Will The Cluster World hold onto an atmosphere for a useful length of time without further intervention?

Question

The scenario goes something like this; a mage pins an earthlike planet's worth of mass (~5.972x10²⁴kg), in the form of rocky asteroids ranging in size from a hundred or so metres to several kilometres in maximum dimension, into a fixed formation. These rocks are held in their relative positions with spacings ranging from just a few metres to a kilometre or so in span. In terms of total volume I'm thinking 70% rocks and 30% air. Based on an elevated average asteroid material density of 3gcm^-3 (official figures use 2gcm^-3 as a generalisation), the total volume of rocks and space for atmosphere/vacuum is approximately 2.84x10²¹m³. Conceptually I'm leaning towards ellipsoid, or possibly egg-like in shape with a denser hemispheric end and a more spread out conical one. Note that at this stage there is no lighting source within the pile so only the outer rocks are going to be warmed by sunlight. There will be a, very small, contribution of internal heat from the asteroids due to the decay of radioisotopes but nothing like Earth's core temperatures. In terms of sunlight I'm thinking earthlike levels of visible light but I'm open to increased IR radiation to make an atmosphere that isn't frozen to the rocks plausible if need be.

What I want to know is whether the mass alone is sufficient to hold a breathable atmosphere over a millennia or two?

I'm not expecting the atmosphere to be "original" to the asteroids, in any way. Rather wondering whether an atmosphere added to the volume in and around the rock pile after it's formation will stick around for a useful length of time, at least 2000 years, without further magical intervention to keep it around. The rock pile is made using magic but then, I want no further magic being involved in the scenario, if possible.

Special thanks to JBH and Robert Rapplean for helping me troubleshoot this question over in the Sandbox and get it up to standard.

As a matter of record, not saying anyone has any answer in any way wrong, I wasn't as specific as I needed to be, but I will want the datum included for follow up. I did originally envision adding sufficient atmosphere for the pile to have 1atm throughout the 30% "empty space" and a thin (Earthlike) atmospheric envelope around it. That's ~8.57x10²⁰m³ of air at STP.

Answer 1

The real problem is that the scenario is not stable due to gravity. Every asteroid would be pulling every other asteroid towards itself. The result would be a net acceleration towards the centre of mass and everything would fall in on itself and collide.

Ignoring that massive issue, if the asteroids could somehow be pinned in place immobile by magic and still exert their gravitational force, then it should work, but the result might not be what you wanted.

The planet Mars was able to retain a considerable atmosphere for many millions of years before it eventually got leached away. Mars is almost a tenth as massive as the Earth, so even allowing for 30% not being rock and the rocks being a lot less dense than the Earth's core it should be able to retain an atmosphere for millions of years.

That said lack of a magnetic field might well increase losses dramatically, but I doubt that the process would be fast enough to remove the atmosphere in just a few thousand years. Calculations would be required to be certain, but given the complexity of the calculations and the magical setting I doubt it would be possible to come to any definitive answer.

The major issue that needs to be considered very carefully is the effect of gravity and how it is supposed to work. With asteroids magically pinned in place but with the asteroids gravity otherwise working as normal, gravity at perhaps 0.5g (extra void/air volume and less dense rocks) would still pull anything on the asteroids towards the centre of the group.

It would behave like a super crazy cave world where all the caves connected up but the rock didn't fall down. So nobody would be able to stand on the underside of the asteroids, they would be pulled off and drawn down towards the the centre of mass until they encountered the top of another asteroid.

Answer 2

There are several reasons why this would likely result in a spectacular failure, but I will focus on just the atmosphere aspect.

The biggest problem I foresee is that the total mass of the asteroids would be equal to Earth but the surface area would be FAR greater, leaving the same amount of atmosphere spread out across a much wider area.

For the sake of simplicity, let's assume the Earth is a sphere with a radius of 6,371km and each asteroid is a sphere of 2km. You would need 32,324,575,351 (~32 billion) asteroids to match the volume of one Earth, and those asteroids cumulatively would have roughly 3,185 times the surface area of the Earth.

