# What would the temperature variation of an outer orbiting planet look like in a binary system?

I am creating a binary star system with a likeness to the Alpha Centauri system with an outer orbiting planet like Proxima Centauri. On an earth similar planet what would the temperature variation look like?

Proxima Centauri orbit Alpha Centauri $$\alpha$$ and Alpha Centauri $$\beta$$ with a semi-major axis of 8700 AU (plus or minus some large estimate) and an eccentricity of 0.5 (also with variation). Let us take the minimum semi-major axis and minimum eccentricity from Wikipedia, 8300 AU and 0.41, respectively.

The ratio of the orbital distance of periapsis (the nearest point to the star) versus distance at apopsis (farthest point from the star) is given by $$\frac{r_p}{r_a} = \frac{1-e}{1+e},$$

While the semi-major axis is

$$a = \frac{r_p + r_a}{2}.$$

Therefore, we solve for periapsis as 4897 AU, and apoapsis as 11703 AU.

The luminosity of the two stars of Alpha Centauri, considered together as a point source, which is appropriate at that distance, is 2.02 times that of the Sun.

Given the average incident radiation on Earth is about 1361 W/m$$^2$$. For this planet, in the place of Proxima Centauri, the average incident radiation at periapsis is 0.115 mW/m$$^2$$, and at apoapsis 0.00201 mW/m$$^2$$.

In both cases, this is significantly less energy than Pluto gets from the Sun, and pluto is a dead ball of ice. Therefore, your planet, in Proxima Centauri's orbit, is a dead ball of ice.

For such an outer orbiting planet, the two suns will be close enough together relative to the planet's distance that the temperature variation will be pretty much the same as in a one-star system. There will be dips in temperature if it is aligned so that one star eclipses the other, and the days will be somewhat longer, but other than that it's pretty much the same. Axial tilt can be anything, but such a planet would most likely have a much more eccentric orbit than Earth.

EDIT: Originally I thought Alpha Centauri A and B were closer together (fractions of an AU), but after reading David Thornley's comment I thought to check Wikipedia and in fact their orbit makes their separation range from 11.3 AU to 35.6 AU. That's roughly the orbits between Saturn and Neptune. Their orbital period is 79.91 years. This means we have a planet that orbits a sun-like star with a smaller star orbiting between Saturn and Neptune (think 2010).

Alpha Centauri B's effect on the planet should be "small" but not negligible. With variations in the order of ~80 years. I have no idea how stable the planet's orbit or habitable zone would be either.

Proxima Centauri's effect on the planet(s) should be very small (as α CVn♦ illustrated in the question comments) and more subtle: with variations in the order of 1/2 million years (its orbital period). Note that I'm assuming David Thornley's configuration in the comments where the planet orbits Alpha Centauri A.

In general scenario: According to this Youtube video where the narrator is using Universe Sandbox 2, my original idea of the Alpha Centauri star system: it would quite a bit. In their simulation, seasonal temperature changes occurred almost daily. This is due to [each star] approaching and receding the planet as they orbited their common barycenter. Because the star's separation was so small, their orbital period was very short, hence the quick but sharp jumps in temperature.

I would recommend watching the video and trying your own simulation, keeping in mind that creating stable orbits can be rather difficult (as per the video). The simulation won't be 100% accurate but it can provide an idea of what can and cannot happen in such a system.