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Some Topics on Solar System Dynamics

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Some Topics on Solar System Dynamics Some topics on Solar System Dynamics Tabaré Gallardo www.fisica.edu.uy/sgallardo/cospar/2015 Facultad de Ciencias Universidad de la República Uruguay COSPAR CBW on Planetary Missions Data Analysis Guaratinguetá, October 2015 Tabaré Gallardo Solar System Dynamics Topics Introduction Gauss equations Secular theory Chaos Kozai-Lidov Mean motion resonances Spin - Orbit resonances Lindblad resonances Tabaré Gallardo Solar System Dynamics Forces in the Solar System Generated by gravity (Newtonian + relativistic) due to Sun, planets, satellites, asteroids. Model: N point masses + perturbations due to non-sphericity. solar radiation radiation pressure (µm): in the direction of the radiation Poynting-Robertson drag (cm): (Doppler) opposite to velocity generates migration to the Sun Yarkovsky effect (from m to km): (thermal inertia) depending on rotation generates migration to or from the Sun sublimation in comets NGF medium: solar wind, gas drag. magnetic fields: Lorentz forces. collisions Tabaré Gallardo Solar System Dynamics Orbital evolution given a Newtonian attraction µ ~¨r = − ^r r2 and some initial conditions (~r;~r_) an orbit is defined: (a; e; i; !; Ω; τ) constant energy " = −µ/2a constant angular moment h = r2f_ The problem: given an extra acceleration F~p or perturbation we want to know a(t); e(t); i(t); ::: Tabaré Gallardo Solar System Dynamics The Tisserand Relation (4) Therefore the comet’s elements before and after the encounter withOrbital Jupiter are evolutionrelated by: WITH close encounters large perturbations Numerical example: drastic orbital changes only numerical methods (clones), statistical studies (Öpik) conserved quantity for RC3BP: Tisserand or Jacobi’s constant no secular evolution a r a C ' p + 2 (1 − e2) cos i = T a ap Tabaré Gallardo Solar System Dynamics Orbital evolution WITH close encounters p v1 ' 3 − T = U U is the encounter velocity with the planet before the gravitational attraction is felt by the particle (that means "at infinity"). The orbital elements (a; e; i) can evolve but T and U remain constant, only the orientation of U~ is modified. It follows that when T > 3 encounters cannot exist. Tabaré Gallardo Solar System Dynamics Orbital evolution WITH close encounters T is a good parameter for classification of small bodies. (from Bertotti et al. 2003) Tabaré Gallardo Solar System Dynamics Orbital evolution WITHOUT close encounters small perturbations small orbital changes NEPTUNO analytical methods ) SOL theoretical predictions conserved quantities: 2000 CR105 "energy", z component of angular moment secular evolution Tabaré Gallardo Solar System Dynamics Gauss equations Tabaré Gallardo Solar System Dynamics Gauss equations: effects on (a; e; i; $; Ω) F~p ) effects on orbital elements T R µ ~¨ ^ ~ r = − 2 r + Fp asteroid r F~ = R^r + T^t + Nn^ SUN p Energy: " = −µ/2a. Variation in energy: d" d~r µ da = F~ · = rR_ + rf_ T = dt p dt 2a2 dt Tabaré Gallardo Solar System Dynamics Gauss equations Instantaneous perturbation: da a3=2 ) = 2 [Re sin f + T(1 + e cos f )] dt pµ(1 − e2) Mean over one orbital period P: da 1 Z