But let's assume that each of these asteroids is created with enough atmosphere to make it comparable to the Earth's atmospheric density. The more dramatic issue is that even if the asteroids are being held in place by magic, their atmospheres probably aren't. Each asteroid would be constantly leeching the atmosphere of its neighbors which I suspect would lead to one of the following happening:

1. The larger asteroids would hog the bulk of the atmospheres causing them to be oversaturated and leaving the smaller asteroids with little to none.

2. The constant shifting of atmospheres would lead to some pretty crazy weather, and by crazy I mean constant gale-force winds particularly in the places where neighboring asteroids are closest to each other.

3. All of the atmospheres would gradually drift to the center of gravity of the entire cluster, leading to a giant ball of compressed air floating somewhere in the middle of space not attached to any particular asteroid.

None of these options are particularly desirable for long-term stability.

Answer 3

Slarty's answer is correct. This may not be a big problem for world-building purposes, it just means that you can't have a situation where you and your friend are standing on opposing surfaces and each looking "up" at each other. (The surface gravity of any individual astroid will be negligible. You suggest the largest will be a few kilometers across; 67P-CG has a surface gravity of about 1/10000g.)

Abion47's third point is correct (their first two are probably not), but remember that the gravity of a system is strongest at its surface. So we would expect the outer asteroids to have no atmosphere but earth-ish gravity. The core would be pitch black, zero-g, and filled with a dense/pressurized (but probably not crazy pressurized) atmosphere.

The core would also be warm! It's hard to say exactly how warm, but since it's not exposed to open space, it can only loose heat by the convection of the atmosphere. If there's no heat coming from radioactive decay, then the equilibrium temperature at the core will be the same as whatever it is at the surface. If radioactive decay is happening normally, then the equilibrium temperature at the core will be more-or-less what it is at Earth's core, depending whether the convecting atmosphere is able to move heat faster or slower than Earth's convecting mantle. In either case, it could take a long time to reach such equilibrium, probably more than a few thousand years.

So if you want this to be science based, then you've got some math to do. And if you want it to be a fun human-habitable 3D environment then you've got a lot of math and fine-tuning to do. Or you could just have the wizard do it.

Answer 4

The short answer is "yes." Anything with the mass and rough density of Earth will hold onto an atmosphere. You're only maintaining 70% of the mass, so you can expect a thinner atmosphere near the edges of the cluster, possibly tapering off to a vacuum before you get entirely out of the cluster.

What you have to consider is how much atmosphere you want it to hold. The shell of habitable atmospheric densities is likely to be very thin. You can move the shell of survivability in and out of the cluster just by increasing the atmospheric mass. I'd have to brush up on calculus to figure out how thin, but intuitively I think it's less than a hundred miles.

We can survive in the range between 1/3 atmosphere and about 30 atmospheres. That's the pressure at which you need to be breathing almost pure oxygen and the pressure at which oxygen is toxic. I'm certain that "life" will exist below that depth, and that would be a fascinating exploration, but descending would be a competitive behavior, like deep scuba diving.

Answer 5

Short Answer:

You should check out the Scale of Science Fiction Hardness. The more plausible and realistic the scientific and technology in a story are, the higher its score on the scale of Science Fiction Hardness.

https://tvtropes.org/pmwiki/pmwiki.php/SlidingScale/MohsScaleOfScienceFictionHardness

It is my opinion that a science fiction - not fantasy - story could be set in your asteroid field if the writer was willing to accept a low score on the scale of science fiction hardness. Thus once the asteroid field is created, it should take comparatively little periodic or constant applicated of super advanced science & technology - or magic - to keep it functioning.

Long Answer:

The asteroid field will have a volume of about 4.1 X 10 to the 21st power cubic meters if you have done your math correctly.

