# Determine the eccentricity of the orbit

A satellite is in orbit at an altitude of 62.5km, with a speed along the earth-relative position vector of 3.98km/s, and a speed perpendicular to the earth-relative position vector of 7.763km/s. What is the eccentricity of its orbit?

• Given those constraints, (and without further acceleration) it is either on an escape vector from Earth, or in a sub-orbital hop to another point on Earth. So since it is 'not in orbit' I'd argue that the question is meaningless. – Andrew Thompson Feb 3 '16 at 9:36
• homework problem? – SE - stop firing the good guys Feb 3 '16 at 10:04
• @AndrewThompson - The magnitude of the velocity vector is 8.72 km/s, less than the 11.1 km/s escape velocity at that altitude. The eccentricity is less than one. This looks like homework, so Carl, it's best if you do this by yourself. Hint: Use the eccentricity vector. – David Hammen Feb 3 '16 at 13:03
• 62.5 km?! It won't be in orbit for very long. Also you didn't say if the 3.98 km/s is up or down. (You don't need that sign for the eccentricity.) If it's down, then it really won't be in orbit for very long. – Mark Adler Feb 3 '16 at 17:48
• Ended up using the vis-viva equation along with locating the semimajor axis distance to finally find eccentricity. Thanks for the help. – Carl Jonson Feb 5 '16 at 0:05

1) Use the given position and velocity values to write the position and velocity vectors, $\vec{r}$ and $\vec{v}$
2) Compute $\vec{h} = \vec{r} \times \vec{v}$ (where $\times$ is the cross product)
3) Compute the eccentricity $\vec{e} = \dfrac{1}{\mu}(\vec{v} \times \vec{h})-\dfrac{\vec{r}}{|\vec{r}|}$