Asteroid Resonances [1]

Kuliah AS8140 Fisika Benda Kecil Tata Surya

dan AS3141 Benda Kecil dalam Tata Surya

Budi DermawanProdi Astronomi 2006/2007

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

Spin-orbit Resonance (1)

o Ex: spin-locked state of the Moon, most natural satellites (Pluto-Charon), binary stellar systems

o 1:1 spin-orbit resonance (synchronous spin state)o 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:

C

ABf

r

GM

2

)(3 ;2cos

3

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

n 22Δ

Spin-orbit Resonance (3)

Case when the orbit is non-circular and small eccentricity

2212 2sin72sin2sin entntentn

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)

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]

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

Secular Resonances (1)

• A planetary precessing ellipse of fixed semimajor axis, ap, eccentricity, ep, and precession rate pp g

• 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

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:

)cos()()()()( 420sec pp

m

p eeCeBeAAa

mH

)(4

1)(

for ,)(812128

1

for ,)(128

1)(

)(8

1)(

)2(2/3

)1(2/32

22

)1(2/32

23

)1(2/3

bC

aabd

d

d

d

aabd

dB

bA

p

p

Secular Resonances (3)

Using the canonically conjugate Delaunay variables - and J = a(1- (1-e2))

pp

m

pm

p

m

pm

p

emaa

Cm

aa

BAm

aa

Ag

JJJgAa

mH

41

21

)( ,

)(4)( ,

)(2

)cos(2)(

0

200sec

Writing the Hamilton’s equations for the Poincaré variables)sin(cos2),( Jyx

)cos( ),sin( 0,00,0 pppp tgxgytgygx 0ppp tg Using

Secular Resonances (4)

Solution: )sin(),cos()(),( 0,0,0

forced ppppp

tgtggg

tytx

At exact resonance (g0 = gp)

)}cos(),{sin()(),( 0,0,resonance pppp tgtgttytx

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:

3

1

max

3

2

max 4

22 ,

22 pe

BA

CeJ

Secular Resonances (6)

Inner edge of MBAs: 6 secular resonance (g0 g6),

g6 28.25/yr mean perihelion precession rate of Saturn’s orbit

iii

i

i

ii i

ii

ii

i

i

Emaa

C

maa

BA

maa

Ag

JJJgH

)6(

0

62

0sec

41

21

)(

,)(4)(

,)(2

)cos(2

Hamiltonian for the 6 resonance (i = a/ai)

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.

Secular Resonances (8)

Empty zones along resonant surfaces

Isolation of groups (Hungaria, Phocaea)

Primordial Excitation & Depletion of MBAs

Canonical variables For east, iast << 1

Petit et al. 2002

Primordial Excitation & Depletion of MBAs

The play of (secular) resonances

Petit et al. 2002