# How to calculate orbital eccentricity from the ratio of satellites' velocities?

I have a problem with working out how to calculate the orbital eccentricity from the ratio of two satellites' linear velocities. We know that two satellites encircles the Earth in two elliptical orbits, which have the same length of semi-major axis. The ratio of satellites' linear velocities in perihelion is 3/2 and the eccentricity of the orbit of the fester satellite is 0,5. I need to find the value of the eccentricity of the second orbit and the ratio of velocities in aphelion. I know that the ratio 3/2 is derived from the formulas:

$V_{perihelion} = \sqrt{\frac{\mu * (1+e)}{perihelion}}$

so for Earth's satellite it will be:

$V_{perihelion} = \sqrt{\frac{398600 * (1+0,5)}{perihelion}}$

and the whole ratio:

$\sqrt{\frac{398600 * (1+0,5)}{perihelion}} / \sqrt{\frac{398600 * (1+e_2)}{perihelion}} = \frac{3}{2}$

How can I get $e_2$ from such euqation with no knowledge about the distance in perihelion nor in aphelion? Thank you

• It looks to me that the value of $\mu$ and $perihelion$ will simply cancel out. You end up in one line with an expression containing just $e_2$ and two lines later you will have your answer. As for the second part, you will find some hints in physicsforums.com/threads/eccentricity-of-orbit.244403 – Floris Nov 19 '14 at 21:54
• You're going about this incorrectly. The two orbits have the same semi-major axis lengths but different eccentricities. This means they will have different periapsis distances. You need a different expression for the velocity. A good place to start is the vis viva equation, $v^2 = mu\left(\frac 2 r - \frac 1 a\right)$. Now substitute $r=a(1-e_1)$ and $r=a(1-e_2)$to represent the fact that the satellites are each at perigee. You should get a nice result that leads to $\mu$ and $a$ canceling on taking the ratio. – David Hammen Nov 20 '14 at 0:13
• thank you very much. indeed, with these formulas the result turned out to be very neat! have a good day! – Gerda Nov 21 '14 at 11:35