# Determining eccentricity of satellite orbit from velocity vectors and altitude

Let's say there's a satellite orbiting earth. At some random point in its orbit, it is observed with a tangential velocity, a radial velocity and some altitude.

Given only this information (along with known values for earth radius, standard grav. parameters etc.), is it possible to determine the eccentricity of the satellite's orbit?

My approach was to determine the specific orbital energy and the specific relative angular momentum at the point observed in its orbit. The former can be found using the vis-viva equation (taking the magnitude of the radial and tangential velocities) but the latter cannot be determined since the random point cannot be assumed to be the peripasis or apoapsis.

• Oops. Yes you are right. Perhaps you should put that sentence in an answer so I can mark it as the answer. – user155876 Sep 24 '15 at 17:22
• OK - comment converted to answer. – Floris Sep 24 '15 at 17:23
• Newton's equation is of second order, therefore knowledge of the initial position and velocity are enough to find the orbit. – auxsvr Sep 24 '15 at 21:07
• Also, the easiest way to find the eccentricity is to calculate the Runge-Lenz vector. – auxsvr Sep 24 '15 at 21:16

Why can't you determine the angular momentum? If you have the velocity vector $\vec v$ and the position vector $\vec r$, angular momentum can be calculated at any point in the orbit: $L=m \vec{r} \times \vec{v}$. In fact since you have the tangential velocity explicitly, it's even easier...