Three Body Mean Motion Resonances Tabar Gallardo Departamento de - - PowerPoint PPT Presentation

▶
three body mean motion resonances
SMART_READER_LITE
LIVE PREVIEW

Three Body Mean Motion Resonances Tabar Gallardo Departamento de - - PowerPoint PPT Presentation

Mar 01, 2024 467 likes •1.31k views

Three Body Mean Motion Resonances Tabar Gallardo Departamento de Astronoma Facultad de Ciencias Universidad de la Repblica Uruguay Luchon, September 2016 Tabar Gallardo Three Body Resonances preliminaries types of three body

slide-1
SLIDE 1

Three Body Mean Motion Resonances

Tabaré Gallardo

Departamento de Astronomía Facultad de Ciencias Universidad de la República Uruguay

Luchon, September 2016

Tabaré Gallardo Three Body Resonances

slide-2
SLIDE 2

preliminaries types of three body resonances (3BRs) semi analytical method numerical studies

dynamical maps induced migration

Tabaré Gallardo Three Body Resonances

slide-3
SLIDE 3

Preliminaries

Tabaré Gallardo Three Body Resonances

slide-4
SLIDE 4

Preliminaries

e = eccentricity a = semimajor axis (in astronomical units) n = mean motion = mean angular velocity =

2π period ∝ 1 a3/2

Two body resonance: k0n0 + k1n1 ≃ 0 with k0, k1 integers.

Tabaré Gallardo Three Body Resonances

slide-5
SLIDE 5

Non resonant asteroid: relative positions

Mean perturbation is radial: Sun-Jupiter

Sun Jupiter

Tabaré Gallardo Three Body Resonances

slide-6
SLIDE 6

Resonant asteroid

Mean perturbation has a transverse component.

Sun Jupiter

Tabaré Gallardo Three Body Resonances

slide-7
SLIDE 7

from Gauss equations

SUN

asteroid

R T

Fperturb = (R, T, N) da dt ∝ (R, T) < da dt >∝ T Non resonant T = 0 ⇒ a = constant Resonant T = 0 ⇒ a = oscillating

Tabaré Gallardo Three Body Resonances

slide-8
SLIDE 8

Critical angle σ

For resonance k0n0 + k1n1 ≃ 0, is defined: σ = k0λ0 + k1λ1 + γ(̟0, ̟1) the λ’s are quick varying angles (mean longitudes) γ(̟0, ̟1) is a linear combination of slow varying angles σ(t) indicates if the motion is resonant or not:

σ(t) oscillating means resonance σ(t) circulating means NO resonance

resonant motion: a(t) is correlated with σ(t)

Tabaré Gallardo Three Body Resonances

slide-9
SLIDE 9

Semimajor axis: width

Nesvorny et al. in Asteroids III Tabaré Gallardo Three Body Resonances

slide-10
SLIDE 10

Two body resonance, restricted case m0 = 0

k0n0 + k1n1 ≃ 0 P1 does not feel the resonance, only P0

Tabaré Gallardo Three Body Resonances

slide-11
SLIDE 11

Two body resonance

k0n0 + k1n1 ≃ 0 Order: q = |k0 + k1| Strength of resonance is approximately ∝ Cm1eq Theories try to obtain expressions for coefficients C Strength is related with amplitude of a(t)

Tabaré Gallardo Three Body Resonances

slide-12
SLIDE 12

Two body resonance, planetary case m0 = 0

k0n0 + k1n1 ≃ 0 both P0 and P1 feel the resonance

Tabaré Gallardo Three Body Resonances

slide-13
SLIDE 13

Two body resonance, planetary case

Observational evidence in extrasolar systems

Tabaré Gallardo Three Body Resonances

slide-14
SLIDE 14

Three body resonances

Tabaré Gallardo Three Body Resonances

slide-15
SLIDE 15

Three body resonance, restricted case m0 = 0

k0n0 + k1n1 + k2n2 ≃ 0

  • nly P0 feels the resonance

Tabaré Gallardo Three Body Resonances

slide-16
SLIDE 16

Three body resonance

Order: q = |k0 + k1 + k2| Strength of resonance is approximately ∝ Cm1m2eq 3BRs are weaker than 2BRs (m1m2 << m1) Theories try to obtain expressions for coefficients C Only planar theories have been developed

