If we have two unmoving equal point masses in the plane (let’s say at ) and release particles from different locations they will swing around the masses in some trajectory. If we colour each point by the mass it approaches closest (or even collides with) we get a basin of attraction for each mass. Can one prove the boundary is a straight line?
User Kasper showed that one can reframe the problem in terms of elliptic coordinates and show that this implies a straight boundary, while User Lineage showed it more simply using the second constant of motion. I have the feeling that there ought to be an even simpler argument. Still, Kasper’s solution show that the generic trajectory will quasiperiodically fill a region and tend to come arbitrarily close to one of the masses.
In any case, here is a plot of the basins of attraction shaded by the time until getting within a small radius around the masses. Dark regions take long to approach any of the masses, white regions don’t converge within a given cut-off time.
The boundary is a straight line, and surrounding the simple regions where orbits fall nearly straight into the nearest mass are the wilder regions where orbits first rock back and forth across the x-axis before settling into ellipses around the masses.
The case for 5 evenly spaced masses for and 0.01 (assuming unit masses at unit distance from origin and ) is somewhat prettier.
As the basins approach ellipses around their central mass, corresponding to orbits that loop around them in elliptic orbits that eventually get close enough to count as a hit. The onion-like shading is due to different number of orbits before this happens. Each basin also has a tail or stem, corresponding to plunging orbits coming in from afar and hitting the mass straight. As the trap condition is made stricter they become thinner and thinner, yet form an ever more intricate chaotic web oughtside the central region. Due to computational limitations (read: only a laptop available) these pictures are of relatively modest integration times.
I cannot claim credit for this fractal, as NiveaNutella already plotted it. But it still fascinates me.
Wada basins and mille-feuille collision manifolds
These patterns are somewhat reminiscent of the classic Newton’s root-finding iterative formula fractals: several basins of attraction with a fractal border where pairs of basins encounter interleaved tiny parts of basins not member of the pair.
However, this dynamics is continuous rather than discrete. The plane is a 2D section through a 4D phase space, where starting points at zero velocity accelerate so that they bob up and down/ana and kata along the velocity axes. This also leads to a neat property of the basins of attraction: they are each arc-connected sets, since for any two member points that are the start of trajectories they end up in a small ball around the attractor mass. One can hence construct a map from to that is a homeomorphism. There are hence just N basins of attraction, plus a set of unstable separatrix points that never approach the masses. Some of these border points are just invariant (like the origin in the case of the evenly distributed masses), others correspond to unstable orbits.
Each mass is surrounded by a set of trajectories hitting it exactly, which we can parametrize by the angle they make and the speed they have inwards when they pass some circle around the mass point. They hence form a 3D manifold where counts the time until collision (i.e. backwards). These collision manifolds must extend through the basin of attraction, approaching the border in ever more convoluted ways as approaches . Each border point has a neighbourhood where there are infinitely many trajectories directly hitting one of the masses. They form 3D sheets get stacked like an infinitely dense mille-feuille cake in the 4D phase space. And typically these sheets are interleaved with the sheets of the other attractors. The end result is very much like the Lakes of Wada. Proving the boundary actually has the Wada property is tricky, although new methods look promising.
The magnetic pendulum
This fractal is similar to one I made back in 1990 inspired by the dynamics of the magnetic decision-making desk toy, a pendulum oscillating above a number of magnets. Eventually it settles over one. The basic dynamics is fairly similar (see Zhampres’ beautiful images or this great treatment). The difference is that the gravity fractal has no dissipation: in principle orbits can continue forever (but I end when they get close to the masses or after a timeout) and in the magnetic fractal the force dependency was bounded, a force rather than the .
That simulation was part of my epic third year project in the gymnasium. The topic was “Chaos and self-organisation”, and I spent a lot of time reading the dynamical systems literature, running computer simulations, struggling with WordPerfect’s equation editor and producing a manuscript of about 150 pages that required careful photocopying by hand to get the pasted diagrams on separate pieces of paper to show up right. My teacher eventually sat down with me and went through my introduction and had me explain Poincaré sections. Then he promptly passed me. That was likely for the best for both of us.
