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Structure in narrow rings: The Scattering approach

Luis Benet, Olivier Merlo

Instituto de Ciencias F´ ısicas Universidad Nacional Aut´

• noma de M´

exico

benet@fis.unam.mx

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Uranus rings

PIA01977 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Uranus rings and shepherds

PIA01976 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

2

Saturn’s F ring

PIA02292 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Saturn’s F ring, Prometheus and Pandora

PIA06143, PIA07523 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Structure in Saturn’s F ring

PIA07522 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Encke gap ringlets

PIA08305 (NASA/JPL/Space Science Institute)

Observations: narrow rings

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Neptune rings and arcs

PIA01493 (NASA/JPL/Space Science Institute)

Open questions

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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We have a first-order understanding of the dynamics and key processes in rings, much of it based in previous work in galactic and stellar dynamics. (...) Unfortunately, the models are often idealized (for example, treating all particles as hard spheres of the same size) and cannot yet predict many phenomena in the detail observed by spacecraft (for example, sharp edges). Non-intuitive collective effects give rise to unusual structures. (...) One such example is the case of shepherding satellites. The F ring is not exactly placed where the shepherding torques would

• balance. Of the Uranian rings, shepherds were found only for the

largest ε (epsilon) ring; even so, they are too small to hold it in place for the age of the solar system. Another issue is that the sharp edges of rings are too sharp! Larry Esposito, Planetary rings (Cambridge University Press, 2006).

Open questions

Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary

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Some open issues are:

• Rings with sharp–edges, narrow and eccentricity
• Multiple ring components: Strands
• Clumps and arcs
• Kinks and bendings
• Stability, life times, origin, ...

Scattering approach to narrow rings

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

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Consider the full N + 1-Hamiltonian in an inertial frame, which can be written as (N = Nmoons + Nring particles)

H =

N

• i=0

1 2Mi | Pi|2 − GM0Mi | Ri − R0|

• −

N

• i<j=0

GMiMj | Ri − Rj| = HKm + Vm−m + HKrp + Vm−rp+Vrp−rp

Scattering approach to narrow rings

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

4

Consider the full N + 1-Hamiltonian in an inertial frame, which can be written as (N = Nmoons + Nring particles)

H =

N

• i=0

1 2Mi | Pi|2 − GM0Mi | Ri − R0|

• −

N

• i<j=0

GMiMj | Ri − Rj| = HKm + Vm−m + HKrp + Vm−rp+Vrp−rp

1st approx.: no interaction among ring particles.

Scattering approach to narrow rings

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

4

Consider the full N + 1-Hamiltonian in an inertial frame, which can be written as (N = Nmoons + Nring particles)

H =

N

• i=0

1 2Mi | Pi|2 − GM0Mi | Ri − R0|

• −

N

• i<j=0

GMiMj | Ri − Rj| = HKm + Vm−m + HKrp + Vm−rp

1st approx.: no interaction among ring particles. 2nd approx.: In the planetary case Mrp ≪ Mm ≪ M0. Thus, we replace the many-body problem by a collection of independent one–particle time-dependent Hamiltonians:

H = 1 2| P|2 + V0(| X|, t) + Veff(| X|, t)

Restricted N-body problem ⇒ intrinsic rotation

Basic ideas

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

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We shall concentrate on: 0. Intrinsic rotation 1. Phase–space regions where scattering dominates the dynamics: Escape to infinity is dominant 2. Organizing centers in phase space (periodic orbits or tori) are stable. 3. An ensemble of non–interacting particles with almost-arbitrary initial conditions Rings are obtained by projecting onto the X − Y space, at fixed time, all dynamically trapped particles

Basic ideas

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

5

Phase space in a co-rotating frame

Consequences

Introduction Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary

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Some structural consequences that follow from the assumptions:

• Scattering dynamics ⇒ rings have sharp edges
• Orbits of organizing centers ⇒ eccentric rings
• Small stable regions in phase space ⇒ narrow rings

The whole scattering approach is robust

The rotating billiard

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

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Ring particles evolution is given by

H = 1 2| P|2 + V0(| X|, t) + Veff(| X|, t)

The simplest case: planar billiard on a Kepler orbit

The rotating billiard

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

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A hard-disk moving on a Kepler elliptic orbit

