Actually, if you just want the hot side, cold side, and little band of habitable, then you might want to just have the planet be tidally locked, which means it takes as long to rotate as it does to orbit the star, so that the same side is always facing the star.

That is a very intriguing idea! I wonder if...somehow the uninhabitable areas could be navigated, sort of like astronauts did on the moon. Fascinating...

Andrew

I've actually changed the world to make the "sunny side" less uninhabitable. It's still a hell to live in, but people do--in fact, most of the population does, since the Monarchist-dictatorship that rules the country has kept news of the equator strip a carefully-guarded secret. I'm still trying to figure out who does live along the equator, though--a privileged upper class that has been "hidden", for some reason, maybe.

A tidally locked world is a fun idea!

On a tide-locked world, I suspect you might need an active agency or some cobha to ensure that the day side--and even the twilight belt--gets any water at all. Because just as the day side would be perpetually blazing, the night side would be perpetually frozen. So any evaporated water would condense and precipitate as snow as soon as it blows to the night side, and get trapped there as ice. (And then the ice would never melt; it never gets any sun) So there needs to be something keeping water in the atmosphere.

Remembering scenes on Ryloth in the YJK series reminded me of this. I'm not sure their make-believe about Ryloth is all very consistent, but they do postulate "heat storms" which make their way onto the night side, boiling water into the atmosphere. Personally, I don't think this would work--the water would just end up getting recycled until it condensed out of range of the heat storms, and then be stuck. Perhaps ice-mining could be a major industry in your world? (But then you would have to explain how life survived before the industry got on its feet)
I think the easiest way to solve the problem would be to establish a reason why it's not always below freezing on the night side...

Yeah, I'm sure I'll have to employ some cobha... Good point about the water.
I actually want it to be always below-freezing on the night side... which is kind of a problem. But I'll figger something out...
Thanks for the thoughts!

Maybe something happens inside the planet? You could enginer a cobha that lets heat travel through the planet from the hot side to the cold side, melt ice there, and bring back water to be evaporated on the hot side. It would be hard to make work, but would be cool if done right.

Another idea (and probably one much easier to implement) would be to use animals and/or plants to move water from from the cold side to the hot side.

Perhaps a bird that lives in the hot side requires water to make an oasis like home (underground?) to attract a mate. He could travel to the ice side, cut out ice chuncks with their beak and return the chuncks to their home in order to collect the water for his survival, his family's survival, and some extra for the ecosystem.

And of course you would include predators, food for the bird, etc... if you choose to use it.

Ok, so I asked about this on a physics forum, and here's my (medium-confidence) interpretation of the information that I gleaned:

1) If the planet has oceans in proportions roughly the same as earth's, the diffusion of heat through the ocean might be enough to prevent the ocean on the night side from freezing, although the landmasses would probably be mostly ice. This would solve the problem of water scarcity--in fact, it makes it quite probable that atmospheric water would be abundant, since the ocean in the middle of the day side would always be boiling/near boiling.

2) If the planet's oceans are too small to be self-sustaining in distributing heat, the water would end up frozen on the night side--but there would still always be a small amount of water in the atmosphere. Ice sublimes even at -133 degrees celsius, so there would be water vapor in the air--and the difference in temperature between the day and night sides would cause a huge Hadley cell to form around the terminator, circulating dry/moist air between the day and night sides. It would be a desert world for certain, but pockets of life might exist where mountains and other geological features cause water to condense in the twilight belt. People could collect water using wind traps like in Frank Herbert's Dune, or by directly mining ice on the night side.

If either of these options works for your world setting, perhaps you could get by without the need for climate-sustaining cobha.

You're most welcome! Out of curiosity, which option do you plan to use for your world?

(As for physics forums...well, there can be some harsh characters on there, true...but if you'd ask my opinion, don't let that stop you! If it's just not your interest, that's a good reason to not waste your time on it. But if you want to learn that stuff for your world building, it's certainly worthwhile. )