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**Asteroid Resonances [1]**

Kuliah AS8140 Fisika Benda Kecil Tata Surya dan AS3141 Benda Kecil dalam Tata Surya Budi Dermawan Prodi Astronomi 2006/2007

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**General Types of Resonance**

Spin-orbit resonance: a commensurability of the rotation period of a satellite with the period of its orbital revolution Secular resonance: a commensurability of the frequencies of precession of the orientation of orbits (direction of perihelion and of the orbit normal) Mean motion resonance: the orbital periods of two bodies are close to a ratio of small integers

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**Spin-orbit Resonance (1)**

Ex: spin-locked state of the Moon, most natural satellites (Pluto-Charon), binary stellar systems 1:1 spin-orbit resonance (synchronous spin state) For a non-spherically shaped satellite (principal moment of inertia: A < B < C, is the orientation relative to the direction of periapse of the orbit, f = f(t) is the true anomaly, and r = r(t) is the distance from the planet), e.o.m:

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**Spin-orbit Resonance (2)**

Rotational symmetry (B = C): no torque from the planet and the satellite’s spin in unperturbed If B C and the orbit is circular, e.o.m is similar to that of the common pendulum The width of the 1:1 spin-orbit resonance (n is the orbital mean motion) is

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**Spin-orbit Resonance (3)**

Case when the orbit is non-circular and small eccentricity Two new terms corresponding to the 1:2 and the 3:2 spin-orbit resonances The width of the 3:2 spin-orbit resonance is a factor (7e/2) smaller than the 1:1 Ex.: the 3:2 spin orbit resonance of Mercury (88d:59d)

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Orbital Resonances (1) Three degrees of freedom: three angular variables [1] the motion of the planet: the frequency revolution around the Sun, [2] orientation of the orbit in space: the slow frequencies of precession of the direction of perihelion and the pole of the orbit plane For a multi-planet system: secular resonances involves commensurabilities amongst [2]; mean motion resonances are commensurabilities of [1]

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Orbital Resonances (2) Most cases: a clear separation of [1] & [2] time scales A coupling between [1] & [2] chaotic dynamics The boundaries (or separatrices) of mean resonances are often the site for such interactions between secular and mean motion resonances Ex. of “hybrid” resonance (a commensurability of a secular precession frequency with an orbital mean motion): the angular velocity of the apsidal precession rate of a ringlet within the C-ring of Saturn is commensurate with the orbital mean motion of Titan the Titan 1:0 apsidal resonance

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Secular Resonances (1) A planetary precessing ellipse of fixed semimajor axis, ap, eccentricity, ep, and precession rate g0 is proportional to the mass of the perturbing planet and is also a function of the orbital semimajor axis of the particle relative to that of the planet Secular resonance occurs when g0 equals gp Effect: to amplify the orbital eccentricity of the particle

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Secular Resonances (2) am = max(a,ap), = min{a/ap, ap/a}, a is the semimajor axis of particle, and p are the longitude of periapse of the test particle and of the planet’s orbit, and Laplace coefficients:

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Secular Resonances (3) Using the canonically conjugate Delaunay variables - and J = a(1- (1-e2)) Writing the Hamilton’s equations for the Poincaré variables Using

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**Secular Resonances (4) Solution: At exact resonance (g0 = gp)**

When g0 gp the non-linear terms limit the growth of the eccentricity For orbits with initial (x,y) = (0,0) the maximum excitation occurs at g0 = gp+3(2/2)1/3, and the maximum amplitude:

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Secular Resonances (6) Inner edge of MBAs: 6 secular resonance (g0 g6), g6 28.25/yr mean perihelion precession rate of Saturn’s orbit Hamiltonian for the 6 resonance (i = a/ai)

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Secular Resonances (7) Specific secular resonance: “Kozai resonance”, or “Kozai mechanism” 1:1 commensurability of the secular precession rates of the perihelion and the orbit normal such that the argument of perihelion is stationary (or librates) Requires significant orbital eccentricity and inclination (causes coupled oscillations) Well known ex.: Pluto whose argument of perihelion librates about 90 deg.

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**Secular Resonances (8) Empty zones along resonant surfaces**

Isolation of groups (Hungaria, Phocaea)

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**Primordial Excitation & Depletion of MBAs**

Petit et al. 2002 Canonical variables For east, iast << 1

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**Primordial Excitation & Depletion of MBAs**

The play of (secular) resonances Petit et al. 2002

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Celestial Mechanics VII

Celestial Mechanics VII

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