# PHYSICS 141/241

## Assignment - Second Set

Due date: 2/7/2020

Mars mission.

The stability of Saturn's rings.

Problem 1 (50 points PHY 141/241)

### Provide a full simulation of the voyage and show if it requires the minimal amount of fuel.

Hohmann transfer orbit tutorial

Problem 2 (50 points PHY 141/241)

In this problem we will study a simple model for the stability of the rings of Saturn. You can investigate mass ratios that lead to stable ring systems vs. unstable ones. If the mass of each ring body is no more than 2.3 times the mass of Saturn divided by the cube of the number of ring particles, then the system can be expected to be stable; otherwise not. See Linear Stability of Ring Systems for the derivation of this inequality.

(a) With the WebGL applet you can investigate mass ratios that lead to stable ring systems vs. unstable ones. In the applet, M is the mass of Saturn (in Earth-masses), m is the mass of an individual ring body, and n is the number of ring bodies. The text field labeled gamma is the ratio m*n^3/M. If this value is smaller than 2.3, the system will be stable. Large values will be unstable. You will find that you don't need to increase m very much to make the system unstable. If you set "warp" to 100, the integrator will show the instability very quickly. Note: the warp parameter only controls how often the screen is updated---large values mean that many time steps of the integrator are performed between each screen update. This makes the simulation run much faster as updating the screen image is more time consuming than a step of the integrator.

(b) Using the Aarseth N-body code, or your own code, reproduce some of your applet simulations with stable and unstable rings. Generate an MPEG movie of a stable and an unstable ring.

Your own code is rewarded by 50 points.

Aarseth code 20 points.

The procedures of animation and movie making are discussed by the TA.

The procedures of running the Aarseth code are discussed in Lab Session IV.

Link to interesting web site on the Saturnian rings:

NASA web site of Cassini mission to Saturn