P da < >= · dt dt P 0 dt da If < dt >6= 0 ) cumulative effect. Tabaré Gallardo Solar System Dynamics Gauss equations Considering that angular momentum is: df h = r2 dt we change the variable: r2 dt = df h da 1 Z 2π da r2 )< >= df dt P 0 dt h Tabaré Gallardo Solar System Dynamics Example 1: a comet with NGF Consider a comet with We obtain: ~ 1 Fp ∼ 2 (R; T; N) da 2 r < >= p T dt (1 − e2) aµ p 2 de 1 − 1 − e < >= T dt epa3µ The radial component is irrelevant, only T matters. Tabaré Gallardo Solar System Dynamics Example 2: Yarkovsky effect for an asteroid da 2 < >= p T dt (1 − e2) aµ prograde rotation: T > 0, then mean da=dt > 0, goes away retrograde: T < 0, then mean da=dt < 0, goes to the Sun Tabaré Gallardo Solar System Dynamics Yarkovsky effect: simulating a family Broz, yarko-site Tabaré Gallardo Solar System Dynamics Yarkovsky effect (Broz et al. 2005) After a collision a family is generated: the smaller fragments (higher magnitude) are the most affected by Yarkovsky (so, the most dispersed). This effect can help us in the determination of the age of the family. Tabaré Gallardo Solar System Dynamics Example 3: gas drag F~p = (R; T; 0) < a_ >/ T < 0 < e_ >/ T < 0 da=dt < 0 fall down de=dt < 0 circularization Tabaré Gallardo Solar System Dynamics Example 4: axisymmetric oblate planet GM h R2 i V(r; φ) = − 1 − J P (sin φ) + ::: r 2 2 r acceleration: ~α = −∇V(r; φ) perturbation: R N F~ = ^r + n^ p r3 r3 T = 0 (symmetry) Tabaré Gallardo Solar System Dynamics Example 4: axisymmetric oblate planet < da=dt >= 0 < de=dt >= 0 < di=dt >= 0 d$=dt > 0, advance of the perihelion dΩ=dt < 0, precession of the nodes It is a very typical orbital behaviour. Tabaré Gallardo Solar System Dynamics Secular Theory Pioneers of the Three-Body Problem Euler Laplace Lagrange Jacobi LeVerrier Hamilton Birkoff Poincaré Tabaré Gallardo Solar System Dynamics Perturbation Theory Consider an asteroid at~r perturbed by a planet at~rp. It is possible to write the equation of motion in the form ~r ~¨r + µ = rR(~r;~r ) r3 p where R is the Disturbing Function. It is possible to transform this equation in another very different form due to Lagrange (+ Euler + Laplace): da 2 @R = dt na @λ p p de 1 − e2 p @R 1 − e2 @R = − 1 − 1 − e2 − dt na2e @λ na2e @$ di tan i @R @R 1 @R = − p 2 + − p dt na2 1 − e2 @λ @$ na2 1 − e2 sin i @Ω Tabaré Gallardo Solar System Dynamics Perturbation Theory: Lagrange’s planetary equations p d$ 1 − e2 @R tan i @R = + p 2 dt na2e @e na2 1 − e2 @i dΩ 1 @R = p dt na2 1 − e2 sin i @i R is a very unfriendly function X R = Ck(a; e; i) cos(σk) k where functions σk(λp; λ, $; Ω) are linear combinations of λp; λ, $; Ω. Tabaré Gallardo Solar System Dynamics Perturbation Theory The λs are quick varying angles, on the contrary $; Ω are slow varying angles. Then: R = RSP($; Ω; λ, λp) + RLP($; Ω) Instead of full R we consider the mean over the quick varying angles λ, λp, then: R ' RLP($; Ω) this part of the disturbing function is the responsible for the long term secular evolution of the system. Tabaré Gallardo Solar System Dynamics Perturbation Theory: secular evolution Taking R ' RLP the first of the Lagrange’s planetary equations becomes: da 2 @R ' LP = 0 dt na @λ p de 1 − e2 @R ' − LP dt na2e @$ ) the semimajor axes of the planets do not change