Evection resonance in Saturn's coorbital moons Dr. Cristian Giuppone1, Lic Ximena Saad Olivera2, Dr. Fernando Roig2
(1) Universidad Nacional de Córdoba, OAC - IATE, Córdoba, Ar (2) Observatorio Nacional, Rio de Janeiro, Brasil
OPS III - Diversis Mundi - Santiago - Chile Evection resonance in Saturn's coorbital moons
MOTIVATION: recent studies in Jovian planets links the evection resonance and the architecture and evolution of their satellites.
Evection resonance in Saturn's coorbital moons
MOTIVATION: recent studies in Jovian planets links the evection resonance and the architecture and evolution of their satellites. GOAL: understand the evection resonance in the frame of coorbital motion of Satellites, through numerical evidence of its importance.
Evection resonance
● The evection resonance is produced when the orbit pericenter of a satellite (ϖ)
precess with the same period that the Sun movement as a perturber (λO) (Brouwer and Clemence, 1961)
The evection angle → Ψ=λO – ϖ
Satellite Sun
Saturn ψ = 0°
Evection resonance
● The evection resonance is produced when the orbit pericenter of a satellite (ϖ)
precess with the same period that the Sun movement as a perturber (λO) (Brouwer and Clemence, 1961)
The evection angle → Ψ=λO – ϖ
Satellite Sun
Saturn ψ = 0°
Evection resonance
● Interior resonance: ● Produced by the oblateness of the planet.
● -3 Located at a ~ 10 RP (a~10 au), and
several armonics are present.
● Importance: tides produce outward migration of inner moons. The moons can cross the resonance and may change substantially their orbital elements (e.g. Nesvorny et al 2003, Cuk et al. 2016)
Evection resonance
● Interior resonance: ● Produced by the oblateness of the planet.
● -3 Located at a ~ 10 RP (a~10 au), and
several armonics are present.
● Importance: tides produce outward migration of inner moons. The moons can cross the resonance and may change substantially their orbital elements (e.g. Nesvorny et al 2003, Cuk et al. 2016)
a (RP) e
a (au) Evection resonance
● Interior resonance: ● Exterior resonance: ● Produced by the oblateness of the planet. ● Produced by external massive perturber (the Sun).
● -3 Located at a ~ 10 RP (a~10 au), and ● Located at a ~ 0.5 RH (a ~ 0.2 au) several armonics are present. and high eccentricity.
● Importance: tides produce outward migration of inner moons. The moons can ● Importance: the outhermost stable cross the resonance and may change resonance in satellites (Yokoyama et substantially their orbital elements (e.g. al 2008, Frouard et al. 2010). Nesvorny et al 2003, Cuk et al. 2016)
a (RP) e
a (au) Evection resonance
● Interior resonance: ● Exterior resonance: ● Produced by the oblateness of the planet. ● Produced by external massive perturber (the Sun).
● -3 Located at a ~ 10 RP (a~10 au), and ● Located at a ~ 0.5 RH (a ~ 0.2 au) several armonics are present. and high eccentricity.
● Importance: tides produce outward migration of inner moons. The moons can ● Importance: the outhermost stable cross the resonance and may change resonance in satellites (Yokoyama et substantially their orbital elements (e.g. al 2008, Frouard et al. 2010). Nesvorny et al 2003, Cuk et al. 2016)
a (RP) a (RP)
Chaotic layer e
a (au) Orbital elements distribution in Saturn satellites
Hyperion e
Iapetus c n
JE Titan a n o s
Rhea e r
r
Dione o i r e t x Tethys E
Interior resonance
Orbital elements distribution in Saturn satellites
regular satellites Hyperion e
Iapetus c n
Titan a n o s
Rhea e r
r
Dione o i r e t x Tethys Irregular satellites E
Interior resonance ● Irregular Satellites: inclined and eccentric orbits. Supposed to be captured. ● Regular Satellites: e~0 in the equatorial plane → formed in a circum-planetary disk irc un the equatorial plane → formed from the spreading of a Ring (Crida & Charnoz, 2014)
Orbital elements distribution in Saturn satellites
regular satellites Hyperion e
Iapetus c n
JE Titan a n o s
Rhea e r
r
Dione o i r e t x Tethys Irregular satellites E
Interior resonance ● Irregular Satellites: inclined and eccentric orbits. Supposed to be captured. ● Regular Satellites: e~0 in the equatorial plane → formed in a circum-planetary disk irc un the equatorial plane → formed from the spreading of a Ring (Crida & Charnoz, 2014)
Telesto (L4) and Calypso (L5) of Saturn-Tethys Helene (L4) and Polydeceus (L5) of Saturn-Dione Evection resonance & Trojans in Saturn Is the evection resonance present for satellites in coorbital motion? Ψ=λO – ϖ
Evection resonance & Trojans in Saturn Is the evection resonance present for satellites in coorbital motion? Ψ=λO – ϖ Choosing an adequate system of coordinates and variables
● m1 and m2 coordinates are referred to m0 ● m3 coordinates referred to the barycenter of subsystem m0,m1,m2 ● N-body equations, BS integrator.
