In this project you will study the motion of three planets moving under the influence of gravitational forces. If the coordinates of the i-th planet at time t are denoted by
then the equations of motion of the i-th planet are
In this equation, is the mass of the i-th planet, and represents the force of planet j on planet i. According to Newton's laws of motion,
Here is the gravitational constant.
Your assignment is to study the solution of these equations in one particular situation.
We will use the following data for the planets:
where the masses are measured in kilograms. The initial conditions are:
To begin with, you are to solve these equations for
The distances are measured in meters, and the times in seconds. (I placed the planets close to each other so that the times would be short; this is artificial.)
The differential equations are stiff, and so must be solved using an appropriate solver. I want you to use ode15s.
C Grade: Solve the equations, and plot the results. For each of the three planets, determine at what time the planet first reverses direction (i.e., at what time the
planet first is moving back toward the initial point). This is a big calculation, so don't run the entire calculation until you are sure that your programs are working correctly.
B Grade: Solve the equations for
Think carefully about how you display your results, since just a single plot of the entire set of trajectories is difficult to understand. Provide detailed views of
``interesting'' portions of the trajectories. You may have to use a variety of viewpoints to display the results effectively. How sensitive are the solutions to perturbations of the initial data?
A Grade: Modify the data so that the equations model the motion of a three-body problem in the solar system. For example, you might use the sun, earth, and moon;
or the sun, Saturn, and Jupiter; or a planet with two moons; etc. You can get data from the Encyclopedia Brittanica, for example, which can be used on-line via the
library's web pages. Create graphical displays of your results.
Preliminary deadline: Monday, November 23
Final deadline: 9:30am, Thursday, December 3