Quote:
Originally Posted by Swift
Ok, let me see if I got this right. For the case of Mc = 107, the accretion rate minus the evaporation rate should be the rate at which the black hole grows, which would be 1.33 x 10-21 kg/second.
If I did my math right, 10 billion years is 3.16 x 1017 seconds. So, as a first approximation (I suppose the rates changes as the BH gets bigger) in 10 billion years, the BH will have gained 0.0004 kg. According to google, the Earth weighs 5.97 x 10 24 kg, so that is 7.03 x 10 -27% of the Earth. Of course the sun will become a red giant in half that time... so what's the problem?
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I did some more calculations. You probably will not like the results. Originally I argued that
dM/dt = 4 * π* v * ρ * R2
where R is the effective capture radius of the black hole. Thus mass as a function of time would be
M = (dM/dt) * t
Now the 4 *
π* v *
ρ part of the equation doesn't change under my scenario, so we can regard it as a constant for our present purposes.
Cw = 4 * π* v * ρ = 4.08 x 108 kg m-2 s-1
where v = 2,500 m s
-1 and
ρ = 13,000 kg m
-3.
First I solve for
CFH's equation (23) substituting the atomic mass of iron atoms (9.27 x 10
-26 kg each), since in my scenario the mBH will be trapped in the Earth's core somewhere.
Then I solve for eq. (22), which I then use to obtain the constant C
em in equation (24) (R = C
em * M
1/4), deriving a value of C
em = 2.89 x 10
-8 m.
Now for a little algebra:
M = (dM/dt) * t
M = (CwR2) * t
M = Cw * (Cem * M1/4)2 * t
M = Cw * Cem2 * M1/2 * t
M * M-1/2 = Cw * Cem2 * t
M1/2 = Cw * Cem2 * t
M = Cw2 * Cem4 * t2
M = 1.16 x 10 -13 * t2
t = (M / 1.16 x 10-13)1/2
Now, according CFH, the mBH will start evolving on a 4-D basis when it reaches a mass on the order of kilograms. Thus setting M = 1 kg, then it would take 2.9 x 10
6 s (< 1 month) to achieve a mass of 1 kg. If we go with an M
c of 10
7 kg, then it would take 9.3 x 10
9 s to achieve M
c. That's about 300 years.
Of course we'll all be dead in half that time. So what's the problem?
CAVEAT: According to
Ord et al. there is a 1 in 1,000 chance that I made an error in my calculations above!