Tabaré Gallardo Three Body Resonances

slide-17
SLIDE 17

Strength and eccentricity

1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 1 0.01 0.1 ∆ρ eccentricity q=0 q=1 q=2 q=3 q=4

for low e strength ∝ eq

Tabaré Gallardo Three Body Resonances

slide-18
SLIDE 18

Three body resonance, planetary case m0 = 0

k0n0 + k1n1 + k2n2 ≃ 0 all three bodies feel the resonance

Tabaré Gallardo Three Body Resonances

slide-19
SLIDE 19

3BR 4n0 − 1n1 − 2n2, planetary case m0 = 0

Tabaré Gallardo Three Body Resonances

slide-20
SLIDE 20

3BR 4n0 − 1n1 − 2n2, restricted case m0 = 0

Tabaré Gallardo Three Body Resonances

slide-21
SLIDE 21

Three body resonance

k0n0 + k1n1 + k2n2 ≃ 0 It is not necessary to have a chain of 2BRs: P0 and P1 not in two body resonance P0 and P2 not in two body resonance P2 and P1 not in two body resonance but...

Tabaré Gallardo Three Body Resonances

slide-22
SLIDE 22

1784: Laplacian resonance

3λEuropa −λIo −2λGanymede ≃ 180◦ 3nEuropa − nIo − 2nGanymede ≃ 0 They are also in commensurability by pairs: 2nEuropa − nIo ≃ 0 2nGanymede − nEuropa ≃ 0 ⇓ It must be the consequence of some physical mechanism.

Tabaré Gallardo Three Body Resonances

slide-23
SLIDE 23

Two types of 3BRs

Three body resonance as... superposition or chain of 2 two-body resonances

nI − 2nE ∼ 0 nE − 2nG ∼ 0 adding: nI − nE − 2nG ∼ 0 ⇒ 3BR order 2 substraction: nI − 3nE + 2nG ∼ 0 ⇒ 3BR order 0

pure: 3BR that are NOT due to 2BR + 2BR.

asteroids + Jupiter + Saturn

Tabaré Gallardo Three Body Resonances

slide-24
SLIDE 24

Asteroids: histogram of a + 2BRs

2 2.2 2.4 2.6 2.8 3 3.2 3.4 log (Strength) a (au) 1:2 Mars 3:1 Jup 2:1 Jup 4:7 Mars 5:2 Jup

Tabaré Gallardo Three Body Resonances

slide-25
SLIDE 25

Asteroids: histogram of a + 3BRs

2 2.2 2.4 2.6 2.8 3 3.2 3.4 log (Strength) a (au) 1-4J+2S 1-4J+3S 2-7J+4S 1-3J+1S 2-7J+5S 2-6J+3S 1-3J+2S 3-8J+4S 2-5J+2S 3-7J+2S 3-8J+5S 2-1M

Tabaré Gallardo Three Body Resonances

slide-26
SLIDE 26

Dynamical evidence from AstDyS

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 3 3.05 3.1 3.15 3.2 3.25 3.3 proper e proper a (au) Tabaré Gallardo Three Body Resonances

slide-27
SLIDE 27

Thousands of asteroids in 3BRs with Jupiter and Saturn

Massive identification of asteroids in three-body resonances

Evgeny A. Smirnov, Ivan I. Shevchenko ⇑

Pulkovo Observatory of the Russian Academy of Sciences, Pulkovskoje Ave. 65, St. Petersburg 196140, Russia Icarus 222 (2013) 220–228

Contents lists available at SciVerse ScienceDirect

Icarus

journal homepage: www.elsevier.com/locate/icarus

Smirnov and Shevchenko (2013)

See next talk!

Tabaré Gallardo Three Body Resonances

slide-28
SLIDE 28

Resonance 1 + 1U−2N, a weird case

263.52 263.53 263.54 263.55 mean a (au) 90 180 270 360 100 200 300 400 500 σ (1+1U-2N) time (Myrs) Tabaré Gallardo Three Body Resonances

slide-29
SLIDE 29

3BRs are WEAK and numerous

Given two planets P1 and P2, an infinite family of 3BRs is defined: n0 = −k1n1 − k2n2 k0 Don’t miss the ”TBR Locator” for Android! Each resonance is defined by (k0, k1, k2) The question is: how strong are they? They are weak because the perturbation that drives the resonant motion is factorized by m1m2. There is a huge number of 3BRs: superposition generates chaotic diffusion.

