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Planetary motion

D.N.C. Lin D epartment o f A stronomy & A strophysics University of California, Santa Cruz. Planetary motion. Lecture 3, AY 222 Apr 9 th , 2012. Class contents. Equation of motion: 1) Star-planet 2 body interaction: time scale 1 year

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Planetary motion

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  1. D.N.C. Lin Department of Astronomy & Astrophysics University of California, Santa Cruz Planetary motion Lecture 3, AY 222 Apr 9th, 2012

  2. Class contents Equation of motion: 1) Star-planet 2 body interaction: time scale 1 year conservation of energy and angular momentum 2) Impulsive planetary perturbation: random small changes on time scale 1 year 3) Planet’s secular perturbation: repeated forcing on a few thousand orbital time scale correlated angular momentum (eccentricity) changes 4) Multiple planets secular perturbation Normal mode response 5) Resonant interaction: repeated forcing on a few hundred orbital time scale correlated energy (semi major axis) changes 6) Multiple (overlapping) resonances interaction: dynamical instability

  3. Copernican perception & Galileo’s many worlds Conceptual bases 3/24

  4. Kepler’s order & Newton’s motion F=ma gravity, angular momentum => first push ? System architecture: a grand design? 4/24

  5. Two body elliptical orbits F=ma; F= - G M1M2r/r3 Central force field: conservation of angular momentum Stationary potential: conservation of energy

  6. Basic mathematics Two bodies in an inertial frame, Fi,j=-GMiMj(ri– rj)/dij3 In a frame centered on the Sun, eq of motion: d2 r/dt2 = -mr/r3 • = - G(Mi+ Mj) includes an indirect term. Nonlinear force: but has analytic solutions Elliptical orbits: semi major axis a, Eccentricity e & longitude of peri apse v Conservation: h= rxdr/dt= [ma(1-e2)]1/2 E=- m/2av Period P= 2p(a3/ m)1/2 Mean motion n=2p/P Solutions to the Kepler’s laws 5/24

  7. Observer’s view * A – Minor, orbiting body * B – Major body being orbited by A * C – Reference plane (e.g. the en:ecliptic) * D – Orbital plane of A * i – Inclination The green line is the node line; going through the ascending and descending node; this is where the reference plane (C) intersects the orbital plane (D).

  8. Elliptical orbits * A – Minor, orbiting body * B – Major body being orbited by A * C – Reference plane, e.g. the * D – Orbital plane of A * E – Ascending node * F – Periapsis * ω – Argument of the periapsis The red line is the line of apsides; going through the periapsis (F) and apoapsis (H); this line coincides wíth the major axix in the elliptical shape of the orbit The green line is the node line; going through the ascending (G) and descending node (E); this is where the reference plane (C) intersects the orbital plane (D).

  9. Restricted 3 body problem Newton’s nightmare: precession of moon’s orbit 3-body problem => more complex dynamics Restricted 3 bodies: M3<< M1,2 , 1 & 2 circular orbit, 2 sources of gravity m1,2= M1,2/(M1+M2) Symmetry & conservation: Time invariance is required for ``energy’’ conservation => rotating frame In a co-rotating frame centered on the C of M, normalized with G(M1+M2)=1 & a12= 1, eq of motion: d2x/dt2 - 2n dy/dt –n2x = -[m1(x+m2)/r13+m2(x-m1)/r23] d2y/dt2+ 2n dx/dt –n2y = -[m1/r13+m2/r23] y d2z/dt2 = -[m1/r13+m2/r23] z where r12= (x+m2)2+y2+z2 and r22= (x+m1)2+y2+z2 Note: 1)additional Corioli’s and centrifugal forces. 2)coordinate can be centered on the Sun 6/24

  10. Equi-potential surface and Roche lope Energy & angular momentum are not conserved. Conserved quantity: Jacobi ``energy’’ Integral CJ = n2(x2 + y2) + 2(m1/r1+m2/r2)-(x2+y2+z2) Roche radius: distance between the planet and L1 rR = (m2/3m1)1/3 a12 (to first order in m2/m1) Hill’s equation is an approximation m1 =1 7/24

  11. Bound & horseshoe orbits, capture zone • and W precession • e and a variations + = epicycle Guiding center Match CJ at L2 and Keplerian orbit at superior conjunction => Da = (12)1/2 rR 8/24

  12. Less restricted 3-body problem More general cases: a) non circular orbit for body 2, b) M2 and M3 are comparable but << M1 Legrange-Laplace planetary equations: a) series expansion of orbital elements, b) time average periodic (rapidly varying) parameters, c) secular (long term) variations of e, v, a, … (derivations in Murray & Dermott ``Solar System Dynamics’’) In a heliocentric (non rotating) frame, gravity on a planet i is 9/24

  13. secular interaction between 2 planets To the lowest order in e, no energy change & only angular momentum exchange: da/dt=0 where Ci is laplacecoef , L is external torque, & N non point-mass potential 10/24

  14. Libration & circulation 11/24

  15. A star plus 3 planets (Mardling & Lin 2002) 12/24

  16. Additional forces 13/24

  17. Orbital elements With Runge-Lenz vector 14/24

  18. Orbit averaging 15/24

  19. Other averaged forces 16/24

  20. Feedback to the outer bodies 17/24

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