# How big should a star be to turn into a black hole?

My initial calculations show that if the radius of a star $$-$$ with a uniform mass density of $$\rho$$ $$-$$ is greater than $$\frac{c}{2\sqrt{\pi G\rho/3}}$$, the star would collapse into a black hole. However, I could not find any reference to check the result. Can anyone help me?

• A uniform density isn't very realistic, you can see the density profile of the Sun here: astronomy.stackexchange.com/a/32734/16685 Aug 21, 2020 at 9:48
• @PM2Ring Yes, I know it. I just presume a uniform density to make the calculations easier. Aug 21, 2020 at 9:53
• Fair enough, but that might be why you can't find a ref to check it against. (BTW, if you derived it from the standard formula for the Schwazschild radius, I think you're missing a 2 inside the square root). Also see physics.stackexchange.com/q/425914/123208 Aug 21, 2020 at 9:57
• have a look at craig wheeler's book on stellar structure and evolution, it will probably answer all your questions. Aug 21, 2020 at 16:18

I guess you have done something like $$R < \frac{2GM}{c^2} = \frac{8\pi G \rho R^3}{3c^2}$$ $$R > \sqrt{\frac{3}{8\pi G\rho}}c$$
The first equation is more useful, because in some circumstances you can assume that $$M$$ is fixed. However it underestimates the upper limit to the radius. Collapse to a black hole becomes inevitable in GR (at least in the Schwarzschild metric) when $$R < 9R_s/8$$, (the Buchdahl limit) and probably a bit higher for realistic equations of state at high densities.