What is the mathematical explanation for the shepherd moons of Saturn?

[ShadowHound]

What exactly is the physics behind the shepherd moons?

Take Pandora and Prometheus of Saturn for example, they keep the F-ring in line by their combined gravitational effects, I went though a lot of books but all of them are quite general hand-waving kind of explanation and nothing mathmatical.

I'm really interested in the math behind the shepherd moons so any help would be greatly appreciated.

 

[meteor]

Maybe, this can help:
http://arxiv.org/abs/physics/0303124
Gravitoelectrodynamics in Saturn's F Ring: Encounters with Prometheus and Pandora
"The dynamics of Saturn's F Ring have been a matter of curiosity ever since Voyagers 1 and 2 sent back pictures of the ring's unusual features. Some of these images showed three distinct ringlets with the outer two displaying a kinked and braided appearance. Many models have been proposed to explain the braiding seen in these images; most of these invoke perturbations caused by the shepherding moons or km sized moonlets imbedded in the ring and are purely gravitational in nature. These models also assume that the plasma densities and charges on the grains are small enough that electromagnetic forces can be ignored. However, Saturn's magnetic field exerts a significant perturbative force on even weakly charged micron and sub-micron sized grains causing the grains to travel in epicyclic orbits about a guiding center. This study examines the effect of Saturn's magnetic field on the dynamics of micron-sized grains along with gravitational interactions between the F Ring's shepherding moons, Prometheus and Pandora. Due to differences in the charge-to-mass ratios of the various sized grains, a phase difference between different size populations is observed in the wavy orbits imposed by passage of the shepherding moons. "

 

[tony873004]

You need to get the book Solar System Dynamics by C.D. Murray and S.F. Dermott. It's heavy in the math. You might be able to view a copy at your local university's library or you can purchase it through Amazon at the below link.