Not to be confused with Ringworld or yesterday’s post, this is about whether a doughnut-shaped planet could exist, but just to clear that up, Ringworld was a concept thought up by Larry Niven for his ‘Known Space’ series of a megastructure consisting of a ring-shaped terrain orbiting a star and given day and night by rectangular shades orbiting further in. It would require as yet undiscovered materials, in other words unobtainium, not to be confused with unobtainium, to be built, although a more diffuse ring of habitats or indeed planets in a single orbit is entirely feasible. This is not that.
Relatively small bodies in this Solar System are representative of the possible shapes which can be achieved by given amounts or quantities of solids, liquids, gases and composite matter made thereof. The fourth state of matter is entirely different but planets are not made of plasma, practically by definition. The smallest approximately object gravitationally obliged to be round is of course the Death Star moon Mimas:
Even this isn’t that close to being perfectly round because of the relatively huge crater Herschel, but it can also be seen that it has a noticeably rough outline in this picture. It’s allowed to have fairly large mountains and deep craters. Mimas is around four hundred kilometres in diameter, although it deviates by about twenty from this, but that’s still pretty round when considering that the deepest trench in our ocean is almost twenty kilometres lower than the highest of our mountains, compared to sea level which is not perfectly spherical itself. Mimas is in absolute terms as close to being round as Earth is, given that our mountains, valleys, trenches, continents and abyssal plains were not scaled down on a Mimas-sized model of our planet.
The next Saturnian moon down from that size appears to be Hyperion:
So far as I can tell, just as Mimas is the smallest roughly round world in our Solar System, so is Hyperion the largest object which is a long way from being round at 360 x 266 x 205 kilometres. It’s within about thirty kilometres of Mimas’s smallest diameter, yet it manages to be rather irregular. It looks more like a pebble of pumice than a planet, and of course it’s neither. Its mean density is only just over half that of water, which is actually lower than pumice, and also lower than that of Mimas, which is about 1.15 times water’s. There must be a complicated relationship between strength, rigidity and density which decides the shape of objects of about Hyperion’s and Mimas’s size.
This house has many Escher prints owing to my family’s joint enthusiasm for the artist. One of my favourites, which I’m sad to say is not on any of our walls, is ‘Double Planetoid’. This shows two intersecting tetrahedra, one completely unaltered by technology, the other completely covered in building. Of this, Escher says:
‘Two regular tetrahedrons that penetrate one another, float through space like a planetoid. The light-coloured one is inhabited by human beings who have completely transformed their region into a complex of houses, trees and roads. The darker tetrahedron has, of course, remained in its natural state, with rocks on which plants and prehistoric animals are living. The two bodies fit together to make a whole but they have no knowledge of each other.’
M C Escher
It would be possible to make a real copy of ‘Double Planetoid’ somewhere in space, at its approximate scale. In fact it would also be possible to scale it up to at least twenty-seven kilometres on a side if it were carved from granite, even if it had Earth’s gravity. However, it probably wouldn’t arise without intelligent manipulation being involved somewhere, except maybe in an infinite Universe or a parallel world somewhere. It would also not generally be possible for ordinary matter to exert sufficient gravity to make a real version of this rather than a model.
I bring this up because clearly an object like Hyperion could be sculpted into a particular shape, although in its case this would probably already be constrained by gravity so it might end up quite rounded off, but one like Mimas couldn’t. Again, there are likely forms something could take other than round, usually just irregular and lumpy, when they’re fairly small, and many of these are seen in asteroids and small moons. 433 Eros, for example, is often described as “sausage-shaped”:
(I’m not sure that’s how I’d describe that). The asteroid Cleopatra is one of several described as dumbbell-shaped:
Once an object is the size of a planet, though, the options for possible shapes closes down a lot. Considering this as an Earthling, the tallest possible cylindrical column of granite is said to be about the height of the Matterhorn, around half that of Everest at 4 478 metres, and the tallest possible pyramid of granite is 13 400 metres, on this planet. However, this can’t be strictly true because if the height of Everest and the depth of the Marianas Trench are added together the total comes to 19 882 metres, so given a wide enough base this can be exceeded. The diameter of the geoid – the shape of this planet defined by the level water would reach given only the influence of gravity and rotation on this planet, which approximately means sea level – varies by over thirty kilometres between the poles and the equator, which is again somewhat more than twice 13.4 kilometres. Twice is fine because we’re talking diameter rather than radius, but more than twice suggests there are other influences, such as rotation. Steel can, if I recall correctly, form a cylindrical column up to thirty kilometres high and there are a few specialised substances which could be used to build a tower which officially reaches into space, but they’re exotic and would have to be specially synthesised.
