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Unformatted text preview: 16.07 Lab I Issued: October 16, 2009 Due: Monday, October 26, 2009 Introduction This laboratory is the first in a series that deal with celestial and spacecraft dynamics. In this lab, you will create a MATLAB code that will allow you to numerically confirm Kepler’s laws as well as simulate the motion of the earth and the other planets in the solar system as well as spacecraft in orbit about or on the way to one of them. Since this lab will focus on the earth and moon system, the presence of the sun will be ignored. In order to accomplish these tasks, your MATLAB code will integrate the equations of motion in time, starting from an initial condition. In later labs, you will add features like spacecraft thrusting, which may allow you to simulate orbital transfer, rendezvous, descent and ascent from planetary surfaces and other more complicated situations. Figure 1: 3 Body coordinate system. Figure 1 shows the coordinate system for a generic problem involving the three planetary/spacecraft bodies. The bodies are labeled as Body 1, Body 2 and Body 3, but these names are just an example. These bodies will turn into the Earth, the Moon and a spacecraft. The important thing is to choose a convention that you understand and can be consistent with. The first step is to formulate the equations we would like to solve with the MATLAB code. In the general case, we desire to calculate the positions and velocities of each body as a result of its interaction with the other members of the system. For our purposes, we only consider the gravitational forces resulting from the mutual attraction of body pairs. Thus, the motion of any of the three moving bodies must consider the gravitational inﬂuence of the other two moving bodies. For instance, the attractive force that Body 2 exerts on Body 1 is given by, F 12 = µ 2 m 1 r 12 . (1) 3 r 12 1 Here, µ 2 = Gm 2 , the gravitational parameter of Body 2, where G is the universal gravitational constant 1 and m 2 is the mass of Body 2. The mass of Body 1 is denoted by m 1 , and r 12 is the magnitude of the distance vector that points from Body 1 to Body 2, r 12 = r 2 − r 1 . Celestial bodies move in three dimensions, so in general you would write your equations for the x, y,and z positions of each body. However, for this lab, we will take the position of the earth, moon and spacecraft system and their motions as confined to the x, y plane. Therefore, the z equations are not required. For numerical purposes it is convenient to write the equation of motion r ¨ = F /m for each body as a system of first order equations. That is, we write, r ˙ = v , (2a) v ˙ = F /m. (2b) Therefore, to describe the motion of a body in two dimensions, four state variables are required – two for position and two for velocity. For three moving bodies confined to planar motion in the x, y plane,we need only four for each body for a total of twelve. We will use a Cartesian coordinate system which means that the equations should be separated...
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- Fall '09