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This is the second part of a question here that I've broken up because they are two very distinct problems. Assuming a perfectly stable ringworld:

Assume a more traditional flat ringworld that is not quite so massive (Niven's first description was of one only a kilometer thick & the mass of Jupiter) and is much closer to the star, well outside the habitable zone. A planet in a discrete orbit "behind" the ringworld has all sunlight occluded by the ring. What kind of heat could this planet receive to make it habitable? Would radiation from the sun spill over the sides? Could the (metallic) ringworld itself, absorbing energy from the sun, radiate an appreciable amount of energy from its dark side into space and create a habitable zone that this planet would be within?

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  • $\begingroup$ You'd need a very large ringworld or a very stable planetary orbit with inclination perfectly matched to the ringworld to block off a whole star. I assume 1km is the thickness perpendicular to the surface? What's the size of the ring above or below the plane of the ring? $\endgroup$ – zstewart May 3 '17 at 19:42
  • $\begingroup$ @zstewart Niven's is a million miles in width, the 1km thickness mentioned is indeed the thickness perpendicular to the flat surface. The ring I imagined could be this 1 million or just the diameter of the planet. The only intended bound on the question is that no direct sunlight reaches the planet's surface or atmosphere. $\endgroup$ – Ross May 3 '17 at 19:51
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    $\begingroup$ So just for fun, minimum size calculations; if the planet is Earth-sized and has an Earth-like orbit, and the ring has a radius similar to Venus' orbit, and the star is sun-sized, bare minimum width to block out the star entirely is 1.2 million kilometers. And that assumes that the orbit of the planet always perfectly aligns with the ring. Any slight orbital perturbation above or below the plane of the ring would bring it into sunlight. If the ring has a smaller radius, it has to be thicker still. $\endgroup$ – zstewart May 3 '17 at 20:31
  • $\begingroup$ aleph.se/andart/archives/2014/02/torusearth.html This link talks about seasons and gravity and various other things about ring worlds $\endgroup$ – Dtb49 May 9 '17 at 13:22
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As Pedro said, geothermal/seismic activity would be a good way of the planet generating its own heat. I want to address the other questions:

Would radiation from the sun spill over the sides?

I'm less able to answer this question, but I think it depends on the ringworld itself. For example, the Earth's magnetosphere deflects a lot of solar radiation around the Earth, so assuming a similar magnetosphere (magnetoroid?) surrounds the ringworld, it's likely that some radiation will be deflected around the ring. However, it's not likely to be enough to heat the planet.

Relevant illustration: http://cdn.inquisitr.com/wp-content/uploads/2016/04/magnetosphere11.jpg

Could the metallic ringworld radiate an appreciable amount of energy from its dark side into space?

If we assume the ringworld is an ideal black body (emissivity of 1), and has a width equal to the diameter of Earth, and transmits the amount power received by the Earth from our own sun from a square on its surface 7914m x 7914m (diameter of the Earth squared):

1.74x10^18 W = (1)*sigma*62.69x10^6 m^2 *T^4

[algebra]

The surface of the Ringworld would have to be around 26450 K. About 4.5x the sun's surface temperature. This is a huge approximation (trying to roughly account for how much of the radiation our planet might receive from the ringworld), but it doesn't seem plausible. If the ringworld in question were considerably larger in size (mostly width) and thus able to absorb and radiate more heat via greater surface area, it might approach the "possible", but I don't think it would be enough on its own to heat a planet.

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Geothermal energy would be the most likely and realistic way for it to get heat. A creative way to do this, although you don't have to, is to make the ringworld's electromagnetic field so strong that it interferes with the other planet's core, causing seismic activity, and therefore heat.

As for livability, you should probably make most of the world ocean and small volcanic islands. Water retains heat much better than air, so most life would hang out near thermal vents on the seafloor, or off the coast of an active volcano where ample residual heat from lava would provide.

I got a C in biology so I'm no expert, but most ecosystems that we know of rely on a flora-fauna equilibrium and on the planet you describe photosynthesis would not be an option. So I'd suggest looking into alternative ways for a planet to get energy. Hope this helps :)

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If you want sunlight to spill over the sides of the ringworld, you may actually resort to mirrors. The sides of the ringworld could be made of a transparent material and it should be a simple task to make it so that the light from the star gets refracted and concentrated on the plane of the planet's orbit. Depending on the angle of refraction and the distance between the ringworld and the planet, though, this might cause sunlight to be directed at the planet's higher latitudes.

Another possibility, if you're willing to sacrifice some space in the ringworld, would be to make the middle of the ringworld transparent. That way some sunlight would fall on the planet following a more or less straight path, as if the ringworld weren't there.

Also, since the ringworld is so close to the star and receiving so much surplus light, you could cover some of it with solar panels and then radiate the energy (in a suitable form, e. g. microwaves) toward the planet for its inhabitants to catch it. This is assuming that your ringworlders have an interest in keeping the planet's dwellers alive and well.

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