# lecture13 - Assumptions of two-body motion Only two bodies...

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Unformatted text preview: Assumptions of two-body motion Only two bodies (satellite and planet), point masses (spherical distribution of mass), no other forces acting, satellite has much smaller mass than planet, so center of mass is essentially at the planets center. L. Healy ENAE404 Spring 2007 Lecture 13 (Mar. 8) 1 Physics of two-body motion Law of universal gravitation r =- r 2 r Consequence: Keplers laws Consequence: Conservation laws, or inte- grals of motion energy = v 2 2- r , angular momentum h = r r , eccentricity vector e = r h - r . L. Healy ENAE404 Spring 2007 Lecture 13 (Mar. 8) 2 Conic section equation From the conservation laws, one can show orbital motion is in a plane, figure in the plane is a conic section, polar relation in the plane between radius r and true anomaly , r = p 1 + e cos , the formula for a conic section in polar co- ordinates. L. Healy ENAE404 Spring 2007 Lecture 13 (Mar. 8) 3 Ellipse quantities b = a q 1- e 2 is the semiminor axis, p = h 2 / = a (1- e 2 ) semilatus rectum, r p = a (1- e ) is the perigee distance, r a = a (1 + e ) is the apogee distance, energy =- 2 a , speed v = s 2 r- 1 a , orbital period T = 2 v u u t a 3 . L. Healy ENAE404 Spring 2007 Lecture 13 (Mar. 8) 4 State and orbital elements Because there are three degrees of freedom, six coordinates must be specified to describe a satellite in orbit. Cartesian state: r , r . Orbital elements: we have a , e for planar motion, and ( t ) giving the satellites angle from perigee at any time, add three more for embedded the plane in 3D space: inclination i right ascension of ascending node , argument of perigee . Singularities in the elements: e = 0, no perigee undefined, i = 0, no node undefined. Many variations of orbital elements; this is one set. Usually some variation of this set....
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lecture13 - Assumptions of two-body motion Only two bodies...

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