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Let's say there is a planet orbiting at L5 behind a star, which is in turn orbiting a more massive star, like so:

binary star with L5 planet

What is the ultimate fate of this planet? I'm predicting that it will spiral into the star, or end up in some sort of horseshoe orbit. Is this prediction right? If so, why? If not, where did I go wrong?

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The stability of this system depends the ratio of masses of the two stars. If the larger star is more than 25 times more massive than the smaller star, then L5 is potentially stable, and this remains the case even if the planet does not have negligible mass

The calculation of the value is done in detail on physics.stackexchange and there you can establish that the key factor in the stability of L5 is the masses of the major and minor bodies, and not the mass of the Trojan.

If the mass of the central star is more than about 25 times1 the orbiting star, then the planet will be stable. If it is less then L5 is not a stable point, and the planet will tend to move away from L5, and ultimately may collide or be ejected: the behaviour will be chaotic.

1 note $ \frac{1}{2}\left(25+ \sqrt{23 \times 27} \right) \approx 25$ cf. section 5.3 (page 14) here

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  • $\begingroup$ Are there any known examples of such a planet? $\endgroup$
    – slowerthanstopped
    Feb 23, 2021 at 21:48
  • $\begingroup$ Not that I'm aware of. It would be rather rare, and probably hard to detect. $\endgroup$
    – James K
    Feb 23, 2021 at 23:48

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