The Dopple World is back, baby

On topic of Gaia BH1, it occurred to me - hiking to the gym and back - whether the G star could support a planet. Maybe if the ten-solar-mass black hole was three times bigger.

If your first instinct is to laugh then, yeah - I've beaten you to it. We don't even have to do any maths to realise that the G star's first Libration point - the Hill Sphere - is (much) less than 0.5 AU. I don't even think it can get away with a Mercury. If this star formed more AU away and was a K type, then we'll consider it. Although if the star's a capture with an eccentric orbit, internal tides should make it warmer, constraining the habitable-zone again.

So then I considered Dopple World. I've generally dismissed the 1 AU equilateral Trojan haloes when Doppling an Earth-mass planet, due to mutual tides again. But here we got a Solar-mass anchor.

We don't really care, here, how a Sun orbiting a black hole has happened; but we do know that it has happened, 1600 light years away/ago. Lagrange (not just Hill) steps in when the mass-ratio exceeds 24.96, so BH1 itself - maxing at 10 - isn't a candidate.

But suppose BH1's central mass be about 25 M_☉. Could a G star form at 1 AU around it? If so, I assume that star quite metallic, if only because forming that monster of a hole would blast out volatiles far far away. Likewise the matter in its orbit. Dopple World will have an iron core like we do. The freakin' magnet should help as its planet oscillates to and from the star. Averaging 1 AU from a G means, well, Earth.

One exception: I expect its day/night cycle to fall into resonance with its orbit around the point because, once more, tides. Here's our Mercurylike, Queen fans.

One serious issue will be comets. Asteroids, meh; I expect Dopple World has cleared the Libration halo in forming. And if our snow line is 4 AU [Ceres forming outside it] then this system's snow line should be 5 AU from the hole. But everyone's got an Oort Cloud and if one of those iceballs gets perturbed, that central mass is going to pull it with absolute Hammer Of Thor force into anything in its way.