PH 322 Celestial Mechanics Fall 1998 Homework Set 7. Due Thursday October 29, 1998

1. Gravity slingshot (simple numerical example). Build a worksheet starting from ellips0 or a variant of it. Assume a small mass at rest at x0=20 and y0=-1 and a planet (mu=0.4) moving in the x-direction at a constant 1 unit of speed. Start the planet at the origin at t=0. The idea is that the planet will come by, 'grab' the particle and 'fling' it into rapid motion.

Notes

• 1. You will need to represent the planet's position as (Vplanetx *t,Vplanety*t).
• 2. Use x(t) and y(t) as always for the particle position.
• 3. Modify rm3 and both equations to include the position of the planet.
• 4. When your numerical solution is done, create a 'ke' function for particle kinetic energy
• 5. Your time interval will need to go from 0 to something like 30.
• 6. Graph position vs time as before.
• 7. Graph ke ( plot(ke,0..30) ).

a) Find the 'final' particle velocity and compare it to the planet velocity (look at ke graph)
b) Repeat but with the particle at y0=-1/2 instead of y0=-1, and again compare particle final speed to that of the planet.

2. Create a worksheet showing how a particle launched from Earth can be put into an orbit such that the particle can be 'run over' by Venus as in the previous problem and gain a lot of energy. Your orbit must be inside Venus's orbit, and not just tangent to it. You must come up with deltaVx and deltaVy, changes to the Earth's velocity when at x=1, y=0.

Notes

• 1. Start from either the orbit equation or a numerical simulation. In both, plot Venus's circular orbit so we can see how the two orbits look.
• 2. If you do a numerical simulation, you can start from ellips0 and put the Sun at the center and then begin with the particle having Earth Vx and Vy. Run this to see that the orbit is a circle of radius 1. After this, add various deltas to Vx and Vy until you find what it takes to intersect Venus's orbit.
• 3. If you go from the orbit equation, you will start by calculating h, p and e for the orbit from initial velocities. With Earth parameters, e should be zero. Then add deltas to Vxo and Vyo and plot the orbit.

3. Start from worksheet Threebody0.mws (the first item in the PH 322 web page) and build a worksheet which lets you see the motion of a particle in the field of the Sun and Jupiter. This is done in a rotating coordinate system.

Notes

• 1. The worksheet is intentionally missing a rotating coordinate system 'force' which you must supply.
• 2. A graph from the intact worksheet for 24 years is intentionally left around, so you can check when yours is working out ok.
• 3. Determine from your plot what the frequency of oscillation is for the small body. Compare this to the frequency from the theory (included in the worksheet).