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# lecture03 - Computation of ellipse quantities Ellipse...

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Computation of ellipse quantities Ellipse quantities may be computed from alge- braic relations we have: b = a 1 - e 2 is the semiminor axis (put x = 0 into Cartesian conic section equation and solve for y ), p = a (1 - e 2 ) is the semilatus rectum , and is the distance from the attracting focus to the orbit perpendicular to the major axis, r p = a (1 - e ) is the perigee geocentric dis- tance, r a = a (1 + e ) is the apogee geocentric dis- tance. (Check the last two yourself with the conic section formula). L. Healy – ENAE404 – Spring 2007 – Lecture 3 (Feb. 1) 1

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Semimajor axis in physical quantities Squaring eccentricity vector gives relation of the semimajor axis a to the speed v and the distance r : e 2 = 1 μ 2 r × h ) · r × h ) - 2 μr r · r × h ) + 1 | ˙ r × h | = vh because ˙ r h , and r · r × h ) = ( r × ˙ r ) · h = h 2 so e 2 = v 2 h 2 μ 2 - 2 h 2 μr + 1 1 - e 2 = h 2 μ 2 r - v 2 μ = p 2 r - v 2 μ p = 1 - e 2 2 r - v 2 μ a = 2 r - v 2 μ - 1 . Check the dimensions of the semimajor axis here. L. Healy – ENAE404 – Spring 2007 – Lecture 3 (Feb. 1) 2
Summary of orbital motion so far By starting with the law of universal gravita- tion, we get 3 conservation laws, and from those, we can get Kepler’s laws and orbit is in a plane, orbit is a conic section, There’s much else we can find out about the orbit – in fact everything except where the satellite is at a given time; “the orbit in time” problem. The orbital period can be solved but we’ll need Kepler’s equation to find the satel- lite’s position for a particular time anywhere along the orbit. For now let’s explore the implication of these two facts. L. Healy – ENAE404 – Spring 2007 – Lecture 3 (Feb. 1) 3

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The orbit in two dimensions Ref.: Sellers Sec. 5.1 We have looked at the orbit in its plane and un- derstand that there are three parameters that describe a satellite’s motion in this plane: semimajor axis a
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