Due date: 2/11/2011
Applications of OpenDX animation and MPEG movie making to planetary orbits.
Applications of the Aarseth N-body code to the stability of Saturn's rings.
Problem 1
(a) Run your leapfrog code for Venus, Earth, Mars, Jupiter, Saturn, orbiting around the Sun over 50 Earth years. Put the Sun fixed at the origin of your coordinate system. Animate the revolving planets in OpenDX.
(b) Make an MPEG movie of the animation.
The procedures of OpenDX animation and MPEG movie making are discussed in Lab Session II.
Problem 2
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 Java 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. Give it a whirl. 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, reproduce some of your applet simulations with stable and unstable rings. Generate an MPEG movie of a stable and an unstable ring.
The procedures of running the Aarseth code are discussed in Lab Session III.
Links to interesting web sites on the Saturnian rings:
NASA web site of Cassini mission to Saturn