Can I eat myself into a black hole?

Question

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?

Answer 1

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.