In a certain star system, Star B orbits Star A. Planet C lies in between the two stars and also orbits Star A. Planet C and Star B share an orbital period, and are at roughly the same places in that period. Consequently, there is no night on Planet C. As Star A sets, Star B rises.
Is such a configuration possible (and if so, plausible)? Could such a planet support life, or would the constant exposure on both sides to radiation make the planet too hot?
The short answer is: No.
To have the same orbital period, they would need to be in the same orbit.
If the planet was in L-4 or L-5 (see Wiki: Lagrange Points)of Star B which is orbiting Star A, there would be very little night but there would still be night.
the only way for this to work is for the planet to be in L-1 between the two stars. the problem is that L-1 is not a stable location and the planet would have drifted off of that position and begun a wild ride orbit long before life could have developed.
Also, I can't imagine that L-1 would be in the habitable zone in any case. There may be a way to jigger the star masses to make that work but the planet still wouldn't stay where it belongs.
The only way for it to work is to find a planet that just happens to be in the proper position now but will soon leave it. Not much chance of life there though.
You could try an Earth-like moon orbiting a gas giant. The gas giant may be reflective enough that it will be almost as bright as the direct sunlight.
Edit:
One possible solution is that the two stars are roughly the same mass and the planet orbits the center of mass of the two stars. I still don't think that would work but someone might be able to math that out. It seems to be too much like balancing on a pin.
"Planet C and Star B share an orbital period" - unfortunately, this won't work. The orbital period of an object is proportional to how far away it is from the object it orbits (more specifically, the oribtal period squared is proportional to the cube of the distance). That means if star B is farther away from A than C is, B will orbit more slowly.
Habitability is much more plausible - the habitable zone of a star depends on how bright the star is. If you somehow had a star on each side of the planet, you'd just need the planet to be at the outer edge (or maybe a little outside it) of the what the habitable zone would be for each star by itself. There is some point at which the stars will be far enough away that the planet is kept at an acceptable temperature only because there is no night.
SHORT ANSWER:
No never, not ever!
LONG ANSWER:
One of the hardest problems of all when designing a fictional solar system is to create one where a planet is warm enough for life, and is habitable for humans, and has either eternal night or eternal day all over the planet's surface.
It is really easy to design a solar system where the closer planets are tidally locked to their star, meaning that their rotation periods and orbital periods are the same. Thus one side of such a planet will always face their star and have eternal day, while the other side of such a planet will always face away from it's star and have eternal night.
it is believed that there should be many billions of such planets in our galaxy. It is uncertain whether such planets could be habitable, but if a significant percentage of such planets are habitable then they should be very numerous in our galaxy.
https://planetplanet.net/2014/10/07/real-life-sci-fi-world-2-the-hot-eyeball-planet/1
http://nautil.us/blog/forget-earth_likewell-first-find-aliens-on-eyeball-planets2
So if you want one side of your planet to have eternal day while the other side has eternal night, that's really easy.
If you want eternal night all over your fictional planet, that can be easy. You can just make your fictional planet a rogue planet that is not in any solar system but is in interstellar space instead, and thus is lit only by dim starlight in its eternal night. Astronomers believe there are many billions of rogue planets in our galaxy, far from the heat of any star and thus with temperatures just a little bit above absolute zero.
Temperatures just a little bit above absolute zero! What if you want to have Earth like life on your planet requiring Earth like temperatures, large multi celled lifeforms, or intelligent natives, or a world habitable for Earth humans? Then you are out of luck with a rogue planet, unless you can imagine or learn about any theories about how a rogue planet could have eternal night and also Earth like temperatures.
https://planetplanet.net/2015/06/04/real-life-sci-fi-world-8-the-free-floating-earth/3
Of course any planet, whether in a solar system or a rogue planet, could have eternal day if an advanced civilization decided to make that happen. They could put a number of gigantic space stations in orbit around the planet. Each space station would have countless gigantic fusion power generators generating vast amounts of electricity to power countless giant lamps pointed at the planet. If enough of those "Sun Satellites" were in orbit around the planet at least one at a time would always be visible from every place of the planet's surface, and every place on the planet's surface would have eternal day.
