What is the natural timescale for making a Dyson shell?

KIC 8462852 (“Tabby’s Star”) continues to confuse. I blogged earlier about why I doubt it is a Dyson sphere. SETI observations in radio and optical has not produced any finds. Now there is evidence that it has dimmed over a century timespan, something hard to square with the comet explanation. Phil Plait over at Bad Astronomy has a nice overview of the headscratching.

However, he said something that I strongly disagree with:

Now, again, let me be clear. I am NOT saying aliens here. But, I’d be remiss if I didn’t note that this general fading is sort of what you’d expect if aliens were building a Dyson swarm. As they construct more of the panels orbiting the star, they block more of its light bit by bit, so a distant observer sees the star fade over time.

However, this doesn’t work well either. … Also, blocking that much of the star over a century would mean they’d have to be cranking out solar panels.

Basically, he is saying that a century timescale construction of a Dyson shell is unlikely. Now, since I have argued that we could make a Dyson shell in about 40 years, I disagree. I got into a Twitter debate with Karim Jebari (@KarimJebari) about this, where he also doubted what the natural timescale for Dyson construction is. So here is a slightly longer than Twitter message exposition of my model.

Lower bound

There is a strict lower bound set by how long it takes for the star to produce enough energy to overcome the binding energy of the source bodies (assuming one already have more than enough collector area). This is on the order of days for terrestrial planets, as per Robert Bradbury’s original calculations.

Basic model

Starting with a small system that builds more copies of itself, solar collectors and mining equipment, one can get exponential growth.

A simple way of reasoning: if you have an area A(t) of solar collectors, you will have energy kA(t) to play with, where k is the energy collected per square meter. This will be used to lift and transform matter into more collectors. If we assume this takes x Joules per square meter on average, we get A'(t) = (k/x)A(t), which makes A(t) is an exponential function with time constant k/x. If a finished Dyson shell has area A_D\approx 2.8\cdot 10^{23} meters and we start with an initial plant of size A(0) (say on the order of a few hundred square meters), then the total time to completion is t = (x/k)\ln(A_D/A(0)) seconds. The logarithmic factor is about 50.

If we assume k \approx 3\cdot 10^2 W and x \approx 40.15 MJ/kg (see numerics below), then t=78 days.

This is very much in line with Robert’s original calculations. He pointed out that given the sun’s power output Earth could be theoretically disassembled in 22 days. In the above calculations  the time constant (the time it takes to get 2.7 times as much area) is 37 hours. So for most of the 78 days there is just a small system expanding, not making a significant dent in the planet nor being very visible over interstellar distances; only in the later part of the period will it start to have radical impact.

The timescale is robust to the above assumptions: sun-like main sequence stars have luminosities within an order of magnitude of the sun (so k can only change a factor of 10), using asteroid material (no gravitational binding cost) brings down x by a factor of 10; if the material needs to be vaporized x increases by less than a factor of 10; if a sizeable fraction of the matter is needed for mining/transport/building systems x goes down proportionally; much thinner shells (see below) may give three orders of magnitude smaller x (and hence bump into the hard bound above). So the conclusion is that for this model the natural timescale of terrestrial planetary disassembly into Dyson shells is on the order of months.

Digging into the practicalities of course shows that there are some other issues. Material needs to be transported into place (natural timescale about a year for a moving something 1 AU), the heating effects are going to be major on the planet being disassembled (lots of energy flow there, but of course just boiling it into space and capturing the condensing dust is a pretty good lifting method), the time it takes to convert 1 kg of undifferentiated matter into something useful places a limit of the mass flow per converting device, and so on. This is why our conservative estimate was 40 years for a Mercury-based shell: we assumed a pretty slow transport system.

Numerical values

Estimate for x: assuming that each square meter shell has mass 1 kg, that the energy cost comes from the mean gravitational binding energy of Earth per kg of mass (37.5 MJ/kg), plus processing energy (on the order of 2.65 MJ/kg for heating and melting silicon). Note that using Earth slows things significantly.

I had a conversation with Eric Drexler today, where he pointed out that assuming 1 kg/square meter for the shell is arbitrary. There is a particular area density that is special: given that solar gravity and light pressure both decline with the square of the distance, there exists a particular density \rho=E/(4 \pi c G M_{sun})\approx 0.78 gram per square meter, which will just hang there neutrally. Heavier shells will need to orbit to remain where they are, lighter shells need cables or extra weight to not blow away. This might hence be a natural density for shells, making x a factor 1282 smaller.

Linear growth does not work

I think the key implicit assumption in Plait’s thought above is that he imagines some kind of alien factory churning out shell. If it produces it at a constant rate A', then the time until it a has produced a finished Dyson shell with area A_D\approx 2.8\cdot 10^{23} square meters. That will take A_D/A' seconds.

