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Asteroid Resonances [2] Kuliah AS8140 Fisika Benda Kecil Tata Surya dan AS3141 Benda Kecil dalam Tata Surya Budi Dermawan Prodi Astronomi 2006/2007

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Mean Motion Resonance (MMR) First order resonance Second order resonance Malhotra 1998

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MMR (2) o Mean motion commensurabilities amongst the Jovian and Saturnian satellites o No exact resonance in the Uranian satellites system o The role of the small but significant splitting of MMR and the interaction of neighboring resonances o Destabilize a previously established resonance MMR lifetimes

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MMR (3): Stability Stable Unstable

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MMR (4) For the p : p + q resonance, the power series expansion of the mutual perturbation potential of a pair of satellites: Subscripts 1 & 2 refer to the inner and outer satellites, ’s are the instantaneous mean longitudes, and are the longitudes of periapse and ascending node For every pair, p, q, there are q + 1 lowest order terms in the eccentricity, and also q + 1 terms in the inclination

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MMR (5) The nominal location of the MMR is defined by: (p + q)n 2 – pn 1 0 But, the resonance is actually split into several sub-resonances defined by each distinct term in the series The locations of the sub-resonances differ by: in frequency: in semimajor axis: If the splitting between neighboring sub-resonances is much greater than the sum of their half-widths, each sub-resonance can be analyzed in isolation When the separation between neighboring resonances is comparable to their widths, the interaction between resonances is strong and a strong instability of the motion occurs most orbits in the vicinity of the resonances are chaotic (very narrow restricted region of stable resonance-locking)

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MMR (6): isolated resonance Ex.: first order interior eccentricity-type p : p + 1 The essential lowest order perturbation terms: With little error, we may evaluate the coefficients A ( ), B ( ), C ( ) at = res = (1+1/p) -2/3 Using the canonical Delaunay variables Hamiltonian function The resonance-induced variations of a and e : a/a p e 2

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MMR (7): isolated resonance The scaled resonance hamiltonian: Poincaré variables The eccentricity is (to lowest order) then proportional to the distance from the origin in the (x,y) plane:

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MMR (8): phase space topology A separatrix (period is unbound) exists for > crit = 1 The separatrix divides the phase space into three zones: an external, an internal, and a resonance zone Malhotra 1998

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MMR (9): resonance width oFor | | >> 1, oscillations in (x,y) are nearly sinusoidal with frequency 3 and amplitude ~ (2/3)| | -1 oIn the vicinity of 0 oscillations are non-sinusoidal with a maximum amplitude of 2 5/3 at = 2 1/3 oJust above = 2 1/3, the amplitude drops to half the maximum oOn the other side of the maximum, the half-maximum amplitude occurs at a value of Thus fwhm 2 for initially circular orbits oThe resonance width and the maximum eccentricity excitation for initially circular orbits:

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MMR (10): resonance width For eccentric orbits (equivalently, large values of ) the resonance width and the frequency of small amplitude oscillations about the resonance: is the (forced) eccentricity at the center of libration Malhotra 1998

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Nesvorný et al MMR (11): chaotic diffusion 2J:1 & 3J:1 MMR

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MMR (12): MBAs global structure Overlapping MMR causes chaotic orbits “Stable chaos”: have strongly chaotic orbits yet are stable on long interval time (three-body resonances) Nesvorný et al. 2002

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MMR (13): MBAs global structure Each resonance corresponds to one V-shaped region except the large first-order MMRs with Jupiter Nesvorný et al. 2002

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MMR (14): TNOs region The 2:3 resonance with Neptune Nesvorný & Roig 2000

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MMR (15): TNOs region The 1:2, 3:4, and weaker resonances Nesvorný & Roig 2001

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