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So I want my planet to have six suns, but I'm not sure how far apart these stars would have to be from each other to not produce adverse effects on the planet that would prevent life from developing. What's a safe distance between each of the stars that would result in a sort of eternal daylight on this planet without rendering it uninhabitable in the long term?
EDIT: The central mass around which these stars orbit is a stellar mass black hole equivalent to about 315 suns.
Real (as it stable enough to be around for us to see) systems are hierarchical binary. You can have two stars close together, another pair close together, and the two pairs forming a higher-order binary.
The problem with a 3rd order would be how much room it needs. The tidal forces of other stuff in the galaxy would tear it apart, if not simply having other stars in the stream get closer than the needed separation.
Consider our nearest neighbor, Proxima. It is not known for sure whether it's attached to the Alpha Cen binary, as the measurement is right on the edge. At the best, it's a temporary attendant, which travels along with it for a fraction of a galactic orbit before deciding it likes another yellow star better and heading off with new partners.
You might consider masses of different size scales: several stars around a black hole, or some big stars and some tiny stars. But big stars are short-lived, so you really can't have anything bigger than our sun for your purpose.
If you want no-night, why not just be in a globular cluster? The night is a bright as our day, with thousands or millions of stars close together!
For eternal daylight, you don't need 6 stars. You only need 2 of them, on opposite sides. This would have full daylight on the sides directly facing the stars and a circle of twilight. If the stars are present directly in front of the poles, then the equator line would be the twilight circle. You might think that the twilight circle would get less sunlight than the day-zones, but the difference would be small, considering that the twilight circle would be getting light from both the stars as opposed to light from single star for the daylight zones. If you place the planet directly in the center of the stars, it is very likely that it would get tidally locked with both the stars.
You can also have 3 stars with the planet between them in triangular-planar form. But this will make gravitational matters very complex and the whole system would probably decay in a couple million years, with horrible consequences for the planet.
Anything with more than 3 stars would be 100% unstable for the planet. It would either be ripped apart, spiral into a star, or be shot out of the system and become a rogue planet.
It all depends on the geometry and the types of stars.
Castor is a 7 star system. It has 3 sets of stars. Castor A and B both have a star somewhat larger and brighter then our sun and dwarf companion. Caster C has only dwarf stars.
Castor Aa and Ab are separated by about 0.022 AU, Castor Ba and Bb are separated by about 0.03 AU. Since both A and B are brighter than our sun (36 times for A, 14 times for B), the habitable zone would be around 6(A) or 4(B) times further away than our system. I.e., a habitable planet would orbit so far away from the Aa Ab pair, that the orbit would be stable. At worst, every few days, the occlusion of the primary by its dwarf companion would result in a dimming as the bright star was obscured.
C is so far away (1100 AU) and dim, that it would nothing more than a star in the night, albeit planet like in that it orbits in about 14,000 years.
A & B have orbit each other in 467 years at a distance of about 110 AU. This means that if you are at B looking A, A would be about 323 times dimmer than our sun from the earth which is also about 1200 times brighter than our moon. So, when you planet is between A and B, the nighttime sun would still be quite bright.
If you construct a rosette of 6 equal suns, you can theoretically orbit at an appropriate rosette in an entirely stable manner with normal day and night cycles. Since each sun would fall in Lagrange points, the rosette should be stable over long time periods even with minor perturbations coming from planets or other stars 110 AU away. With a large central black hole, the rosette is even more stable, though orbital speeds of everything else is necessarily considerably faster.
For many years I would have answered no. It is perfectly possible for a planet to orbit around all six stars in a sextenary system. And it is perfectly possible for a planet in a sextenary system to orbit around one or two of the stars and be in the habitable zone.
But it is impossible for any normal star system to have enough stars close enough to a planet and in enough different directions for every part of that planet to have eternal day.
It is easy enough for a planet to have one hemisphere in eternal light and another hemisphere in eternal darkness.
It is easy enough to design a solar system where every part of a planet is in constant brighter or dimmer light from nearer or farther stars in that system for decades, centuries, or millennia of Earth time. Time enough that it might have lasted for all of recorded history on that planet, until it begins to change.
But it is impossible to design a star system where a planet will have constant daylight for millions and billions of years. No possible multiple star system can exist where a planet has eternal light all over its surface for billions of years, long enough for intelligent life to develop.
Or so I thought, until reading a couple of posts at Sean Raymond's site PlanetPlanet.
https://planetplanet.net/2018/02/02/real-life-sci-fi-world-11-kalgash-a-planet-in-permanent-daytime-from-asimovs-nightfall/1
https://planetplanet.net/2018/03/21/asimov-kalgash-take2/2
So he has figured out a plausible way for a planet to have eternal light for billions of years.
Binary systems can be stable, but I'm not even sure this would allow a much smaller third body (a planet) to exist in a stable orbit for long enough periods to maintain a stable atmosphere required for habitable conditions. Even in the Castor binary system which the other answers cite, the distance between the stars is larger than 10 light years, which probably means that a body orbiting both stars's center of gravity will not be habitable, and planet that "switches" between orbits will have periods of having a single sun and a secondary star indistinguishable from the rest followed by periods of intense non-habitability as it is far away from both (so no double sun like in Tatooine).
Systems with three stars or more are not stable (except for in some very restricted conditions for the three body problem), meaning that the movement of the objects is chaotic and will eventually result in either a body leaving the system or crashing into another -- so not the best for a habitable planet.
The binary system with a planet is technically a three-body problem (the two suns and the planet)
In fact, this is the plot of a pretty good Chinese science fiction book, called The Three-Body Problem
