This equation computes the velocity, `V_"cs"` for a satellite in a circular orbit given the orbit radius, r.

This is assuming a spherical Earth model and uses the fact that for a circular orbit the semi-major axis equals the radius of the orbit, which is constant. The picture below shows the distance r for an elliptical orbit and its easy to see that the distance r of the satellite from the Earth changes throughout the orbit.  Bot this equation makes the simplifying assumption that the orbit is perfectly circular, which is just a special case of an elliptical orbit.

`V_"cs"= 7.905366 (R_E/r)^(1/2)`, where `V_"cs"` is in km/sec.

Notes

For many analyses the spherical Earth approximation is adequate.  See the WGS-84 constants to see some of the the variations in Earth models.

Note that the orbit's radius, r, must be greater than the radius of the Earth `R_e` or else we would be hearing a very strange rumbling beneath or feet.