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There is much fiction on binary stars with planets, for whatever reason, and this keeps supporting the popular impression that those systems are frequent. Indeed, the methods for identifying exoplanets and binary stars are somewhat the same. Are there actual observations of a binary star system with at least another planet (habitable or not)?

I know that binary star systems are rather frequent, and I am wondering whether there is any hard-science based statistics on how probable it is that systems of two or more stars have planets? As non-expert, I have the impression that not all configurations of binary star systems seem to be in favor of planets, see e.g. Roche lobe.

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    $\begingroup$ There is both prolific observational evidence of such planets: en.wikipedia.org/wiki/Circumbinary_planet , and theoretical orbital mechanical analysis showing large regions of long term, stable planetary orbits in binary star systems: arxiv.org/pdf/astro-ph/9809315.pdf . So they are highly probably and detected at significant rates. $\endgroup$
    – Connor Garcia
    Dec 16 '20 at 19:24
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    $\begingroup$ The phrase "Circumbinary Planets", I think only applies to planets that orbit around the whole binary star system. I am not sure if there is another name for a planet that orbits only one of the stars. $\endgroup$
    – Connor Garcia
    Dec 16 '20 at 20:27
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    $\begingroup$ @ConnorGarcia A proposal: astronomy.stackexchange.com/a/15003/6 $\endgroup$
    – called2voyage
    Dec 17 '20 at 18:41
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There are only about a dozen circumbinary exoplanetary systems known - the first discovered was Kepler 16b.

The statistics of circumbinary planets is a relatively unstudied area. One of the major uncertainties is the inclination distribution of the exoplanets. If they are confined to the orbital plane of the binary system, then transiting circumbinary planets are easier to detect and so their observed frequency translates to a lower actual frequency than if their orbits are scattered around the binary orbital plane. Armstrong et al. (2014) use the Kepler survey results and find that if circumbinary exoplanets are coplanar with the binary, then the frequency of giant exoplanets within 300 day orbits is about 10%, and consistent with the frequency of similar exoplanets around single stars. On the other hand if their orbital inclinations were random with respect to the binary, then the exoplanet frequency could be a factor of 5 higher.

On the other hand we do know that circumbinary planets around very short period binaries (those with periods less than 7 days) are rare Fleming et al. 2018). This probably has something to do with orbital evolution of the binary system in these cases.

Exoplanets that orbit around one of the stars in a binary system are better studied. If the stars are widely separated then they have little influence on each other or on planetary systems that orbit within say 1/10 of the stellar separation. Exoplanets on orbits wider than about 1/5 of the separation are expected to be unstable.

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The question is one which could have been asked a generation ago by someone who kept up with astronomical discoveries, but not more recently by someone familiar with the search for exoplanets.

In short, the answer is yes, binary systems can have planets and some have been discovered.

I don't know much about theoretical calculations about how probable it is for binary stars to have planets.

Over four thousand exoplanets have been discovered so far, which is a very small sample considering that there are hundreds of billions of stars in the Milky Way Galaxy. Thus the statistics of which percentage of a particular type of stars has exoplanets do not yet have a very large numerical basis.

Exoplanets have been found orbiting both single stars and binary stars.

Some of those exoplanets in binary systems orbit one star in the system, with the other star much farther away, in what is called an S-type orbit. Other exoplanets in binary systems orbit both of the stars at a distance several times the separation of the two stars, in what are called circumbinary or P-type orbits.

PH1-b, or Kepler-64B, is an exoplanet in the quadruple star system Kepler-64.

The giant planet is Neptune-sized, about 20-55 Earth-masses (M⊕). It has a radius 6.2 times that of Earth's. The star system is 7200 light years[10] from Earth.[3]11[12] The planet orbits a close binary, with a more distant binary orbiting at a distance, forming the quadruple star system.

https://en.wikipedia.org/wiki/PH1b[1]

The first exoplanet in a binary system orbiting one of the stars in a S-Type orbit to be discovered is 55 Cancri b, discovered in 1996. It should not be confused with the other star in the system, 55 Cancri B.

https://en.wikipedia.org/wiki/55_Cancri_b[2]

The first exoplanet in a binary system orbiting both of the stars in a circumbinary or P-Type orbit to be discovered 1s PSR B1620−26 b, discovered in 1993 and confirmed in 2003. However, it orbits a pulsar and a white dwarf, not main sequence stars.

https://en.wikipedia.org/wiki/PSR_B1620%E2%88%9226[3]

The first exoplanet in a binary system orbiting both of the stars in a circumbinary or P-Type orbit, with both of the stars being main sequence stars, to be discovered is Kepler-16b, discovered in 2011.

https://en.wikipedia.org/wiki/Kepler-16b[4]

The first binary star with multiple exoplanets to be discovered is Kepler-47, with three planets in circumbinary or P-type orbits around the pair of stars Kepler-47 A & B, in 2012.

Prior to the discovery of the Kepler-47 planetary system by Jerome Orosz, his colleagues, as well as astronomers from Tel-Aviv University in 2012,[7] it was thought that binary stars with multiple planets could not exist.4 It was believed that gravitational perturbations caused by the orbiting parent stars would cause any circumbinary planets to collide with each other or be ejected out of orbit, either into one of the parent stars or away from the system.4 However, this discovery demonstrates that multiple planets can form around binary stars, even in their habitable zones; and while the planets in the Kepler-47 system are unlikely to harbor life, other planets orbiting around binary star systems may be habitable and could support life.4 Because most stars are binary, the discovery that multi-planet systems can form in such a system has impacted previous theories of planetary formation, and could provide more opportunities for finding potentially habitable exoplanets.6

https://en.wikipedia.org/wiki/Kepler-47[5]

As more and more exoplanets in different types of star systems are discovered, statistics about the percentages of stars of various types which have planets will become more and more accurate.

Added 12-17-2020

I forget to suggest that you try to get in touch with astrophysicist Sean Raymond at the PlanetPlanet blog, who has a hobby of trying to make solar systems in various science fiction stories scientifically plausible, and designing imaginary solar systems. I suspect that he would be up to date on the latest theories of planetary formation and how they affect the probability of planets in a binary star system.

https://planetplanet.net/about/[6]

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    $\begingroup$ This answers half the question. Unless your answer is that nobody knows how frequently planets are found around binary stars? $\endgroup$
    – ProfRob
    Dec 17 '20 at 8:15
  • $\begingroup$ @Rob Jeffries I don't know enough to say whether "nobody knows". But I just added a suggestion for someone to ask. $\endgroup$ Dec 17 '20 at 18:20

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