces so the dark matter particles can't dissipate energy and can't settle into the Sun. Assuming there is dark matter in the Solar System it will be oscillating about the Sun. In principle weak interactions will eventually dissipate enough energy for the dark matter to become gravitationally bound within the Sun, but it's going to take a long time.

The average density of matter is astonishingly low. At present the total density is around 5 protons per cubic metre, so the dark matter density is only 1 proton per cubic meter (and the average baryon density is 1 proton per five cubic metres!). There will be variations in the dark matter density caused by quantum fluctuations during inflation, and indeed these are thought to have been critical in seeding formation of the first galaxies. However on sub-galactic scales the dark matter density fluctuations are so small we can ignore them.