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[Answered myself below] I've seen similar questions have been asked before, but the details are slightly different.

Say you have a tidally-locked, roughly Earth-mass rocky planet in the habitable zone of an M-type red dwarf. That star is in a stable orbit with another star, a G-type star similar to our own sun (apologies, I don't know the orbital mechanics well enough to give exact distances and such here). Just to clarify, this is called an S-type planet, i.e. a planet in a binary star system which only orbits one of the stars. From what research I've done, this is a fairly common type of arrangement in our galaxy.

Obviously, the planet would have some kind of variation in illumination over time, but what kind of distances and luminosity would be required for it to have a "day/night" cycle broadly recognizable to humans (think within 18-36 hours)? Or would the cycle always take place moreso over the span of months/years in such an arrangement?

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  • $\begingroup$ If you don't give any distances, what kind of answer do you expect from us other than a "it depends on the distances"? $\endgroup$
    – L.Dutch ♦
    Commented Mar 16, 2023 at 16:12
  • $\begingroup$ When you say the two stars are orbiting one another, are they close together and the planet orbiting both of them, or far apart and the planet orbits just the G-type? $\endgroup$
    – Cadence
    Commented Mar 16, 2023 at 16:22
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    $\begingroup$ @Cadence since the planet is tidally locked in the habitable zone, it is obviously orbiting the m type only. $\endgroup$
    – ths
    Commented Mar 16, 2023 at 18:03
  • $\begingroup$ @L.Dutch I just edited the question a little bit. I guess what I'm really asking would be what kinds of distances/luminosity would be required to create a broadly Earthlike day/night cycle. Or is that impossible within this arrangement? $\endgroup$
    – DMacc1917
    Commented Mar 16, 2023 at 18:15
  • $\begingroup$ @Cadence I clarified the question, but the planet only orbits the M-type $\endgroup$
    – DMacc1917
    Commented Mar 16, 2023 at 18:17

2 Answers 2

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I think I may have answered this on my own. The planet's day/night cycle would be governed by how quickly it orbits the M-type and it would have the following phases:

On the "hot" side of the planet

  • Night 1: hot side facing away from G-type
  • Day 1: G-type visible on hot side
  • Night 2: M-type blocks G-Type
  • Day 2: G-type visible

On the "cold" side of the planet

  • Day: cold side facing G-type
  • Night: cold side facing away from G-type

Considering some habitable planets around red dwarfs (like Trappist 1) have very short orbital periods (on the order of a few Earth days), the day/night cycle on the hot side would be odd but perhaps broadly recognizable to humans. On the cold side, you'd have a pretty clear day and night, but they'd be longer than Earth's. Even for a planet with a mere 4 Earth day orbital period, the days and nights on the cold side would be twice as long as Earth's.

Please let me know if I've gotten anything significantly wrong here.

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    $\begingroup$ You seem to look at TRAPPIST-1d. TRAPPIST-1 is slightly larger than Jupiter. Planet d orbits at it at 0.02227 AU, orbital period is 4 days. With these parameters angular size of TRAPPIST-1 as seen from d is less than 3 degrees. Considering short orbital period Night 2 will be ~46 minutes long. Also, "There is no clear evidence that any of the planets have an atmosphere and it is unclear whether the planets could retain an atmosphere given TRAPPIST-1's radiation emission". Planet d is in habitable zone but unlikely to have atmosphere. It seems that this system is more trouble than it worth tbh. $\endgroup$
    – D'Monlord
    Commented Mar 17, 2023 at 14:20
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    $\begingroup$ It is an interesting thought experiment, but does this orbital mechanics significantly affect the plot? At this point your planet is nearly identical to moon of Jupiter if it was much closer to the Sun. $\endgroup$
    – D'Monlord
    Commented Mar 17, 2023 at 14:24
  • $\begingroup$ The tidal locking to the red dwarf is a pretty significant part of the plot, as I want the planet to have the distinct cold and hot sides with that narrow habitable band. $\endgroup$
    – DMacc1917
    Commented Mar 17, 2023 at 14:29
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    $\begingroup$ I see. It looks like a difficult system to me but we don't really know that much about such systems, it is mostly guesses. Very few readers would be pedantic enough to try fact checking the world. I wouldn't. Good luck. $\endgroup$
    – D'Monlord
    Commented Mar 17, 2023 at 15:15
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I would say months/years. G-type must be at least 1 AU away from your planet at any time, preferably much farther, otherwise G-type star becomes a major energy contributor for already problematic planet.

Even if the M-type star is just at 1 AU from G-type star, it makes full circle around it in 1 year. If it is at 4 AU - 2 years, etc. So, rotation of M-type and planet around G-type star can't give it day/nigh cycle, it is just way too slow.

The only hope is rotation of the planet around its axis. But it is also extremely slow (tidal lock), it makes 1 turn around axis in the same time as planet completes 1 orbit around the M-type. No meaningful day/time here either.

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  • $\begingroup$ Check out my answer below. I think I figured it out, but let me know if I'm totally wrong $\endgroup$
    – DMacc1917
    Commented Mar 17, 2023 at 13:21

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