11 October 2007

The "fleshball" -- a crude upper bound on human population

What bounds are there on the growth of human population?

Well, there are a lot of obvious ones. We might kill ourselves with global warming or nuclear war or a host of other things. We're still stuck on this planet. It might turn out that there's some race of killer aliens that will vaporize us when they discover we exist, perhaps because they want to demolish our planet to build a hyperspace bypass.

But let's say we survive all that. We'll still run into trouble when we can't find space for people to live.

And at some point, if we reproduce like we do now into the indefinite future, the sphere which holds all human flesh will be expanding faster than the speed of light.

Let's say you could pack all the humans that currently exist into a giant sphere. The volume of that sphere would be just Vn, where V is the volume of a human being and n is the number of humans; its radius can be obtained by solving the equation $Vn = {4\over 3} \pi x^3$, and we get $x = \left({3Vn/4\pi}\right)^{1/3}$. With current numbers of n = 6.6 billion, V = 0.070 m3 (I'm assuming the average human being weighs 70 kilograms and has the density of water), we have x = 480 meters. That's right -- all of humanity could fit into a ball not quite a kilometer across. Kind of puts things in perspective, doesn't it?

Now let's let n be a function of time. Let's say the human population grows exponentially, so we have $n(t) = n_0 e^{rt}$. Here, n0 is the population at time zero, and r is the rate of population growth; this is currently about 0.012/year.

Then we have
x(t) = \left( {3V n_0 e^{rt} \over 4\pi} \right)^{1/3}

and so the rate of change of the radius of the giant flesh ball, with respect to time, is
{k^{1/3} \over 3} r e^{rt/3}

where k = 3V n_0 \over 4\pi.

The time at which the fleshball will be expanding at the speed of light is just the time when this expression for the derivative is equal to c. Setting it equal to c and solving for t gives
t = {3 \over r} \log {3c \over r k^{1/3}}
Now, we have r = 0.012/yr; that's about 4 × 10-10/sec. c is of course the speed of light, 3 × 108 m/s. And k is about 1.1 × 108 m3. Plugging in numbers gives t = 2.7 × 1011 s, or about nine thousand years from the present.

The radius of the ball at that time will be 3c/r, or 2.25 × 1018 meters; that's 237 light years. (Not surprisingly, the radius of the ball at the time that it hits the speed c doesn't depend on the volume of an individual person; as far as the ball is concerned we're just one giant blob that grows at slightly over one percent per year.)

And the number of people in the ball is
27 n_0 c^3 \over kr^3

which is 6.8 × 1056.

Somehow I don't think we need to worry about this. Why? A proton weighs 1.7 × 10-27 kg; about half of a person's mass is protons, so each person contains about 2 × 1028 protons. Thus this many people would contain about 1085 protons. But the number of protons in the universe is believed to be somewhere under 1080. So we'll run out of mass in the universe well before we hit this threshhold.

Besides, the whole idea of humanity being a solid ball of flesh wasn't so appealing anyway.

1 comment:

John Armstrong said...

the whole idea of humanity being a solid ball of flesh wasn't so appealing anyway

<snark>Indeed. Many humans are already solid balls of flesh, and I find them singularly unappealing</snark>