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Recursive star systems like this are tricky I did an analysis in this question about habitable zones within habitable zones and how they interact - the habitable zone concept breaks down from a nice series of circles into something much more complex. It basically gets replaced with a 4d function "how habitable is this spot at this time", and you ...

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Moon made of aerogel with a total mass comparable to the miserably small asteroids that Mars has for "moons", painted in a high-tech reflective coating. The low-density material should make it big enough for the relatively small mass. Preventing meteorite strikes from damaging, deforming, and ultimately knocking the ultra-light secondary moon out ...

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Quite a few of oportunities here: A body in an orbital resonance with the main moon - be it 1:2, 2:3, whatever - it may be both inside and outside the main moon's orbit. A body in L4 or L5 Lagrange point of Earth/Moon system A body in L4 or L5 Lagrange point of Sun/Earth system A body with 1/100 mass of the main moon is still pretty much visible. The ...

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You can combine a small body, like Mars satellites Phobos and Deimos, with high albedo because it happens to be made of metal or a highly reflective material if you want to make them more visible. Deimos has a mean radius of 6.2 km (3.9 mi) and takes 30.3 hours to orbit Mars. Deimos is 23,460 km (14,580 mi) from Mars, much further than Mars's other moon, ...

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Given the information you've provided, there's no way to compute the eccentricity of the orbit; with this setup, it's a free parameter that you can set to whatever you want (although of course we need $e<1$ for a bound orbit). It would be possible to determine if we had some additional information, such as the energy of the planet $E$ and its angular ...

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Short Answer: Such a settng would be impossible to happen naturally. But if your story is sufficently soft on the Mohs Scale of Science fiction Hardness - TV tropes warning - https://tvtropes.org/pmwiki/pmwiki.php/Main/MohsScaleOfScienceFictionHardness[1] Or if your story isn't science fiction at all but a fantasy which happens to be set on another planet (...

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The L1 point is a point of unstable equilibrium, as stated already by Alexander. This means that a body without active control of its orbital parameters would soon move out of that point due to gravitational perturbations. Moreover, a body in L1 would not be orbiting the planet but the star, which goes against your requirement of "a moon orbiting a ...

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L1 Lagrange point is unstable, so it would be too much to expect a moon to appear at that point naturally or stay in it for many million years, but otherwise yes, this setup is possible.

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