Current solar cell factories produce on the order of a few hundred MW of solar cells per year; assuming each makes about 2 million square meters per year, we need 140 million billion years. Making a million factories merely brings things down to 140 billion years. To get a century scale dimming time, A' \approx 8.9\cdot 10^{13} square meters per second, about the area of the Atlantic ocean.

This feels absurd. Which is no good reason for discounting the possibility.

Automation makes the absurd normal

As we argued in our paper, the key assumptions are (1) things we can do can be automated, so that if there are more machines doing it (or doing it faster) there will be more done. (2) we have historically been good at doing things already occurring in nature. (3) self-replication and autonomous action occurs in nature. 2+3 suggests exponentially growing technologies are possible where a myriad entities work in parallel, and 1 suggests that this allows functions such as manufacturing to be scaled up as far as the growth goes. As Kardashev pointed out, there is no reason to think there is any particular size scale for the activities of a civilization except as set by resources and communication.

Incidentally, automation is also why cost overruns or lack of will may not matter so much for this kind of megascale projects. The reason Intel and AMD can reliably make billions of processors containing billions of transistors each is that everything is automated. Making the blueprint and fab pipeline is highly complex and requires an impressive degree of skill (this is where most overruns and delays happen), but once it is done production can just go on indefinitely. The same thing is true of Dyson-making replicators. The first one may be a tough problem that takes time to achieve, but once it is up and running it is autonomous and merely requires some degree of watching (make sure it only picks apart the planets you don’t want!) There is no requirement of continued interest in its operations to keep them going.

Likely growth rates

But is exponential growth limited mostly by energy the natural growth rate? As Karim and others have suggested, maybe the aliens are lazy or taking their time? Or, conversely, that multi century projects are unexpectedly long-term and hence rare.

Obviously projects could occur with any possible speed: if something can construct something in time X, it can in generally be done half as fast. And if you can construct something of size X, you can do half of it. But not every speed or boundary is natural. We do not answer the question of why a forest or the Great Barrier reef have the size they do by cost overruns stopping them, or that they will eventually grow to arbitrary size, but the growth rate is so small that it is imperceptible. The spread of a wildfire is largely set by physical factors, and a static wildfire will soon approach its maximum allowed speed since part of the fire that do not spread will be overtaken by parts that do. The same is true for species colonizing new ecological niches or businesses finding new markets. They can run slow, it is just that typically they seem to move as fast as they can.

Human economic growth has been on the order of 2% per year for very long historical periods. That implies a time constant \ln(1.02)\approx 50 years. This is a “stylized fact” that remained roughly true despite very different technologies, cultures, attempts at boosting it, etc. It seems to be “natural” for human economies. So were a Dyson shell built as a part of a human economy, we might expect it to be completed in 250 years.

What about biological reproduction rates? Merkle and Freitas lists the replication time for various organisms and machines. They cover almost 25 orders of magnitude, but seem to roughly scale as \tau \approx c M^{1/4}, where M is the mass and c\approx 10^7. So if a total mass $M_T$ needs to be converted into replicators of mass M, it will take time t=\tau\ln(M_T)/\ln(2). Plugging in the first formula gives t=c M^{1/4} \ln(M_T)/\ln(2). The smallest independent replicators have M_s=10^{-15} kg (this gives \tau_s=10^{3.25}=29 minutes) while a big factory-like replicator (or a tree!) would have M_b=10^5 (\tau_b=10^{8.25}=5.6 years). In turn, if we set M_T=A_D\rho=2.18\cdot 10^{20} (a “light” Dyson shell) the time till construction ranges from 32 hours for the tiny to 378 years for the heavy replicator. Setting M_T to an Earth mass gives a range from 36 hours to 408 years.

The lower end is infeasible, since this model assumes enough input material and energy – the explosive growth of bacteria-like replicators is not possible if there is not enough energy to lift matter out of gravity wells. But it is telling that the upper end of the range is merely multi-century. This makes a century dimming actually reasonable if we think we are seeing the last stages (remember, most of the construction time the star will be looking totally normal); however, as I argued in my previous post, the likelihood of seeing this period in a random star being englobed is rather low. So if you want to claim it takes millennia or more to build a Dyson shell, you need to assume replicators that are very large and heavy.

[Also note that some of the technological systems discussed in Merkle & Freitas are significantly faster than the main branch. Also, this discussion has talked about general replicators able to make all their parts: if subsystems specialize they can become significantly faster than more general constructors. Hence we have reason to think that the upper end is conservative.]

Conclusion

There is a lower limit on how fast a Dyson shell can be built, which is likely on the order of hours for manufacturing and a year of dispersion. Replicator sizes smaller than a hundred tons imply a construction time at most a few centuries. This range includes the effect of existing biological and economic growth rates. We hence have a good reason to think most Dyson construction is fast compared to astronomical time, and that catching a star being englobed is pretty unlikely.

I think that models involving slowly growing Dyson spheres require more motivation than models where they are closer to the limits of growth.

18 thoughts on “What is the natural timescale for making a Dyson shell?