Earth has a volume of 1.08321 times 10 to the 12th power cubic kilometers. Since there are 10 to the 9th power cubic meters in a cubic Kilometer, Earth has a volume of 1.08321 times 10 to the 21st power cubic meters.

So your asteroid field should have a volume of 3.7850462 Earth's volume. Assuming that its perfectly spherical to make the math easier, the asteroid field should have a radius of approximately 1.558441 of Earth's radius with a mass of approximately 1.0 Earth Mass.

According to this surface gravity calculator, https://www.omnicalculator.com/physics/acceleration-due-to-gravity, with a mass of 1 Earth mass and a radius of 1.558441 Earth radius the outer surfaces of the outermost asteroids should have a surface gravity of 0.4123 g, Earth's surface gravity.

According to this escape velocity calculator https://www.omnicalculator.com/physics/escape-velocity with a mass of 1 Earth mass and a radius of 1.558441 Earth radius the asteroid field should have an escape velocity of 8.96 kilometers per second - which is 0.8010012 of Earth's 11.186 kilometers per second.

So in this case the escape velocity is 1.9427 times as strong, relative to Earth's, as the surface gravity is relative to Earth's. Thus the surface gravity and the escape velocity of an astronomical object should be calculated separately, since they don't vary to the same degree.

The surface gravity is important for how comfortable a world will be for humans to live on the surface, while the escape velocity will be important in determining how well a planet - or in this case an asteroid field - can keep its atmosphere.

According to pages 34 to 35 in Habitable Planets for Man, Stephen H. Dole, 1964,

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

There is a rule of thumb formula to calculate how long a world would take to lose enough of its original air for the remaining amount to be only 1/e, or 0.3678 of the original amount.

It depends on the ratio of the world's escape velocity divided by the root-mean-square velocity of gas particles in the exosphere, the super thin outer layer of the atmosphere, where gas particles escape from.

Table 5 shows how long a planet would take to go down to 0.3678 of the original gas at various ratios.

At ratios of 1 or 1 the time span is instant.

At a ratio of 3 the time span is a few weeks.

At a ratio of 4 the time span is several thousand years.

At a ratio of 5 the time span is about a hundred million years.

At a ratio of 6 the time span is "infinite".

So a comparatively minor change in the ratio can change the time it takes for a world to lose its atmosphere into space from instantly to infinite.

According to calculations, the outermost asteroids in your asteroid field would have an escape velocity of 8.96 kilometers per second.

If the root-means-square velocity of oxygen particles is less than 1.49333 kilometers per second, your asteroid field can retain 0.3678 of them for an infinite time.

If the root-means-square velocity of oxygen particles is less than 1.792 kilometers per second, your asteroid field can retain 0.3678 of them for about a hundred million years.

If the root-means-square velocity of oxygen particles is less than 2.24 kilometers per second, your asteroid field can retain 0.3678 of them for several thousand years.

According to page 54, the exosphere temperatures in Earth's exosphere are from 1000 to 2000 degrees K. Dole says that if the exosphere temperatures are no more 1000 degrees K, the speed of atomic oxygen in the exosphere should be about 1.25 kilometers per second, and an escape velocity of about 6.25 kilometers per second should be enough to retain 0.3678 of the original oxygen after about a hundred million years.

But what if the temperature in the exosphere is higher producing a greater velocity of the oxygen? The asteroid field might lose oxygen more rapidly.

The temperature of gas particles in the exosphere of a world is believed to be caused by hard ultraviolet radiation from the star, and not by the other frequencies of radiation from the star.

So maybe the asteroid field could orbit a star which is cooler than the Sun, and emits much less ultraviolet radiation, thus lowering the temperature in the exosphere and he velocities of the gas particles.

Maybe there is a giant sunshade between the star and the asteroid field, transparent to most wavelengths but opaque to ultraviolet, thus keeping the exosphere temperature lower.

Maybe the asteroid field is in interstellar space, light years from the nearest star, and has an artificial "Sun" to illuminate it, one which doesn't emit much ultraviolet.

