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Okay, so the idea is that humans have found a method to terraform planets by way of an intentionally engineered grey goo scenario. The bots would be used to photosynthesize a sustainable atmosphere by operating as an algae stand-in before moving on to other tasks once that was completed. The bit I need help with is figuring out how long it would reasonably take to cover an area with the nanobots after landing a seed probe, assuming exponential growth.

Ideally, I'd like an estimate for both city-sized, continent, and entire planet, but the only one I actually need is the timeframe for the full planet. For the sake of convenience, assume a relatively level terrain without statistically significant high or low points, with no liquid water and a total surface area effectively identical to that of Earth.

Replication occurs at a rate of once per minute with a starting material base of approximately a trillion nanites. For the sake of the question, assume only one seed per planet.

EDIT — After looking through the answers, I see a few ways to improve the question. First up, it seems I was quite a bit overly generous with the replication rate. With that in mind: Replication occurs at about half the standard rate of natural mitosis, so between one to one and a half hours.

Next, energy source: mostly solar. Capable of using thermal, petroleum, natural gas, or radiation with access to any those materials. For the purpose of the question, assume energy is not a constraint and that replication can occur at maximum rate throughout.

Finally, locomotion: early phase of replication involves forming larger structures capable of rapid travel and dispersal of nanites. About the size and speed of an ant. Seed material divides entirety of it's original mass into producing these objects, which then proceed to travel outward radially from central mass laying a trail of nanites in their wake. This continues until enough mass has been lost that it is no longer efficient to continue, at which point the remaining nanites revert to early stage routines focusing on building up enough seed material to repeat the process.

The actual growth pattern will involve large rapid advances, followed by stopgap periods while the gains are consolidated and the areas between the 'spokes' are filled in, before repeating. Nanites located in the center that are no longer capable of efficiently contributing to expansion will focus on energy acquisition and beginning the actual terraforming process.

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    $\begingroup$ en.wikipedia.org/wiki/Gray_goo for anyone wondering what the question is asking. $\endgroup$
    – Green
    Jul 7 at 20:38
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    $\begingroup$ How long does it take one to self-replicate? How many does a terraforming operation deploy? Decide those and you'll have an answer; otherwise it's a guess. $\endgroup$
    – rek
    Jul 7 at 21:13
  • $\begingroup$ VTC:Needs-Details. @rek is right, without the initial starting count and rate of replication, this is impossible to answer. It's worth noting that "exponential growth" is meaningless because that speed is absorbed into the circumference of the increasing circle of influence (simplifying a bit) as that circumference grows. The estimate of coverage is likely only geometric and could be estimated by the time required to draw a "straight" line from the starting point to the opposite side of the planet. \@jbh me when you edit your Q and I'll retract my vote. $\endgroup$ Jul 7 at 22:53
  • $\begingroup$ Added extra info as requested. $\endgroup$
    – Aristo-TTL
    Jul 8 at 2:33
  • $\begingroup$ Two more pieces of information: what is the specific size of a nanite and how fast can it move? @jbh me after that and I'll retract. $\endgroup$ Jul 8 at 7:34
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A nanobot is generally somewhere in the size range of 0.1-10μm. For simplicity sake, I will assume that your nanobots are somewhere in the middle with about a 1μm^2 cross section. So, assuming you start with 1 trillion nanobots, that means that your initial coverage area is at most going to be an area about 64cm across and one layer of nanobots thick.

While logic tells us growth should be exponential, this is not necessarily true depending on how the goo disperses itself.

First let's assume that it spreads like a bacterial colony: Even if it grows in all 3 dimensions, only the nanobots that can make contact with the raw materials of the ground will be able to replicate at any given time. At first, your nanobots replicated at the middle of a goo will be able to crawl to the edge allowing an outward expansion to seem exponential as you maintain only a minimally thin layer of bots so that all bots can replicate all the time, but eventually you will hit a maximum rate of expansion limited by how fast your bots can crawl.

The fastest single celled organisms seem to be able to traverse at speeds of about 1mm/sec. If they can reproduce at a rate of once per minute, then that means that only the outer 60mm of any given grey goo colony can meaningfully contribute to the colony's rate of expansion. This means that you can estimate the time it takes to cover a planet as half of it's circumference in mm/sec. An Earth sized world has a circumference of ~40,000 km; so, that means that your grey goo will take about 634 years under its own locomotion to spread to the whole planet.