Evection resonance & Trojans in Saturn Is the evection resonance present for satellites in coorbital motion? Ψ=λO – ϖ Choosing an adequate system of coordinates and variables
● m1 and m2 coordinates are referred to m0 ● m3 coordinates referred to the barycenter of subsystem m0,m1,m2 ● N-body equations, BS integrator.
Saturn m m 2 Sun 0 Two evection angles 60° Ψ1=λO – ϖ1
Ψ2=λO – ϖ2 m3 m1 Evection resonance & Trojans in Saturn Is the evection resonance present for satellites in coorbital motion? Ψ=λO – ϖ Choosing an adequate system of coordinates and variables
● m1 and m2 coordinates are referred to m0 ● m3 coordinates referred to the barycenter of subsystem m0,m1,m2 ● N-body equations, BS integrator.
Coorbital resonance at L4 (or L5) TROJAN configuration: o o (σ, Δϖ ) = (λ2-λ1,ϖ2-ϖ1)=(60 , 60 ) Initial conditions near to periodic coorbital orbits
n1~ n2 and e1~ e2 (Giuppone et al 2011)
Saturn m m 2 Sun 0 Two evection angles 60° Ψ1=λO – ϖ1
Ψ2=λO – ϖ2 m3 m1 Evection resonance & Trojans in Saturn – INTERIOR - Numerical Analysis
Single
Satellite Sun Saturn ψ = 0°
aΔ (ReP)
Δe
Evection resonance & Trojans in Saturn – INTERIOR - Numerical Analysis
ψ = 0°, ψ = -60°, λ = 0° Single 2 1 3
Δϖ = ϖ2-ϖ1=60°
σ = λ2-λ1=60° Satellite Sun Sun m2 Saturn ψ = 0° m 1 60°
a (R ) a (RP) P 2 e = 1 e
Δe Δe
a1=a2 (au) The resonance is present almost at same location than when single satellite is considered Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
a (R ) P Δe White:unstable in ~ 5 104 y o m2/m1=1, ψ2=0 2 e = 1 e
a1=a2 (au)
Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
a (R ) P Δe White:unstable in ~ 5 104 y o m2/m1=1, ψ2=0 2 e = 1 e
a1=a2 (au)
● Trojan configurations can not exist beyond 0.08 au because Solar perturbation provokes that Δϖ librates, breaking the resonant protection.
Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
a (R ) P Δe White:unstable in ~ 5 104 y o m2/m1=1, ψ2=0 2 e = 1 e
a1=a2 (au)
● Can we have a trojan pair of satellites in the region of exterior evection resonance?
Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
o Instead of setting Ψ2=0 , we choose to change the orientation of trojan-pair orbits with respect to o the Sun (initially at λO=0 ). Short integrations (T=100 y), guessing initial eccentricity similar to exterior resonance in single systems (ei=0.4). Sun Saturn
m2 m0 m3 60°
m1
m2
° Sun 0 6 n r 1 tu m a S 0 m m3 Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
o Instead of setting Ψ2=0 , we choose to change the orientation of trojan-pair orbits with respect to o the Sun (initially at λO=0 ). .
Δψ2 Δψ2
Ψ1=λO – ϖ1
Ψ2=λO – ϖ2= – ϖ2
Ψ1 librates
a1=a2 (au) Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
o Instead of setting Ψ2=0 , we choose to change the orientation of trojan-pair orbits with respect to o the Sun (initially at λO=0 ). .
Δψ2 Δψ2 Δψ1
Ψ1=λO – ϖ1
Ψ2=λO – ϖ2= – ϖ2
Ψ1 librates
Ψ2 librates
a1=a2 (au) a1=a2 (au) No regions where both evection angles librates Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
● o o Condition 1. Ψ1 librates, Ψ2 circulates. ϖ2=75 (Ψ2 = 285 )
σ ● Ψ2 e2→ chaotic nature ● Survival time ~ 106 y.
