# Strange Airy Black Hole

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?

• If the math works that way, then yes. – Kyle Kanos Jan 13 '15 at 4:30
• 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. – jabirali Jan 13 '15 at 5:12
• I used Neptune, not Pluto. – Serban Tanasa Jan 13 '15 at 5:15
• 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. – jabirali Jan 13 '15 at 5:18
• 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. – 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.