But this math does not tell the whole story here... If your planet has wind and rivers, these could pick up nanobots and spread them much more quickly than they could crawl. A large river could spread nanobots over thousands of miles in just a few months. A large storm like a hurricane could even spread bots over the area of a small country in just a few days. These weather events will ensure that instead of just having 1 slowly expanding blob, that you would have billions of tiny colonies spread over a larger area.

If a weather event were to deposit just 1 nantite by itself somewhere, within 30 minutes it could grow to a radius of 60mm reaching its maximum rate of expansion of 1mm/sec. This means that weather could account for speeding up the process to the point that large portions of your world could be completely covered within the first few years. The more weather your world has, the faster the bots can spread.

Furthermore, there is the question of how intelligently these nanobots can work together. Macroscopic organisms can move much faster than microbes because we use the compounding locomotion of cells chained together. So, if you have a thousand nanobots all pushing in chain, at a speed of 1mm/sec then the outer bots will actually be moving apart at a speed of 1m/sec. This means that a colony designed to push itself apart can actually open up room for its central bots to replicate allowing it to grow at a speed limited only by the toughness of your nanobots and how fast it can move it's parts. While microbes can only traverse at very low speeds, some can expand and contract at accelerations up to 200m/sec^2... so, the only real limit here is how much cross-sectional stress the bots can take as they expand. This becomes much harder to estimate, but if we assume these bots are roughly comparable to the toughness of macroscopic organisms, then we can assume that they can probably push apart about as fast as a macroscopic organism can push off the ground to run... so following this model, it would not be unfeasible to assume the goo's growth rate could cap off at rates in excess of 100km/hr.

enter image description here

If this is the case, then your goo could spread to cover the whole planet under it's own power in about 8-9 days assuming the technology behind them can actually sustain a 1 minute per generation rate of replication the whole time. That said, MolbOrg brings up a good point in comments that power and resource scarcity will make 1 minute per generation replication pretty much impossible with any sort of near-future technologies.

For this to work you need to assume your nanites are powered by some manner of handwavium and that they can replicate themselves from pretty much any form of matter.

To maintain this speed, the nanobot colony will need to achieve a radius of about 1.7 km and move along the ground in a rippling pattern so that no cross section has to actually support the lateral weight of the whole 1.7km body of nanobots. Since this is a terraforming goo, it is also worth noting that the big moving mass of replicators do not have to cover the entire planet. Instead they will form what looks like a tsunami of bots that spreads in a 1.7km ring around the world, leaving behind only enough bots to do the terraforming, and recycling the rest to replenish what was taken from the soil to make your grey goo.

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    $\begingroup$ Generally not bad, at least interesting, but that assumotikn "like a bacterial colony" it really got u in a wrong direction, you should have started with the "how intelligently these nanobots can work together." and you probably would notice energy problem and not come with the 8-9 days which is just unrealistic cuz energy source is sun illuminatiin and it isn't that rich, I guess, needs calculations. Main objection the thing isn't restricted by locomotion, but more by bill of materials and energy for construction. $\endgroup$
    – MolbOrg
    Jul 8 at 18:21
  • $\begingroup$ @MolbOrg That is a valid concern; so, I've added mention of it to the answer. However, you can not automatically assume that they are solar powered and require scarce resources. The OP stated "Replication occurs at a rate of once per minute" and makes no speculations about how they reproduce so fast or how advanced the tech is. Not knowing how sophisticated these things are is part of why I included both methods of organization to establish upper and lower boundaries of how long it could take. $\endgroup$
    – Nosajimiki
    Jul 8 at 19:02
  • $\begingroup$ Wrote my take on the thing, I think you can deciper it and word it better or get some inspirations. OP with 2 minutes is unfortunatly not qualified and basically was forced to squese that by JBH under of treat of q clousure. Clearly he does not know(not a surprise, no one knows), and I think a good answer has to adress that and find some way to put some reasonable estimations, which is basically the essence of OPs q. Subject of q sure is poorly specified, namely what those nanomachines are capable of, but it also is why it is a q for the OP. $\endgroup$
    – MolbOrg
    Jul 8 at 21:11
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There's a good Skeptoid episode about nanobots, which includes a link to a scientific paper with the wonderful title of Some Limits to Global Ecophagy by Biovorous Nanoreplicators, with Public Policy Recommendations which goes into the question in some detail. The author considers a number of possible scenarios, all of which would proceed differently, but the abstract provides a pretty good overall answer to your question:

The maximum rate of global ecophagy by biovorous self-replicating nanorobots is fundamentally restricted by the replicative strategy employed; by the maximum dispersal velocity of mobile replicators; by operational energy and chemical element requirements; by the homeostatic resistance of biological ecologies to ecophagy; by ecophagic thermal pollution limits (ETPL); and most importantly by our determination and readiness to stop them. Assuming current and foreseeable energy-dissipative designs requiring ~100 MJ/kg for chemical transformations (most likely for biovorous systems), ecophagy that proceeds slowly enough to add ~4°C to global warming (near the current threshold for immediate climatological detection) will require ~20 months to run to completion; faster ecophagic devices run hotter, allowing quicker detection by policing authorities. All ecophagic scenarios examined appear to permit early detection by vigilant monitoring, thus enabling rapid deployment of effective defensive instrumentalities.

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    $\begingroup$ Can you include the relative part of the links in the body of your reply? Without them this is not an answer. $\endgroup$
    – rek
    Jul 8 at 13:01
  • $\begingroup$ Okay, editing now. $\endgroup$ Jul 8 at 17:22
  • $\begingroup$ Good cite, like firrst sentence and first half of it, it adreses exactly the thing the question is lacking or what OP failed to consider $\endgroup$
    – MolbOrg
    Jul 8 at 18:02
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The main factors here is how fast the nanobots can move and their efficiency at converting foreign materials.

This, of course, has probably been taken into account by the nanobot's creators. A single nanobot, or the smallest viable nanobot group (they probably need to collaborate in order to reproduce - no way a single nanobot can instantiate a Von Neumann replicator), will surely be able to crawl.

Larger aggregations of nanobots could be able to fly, by organizing in large thin sheets and flapping; roll, organizing into wheels; or run, by forming stick-like "leg" configurations and moving that way. Maximum speed will be probably below 50 km/h, except maybe on highways or steep descents (the "wheel" will probably disintegrate into nanobot groups, but that's okay - it might even be a feature).

I would expect the gray goo infection to spread more or less at that speed, the "infectors" spreading replicator groups. To prevent those groups growing too small to continue the locomotory phase, at least the ground-hugging infectors would probably gobble up material on the run, accumulating it internally (in the "hub" of the "wheel" for example) and using it to produce new replicator groups. If some needed material (rare earths, some metals, etc.) was not found, the mobile groups would either stop and become sessile for as long as required, or backtrack and hunt.

This might also be a way of containing the infection or keeping it out of some areas: a large volume of material useless for replication (e.g. saltwater, around a floating island - the floating part is against burrowing nanobots) that could nonetheless allow for spotting approaching infectors. The nanobots would then switch to passive floating groups, sort of "seeds", waiting to enter in contact with the vulnerable internal volume. This could be defended with something capable of disabling the seeds or impervious to their attack (hard UV light, EMP armor, boron nitride ceramics).

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  • $\begingroup$ First half, the cooperation part is not bad, rather good compared to other answers. $\endgroup$
    – MolbOrg
    Jul 8 at 18:24
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It depends on logistics, but probably a few years.

Your nanites infest pretty generic silicon planets. I'll suppose the surface, lacking water, is mostly SiO2, and that's all the nanites care about. They create an atmosphere of O2 with standard pressure 101.325 J/L = 101325 N / m^2 = 10.1325 N/cm^2. For Earthlike planets (close to 10 m/s^2 gravity) we'll say that's 1 kg / cm^2. Silicon dioxide is 2.3 g/cm^3, of which 1.2 g/cm^3 is oxygen, so we need to dig down 830 cm to get a kilogram of oxygen. In the process, we're going to release nearly a kilogram of silicon as nanites. Earth has 5E+8 km^2 of area = 5E+18 cm^2, so we need to make 5E+18 kg of nanites. (Technically we could just throw out most of the silicon, but it's not going to throw itself out unless it's nanites)

The nanites are well-patterned silicene, with a "negligible" amount of oxygen to insulate circuits. They are very long and narrow and absorb solar energy as a basal energy source. They can also emit light in coherent beams that are directed at one another as communication and energy source.