Ψ1
Evection resonance & Trojans in Saturn – Exterior - Numerical Analysis
● o o Condition 1. Ψ1 librates, Ψ2 circulates. ϖ2=75 (Ψ2 = 285 )
σ ● Ψ2 e2→ chaotic nature ● Survival time ~ 106 y.
Ψ1
● o o Condition 2. Ψ1 circulates, Ψ2 librates. ϖ2=0 (Ψ2 = 0 )
Ψ ● 1 σ e1→ chaotic nature ● Survival time ~ 103 y.
Ψ2
Evection resonance & Trojans in Saturn – Tidal Evolution
Evection resonance & Trojans in Saturn – Tidal Evolution There is evidence that many trojan systems can form in regions closer to the planet than the position of the interior evection (Crida et al 2017) Interior Evection resonance seems to be important in the past evolution of regular moons of Saturn (Cuk et al 2016).
Evection resonance & Trojans in Saturn – Tidal Evolution There is evidence that many trojan systems can form in regions closer to the planet than the position of the interior evection (Crida et al 2017). Interior Evection resonance seems to be important in the past evolution of regular moons of Saturn (Cuk et al 2016). We can include an additional perturbative term, that mimics Tidal evolution of trojan pairs
Evection resonance & Trojans in Saturn – Tidal Evolution There is evidence that many trojan systems can form in regions closer to the planet than the position of the interior evection (Crida et al 2017). Interior Evection resonance seems to be important in the past evolution of regular moons of Saturn (Cuk et al 2016). We can include an additional perturbative term, that mimics Tidal evolution of trojan pairs. We run experiments with different folding times, starting with the same I.C.s (~ Dione + Helene) . In the adiabatic regime the system will evolve along the family of periodic orbits.
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Faster migration: e2 The resonance provokes a tiny jump in eccentricity and breaks the libration of Δ.ϖ e1
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Faster migration: e2 The resonance provokes a tiny jump in eccentricity and breaks the libration of Δ.ϖ e 1 The system cross the resonance, and continues with high amplitude of oscillation Δϖ until reach the edge of stable region at a~0.2
Δϖ
σ
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Faster migration: e2 The resonance provokes a tiny jump in eccentricity and breaks the libration of Δ.ϖ e 1 The system cross the resonance, and continues with high amplitude of oscillation Δϖ until reach the edge of stable region at a~0.2
Δϖ
) g e d (
2 ψ
,
1
σ ψ
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Slower migration: e 2 The resonance provokes a tiny jump in eccentricity.
e1
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Slower migration: e 2 The resonance provokes a tiny jump in eccentricity, and after that the less massive satellite got captured in o. resonance ψ2 around 90 The angle Δϖ started to circulate and σ → 180o e 1 The system survives until a collision 6 occurs at a~0.03 (~aIAPETUS) at 10 y
Δϖ
σ
Evection resonance & Trojans in Saturn – Interior - Numerical Analysis – ADIABATIC MIGRATION
Slower migration: e 2 The resonance provokes a tiny jump in eccentricity, and after that the less massive satellite got captured in o. resonance ψ2 around 90 The angle Δϖ started to circulate and σ → 180o e 1 The system survives until a collision 6 occurs at a~0.03 (~aIAPETUS) at 10 y
Δϖ
) g e d (
2 ψ
,
σ 1 ψ
Summary SATURN: Evection resonance & Trojans
Summary SATURN: Evection resonance & Trojans Interior evection resonance (caused by the oblateness):
● o o The trojan configuration is possible, a~8.2 RP when e~0 at ψi =0 and ψi =180 ≡ single satelites
Summary SATURN: Evection resonance & Trojans Interior evection resonance (caused by the oblateness):
● o o The trojan configuration is possible, a~8.2 RP when e~0 at ψi =0 and ψi =180 ≡ single satelites
Exterior evection resonance (caused by the Sun):
● Ψi librates in two regions, depending on relative orientation to the Sun, a~0.4 RH,,0.2 < ei < 0.4 . 3 1) Ψ1 circulates and Ψ2 librates. T ~ 10 years 6 2) Ψ1 librates and Ψ2 circulates. T ~ 10 years → Stability of trojan configurations strongly depends on Sun relative orientation
Summary SATURN: Evection resonance & Trojans Interior evection resonance (caused by the oblateness):
● o o The trojan configuration is possible, a~8.2 RP when e~0 at ψi =0 and ψi =180 ≡ single satelites
Exterior evection resonance (caused by the Sun):
● Ψi librates in two regions, depending on relative orientation to the Sun, a~0.4 RH,,0.2 < ei < 0.4 . 3 1) Ψ1 circulates and Ψ2 librates. T ~ 10 years 6 2) Ψ1 librates and Ψ2 circulates. T ~ 10 years → Stability of trojan configurations strongly depends on Sun relative orientation Tidal migration trojan configurations: ● Fast migration: the system cross the interior resonance modifying e,i ● Slow migration: less massive companion evolve in the resonance until a collision occurs. Could the impacts of ancients own trojans modify the surface or even the orbits of Rhea?...Titan? ...Hyperion?
