Alpha Centauri A and B orbit from 11.2 AU (roughly the distance to Saturn) and 36 AU (Pluto).
70 Ophiuchi A and B have a similar orbit, estimated at from 11.7 AU to 35 AU.

Cygnus A and B are a little more distant, ranging from 48 AU to 84 AU.

I guess my question is; How likely are there to be stable orbit planets around binaries like Alpha Centauri (A or B) or 70 Ophiuchi given their proximity?

For example, let’s assume that Sol has a similar companion, hence we have no planets beyond Mars (for a start). How stable, or how severely affected, would the orbit of Earth be?

There are three stable configurations: The two suns can be very distant, with a planet orbiting one of them at a distance much less than the suns’ separation, and essentially unaffected by the other. The two suns can be very close, with a planet orbiting both of them, at a distance much greater than the suns’ separation. Or, if the two suns have significantly different masses, you can put a planet in one of the Trojan points, forming an equilateral triangle. Anything else will be unstable, and end up in a big mess.

By the way, I wouldn’t call alpha Cen a “close binary system”, except in comparison to something like Proxima Cen. Now, W UMa, that’s a close binary.

There are some exceptions to this based upon exact orbital resonances; see this site for some animations of stable planetary orbits that fall between two close-orbiting binaries. However, notice that the webpage author specfies that he starts with circular orbits for the stars, whereas any real-world orbit will have some eccentricity which creates enough variation from period to period that you’ll get some kind of positive feedback to the system. Most families of resonance orbits that are not periodically uniform are at best metastable and are easily perturbed by any outside influence, or even from cumulative differences between orbits. The simulations I’ve run of multi-body systems in Matlab and Mathematica in which I didn’t truncate floating point errors (small cumulative errors in calculations) tend to indicate instability of most 3-body systems over the long term unless you have very large differences in the relative masses (M1>>M2>>M3… where >> is a ratio of more than 25:1) or the difference between apoapsis of two bodies is at least three orders of magnitude less than periapsis of the third to the first two, or in other words, one body never transitions between the sphere of influence of the other two bodies (which is just another way of stating what Chronos just wrote). For the first case (where M1 and M2 are stars and M3 is a planet in orbit of M2, which both orbit M1), an approximate distance ratio is R[sub]23[/sub]/R[sub]12[/sub] ~ [M[sub]2[/sub]/M[sub]3[/sub]][sup]2/5[/sup]. For the second scenario, if the sum of the specific orbital energy of M3 about M1 and about M2 at either conjunction (alignment of major axes) is positive or nearly positive, then it is unstable or metastable (easily perturbed).

As a point of note, “a big mess”, while being chaotic (i.e. the system dynamics change rapidly and unpredictably in their specific motions) is not necessary unstable. It is certainly possible to have a multi-body system in which orbits change, but the specific orbital energies and energy transfers are low enough that the system is in a regime of dynamic stability over a long period of time as long as it is not subjected to outside influences. This is especially true if you have opposing forces (like gravity and electrodynamic forces) that that can damp out large impulses, which is why the atmosphere stays stuck to the Earth rather than trailing off behind the planet. However, if you have only a single attractive force and two or more bodies of similar mass that are close enough to dominate system dynamics, they’ll eventually couple together to throw smaller objects outward.

So then, with Alpha Centauri A being 1.1 times the mass of Sol, and Alpha Centauri B being 0.9 Sol, and an eliptical orbit ranging from 11.2 to 35.6 AU; the odds of stable orbit planets within 1-2 AU of either star is pretty low, right?

It should be, although I see a bunch of pop sci articles online that posit that the Alpha Centauri system could develop stable Earth-sized worlds. I’d have to go back and model the system specifically to see if there is a stable orbital regime in zone in which liquid water could exist, but even if it did, I would expect perturbations from the closest approach of the two stars would make even a sustainable orbit deviate significantly from a conic.