Tabaré Gallardo Three Body Resonances

slide-30
SLIDE 30

Multiplet resonances and chaos

σ = k0λ0 + k1λ1 + k2λ2+k4̟0 + k5Ω0

Figure 8. Separatrices of four multiplet resonances of the 6 1 − 3 three-body resonance.

Nesvorny and Morbidelli (1999) Tabaré Gallardo Three Body Resonances

slide-31
SLIDE 31

Chaotic diffusion: growing e

Morbidelli and Nesvorny (1999) Tabaré Gallardo Three Body Resonances

slide-32
SLIDE 32

Chaotic diffusion in the TNR: growing e

Nesvorny and Roig (2001) Tabaré Gallardo Three Body Resonances

slide-33
SLIDE 33

Three body resonances as...

Chains of two body resonances

Galilean satellites (Sinclair 1975, Ferraz-Mello, Malhotra, Showman, Peale, Lainey...) Callegari and Yokoyama (2010): satellites of Saturn Extrasolar systems (Libert and Tsiganis 2011; Martí, Batygin, Morbidelli, Papaloizou, Quillen...)

Pure three body resonances

Lazzaro et al. (1984): satellites of Uranus Aksnes (1988): zero order asteroidal resonances Nesvorny y Morbidelli (1999): theory Jupiter-Saturn-asteroid Cachucho et al. (2010): diffusion in 5J -2S -2. Quillen (2011): zero order extrasolar systems Gallardo (2014), Gallardo et al. (2016): semianalytic Showalter and Hamilton (2015): Pluto satellites 3nS − 5nN + 2nH ∼ 0

Tabaré Gallardo Three Body Resonances

slide-34
SLIDE 34

Disturbing function

Disturbing function for resonance k0 + k1 + k2: R = k2m1m2

  • j

Pj cos(σj) σj = k0λ0 + k1λ1 + k2λ2 + γj γj = k3̟0 + k4̟1 + k5̟2 + k6Ω0 + k7Ω1 + k8Ω2 Pj is a polynomial function depending on the eccentricities and inclinations which its lowest order term is Ce|k3|

0 e|k4| 1 e|k5| 2

sin(i0)|k6| sin(i1)|k7| sin(i2)|k8|

Tabaré Gallardo Three Body Resonances

slide-35
SLIDE 35

Theories are complicated... it is necessary to consider several Pj cos(σj) with several terms in Pj calculation of the Cs is not trivial

  • nly planar theories exist

To avoid the difficulties of the analytical methods we proposed to calculate R numerically.

Tabaré Gallardo Three Body Resonances

slide-36
SLIDE 36

Semi analytical method

Tabaré Gallardo Three Body Resonances

slide-37
SLIDE 37

Atlas of three body mean motion resonances in the Solar System

Tabaré Gallardo

Departamento de Astronomía, Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay

Icarus

journal homepage: www.elsevier.com/locate/icarus

Tabaré Gallardo Three Body Resonances

slide-38
SLIDE 38

Method

Disturbing function is a mean over all possible resonant configurations. The point: the disturbing function R must be calculated with the perturbed positions. We cannot assume unperturbed ellipses for the three orbits.

Tabaré Gallardo Three Body Resonances

slide-39
SLIDE 39

Method

For a given resonance: consider a large sample of configurations verifying the resonant condition (σ = constant) calculate the mutual perturbations ∆r0, ∆r1, ∆r2 calculate the effect ∆R due to (∆r0, ∆r1, ∆r2) integrate all ∆R and obtain ρ(σ) repeat for several σ ∈ (0, 360) obtaining ρ(σ)

Tabaré Gallardo Three Body Resonances

slide-40
SLIDE 40

Method

Then, being in a resonant configuration

∆r1 ∆r2 ∆r0 SUN

asteroid

and these ∆r generate the ∆R

Tabaré Gallardo Three Body Resonances

slide-41
SLIDE 41

Disturbing function ∼ ρ(σ)

90 180 270 360 ρ(σ) σ 1-3J+1S, e=0.01

Tabaré Gallardo Three Body Resonances

slide-42
SLIDE 42

Asymmetric equilibrium points

Tabaré Gallardo Three Body Resonances

slide-43
SLIDE 43

Zero order resonance, e ≃ 0,05, a ≃ 3,8

90 180 270 360 20 40 60 80 100 sigma (1-2J+1S) time (thousand yrs)

Numerical integration of full equations of motion.

