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Unformatted text preview: Orbital dyanmics : Calculus Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 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 π := 2 π 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 , ρ := radius , s := speed . Hence s · ‘ = ρ · π . 1 : Twice time-differentiating this circular motion, we see that the inward acceleration of the object is Accel. of motion = s 2 /ρ . 2 : Newton tells us, at distance ρ from a SoG, that Accel. from Grav = K /ρ 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,ρ,‘-orbit satisfies K = s 2 · ρ (1) = ρ 3 ‘ 2 · π 2 , i.e, ρ 3 = K ‘ 2 · π 2 . 3 : ♥ 1 If the SoG comes from a mass m , then K equals m times the Universal Gravitational Constant....
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
- Fall '07