Due date: 2/09/2016
Applications of animation and MPEG movie making to planetary orbits.
Applications of the Aarseth N-body code or your own code to the stability of Saturn's rings.
Problem 1 (30 points)
(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. You can use any platform for animation, like Matlab based procedures.
(b) Make a stand alone MPEG movie of the animation, or use some other equivalent media, like AVI.
The procedures of animation and movie making are discussed by the TA.
Problem 2 (20 bonus points for PHY 141 and 30 bonus points for PHY 241)
Problem 3
(own code: 70 points for circular orbit plus 30 bonus points for elliptic Kepler orbit)
(using Aarseth code earns half of the points)
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, 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 will earn double points!
The procedures of running the Aarseth code are discussed in Lab Session IV.
Links to interesting web sites on the Saturnian rings:
NASA web site of Cassini mission to Saturn
Professional ring simulation