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What would the Earth be like if it was the...
What would the Earth be like if it was the shape of a donut?
According to the laws of physics, a planet the shape of a donut, or toroid, could actually exist, but it’s extremely unlikely to ever form naturally. But what if an advanced alien civilization decided to build one? What properties would a toroid-Earth exhibit? And what would life be like?
Can Toroid Planets Exist?
For all practical purposes planets are liquid blobs with no surface tension: the strength of rock is nothing compared to the weight of a planet. Their surfaces will be equipotential surfaces of gravity plus centrifugal potential. If they were not, there would be some spots that could reduce their energy by flowing to a lower potential. Another obvious fact is that there exists an upper rotation rate beyond which the planet falls apart: the centrifugal force at the equator becomes larger than gravity and material starts to flow into space.
The equilibrium shapes of self-gravitating rotating ellipsoidal planets have been extensively analyzed. Similarly, equilibrium states of self-gravitating toroid shapes have been examined. One can at least in theory spin up an ellipsoidal planet into a ring, although there is plenty of potential for complex wobbles that destabilize the whole system and it looks like there is a “jump” to the ring state. The ring may itself be unstable, in particular to a “bead” instability where more and more mass accumulates at some meridians than others, leading to breakup into two or more orbiting blobs. There is also a lower rotation rate where the ring become unstable to tidal forces and implodes into a “hamburger" or ellipsoid. So the total mass and angular momentum needs to be in the right region from the start.
It looks like a toroid planet is not forbidden by the laws of physics. It is just darn unlikely to ever form naturally, and likely will go unstable over geological timescales because of outside disturbances. So if we decide to assume it just is there, perhaps due to an advanced civilization with more aesthetics than sanity, what are its properties?
Toroid Gravity
The case of a very large main radius torus is essentially a cylindrical planet. In this case the gravitational force falls off as 1/r, where r is the distance from the axis. The total force on any section will be proportional to the total mass and the gravitational force, so the overall force will be constant as we increase R. Adding some rotation will balance it. As long as the surface gravity is big enough this will overcome the centrifugal acceleration and stuff will indeed stay down.
Why doesn’t the planet get squashed into a plane disk? The rotational pull tries to flatten the planet, but it must act against the local gravity field which tries to turn it into a ball (or cylinder).
I will look at a chubby toroid of one Earth mass and a small central hole (“Donut”), and a wider hoop-like toroid with 6 Earth masses but more earth-like gravity (“Hoop”).
Donut Earth
Donut has a hubward/interior equator 1,305 km from the center, and a rimward/exterior equator 10,633 km away. The equatorial diameter is 9,328 km. The planet extends 1,953 km from the equatorial plane, with a north-south diameter of 3,906 km. The ratio of the diameters is 2.4. The north-south circumference is 21,587 km (0.54 times Earth), while the east-west circumference is 66,809 km (1.7 of Earth). The total area 8.2*108 km2, 1.6 times Earth. The total volume is 1.1*1012 km3, within 1% of Earth (after all, Donut was selected as a roughly one Earth mass world). The Volume/area = 1300, 61% of Earth: there is more surface per unit of volume.
One day is 2.84 hours long.
Hoop Earth
The planet extends 4,070 km from the equatorial plane, with a north-south diameter of 8,141 km. The cross-section has roughly the 4:3 ratio of an old monitor. The center of mass circle is 14,294 km from the center. The north-south circumference is 30,794 km (0.77 of Earth) while the east-west circumference is 125,270 km (3.1 times Earth). The total area is 2.5*109 km2, 4.9 times Earth, and the total volume 6.5*1012km3, 6 times Earth. The volume/area = 1500, 70% of Earth.
The day is 3.53 hours long.
Light
First, the nights and days of these worlds are very short. There is not much time for the environment to cool down or heat up during the diurnal cycle. What really matters is how much light they get over longer periods like seasons. Assuming these worlds orbit at an Earth-like distance from a Sun-like star, these are long enough to matter.
Torus-shaped worlds have an outer rim that is not too different from a normal ellipsoidal planet. Days occur with a sunrise at the eastern horizon and a sunset at the western horizon. The sun moves along a great circle that slowly shifts north and south over the year, giving seasons. However, on the interior side things are different. Here other parts of the planet can shadow the sun: to a first approximation we should expect far less solar energy.
Geosphere
The surface area is larger than on Earth, and the volume/area ratio is smaller (for Donut the ratio is 1,300 km, for Hoop 1,500 km, for Earth 2,124 km). One might hence suspect that more thermal energy is leaking out, reducing volcanism and plate tectonics. However, even a small amount of tidal heating due to influences from the sun might release plenty of energy stored in angular momentum. In the case of Hoop there are also 6 times more radioisotopes inside the planet than on Earth but only 5 times more surface area.
Gravity affects the height of mountains. On Hoop the difference is not enormous compared to Earth, but on Donut mountains at the poles can be 1.5 times higher (maximum around 12 km) and near the equators 3 times higher (24 km). Combined with the ruggedness near the hole this might make for some dramatic landscapes.
The fast rotation will likely produce a strong magnetic field; unlike on Earth the polar regions will not have auroras since the field lines will not intersect the surface… I think — figuring out dynamo currents in a toroid iron core sounds fun but is beyond me.
Atmosphere
On torus worlds the rotation rate is 8 times faster than on Earth and the velocity differences are larger. Air hence tends to be twisted around far more, producing a more banded zonal climate than on Earth. Exactly how banded is hard to tell without detailed atmospheric calculations, but it is likely more like on Jupiter than on Earth. This in turn means that heat transfer is less effective: the temperature differences between the hot and cold regions will be bigger.
Like on Earth cyclones can form at the mid-latitudes. Stronger Coriolis forces would make tighter hurricanes, about four times smaller. However, they would tend to last longer on Donut. Wind speeds depend on the temperature difference between the top of the atmosphere and the ocean, which could vary a great deal across the year.
Hydrosphere
The amount of water on either world is not vastly different from Earth, although Hoop’s 6 times greater mass with merely 5 times greater area would provide it with 20% more water volume from the initial accretion (so for the same coverage the oceans would be 20% deeper).
The big seasonal temperature swings will be more pronounced far from the moderating influence of oceans: continents near the poles will be more extreme than equatorial ones. Whether they can maintain ice caps throughout polar summer depends on their layout and the background temperature.
The low gravity near the equator will make some tall waves on Donut: they can be expected to be three times taller than on Earth. Waves at Donut’s poles are still 150% of the ones on Earth. Hoop is closer to normal (133% taller at the equator, 90% height at the poles).
Moons
A moon orbiting exactly in the equatorial plane in a circular orbit would just look like it came from a spherical planet of some intermediate density. However, if it orbited in slightlyeccentric orbit things would change. And as soon as the orbit becomes slightly tilted things turn even more complicated – now the moon will feel the flatness.
A nearly polar orbit has even more precession, not just making it rosette around in a plane but also slowly precessing the plane. The moon could appear in the sky in any constellation.
What about orbits through the hole? The exact center is an unstable point. Place a moon there, and any kick will make it fall out. But there are orbits through the center that look stable (or rather, give them a kick and they turn into another similar-looking orbit rather than fall down). The simplest is just a moon bobbing up and down through the hole.
Summary
Torus-worlds are unlikely to exist naturally. But if they did, they would make awesome places for adventure. A large surface area. Regions with very different climate, seasons, gravity and ecosystems. Awesome skies on the interior surface. Dramatic weather. Moons in strange orbits.
We better learn how to make them outside of simulations!
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