Effects of a binary star system on a tidally locked planet

This is basically a world-building question, but I'm looking for a scientifically based answer, so I'm posting here.

Imagine a fairly earth-like planet- Lets call it Planet X. Planet X has an earth-like compisition, size, gravity, and such.

Planet X is also tidally locked to its sun, Sun 1. It's an eyeball planet, a scorching desert on one side, freezing cold on the other, with a narrow habitable band inbetween.

However, this is a binary star system. There are two suns- Sun 1 and Sun 2. Planet X orbits Sun 1, with Sun 2 orbiting at a distance. (or, possibly, Sun 1 orbits Sun 2.)

My question is this: Is there any way for this to happen, that Sun 2 is close enough to melt (some) of the ice on the backside of Planet X when it faces Sun 2? That this could cause a weather cycle on Planet X, when the floods come from the locked away ice melting?

The "three-body" problem and its stability are still an unsolved problem in general. There are papers that do long term integrations of planet orbits in binary systems to study their stability - an example is De Cesare & Capuzzo-Dolcetta (2021).

Whether there is stability depends on the relative masses of the three components, their semi-major axes and the inclinations of their respective orbits.

A general rule-of-thumb is that instability will ensue if the ratio of the separations of the outer to the inner system is $$<5$$. In other words, the binary star separation needs to be more than 5 times larger than the planet is from the star it is orbiting. But that factor of 5, could be as much as 10 in some circumstances.

The flux received by the planet from the "secondary" would therefore be at least 25 times less than that received from the "primary", if those stars had equal mass.

Your best bet to make the numbers work is to switch the stars around so that the planet orbits a low-mass star and that pair orbits a high-mass star. The reason is the very non-linear relationship between luminosity and mass ($$L \propto M^{3.5}$$) for main sequence stars. That would mean the tidally-locked star-planet system could be relatively distant from the more massive star but still receive a considerable flux.

e.g., Suppose planet X orbits a very low-mass star like Trappist-1. If its eccentricity is low and it was say 0.02 au from the star then it might be tidally locked, with one side "fried" (e.g. Makarov et al. 2018). There would then be no problem for that system to orbit a solar-type star at a distance of 1 au, because the ratio discussed above would be 50.

• Ok, I think I understand. So is this feasable: Sun2 is a sun like ours, with Sun1, a red dwarf, orbiting it- with Planet X orbiting Sun1? Going from the Trappist-1 system, (specificially TRAPPIST-1E), Planet X would only be 0.029 AU from Sun1(a red dwarf with 9% of earths mass), with a 9 day 'year'? Could Sun1 orbit Sun2 close enough that the light from Sun2 would have a noticable impact on the planet? Aug 10 at 22:13
• @user1642529 Yes, there would be no problem having a Trappist-1 like system orbiting a solar-type star at 1 au. Aug 11 at 6:28
• @ProfRob A minor correction regarding TRAPPIST-1. In the system, the innermost planet is likely to be not synchronised but pseudo-synchronised. Sorry for this excessive pedantry :) Aug 16 at 23:23
• @Michael_1812 Trappist 1 is a star. " A close in planet to a very low-mass star like Trappist-1 might be tidally locked." is what I wrote. The inner planet of Trappist-1 also isn't at a separation of 0.02 au. Aug 17 at 6:24
• @ProfRob Yes, it might be tidally locked -- but it may also be in a pseudo-synchronous rotation state. For details, please see iopscience.iop.org/article/10.3847/1538-4357/aab845 Aug 17 at 20:42