Asteroid Resonances [2]

Kuliah AS8140 Fisika Benda Kecil Tata Surya

dan AS3141 Benda Kecil dalam Tata Surya

Budi DermawanProdi Astronomi 2006/2007

Mean Motion Resonance (MMR)

First order resonance

Second order resonance

21

32

~/Δ enn

21

2 ~/Δ enn

Malhotra 1998

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

MMR (3): Stability

Stable

Unstable

MMR (4)For the p : p + q resonance, the power series expansion of the mutual perturbation potential of a pair of satellites:

q

r

rqrpr

q

r

rqrprqp

rqrpqpiiDa

mm

rqrpqpeeCa

mmV

0211221

2

21

0211221

2

21,

Ω)(Ω)(cos

)()(cos

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

MMR (5) The nominal location of the MMR is defined by: (p + q)n2 – pn1 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:

121

2121

/~/Δ

ΩΩ~ and ~

naa jj

• 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)

MMR (6): isolated resonance

Ex.: first order interior eccentricity-type p : p + 1 The essential lowest order perturbation terms:

Mmaa

ptnpeCeBeAmV

/ ,/

)1(cos)()()(/

221

242

1

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 pe2

MMR (7): isolated resonance

The scaled resonance hamiltonian:

sin,cos2,

24

1

2

3cos223

222222

Iyx

xyxyxIIIH

Poincaré variables

The eccentricity is (to lowest order) then proportional to the distance from the origin in the (x,y) plane:

223

1

23

)(yx

p

Ce

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

MMR (9): resonance widtho For | | >> 1, oscillations in (x,y) are nearly sinusoidal with

frequency 3 and amplitude ~ (2/3)| |-1

o In the vicinity of 0 oscillations are non-sinusoidal with a maximum amplitude of 25/3 at = 21/3

o Just above = 21/3, the amplitude drops to half the maximumo On the other side of the maximum, the half-maximum

amplitude occurs at a value of 0.42. Thus fwhm 2 for initially circular orbits

o The resonance width and the maximum eccentricity excitation for initially circular orbits:

31

32

2max 3

)(42 ;)(9Δ

p

CenCpnp

MMR (10): resonance widthFor eccentric orbits (equivalently, large values of ) the resonance width and the frequency of small amplitude oscillations about the resonance:

neCp

neCn

21

21

)(3

)(34Δ

20

is the (forced) eccentricity at the center of libration

e

Malhotra 1998

Nesvorný et al. 2002

MMR (11): chaotic diffusion

2J:1 & 3J:1 MMR

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

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

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

MMR (15): TNOs regionThe 1:2, 3:4, and weaker resonances

Nesvorný & Roig 2001