Answer Happy

The Three Body Problem Let us consider the motion of a spacecraft in the gravitational field of two bodies (such as the Earth and the Moon). The equations governing the motion of the spacecraft form a system of second order differential equations. The system is illustrated below Spacecraft Earth Moon + +y Centre of Mass E" M Figure 1: The Three Body Problem. E and M are the distances the Earth and Moon are from the centre of mass respectively. Both bodies impose a force on the spacecraft according to Newton's gravitational law, but the mass of the spacecraft is too small to significantly affect the motion of the bodies. We therefore neglect the influence of the spacecraft on the two stellar bodies. Our coordinate system has its origin at the centre of mass of the Earth and Moon. The governing equations are given by M(x+E) EI-M) d12 d! re fy dr My Ey di? dr where re = V(x+E)? +v.ru Vir - M)2 + y2, and M = 1- E. The Earth and Moon are assumed to have circular orbits around the centre of mass of the system. To simplify the problem, we therefore consider a coordinate system which is rotating with the Earth and Moon. In this system, the Earth does not move and is located at (-E.0) and the moon is located at (M.0). The governing equations contain terms in which correspond to the gravitational force. (Ok, in the governing equations it is actually but this term is still associated with the gravitational force, ask me if you would like an explanation.) The equations also contain terms and which correspond to the Coriolis force, as well as terms and y, which correspond to the Centrifugal force. Before you try and solve this system of second-order differential equations we must reduce it to a set of four first order equations. To do this, we define dy = = 2= dt =y. and 2 In vector form, we get dt Z= = 23
Therefore, 22 2:4+ E-M ż 8-01-11 = = where re = 1 + E)? +TM= E = 0.012277 - M)2+ and M = 1- E. Set the value Your job as modellers is to write a MATLAB program which prompts the user for the length of time you wish to solve the spacecraft motion for the initial position and the initial velocity of the spacecraft. Use the following values to test your program: Time from: t=0 to 1 = 24 Initial Position: (0) = 1.15, y(0) = 0. d0) = 0, de Initial Velocity: (0) = 0.008688 d! After establishing these values your program will solve the given system of equations and plot the solution (the trajectory of the spacecraft) in a figure window. The solution for the test parameters above is plotted in the figure below. You should be able to reproduce this plot with you program. The story of the spacecrutin rating ordination 21 © - 20 Figure 2: Solution to the Three Body Problem for the test parameters,