They propel themselves and absorb additional energy by "electropulsion". Each nanite has the ability to eject an electron from its semiconductor matrix, accelerating it along the long axis. These rapidly moving electrons serve as reaction mass, passing short distances where they are almost invariably absorbed by another nanite.

The control of electrical potential by sending or receiving electrons (they can request them via their light-based internet) means that they can create electrostatic structures, pressing themselves apart by charge. This is how they can climb rapidly to the ionosphere to draw its power, for example. It also allows them to create an "electrolev train" for nanites to skate effortlessly long distances in air currents that are controlled by charge differentials.

If limited to solar energy, the nanites can only absorb (roughly) 1000 W/m^2, or 0.1 J/s cm^2. I'll possibly abuse the heat of formation of -910 kJ/mol to suppose we need about 15 MJ to make the nanites we need, or 1.5E+8 seconds of sunlight, or 4.8 years. Ouch! Well, if solar energy could strip the rock off a planet easily, we'd all be in big trouble I guess. Based on this, I suppose the other statistics on replication are irrelevant.

So the answer really hinges on what energy sources they can pull out. Wind is obvious - harvested at every level, all the time, it could be very substantial, but how well you can do that with an electrostatic network of nanites isn't clear to me.

It is possible that the nanites can absorb electrical potential in the atmosphere, the ionosphere, even Van Allen radiation belts, by relaying electrons; however, doing so effectively without losing too much energy in resistance is no easy feat of design.

The planet may provide some built-in sources of chemical energy, though I doubt they will amount to much. Nonetheless, nanites can burrow and seek out mineral resources. But with the oxygen as the end product, burning abiotic petroleum, if it exists, would do no good!

I'm going to go with "a few years" here.

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  • $\begingroup$ 4 times of 4.8, cuz, surface recieving energy is a circle(crossection of a sphere), and surface of a plant is 4 times of surface of that circle. $\endgroup$
    – MolbOrg
    Jul 12 at 19:04
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Nanomachines, Son

General depiction of GG apocalyptic scenarios misses 2 or 3 important points.

  1. Not every material can be used for nanomachines construction - bill of materials
  2. Construction, extraction, etc requires energy, which has to be obtained from the external environment, and no, it can't be magic of sucking heat just cuz those are nanomachines son - no of such activity can break rules we well aware on a macroscopic scale for just because it is some nanomachines.
  3. This one can be attributed to the inertia of thinking, when people get a hand on the idea of nanomachines they try to solve all and every problem with it, totally missing the fact that they are a good means to create macroscopic constructions - machines, tools, etc - things which we are more familiar with, especially if it better suits for a task. What they really add as a tool, the nanomachines, is the flexibility and ease to do so on all scales from nano to macro.

What that means

Reading currently present answers and it seems everyone missed point 2 and start to consider "a problem" of locomotion, which is not a problem because of point 3.

One answer attempts to assume some degree of cooperation, but let's put few sentences of how it really should be.

You do not do nanomachines for purpose of them procreating, there is no use in them restructuring some materials in themselves for purpose of restructuring. I mean, once you do that for the first time - sure it is cool and all that, but the next step(actually way before you succeed with the first one) is to consider and design things in the way you can practically use. So GG has to have a practical use, and it is not such a bad idea to apply those existing useful uses for planet terraforming as in OP's case.

Idk, let have some simple example(s) what useful things nanomachines can do, what practical things may look like, and at the same time for them to be less like universal matter replicator(which they can't be anyway).

I guess many may be familiar with the terms additive and subtractive manufacturing - additive is your regular 3d printer(and other more factory type approaches like that) processes, subtractive it basically cutting and metal chips production.

Nanomachines can combine both, by being just one subtractive manufacturing process.

  • not meaning they can't be additive as well, and it generally, I guess, is the notion about them being just additive ones, but they do not have to be, and there are all sorts of problems for them to be additive ones. But as better and more flexible cutters they really can change things, spare a lot of energy, solve a lot of problems.