V0(| X|, t) = Veff(| X − Rd(t)| > d) = Veff(| X − Rd(t)| ≤ d) = ∞ Rd = a(1 − ε2) 1 + ε cos φ R2

d ˙

φ = a(1 − ε2)1/2

Simpler periodic orbits: Consecutive radial–collision orbits

Circular case: Periodic orbits and stability

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

8

Radial periodic orbits:

Jn (R − d)2 = 2 cos2 θ + ∆φ sin(2θ) (∆φ)2

with ∆φ = (2n − 1)π + 2θ Stability:

Tr DPJ = 2 + (∆φ)2(1 − tan2 θ) d/R −4(1 + ∆φ tan θ) d/R

Changes of stability at Tr DPJ = ±2

Circular case: Periodic orbits and stability

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

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J/(r − d)2 = 0.29325 J/(r − d)2 = 0.29218 p = −d − R cos α − v sin(α − θ)

Occurrence of rings

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

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Occurrence of rings

Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary

9

Strands and Arcs

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

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• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X b)

ε = 0.00165

Strands and Arcs

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

10

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X b)

ε = 0.00167

Strands and Arcs

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

10

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X c)

ε = 0.00168

Strands and Arcs

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

10

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X d)

ε = 0.001683

Phase-space volume of trapped regions

11

ε = 0

4.08 4.1 4.12 4.14 4.16 <∆t> 100 200 300 400 500 600 700 Ν(<∆t>) 1:1 1:2 1:3 1:4 1:5 1:6 a)

Stability resonance: eiα, cos(α) = 2Tr DPJ, αp:q/(2π) = p/q

Phase-space volume of trapped regions

11

ε = 0

0,796 0,798 0,8 0,802 0,804 p 3,12 3,13 3,14 3,15 3,16 α W

s

W

u

0,796 0,798 0,8 0,802 0,804 p 3,12 3,13 3,14 3,15 3,16 α W

s

W

u

0,796 0,798 0,8 0,802 0,804 p 3,12 3,13 3,14 3,15 3,16 α W

s

W

u

Phase-space volume of trapped regions

11

ε = 0

3.12 3.13 3.14 3.15 3.16 0.796 0.798 0.8 0.802 0.804 α p I1 I2 I3

Phase-space volume of trapped regions

11

ε = 0

4.08 4.1 4.12 4.14 4.16 <∆t> 100 200 300 400 500 600 700 Ν(<∆t>) a)

• 0.34
• 0.32
• 0.3
• 0.28

X

• 0.2
• 0.1

0.1 0.2 Y b)

Excitation of stability resonances separates regions in phase space

Mean-motion Resonances

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

12

ε = 0.00165 ε = 0.00167

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X b)

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X c)

Mean-motion Resonances

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

12

ε = 0.00165 ε = 0.00167

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X b)

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 0.3004
• 0.3002
• 0.3
• 0.2998
• 0.2996

Y X c)

4,088 4,0882 4,0884 4,0886 4,0888 <∆t> 50 100 150 200 250 300 Ν(<∆t>) b) 1 4 9 : 2 2 9 9 5 : 1 4 6 1 3 6 : 2 9 4,088 4,0882 4,0884 4,0886 4,0888 <∆t> 50 100 150 200 Ν(<∆t>) c) 1 4 9 : 2 2 9 9 5 : 1 4 6 1 3 6 : 2 9

Mean-motion Resonances

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

12

• 17
• 16
• 15
• 14
• 13
• 12
• 11
• 10
• 9

4.08 4.09 4.1 4.11 4.12 4.13 4.14 4.15 4.16 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 ∆t

ε

Dynamics

Introduction Scattering approach Occurrence of rings Structure in rings Strands and Arcs Phase-space volume M-M Resonaces Dynamics Summary

13

Summary

Introduction Scattering approach Occurrence of rings Structure in rings Summary

14

• Using a scattering obtain consistently sharp–edged narrow

eccentric rings. Scattering dynamics ⇒ rings with sharp edges Eccentric orbits as organizing centers ⇒ eccentric rings Small stable regions in phase space ⇒ narrow rings

• For more than two degrees of freedom, rings may have several

components, strands. These appear by exciting certain stability resonances which lead to instabilities.

• Arcs (clumps) are related to mean-motion resonances within

thin strands.

Outlook (work in progress)

15

CRTBP µ = 2.086 × 10−6

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 1.02
• 1.015
• 1.01
• 1.005
• 1

px x

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

• 1.02
• 1.015
• 1.01
• 1.005
• 1

px x

Outlook (work in progress)

15

Ring obtained using a realistic consistent gravitational restricted 5 body problem