Eccentricity of Elliptical Orbit

[cscott]

A particle moves in an elliptical orbit in an inverse-square law central force field. If the
ratio of the maximum angular velocity to the minimum angular velocity of the particle
in its orbit is n, then show that the eccentricity of the orbit is

$$\epsilon = \frac{\sqrt{n}-1}{\sqrt{n}+1}$$

Not sure where to go with this. I tried finding total energy and angular momentum in terms of max/min angular velocity and radius but can't get anywhere

 

[dynamicsolo]

At what points in the orbit are the maximal and minimum angular (or, for that matter, linear) velocities attained? At what distances from the "massive body" (what the particle is orbiting around -- assumed to be "infinitely massive" here) is the particle at those moments? (You don't need values here -- just identify those places on the orbit and label them appropriately.)

Now for the critical part. Angular momentum is conserved. What angle does the velocity makes to the radial vector from the massive body at those moments (and no others)? Express the angular momentum in terms of radial distance and velocities for those two moments and set them equal. What is the relationship between these two angular (or linear) velocities and the two distances from the massive body?

Having found how the ratio of angular velocities, called n here, relates to those distances, how do those distances fit into the expression for the eccentricity of an ellipse?

That would be the full derivation of the answer. If you already know how n relates to the ratio of distances, it's a short step to getting to the eccentricity expression...

 

[cscott]

Thanks, got it. Silly of me for starting with eccentricity in terms of energy and angular momentum instead of geometry.

 

[tiny-tim]

… geometry …

Hi cscott!

Consider it geometrically …

Hint: if F is a focus of the ellipse, and P and Q are the ends of the major axis, what is PF/QF as a function of e?

And then what is n as a function of PF/QF?