Three Body Mean Motion Resonances
Tabaré Gallardo
Departamento de AstronomíaFacultad de Ciencias
Universidad de la RepúblicaUruguay
Luchon, September 2016
Tabaré Gallardo Three Body Resonances
preliminaries
types of three body resonances (3BRs)
semi analytical methodnumerical studies
dynamical mapsinduced migration
Tabaré Gallardo Three Body Resonances
Preliminaries
Tabaré Gallardo Three Body Resonances
Preliminaries
e = eccentricitya = semimajor axis (in astronomical units)n = mean motion = mean angular velocity = 2π
period ∝1
a3/2
Two body resonance: k0n0 + k1n1 ' 0with k0, k1 integers.
Tabaré Gallardo Three Body Resonances
Non resonant asteroid: relative positions
Mean perturbation is radial: Sun-Jupiter
Sun Jupiter
Tabaré Gallardo Three Body Resonances
Resonant asteroid
Mean perturbation has a transverse component.
Sun Jupiter
Tabaré Gallardo Three Body Resonances
from Gauss equations
SUN
asteroid
R T Fperturb = (R,T,N)
dadt∝ (R,T)
<dadt>∝ T
Non resonantT = 0⇒ a = constant
ResonantT 6= 0⇒ a = oscillating
Tabaré Gallardo Three Body Resonances
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 anglesσ(t) indicates if the motion is resonant or not:
σ(t) oscillating means resonanceσ(t) circulating means NO resonance
resonant motion: a(t) is correlated with σ(t)
Tabaré Gallardo Three Body Resonances
Semimajor axis: width
Nesvorny et al. in Asteroids III
Tabaré Gallardo Three Body Resonances
Two body resonance, restricted case m0 = 0
k0n0 + k1n1 ' 0 P1 does not feel the resonance, only P0
Tabaré Gallardo Three Body Resonances
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
Two body resonance, planetary case m0 6= 0
k0n0 + k1n1 ' 0 both P0 and P1 feel the resonance
Tabaré Gallardo Three Body Resonances
Two body resonance, planetary case
Observational evidence in extrasolar systems
Tabaré Gallardo Three Body Resonances
Three body resonances
Tabaré Gallardo Three Body Resonances
Three body resonance, restricted case m0 = 0
k0n0 + k1n1 + k2n2 ' 0 only P0 feels the resonance
Tabaré Gallardo Three Body Resonances
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
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
Three body resonance, planetary case m0 6= 0
k0n0 + k1n1 + k2n2 ' 0 all three bodies feel the resonance
Tabaré Gallardo Three Body Resonances
3BR 4n0 − 1n1 − 2n2, planetary case m0 6= 0
Tabaré Gallardo Three Body Resonances
3BR 4n0 − 1n1 − 2n2, restricted case m0 = 0
Tabaré Gallardo Three Body Resonances
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
1784: Laplacian resonance
3λEuropa−λIo−2λGanymede ' 180
3nEuropa − nIo − 2nGanymede ' 0
They are also in commensurabilityby pairs:
2nEuropa − nIo ' 0
2nGanymede − nEuropa ' 0
⇓
It must be the consequence of some physical mechanism.
Tabaré Gallardo Three Body Resonances
Two types of 3BRs
Three body resonance as...
