How would one work out the habitable zone of a binary system

[Bluewolf]

I've been using Universe Sandbox lately and I've been a fan of creating binary systems. I must ask though, how would one work out the habitable zone of that system? I am not that knowledgeable with this part of astronomy, thus why I am asking. Though I want kinda want to know for any star combination, lets just say two stars like Alpha Centuri A and Alpha Centuri B.

[Simon_Jester]

The first step is to work out the zones where stable orbits exist. That's the really tricky bit about binary star systems- whether a planet can follow a stable orbital track in close orbit around one of the two stars, without the other star's gravity distorting the orbit and causing its motion to become chaotic.

For Alpha Centauri, this is fairly viable: the two major suns are far enough apart to permit stable orbits in the inner system of each star. There are other problems, but they're probably beyond the scope of what an amateur can find the resources to take into account.

http://en.wikipedia.org/wiki/Alpha_Cent ... of_planets

[Eternal_Freedom]

There is the other extreme, where the two stars are close enough that you can treat them as one star for orbit/habitable zone.

[Feil]

Average[r12]*I1 + Average[r22]*I2 ~= 1AU2*I0

rn is the distance to star n. In is the luminosity of star n. I0 is the luminosity of Sol.

You can take the average distance r^2 by integrating around the orbital path in polar coordinates to compute the area and dividing by pi.

[Simon_Jester]

There is the other extreme, where the two stars are close enough that you can treat them as one star for orbit/habitable zone.

To make that work they have to be very close together relative to the orbital radius of the planet. I'm dubious of whether that's practical, unless someone's done the simulations on it.

[Eternal_Freedom]

I cannot recall the exact details, but I think it works as long as the stars are hotter and brighter than the Sun.

That way the habitable zone orbits are far enough out.

Of course, the OP did not specify what stars are in this binary. If they aren't both G-type main sequence stars things can get all sorts of crazy

[Feil]

There is the other extreme, where the two stars are close enough that you can treat them as one star for orbit/habitable zone.

To make that work they have to be very close together relative to the orbital radius of the planet. I'm dubious of whether that's practical, unless someone's done the simulations on it.

If one takes as the condition for the life zone that the average radiance must be earthlike (the formula I gave), then to count the stars as one star with the total luminosity of both stars, the only necessary condition is that the path of the planet relative each star must always be concave-down. Put otherwise, the orbit of the planet must never cross the orbit of the secondary star around the primary. (It doesn't matter which star you take for a reference point.) In short, it must be an outer planet with a stable orbit, or you cannot treat the stars as a single star (although you could still conceivably be in the life zone). This is trivial to check, either by graphical inspection or by computing the second derivative.

If one requires that the radiance must be earthlike at all times, then they do indeed have to be so close together relative the orbital radius that you can treat them as one star for all purposes, or the planet must be in one of the lagrange points of the two stars. My formula will still work: it is necessary, but not sufficient, in this case. I cannot conceive the possibility of another orbit which would satisfy the earthlike-at-all-times condition but a lagrange point or a rplanet>>rsecondary star, but I have no proof that it doesn't exist.

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Average radiance being earthlike gives you a planet which can have surface water and will necessarily have surface water sometime every year, absent other factors (like passing close enough to a star in its orbital period to burn off all the atmosphere, for instance).

Radiance being continuously earthlike gives you a planet which can sustain surface water indefinitely - again, in the absence of other factors. (The moon, for instance, is a lifeless rock under hard vacuum, always has been, and likely always will be.)

Neither is anything resembling a guarantor of planetary habitability or even the presence of liquid water. Nor is being outside of the life zone an indicator of non-habitability or the absence of liquid water. Tidal heating, for example, is responsible for the widely-theorized existence of liquid water below the surface of Europa, far outside of Sol's life zone; and Europa may very well support some variety of life.

[Feil]

I just noticed that the exponents in my first post in this thread have the wrong sign. They should all be r-2, not r2

[Omeganian]

Do trojan points work?

[Feil]

Yes - the Trojan points, also known as the L4 and L5 Lagrange points, should work.