Can a binary star optically "orbit" a planet? I was watching a science fiction anime from the mid 70's. In it, there was a star orbiting a planet. Since this seemed impossible, and I had never heard of it, I looked on the internet to see if this could be possible in a very unlikely situation or in theory. 

On a similar question on Quora, after many no, no, no, no answers from astronomers/astrophysicists, some of them with very detailed models and theories with different scenarios and equations, one answer read,


  Absolutely.
  
  If there were two stars of identical mass orbiting each other in equilibrium, then you could throw a planet of any size at the barycenter of the two stars. Then optically you'd have not just one, but two stars orbiting a planet. The planet would be equidistant from the two stars at all moments in time, and would experience identical gravitational pull in two diametrically opposite directions and hence never deviate from the barycenter. This system would not be in a stable equilibrium though, and a passing celestial object of any mass could disrupt this system. Hence, this is probably something that is unlikely to be observed.


Is this possible?
Planets don't orbit stars. Stars don't orbit planets.

Whenever there are two bodies bound by gravity, they are both orbiting their common center of mass. For example, both the Earth and the Moon orbit their common center of mass - but that's pretty close to the center of the Earth actually, so it seems like the Moon orbits the Earth.

For a star to seem to orbit a planet, that would mean the planet is much heavier than the star. As far as we know, that's impossible. Their common center of mass would be much closer to the star, so it would seem like the planet is orbiting the star, as usual.



If you have two stars orbiting each other (actually, orbiting their common center of mass) very closely, then you could have a planet circling around both. If the planet is circling around too closely, its orbit would not be stable.

But if the planet's orbit is far enough from the pair of stars, the orbit could be stable a very long time. The minimum distance it at least 2x ... 4x the distance between stars, ideally much bigger. We have discovered such planets. Kepler-47c is a gas giant in the circumbinary habitable zone of the Kepler-47 system.

This wiki page has more details:

https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems



EDIT: If you have a closely-bound binary star, with a planet right in between them, in the barycenter of the two stars, that's not a stable system. Any slight perturbation would pull the planet out of there, and then the net force would be pulling it further out. It would eventually fall into one of the stars. It's not self-stabilizing.

I can't even imagine any conceivable mechanism that would put the planet in there to begin with.