superposition or chain of 2 two-body resonances
nI − 2nE ∼ 0nE − 2nG ∼ 0adding: nI − nE − 2nG ∼ 0⇒ 3BR order 2substraction: nI − 3nE + 2nG ∼ 0⇒ 3BR order 0
pure: 3BR that are NOT due to 2BR + 2BR.asteroids + Jupiter + Saturn
Tabaré Gallardo Three Body Resonances
Asteroids: histogram of a + 2BRs
2 2.2 2.4 2.6 2.8 3 3.2 3.4
log
(S
treng
th)
a (au)
1:2
Mars
3:1
Jup
2:1
Jup
4:7
Mars
5:2
Jup
Tabaré Gallardo Three Body Resonances
Asteroids: histogram of a + 3BRs
2 2.2 2.4 2.6 2.8 3 3.2 3.4
log (
Str
ength
)
a (au)
1-4
J+
2S
1-4
J+
3S
2-7
J+
4S
1-3
J+
1S
2-7
J+
5S
2-6
J+
3S
1-3
J+
2S
3-8
J+
4S
2-5
J+
2S
3-7
J+
2S
3-8
J+
5S
2-1
M
Tabaré Gallardo Three Body Resonances
Dynamical evidence from AstDyS
0
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
prop
er e
proper a (au)
Tabaré Gallardo Three Body Resonances
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
a r t i c l e i n f o
Article history:Received 15 June 2012Revised 29 October 2012Accepted 29 October 2012Available online 10 November 2012
Keywords:AsteroidsAsteroids, DynamicsCelestial mechanics
a b s t r a c t
An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body meanmotion resonances exist, in which an asteroid and two planets participate. Identification of asteroidsin three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvornyand Morbidelli (Nesvorny D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visualanalysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances.We develop specialized algorithms and software for massive automatic identification of asteroids in thethree-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of thetime behaviour of resonant arguments. In the computation of orbits, all essential perturbations are takeninto account. We integrate the asteroidal orbits on the time interval of 100,000 yr and identify main-beltasteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in thetwo-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of 250,000 objects fromthe ‘‘Asteroids – Dynamic Site’’ (AstDyS) database. The percentages of resonant objects, including extrap-olations for higher-order resonances, are determined. In particular, the observed fraction of pure-reso-nant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is 0.9% of the whole set; and, using a higher-order extrap-olation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders isestimated as 1.1% of the whole set.
2012 Elsevier Inc. All rights reserved.
1. Introduction
A substantial role of resonances in the dynamics of asteroids be-came evident with the discovery of resonant ‘‘gaps’’ in the asteroidbelt by D. Kirkwood in 1867. The deepest minima in the distribu-tion of asteroids in the semimajor axes of their orbits correspondto the mean motion resonances 2/1, 3/1, 4/1, 5/2, and 7/3 withJupiter. Mean motion resonance represents a commensurabilitybetween the mean frequencies of the orbital motions of an asteroidand a planet. Apart from the mean motion resonances, so-calledsecular resonances (Murray and Dermott, 1999; Morbidelli,2002), representing commensurabilities between the precessionrates of the orbits of an asteroid and a planet, are important informing the dynamical structure of the asteroid belt.
There are two important classes of the mean motion reso-nances: apart from the usual (two-body) mean motion resonancesof an asteroid and a planet, an appreciable role in the asteroidaldynamics is played by so-called three-body mean motion reso-nances (Murray et al., 1998; Nesvorny and Morbidelli, 1998;Nesvorny and Morbidelli, 1999; Morbidelli, 2002). In the latter
case, the resonance represents a commensurability between themean frequencies of the orbital motions of an asteroid and twoplanets (e.g., Jupiter and Saturn):
mJ_kJ þmS
_kS þm _k 0; ð1Þ
where _kJ, _kS; _k are the time derivatives of the mean longitudes ofJupiter, Saturn, and asteroid, respectively, and mJ;mS;m areintegers.
In view of the ‘‘overdensity’’ of the three-body resonances in thephase space of the asteroidal motion, Nesvorny and Morbidelli(1998) asserted that ‘‘the three-body mean motion resonancesseem to be the main actors structuring the dynamics in the mainasteroid belt’’.
Chaotic behaviour, which is often present in the dynamics ofcelestial bodies, is usually due to interaction of resonances (as inany Hamiltonian system, see Chirikov, 1979), but not always it isknown which are the interacting resonances that give rise to chaos.It is especially difficult to identify three-body resonances. How todistinguish between resonant and non-resonant motions? To solvethis problem, a ‘‘resonant argument’’ (synonymously ‘‘resonantphase’’ or ‘‘critical argument’’) is introduced. It is a linear combina-tion of some angular variables of a system under consideration; inthe planar asteroidal problem it is given by
0019-1035/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.icarus.2012.10.034
⇑ Corresponding author. Fax: +7 812 7231922.E-mail address: [email protected] (I.I. Shevchenko).