The planet with the most obvious deviation from spherical in this Solar System is Jupiter, which has a polar diameter of 133 708 kilometres but an equatorial one of 139 820, which is a variation of 4.5%. This is because Jupiter is a substantially fluid body consisting of liquids and gases, and because it spins very fast. In terms of velocity the planet’s equator is moving over thirty times faster than Earth’s and it’s also over three hundred times our mass. However, Jupiter happens not to be the least spherical planet in our neighbourhood. That honour goes to Saturn, whose rings may disguise the fact. Saturn has an equatorial diameter of 116 460 kilometres and a polar one of 108 728, which is a variation of over seven percent. This may be connected to the fact that it’s also the least dense planet.
It’s established, then, that a planet can be tangerine-shaped rather than spherical if it’s sufficiently fluid. These two examples are also large and rotate fast. Earth is not like that, but it’s theorised that there are planets out there in the Universe which are mainly made of water, or which have extremely deep oceans. These could presumably assume such a shape, which on a planet the size of Earth would be like having a stationary wave almost a thousand kilometres high circling the equator, and in fact this would even be noticeable from the surface as it’s close to a gradient of one in ten. This could be described as a tangerine-shaped planet, but I have to say I don’t find that idea very interesting. The shape is officially called an “oblate spheroid”. There are stars which are markèdly flattened in this way, such as VTFS 102 in the Large Magellanic Cloud’s Tarantula Nebula, which is three times wider at the equator than at the poles, but stars are not made of solids, liquids or gases.
Another well-known variation of a spheroid is the rugby-ball shape, or prolate spheroid, and there are also stars of this shape. Some binary stars orbit each other so closely that they are mutually distorted into elongated shapes of this kind, and I don’t know this but it seems possible to me that this is also because of how fast they’re whizzing round. The question arises of whether a planet could have such a shape. Larry Niven, again, imagined such a planet in the Sirius system he called Jinx, whose “poles”, so to speak, were effectively vast plateaux rising out of the atmosphere at each end, and the “equator” was a high gravity area. The humans living near the equator needed to be very strong and muscular to cope. I don’t feel convinced that this is possible for a largely solid planet, but just as Saturn and Jupiter can get squished by their rotation I can see that if there is a system somewhere with a double gas giant, this might be what shape those planets would assume. The same might even apply to double deep ocean planets.
Other possibilities are very limited. For instance, an egg-shaped planet flat at one end and pointed at the other is difficult to envisage. However, there is one possibility which, oddly, is very far from being spherical but is still possible.
I’ve mentioned the periodical ‘Manifold’ on here a couple of times. This was a mathematical magazine published by the University of Warwick, one of my almæ matres, from 1968 to 1980, one of whose claims to fame is that it invented the game ‘Mornington Crescent’. I used to read it back then, and one of its many whimsies was a fictional toroidal planet whose name escapes me, with six cities all joined together by an underground railway. This is a reference to a well-known mathematical puzzle involving three houses all of which need a water, gas and electricity supply but none of the pipes could cross each other. This is impossible to arrange on a flat surface but works fine on a torus.
When I read this, I mused that it was a shame that such a planet could never exist, and I started working out things like I did above, with the likes of pyramids as very high mountains and various irregularities in its surface ruling it out. I then realised that I couldn’t actually find a reason for such a planet not to exist, and just assumed I didn’t have the mathematical prowess to work out why it couldn’t.
