Mathematics of a “Cold” Space-Faring Society

Ultimately, civilization will probably use fusion reactors for power. Currently, this is still far out of the reach of humanity, but benefits that come from better technology (particularly material science and even more particularly semiconductors) will yield improvements in fusion vastly disproportionate to the investment. Here is a great talk on the current state-of-the-art. In summary, the Iter project remains a gigantic political clusterfuck, but there seems to be an inevitability that fusion will displace all other power sources in the end. This is even more true as we expand toward the outer solar system, where solar power becomes diffuse to be almost useless.

At some sufficiently advanced point, we might as well take it for granted that fusion power has been mastered, and this point will probably be before a Kardashev scale type II civilization. That implies that something like a Dyson sphere/swarm won’t even be relevant at the point that someone would have the capabilities to make it. However, the constraints of such a society remain poorly articulated. If fusion power is available, then isn’t energy just a “solved” problem? Since the fuel is abundant, you won’t even really run out.

It turns out, things aren’t so simple in the real world. No matter how easy it is to make heat and convert it into work, you still have to put all that heat somewhere — reject it into a radiator of some sort. Even today, on Earth, power plants spend a significant fraction of their construction and operational costs on the cooling system if they are purely thermal power plants. I argue that the balance will be even more lopsided in space. While fusion power can be made cheap by technological advances, the cooling part is limited by total gross area and temperature, which will all resist favorable technological scaling that the energy source may have. So in the long run, the cooling part might comprise the vast majority of the cost of energy production of a space-faring society.

If you’re still not convinced of the need to rethink this matter — consider that the general efficacy of the energy production system will be determined almost entirely by the radiator temperature. I will prove this empirically here, and base everything on a Carnot cycle for a reference. When producing and using energy in deep space, the following is the system:

You can see 2 thermal cycles. That’s because the environment for the habitat (E) is warmer than the temperatures you can obtain by radiators within an arbitrary part of interstellar space. That means you can run a thermal cycle from the environment to space, recovering some of the energy you used. You then re-use that energy and it counts toward the useful work you can do. All of the useful work is ultimately converted into heat within the habitat. The energy flowing through the system is that produced by the fusion power plant, Q prime. You can start from the fundamental Carnot cycle and derive the following for yourself about the described system:

I took the liberty to provide a simplification at the end, because it’s roughly true. A fusion plant steam generator could be at 2,000 K or higher. Room temperature is something like 293 K, and space can be as cold as 4 K. Seeing orders-of-magnitude differences, we can comfortably just consider the final simplified version for our discussion.

I will discuss that conclusion, but first, let’s consider another approach to get roughly the same thing.

You may or may not like the idea of computronium, but it’s a useful tool for this kind of analysis. Now, the futuristic idea is that an advanced society can be approximated as a network of computers.

This raises the possibility of another method of analysis. That is to apply Landauer’s principle to the thermodynamic cycle at work. The math behind this is astoundingly simple. You have the energy coming from the fusion plant as Q prime. Next, a single computation requires k T ln(2) units of energy to complete. Thus, divide the energy flow available by the energy to produce the computation, and you have a proxy for the number of computations you can perform.

To keep this discussion qualitative, I just used proportionality instead of putting in the full expression. Once again, we see the heat reservoir temperature (Tc) in the denominator, although in an even simpler form here.

The bottom line is that the energy you can get for useful work is inversely proportional to the temperature you can attain. Why is such a blatantly fact not obvious to our own intuition? Why do the energy accounting agencies count anything in terms of energy, completely neglecting the heat sink temperature? Because on Earth, this discussion isn’t important. There is a fairly constant ambient temperature on this planet. In space, however, it becomes an extremely major issue. For a large city in space, it becomes critically important. For an advanced space-faring civilization, it may be so important as to be equated to their life-blood.

What’s more, the current plate of known options do not look great. Building coolant channels (pipes) to move heat out to a radiator and back will probably perform terribly in this optimization problem of maximizing energy production. The options as I see them are:

  1. Find comets and icy rogue plants, and consume their materials as fodder for your cooling systems without any attempt to re-cool them. This is using the “fossil cold” to get absurd efficiency.
  2. Deploy enormous megastructures that expose a large surface to space, far away from a star.
  3. In a controlled manner, drop large balls of metal or some other material into orbits of free-fall in interstellar space. On a regular schedule, visit them in your mobile habitat, deposit your heat, and move on to the next one.

I find option #1 to be inevitable, but it can be depleted. Eventually, and advanced society may “eat” all of the comets in their system, and only then will they be forced to go with one of the more expensive options. Stationary habitats (on a planet or asteroid) may have no choice but to use the expensive #2 option, but in the long-run, I think #3 is the most fascinating and the best performer.

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