Icarus 222 (2013) 220–228
Contents lists available at SciVerse ScienceDirect
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Smirnov and Shevchenko (2013)
See next talk!
Tabaré Gallardo Three Body Resonances
Resonance 1 + 1U−2N, a weird case
263.52
263.53
263.54
263.55
mean a
(au)
0
90
180
270
360
0 100 200 300 400 500
σ (1
+1U
-2N
)
time (Myrs)
Tabaré Gallardo Three Body Resonances
3BRs are WEAK and numerous
Given two planets P1 and P2, an infinite family of 3BRs isdefined:
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 resonantmotion is factorized by m1m2.
There is a huge number of 3BRs: superposition generateschaotic diffusion.
Tabaré Gallardo Three Body Resonances
Multiplet resonances and chaos
σ = k0λ0 + k1λ1 + k2λ2+k4$0 + k5Ω0
270 D. NESVORNY AND A. MORBIDELLI
Figure 8.Separatrices of four multiplet resonances of the 6 1− 3 three-body resonance.
7. Conclusion
In this paper we have developed an analytic model of three-body resonances amongthe mean motions of an asteroid, Jupiter and Saturn. Our model is restricted to theplanar case, but turns out to allow good predictions of the main characteristics of theresonances, namely, for what concerns the amplitude of semimajor axis oscillations,the typical period of libration/circulation of the critical angles and of the Lyapunovexponent. It also allows to understand on a simple basis why most of the resonantphase space is chaotic. We show that a three-body resonance splits in a naturalway into a multiplet of resonances with different critical angles. These multipletresonances overlap each other, generating global chaos.
Nesvorny and Morbidelli (1999)
Tabaré Gallardo Three Body Resonances
Chaotic diffusion: growing e
Morbidelli and Nesvorny (1999)
Tabaré Gallardo Three Body Resonances
Chaotic diffusion in the TNR: growing e
Nesvorny and Roig (2001)
Tabaré Gallardo Three Body Resonances
Three body resonances as...
Chains of two body resonancesGalilean satellites (Sinclair 1975, Ferraz-Mello, Malhotra,Showman, Peale, Lainey...)Callegari and Yokoyama (2010): satellites of SaturnExtrasolar systems (Libert and Tsiganis 2011; Martí, Batygin,Morbidelli, Papaloizou, Quillen...)
Pure three body resonances
Lazzaro et al. (1984): satellites of UranusAksnes (1988): zero order asteroidal resonancesNesvorny y Morbidelli (1999): theory Jupiter-Saturn-asteroidCachucho et al. (2010): diffusion in 5J -2S -2.Quillen (2011): zero order extrasolar systemsGallardo (2014), Gallardo et al. (2016): semianalyticShowalter and Hamilton (2015): Pluto satellites3nS − 5nN + 2nH ∼ 0
Tabaré Gallardo Three Body Resonances
Disturbing function
Disturbing function for resonance k0 + k1 + k2:
R = k2m1m2
∑j
Pj cos(σj)
σj = k0λ0 + k1λ1 + k2λ2 + γj
γj = k3$0 + k4$1 + k5$2 + k6Ω0 + k7Ω1 + k8Ω2
Pj is a polynomial function depending on the eccentricities andinclinations 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
Theories are complicated...
it is necessary to consider several Pj cos(σj)
with several terms in Pj
calculation of the Cs is not trivial
only planar theories exist
To avoid the difficulties of the analytical methods we proposed tocalculate R numerically.
