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Unformatted text preview: Special cases have usable solutions There are two approximately solvable cases of interest to satellite orbit study. The perturbative (lunisolar or planetary per turbation) approximation for earthorbiting satellites we have seen. Restricted three body problem, applicable to interplanetary missions. The full, exact three body problem is unsolv able and has been the subject of intense scrutiny by celestial mechanicians and mathematicians for centuries. One body by itself determines the twobody motion; this is its sphere of influence . As you get further away from it, third body induces noticeable perturbations . As you go further, near Lagrange points, you need a special three body analysis. L. Healy ENAE404 Spring 2007 Lecture 29 (May 10) 1 Third body forces Recall the twobody equations of motion from Lecture 1, r + r 2 r = 0 . If we now add a third object, say the sun, the whole picture changes. From the satellites point of view, r + r sat r 2 sat + r sat r 2 sat = 0 where r is the acceleration in some inertial frame, and r sat is the vector from the center of the earth to the satellite and r sat is the vector from the center of the sun to the satellite. L. Healy ENAE404 Spring 2007 Lecture 29 (May 10) 2 Restricted 3 body problem Even though its not solvable, it is still possible to gain some understanding of third body in fluences beyond the perturbation regime. The restricted 3 body problem is applicable in a wide variety of realistic situations for space craft. We assume: there are two large bodies whose relative motion is circular, e.g. earthmoon, earth sun; the third body (artificial satellite) has neg ligible mass, so it does not affect the other two bodies motion. (optionally) Coplanar motion of all three objects. L. Healy ENAE404 Spring 2007 Lecture 29 (May 10) 3 Coordinate system center When doing calculations based on the two body equation of motion, we picked a coordinate frame IJK centered on the earth. But the two body motion is only a simple, solvable form (conic sections) when encountered in the cen ter of mass system . Because the mass of an artificial satellite is much less than that of the earth, this essentially coincides with the center of the earth, so everything works out....
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This document was uploaded on 04/27/2011.
 Spring '11

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