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Say we have a planet that takes 8 Earth days to fully rotate on the axis. At a latitude of 45 degrees, during the winter solstice, its day lasts 1 Earth day, or 24 hours, and its night lasts 7 Earth days, or 1 week. What axial tilt will result in a day length of 1/8 of the axial rotation at an latitude of 45 degrees?

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43-44 degrees

Looking at Earth, even when you get to a latitude of 60 degrees there are still about five and a half hours of daylight (ie almost a quarter of the day length). In order to get the daylight length down to 1/8 of the rotation period, it is necessary to be within a degree or so of the Arctic/Antarctic Circle. Which puts the Arctic/Antarctic circles at 46-47 degrees north and south respectively, with a corresponding tilt of 43-44 degrees.

Exactly how close depends on the definition of "daylight" - does the sun need to be above the horizon for the entire 1/8 of the rotation or does a period of pre-dawn/dusk light count? Is there a mountain range immediately to the north/south as applicable that will block low-angle sunlight?

Note that this puts the tropics within 2-4 degrees of the arctic/antarctic circle! It may be worth looking at the answers to this question about a planet with an axial tilt of 60 degrees to consider the impacts on your proposed planet's environment, even though it is not quite as extreme.

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    $\begingroup$ I appreciate the acceptance, but generally it is best to wait 24 hours before accepting an answer. This gives an opportunity for people to comment on my answer - it's possible that I have made a mistake - or for someone to provide a better answer. With people on this site from all over the world, we need to allow for all timezones. $\endgroup$ Jun 13, 2023 at 2:04
  • $\begingroup$ The green checkmark tyranny must be culled in the root. The users could vote on a meta to have the rule implemented by the system. If not a full block, at least a warning to the OP stating that they should wait for more answers. $\endgroup$ Jun 13, 2023 at 13:14

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