Tabaré Gallardo Three Body Resonances
Semi analytical method
Tabaré Gallardo Three Body Resonances
Author's personal copy
Atlas of three body mean motion resonances in the Solar System
Tabaré GallardoDepartamento de Astronomía, Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
a r t i c l e i n f o
Article history:Received 22 July 2013Revised 14 December 2013Accepted 17 December 2013Available online 27 December 2013
Keywords:Asteroids, dynamicsCelestial mechanicsResonances, orbitalKuiper belt
a b s t r a c t
We present a numerical method to estimate the strengths of arbitrary three body mean motion reso-nances between two planets in circular coplanar orbits and a massless particle in an arbitrary orbit. Thismethod allows us to obtain an atlas of the three body resonances in the Solar System showing where arelocated and how strong are thousands of resonances involving all the planets from 0 to 1000 au. This atlasconfirms the dynamical relevance of the three body resonances involving Jupiter and Saturn in the aster-oid belt but also shows the existence of a family of relatively strong three body resonances involving Ura-nus and Neptune in the far Trans-Neptunian region and relatively strong resonances involving terrestrialand jovian planets in the inner planetary system. We calculate the density of relevant resonances alongthe Solar System resulting that the main asteroid belt is located in a region of the planetary system withthe lowest density of three body resonances. The method also allows the location of the equilibriumpoints showing the existence of asymmetric librations (r – 0 or 180). We obtain the functional depen-dence of the resonance’s strength with the order of the resonance and the eccentricity and inclination ofthe particle’s orbit. We identify some objects evolving in or very close to three body resonances withEarth–Jupiter, Saturn–Neptune and Uranus–Neptune apart from Jupiter–Saturn, in particular the NEA2009 SJ18 is evolving in the resonance 1 1E 1J and the centaur 10199 Chariklo is evolving underthe influence of the resonance 5 2S 2N.
2013 Elsevier Inc. All rights reserved.
1. Introduction
Fifteen years ago, concentrated over a period of about a year, asuccession of papers were published (Murray et al., 1998; Nesv-orny and Morbidelli, 1998, 1999; Morbidelli and Nesvorny, 1999)showing an intense theoretical and numerical work on three bodymean motion resonances (TBRs) involving an asteroid and twomassive planets. These papers, that have their roots on earlierworks devoted to the study of an asteroid in zero order TBRs(Wilkens, 1933; Okyay, 1935; Aksnes, 1988), stated the relevancethat the TBRs involving Jupiter–Saturn and also Mars–Jupiter havein the long term stability in the asteroid’s region. In spite of beingweaker than the two body resonances they are much more numer-ous generating several dynamical features in the asteroidal popu-lation, like concentrations for some values of semimajor axes,anomalous amplitude librations and chaotic evolutions (Nesvornyand Morbidelli, 1998). In particular, both borders of the main aster-oid belt exhibit chaotic diffusion due to the superposition ofseveral weak two-body and TBRs (Murray and Holman, 1997;Morbidelli and Nesvorny, 1999). More recently, Smirnov andShevchenko (2013) in a massive numerical integration of 249,567asteroids by 105 years and looking at the time evolution of the
critical angles, identified thousands of asteroids in TBRs with Jupi-ter and Saturn, concluding there are more asteroids in TBRs than intwo body resonances.
The approximate nominal position in semimajor axis of theTBRs taking arbitrary pairs of planets is very simple if we ignorethe secular motion of the perihelia and nodes of the three bodies.When these slow secular motions are taken into account each ofthe nominal TBRs split in a multiplet of resonances all them verynear the nominal one (Morbidelli, 2002). The challenge is to obtainthe strength, width or libration timescale that give us the dynam-ical relevance of these resonances. Analytical planar theoriesdeveloped by Murray et al. (1998) and Nesvorny and Morbidelli(1999) allowed to describe and understand the dynamics of theTBRs involving Jupiter and Saturn in the asteroidal region. Thesetheories are appropriated to study in detail specific resonanceswith Jupiter and Saturn but its application to any arbitrary reso-nance involving any planet is not trivial, which possibly explainsthe absence of papers published on this topic since these yearswith the exceptions of a few ones devoted to specific scenarios(Guzzo, 2005; de La Fuente Marcos and de La Fuente Marcos,2008; Cachucho et al., 2010; Quillen, 2011).
