I've lately read Isaac Asimov's essay collection The Stars in their
Courses

(collected non-fiction essays from *The Magazine of Fantasy and
Science Fiction*, May 1969 to September 1970), right after reading Sir James
Jeans' book of the same title from 1931. Asimov ends (not uncharacteristically)
on the topic of population growth, computing how long then-current growth rates
(doubling every 47 years, from 3.5 giga people in 1970) would take to,
variously:

- fill the world's total surface area to Manhattan island's daily peak
population density (0.2 giga mile
^{2}@ 2.2 mega people / 22 / mile^{2}= 20 tera people); 585 years - fill ten planets per star, throughout a hundred milliards of galaxies comparable to our own, each to the same density (20 tera people × 10 planets / star × 135 giga star / galaxy × 0.1 tera galaxy = 2.7e36 people); 4200 years
- convert all the mass in the universe into human flesh; 6700 years.

His point being that, no matter how optimistic the rationale of our
estimate, the answer is finite. He then goes on to point out the impossibility
of any of this and compute some more conservative estimates, predicting famines
in India and Indonesia by 1980. I note that, in 2000, a population of 6.1 giga
people was estimated to be growing at either 1.4% / year, implying a doubling
time of about 50 years, or 1.2% / year, implying a doubling time of about 58
years (different sources quote different growth rates). It's worth noting that,
if we wind time *backwards*, the doubling times cited thus far would
imply that there were only two people sometime between 170 AD and 520 AD. Until
the 1800s disease was a regular killer; before we used canals, and later trains,
to transport food, famine worked hand in hand with disease.

Now, a few years back, my friend Aubrey pointed out that the forward light cone of humanity only grows in radius proportional to time, so that the total resources within the sphere of human influence (which necessarily falls inside that light cone) must eventually be over-taken by any exponential growth in human population. In practice our sphere of influence cannot grow anything like as fast as our future light cone, but we can at least tighten Asimov's two most generous estimates by using it as an inescapable upper bound. For that, I'll need some estimate of the matter density out to at least several hundred light years from our home solar system.

The matter density of our galaxy is known to die off with radius; from the observed fact that speeds of stars' orbits around the galaxy are roughly constant from about a third of our sun's distance from the centre to about ten times as far out as our sun, it may be inferred that the mass closer to the centre than any given radius grows in proportion to that radius (this is a shocking fact, if you think about it). Since this remains true out to about ten times as far out as our orbit, we may suppose that the total mass of the galaxy is about ten times that inferred from the speed and radius of our orbit. However, I'm only interested here in its density in our neighbourhood. The mass inwards from us is about 9e10 times the mass of our solar system; which would imply about 150 giga solar systems within the 50 k ly radius that contains most of the visible galaxy, which is at least compatible with the 200 giga stars I gather make up the Galaxy.

The Galaxy in our neighbourhood is about 2 k ly thick (it's about ten times
as thick near the core).
Since mass inwards from any given radius is proportional to radius, the mass at
radii within 1 k ly of our roughly 28 k ly orbital radius must be about 1/14 of
the total inwards from us. This is spread out as a circular ring, 2 k ly broad
and 2 k ly thick, with radius 28 k ly, so length about 88 k ly. A 2 k ly length
of that ribbon then has 1/14/44 of the total Galactic mass inwards from us,
yielding a density of 9e10/14/44/8 solar systems per cubic k ly; that's about 18
solar systems in each cube, near us, of side ten light years; or about 20 times
Jupiter's mass in each cubic light year. (This is about six times the
average density

I've seen quoted
elsewhere, which I suppose includes much of the space outside the Galaxy's
disk, so I don't feel a need to correct my estimate of *local* density.)
Note that this density is an average of a very lumpy distribution; our solar
system contains all the mass this density would put within 2.36 light years of
it; that's more than half way to the nearest other star. Still, it'll do as a
density estimate for now.

Measure time forwards from the moment (estimated to have been in the year 2000) when the human population was 6.1 giga people; assume it doubles every 50 years; the future human population at time t.year will then be 6.1e9 × power(t/50, 2). The most ludicrously generous estimate would let us turn all the mass within a distance t ly of Earth into humans: assuming 50 kg per person we must then solve for when 30.5e10 kg.power(t/50, 2) catches up with 36.33e27.kg.(4.π/3).power(3,t). Now, 36.33e27.(4.π/3) / 30.5e10 gives us nearly 5e17 times the mass of our present human population as the mass within one light year of us; we need to solve for when its log is equal to t.log(2)/50−3.log(t). This will happen within 4773 years. That's long enough that it's relied on the matter density remaining constant out to nearly 5 k ly and we can only really assume that out to about 2 k ly; in one direction, you hit the edge of the Galactic disk at that point. Still, it's a start.

Now, in practice, most of the mass we were discussing is in stars and not very accessible to us; nor, indeed, is the mass locked up in gas giants. The inner planets have about six millionths of the solar system's mass; adding all the asteroids and other rubble in, we might be able to stretch to claiming that as much as ten millionths of the mass density above is available to us. That reduces mass available within one light year to 5e12 times the mass of our present human population so we'd hit the limit 874 years earlier, within 3900 years. But, as Asimov points out in his more conservative estimates, at each step in a food chain, one layer can only have a mass of about one tenth that of the layer it eats. So only one tenth of the notionally available mass is available for humans; that shaves another 176 years off; time would run out within 3722 years.

