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Assuming I have not made a basic math error, an Earthlike world orbiting a red dwarf like Proxima would put an object at the star-planet L2 point in permanent shadow. I get about 18 degrees for the apparent width of the planet from L2 vs about 10 degrees for the star.

This is interesting from a writing POV because stars like Proxima are prone to flares that can deliver about two hundred times the LD50 dose of x-rays at the distance an Earthlike planet would have to orbit at. Atmosphere will block that nicely but ships on orbit would need shielding. 12,000 km of rock in the form of a nearby world would seem to qualify as shielding.

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