Orbital Perturbations: Solving for Equations of Motion in Elliptical Orbits

[Konchock Dawa]

Homework Statement

Say I have some planet in a circular orbit around a star, and I give it a small radial push (directly toward or directly away from the star). How would I describe the new orbit? I.e. how would I determine the equations of motion? h

Homework Equations

Kepler orbital radius: r(φ)=c/(1+εcos(φ)) where c=(L^2)/(γμ).

Also possibly relevant, orbital energy: E=(γ^2)μ((ε^2)-1)/(2L^2)

The Attempt at a Solution

My intuitive hypothesis is that the new orbit should be elliptical. A circular orbit has constant radius r(φ)=c, and we have already proved in this class that the radius after a small push will oscillate with a period equal to the orbital period, so it seems like if we just add a sinusoidal term, r(φ)=c+Asin(φ) we should be able to get the equation in the elliptical form r(φ)=c/(1+εcos(φ)) but that doesn't seem to work. I have a hunch that I'm supposed to do some sort of a Taylor expansion, but I don't know which function I would expand. It also seems like, since a push would change the angular momentum L, we should be able to replace L with L+δ and again get something of the form r(φ)=c/(1+εcos(φ)). Is any of this heading in the right direction? Have spent many hours on this problem and could use a hint.

 

[phyzguy]

If the push is radial, then I think it won't change the angular momentum. A radial force does not result in any torque, and if there is no torque, there can be no change in angular momentum. So only the orbital energy will change. If you assume E changes and L does not, then you can compute the new value of eccentricity.