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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?

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  • $\begingroup$ A uniform density isn't very realistic, you can see the density profile of the Sun here: astronomy.stackexchange.com/a/32734/16685 $\endgroup$
    – PM 2Ring
    Aug 21, 2020 at 9:48
  • $\begingroup$ @PM2Ring Yes, I know it. I just presume a uniform density to make the calculations easier. $\endgroup$ Aug 21, 2020 at 9:53
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    $\begingroup$ 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 $\endgroup$
    – PM 2Ring
    Aug 21, 2020 at 9:57
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    $\begingroup$ have a look at craig wheeler's book on stellar structure and evolution, it will probably answer all your questions. $\endgroup$ Aug 21, 2020 at 16:18

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I guess you have done something like $$R < \frac{2GM}{c^2} = \frac{8\pi G \rho R^3}{3c^2}$$ $$ R > \sqrt{\frac{3}{8\pi G\rho}}c$$

But I cannot see how this inequality is useful. It is just a statement that if the radius of a star is less than the Schwarzschild radius, then it is a black hole, not that it will collapse and become one. It has no diagnostic value because the density of the star obviously changes when you change its radius.

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.

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Check out the Tolman-Oppenheimer-Volkoff limit. The collapse to a black hole depends on mass

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If a star is collapsing into a black hole, it is reasonable to conclude that its density is not constant. As a star runs out of fuel, its temperature decreases, which means molecules have less energy and can resist gravity less. This means the star will compress, and density will increase. So just because a star is currently not dense enough to form a black hole doesn't mean it can't turn into a black hole later. According to this site, it take about three times the mass of our sun to form a black hole.

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