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MIT16_07F09_lab2 - 16.07 Lab I I Issued Monday November 2...

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16.07 Lab II Issued: Monday, November 2, 2009 Due: Monday, November 23, 2009 Introduction In this second numerical laboratory exercise, you will build upon the simulation developed in the first lab. Here, you will use your capability to change from the Earth-fixed inertial reference frame, to the Earth-Moon rotating reference frames, in which the Earth and Moon are fixed in the frame of reference. This frame change will help you explore different orbital phenomena within the framework of the restricted three-body problem, such as free-return trajectories to the moon, the behavior of orbits in the Earth-Moon neighbor as well as the behavior of spacecraft near the Lagrange points, both stable (L4, L5) and unstable (L1, L2). In the restricted three-body problem, we consider the motion of three bodies, but do not consider the gravitational force from the smaller body, the spacecraft, on the two large bodies, called the primaries. In lab 1, you used the radius of the earth and your length scale, and 1 day (24 hours) as your time scale. This results in the following values for the important parameters of the system. Table 1: Non-Dimensional Physical and Orbital Data µ earth 11468 µ moon 141 R-distance between moon and earth 60.269 r 0 -distance between earth and origin .7324 r 1 -distance between moon and origin 59.53 Ω-rotation rate of coordinate system .230325 If you were successful in getting the earth and the moon to remain fixed in your rotating coordinate system by adjusting slightly the value of Ω, you may place the earth and moon at these fixed points and only consider the motion of the spacecraft in the gravitational fields of the earth and moon in the rotating coordinate system using this value of Ω. (Since we are going to ignore the effects of the spacecraft on the earth and moon, we leave the earth and moon at these fixed positions.) If you did not obtain fixed positions for the earth and the moon in the rotating coordinate system, get cracking. I don’t think Lab 2 will behave without this. 1
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Problems In the following problems, the items you must turn in—either the written answers to questions, or required plots—are highlighted in bold. The ordering of these deliverables in your report should match the ordering that appears below. 1.Earth Fixed Satellites In this problem, we revisit the restricted three-body problem simulation from Lab I. Our first task is to place a satellite in a circular orbit at a radius R from the center of the earth of magnitude R = 2 R earth . The proper choice of boundary conditions for this calculation is subtle. As shown in the figure, in an inertial coordinate system, placing a satellite in earth orbit requires matching the boundary condition with the rotation of the earth-moon system about its center of mass. Consider for simplicity only the insertion point directly beyond the earth, on the line joining the earth-moon centers, the line of symmetry.
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