Everything which we 3d print today, can be carved out by nanomachines. They do not have to produce a lot of chips to shape blanks into parts, like EDM machine they can do microscopic cuts of any configuration(EDM is good 0.2 mm cuts, but nm can do better 2 orders of magnitude with any configuration of cut surface which EDM can't so as no other tech can). This could be quite a process that could replace all(99%)of current manufacturing processes, not relying on some magical nanoscale to nanoscale interaction of a tool(nanomachines, gg, nm) and the raw materials, blanks, etc.

So there is nothing strange in objectives like carving out stone gears and wheels and making reflectors and Stirling engines early on to address the locomotion problem - I mean mostly there is a myriad of ways to solve that "problem" - airplane, boats, submarines, zeppelins, etc etc. An airplane that can be of a size and mass of typical foam model and be capable to seed a big territory, or it may be a thousand tonnes load zeppelin fleet - like big flat bubbles floating and absorbing energy.

EROEI is a key in every bootstrapping expansion.

EROEI is an important aspect of any expansion. It really meaningless to set a 2minute replication time, and btw quite fast bacterias do that in half-hour, and it is a more reasonable number if you like to handwave the time.

An important question is how much energy it needs to invest in material processing and nanobots construction and how much energy it can bring in return.

Unfortunately, it depends on the technology of those nanobots, and thus unless we get hands-on specs of such technology we are bound to make assumptions. However, those assumptions do not necessarily have to be unreasonable. And it quite helpful to break that box of mentality everything has to be done by nanomachines and adopt the notion that nanomachines are the grease in the processes and can look like or be like macro processes we know.

Doing that(breaking the box) in the right way still lands us in assumption territory, even if we would like to use existing technologies to guesstimate upper limits of the processes, using photovoltaics as an example, because it hard to find real numbers about those and it depends.

But in general, there are numbers like we can increase powerplant energy production, double it in 2-3 years. For multiple reasons, like less waste in cutting wafers(2-3-4 times of improvement here alone), in using less metal, higher utilization of materials laying around, less energy to make brick(just for lack of better words and example) materials with less energy, less energy to make cement-like stuff, better reuse of that cement-like stuff(almost 0 energy to do so), no regulations because of different goals, etc it may be quite reasonable to assume that number of 2-3 years can be significantly lowered, but by how much exactly is unknown, I would guess the order of magnitude easily, would be not surprised by two orders of magnitude.

Still, plenty of guestimations on the road, let's hope OP manages to do that in an interesting way on his own or through more questions on WB.

potential, reasonable scenario, upper limit, and bill of materials

  • it sure IMHO, keeping in mind particular technology, 2d-nanomachines, which specs it out of scope.
  • also to mention the bill of materials problem, which I forgot to do so earlier

The technology of nanomachines, which I keep in mind, uses mostly carbon nanotubes, and may be doped with some other materials, but nothing exotic like rare earth materials, but more like your typical silicone stuff.

  • that notion that nanomachines can be made of any material, engulf or convert a whole planet in nanomachines globe, using 100 percent of it - it has no bearing in reality - a hint? Chemistry.

Based on a space-related estimation, where I used existing technologies, somewhere in my answers, double happens in 3-5 days(without nanotech), upper limit which way too generous is how things are rolling today on the planet a double in 2-3 years.

So it needs to estimate how much energy can 1kg of nanotech worth - chemistry is helpful here idk some random number of 12kWh/kg (on pair with oil) probably is stored in that 1kg.

One of the last things to figure out or assume is the efficiency of extraction of energy from the environment, like sun light, let's say 10-20 percent, and efficiency of use of that energy to extract bill of materials from the environment and use to store/recombine it in nanomachines, let's go with the same 20 percent.

The last thing is how much energy can be produced by that 1kg of nanomachines, quite a question, let's estimate it by silicon wafer 50 microns thick, with 10 percent of efficiency, 2.33 grams/cm3, soo 1kg is enough for about 400mL of the thing, and smeared with a thickness of 0.005cm (why google gets things in g per cm, soo wrong, google use metric units it kg per cubic meter; SpaceX use meters per second) that gives us 80'000 square centimeters and back to the metric world it is 8 sq meters.

  • 50 micron is quite a generous assumption, 10 could be enough, but....