Well, it turns out that it can, in the sense that if a toroidal object of Earth’s volume, mass and composition could be formed in the first place, it wouldn’t be susceptible to collapsing into a spheroidal form. The above shape, surprisingly, is gravitationally stable. Incidentally this would also apply to water in free fall, so a spinning doughnut shaped swimming pool in space made entirely of water is completely feasible, under a pressurised atmosphere of course. There’s a fairly easy way of understanding this. It’s already been shown that tangerine-shaped planets exist, which are largely fluid and flattened by spinning. This is a kind of limiting case of that situation. If a fluid planet ended up spinning fast enough, not only would it become flattened but its matter would be completely pulled away from its axis of rotation. Most planets can be considered either to start out as fluid, i.e. they are either actually liquid, such as made of magma, or of sufficiently small lumps of matter that they behave on a planetary scale as if they were, just as an actual liquid consists of molecules or a heap of sand or dust can flow like water, have surface waves and even drown people. The difficulty is in imagining a scenario where this would actually happen on its own. The alternative is simply to say it’s being done by intelligent life but then the imagination falters a little as well, because how powerful would a civilisation have to be to have the resources to make its own planets? Also, why?
Nonetheless, however it came into being, once it was there it could continue as long as any other planet in its current shape, and this is a little surprising because the deviation from a sphere in this case is extreme. I also have to admit to a little confusion and have to insert an explanatory note. I can’t honestly tell whether this shape is sustainable simply due to the planet’s gravity or whether it would also need to be rotating fast with the axis passing “vertically” through the hole. I suspect the former is the case, but even if it isn’t, the second case would guarantee that it’s possible, although it’s not clear how fast it would have to be spinning. That would also depend on the proportions of the torus. Now for the explanatory note. Thus far I’ve assiduously avoided using the words “centrifugal force” because that doesn’t exist as such, as is well-known, but it can be quite awkward to express oneself without using those words. What is in fact happening in this situation is that the mass of the planet is constantly “trying” to move in an infinite number of straight lines, all tangent to its surface at the outer equator, but is pulled away from that path owing to the electromagnetic and gravitational forces holding it together.
It’s also very unclear how big this planet would be. According to the second picture at the top of this post, the north-south distance across Afrika is rougly equivalent to the width of the “tube” of the torus, the same distance for Australia is its thickness and the hole is about the same size as Australia again. Hence that version of a toroidal Earth is 3 000 kilometres thick, 7 000 kilometres wide on either side and has a hole 3 000 kilometres in diameter. This raises two questions for me about how to calculate the volume and surface area of a torus and also what to call the different features of the shape. Strictly speaking, the shape in the picture is not a torus because it’s not circular but elliptical in cross-section. The distance from the centre of the hole to the outer edge is called the “major radius”, R, and that from the centre to the inner edge is the “minor radius”, r. There’s also the aspect ratio, which is R/r. Strictly speaking, not only is the above not a torus (although the blue image is), but even if it was, it would only be a particular kind, namely the ring torus. There are also horn and spindle tori. A horn torus has its circular cross-sections touch at the centre, so strictly speaking has no hole, and a spindle torus has the circles overlapping. Both of these shapes are slightly more achievable for a planet than the ring in terms of events happening without intelligent intervention.
The formulæ for surface area and volume are respectively:
where p=R and q=r. This suggests several “equivalences”. One is the size of a torus with the same surface area as Earth’s, another the volume of such a torus, another the size of a torus with the same volume as Earth’s and another the surface area of that torus. All of these are also dependent on R and r, and thus the aspect ratio. I’m not going to address these immediately.
The torus in the second picture has basically the same continents as the real world, but Antarctica seems to be missing. In fact it can be concluded that the polar regions are missing altogether. However, there are two circles corresponding to the poles and of course two further circles corresponding to the Equator. Assuming the planet is held together by its rotation and doesn’t have constant daylight anywhere on its surface, i.e. not rotating with the hole axis facing the Sun, the “polar” regions are pretty close to lands which are equatorial on the real Earth in that image, although the other side has another circle which in the Arctic. Meanwhile there are inner and outer equators, and the outer passes through the Mediterranean. Assuming no axial tilt the inner equator is in eternal darkness and therefore colder than Antarctica, which would take a lot of water out of circulation and probably cool the whole planet. If it’s tilted at the same angle as we are, on the other hand, it would be exposed to sunlight some of the time and the “polar” circles would also have seasons, half a year of night and half of day and so forth, as they have here.