Summary SATURN: Evection resonance & Trojans Interior evection resonance (caused by the oblateness):
● o o The trojan configuration is possible, a~8.2 RP when e~0 at ψi =0 and ψi =180 ≡ single satelites
Exterior evection resonance (caused by the Sun):
● Ψi librates in two regions, depending on relative orientation to the Sun, a~0.4 RH,,0.2 < ei < 0.4 . 3 1) Ψ1 circulates and Ψ2 librates. T ~ 10 years 6 2) Ψ1 librates and Ψ2 circulates. T ~ 10 years → Stability of trojan configurations strongly depends on Sun relative orientation Tidal migration trojan configurations: ● Fast migration: the system cross the interior resonance modifying e,i ● Slow migration: less massive companion evolve in the resonance until a collision occurs. Could the impacts of ancients own trojans modify the surface or even the orbits of Rhea?...Titan? ...Hyperion? EXOMOONS & TROJAN COMPANIONS IN EXOPLANETS ● Resonant removal of primordial single exomoons will occur while the planets migrates inward due to the interior evection resonance depending on their initial locations and composition (Spalding et al., 2016)…. trojan companions will also suffer this effect and might reduce drastically the kind of trojans that we could expect to detect. THANKS !!!
Conclusions:
Interior evection resonance. Trojan pair of moons are present inside.
D Trojan configurations are io n destroyed if smooth e migration occurs. Could we infer Q* value ?
Rhea
Thetys
Conclusions:
Interior evection resonance. Exterior evection resonance. Trojan pair of moons are e ~ 0.4 present inside. No single moons in prograde D motion are located here. Trojan configurations are io n destroyed if smooth e Exterior evection resonance migration occurs. in retrograde case. Could we infer Q* value ? Numerical experiments suggest that trojan pairs can Rhea exist
Thetys
Evection resonance
● Consider a planetocentric system with a Satellite and the Sun as perturber, the
evection Ψ=λO - ϖ, is associated to the conmensurability between ϖ and λO (Brouwer and Clemence, 1961)
● Hénon (1969, 1970) studied the periodic orbits and Hamilton & Krivov (1997) identified the differences between prograde (i~0o → Ψ librates araound 0o or 180o) and retrograde resonance (i~180o → Ψ librates araound 90o or 270o).
● Yokoyama et al. 2008 & Frouard et al 2010 presented Analitycal developments of Hamiltonian in order to understand the resonance..
● Nesvorny et al (2003) and Cuk et al (2016) demonstrated the importance of evection resonance sculpting moons architecture in the Jovian planets.
Conclusions:
● Numerical retrograde evection resonance in a single moon was found, librating with high amplitude around Ψi = 180o.
● Trojan pair of moons can exist within the interior evection resonance at almost the same a than single systems.
● o No trojan configuration is stable for a>0.08 au, when Ψ2 = 0 .
● We identify two regions where Ψi librates depending on initial value of ϖ2
and ei ~ 0.4 (No region where both Ψi librate) o . 3 .ω2= 0 Ψ1 circulates and Ψ2 librates. T < 10 years o . 6 .ω2= 75 Ψ1 librates and Ψ2 circulates. T ~ 10 years ● Average the analytical hamiltonian to reduce d.o.f is not possible.
● Further work is necessary to understand both retrograde evection resonance in single systems and prograde evection resonance in the trojan case.
Evection resonance
● Interior resonance: ● Exterior resonance: ● Produced by the oblateness of the planet. ● Produced by external massive perturber (the Sun).
● -3 Located at a ~ 10 RP (a~10 au), and ● Located at a ~ 0.5 RH (a ~ 0.2 au) several armonics are present. and high eccentricity.
