# How do you calculate length of day in a multiple-star system?

How can you calculate the length of day for a planet (given a particular latitude and longitude on that planet) orbiting multiple stars?

What about a planet that is orbiting one star, but that star has another star orbiting it?

• Off the top of my head: even in a multiple star system, the stars are far enough apart that only one can really be considered a planet's "sun". The others would just appear as bright stars, but much fainter than our own moon.
– user21
Commented Jun 21, 2015 at 16:40
• en.wikipedia.org/wiki/Nightfall_(Asimov_short_story_and_novel) Commented Jun 23, 2015 at 21:43

## 1 Answer

Are you asking how to measure a day for an exoplanet? Because that is pretty hard given that they are far away and hard to image their surfaces. But, if you are asking how to define the day, say for simulations, then that is pretty easy, actually. From Wikipedi's entry on Day.

For a given planet, there are three types of day defined in astronomy:

stellar day - an entire rotation of a planet with respect to the distant stars

sidereal day - a single rotation of a planet with respect to the vernal equinox

mean solar day - average time of a single rotation of a planet with respect to the sun as the central star

The first two have to do with the spin of the planet with respect to the remote stars and therefore the complexity of the stellar system does not enter, to first approximation. The 'stellar day' depends on how much time it takes for some remote galaxy, for instance, to come back to the same point in the sky as seen by an observer on the planet. The 'sidereal day' depends on how much time it takes for the direction that is at the extension of the equator of the planet with the plane of the orbit (pick 1 of the 2 points) to come back to the same point in the sky. So presuming that the spin axis and orbital plane are not changing much (and conservation of angular momentum law helps here), this is all well defined. If the planet's orbital plane is changing rapidly then there simply is no well defined sidereal day, but still there would be a valid stellar day.

The 'mean solar day' is the average time it takes for the nearest star to return to the same spot in the sky. Generally, even in a binary or triple system, the planet is close enough to one that this is well defined. This is because, if not, the planet is probably not in a stable orbit and will quickly leave the system. However, if the stars are in a tight configuration and the planet's orbit is wide and far enough away from them, it could be stable. Then the simplest definition for 'mean solar day' would use the center of mass of the stellar system instead of the nearest star (which would change over time).

There are a few stable complicated orbits, like a figure 8 orbit, where the planet can switch back and forth between two stars. Unlikely, but theoretically possible. In such a case, the mean solar day is ill-defined, but the stellar and sidereal day are well defined.

• I was hoping for a way of calculating the length of day (defined as being the time when a sun is above) in minutes or hours for a given position on a given planet. Commented Jun 22, 2015 at 20:43
• So you want the daylight duration, not the length of a day. A formula for the general case would be immensely complicated and the answer would be different each day. For instance, if the orbital period is shorter then the rotation period, this can be really hard. A precise solution would require a full up simulation. There are approximations that can usually be made, but they differ depending on the particular situation. Commented Jun 23, 2015 at 3:14