In this paper, from a different approach, we will obtain a globalview of all dynamically relevant TBRs involving all the planets ta-ken by pairs along all the Solar System. Our method is notanalytical but numerical and it is based on an estimation of the
0019-1035/$ - see front matter 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.icarus.2013.12.020
E-mail address: [email protected]
Icarus 231 (2014) 273–286
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Tabaré Gallardo Three Body Resonances
Method
Disturbing function is a mean over all possible resonantconfigurations.
The point: the disturbing function R must be calculated with theperturbed positions.
We cannot assume unperturbed ellipses for the three orbits.
Tabaré Gallardo Three Body Resonances
Method
For a given resonance:
consider a large sample of configurations verifying the resonantcondition (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
Method
Then, being in a resonant configuration
∆r1
∆r2
∆r0
SUN asteroid
and these ∆r generate the ∆R
Tabaré Gallardo Three Body Resonances
Disturbing function ∼ ρ(σ)
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.01
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.1
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.2
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.3
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.4
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.5
Tabaré Gallardo Three Body Resonances
Asymmetric equilibrium points
Tabaré Gallardo Three Body Resonances
Zero order resonance, e ' 0,05, a ' 3,8
0
90
180
270
360
0 20 40 60 80 100
sigm
a (
1-2
J+1
S)
time (thousand yrs)
Numerical integration of full equations of motion.
Tabaré Gallardo Three Body Resonances
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.01
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.1
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.2
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.3
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.4
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.5
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 equilibriumpoint
Tabaré Gallardo Three Body Resonances
Strength, S
We numerically obtain ρ(σ)
We define Strength
S =12
∆ρ(σ)
For planetary case we have 3 strengths
Si =12
∆ρi(σ)
Codes: www.fisica.edu.uy/~gallardo/atlas
Tabaré Gallardo Three Body Resonances
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
0 1 2 3 4 5 6 7 8 9 10 11 12 13
∆ρ
order q
log(∆ρ) ∝ −q
Tabaré Gallardo Three Body Resonances
Planetary case. Dependence on e0. Case q = 4.
1e-014
1e-012
1e-010
1e-008
1e-006
0.0001
0.01 0.1
stre
ngth
e0
2P0 - 1P1 + 3P2
e1=e2=0
S0S1S2
Tabaré Gallardo Three Body Resonances
Planetary case. Dependence on e0. Case q = 0.
1e-014
1e-012
1e-010
1e-008
1e-006
0.0001
0.01 0.1
stre
ngth
e0
6P0 - 1P1 - 5P2
S0S1S2
Tabaré Gallardo Three Body Resonances
Dynamical maps
Tabaré Gallardo Three Body Resonances
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
mea
n a
(au)
time (yr)
secular
chaotic
resonant
Tabaré Gallardo Three Body Resonances
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
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
Tabaré Gallardo Three Body Resonances
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
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
490 Veritas
Tabaré Gallardo Three Body Resonances
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
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
490 Veritas
Tabaré Gallardo Three Body Resonances
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
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
9864
C+L L C-
Tabaré Gallardo Three Body Resonances
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
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
Tabaré Gallardo Three Body Resonances
2BR 6P0 − 13P2 and 3BR 5P0 − 1P1 − 4P2.