with time... the planetary system do not shrinks nor expands That was a very impacting result of the XVIII century due to Euler, Lagrange and Laplace. In fact a(t) = asec+ small amplitude oscillations. It is also possible to show that e and i do not grow systematically but oscillate. Tabaré Gallardo Solar System Dynamics Perturbation Theory: Delaunay canonical variables Working in canonical variables (M; !; Ω; L; G; H) where: p L = µa, momentum conjugate of M G = pµa(1 − e2), momentum conjugate of ! H = pµa(1 − e2) cos i, momentum conjugate of Ω dM @H dL @H = = − dt dL dt dM d! @H dG @H = = − dt dG dt d! dΩ @H dH @H = = − dt dH dt dΩ v2 µ µ µ2 H = − + R = − + R = − + R 2 r 2a 2L2 Tabaré Gallardo Solar System Dynamics Perturbation Theory: Delaunay canonical variables µ2 H (−; !; Ω; L; G; H) = − + R sec 2L2 sec dL @H = − sec = 0 dt dM p ) L = µa = constant that means a = constant Then, secular evolution ) a = constant. Tabaré Gallardo Solar System Dynamics Numerical integrations of the exact equations Given a problem write equations~¨r = ~α for all bodies design an algorithm to calculate~r(t + ∆t) from~r(t) write in some computer language run in a computer we obtain~r(t);~r_(t) and a(t); e(t); i(t);::: Tabaré Gallardo Solar System Dynamics Numerical integrations of the exact equations 1.6 Venus 1.5 Earth Mars 1.4 1.3 1.2 a (au) 1.1 1 0.9 0.8 0.7 0 1 2 3 4 5 time (Myr) Tabaré Gallardo Solar System Dynamics Numerical integrations: detail on Mars zoom a(t) = asec+ small amplitude oscillations 1.52385 1.5238 1.52375 1.5237 a Mars (au) 1.52365 1.5236 1.52355 0 100 200 300 400 500 time (yrs) Tabaré Gallardo Solar System Dynamics Eccentricities 0.14 Venus Earth 0.12 Mars 0.1 0.08 0.06 eccentricity 0.04 0.02 0 0 1 2 3 4 5 time (Myr) Tabaré Gallardo Solar System Dynamics Inclinations 9 Venus 8 Earth Mars 7 6 5 4 i (degrees) 3 2 1 0 0 1 2 3 4 5 time (Myr) Tabaré Gallardo Solar System Dynamics Secular evolution There are 3 methods to obtain Rsec (or Hsec) from R: canonical transformations scissors (just dropping SP terms) numerical averaging of the exact R: 1 Z 2π Z 2π Rsec = 2 R dλ dλp 4π 0 0 Tabaré Gallardo Solar System Dynamics Asteroid: secular evolution for small e; i assuming e; i small define I = sin(i=2) the secular equations become de 1 @R d$ 1 @R = − = dt na2e @$ dt na2e @e dI 1 @R dΩ 1 @R = − = dt na2I @Ω dt na2I @I with a = constant Tabaré Gallardo Solar System Dynamics Asteroid: secular evolution Changing variables (e; $; I; Ω) ! (h; k; p; q) h = e sin $ p = I sin Ω k = e cos $ q = I cos Ω ) R(h; k; p; q) New equations: dh @R dp @R = C = C dt @k dt @q dk @R dq @R = −C = −C dt @h dt @p 1 With C = na2 and discarding high order terms in R we obtain the solution =) Tabaré Gallardo Solar System Dynamics Asteroid: secular evolution h = e sin $; k = e cos $ h(t) = eproper sin(At + β) + hforced k(t) = eproper cos(At + β) + kforced with A(a), proper frequency Murray and Dermott 1999 Osculating eccentricity: p 2 2 eosc(t) = h + k Tabaré Gallardo Solar System Dynamics Asteroid: secular evolution p = I sin Ω; q = I cos Ω p(t) = Iproper sin(Bt + γ) + pforced q(t) = Iproper cos(Bt + γ) + qforced with B(a), proper frequency Murray and Dermott 1999 Osculating inclination: p 2 2 Iosc(t)
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