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Several years ago a discovery was made of planet orbiting a star of a binary system (two stars orbiting each other). Since binary star systems are plentiful in our galaxy, I presume we will be discovering even more such planets.

However, as far as I know, no planet has been discovered that orbits both stars of a binary star system. Question, is this feasible and likely that a planet would orbit both stars of a binary system?

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A planet in such an orbit is called a circumbinary planet. Since planetary systems originate from a rotating disk of matter, and since binary stars may also originate that way, the possibility of ending up with two stars and one or more planets all orbiting in a common plane seems intuitively plausible, and some candidates have been reported. The paper [1] says:

Following the first detection of a circumbinary planet with the Kepler space telescope, namely Kepler-16b, eight more binary star systems with a planet on a P-type orbit have been discovered. All these systems show striking similarities. They are all very flat, meaning that the binary and the planet orbit are in the same plane, suggesting that these planets formed in a circumbinary disc aligned with the orbital plane of the central binary. Furthermore, in all systems, the innermost planet (so far only Kepler-47 is known to have more than one planet) is close to the calculated stability limit...

Another theoretical analysis of instabilities in the orbits of circumbinary planets is presented in [2], whcih says:

Consider a planar three-body system of gravitating bodies: a central massive binary and a much less massive particle orbiting around the binary. Thus, the particle’s orbit is circumbinary. ... the circumbinary orbits cannot be permanently circular. This phenomenon provides a natural universal mechanism of internal tidal friction and heating in circumbinary planets (CBP)... In this article, we describe and consider the planetary escape process that takes place as a result of this shrinkage. Indeed, a particle in the slowly shrinking circumbinary orbit enters eventually the chaotic zone around the central binary, and therefore escapes. Once in the zone, the particle escapes inevitably... We show that the effect of tidal decay may explain, at least partially, the observed lack of CBP of close-enough (with periods <$ 5$ days) stellar binaries.

A recent search for circumbinary planets is reported in [3], which says:

We present the full survey results of the Search for Planets Orbiting Two Stars (SPOTS) survey, which is the first direct imaging survey targeting CBPs. The SPOTS observational program comprises 62 tight binaries that are young and nearby, and thus suitable for direct imaging studies... Results from SPOTS include the resolved circumbinary disk around AK Sco, the discovery of a low-mass stellar companion in a triple packed system, the relative astrometry of up to 9 resolved binaries, and possible indications of non-background planetary-mass candidates around HIP 77911. We did not find any CBP within 300 AU...

An older report [4] says:

Ranked near the top of the long list of exciting discoveries made with NASA's Kepler photometer is the detection of transiting circumbinary planets. In just over a year the number of such planets went from zero to seven, including a multi-planet system with one of the planets in the habitable zone (Kepler-47).

Some other recent references are cited in the the introduction of [5], which says this:

One of the exotic type of planetary systems detected by the Kepler mission are transiting circumbinary planets (CBPs), i.e., planets in nearly-coplanar orbits around a stellar binary (and nearly coplanar with the plane of the sky), temporarily blocking the binary’s light and giving a generally complex light curve [references]. Currently, 10 confirmed Kepler CBPs are known in 9 binary systems (see Table 1 for an overview [with references]).


References:

[1] "Migration of planets in circumbinary discs," https://arxiv.org/abs/1806.00314

[2] "Tidal decay of circumbinary planetary systems," https://arxiv.org/abs/1808.02090

[3] "SPOTS: The Search for Planets Orbiting Two Stars. III. Complete Sample and Statistical Analysis," https://arxiv.org/abs/1807.08687

[4] "Recent Kepler Results On Circumbinary Planets," https://arxiv.org/abs/1308.6328

[5] "Stability of exomoons around the Kepler transiting circumbinary planets," https://arxiv.org/abs/1806.06075

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Generally speaking, binaries cannot have shared planets orbiting at close distances. Neither dynamics nor kinematics holds.

If a binary has a common planet, the planet must be far enough, that is, the radius of the circle is large enough to be far greater than the distance between the two stars. This approximates two stars as one star at the center of mass.

The acceleration of the winding provides centripetal force, so the system can be stable. It's impossible to circle two stars at close range.

Each star of the binary can have its own planet, in this case, the planet need to close to its own star and far away to other star.

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    $\begingroup$ Dear Cang Ye. It is usually frown upon to directly copy-paste identical answers. (The problem is if everybody start to copy-paste identical answers en mass.) $\endgroup$ – Qmechanic Jul 10 at 3:51

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