How far above the galactic plane could an Earth-like planet develop?
I was zooming around on Space Engine and came across the moon of a gas giant that was hovering well above the galactic plane. I'm admittedly not well-versed in under what conditions an Earth-like planets can form, but it got me wondering... is it possible for a terrestrial planet capable of life to form so high up above the galactic plane? If so, would the night sky (sans light pollution) be accurate to what we're seeing above, or would the atmosphere play a role in masking it?
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This is actually a really interesting question. While there are many stars with orbits outside the galactic plane ("halo stars"), these tend to be old and metal-poor, as star formation in the halo has long since ended. This means that assumptions we make about exoplanets around stars in the disk might not hold in the halo. Unfortunately, it's hard to study exoplanets around these stars, simply because they tend to be far from Earth and thus comparatively hard to detect.
Fortunately, many of these stars pass through the galactic plane over the course of an orbit, and an excellent example is Kapteyn's star, a red dwarf lying less than 13 light-years from Earth. It currently lies in the disk of the Milky Way, but kinematically it's part of a group of halo stars that just happen to be in the plane at the moment. It is currently suspected to host one and possibly two planets (Anglada-Escude et al. 2014), both of which are potentially super-Earths.$^{\dagger}$ This would seem strong evidence in favor of halo stars developing planets.
It's believed that Kapteyn's star is a former member of a globular cluster or dwarf galaxy, now torn apart by tidal forces. As such, it seems to be a reasonable chemical representative of the greater halo population, including native Milky Way stars. Theoretical models of low-metallicity systems support the idea that halo stars can form Earth-like planets, and if the system is confirmed, we would seem to have observations on our side, too.
Exoplanets can almost certainly form around halo stars - and so the answer to your question is presumably that a planet can develop as high above the planet as a star can form.
Regarding the night sky on this planet: well, I think we've talked about that kind of scenario quite a lot on this website. You'd see a couple of orders of magnitude fewer stars, for starters, once you get far enough out of the disk. It also seems likely that you'd be able to see a fair portion of the galaxy - not a bad view. It would certainly be asymmetric, though, with the disk on one side and intergalactic space on the other.
$^{\dagger}$As I understand it, exoplanets around metal-poor stars are expected to be biased to lower masses (i.e. terrestrial, Earth-like planets).
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