# Freefall timescale for a Jeans-unstable hydrogen cloud of mass M

I'm working on a project (personal, not academic) that involves calculating the collapse timescales for protostars which will end up becoming stars of varying given masses. I'm treating the pre-main sequence part of the star's life separately, so I'm only concerned (for now) with the initial period of rapid collapse of the protostellar dust cloud.

My initial idea was to calculate the density of a gas cloud that would have a Jeans mass equal to the mass I'm interested in, and then use that to calculate the freefall time for a cloud with that density. I'm working with the equations found here, which I believe are correct but I'm no astrophysicist so I may not notice mistakes if the page has any. For my purposes, treating the interstellar medium as 100% H$$_2$$ at a uniform temperature of 20K is a good enough approximation.

Given the above, I derived:

$$n\approx\frac{4.2\times10^{69}\frac{kg^2}{m^3\cdot K^3}T^3}{M_J^2}\Rightarrow t_{ff}\approx\frac{84580\frac{s\cdot kg^{\frac{1}{2}}}{m^{\frac{3}{2}}}}{\sqrt{\frac{56.2\times10^{45}\frac{kg^3}{m^3\cdot K^3}}{M_J^2}}}$$

Everything is in SI units. So, for example, say I know that I want a star which will be one solar mass when it reaches ZAMS. I take $$M_J$$ to be one solar mass, and I find that I get a cloud density of about 8.48$$\times$$10$$^{12}$$ molecules per cubic meter, which I believe is quite reasonable for a molecular cloud, and a freefall time of about 23,000 years (which seems a bit short to me, but again I'm no astrophysicist).

For stars with more extreme masses, I don't have any idea if my results are reasonable. Here're a couple of examples: if I want a star with 0.07 solar masses, I get a freefall time of only about 6000 years, from a cloud of density 1.73$$\times$$10$$^{15}$$ molecules per cubic meter. This seems unrealistically dense for a gas cloud, but maybe you could get densities like that during fragmentation of a larger Jeans-unstable region? For a star which will be 150 solar masses, I get a freefall time of 280,000 years (which is more like what I was expecting for one solar mass) and a cloud density of 3.77$$\times$$10$$^8$$ molecules per cubic meter.

My questions are basically: is this approach reasonable, and do these results seem at least passingly plausible? If this approach is sound, I've already derived equations for radius, luminosity, and effective temperature of the collapsing cloud as functions of time and mass, so I'm hoping my work to this point hasn't been completely wasted, but I'd be happy to use another approximation method if someone could suggest a better one.

• Note that on the page you have linked, on the second yellow box from the bottom, the author has recast the free-fall time into a form of 'typical values'. Those may look messy at first, but they're often used in astrophysics exactly because they immediately answer your question: What a typical value would look like. – AtmosphericPrisonEscape Nov 20 '19 at 16:45
• Good point! Checking a range of values for density from 10$^2$ to 10$^6$ molecules per cubic centimeter (matching units to the boxed equation), I'm seeing freefall times between 10Myr and 100,000yr. Those are pretty different from what I was getting, which makes me more confident that I've done something wrong. I'm about pretty confident it's not the units, so there must be something else flawed about my approach... – realityChemist Nov 20 '19 at 16:54