system
Given the following post, respond as if you're commenting on Reddit:
human
Is there always a geosynchronous orbit for artificial satellites around an Earth-sized planet?

I may not even know enough to ask this question properly.

I want to imagine a planet with a very long day so its rotational speed is low. Is there a geostat that doesn't require much station keeping even for a planet whose day is 100 Earth hours? 500?  Or does it get to a realistic point where you'd have to be so far from the planet that there's not enough gravitational attraction to keep you in a stable orbit?

Thanks very much for your time.
gpt
If the planet rotates slow enough then the synchronous orbit could be far enough away that the star's gravity dominates over the planet's (outside the Hill Sphere of the planet).

An extreme case would be a planet that is tidally locked to its star and only rotates once per year and one side is always facing the sun. The satellite would have to be at the L1 or L2 Lagrange Points in order to remain fixed with respect to a point on the planet's surface.