Appendix: Matlab code
showPlot=0; % plot individual trajectories
randMass = 0; % place masses randomly rather than in circle
RTRAP=0.0001; % size of trap region
tmax=60; % max timesteps to run
S=1000; % resolution
s = rng(3);
hitN = X*0; % attractor basin
hitT = X*0; % time until hit
closest = X*0+100;
closestN=closest; % closest mass to trajectory
tic; % measure time
[t,u,te,ye,ie]=ode45(@(t,y) forceLaw(t,y,N,PX,PY), [0 tmax], [X(a,b) 0 Y(a,b) 0],odeset('Events',@(t,y) finishFun(t,y,N,PX,PY,RTRAP^2)));
if (d2<mind2) mind2=d2; hitN(a,b)=k; end
if (closest(a,b)==sqrt(d2)) closestN(a,b)=k; end
elapsedTime = toc
% Make colorful plot
function dudt = forceLaw(t,u,N,PX,PY)
%dudt = zeros(4,1);
dudt(1) = u(2);
dudt(2) = 0;
dudt(3) = u(4);
dudt(4) = 0;
% for k=1:N
% Are we close enough to one of the masses?
function [value,isterminal,direction] =finishFun(t,u,N,PX,PY,r2)
The question is of course ill-defined, since “largest”, “possible”, “inhabitable” and “world” are slippery terms. But let us aim at something with maximal surface area that can be inhabited by at least terrestrial-style organic life of human size and is allowed by the known laws of physics. This gives us plenty of leeway.
Piled higher and deeper
We could simply imagining adding more and more mass to a planet. At first we might get something like my double Earths, ocean worlds surrounding a rock core. The oceans are due to the water content of the asteroids and planetesimals we build them from: a huge dry planet is unlikely without some process stripping away water. As we add more material the ocean gets deeper until the extreme pressure makes the bottom solidify into exotic ice – which slows down the expansion somewhat.
Adding even more matter will produce a denser atmosphere too. A naturally accreting planet will acquire gas if it is heavy and cold enough, at first producing something like Neptune and then a gas giant. Keep it up, and you get a brown dwarf and eventually a star. These gassy worlds are also far more compressible than a rock- or water-world, so their radius does not increase when they get heavier. In fact, most gas giants are expected to be about the size of Jupiter.
If this is true, why is the sun and some hot Jupiters much bigger? Jupiter’s radius is 69,911 km, the sun radius is 695,800 km, and the largest exoplanets known today have radii around 140,000 km. The answer is that another factor determining size is temperature. As the ideal gas law states, to a first approximation pressure times volume equals temperature: the pressure at the core due to the weight of all the matter stays roughly the same, but at higher temperatures the same planet/star gets larger. But I will assume inhabitable worlds are reasonably cold.
Planetary models also suggest that a heavy planet will tend to become denser: adding more mass compresses the interior, making the radius climb more slowly.
The central pressure of a uniform body is . In reality planets do not tend to be uniform, but let us ignore this. Given an average density we see that the pressure grows with the square of the radius and quickly becomes very large (in Earth, the core pressure is somewhere in the vicinity of 350 GPa). If we wanted something huge and heavy we need to make it out of something incompressible, or in the language of physics, something with a stiff equation of state. There is a fair amount of research about super-earth compositions and mass-radius relationships in the astrophysics community, with models of various levels of complexity.
Taking the constants (table 4) corresponding to iron gives a maximum radius at the mass of 274 Earths, perovskite at 378 Earths, and for ice at 359 Earths. We should likely not trust the calculation very much around the turning point, since we are well above the domain of applicability. Still, looking at figure 4 shows that the authors at least plot the curves up to this range. The maximal iron world is about 2.7 times larger than Earth, the maximal perovskite worlds manage a bit more than 3 times Earth’s radius, and the waterworlds just about reach 5 times. My own plot of the approximation function gives somewhat smaller radii:
Mordasini et al. have a paper producing similar results; for masses around 1000 Earth masses their maximum sizes are about 3.2 times for a Earthlike 2:1 silicate-to-iron ratio, 4 times for an 50% ice, 33% silicate and 70% iron planet, and 4.8 times for planets made completely of ice.