Tabaré Gallardo Three Body Resonances

slide-44
SLIDE 44

90 180 270 360 ρ(σ) σ 1-3J+1S, e=0.01

large variations of ρ with σ is indicative of a strong resonance small variations of ρ with σ is indicative of a weak resonance an extreme of ρ(σ) at some σ means there is an equilibrium point

Tabaré Gallardo Three Body Resonances

slide-45
SLIDE 45

Strength, S

We numerically obtain ρ(σ) We define Strength S = 1 2∆ρ(σ) For planetary case we have 3 strengths Si = 1 2∆ρi(σ) Codes: www.fisica.edu.uy/~gallardo/atlas

Tabaré Gallardo Three Body Resonances

slide-46
SLIDE 46

Strength and order: 3BRs with Jupiter and Saturn

1e-018 1e-016 1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 1 2 3 4 5 6 7 8 9 10 11 12 13 ∆ρ

  • rder q

log(∆ρ) ∝ −q

Tabaré Gallardo Three Body Resonances

slide-47
SLIDE 47

Planetary case. Dependence on e0. Case q = 4.

1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 0.1 strength e0 2P0 - 1P1 + 3P2 e1=e2=0 S0 S1 S2 Tabaré Gallardo Three Body Resonances

slide-48
SLIDE 48

Planetary case. Dependence on e0. Case q = 0.

1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 0.1 strength e0 6P0 - 1P1 - 5P2 S0 S1 S2 Tabaré Gallardo Three Body Resonances

slide-49
SLIDE 49

Dynamical maps

Tabaré Gallardo Three Body Resonances

slide-50
SLIDE 50

Variations in mean a

1.86098 1.861 1.86102 1.86104 1.86106 1.86108 1.8611 1.86112 10000 12000 14000 16000 18000 20000 mean a (au) time (yr) secular chaotic resonant Tabaré Gallardo Three Body Resonances

slide-51
SLIDE 51

take set of initial values (a, e) integrate for some 10.000 yrs calculate the mean < a > in some interval calculate the variation ∆ < a > (running window) surface plot of ∆ < a > (a, e)

Model: real SS. Initial i = 0 1.86 1.861 1.862 1.863 1.864 1.865 1.866 1.867 1.868 1.869 1.87 initial a 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 initial e

  • 8.5
  • 8
  • 7.5
  • 7
  • 6.5
  • 6
  • 5.5
  • 5
  • 4.5
  • 4
  • 3.5

Tabaré Gallardo Three Body Resonances

slide-52
SLIDE 52

Resonance 2 - 5J + 2S. Model: real Solar System

Resonance 2 - 5J + 2S 3.166 3.168 3.17 3.172 3.174 3.176 initial a (au) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 initial e

  • 6
  • 5.5
  • 5
  • 4.5
  • 4
  • 3.5
  • 3
  • 2.5

490 Veritas

Tabaré Gallardo Three Body Resonances

slide-53
SLIDE 53

Resonance 2 - 5J + 2S. Model: J+S with circular orbits

Resonance 2 - 5J + 2S. J+S with e=i=0. 3.166 3.168 3.17 3.172 3.174 3.176 initial a (au) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 initial e

  • 7
  • 6.5
  • 6
  • 5.5
  • 5
  • 4.5
  • 4
  • 3.5
  • 3
  • 2.5

490 Veritas

Tabaré Gallardo Three Body Resonances

slide-54
SLIDE 54

Resonance 1 - 3J + 2S. Model: real Solar System

9864 (1991 RT17) at 1 - 3J + 2S. Model: real Solar System 3.072 3.074 3.076 3.078 3.08 3.082 3.084 initial a (au) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 initial e

