# A External satellite resonance

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1. Aug 1, 2017

### Kelly Lam

I'm working through a paper "Dynamics of Planetary Rings" by Goldreich and Tremaine (http://www.annualreviews.org/doi/pdf/10.1146/annurev.aa.20.090182.001341). I'm working through p.22 about expanding the potential of an external satellite as a double Fourier series. The external satellite is in epicyclic oscillation with coordinates given by $r = a(1 - ecos(\kappa t))$ and $\theta = \Omega t + \frac{2\Omega e}{\kappa} sin(\kappa t)$ (these can be found in p.2 of http://articles.adsabs.harvard.edu//full/1980ApJ...241..425G/0000426.000.html). $\kappa$ is the epicyclic frequency and $\Omega$ is the circular orbital frequency at $r=a$.

I'm struggling to see how to the lowest order in e the eccentricity of the satellite, the first coefficient (m,0) is given as that in the paper.

2. Aug 6, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.