The upper size limit is set by the appearance of degenerate matter. Electrons are not allowed to be in the same energy state in the same place. If you squeeze atoms together, eventually the electrons will have to start piling into higher energy states due to lack of space. This is resisted, producing the degeneracy pressure. However, it grows rather slowly with density, so degenerate cores will readily compress. For fully degenerate bodies like white dwarves and neutron stars the radius declines with increasing mass (making the largest neutron stars the lightest!). And of course, beyond a certain limit the degeneracy pressure is unable to stop gravitational collapse and they implode into black holes.
What about rotating the planet fast enough? As Mesklin in Hal Clement’s Mission of Gravity demonstrates, we can have a planet with hundreds of Gs of gravity at the poles, yet a habitable mere 3 G equator. Of course, this is cheating somewhat with the habitability condition: only a tiny part is human-habitable, yet there is a lot of unusable (to humans, not mesklinites) surface area. Estimating the maximum size becomes fairly involved since the acceleration and pressure fields inside are not spherically symmetric. A crude guesstimate would be to look at the polar radius and assume it is limited by the above degeneracy conditions, and then note that the limiting eccentricity is about 0.4: that would make the equatorial radius 2.5 times larger than the polar radius. So for the spun-up ice world we might get an equatorial radius 12 times Earth and a surface area about 92 times larger. If we want to go beyond this we might consider torus-worlds; they can potentially have an arbitrarily large area with a low gravity outer equator. Unfortunately they are likely not very stable: any tidal forces or big impacts (see below) might introduce a fatal wobble and breakup.
So in some sense the maximal size planets would be habitable. However, as mentioned above, they would also likely turn into waterworlds and warm Neptunes.
Getting a solid mega-Earth (and keeping it solid)
The most obvious change is to postulate that the planet indeed just has the right amount of water to make decent lakes and oceans, but does not turn into an ocean-world. Similarly we may hand-wave away the atmosphere accretion and end up with a huge planet with a terrestrial surface.
Although it is not going to stay that way for long. The total heat production inside the planet is proportional to the volume which is proportional to the cube of the radius, but the surface area that radiates away heat is proportional to the square of the radius. Large planets will have more heat per square meter of surface, and hence have more volcanism and plate tectonics. That big world will soon get a fair bit of atmosphere from volcanic eruptions, and not the good kind – lots of sulphuric oxides, carbon dioxide and other nasties. (A pure ice-Earth would escape this, since all hydrogen and oxygen isotopes are short lived – once it solidified it would stay solid and boring).
And the big planet will get hit by comets too. The planet will sweep up stuff that comes inside its capture cross section where is the geometric cross section, the escape velocity and the original velocity of the stuff. Putting it all together gives a capture cross section proportional to : double-Earth will get hit by times as much space junk as Earth. Iron-Earth by 53 times as much.
So over time the planet will accumulate an atmosphere denser than it started. But the impact cataclysms might also be worse for habitability – the energy released when something hits is roughly proportional to the square of the escape velocity, which scales as . On Double-Earth the Chicxulub impact would have been four times more energetic. So the mean energy per unit of time due to impacts scales like . Ouch. Crater sizes scale as where is the energy. So for our big worlds the scars will scale as . Double-Earth will have craters 70% larger than Earth, and iron-Earth 121% larger.
Big and light worlds
Surface gravity scales as . So if we want R to be huge but g modest, the density has to go down. This is also a good strategy for reducing internal pressure, which is compressing our core. This approach is a classic in science fiction, perhaps most known from Jack Vance’s Big Planet.
Could we achieve this by assuming it to be made out of something very light like lithium hydride (LiH)? Lithium hydride is nicely low density (0.78 g/cm3) but also appears to be rather soft (3.5 on the Mohs scale), plus of course that it reacts with oxygen and water, which is bad for habitability. Getting something that doesn’t react badly rules out most stuff at the start of the periodic table: I think the first compound (besides helium) that doesn’t decompose in water or is acutely toxic is likely pure boron. Of course, density is not a simple function of atomic number: amorphous carbon and graphite have lower densities than boron.
