Imagine a sphere, with a radius about the size of the Solar system. Somehow fill it with air at 100KPa (Earth surface density). $1 km^3$ of air is about 1.25 billion kg. A Solar system radius is about 4.5 billion km, yielding a sphere with a volume of $3.82\times10^{29} km^3$, for a total mass of about $4.7\times 10^{38} kg$

Looking at the Schwartzchild radius $R=\frac{2GM}{c^2}$ suggests that such an air-filled sphere would be close to becoming black hole (by a few orders of magnitude). Is that actually possible?

  • $\begingroup$ If the math works that way, then yes. $\endgroup$ – Kyle Kanos Jan 13 '15 at 4:30
  • $\begingroup$ Are you sure about those numbers? A quick and dirty calculation I did using WolframAlpha suggests that a solar system filled with air at 1 atm pressure has a radius that is 3.4 times larger than its Schwartzschild radius, implying that its volume has to shrink by a factor $3.4^3=39$ to become a black hole. $\endgroup$ – jabirali Jan 13 '15 at 5:12
  • $\begingroup$ I used Neptune, not Pluto. $\endgroup$ – Serban Tanasa Jan 13 '15 at 5:15
  • $\begingroup$ The updated calculation using Neptune instead of Pluto ends up giving a radius $6.2$ times bigger than the Schwartzschild radius, meaning that the volume of your gas cloud has to shrink by a factor $6.2^3=238$ to become a black hole. $\endgroup$ – jabirali Jan 13 '15 at 5:18
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    $\begingroup$ Perhaps more interesting, you could also calculate at which size a 1 atm gas cloud becomes a black hole, which seems to be at a radius of 75 au, which is just 1.85 times the average orbital radius of Pluto. $\endgroup$ – jabirali Jan 13 '15 at 5:29

The density of a black hole is the mass divided by volume: $$ \rho=\frac M V=\frac M{4/3\pi R^3}=\frac{3M}{4\pi}\left(\frac{c^2}{2GM}\right)^3=\frac{3c^6}{32\pi G^3 M^2}\approx1.9\times10^{19}\text{ kg}/\text{m}^3\left(\frac{M_\odot}M\right)^2 $$ This means that the average density of the black hole decreases as the mass increases! We can also solve for the density as a function of radius: $$ \rho=\frac{3c^2}{8\pi GR^2}\approx 7200\text{ kg}/\text{m}^3\left(\frac{1\text{ AU}}R\right)^2 $$ Again, we see that the larger the black hole gets, the less dense it is. In fact, a water-density black hole has a radius of about 2.6 AU and around 140 million solar masses. An air-density black hole has a radius of around 77 AU and about 3.9 billion solar masses.

So you're close, but a little small. The Schwarzschild radius of your air mass is 0.71 billion km, smaller than the 4.5 billion km radius. Since the mass is partly outside the Schwarzschild radius, it is not dense enough to be a black hole.

However, since stars can only get around 100 or 200 solar masses, your gas cloud would definitely collapse into a black hole.


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