Planets also loose atmosphere from other processes. One such process is particles from the solar (or stellar) wind striking the atmosphere and knocking gas particles off into outer space. Since those particles are charged, Earth's magnetic field deflects most of them away from the atmosphere, slowing down atmospheric losses.

Maybe the wizard will have to build a gigantic magnetic field generator to protect the asteroid field's atmosphere from any stellar wind. Unless it is in interstellar space and has an artificial "sun" which would be designed to not produce a stellar wind.

I also note that the planet Venus has a lightly lower escape velocity than Earth, has no magnetic field, and is closer to the Sun and faces a stronger solar wind. So it should lose atmosphere due to solar wind much faster than Earth does. And yet the atmosphere of Venus left after Billions of years is many times thicker and more massive than Earth's atmosphere.

I note that an atmosphere will be denser the closer it is to the surface of a planet. Considering that the surface of the asteroid field surface has gaps between the asteroids, the atmosphere should extend down to the center of the asteroid field, getting denser and denser with depth. If there is an Earthlike atmospheric pressure near the outer edge of the asteroid field the total amount of gas in the atmosphere would be be immense.

Another problem would be keeping the asteroids in their relative positions. If they are fee to move, their mutual gravity will relatively soon pull them together to make a single planet about the mass of Earth.

In a science fiction story with a rather low hardness score, tractor beams, which attract matter, and pressor beams, which push matter away, can be used to lock each one of the billions of asteroids in position relative to the others.

Of course each of the tractor and pressor beams would be generated by machinery which would use energy generated by giant fusion power generators. That machinery would have to be tended and maintained by hordes of service robots - or by demons in a fantasy story.

Another possibility is a shellworld. There are several types of hypothetical shellworlds. One type is:

A planet or a planetoid turned into series of concentric matryoshka doll-like layers supported by massive pillars.1 A shellworld of this type features prominently in Iain M. Banks' novel Matter.

So I can picture your asteroid field consisting of countless thousands of layers of geodesic spheres, each with an asteroid where ever several beams come together, and with other, vertical beams, supporting each of the asteroid vertically.

That would require less energy for tractor and pressor beams being on permanently and constantly using energy. Of course it would require very strong materials. Possibly if carbon nanofibers could be produced in the right size and quantity they could make the beams.

At most of the thousands of levels there would not be solid shells but merely beams between the asteroids. But there could be a shell above the outermost asteroids to hold the atmosphere in, and another shell below the level of inhabitable asteroids, to keep the atmosphere from falling to the center of the asteroid field, and thus immensely reducing the total amount of atmosphere needed.

Or there could be only one level of asteroids, connected by tractor or pressor beams or by beams of super strong materials, with an outer shell above the asteroids to keep the atmosphere in, and an inner shell below the asteroids to keep the atmosphere from falling.

With only one level of asteroids, instead of countless thousands, an asteroid field of the same size would contain only a tiny fraction of the total number and mass of asteroids as before. Or with the same number and mass of asteroids the hollow field of asteroids could be much larger.

And possibly the asteroid field with one layer could be in a cylindrical shape. If the cylinder rotated at the proper speed the forces on the connecting beams pulling out could balance the forces pulling inward, reducing the stress on the energy beams or solid structural beams holding the asteroid field together.

Thus the asteroid might resemble a giant O'Neil space habitat but with spaces between the parts.

I note that if there are solid beams of matter between the different asteroids they could contain roads and people might not need to fly between asteroids.

So I have done my best to think of ways to make your asteroid field plausible in a science fiction story, and thus to reduce the amount of magic needed for it in a fantasy story.