Still, people and plants need some place to live. In practice, we'd need to build cylinder ships out of some of the rubble – this maximises the habitable surface area per tonne of rubble. Indeed, the above implicitly assumed we'd tear apart all the planets from Earth's size down for use as raw materials: most of that rubble is made of the wrong chemicals for life, but would do fine for building cylinder ships. So, absent large-scale transmutation of elements, most of the notionally available mass isn't much use for making life out of; we may as well build habitat and life-support equipment out of it. That'll reduce the practically availabile mass by another factor or two of ten; each factor of ten will shave off another 176 or so years.

Of course, while we're assuming we can tear Earth-sized planets apart, we may as well consider what'd happen if we could tear apart gas giants while we're at it; that increases our available mass by a factor of a few hundred and gives us enough rubble to build Dyson spheres around all the stars; which will claw back a few factors of ten, gaining us back a few 176 year bites.

Still, tearing apart bodies the size of The Earth, or even smaller planets, is a bit beyond us. Maybe we'll be able to do that in three thousand years' time, but we have to survive that long to do so. With the technology we can realistically hope to have in the next century or two, we have to make do with the habitable surfaces of the planets plus what we can build out of the asteroids and kindred rubble we find around the solar system. I'll assume the asteroids, comets and kindred rubble in our solar system collectively have mass somewhere between that of The Earth and The Moon; and that we can make strong enough materials that somewhere between twenty and fifty metres will be thick enough for the skin of an environment whose inner surface experiences natural gravity outwards. [Even if you can make materials strong enough to do much better than that, you still want an outer crust to cope with space's perpetual bombardments of various kinds – an ablation shield.] That'll suffice to build cylinder ships with total habitable surface area somewhere between a thousand and a million times as large as that of the Earth, so it's quite good going. All the same, rock's five or six times as dense as life and we need dozens of cubic metres of it per square metre of habitat; and we only get something like a tenth as much mass to play with because there's much less rubble than planets. That burns another few factors of ten, each costing us 176 years or so. All the same, we'd still have three millennia.

So, fine, each factor of ten on our estimate of the available mass changes the available time by a couple of centuries. What other limitations should we consider ?

In reality, moving at the speed of light is unrealistic. Even moving at one tenth that speed would be extremely hard, both in terms of brute energy requirements and in terms of the sophisticated control systems needed to avoid colliding with rubble in the interstellar space – at those speeds, a small pebble makes a big crater in whatever it hits. The effect of going slower by a factor of ten is to reduce the mass within our reach, in a given time, by a factor of a thousand (the cube of ten); that costs a bit over half a millennium. Our most distant probes have travelled a few light hours in a few decades: even a hundred times as fast as them is a fraction of a percent of the speed of light. Since our expanding population is relying on its outward movement to increase its supply of resources, it actually has to stop and collect those resources; so in practice it'd be hard pressed to maintain expansion at even one ten thousandth of the speed of light. If we ignore all the other mass reducing factors above, this alone suffices to reduce the time available by over 2100 years, to 2652 years.

Now, the inconvenient detail: that reduction makes the situation more sensitive to the factors of ten discussed before. Each now costs more; and they re-inforce one another. At one ten thousandth of the speed of light, turning all the mass into human flesh would grant us 2652 years; allowing for only turning all the mass in Earth-like planets and smaller rubble to human flesh, we're left with only 1686 years, a loss of almost a millennium. A factor of 10 to make space for plants and a factor of 50 to build cylinder ships lose us another half millennium, leaving only 1156 years. At one ten thousandth of the speed of light, that won't get us to the next star system. Indeed, even the 2652 years that total mass conversion promised would only get us one sixteenth of the way to the next star system. We have to curb our population growth while still confined to our own solar system.

The above estimates suggest we have, within our solar system, scope for
several times the space presently on Earth. Still, if we consider Asimov's
first estimate above, and allow that actually only one tenth of the habitable
area can be given over to humans, we'd need ten times the Earth's surface to be
habitable just to continue our present growth for 585 years. Even my optimistic
opinion doesn't expect us to get adept at nurturing life in dead rubble in
anything less than several centuries; it'd be a stretch to have *any*
cylinder ships ready in that time. Earth will be Manhattan/10

full in
393 years, requiring an emigration rate of nearly 900 people per second, even if
we have somewhere to send them. The power requirement to lift that exodus would
be 2.75 tera Watt – all day, every day. Such a world would not be
sustainable: the only sustainable solution would be to halt population growth.
We have to solve our population growth problem on our home planet.

The astronomical expansion in resources available via space travel can't solve that problem for us. That's not to say anything against building cylinder ships: only that we also need to halt our population growth. Habitats in space will give life more places to survive in the event of a crisis in any one of its homes; and they'll make it economic to feed folk living above Earth's gravity well, thereby making a reality of various opportunities which are, presently, too expensive to explore. Once we have control of fusion power, cylinder ships can travel to other solar systems, both as explorers and as colonists; again, this will be no help to the population problem, but it will improve life's chances of long-term survival; and make a reality of opportunities we can, today, only dream of.

- More on population issues.
- Some handy data on global demographics.