Very nice work SpaceTiger, but I still have a few questions, comments below.
SpaceTiger said:
...The "effective radius" that you're talking about (in other words, the distance from the black hole an object has to get to be pulled in by the gravity) depends on the velocity of the incoming object. It can be approximated by saying that an object will fall in when the potential energy is comparable to the kinetic energy. Thus:
\frac{1}{2}mv^2=\frac{GMm}{r}
r \sim \frac{2GM}{v^2}
The typical velocity of particles on the surface of the Earth can be approximated via
kT=\frac{1}{2}m_pv^2
v \sim \sqrt{\frac{2kT}{m_p}}
where m_p is the mass of the proton. For T \sim 300 K, this gives \sim 2 km/s. Plugging this into the original equation, along with M=1 kg, we get \sim 3 \times 10^{-15} cm. This is still seven orders of magnitude smaller than the Bohr radius (approximate radius of an atom). ...Furthermore, this "capture" radius isn't even really that, it's just the radius at which the black hole's gravity has a noticable effect. I suspect that you would need a three-body interaction to bind the nucleon to the black hole. Otherwise, its path would just be bent by the gravitational field and it would pass right on by...
If I understand you correctly, you have evaluated the distance from the BH where the depth of its potential well (infinity being zero potential) is equal to the KE of the typical room temperature proton. Proceeding on the assumption this is correct:
1)First I note that "equal partition of energy" should apply and that the typical oxygen molecule (A= 32proton masses) will be moving almost 6 times slower than your proton.
2)Even if the velocity of the proton were directly away from the BH, since by assumption it is equal to the depth of the potential well, it would just be able to separate from the BH. But of course in the typical case the velocity will not be directly away form the BH. A more reasonable case is that the proton/molecule is in a bound orbit about the black hole, although this does not make much sense either as it is much bigger than your 10^-15cm.
I don't know exactly how to think about this, but bet that the gradient rips molecules apart, and probably even the outer shell electrons of the oxygen atom nearer the BH also are stripped away. Thus in addition to gravity, we are going to have strong electrical forces acting on the +ion oxygen nucleus and a "cloud" of electrons orbiting the BH. My poor classsical mind is not up to doing this right, but you get the idea, which I will try to sumarize as follows:
Because the gravity gradient rips up atoms, makes electrons clouds orbiting the BH, etc. we have opportunity for disapative events (even some EM radiation? - probably very harse UV continium, not lines). Even if some strange "BH / O+ ion" molecule forms which is stable against these disapative processes, (except for line radiation), it is not likely that it will remain stable when others join this "soup of orbiting charges"
3) I forget my statistics, but think the Maxwellian distribution's "hot tail" contributes a relative large amount to the average energy compared to the number of particles in it. That is, unless I am wrong, the median velocity is significantly less than you have calculated. In any case, there are lots of relatively slow moving oxygen molecules for the HB to rip appart and eat.
Again, you probably are right, but I am still not willing to throw in the towel. I think several OOM still more distant from the BH than your now relaxed 10^-15cm there are molecules that just happen to be headed roughtly towards the BH and that although they will initially not feel much of its gravity - little effect on their trajectory, but as they come closer (on their own momentum, not falling in yet) they will get close enough to get ripped up by the gradient, disapatively interact with the electons they once owned and feed that hungry BH.I beet you do not really deny this is a posibility, but may think it takes too long. If this is the case, we need to know how the presence of an atmosphere modifies the rate of "vacuum polarization," if it does. I still would like to look into this alittle and need to know where you 10^-16 second life time comes from.
I am prety confident that you error by setting the KE equal to the potential well depth to evaluate the effective "capture cross section" of BH in Browian motion air, but you certainly are correct to note that on these small scales, most of what you have arround is "nothing but empty space."
What do you think? Should I throw in the towel, or do I still have a point?
) would probably be a huge problem too. It would probably still be easier than dragging 3 SM's of material together, but it's many, many years down the road, if it's possible at all.
