On April 10th scientists from around the world came together to show the first ever picture of a blackhole. This was really cool. A strange thing to learn from all of the science tweets that came about as a result of the black hole news was a twitter thread where the author went into how the density of a black hole can be surprisingly low, like lower than the density of air at sea level. This is because the radius of a black hole is proportional to the mass of a black hole (the thread I linked will do a better job of explaining why this is the case), accepting this fact means that when you double the mass of the blackhole, you've doubled the radius, but the volume of will have gone up 8X. This then asks the question, is there a mass of a black hole that would be so massive that its density would be less than that of the interstellar void?
Any questions/feedback are welcome.
The goodnews is that there is already a solid starting point with the Wikipedia entry on the Schwarzschild radius, in this article there is a hand table that highlights the various densities of blackholes made from various amounts of stuff.
| Observable universe[7] | 8.8×1052 kg | 1.3×1026 m (13.7 billion ly) | 9.5×10−27 kg/m3 |
| Milky Way | 1.6×1042 kg | 2.4×1015 m (~0.25 ly) | 0.000029 kg/m3 |
| SMBH in NGC 4889 | 4.2×1040 kg | 6.2×1013 m | 0.042 kg/m3 |
| SMBH in Messier 87[9] | 1.3×1040 kg | 1.9×1013 m | 0.44 kg/m3 |
| SMBH in Andromeda Galaxy[10] | 3.4×1038 kg | 5.0×1011 m | 640 kg/m3 |
| Sagittarius A* (SMBH) | 8.2×1036 kg | 1.2×1010 m | 1.1×106 kg/m3 |
| Sun | 1.99×1030 kg | 2.95×103 m | 1.84×1019 kg/m3 |
| Jupiter | 1.90×1027 kg | 2.82 meters | 2.02×1025 kg/m3 |
| Earth | 5.97×1024 kg | 8.87×10−3 m | 2.04×1030 kg/m3 |
| Moon | 7.35×1022 kg | 1.09×10−4 m | 1.35×1034 kg/m3 |
| Human | 70 kilograms | 1.04×10−25 m | 1.49×1076 kg/m3 |
| Big Mac | 0.215 kilograms | 3.19×10−28 m | 1.58×1081 kg/m3 |
| Planck mass | 2.18×10−8 kg | 3.23×10−35 m | 1.54×1095 kg/m3 |
With this starting point we now need to know what the density of the interstellar space is, this number slightly varies from source to source, but seems to ballpark between 0.1 and 1 particle per cm^3, for sake of consistency (and the fact that its the most common element) we will assume that these particles are hydrogen atoms. Therefore the average cubic meter of the interstellar void has between one hundred thousand and one million hydrogen atoms. A single hydrogen atom has a mass of about 1.66 *10^-27 kg. This gives a density of the interstallar medium of between 1.66*10^-22 kg/m^3 and 1.66*10^-21kg/m^3.
Now we need to determine how much mass our black hole needs to have to achieve this rather rarified density. Right off the bat we know we can bound the mass somewhere between the Milky Way and the Observable universe. As the Milky Way mass blackhole is too dense, and the Observable universe is roughly the density of 6 hydrogen atoms/cubic meter.
Plugging the Equation [3*C^3]/[32*π*G^3*M2] into Excel (where C is the speed of light in meters/second, G is the gravitational constant, and M is the mass of the object)
Now we need to determine how much mass our black hole needs to have to achieve this rather rarified density. Right off the bat we know we can bound the mass somewhere between the Milky Way and the Observable universe. As the Milky Way mass blackhole is too dense, and the Observable universe is roughly the density of 6 hydrogen atoms/cubic meter.
Plugging the Equation [3*C^3]/[32*π*G^3*M2] into Excel (where C is the speed of light in meters/second, G is the gravitational constant, and M is the mass of the object)
We are able to determine that the blackhole would need to have a mass of 6.63*10^50 kilograms of mass to achieve our lower density of 1.66*10^-22 kilograms/m^3, and 2.10*10^50 kukigrams to get to 1.66*10-21 kilograms/m^3.
The radius of our blackhole would be between 3.11*10^23 and 9.84*10^23 meters in radius. In light years that would be between 32.9 and 104 million light years. Otherwise really effing big
Any questions/feedback are welcome.