If this planet maintains its shape through rotation, there will probably be strong winds and ocean currents everywhere. There’s also an important topological difference between a spheroidal and a toroidal Earth. Topologically, considering the troposphere (the bit with the weather in it) as a single layer, there must always be at least two locations on Earth where there is no wind. This is not so on a toroidal planet because the hypothetical still spots could be lined up to be in the hole. Ocean currents are like this in the real world, because the land punches holes in the ocean in which potential still points could be located. If you go high enough in Earth’s atmosphere, the air is no longer dragged along by our rotation, so perhaps a toroidal Earth could have a relatively calm troposphere like ours is.
Apparent gravity would also vary across its surface. The rapid spin would act against gravity at the outer equator and in favour of it at the inner one. Some time ago, Alfred Wegener attributed continental drift to the centrifugal effect he called Pohlflucht. Arguably, as it depends on how rigid the planet is, Pohlflucht could be a reality on this world. Perhaps the continents actually would cluster around the outer equator. If they did, though, they would have to be quite mountainous to prevail over the water, which would be pulled into a belt in the same region. This, however, might actually be so because the lower gravity would favour higher mountains in that area. It seems to be shaping up into a situation where the inner region is a cold, flat desert, there are two strips of land either side of the outer equator along with a tendency for continents to move towards the outer equator where they form fold mountains which are, however, submerged under a deep ocean, which resembles the Tethys of our prehistoric past.
There needs to be a moon of some kind to generate a protective magnetic field. This could orbit at the outer equator. The toroidal magnetosphere thus formed would be a different shape than the real one.
From the surface, there are conventional horizons to the north and south, but to the east and west the vista depends on where you are. On the outer equator, the situation is pretty much as it is here. Near the “polar” circles, the planet is effectively flat along one circumference, and the landscape or seascape (snowscape more likely) disappears into the haze of the atmosphere. The inner equator offers the most spectacular view. During the day, the sides climb upwards into curved, hornlike shapes which gradually plunge into night, forming an overhead arc. At night, the situation is the same but there would be a visible daylit sector which would first recede up the horn, travel across the sky and then descend towards the observer until dawn. On this inner surface, gravity would be high, so the view might be nice but it would also be quite uncomfortable or even uninhabitable. I’m assuming here that there would be an axial tilt.
There’s a limit to the relative size of the hole. The narrower the ring, the less stable the planet. Both of the first two illustrations are viable, but a more traditional banded ring shape would be highly volcanic because it would tend to flex and crack under the forces maintaining its shape. Hence a doughnut shape is best. Even then the day would only last about three hours. A moon might move in a straight line in and out of the hole, or it could follow an ∞-shaped orbit.
The remarkable thing about this scenario is, of course, that it isn’t impossible. There could never be a tetrahedral or cube-shaped planet and the largest conceivable regular polyhedral planet would probably be something like a dodecahedron perhaps somewhat larger than Mimas but still much smaller than Earth or even Mercury, because the vertices would effectively be high mountains. In fact planets are in a sense polyhedral because they aren’t perfectly smooth spheroids on one scale, although on a smaller scale the jagged peaks and steep valleys would be rounded – this is a fractal issue, because on a smaller scale still they’d be jagged again, and so on. However, they are also very close to being smooth. As far as I can tell, the only possible shape a planet could be which is radically different from a sphere is a torus. What isn’t clear is whether it could ever happen on its own. I can easily believe that there are occasionally asteroids which have holes all the way through the middle, although as far as I know there are no known examples in this Solar System. A very rapidly spinning protoplanet could form into a torus, and the question then arises of what could cause it to spin so rapidly. Perhaps if it were high in iron and close to a neutron star this could happen, but it would be unlikely to be habitable. A non-habitable toroidal planet is unsurprisingly much easier to devise than a habitable one. However, given the will, the technology and the access to resources, nothing at all seems to stop an intelligent technological culture from making such a planet on a whim, or perhaps as a work of art. Isn’t that amazing?
A Box Of Chocolates 