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This was a humorous thought experiment that occurred while chatting about black holes. The person that I was talking to assumed that a black hole required a specific density to be achieved. I pointed to the formula for the Schwarzschild radius. This suggests that low density black holes are possible if they are large enough. I'll assume simple spherical, non-rotating, uncharged black holes.

$$ r = \frac{2 G M}{c ^ 2} $$

Using $\rho$ as the the average density.

$$ M = \frac{4}{3} \pi r^3 \rho $$

$$ \rho = \frac{3}{8 \pi}\frac{c^2}{G r^2} $$

Which can be as small as you like by having $r$ large enough.

So, assuming that the average density of a person is approximately that of water, I calculate that if I achieved a radius of approximately $4 \times 10^8 km$ then I would become a black hole. If I was centred at the Sun then I would extend into the asteroid belt. Of course, people are not usually spherical though if you get this large then it is probably a good approximation. There may also be some health issues and practical issues with obtaining sufficient food, oxygen, etc but they are off-topic in this group.

Am I right, is a black hole with the same average density as water possible?

Apart from the biological and nutrient issues, what others might I face? I guess that I will suffer severe problems long before I reach this size. Would I collapse and become a star (quite literally)?

Alternatively, if an extremely large number of people gathered in a huge group hug, could they become a black hole?

Update

Thanks to the comments from kleingordon and dmitry-brant, my main question is answered. This leaves:

  1. Is my calculation right? (To 1 significant digit)

  2. Would I become a star or face some other calamity before becoming a black hole?

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  • $\begingroup$ An amusing comment discussion has been moved to chat. $\endgroup$ – rob Jul 31 '18 at 15:48
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As you suggest, long before you got that large gravitation would become dominant. One of the early what ifs is about what you get if you take a mole (ie $6\times 10^{23}$) of moles (small animals) which results in something a little larger than the Moon, and the answer is not very pretty (at least not for the moles).

This is indeed why stars happen: if you get very large collections of matter, they collapse in on themselves gravitationally, fusion starts in their cores (you are mostly carbon, which would fuse I presume). I don't know if anyone has done the sums describing the formation of stars whose precursors were very large humans: I suspect not. But this would happen long before you got large enough to reach your Schwarzschild radius.

In fact, long before that you would hit another problem: surface to volume and temperature. A grovel over the internet says that humans dissipate about $100\,\mathrm{W}$, so if the average human has a mass of $60\,\mathrm{kg}$, then this is about $1.6\,\mathrm{W/kg}$ or (assuming humans are about as dense as water) about $1.6\,\mathrm{kW/m^3}$. If you assume humans are spherical then you get surface flux, I think, of about $550 r\,\mathrm{W/m^3}$, where $r$ is the radius of the human (the units are right here: r has the dimensions of length so the flux comes out as $\mathrm{W/m^2}$). And since humans are approximately black bodies, this corresponds to a surface temperature of

$$\left(\frac{550r}{\sigma}\right)^{\frac{1}{4}}$$

So this is going like $r^{1/4}$ which smells right to me. Plugging in some numbers, a $100\,\mathrm{m}$ radius human is at around $1000\,\mathrm{K}$. So you're dead well before you get that big.

So no, you can't eat yourself into a black hole: you die of heat exhaustion before gravity even becomes significant.

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    $\begingroup$ Thanks. A mole of moles, I like that. I think that I am rather more oxygen than carbon but both are well before iron so should support fusion. Maybe the hydrogen in me would ignite first. The heat calculation is interesting as another limit. The most obvious limit is my heart, it would probably struggle to pump blood around such a large body. Imagine what my blood pressure would need to be. It seems fairly certain that is my first limit but not as interesting to speculate on as the others beyond it. $\endgroup$ – badjohn Jul 4 '18 at 15:50
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    $\begingroup$ As a percent of mass, oxygen dominates. As an atomic percentage, hydrogen dominates. $\endgroup$ – Bob Jarvis Jul 5 '18 at 3:24
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    $\begingroup$ Humans dissipate a lot more if you place them in a cold vacuum. Around 1 kW. See this. Human bodies are regulating the temperature, so I'm not sure if your calculation is applicable. $\endgroup$ – Arsenal Jul 5 '18 at 11:21
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    $\begingroup$ @badjohn Also, adult human metabolism per unit volume pumps out more heat than the core of the Sun, which only produces as much heat per unit volume as a reptile or an active compost heap. See en.wikipedia.org/wiki/Solar_core#Energy_conversion $\endgroup$ – PM 2Ring Jul 5 '18 at 11:32
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    $\begingroup$ @Arsenal: the actual numbers don't really matter: human metabolism creates heat and there's a lower bound on how much heat per unit volume it needs to dump. That means there's an upper bound on how big a human can be, and (unless the metabolic heat production is absurdly low) this will be absurdly smaller than the Schwarzschild radius. Note the other comment which points out that the Sun generates less heat per volume than a human. $\endgroup$ – tfb Jul 5 '18 at 14:07

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