That gives us a 12 kWh investment can produce, on sunny days, something like 1kW per hour with 10 percent efficiency and double with 20. With 20 percent efficiency of energy usage, to make bots, it lands us at 30-60 sunny hours which probably, something like 5-10 days for a double.

Taking 10 days, starting with 1 m2, having planet size the earth(water btw isn't a problem for propagation) gets us to about 49 cycles (ln(surface area)/ln(2)) or 490 days. (Or whatever number of days(minutes, hours) per cycle multiplied by 49)

To build a 10's of microns layer on the surface of the planet, with 100 percent coverage. It is not enough for purposes of terraforming, but at this point, the system reaches the bottleneck of energy production. And if there still are sufficient sources of carbon in the air, then it can grow linearly, let's take that 50-micron number, so 5 microns per day. In another year 1.5mm thickness coverage if we do not break the box mentioned in point 3 from the intro, it still can be not enough, otherwise it quite a mighty force.

energy sources

Under crust, magma can be quite a potent energy source, and nanomachines can be the tool to reach it and work with it.

On Venus-like planets, the atmosphere can be used as quite a potent energy source.

In both of those cases it is nice that it is stored accumulated energy and it basically depends on your abilities to extract it, meaning to lift the bottleneck, but at the same time affecting the whole planet, with desirable(venus) or not desirable effects(earth).

Potential organic matter present on a planet can be used as an energy source.

Uranium and friends can be used but it not so convenient and not worth it in long run, but if one lands on it, yeah why not, at the beginning.

Fusion can be if conditions are right and technology allows it and bill of materials for it accessible enough - there are some interesting fusion approaches that do not use superconductors as an example, nanomachines like fusion, but it depends.

It does not take that long to launch things in space, a city-size structure, with nanomachines grease can do that easily, and then energy bottleneck is basically non-existant in the way it was mention here, limits will have a more complex configuration, still, it will be about energy. Getting in space may be beneficial at cycle 20 or something like that(starting with 1 m2), if double-time is measured in years it can save some time, and it makes sense after cycle 49 with a short double time.

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Okay, as of right now, we still don't have enough information for a "Spherical cow in a vacuum" calculations. (These would not get us the correct answer. In this case, we'd be assuming the final mass of nanites = the initial mass of the planet.)

The reason that wouldn't work is because the nanites will not have the materials for unrestrained exponential growth. In the center of the mass, the nanites will not have access to material. Only nanites on the edge of the mass, where there is suitable raw material, will be able to reproduce.

On the large scale, what we will eventually see is a ring of nanites on the edge of the nanite mass that are close enough to the edge that any child nanites they produce will make it to the edge before the expands. The thickness of this ring is going to depend on the relative ratio of reproduction speed to transport speed. If transport speed is higher, then nanites far from the border can contribute child nanites to the border of the mass effectively. If reproduction speed is sufficiently higher, then only the nanites on the edge of the mass will effectively contribute, as by the time any other nanites make it to what used to be edge, it will be even further away.

In the case of the transport-dominated situation, the time it takes is approximately half the circumference of the planet, divided by the transport speed. The entire process will take slightly longer, as the nanite mass needs to grow enough so that the ring of effective reproduction is big enough to service the perimeter.

In the case of the reproduction-dominated situation, the time it takes is half the diameter of the planet, divided by the size of the nanite, multiplied by the reproduction time.

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  • $\begingroup$ "we'd be assuming the final mass of nanites = the initial mass of the planet." - lol, so wrong, maybe, lol. I mean u can have micro machines build from planet material, micro and mini, and macro - but assuming nanites to be able to utilize the whole planet for themselfs - very unlikely, at least not how we can it now. $\endgroup$
    – MolbOrg
    Jul 8 at 18:08
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Wind is probably gonna be the main dispersal method.

Nanobots aren't gonna be that fast moving, as microbes, and so wind and natural currents will be the main things moving them. Once they reach a location they'll quickly explode in numbers and probably spread out at a rate of a few centimeters an hour.

Wind moves around 10 kilometers per hour, and the earth has a circumference of 40000 kilometers. As such, it would take around 2000 hours, or a quarter of a year, to spread to most places worldwide. Once they reach any point their rapid replication will let them quickly convert everything, spreading out at a very slow speed.

A city is gonna be slower to convert, because it has lots of wind barriers. It might be a few weeks or months depending on the size to get to everywhere.

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