Science and Computers -- PHY307/607

Chaotic Rotational Motion of Hyperion

1. Login to SUnix and create a subdir of PHY307 called Hyperion.Set its permissions appropriately. Download the source codes for the applet Hyperion from the PHY307 pages to this new directory.
2. Create a file Hyperion.html which uses an applet tag to load the file Hyperion.class. Set the width and height parameters to 400.
3. Compile the Java source codes inside the directory Hyperion. Set the permissions of the .class and .html files.
4. Point Netscape at the file Hyperion.html. The applet shows a graph of the rotation rate of Hyperion against time. Make a sketch of the graph. For this initial speed the orbit of Hyperion is circular. Is the motion chaotic ?
5. Now change the initial speed to 5.0 (this makes the orbit elliptical). Sketch the rotation rate now. Is it chaotic now ?
6. Now edit the code DrawSim.java. Change the variable xplot to true and set the variable vplot to false. The applet will now plot not the rotation rate but the angle of rotation.
7. Recompile the code and reload Netscape. Sketch the plot you see. Why does the curve never go above 3.2 ish ?

Jupiter's gravitational pull on the Earth

1. Link to the ThreeBody applet off the PHY307 lab page and hit Go. Initially we set Jupiter's mass to zero. Thus you should see that the blue point (Earth) orbits in a elliptical path about the yellow spot representing the Sun. Jupiter (the red circle) orbits further out.
2. Set the mass of Jupiter to 0.04 (these are units in which the mass of the Sun is roughly 39). Sketch the orbit now. You should see that Earth's orbit is now affected by Jupiter's motion. The ellipse tries to rotate in space - this is called precession of the orbit and indeed occurs for the Earth.
3. What happens for large mass ? You should see that for a sufficiently large gravitational force from Jupiter the Earth's orbit is completely disrupted. A similar effect has forbidden a planet between Jupiter and Mars -- the asteroid belt is all that is left of this planet ...

Orbits around a black hole

1. Create a directory called Orbit under PHY307. Download the source code for the ThreeBody applet to this directory.
2. Edit the file Simulation.java. Locate the method f() near the end of this file. Replace the line

   
   dy[5]=-GM_S*yt[4]/Math.pow(dum2,3.0)-GM_J*(yt[4]-yt[0])/Math.pow(dum3,3.0);
   

   by

   
   dy[5]=-GM_S*yt[4]*(1.0/Math.pow(dum2,3.0)+A/Math.pow(dum2,4.0));
   

   Similarly replace the line

   
   dy[7]=-GM_S*yt[6]/Math.pow(dum2,3.0)-GM_J*(yt[6]-yt[2])/Math.pow(dum3,3.0);
   

   by

   
   dy[7]=-GM_S*yt[6]*(1.0/Math.pow(dum2,3.0)+A/Math.pow(dum2,4.0));
   

3. Before the lines double dum1,dum2,dum3 add the line double A=0.01. The size of A depends on the strength of the gravitational force at the orbital radius. The value for the Sun is rather small but for a moderate black hole can be rather large.
4. Finally find the variables v_E and x_E near the top of the file and set them to 4.0 and 1.5 respectively (this sets the Earth's initial speed and position). Also delete the line of code plot.addXTo.Screen_J( ... which is responsible for drawing Jupiter's orbit to the screen (now irrelevant).
5. Recompile this new code, set the file permissions as per usual and create a file Orbit.html which contains the line

   < applet code=ThreeBody.class width=300 height=300 >
   < /applet >

   Set the file permissions on the .class and .html files and point Netscape at this applet. Sketch the orbit you see. This precession of the orbit due to Einstein's theory can even be seen in the Solar System -- the calculated precession of Mercury agrees perfectly with the predictions of General Relativity and was one of the early successes of the theory.

1. Hand in written answers to the questions posed in the lab.