A Simpler case
A simpler case would be to put a single toroidal planet around the Sun. Basically a mix of Niven Ring and a hoopworld .If we wanted to aim at Earth-like properties, we could set the distance and rotational period to the same values as those of Earth's (1 au, ie. 1.496×10⁸ km, and a rotation per year), because the mass of the orbiter is irrelevant.
How thick does it have to be to have Earth's gravity? Since the major radius of our torus is much larger than the minor radius, let's model it as an infinite cylinder, the gravity of which is
$$ g = {2 \lambda G \over x} = {2 \pi r_{minor}^2 \rho G \over x} $$
where $\lambda$ is density per length, $r_{minor}$ the radius of the cylinder, and $x$ the distance from the center. At the surface $x=r_{minor}$, so
$$ g=2 \pi G \rho r_{minor}$$
where $\rho$ is the average density of the cylinder, and $G$ the gravitational constant. We get
$$r_{minor} = {g_{Terra} \over {2\pi G}} ≃ 4240 \mathrm{ km} $$
For comparison, the radius of Earth is about 6371 km.
Problems with rotation
What if we want to have day and night on this world? How much of a strain would it put on the surface as the shorter noon side rolls out to face outer space? In absolute terms the difference is$$ l_{outer}-l_{inner} \\
= 2\pi(r_{major}+r_{minor})- 2\pi(r_{major}-r_{minor}) \\
= 4\pi r_{minor} \\
= 4\pi 4240 \mathrm{ km}\\
≃ 53280 \mathrm{ km}
$$
...which sounds like a lot, but compared to the major circumference of the torus, is only
$${\Delta l \over 2\pi r_{major} }≃ 0.0057 \% $$
...which doesn't sound a lot, as does 5.7 cm per a kilometer, but when you put it in terms of the Earth's circumference, you get the equivalent of 2.27 kilometers of stretching each day on the equator, which is definitely a lot. Floor would be lava in that world.
I don't know how to wave that away. Given that our hypothetical architects could put this kind of a world circling a sun in the first place, they'd probably have a solution. Maybe the crust would be of a substance that can take deformation in a stable way, maybe they'd have slices of deformable material every x kilometers along the torus etc, etc.
How about gyroscopic effects? The tube would want to continue to roll around its axis while the year goes along, trying to chop the whole ring into small independent pieces. Maybe.
Mass
Mass itself is another problem. The total mass of this single hoop world would be$$\pi r_{minor}^2 \cdot 2\pi r_{major} \cdot \rho$$
which gives 2.93×10²⁹ kg. There is little hope when looking at what kind of masses we have at our disposal (assuming that the case would be about the same for other solar systems). You'd need a star's worth of mass, and to convert it into heavier elements. Cue magic. Maybe you'd implement the construction in an older universe where heavier elements would be more abundant.
stevebowers of Orion's Arm suggests building the ring at a young star like T Tauri with the accretion disk still in place, and I feel pretty stupid for not thinking about that on my own.
More Complex Configurations
Letting go of our already overstretched suspension of disbelief, let's take the scenario further. Two torī orbiting each other like twin planets, both in a helical arrangement, rotating daily. Intuitively forming a helix would take some strain off the torus: instead of having to alternate between the extremes of the daily perihelion and aphelion (terms to be taken literally), the circumference would remain nearer constant if a parts of the same torus would be nearer while other parts would be farther from the sun. In practice, however, it would be probably more correct to think of the worlds as liquid and disregard any intuitions about solid hoops, like having a variable local orbital speed in different phases of the daily cycle.Putting aside the problems with seismology, the arrangement itself seems sound to me. Extremely delicate, but critically stable. Left alone it would form lumps from the atmospheres, seas, and ultimately solid masses gathering around even the most infinitesimal of local mass anomalies. Maybe the torī wouldn't have liquid cores and maybe seas would have a maximum width to reduce downright flowing into a string of pearls. If the solid mass could be kept in check, I assume the atmosphere and seas would play along.
stevebowers suggested to make the whole thing out of water, which would address (in part) both the problem of material and stretching. It would be far easier to burn hydrogen and oxygen into water than to transmute or haul heavier elements from god knows where. You could cover the water ring with rafts where needed. Exotic phases of water or a 'regular' solid core could be found in the center, as with any waterworld.
How to fight the subtle disturbances in the orbits? Carefully operated active dampening masses rolling around the surface? This is beyond me.
The failure of this kind of world, unraveling in time and space, would provide interesting apocalyptic sci-fi scenarios. A ring of uneven worlds, sections without air, sections with massive bulging oceans. A looming chaos in the sky: a catastrophically failing segment drawing a red line of glowing lava across the Milky Way, getting closer and closer by the decade. You'd be in a hurry to sail across that blob of an ocean or traverse the million kilometer Desert of Very Little Air to buy yourself some time. Or develop a space program and hop to the other hoop with a bit less destruction going on. Or build a space station in the exact middle of the two hoops.