  1. It is possible that we are seeing Tabby’s Star in the midst of a millennial project to enclose a Dyson Sphere around it. Since it has a lifetime of several million millennia, the odds that we are in this privileged moment are several million to one against.
    But all the natural explanations are kooky as well. I doubt this star is First Contact, but it has well earned its nickname of WTF-001.

  2. It’s the last part pf the process that would need to slow down considerably from exponential growth. Once you’ve established a nice population of power collectors around this star, all that collected power can be directed towards the planet in order to disassemble it; but this would cause the planet to heat up, and any disassemblers working there would melt. Simply boiling the planet would result in a cloud of vapour which would be difficult to harvest.

    Hmm; perhaps we could borrow an idea from Douglas Adams; a hollow shell far above the planet, supported (and cooled) by orbital rings; the planet boils merrily inside, and condenses on the inside of the shell, where it can be harvested at a lower temperature.

    1. Sublimating planets! 🙂

      I think a good method is to construct mass flows of capsules launched and turned electromagnetically. The ones going upward bring up mass and heated coolant, the ones going downward bring down coolant from big radiator installations in the shadow of the collectors.

      In practice one would start with all the small objects like asteroids, since they have nearly no binding energy. Once they are done one would already have a fairly big shell to use.

  3. Really, we are like Francis Godwin or Cyrano de Bergerac trying to figure out how to get to the Moon (or at best Jules Verne or H G Wells) when it comes to building Dyson Spheres. Maybe a month is a good timescale for the dimming of the star. Maybe a millennium. So I don’t think the speed of KIC 8462852’s dimming is an argument against it being a Dyson Sphere. I have heard someone say the asymmetry of the dimming is an argument against it being a DS. I dunno, as I said, we are too ignorant of how to build a DS to be really sure about that. It is the improbability of us being so temporally/chronolgically privileged that makes me skeptical.
    There are three prospects. Either KIC 8462852 is a boring natural phenomenon (like Martian Canals being an optical illusion), a bizarre natural phenomenon (like “LGM-1” being the first discovered pulsar), or it is ETI (no historical analog, but bajillions of SF ones). I think prospect 1 is ruled out by now. I expect prospect 2, but am still hoping for prospect 3.

    1. I gave arguments in my post.

      The problem with “we are not advanced enough to understand” claims is that they do not have any explanatory power. They merely tell us that we should be cautious about believing arguments too strongly. At most they suggest we should keep closer to our priors.

  4. According to Wikipedia the next “scheduled” dimming is May 2017.
    One of two things will happen…it will/won’t dim.
    If it does, we will be looking closely this time (spectra, etc.).
    If it doesn’t, a lot of models (involving 750 day orbit or period) are ruled out.
    15 months…how I wish it were 15 hours! 🙂

  5. Please forgive my commenting again, I am an aspie…when obsession strikes, it strikes 🙂
    I thought of a way of making seeing the construction of a Dyson Sphere merely unlikely, not *preposterously* unlikely…if it is a common event in the Milky Way. But this would imply that there are at least several billion DSs in the MW.
    Do you know if there is an upper bound on the frequency of DSs in our galaxy, especially Matryoshka Dyson Spheres which might be harder to detect?

    1. Obviously we see the galaxy, so not *every* star can be a DS. Let’s suppose a fraction F1 of a star’s lifespan is taken up by Dyson construction, this tends to happen at a random age, and a fraction F2 of stars are ever converted. Then at the moment, we should expect out of N stars to see [(1-F2) + F2*(1/2 – F1)]*N visible stars, F2*(1/2 – F1)*N Dyson shells, and F2*F1*N construction processes. So out of the visible stars there would bne F1*F2/(1-F2+F2/2-F1*F2)=F1*F2/(1+(F1-1/2)*F2) construction processes.

      Now, F2 cannot be large, or we would notice a lot of missing mass that wouldn’t fit the dark matter searches (when Robert Bradbury suggested M-brains was to blame MACHOs was still a viable possibility of dark matter, but it has been ruled out through gravitational lensing). I am uncertain of the exact bound, but I think if more than 1% of stars were converted we would really notice. F1 looks like it should also be a small number, since a century or even a million years out of a multi-billion year lifespan is less than 10^-3. So that gives an estimate of 0.028% of all stars visible right now having construction. I think that is close to ruled out by the number of stars in the Kepler field (223,000) – there would be about 62 of them showing construction. So clearly these numbers have to be lower.

    1. The best reason I know for a partial DS is a Shkadov thruster: reflect sunlight to steer the star (slowly).

      A full DS seems to be reasonably stable, especially if you have a bit of station-keeping from the orbiting panels using solar sail “rudders”.

    1. The Kepler light curve also has an odd knee, tricky to fit to an ongoing fine-grained construction process.

    1. I am not super-convinced; mirror systems can look arbitrarily confusing. So they can clearly explain a very wide range of light curves, and the Bayes update becomes rather weak. The ring paper is cool, though.

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