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So the idea is you have an Earth-sized planet with a moon orbiting an M-class star (let's give it .25 solar masses) at 0.2 AU, with an orbital period of 63 days and a 24-hour day. Its orbit has an eccentricity comparable to Mercury, because it was captured by the M-class star early in the solar system's history. The M-class star in turn orbits a K-class star with .75 solar masses at 1.54 AU, with an orbital period of 566.5 days.

Is there a way for me to calculate the period during which the K-class star is visible from the planet's sky, sort of like a secondary "day?"

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    $\begingroup$ Is the 24-hour figure the sidereal day or the mean synodic day? How precise do you need the result to be? Because, on first approximation, the day is 24 hours for both stars, since it mainly depends on the planet's rotation $\endgroup$
    – Prallax
    Aug 21, 2022 at 6:55

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While the situation is not exactly the same, the complexity is similar to calculating when the Moon is visible from a point on Earth.

Just as the Earth rotates once a day (or 23hr56 min), and the moon rises about 50 minutes later each day, so after a month the moon is back to where it started. 50 minutes × 29 (days/month) = (roughly) 24 hours.

So the K class star would rise a little later (or perhaps a little earlier) each day, relative to the stars or relative to the rising of the M class star.A very rough estimate is about twenty minutes on average (20min×63=24hr)

To calculate it more accurately your planet's astronomers would have to develop something like our "lunar theory" to describe the various perturbations of your planet's orbit.

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