A carbon planet is actually not too weird. There are exoplanets that are believed to be carbon worlds where a sizeable amount of mass is carbon. They are unlikely to be very habitable for terrestrial organisms since oxygen would tend to react with all the carbon and turn into carbon dioxide, but would have interesting surface environments with tars, graphite and diamonds. We could imagine a “pure” carbon planet composed largely of graphite, diamond and a core of metallic carbon. If we handwave that on top of the carbon core there is some intervening rock layer or that the oxidation processes are slow enough, then we could have a habitable surface (until volcanism and meteors get it). A diamond planet with 1 G gravity is would be km. We get a 1.6 times larger radius than earth this way, and 2.5 times more surface area. (Here I ignore all the detailed calculations in real planetary astrophysics and just assume uniformity; I suspect the right diamond structure will be larger.)
A graphite planet would have radius 16,805 km, 2.6 times ours and with about 7 times our surface area. Unfortunately it would likely turn (cataclysmically) into a diamond planet as the core compressed.
Another approach to low density is of course to use stiff materials with voids. Aerogels have densities close to 1 kg per cubic meter, but that is of course mostly the air: the real density of a silica aerogel is 0.003-0.35 g/cm3. Now that would allow a fluffy world up to 1837 times Earth’s radius! We can do even better with metallic microlattices, where the current record is about 0.0009 g/cm3 – this metal fluffworld would have a radius 39,025,914 km, 6125 times Earth, with 3.8 million times our surface area!
The problem is that aerogels and microlattices do not have that great bulk modulus, the ability to resist compression. Their modulus scales with the cube or square of density, so the lighter they are, the more compressible they get – wonderful for many applications, but very bad for keeping planets from imploding. Imagine trying to build a planet out of foam rubber. Diamond is far, far better. What we should look for is something with a high specific modulus, the ratio between bulk modulus and density. Looking at this table suggests carbon fiber is best at 417 million m2/s2, followed by diamond at 346 million m2/s2. So pure carbon worlds are likely the largest we could get, a few times Earth’s size.
We can do better if we abandon the last pretence of the world being able to form naturally (natural metal microlattices, seriously?).
Consider roofing over the entire Earth’s surface: it would take a fair amount of material, but we could mine it by digging tunnels under the surface. At the end we would have more than doubled the available surface (roof, old ground, plus some tunnels). We can continue the process, digging up material to build a giant onion of concentric floors and giant pillars holding up the rest. The end result is akin to the megastructure in Iain M. Banks’ Matter.
If each floor has material density kg/m2 (lets ignore the pillars for the moment) and ceiling height h, then the total mass from all floors is . Moving terms over to the left we get . If N is very large the N^3/3 term dominates (just consider the case of N=1000: the first term is a third of a billion, the second half a million and the final one 166.6…) and we get
with radius .
The total surface area is
So the area grows proportional to the total mass (since N scales as ). It is nearly independent of ( scales as ) – the closer together the floors are, the more floors you get, but the radius increases only slowly. Area also scales as : if we just sliced the planet into microthin films with maximal separation we could get a humongous area.
If we set meters, kg per square meter, and use the Earth’s mass, then , with a radius of 20,000 km. Not quite xkcd’s billion floor skyscraper, but respectable floorspace: square meters, about 23 million times Earth’s area.
If we raise the ceiling to meters the number of floors drops to 660,000 and the radius balloons to 65,000 km. If we raise them a fair bit more, kilometres, then we reach the orbit of the moon with the 19,000th floor. However, the area stubbornly remains about 23 million times Earth. We will get back to this ballooning shortly.
Keeping the roof up
The single floor shell has an interesting issue with gravity. If you stand on the surface of a big hollow sphere the surface gravity will be the same as for a planet with the same size and mass (it will be rather low, of course). However, on the inside you would be weightless. This follows from Newton’s shell theorem, which states that the force from a spherically symmetric distribution of mass is proportional to the amount of mass at radii closer to the centre: outside shells of mass do not matter.
This means that the inner shells do not have to worry about the gravity of the outer shells, which is actually a shame: they still weigh a lot, and that has to be transferred inwards by supporting pillars – some upward gravity would really have helped construction, if not habitability. If the shells were amazingly stiff they could just float there as domes with no edge (see discussion of Dyson shells below), but for real materials we need pillars.
