Things to do

Write a C program to call finteg to study the motion of a spacecraft in the Earth-Moon environment. Do the calculation in the inertial (centre-of-mass) frame of reference. The problem to be solved is to find initial conditions (position and velocity) for the spacecraft near the surface of Earth such that it will fly to the Moon and be swung back by the Moon's gravity so that it returns to the vicinity of Earth with no further rocket engine assistance i.e. purely by gravitational forces. For convenience you could always start at a point on the line joining Earth and Moon. For the actual Earth-Moon mass ratio of 81 the required initial conditions are quite critical. For example a solution can be found for an initial distance from the Earth centre of 6800km and velocity but the initial direction needs to be specified to within about 1 degree and the velocity to 4 significant figures. It is easier to solve the problem if the Earth-Moon mass ratio is made less. Why is this ?

Your program should generate a plot of the position of Earth, Moon and spacecraft in the inertial frame of reference. It is easiest to plot in Cartesian coordinates. If any object has polar coordinates the corresponding Cartesian coordinates are readily generated as and . But remember to generate the Cartesian coordinates as variables of type float for plplot.

1. Find a solution (i.e. initial conditions) for a spacecraft starting on the surface of Earth (radius 6400km). Remembering that the Earth is rotating, what extra kinetic energy must be given to the spacecraft initially, as an impulse, so that it goes to the Moon and back (to somewhere close to Earth) purely under gravitational influence ?
2. Keep track of the total mechanical energy of the spacecraft (T+V). Is this energy conserved over the flight path ? If not, is this surprising since this is a conservative system (gravity is the only force acting) ? How can this be explained ?
3. Extend your program so that you can choose to plot with an origin at the centre of the Earth or at the centre of the Moon. This is simply done using Cartesian coordinates for spacecraft, Earth and Moon.
4. Devise a scheme for putting the spacecraft in orbit around the moon. For this you must be allowed to brake the craft i.e. you need to detect when the craft is suitably close to the Moon and then you are allowed to change the velocity in an arbitrary way so that it is in orbit round the Moon. The radius of the Moon is 1738km. Don't actually collide with the Moon. For some particular case find the work that has to be done by the rocket motor to put the craft in orbit round the Moon.
   

   Because of the relatively great mass of the Earth vs the Moon you may find that the Earth has a surprisingly great effect on the orbit of a craft bound to the Moon. How close to the lunar surface does the spacecraft have to stay for its motion to be dominated by the Moon?