For a gas giant to have a habitable Earth-like moon, in a P-type circumbinary orbit with the stellar classes of the 2 stars being A7 type III and F2 type IV separated by a distance of 2AU to 3AU, what would be the minimum AU of the gas giant from the binary to support that Earth-like (tropical) moon?
I should comment on Foo Bar's answer but I can't log in to my account for some reason and have to act like a guest.
Foo Bar may be wrong to suggest that a habitable planet could have a Trojan obit around two stars.
It is possible for a habitable planet to be the object B in a Trojan system, because an asteroid in a Trojan obit relative to Earth has been discovered, as well as Mars and Venus Trojans.
But is it possible for a habitable planet to be the object C in a Trojan system with two stars as objects A and B?
There is a maximum mass for a star to remain on the main sequence long enough for its planet to become habitable.
There is a minimum mass for a celestial body to have to be classified as a star.
And I believe that the maximum mass is less than ten times the minimum mass.
Look at the relative masses of the three bodies A, B, and C in known examples of Trojan orbits in our solar system.
Take Trojan asteroids associated with planets. The Sun, A, is hundreds or thousands of times as massive as the planet B, while the planet, B, is many thousands of times as massive as any of its Trojan asteroids, C.
In the Saturn moon system Saturn, A, is thousands of times as massive as its moons B, Tethys and Dione, and Tethys and Dione are each many thousands of times as massive as their Trojans C, Telesto & Calypso, and Helene & Polydeuces.
Thus in the examples of Trojan orbits in our solar system, A is many times as massive as B, and B is many times as massive as C.
And I read somewhere that it is dynamically necessary for there to be such mass ratios between A, B, and C in a Trojan system.
There are some experts in celestial mechanics in this board. Perhaps they can demonstrate whether it is possible for a habitable planet to be object C in a Trojan system with two stars as objects A and B.
Obviously, it depends on the masses of the two stars, and the distance between them. Let's assume both of them are Sun-sized or smaller, and the gas giant is Jupiter-sized.
If the stars are very close together, perhaps 0.1 AU, then you could have a stable planetary orbit at 1 AU without any trouble.
Another stable option is if the stars are 1 AU apart, and the planet is at the shared L4/5 point, forming an equilateral triangle. Technically this is a P-type orbit, since the system center of gravity is closer to the stars than the planet.
Other arrangements would probably require the planet to be farther away, otherwise it would risk being perturbed by the closer star when the three bodies are aligned.