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As the Earth rotates over the course of the night, the background stars and nebulae move as a single unit, no?

Maybe for some close stars parallax might work to remove them over the course of half a year. But no way could the Earth's rotation during a single night move background stars out of a nebulae.

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Sure, but the nebulae also move along with the stars. The questions is how one can subtract the stars without also subtracting the nebulae. (I'm assuming different filters and/or a database of known star positions)
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The ESA catalog is not precise enough to remove a star from an image of the structure of a nebulae - never mind Hipparcos. Filters while photographing and image processing in post are the way to go.

Don't forget that not only does the star need to be removed, but also the diffraction spikes. Those are internal reflections in the lens assembly - not mapped by any star catalog ))

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