Tabaré Gallardo Three Body Resonances
For larger m1
Tabaré Gallardo Three Body Resonances
Excited orbits
Tabaré Gallardo Three Body Resonances
Galilean satellites
Tabaré Gallardo Three Body Resonances
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
stre
ngth
S0
a (au)
Io Gan
ymed
e
Eur
opa
3-1-
22-1-
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-GaEu-Ga-CaIo-Eu-Ca
Tabaré Gallardo Three Body Resonances
Dynamical map: ∆a(a, e)
0.00435 0.0044 0.00445 0.0045 0.00455
initial a (au)
0
0.02
0.04
0.06
0.08
0.1in
itial
e
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
E
∆a
Tabaré Gallardo Three Body Resonances
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 amplitudeoscillations
narrow distribution: small amplitude oscillations
Tabaré Gallardo Three Body Resonances
Tabaré Gallardo Three Body Resonances
Inducing migration
Tabaré Gallardo Three Body Resonances
Capture in a chain of 2BRs
2×(3P0−5P2)+(9P0−5P1) = 15P0−5P1−10P2 = 3P0−1P1−2P2
2.052.072.092.112.132.15
a2
1.46
1.47
1.48
1.49
1.5
a0
0.98
0.985
0.99
0.995
1
a1
0
90
180
270
360
0 50000 100000 150000 200000 250000 300000
σ
time (yrs)
Tabaré Gallardo Three Body Resonances
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
0
90
0 50000 100000 150000 200000 250000 300000
sigm
a
time (yrs)
Tabaré Gallardo Three Body Resonances
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
090
180270360
0 10 20 30 40 50
σ
time (yrs)
Tabaré Gallardo Three Body Resonances
Galilean migration
Tabaré Gallardo Three Body Resonances
Migration: 2BRs versus 3BRs
two body resonances
nI − 2nE ' 0
∆nE ' 0,5∆nI
The two bodies migrateboth inwards or bothoutwards.
three body resonances
3nE − nI − 2nG ' 0
3∆nE − 2∆nG ' ∆nI
In 3BRs bodies can migrate in differentdirections while trapped in resonance.
Tabaré Gallardo Three Body Resonances
Galilean migration: critical angles
Tabaré Gallardo Three Body Resonances
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 of2BRs 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
Thanks!Merci!
See you atMontevideo!
Tabaré Gallardo Three Body Resonances
Appendix
Tabaré Gallardo Three Body Resonances
Disturbing function for the asteroid
R(~r0, ~r1, ~r2) = R(λ0, λ1, λ2) = R01 + R02
being
Rij = k2mj(1rij−~ri · ~rj
r3j
)
resonance condition:
λ0 = (σ − k1λ1 − k2λ2 + (k0 + k1 + k2)$0) /k0
=⇒ λ0 = λ0(σ, λ1, λ2, $0)
Tabaré Gallardo Three Body Resonances
Averaging
R(σ) =1
4π2
∫ 2π
0dλ1
∫ 2π
0R(λ0, λ1, λ2
)dλ2
R = R01(λ0, λ1) + R02(λ0, λ2), both independent of σ !!
we cannot calculate R01 + R02 using the unperturbedKeplerian positions
Tabaré Gallardo Three Body Resonances
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 perturbeddisplacements of the three bodies in a small interval ∆t.
Tabaré Gallardo Three Body Resonances
Method
Then, being in a resonant configuration
∆r1
∆r2
∆r0
SUN asteroid
and these ∆ generate the ∆R
Tabaré Gallardo Three Body Resonances
Approximate mean resonant disturbing function ρ(σ)
The integral of Ru = R01 + R02 is independent of σ, then we onlyneed to calculate ρ(σ) defined by
ρ(σ) =1
4π2
∫ 2π
0dλ1
∫ 2π
0∆R dλ2
always satisfying the resonant condition λ0(σ, λ1, λ2, $0).
Tabaré Gallardo Three Body Resonances
Method
For a given resonance:
consider a large sample of configurations verifying the resonantcondition (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
Disturbing function ∼ ρ(σ)
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.01
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.1
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.2
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.3
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.4
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.5
Tabaré Gallardo Three Body Resonances
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.01
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.1
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.2
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.3
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.4
0 90 180 270 360
ρ(σ)
σ
1-3J+1S, e=0.5
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 equilibriumpoint
Tabaré Gallardo Three Body Resonances
Density of resonances versus density of asteroids
0
50
100
150
200
250
1 1.5 2 2.5 3 3.5 4 4.5
dens
ity o
f 3-b
ody
and
2-bo
dy r
eson
ance
s
a (au)
3BRs2BRs
asteroids
Tabaré Gallardo Three Body Resonances
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