What is geostationary orbit radius?

Question

I'm asking this apparently "general reference" question for the simple reason: I was unable to find whether the quoted everywhere "35,786 kilometers (22,236 mi) above the Earth's equator" means "radius" or "altitude above equator." (yes, for it to be geostationary it must be located above the equator, but I'm really not sure if the number includes Earth radius or not).

With Earth radius at Equator equal 6,378 km that's a considerable difference. So, is the orbital radius 35,786km, and altitude 29,390 km or is the altitude 35,768 and radius 42,164 km?

Answer 1

It's pretty easy to calculate. For geostationary orbits, the orbital period $T$ should be equal to the rotational period of the Earth $\Omega_E$:

$ \matrix { T &=& 2 \pi \sqrt{a^3/\mu} \\ \Omega_E &=& 1\ \mathrm{stellar\ day} } $

$ \ \ T=\Omega_E \rightarrow a = \sqrt[3]{\mu \cdot \frac{\mathrm{day}^2}{4\pi^2} } = \sqrt[3]{398600.44 \cdot \frac{86164.099^2}{4 \pi^2}} \approx 42164\ \mathrm{km}. $

Note that this equals the semi-major axis of the orbit, which means that if you want the altitude, you'll have to subtract Earth's equatorial radius:

$ h = a - R_E = 42241 - 6378 = 35786 \ \mathrm{km} $

Answer 2

The altitude is about 36000 km, so the radius of the geostationary orbit is about 42000 km (see, e.g., http://en.wikipedia.org/wiki/Geostationary_orbit ).