How many pillars do we need? Let’s switch the meaning of to denote mass per cubic meter again, making the mass inside a radius . A shell at radius needs to support the weight of all shells above it, a total force of (mass of the shell times the gravitational force). Then .
If our pillars have compressive strength per square meter, we need square meters of pillars at radius : a fraction of the area needs to be pillars. Note that at some radius 100% of the floor has to be pillars.
Plugging in our original m, kg per cubic meter, meter world, and assuming GPa (diamond), and assuming I have done my algebra right, we get km – this is the core, where there is actually no floors left. The big moonscraper has a core with radius 46 km, far less.
We have so far ignored the weight of all these pillars. They are not going to be insignificant, and if they are long we need to think about buckling and all those annoying real world engineering considerations that actually keep our buildings standing up.
We may think of topological shape optimization: start with a completely filled shell and remove material to make voids, while keeping everything stiff enough to support a spherical surface. At first we might imagine pillars that branch to hold up the surface. But the gravity on those pillars depend on how much stuff is under them, so minimizing it will make the the whole thing lighter. I suspect that in the end we get just a shell with some internal bracing, and nothing beneath. Recall the promising increase in area we got for fewer but taller levels: if there are no levels above a shell, there is no need for pillars. And since there is almost nothing beneath it, there will be little gravity.
Single shell worlds
Making a single giant shell is actually more efficient than the concentric shell world. – no wasted pillars, all material used to generate area That shell has and area (which, when you think about units, is the natural answer). For Earth mass shells with 500 kg per square meter, the radius becomes 31 million km, and the surface area is square meters, 23 million times the Earth’s surface.
The gravity will however be microscopic, since it scales as – for all practical purposes it is zero. Bad for keeping an atmosphere in. We can of course cheat by simply putting a thin plastic roof on top of this sphere to maintain the atmosphere, but we would still be floating around.
Building shells around central masses seems to be a nice way of getting gravity at first. Just roof over Jupiter at the right radius ( 113,000 km) and you have a lot of 1 G living area. Or why not do it with a suitably quiet star? For the sun, that would be a shell with radius 3.7 million km, with an area 334,000 times Earth.
Of course, we may get serious gravity by constructing shells around black holes. If we use the Sagittarius A* hole we get a radius of 6.9 light-hours, with 1.4 trillion times Earth’s area. Of course, it also needs a lot of shell material, something on the order of 20% of a sun mass if we still assume 500 kg per square meter.
A small problem is that the shell theorem means the shell will not remain aligned with the internal mass: there is no net force. Anything that hits the surface will give it a bit of momentum away from where it should be. However, this can likely solved with dynamical corrections: just add engines here and there to realign it.
A far bigger problem is that the structure will be in compression. Each piece will be pulled towards the centre with a force per m^2, and to remain in place it needs to be held up by neighbouring pieces with an equal force. This must be summed across the entire surface. Frank Palmer pointed out one could calculate this as two hemispheres joined at a seam, finding a total pressure of . If we have a maximum strength the maximal radius for this gravity becomes . Using diamond and 1 G we get km. That is not much, at least if we dream about enclosing stars (Jupiter is fine). Worse, buckling is a real problem.
Eder also points out that one limit of the size of such worlds is the need to radiate heat from the inhabitants. Each human produces about 100 W of waste heat; this has to be radiated away from a surface area of at around 300K: this means that the maximum number of inhabitants is . For a bubbleworld this is people. For Earth, it is people.
If we accept volume instead of area, we may think of living inside such bubbles. Karl Schroeder’s Virga books come to mind, although he modestly went for something like a 5,000 mile diameter. Niven discusses building an air-filled volume around a Dyson shell surrounding the galactic core, with literally cubic lightyears of air.