● Importance: after circumplanetary disk dissipation, tides produce outward ● Importance: the outhermost stable migration of inner moons. The moons can resonance in satellites (Nesvorny cross the resonance and may change 2003). substantially their orbital elements (e.g. Nesvorny et al 2003, Cuk et al. 2016)
a (RP) a (RP)
Chaotic layer
Evection resonance & Trojans in Saturn – Analytical Model Choosing adequate canonical set of coordinates. + HJ2 Oblateness contribution
Sun movement, relative Coorbital expansion to baricenter of m ,m ,m (Robutel & Pousse,2013 ; 0 1 2 Giuppone & Leiva 2016) (similar to Brouwer and Clemence)
Evection resonance & Trojans in Saturn – Analytical Model Choosing adequate canonical set of coordinates. + HJ2 Oblateness contribution
Sun movement, relative Coorbital expansion to baricenter of m ,m ,m (Robutel & Pousse,2013 ; 0 1 2 Giuppone & Leiva 2016) (similar to Brouwer and Clemence)
rd ● Expansions up to order 3 in ei, planar case, Sun in circular orbit → 5 d.o.f
● Average on fast angle → 4 d.o.f
● Canonical transformation in order to identify evection terms (ψ) in the model
Evection resonance & Trojans in Saturn – Analytical Model Choosing adequate canonical set of coordinates. + HJ2 Oblateness contribution
Sun movement, relative Coorbital expansion to baricenter of m ,m ,m (Robutel & Pousse,2013 ; 0 1 2 Giuppone & Leiva 2016) (similar to Brouwer and Clemence)
rd ● Expansions up to order 3 in ei, planar case, Sun in circular orbit → 5 d.o.f
● Average on fast angle → 4 d.o.f
● Canonical transformation in order to identify evection terms (ψ) in the model
Reduce and additional d.o.f? Minimize in the 4D phase-space? The model must still be tested, but numerical experiments help to understand the dynamics Evection resonance & Trojans in Saturn – Analytical Model
Architecture of migrating exoplanets Spalding et al. (2016) studied resonant removal of exomoons during planetary migration, motivated by the potential for upcoming exomoons detections (Kipping 2009; Kipping et al. 2009, 2012, 2015).
How the present-day configurations of exomoons might have been sculpted by dynamical interactions playing out during the epoch of planet formation?
Architecture of migrating exoplanets Spalding et al. (2016) studied resonant removal of exomoons during planetary migration, motivated by the potential for upcoming exomoons detections (Kipping 2009; Kipping et al. 2009, 2012, 2015).
How the present-day configurations of exomoons might have been sculpted by dynamical interactions playing out during the epoch of planet formation?
Idea: Inward migration of an oblate planet with a moon, starting from circular lunar orbit.
→ At far heliocentric distance apsidal precesion is more rapid than planetary mean motion
Architecture of migrating exoplanets Spalding et al. (2016) studied resonant removal of exomoons during planetary migration, motivated by the potential for upcoming exomoons detections (Kipping 2009; Kipping et al. 2009, 2012, 2015).
How the present-day configurations of exomoons might have been sculpted by dynamical interactions playing out during the epoch of planet formation?
Idea: Inward migration of an oblate planet with a moon, starting from circular lunar orbit.
→ At far heliocentric distance apsidal precesion is more rapid than planetary mean motion
→ As the planet migrates inward the orbital frequency increases.
→ at a_res both
frequencies are equal,ares Architecture of migrating exoplanets Spalding et al. (2016) We can detect systems with moons locked at evection resonace depending on parameters of inward migration.
* Initial planet location a0p
* Characteristic time-scale
Also some moons can be lost, converted in planetocentric impactors.
Architecture of migrating exoplanets Spalding et al. (2016) We can detect systems with moons locked at evection resonace depending on parameters of inward migration.
* Initial planet location a0p
moon capture* Characteristic time-scale in evection Also some moons can be lost, converted in planetocentric impactors.
Final position 1 AU → moon lost 10 r
Final position 1 AU → moon lost 6.8 r
Where can we locate exomoons? Fiducial Jupiter-Io systems.
Where can we locate exomoons? Fiducial Jupiter-Io systems.
a = aIO = 4.2 RP
p Where can we locate exomoons? Fiducial Jupiter-Io systems.
a = aIO = 4.2 RP
2
adiabatic
1
p Where can we locate exomoons? Fiducial Jupiter-Io systems.
a = aIO = 4.2 RP a = 7.1 RP
p p Where can we locate exomoons? Fiducial Jupiter-Io-Io systems.
ai = 7.1 RP
Δψ