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If orbiting in the L2 lagrange point of a large planet (doesn't actually have to be a hot Jupiter, just large enough to fully shadow the smaller planet), how close could you get to the host star before the planet is no longer habitable?

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If possible, I'd like to ignore any stability issues. The L2 lagrange point of Earth is far too unstable for a natural body; the satellites we put there must regularly make slight adjustments to stay in place. I'm not sure if substantially increasing the size of the large planet would help with stability (or even make it worse) or if there is some way to mitigate this, but that is another issue.

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  • $\begingroup$ Essentially, what you're asking is: 'Ignoring the problem of stability, is there ever a set of circumstances that an object at a planet's L2 point would be naturally inhabitable?', is that correct? $\endgroup$ – Halfthawed Jan 6 at 20:29
  • $\begingroup$ To satisfy your secondary curiosity: The stability issue shouldn't be any different based on the size of the satellite in L2. The engineering difficulty of artificially keeping the object in orbit will be scaled up... but the stability will be the same. (For the curios, that stability is like a beach ball resting on a ridge between mountain peaks... It won't go forward or back, but it will go to one side, and even an inch to the side will be quickly amplified.) Addressing the primary question: If L2 is in the Goldilocks Zone, you're fine. $\endgroup$ – Ghedipunk Jan 6 at 20:36
  • $\begingroup$ @Halfthawed I'm making the assumptions that it is habitable at some distance and that the shadow of the larger planet will provide some cooling, or "shade" if you will. Could it be closer to the star than the habitable zone and if so, how much? $\endgroup$ – WillRoss1 Jan 6 at 20:41
  • $\begingroup$ You're actually asking two questions here, I think. Question 1 is how close can the Gas Giant be to its parent star before proximity to the star creates problems for the Gas Giant. Question 2 is whether a planet in an L2 point like this can be habitable in the first place. It won't be getting any energy from the star at all, so it will only be habitable if it's getting enough energy directly from the Gas Giant to support life. $\endgroup$ – Morris The Cat Jan 6 at 20:46
  • $\begingroup$ @MorrisTheCat Problems for the gas giant aren't a concern, just for the smaller planet. Of course, being habitable in the first place is a prerequisite for finding the point at which it is longer habitable. You bring up an excellent point about receiving no energy from the star. If the larger planet blocks all the star's light and heat, does that mean (all other things aside) the smaller planet would be an ice world regardless of how close the two got to the star? Or would there be a way for the larger to radiate enough energy to sufficiently warm the smaller? $\endgroup$ – WillRoss1 Jan 6 at 21:40
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If we focus on stellar radiation issue only, then the answer is "very close".

Two things are at play here - distance to Lagrangian point L2 and size of planetary shadow cone. Depending on our play with system parameters, the L2 moon can be either fully or partially in planetary Umbra, or in the Antumbra. Unless the moon in completely inside the umbra, we can declare in potentially habitable.

Distance to $L_2$ can be approximated as:

$$r \approx D \sqrt[3]{\frac{M_p}{3 M_s}}$$

where D is the distance between planet and star, $M_p$ is the mass of a planet and $M_s$ is the mass of a star.

Distance to the apex of the umbra can be calculated as:

$$D_a = \frac{D × R_p}{R_s - R_p}$$

Where $D_a$ is the height of the shadow cone, D is the distance between planet and star, $R_p$ is the radius of the planet and $R_s$ is the star's radius.

Let's assume that our moon is approximately at the apex ($r = D_a$).

$$\frac{D × R_p}{R_s - R_p} \approx D \sqrt[3]{\frac{M_p}{3 M_s}}$$

or

$$\frac{R_p}{R_s - R_p} \approx \sqrt[3]{\frac{M_p}{3 M_s}}$$

You see that distance between gas giant and star is irrelevant here. As long as we put appropriate numbers for their masses and diameters, the moon at $L_2$ point can always be at the apex.

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