  • 5.5
  • 5
  • 4.5
  • 4
  • 3.5
  • 3
  • 2.5
  • 2

9864 C+ L L C-

Tabaré Gallardo Three Body Resonances

slide-55
SLIDE 55

Resonance 1 - 3J + 2S. Model: J+S with circular orbits

Resonance 1 - 3J + 2S. Model: only J + S with e=i=0 3.072 3.074 3.076 3.078 3.08 3.082 3.084 initial a (au) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 initial e

  • 6.5
  • 6
  • 5.5
  • 5
  • 4.5
  • 4
  • 3.5
  • 3
  • 2.5

Tabaré Gallardo Three Body Resonances

slide-56
SLIDE 56

2BR 6P0 − 13P2 and 3BR 5P0 − 1P1 − 4P2.

Tabaré Gallardo Three Body Resonances

slide-57
SLIDE 57

For larger m1

Tabaré Gallardo Three Body Resonances

slide-58
SLIDE 58

Excited orbits

Tabaré Gallardo Three Body Resonances

slide-59
SLIDE 59

Galilean satellites

Tabaré Gallardo Three Body Resonances

slide-60
SLIDE 60

3BRs near Europa

1e-006 1e-005 0.0001 0.001 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007 0.0075 strength S0 a (au) Io Ganymede Europa 3-1-2 2-1-1 5-2-3 4-1-3 3-2-1 5-3-2 7-2-5 5-1-4 4-3-1 6-1-5 Io-Eu-Ga Eu-Ga-Ca Io-Eu-Ca Tabaré Gallardo Three Body Resonances

slide-61
SLIDE 61

Dynamical map: ∆a(a, e)

0.00435 0.0044 0.00445 0.0045 0.00455 initial a (au) 0.02 0.04 0.06 0.08 0.1 initial e

  • 8.5
  • 8
  • 7.5
  • 7
  • 6.5
  • 6
  • 5.5
  • 5
  • 4.5
  • 4

E ∆a

Tabaré Gallardo Three Body Resonances

slide-62
SLIDE 62

Maps for critical angles

take set of initial values (a, e) integrate for some 1.000 yrs calculate the distribution of σ between 0 and 360 uniform or wide distribution: circulation or large amplitude

  • scillations

narrow distribution: small amplitude oscillations

Tabaré Gallardo Three Body Resonances

slide-63
SLIDE 63

Tabaré Gallardo Three Body Resonances

slide-64
SLIDE 64

Inducing migration

Tabaré Gallardo Three Body Resonances

slide-65
SLIDE 65

Capture in a chain of 2BRs

2×(3P0−5P2)+(9P0−5P1) = 15P0−5P1−10P2 = 3P0−1P1−2P2

2.05 2.07 2.09 2.11 2.13 2.15 a2 1.46 1.47 1.48 1.49 1.5 a0 0.98 0.985 0.99 0.995 1 a1 90 180 270 360 50000 100000 150000 200000 250000 300000 σ time (yrs)

Tabaré Gallardo Three Body Resonances

slide-66
SLIDE 66

Migration while inside a pure 3BR (4-1-2)

0.006 0.007 0.008 a2-3.63 0.0082 0.0084 0.0086 a0-2.12

  • 1e-005

1e-005 3e-005 5e-005 a1-1.0

  • 180
  • 90

90 50000 100000 150000 200000 250000 300000 sigma time (yrs)

Tabaré Gallardo Three Body Resonances

slide-67
SLIDE 67

Inducing migration on Io

0.012544 0.012545 0.012546 0.012547 0.012548 Callisto 0.007136 0.007137 0.007138 0.007139 Ganymede 0.00447 0.004471 0.004472 0.004473 0.004474 Europa 0.002808 0.002809 0.00281 0.002811 Io 90 180 270 360 10 20 30 40 50 σ time (yrs)

Tabaré Gallardo Three Body Resonances

slide-68
SLIDE 68

Galilean migration

Tabaré Gallardo Three Body Resonances

slide-69
SLIDE 69

Migration: 2BRs versus 3BRs

two body resonances nI − 2nE ≃ 0 ∆nE ≃ 0,5∆nI The two bodies migrate both inwards or both

  • utwards.

three body resonances 3nE − nI − 2nG ≃ 0 3∆nE − 2∆nG ≃ ∆nI In 3BRs bodies can migrate in different directions while trapped in resonance.