Answer 6

The hard part is keeping the atmosphere roughly uniform. If you evenly atomize the rock, so that the local gravity of each lump is minimal and the lumps are evenly spaced, then the atmosphere falls down to the lowest level. If you're, say, 10% of the way out, assuming internal gravity roughly proportional to radius (r cubed over r squared), using the 0.4 g figure from M.A. Golding's answer, that's 0.04 g. The scale height is 1/0.04 = 25 times bigger than that on Earth. But Earth's scale height is only 8 km, so that's 200 km, and if your planet's radius is roughly 10000 km, you have 5 times that scale height (factor of e increase in air pressure) in the first 10% of the radius going outward. So the air ends up as a little lake of supercritical gas inside the planet, and vacuum all around it.

We can fix that by putting more mass closer to the surface, but it cancels out in a shell and has no actual reverse gravity pulling the air outward. We could make a hollow Earth full of air, all at the same pressure below the shell of rock, but if we then try to add any rocks inside that they start hogging the air if they add up to enough mass to matter. That makes for a pretty rationed amount of stone although, having been broken up, it's a vast surface area. But it's also all at near zero gee.

My best speculation off the top of my head is a planet a bit more like a Volvox. It has a shell on the outside, with near vacuum external atmosphere, which serves as radiation shield. That is in thermal equilibrium with air inside (which for a true Earth orbit would be cold, 255 K I think, due to lack of greenhouse gas, unless there are other heat sources) Then inside you clump your stones into mini-planets, not big enough to generate a substantial scale height or escape velocity, but enough to have significant gravity and perhaps a little bonus air around each one. The bigger clumps are further out and the smaller clumps inside, so that the gravitational potential is the same between the surface of any one of them, so none of them hogs up all the air for itself.

Answer 7

You seem to have some problems with the atmosphere.

The surface area of this world, exposed to its atmosphere, is vastly greater than the surface area of Earth. Said surface is composed of asteroid rock, and if that's been baking in sunlight inside the frost line for a few billion years, as our asteroid belt has, it's going to absorb gasses really fast. Much of your atmosphere is going to get bound to the surface of the asteroids, and this will happen much faster than losses by gasses escaping from the world's gravity.

For extra entertainment value, the oxygen will be combining with the minerals that form the surface, just as oxygen freed by early life did on Earth. Oxygen didn't start to accumulate in Earth's atmosphere until the Great Oxidation Event about 2.4 billion years ago, when all the exposed minerals that would react with oxygen were satisfied. So oxygen will get absorbed selectively, making the atmosphere less suitable for life.

Also, unless the cluster is quite tightly packed, you're going to need quite a lot of atmosphere, far more than Earth has, to get some atmosphere on the outside of the world, where sunlight can power life.

It seems very unlikely that plant life on the surface of this world can liberate more oxygen faster than the interior surface of the world can absorb it. The absorption rate is proportional to the volume of the world, and plants can only live on the outside, where there is sunlight to power photosynthesis.

There are at least two more problems, but they're easier to solve with magic, if the wizard can create matter from nowhere on a planetary scale:

• Where does a whole planet's worth of asteroid mass come from? Our solar system doesn't have anything like that much mass in asteroids.

• Where did the wizard get that much breathable atmosphere from? An ordinary planet with life doesn't have early enough to supply this constructed world, even if you're willing to kill all the life on the donor world by stealing its atmosphere.

Answer 8

This asteroid collection may have a strong magnetic field. A wizard created it? If need be, he made sure every asteroid has a nice big lodestone (magnetic iron) in its core - and they all align. There you have your magnetic field protecting the atmosphere from solar wind. And nice aurora borealis at whatever height the upper part of that atmosphere is. The field will be useful for navigating the inside.

Some radioactive heating of the inner parts is a good thing: there will be convection winds moving air up and down keeping the atmosphere mixed. So no problem with stale air in the deep, it will be flushed to the upper layers where plants can renew oxygen. The oxygen need not be eaten by rock - your wizard may make inert types of rock.

Water, if any, will reside in a dark central ocean. With convection winds bringing moisture up and create rain. And millions of waterfalls, as water go over the edge of one asteroid and falls onto the next one.