Smallest Black Hole to 'heat' a Gas Giant
So I want a large power source to heat my new real estate on the Jovian moons. I create a micro black hole (MBH), maybe a few kg, maybe more and drop it into Jupiter. I expect the following to happen [...] the MBH absorbs mass and emits Hawking radiation, Mass falling towards the MBH will heat up due to the high pressure near the MBH [...]
Or so I assumed in this question. However, it was pointed out to me in comments that such a small black hole is really, really small and that it emits enough Hawking radiation to effectivly push away all mass, meaning no accretion will happen and the MBH will evaporate.
Hawking radiation and resulting pressure will be lower with higher MBH mass, but that also means at some point Hawking radiation won't heat my gas giant - I still want a small MBH.
What is the smallest MBH that will, when shot into a gas giant, accrete mass fast enough for it to not evaporate?
I think this means that radiation pressure from Hawking radiation should be smaller than surrounding pressure, we may help by shooting the MBH to get some impact pressure in front.
It is entirely possible that the resulting MBH is too large to heat the gas giant via Hawking radiation.
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I went about this a bit differently than kingledion, and got a different answer (off by $\sim6$ orders of magnitude!). The difference is that I assumed that there would be accretion no matter what the mass of the black hole is; this is incorrect because accretion would probably be prevented because of pressure by Hawking radiation. I'll keep this answer here for posterity, and also to show that even if you ignore kingledion's key pressure assumption, there's still an even lower limit - and thus his solution still works. A black hole of $\sim10^{16}\text{ kg}$ would certainly be able to heat the gas giant.
The power emitted by a black hole from Hawking radiation is $$P=\frac{\hbar c^6}{15360\pi G^2M^2}=-c^2\dot{M}_H$$ where $\dot{M}_H$ is the change in mass of the black hole from Hawking radiation. Let's assume that the black hole also accretes mass; the equation for Bondi accretion should give us a good estimate: $$\dot{M}_B\simeq \frac{\pi\rho G^2M^2}{c_s^3}$$ where $\rho$ is the density and $c_s$ is the speed of sound. The central density of Jupiter is roughly $5\text{ g cm}^{-3}$, or $5000\text{ kg m}^{-3}$. We can find $c_s$ as $$c_s=\sqrt{\frac{K}{\rho}}$$ where $K$ is the bulk modulus - about $125\text{ GPa}$. This gives us $c_s\simeq5000\text{ m/s}$. We then set $$\dot{M}_H+\dot{M}_B=0$$ and solve $$\frac{\hbar c^4}{15360\pi G^2M^2}=\frac{\pi\rho G^2M^2}{c_s^3}\to M=\left[\frac{\hbar c^4c_s^3}{15360\pi^2\rho G^4}\right]^{1/4}$$ Plugging things in, we have $$M=\left[\frac{\hbar c^4(5000\text{ m/s})^3}{15360\pi^2(5000\text{ kg m}^{-3})G^4}\right]^{1/4}=5.158\times10^{10}\text{ kg}$$ This is, as I said, off from kingledion's result by a factor of one million.
There are only two things that could really be varied - the bulk modulus and the density. If we move the other factors out, we see that $$M=7.295\times10^8\text{ kg}^{5/4}\text{ s}^{3/4}\text{ m}^{-3/2}\left(\frac{K^{3/2}}{\rho^{5/2}}\right)^{1/4}$$ Even raising $K$ by an order of magnitude and lowering $\rho$ by an order of magnitude only multiplies our result by $10$.
This demonstrates the power of radiation pressure! It raises the lower limit by six orders of magnitude, which is pretty incredible. Be careful of what physical assumptions you make.
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