Sunday, May 23, 2010

Malthusian Astronomy

It's handwavy math time again!

Two weeks ago, I was on the road from Mississippi to Virginia, so I had a lot of time to think. One of the things that crossed my mind was the maximum possible human population of the Milky Way galaxy.

To do this calculation, I started with a set of very simplifying assumptions:

  1. A human being requires a particular quantity of energy to survive every day.

  2. Human beings can fuse hydrogen into food energy with 100% efficiency.

  3. The human population has grown exponentially and will continue to do so.

These assumptions flaw the estimation, but they effectively provide a ceiling on the number of humans who can live in the Milky Way Galaxy. Waste will only serve to reduce the maximum number of people, as would less than exponential growth because time would be spent using energy and not contributing to more people, energy that would go into sustaining the unborn.

So to start, we have to figure how much energy is used by the human race daily. Effectively, that is:
Energy used per one person per day X Number of People = Energy used per day

To give these quantitative values, I'll have to make some more assumptions. For the energy used per person per one day, I'm going to use 2000 kilocalories--that quantity for which recommended values in the nutrition facts are based. (Kilo)calories are an actual measure of energy, which is why they can be used for this purpose. 2000 kilocalories works out to about 8.3 million Joules. Joules are the scientific measure of energy and will come up again.

For the number of people, we are assuming exponential growth. This takes the form of:

Population(t) = (Initial population) X e^(rt)

This may look vaguely familiar, even if you haven't had a lot of science--this is the same as the continuously compounded interest formula--"Pert"--taught in high schools.

The population formula tells us the number of people, making the energy used per day formula:

Energy used per day = (8.3 X 10^6 Joules/Person) X ((Initial Population) X e^(rt))

I've failed to fill in some quantities here--particularly, initial population and "r." For these, I go back to my assumptions, where I stated, "the human population has grown exponentially and will continue to do so." To fulfill this and fill in the missing variables, I will use two known dates and populations for the two unknowns--the population in 1950, roughly 2.5 billion, and the population in 2010, roughly 6.7 billion. January 1, 1950, then, is t=0, and January 1, 2010 is t=60*365. This solves for the two unknowns and gives us completely the energy used per day as a function of day.

Initially, when looking at this problem, I figured I'd be able to use a definite integral to solve it. Integrals are mathematical operations that add up infinitely small pieces of something between two points. This would be an easy integral, but it's not a valid place for one. This is energy used per day, not energy used per infinitely small piece of time, so the result wouldn't answer our question.

An integral won't work, but summation will. Instead of integrating the energy used everyday, simply add up the energies of each day, resulting in the total energy used. Because adding the energy of each day is a tedious exercise, we'll make a computer do it.

With all of this, I have an approximation of the total energy consumed at a particular time. However, I don't have an expression for the energy remaining or, at least, for the energy there in the first place. The energy remaining is:

Energy Available - Energy Used = Energy Remaining

Our fundamental question--when will humanity consume all the energy--in terms of this expression is "when will energy remaining equal zero?" We're relatively close--energy used we will know with brute force computer approximation. With energy available, though, we have to turn to astronomy.

To simplify our question, I'm going to treat energy available as energy available by combining all atoms of hydrogen in the Milky Way. About 17.6 MeV are produced in the creation of one helium nucleus. Scaling this to the entire Milky Way, along with a few unit conversions, there are about 4.34 X 10^53 Joules available. That's a lot of energy.

Exponential growth, however, is very powerful. Einstein once said that "compound interest is the most powerful force in the universe." It's the exponential that does it. Putting all of the information into the computer yielded the following date for such a fantastic Malthusian disaster:

Decimal year: 6417.92054795 Population: 2.43421421321e+42 Energy remaining: 1.98606230643e+50
Decimal year: 6417.92328767 Population: 2.43432759022e+42 Energy remaining: 1.78401311644e+50
Decimal year: 6417.9260274 Population: 2.4344409725e+42 Energy remaining: 1.58195451572e+50
Decimal year: 6417.92876712 Population: 2.43455436006e+42 Energy remaining: 1.37988650384e+50
Decimal year: 6417.93150685 Population: 2.4346677529e+42 Energy remaining: 1.17780908035e+50
Decimal year: 6417.93424658 Population: 2.43478115103e+42 Energy remaining: 9.75722244812e+49
Decimal year: 6417.9369863 Population: 2.43489455444e+42 Energy remaining: 7.73625996793e+49
Decimal year: 6417.93972603 Population: 2.43500796312e+42 Energy remaining: 5.71520335854e+49
Decimal year: 6417.94246575 Population: 2.43512137709e+42 Energy remaining: 3.69405261556e+49
Decimal year: 6417.94520548 Population: 2.43523479635e+42 Energy remaining: 1.67280773459e+49
Decimal year: 6417.94794521 Population: 2.43534822088e+42 Energy remaining: -3.4853128875e+48

The exponential growth allowed, in effect, for daily energy consumption to reach the same order of magnitude as the amount of energy available in the Milky Way.
There are, of course, many flaws in this, as stated before--lots of hand wavy estimates, neglected factors. A more precise mathematical model could be constructed, but why bother? The forces that guide population growth over time are very complex, often escaping the mathematical domain--forces like culture, technological capacity.

One thing from this can be said with certainty--the population of the human race residing within the Milky Way will never exceed 10^42.5 people. There simply is not enough resources to create and sustain such a large population. To accomplish this, the human race would have to harvest every atom of hydrogen--from vast fields of low-density, monatomic hydrogen to stars themselves--without using any energy and convert it directly into food without any loss of energy. Thermodynamics directly forbids this, so this is impossible. Once again, this estimate is a ceiling.

10^42.5 may sound small, especially considering that there's a possible cap on population in the spacefaring distant future. Here, however, our notation betrays us. Writing out 10^42.5 looks something like this:

There are currently 6,700,000,000 people, so 10^42.5 really is a lot of people. At the same time, it's strange to consider that there is a physical limit of population. Sure, on the Earth there are limitations regarding space and arable land, but aren't the depths of space effectively boundless? Not really--nearly any "effective" boundlessness can be caught up to with exponential growth at some point. It's only a matter of time.

Something I hadn't realized before was that, even in exponential growth, it's not the beginning that counts. Early on, during the simulation, the "energy remaining" column remained static. The energy consumed per day was trivial compared to the energy remaining in the galaxy. That is our current situation; the energy we consume really is small--in the scientific sense--compared to that available. It's only when the population is on the same order of magnitude of that of the galaxy where problems really kick in. Consuming energy in this same manner with our current population would last billions of years. With the rate of growth shown here, we lasted for thousands. It is the end, where the population is massive, where the situation really is hopeless.

We don't have to look at this from fatalistic perspective. We don't need to reproduce haphazardly into crises. Now is our chance to adapt our culture and way of living into one of survival. If the human spirit proves to be less inclined to live in quantity, then we may have a shot at living forever.


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