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# lecture29 - Special cases have usable solutions There are...

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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 earth-orbiting 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 two-body 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

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Third body forces Recall the two-body equations of motion from Lecture 1, ¨ r + μ r 2 ˆ r = 0 . If we now add a third object, say the sun, the whole picture changes. From the satellite’s point of view, ¨ r + μ ˆ r sat r 2 sat + μ ˆ 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 it’s 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. earth-moon, 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

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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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lecture29 - Special cases have usable solutions There are...

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