Isaac Asimov wrote a famous science fiction story "Nightfall" where the planet Kalgash had eternal day over its entire surface. There were six stars in the Kalgash system, and so there were always at least one or two of them above the horizon everywhere on the planet.
Except that once every two thousand Kalgash years the orbits of Kalgash and the stars would put only one star on one side of Kalgash and the other five stars on the other side of Kalgash. And at the same time, a large moon of Kalgash would eclipse the only star on that one side of Kalgash and plunge that side of Kalgash into the first darkness in 2,000 years. And the eclipse would last so long that the rotation of Kalgash would turn the whole half of Kalgash that had been facing the other five stars at the beginning of the eclipse to face the one single star that was eclipsed, and so that other side of Kalgash would also face hours of darkness for the first time in 2,000 Kalgash years.
Would the Kalgash system be possible?
Astronomers have identified many multiple star systems, including some, like Castor, that have six stars, and even two with seven stars. s astronomers have learned a lot about the orbital dynamics of multiiple star systems and the possible orbits of hypothetical habitable planets in them, etc.
PlanetPlanet is a blog 'about where planets come from and where they are going". And it has a section where the blogger Sean Raymond discusses the plausibility of various solar systems in science fiction. And that includes two posts where Raymond tries to design a workable version of the Kalgash system:
https://planetplanet.net/2018/02/02/real-life-sci-fi-world-11-kalgash-a-planet-in-permanent-daytime-from-asimovs-nightfall/4
https://planetplanet.net/2018/03/21/asimov-kalgash-take2/
And you can see that Raymond had a lot of trouble designing a realistic Kalgash system.
And you could take some of his proposed solutions as the basis for a habitable world which is always in eternal day all over the planet. Even though it seems extremely improbable for some of those solar systems to form naturally.
See answers to this question: Is a habitable planet in a sextenary star system possible?5
Or you could have your fictional solar system happen by chance to pass through a "stellar nursery" as both orbit around the center of the galaxy. It could take your solar system many thousands or even millions of years to pass through the "stellar nursery". A "stellar nursery" is a giant nebula of dust and gas that is condensing to form stars and solar systems. And as the stars form and begin to shine some of their light should be reflected off dust and scattered in all directions, and some of their light should make the gas emit its own light.
And if your solar system has been light years deep within that "stellar nursery" for thousands or millions of years, maybe the light from all around the solar system might be intense enough to make a difference on the planet. It is possible that, for example, if the background light from the gas and dust in the dense nebula is at least one percent of the intensity of the light of the planet's star, when the planet's star sets the sky won't get as dark as it would on Earth and the sky will remain blue, though much darker, and other stars will not be visible, and it will seem more like a very dim Earth day than like an Earth night.
But I have not calculated whether it would be possible for such a dense and bright nebula to illuminate the planet that much and I don't know if that is possible.
Another possibility is that a globular star cluster is passing though the galactic disc, and its orbit and the orbit of your solar system happen to intersect. It could take your solar system many thousands of years to pass through the dense central region of that globular star cluster, and for all that time the combined light from the thousands of closest stars might possibly be intense enough to make the night on your planet look more like a very dim day on Earth.
But again, I have not made calculations to see whether the light in the core of a globular star cluster could be intense enough. Possibly you could get some other user here to do such calculations for you.
I think yes.
Start with Earth (planet C) and sun (star A). You might make Star A a little more massive and move Planet C's orbit out some.
Star B is a little star - equal to 75 Jupiters. It is far away, in approximately the orbit of Neptune. It is more massive than a planet and so it can (must) move much faster than Neptune does. Also you need it to move fast to complete its orbit in the time planet C completes its much smaller circle. At the same time you need it to stay far away from Planet C or its proximity is going to perturb the planet's orbit.
A thing to remember about orbits: X does not orbit Y. X and Y orbit their common center of gravity. When X (e.g. star) is of hugely greater mass than Y (e.g. planet) then their center of gravity is usually still within X. But if you have 2 stellar mass objects then the center of mass might not be within one of them.
