The location of a geostationary moon in the sky?

Question

On a planet with a geostationary moon, how can I determine the position in the sky of the moon depending on the position on the planet surface?

For example, Pluto's Charon. The moon wouldn't move across the sky standing still but if you were to move where you were on the planet, where would the moon appear to be? I imagine someone standing directly in line with it at the equator would see it somewhere else than a person standing in the southern hemisphere to the east. And as you drifted out of its view (east to west or north to south) would it slowly disappear beneath the horizon?

I'm sure this could be figured out studying one of our own satellites but I wasn't sure how to approach that.

Answer 1

Simple geometric considerations can help you here.

Imagine, for the sake of simplicity, that the moon is on the same plane of the equator and on the meridian where you take the latitude 0. If you measure your latitude starting from 0 at the equator and the observer is at a latitude $\theta$, the moon will appear on the sky at an angle $90 - \theta$

If you look at the picture above, an observer on the equator (plane a) will see the moon at 90 degrees in the sky, in other word at its zenith.

An observer closer to the pole (plane b) will see the moon much lower on the horizon, until at the pole the moon will be exactly on the horizon.

Same goes with the longitude.

Mind that when the angle exceeds the 90 degrees the moon won't be visible at all, because it will be hidden by the planet. This happens on the farthest side of our Moon, which never sees Earth.

Answer 2

A geostationary satellite orbits in the same plane as the equator with a period equal to the rotational period of the planet. In other words, from anywhere on earth, the satellite appears fixed in one position in the sky, never moving - it is always directly above the same point on the equator. If you happen to be standing on that point, the satellite will be directly overhead. If you move away from that point, the satellite will dip lower and lower, until the planet itself obscures your view of the satellite. If you're directly north of the geostationary point on the surface, the satellite will be in the sky to your south, and conversely, if you're toward the south, the satellite will appear in the north. If you go west, the satellite will set in the east, and if you go east, the satellite will set in the west.