Tabaré Gallardo Three Body Resonances

slide-70
SLIDE 70

Galilean migration: critical angles

Tabaré Gallardo Three Body Resonances

slide-71
SLIDE 71

Conclusions

restricted and planetary cases weak but numerous (chaotic diffusion) zero order resonances are the strongest, especially at e ∼ 0 for excited orbits high order 2BRs dominate there are pure 3BRs and chains of 2BRs is easiest to capture planetary (satellite) systems in a chain of 2BRs than in a pure 3BR migration in a 3BRs generates positive AND negative ∆a lot of work must to be done to understand the structure in (a, e, i)

Tabaré Gallardo Three Body Resonances

slide-72
SLIDE 72

Thanks! Merci! See you at Montevideo!

Tabaré Gallardo Three Body Resonances

slide-73
SLIDE 73

Appendix

Tabaré Gallardo Three Body Resonances

slide-74
SLIDE 74

Disturbing function for the asteroid

R( r0, r1, r2) = R(λ0, λ1, λ2) = R01 + R02 being Rij = k2mj( 1 rij − ri · rj r3

j

) resonance condition: λ0 = (σ − k1λ1 − k2λ2 + (k0 + k1 + k2)̟0) /k0 = ⇒ λ0 = λ0(σ, λ1, λ2, ̟0)

Tabaré Gallardo Three Body Resonances

slide-75
SLIDE 75

Averaging

R(σ) = 1 4π2 2π dλ1 2π R

  • λ0, λ1, λ2
  • dλ2

R = R01(λ0, λ1) + R02(λ0, λ2), both independent of σ !! we cannot calculate R01 + R02 using the unperturbed Keplerian positions

Tabaré Gallardo Three Body Resonances

slide-76
SLIDE 76

The idea

We adopt the following scheme: R(λ0, λ1, λ2) ≃ Ru + ∆R Ru is R calculated at the unperturbed positions of the three bodies (useless!) ∆R stands from the variation in Ru generated by the perturbed displacements of the three bodies in a small interval ∆t.

Tabaré Gallardo Three Body Resonances

slide-77
SLIDE 77

Method

Then, being in a resonant configuration

∆r1 ∆r2 ∆r0 SUN

asteroid

and these ∆ generate the ∆R

Tabaré Gallardo Three Body Resonances

slide-78
SLIDE 78

Approximate mean resonant disturbing function ρ(σ)

The integral of Ru = R01 + R02 is independent of σ, then we only need to calculate ρ(σ) defined by ρ(σ) = 1 4π2 2π dλ1 2π ∆R dλ2 always satisfying the resonant condition λ0(σ, λ1, λ2, ̟0).

Tabaré Gallardo Three Body Resonances

slide-79
SLIDE 79

Method

For a given resonance: consider a large sample of configurations verifying the resonant condition (σ = constant) calculate the mutual perturbations ∆r0, ∆r1, ∆r2 calculate the effect ∆R due to (∆r0, ∆r1, ∆r2) integrate all ∆R and obtain ρ(σ) repeat for several σ ∈ (0, 360)

Tabaré Gallardo Three Body Resonances

slide-80
SLIDE 80

Disturbing function ∼ ρ(σ)

90 180 270 360 ρ(σ) σ 1-3J+1S, e=0.01

Tabaré Gallardo Three Body Resonances

slide-81
SLIDE 81

90 180 270 360 ρ(σ) σ 1-3J+1S, e=0.01

large variations of ρ with σ is indicative of a strong resonance small variations of ρ with σ is indicative of a weak resonance an extreme of ρ(σ) at some σ means there is an equilibrium point

Tabaré Gallardo Three Body Resonances

slide-82
SLIDE 82

Density of resonances versus density of asteroids

50 100 150 200 250 1 1.5 2 2.5 3 3.5 4 4.5 density of 3-body and 2-body resonances a (au) 3BRs 2BRs asteroids Tabaré Gallardo Three Body Resonances