# Determining orbital height of a satellite just by observing angular velocity

A few days ago I obervered a satellite with my telescope. But when I later looked at Stellarium (after adding all available satellite databases), there was no satellite passing the spot that I observed at the given time (one was close but way too fast and should not have been visible). Since I knew how big my FOV was and had roughly counted how many seconds it took to cross it, I thought it should be possible to determine the height of the satellite's orbit (assuming it's circular). I had a few different approaches, one was to set the formula for orbital velocity $$v = \sqrt{\frac{GM}{R+h}}$$ equal to the formula for apparent size (per second to get velocity) $$v = 2h \cdot tan(a/2)$$ [M=mass of earth, R=radius of earth, h=orbital height, a=angle the satellite moves in one second measured in degrees] Using WolframAlpha, I got a very long formula as a result with cot and stuff but in the end it turned out it wasn't right, as I checked with other satellites, it gave wrong heights, and for LEO objects it gave an error since the result would be imaginary. So I tried a different approach: $$\sqrt{\frac{GM}{R}} = \frac{(2\pi R)}{(2\pi/\omega)}$$ [R=total radius of satellite orbit, w=angular velocity in rad/s] (Basically orbital velocity using gravity = orbital length depending on radius / time it takes for a $$2\pi$$ (i.e. one complete) orbit)

In the end you should get something like $$R = \sqrt[3]{GM/\omega^2}$$ And to get the height just subtract the earth's radius It seems to work out better, as it can return solutions in a LEO, but it still gives the wrong solutions.

• If the satellite passes nearly overhead, then it defines an angle within your field of view: θ = vt/h. But for a circular orbit: $v^2$ = GM/R = GM/($R_o$ + h) where $R_o$ is the radius of the earth and h is the altitude of the satellite. Combine to eliminate v and solve for h. (This assumes you know the angle for the field of view in radians.) Commented Jan 5, 2022 at 15:20