The ultimate limit is avoiding Jeans instability: sufficiently large gas volumes are unstable against gravitational contraction and will implode into stars or planets. The Jeans length is
where is the mass per particle. Plugging in 300 K, the mass of nitrogen molecules and air density I get a radius of 40,000 km (see also this post for some alternate numbers). This is a liveable volume of cubic kilometres, or 0.17 Jupiter volumes. The overall calculation is somewhat approximate, since such a gas mass will not have constant density throughout and there has to be loads of corrections, but it gives a rough sense of the volume. Schroeder does OK, but Niven’s megasphere is not possible.
Living on surfaces might be a mistake. At least if one wants a lot of living space.
Bigger than worlds
The locus classicus on artificial megastructures is Larry Niven’s essay Bigger than worlds. Beside the normal big things like O’Neill cylinders it leads up to the truly big ones like Dyson spheres. It mentions that Dan Alderson suggested a double Dyson sphere, where two concentric shells had atmosphere between them and gravity provided by the internal star. (His Alderson Disk design is ruled out for consideration in my essay because we do not know any physics that would allow that strong materials.) Of course, as discussed above, solid Dyson shells are problematic to build. A Dyson swarm of free-floating habitats and solar collectors is far more physically plausible, but fails at being *a* world: it is a collection of lot of worlds.
One fun idea mentioned by Niven is the topopolis suggested by Pat Gunkel. Consider a very long cylinder rotating about its axis: it has internal pseudogravity, it is mechanically possible (there is some stress on the circumferential material, but unless the radius or rotation is very large or fast we know how to build this from existing materials like carbon fibers). There is no force between the hoops making up the cylinder: were we to cut them apart they would still rotate in line.
Now make the cylinder km long and bend it into a torus with major radius R. If the cylinder has radius r, the difference in circumference between the inner and outer edge is . Spread out around the circumference, that means each hoop is subjected to a compression of size if it continues to rotate like it did before. Since R is huge, this is a very small factor. This is also why the curvature of the initial bend can be ignored. For a topopolis orbiting Earth in geostationary orbit, if r is 1 km the compression factor is ; if it loops around the sun and is a 1000 km across the effect is just . Heat expansion is likely a bigger problem. At large enough scales O’Neill cylinders are like floppy hoses.
The area would be . In the first case 0.0005 of Earth’s area, in the second case 1842 times.
The funny thing about topopolis is that there is no reason for it to go just one turn around the orbited object. It could form a large torus knot winding around the object. So why not double, triple or quadruple the area? In principle we could just keep going and get nearly any area (up until the point where self-gravity started to matter).
There is some trouble with Kepler’s second law: parts closer to the central body will tend to move faster, causing tension and compression along the topopolis, but if the change in radial distance is small these forces will also be small and spread out along a enormous length.
Unfortunately topopolis has the same problem as a ringworld: it is not stably in orbit if it is rigid (any displacement tends to be amplified), and the flexibility likely makes things far worse. Like the ringworld and Dyson shell it can plausibly be kept in shape by active control, perhaps solar sails or thrusters that fire to keep it where it should. This also serves to ensure that it does not collide with itself: effectively there are carefully tuned transversal waves progressing around the circumference keeping it shaped like a proper knot. But I do not want to be anywhere close if there is an error: this kind of system will not fail gracefully.
Diamond 1G planet
Graphite 1G planet
Aerogel 1G planet
Microlattice 1G planet
Sag A roof
1 AU ring
Why aim for a large world in the first place? There are three apparent reasons. The first is simply survival, or perhaps Lebensraum: large worlds have more space for more beings, and this may be a good thing in itself. The second is to have more space for stuff of value, whether that is toys, gardens or wilderness. The third is to desire for diversity: a large world can have more places that are different from each other. There is more space for exploration, for divergent evolution. Even if the world is deliberately made parts can become different and unique.
Planets are neat, self-assembling systems. They also use a lot of mass to provide gravity and are not very good at producing living space. Artificial constructs can become far larger and are far more efficient at living space per kilogram. But in the end they tend to be limited by gravity.
Our search for the largest possible world demonstrates that demanding a singular world may be a foolish constraint: a swarm of O’Neill cylinders, or a Dyson swarm surrounding a star, has enormously much more area than any singular structure and few of the mechanical problems. Even a carefully arranged solar system could have far more habitable worlds within (relatively) easy reach.