There are some interesting choices to be made with the configuration of these worlds. They could be asymmetrical (like Earth and Moon) or Roche-like (with a different profile thanks to a linear falloff of gravity). There could be 3 of them, or 3 in a Rocheworld configuration, touching or not. 2 or 3, the individual rings could rotate in either direction around the sun: the direction would have no bearing on their mutual rotation around each other.
Again, strange possibilities for sci-fi authors: imagine 2 tidally locked hoops going the same direction, with space stations in the middle, elevators to both worlds. Or different directions, with a skyhook spinning in the middle on the ecliptic plane, touching down twice a day along the torī, but returning to the same spot twice a year. Or almost touching Roche torī with inner edges hurling by, creating humongous storms and eventually touching, releasing a fresh new hell on the tortured denizens.
I choose rather arbitrarily 2 locked worlds of identical mass, orbiting the sun in opposing directions, and each other once per day. Having a different sky each day makes it more interesting. Something like this (not to scale):
Or this:
Distance
Masses and orbital period known, what is the distance between them? I'm on dodgy ground here, but here goes. Please comment if you can fix anything on this post.$$a_{centripetal}=g_{infinite cylinder}\\
\omega^2 {x\over2} = 2 \pi r^2 \rho { G \over x}\\
\omega^2 x^2 = 4\pi r^2 \rho G\\
x = \sqrt{4\pi r^2\rho G \over \omega^2}
$$
where $x$ is the distance between the 2 worlds and $\omega$ is the angular velocity (radians per time). $x/2 $ because the body is rotating around the center of the masses. This gives 125 300 km, about the third of the distance between Earth and the Moon (380 995 km). In degrees, the other torus would be 3.878° wide (Moon: 0.5167°).
Magnetic Field
How about a magnetic field? If the torī would have a net charge and swirl in opposite directions, they would be analogous to parallel conductors with a current flowing opposite ways. They'd generate a magnetic field with field lines roughly aligning with their surfaces. If you took a compass and walked North, you'd keep going round the world.
But would they have a net charge? I haven't been able to find a good answer to whether planets accumulate a charge from the mainly positive particles of the solar wind, or would any buildup of a charge attract instantly an equal amount of opposite particles. Surely there would be an imbalance, but I don't have the slightest clue if it would be enough to create a magnetic field substantial enough to shield the surface from radiation. If it did, there probably wouldn't be any Aurora Borealis, since the field lines wouldn't pass through the atmosphere, but would go through the space between the torī. Would there be an analogue of Van Allen belts, and where? Streaming between the worlds and their outer rims?
Lighting
If you live on the twin-side of your world, the twin will block the sun at noon, but also make most of your night very bright, shining at the sky. If you live on the dark side, your sky look pretty much like a normal planet's sky, with the exception of the horizon rising into thin shimmering lines in 2 directions.
The relative amount of light by latitude looks like this:
<-- spacewards twinwards -->
Thanks to the noon 'eclipse' at twin side, it gets slightly less light in total. About 60° and 300° (60 degrees from the twin side) gets most light from the additional nightly twin glow.
Since there isn't as big of an temperature gradient as with a spherical planet with incidence angles from 0° to 90° and lighting is quite homogenous too, there won't be dramatically different climates. Slightly warmer night glow areas might create a low pressure zones, creating upward wind that turns twin- and spacewards and eventually after having cooled, return near the ground. The differences are so slight that I doubt there would be as distinct cells of circulation as on Earth.
There wouldn't be an orbital mechanical way to introduce seasons. Axial tilt makes no sense for a torus world. Nor could you have its orbit be elliptic to generate seasons: the torus would have to move faster at perihelion than at aphelion, demanding the whole thing be made from rubber.
But since we are talking megastructures, you could just have a swarm of statites dimming the sun for half an year.
What kind of a sky would the hoops have, besides having each other hanging overhead for those who live in the twinward side? Assuming similar atmosphere as with Earth, the sky above and in the direction of the tangent of $r_{minor}$, but in the direction of the $r_{major}$ tangent, ie. looking at the world extending to space, the 'horizon' would seem redish, just like during sunrise and sunset. This is because there will be more atmosphere for light to travel through, and higher frequence (blue) light scatters away, leaving reds. Something like this:
Summary
All in all as I looked into this, the option b) (being downright ridiculous in practice) became more and more apparent. In any case it's good food for thought and provides - as any megastructure - interesting scenarios for science fiction, given a small star's worth of unobtainium and some leeway with what is practically possible or not.I would really appreciate corrections, comments, and further ideas and calculations (for example on weather) on this.
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