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Unformatted text preview: Orbital dyanmics : Calculus Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA email@example.com Webpage http://www.math.ufl.edu/ squash/ 4 October, 2009 (at 14:11 ) ( Also ~ /Problems/Analysis/Calculus/tunnel.latex ) N.B. The following notes are preliminary. Abbreviations. For common numbers, use := 2 and 2 := 2 . ( Mnemonically: The angle of a Circle and root 2. ) Use i.p.t for is proportional to, with as sym- bol. Use SoG for Source of Gravity; the center of an inverse-square rotationally symmetric grav- ity field. Language: A planet rotates about its axis, and revolves about the Sun. Notation. Planet Pal has D := Day , R := Radius , A := Sur-Acc . Pal is in orbit about sun Sol , with Y := Year , U := OrbitalRadiu s . For a constant-speed object traveling in a circle we use := period , := radius , s := speed . Hence s = . 1 : Twice time-differentiating this circular motion, we see that the inward acceleration of the object is Accel. of motion = s 2 / . 2 : Newton tells us, at distance from a SoG, that Accel. from Grav = K / 2 , 2 : where K is a constant of proportionality that depends on the planet; it has units d 3 / t 2 . 1 So (2) & (2 ) show that an s,,-orbit satisfies K = s 2 (1) = 3 2 2 , i.e, 3 = K 2 2 . 3 : 1 If the SoG comes from a mass m , then K equals m times the Universal Gravitational Constant....
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