# 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.

# Halloween explanation of Fermi question

John Harris proposed a radical solution to the KIC 8462852 problem: it is a Halloween pumpkin.

A full Dyson sphere does not have to be 100% opaque. It consists of independently orbiting energy collectors, presumably big flat surfaces. But such collectors can turn their thin side towards the star, letting past starlight. So with the right program, your Dyson sphere could project any pattern of light like a lantern.

Of course, the real implication of this is that we should watch out for trick-or-treating alien super-civilizations. By using self-replicating Bracewell probes they could spread across the Milky way within a few million years: they ought to be here by now. And in this scenario they are… they are just hiding until KIC 8462852 suddenly turns into a skull, and suddenly the skies will swarming with their saucers demanding we give them treats – or suffer their tricks…

There is just one problem: when is galactic Halloween? A galactic year is 250 million years. We have a 1/365 chance of being in the galactic “day” corresponding to Halloween (itself 680,000 years long). We might be in for a long night…

# Likely not even a microDyson

Right now KIC 8462852 is really hot, and not just because it is a F3 V/IV type star: the light curve, as measured by Kepler, has irregular dips that looks like something (or rather, several somethings) are obscuring the star. The shapes of the dips are odd. The system is too old and IR-clean to have a remaining protoplanetary disk, dust clumps would coalesce, the aftermath of a giant planet impact is very unlikely (and hard to fit with the aperiodicity); maybe there is a storm of comets due to a recent stellar encounter, but comets are not very good at obscuring stars. So a lot of people on the net are quietly or not so quietly thinking that just maybe this is a Dyson sphere under construction.

I doubt it.

My basic argument is this: if a civilization builds a Dyson sphere it is unlikely to remain small for a long period of time. Just as planetary collisions are so rare that we should not expect to see any in the Kepler field, the time it takes to make a Dyson sphere is also very short: seeing it during construction is very unlikely.

# Fast enshrouding

In my and Stuart Armstrong’s paper “Eternity in Six Hours” we calculated that disassembling Mercury to make a partial Dyson shell could be done in 31 years. We did not try to push things here: our aim was to show that using a small fraction of the resources in the solar system it is possible to harness enough energy to launch a massive space colonization effort (literally reaching every reachable galaxy, eventually each solar system). Using energy from already built solar captors more material is mined and launched, producing an exponential feedback loop. This was originally discussed by Robert Bradbury. The time to disassemble terrestrial planets is not much longer than for Mercury, while the gas giants would take a few centuries.

If we imagine the history of a F5 star 1,000 years is not much. Given the estimated mass of KIC 8462852 as 1.46 solar masses, it will have a main sequence lifespan of 4.1 billion years. The chance of seeing it while being enshrouded is one in 4.3 million. This is the same problem as the giant impact theory.

# A ruin?

An abandoned Dyson shell would likely start clumping together; this might at first sound like a promising – if depressing – explanation of the observation. But the timescale is likely faster than planetary formation timescales of $10^5$$10^6$ years – the pieces are in nearly identical orbits – so the probability problem remains.

But it is indeed more likely to see the decay of the shell than the construction by several orders of magnitude. Just like normal ruins hang around far longer than the time it took to build the original building.

# Laid-back aliens?

Maybe the aliens are not pushing things? Obviously one can build a Dyson shell very slowly – in a sense we are doing it (and disassembling Earth to a tiny extent!) by launching satellites one by one. So if an alien civilization wanted to grow at a leisurely rate or just needed a bit of Dyson shell they could of course do it.

However, if you need something like $2.87\cdot 10^{19}$ Watt (a 100,000 km collector at 1 AU around the star) your demands are not modest. Freeman Dyson originally proposed the concept based on the observation that human energy needs were growing exponentially, and this was the logical endpoint. Even at 1% growth rate a civilization quickly – in a few millennia – need most of the star’s energy.

In order to get a reasonably high probability of seeing an incomplete shell we need to assume growth rates that are exceedingly small (on the order of less than a millionth per year). While it is not impossible, given how the trend seems to be towards more intense energy use in many systems and that entities with higher growth rates will tend to dominate a population, it seems rather unlikely. Of course, one can argue that we currently can more easily detect the rare laid-back civilizations than the ones that aggressively enshrouded their stars, but Dyson spheres do look pretty rare.

# Other uses?

Dyson shells are not the only megastructures that could cause intriguing transits.

C. R. McInnes has a suite of fun papers looking at various kinds of light-related megastructures. One can sort asteroid material using light pressure, engineer climate, adjust planetary orbits, and of course travel using solar sails. Most of these are smallish compared to stars (and in many cases dust clouds), but they show some of the utility of obscuring objects.

Duncan Forgan has a paper on detecting stellar engines (Shkadov thrusters) using light curves; unfortunately the calculated curves do not fit KIC8462852 as far as I can tell.

Luc Arnold analysed the light curves produced by various shapes of artificial objectsHe suggested that one could make a weirdly shaped mask for signalling one’s presence using transits. In principle one could make nearly any shape, but for signalling something unusual yet simple enough to be artificial would make most sense: I doubt the KIC transits fit this.

# More research is needed (duh)

In the end, we need more data. I suspect we will find that it is yet another odd natural phenomenon or coincidence. But it makes sense to watch, just in case.

Were we to learn that there is (or was) a technological civilization acting on a grand scale it would be immensely reassuring: we would know intelligent life could survive for at least some sizeable time. This is the opposite side of the Great Filter argument for why we should hope not to see any extraterrestrial life: life without intelligence is evidence for intelligence either being rare or transient, but somewhat non-transient intelligence in our backyard (just 1,500 light-years away!) is evidence that it is neither rare nor transient. Which is good news, unless we fancy ourselves as unique and burdened by being stewards of the entire reachable universe.

But I think we will instead learn that the ordinary processes of astrophysics can produce weird transit curves, perhaps due to weird objects (remember when we thought hot jupiters were exotic?) The universe is full of strange things, which makes me happy I live in it.

[An edited version of this post can be found at The Conversation: What are the odds of an alien megastructure blocking light from a distant star? ]