GoBack Structure in narrow rings: The Scattering approach Luis - PowerPoint PPT Presentation

GoBack

Structure in narrow rings: The Scattering approach Luis Benet, Olivier Merlo Instituto de Ciencias F´ ısicas Universidad Nacional Aut´ onoma de M´ exico benet@fis.unam.mx 1

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Uranus rings PIA01977 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Uranus rings and shepherds PIA01976 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Saturn’s F ring PIA02292 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Saturn’s F ring, Prometheus and Pandora PIA06143, PIA07523 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Structure in Saturn’s F ring PIA07522 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Encke gap ringlets PIA08305 (NASA/JPL/Space Science Institute) 2

Observations: narrow rings Introduction Observations Open questions Scattering approach Occurrence of rings Structure in rings Summary Neptune rings and arcs PIA01493 (NASA/JPL/Space Science Institute) 2

Open questions Introduction We have a first-order understanding of the dynamics and key Observations processes in rings, much of it based in previous work in galactic and Open questions stellar dynamics. (...) Unfortunately, the models are often idealized Scattering approach (for example, treating all particles as hard spheres of the same size) Occurrence of rings and cannot yet predict many phenomena in the detail observed by Structure in rings spacecraft (for example, sharp edges). Non-intuitive collective Summary 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). 3

Open questions Introduction Some open issues are: Observations Open questions Rings with sharp–edges, narrow and eccentricity � Scattering approach Multiple ring components: Strands � Occurrence of rings Structure in rings Clumps and arcs � Summary Kinks and bendings � Stability, life times, origin, ... � 3

Scattering approach to narrow rings Consider the full N + 1 -Hamiltonian in an inertial frame, which can Introduction Scattering approach be written as ( N = N moons + N ring particles ) Hamiltonian Basic ideas � 1 N N P i | 2 − GM 0 M i GM i M j Consequences � | � � � H = − | � R i − � | � R i − � Occurrence of rings 2 M i R 0 | R j | i =0 i<j � =0 Structure in rings = H K m + V m − m + H K rp + V m − rp + V rp − rp Summary 4

Scattering approach to narrow rings Consider the full N + 1 -Hamiltonian in an inertial frame, which can Introduction Scattering approach be written as ( N = N moons + N ring particles ) Hamiltonian Basic ideas � 1 N N P i | 2 − GM 0 M i GM i M j Consequences � | � � � H = − | � R i − � | � R i − � Occurrence of rings 2 M i R 0 | R j | i =0 i<j � =0 Structure in rings = H K m + V m − m + H K rp + V m − rp + V rp − rp Summary 1st approx.: no interaction among ring particles. 4

Scattering approach to narrow rings Consider the full N + 1 -Hamiltonian in an inertial frame, which can Introduction Scattering approach be written as ( N = N moons + N ring particles ) Hamiltonian Basic ideas � 1 N N P i | 2 − GM 0 M i GM i M j Consequences � | � � � H = − | � R i − � | � R i − � Occurrence of rings 2 M i R 0 | R j | i =0 i<j � =0 Structure in rings = H K m + V m − m + H K rp + V m − rp Summary 1st approx.: no interaction among ring particles. 2nd approx.: In the planetary case M rp ≪ M m ≪ M 0 . Thus, we replace the many-body problem by a collection of independent one–particle time-dependent Hamiltonians: H = 1 P | 2 + V 0 ( | � 2 | � X | , t ) + V eff ( | � X | , t ) Restricted N -body problem ⇒ intrinsic rotation 4

Basic ideas Introduction We shall concentrate on: Scattering approach 0. Intrinsic rotation Hamiltonian Basic ideas 1. Phase–space regions where scattering dominates the Consequences dynamics: Escape to infinity is dominant Occurrence of rings Structure in rings 2. Organizing centers in phase space (periodic orbits or tori) are Summary 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 5

Basic ideas Introduction Phase space in a co-rotating frame Scattering approach Hamiltonian Basic ideas Consequences Occurrence of rings Structure in rings Summary 5

Consequences Introduction Some structural consequences that follow from the assumptions: Scattering approach Scattering dynamics ⇒ rings have sharp edges � Hamiltonian Basic ideas Orbits of organizing centers ⇒ eccentric rings � Consequences Occurrence of rings Small stable regions in phase space ⇒ narrow rings � Structure in rings Summary The whole scattering approach is robust 6

The rotating billiard Introduction Ring particles evolution is given by Scattering approach Occurrence of rings H = 1 P | 2 + V 0 ( | � Rotating billiard 2 | � X | , t ) + V eff ( | � X | , t ) Periodic orbits Occurrence of rings Structure in rings The simplest case: planar billiard on a Kepler orbit Summary 7

The rotating billiard Introduction A hard-disk moving on a Kepler elliptic orbit Scattering approach Occurrence of rings Rotating billiard V 0 ( | � X | , t ) = 0 Periodic orbits Occurrence of rings V eff ( | � X − � R d ( t ) | > d ) = 0 Structure in rings V eff ( | � X − � R d ( t ) | ≤ d ) = ∞ Summary a (1 − ε 2 ) R d = 1 + ε cos φ d ˙ a (1 − ε 2 ) 1 / 2 R 2 φ = Simpler periodic orbits: Consecutive radial–collision orbits 7

Circular case: Periodic orbits and stability Introduction Radial periodic orbits: Scattering approach Occurrence of rings ( R − d ) 2 = 2 cos 2 θ + ∆ φ sin(2 θ ) J n Rotating billiard Periodic orbits (∆ φ ) 2 Occurrence of rings with ∆ φ = (2 n − 1) π + 2 θ Structure in rings Summary Stability: 2 + (∆ φ ) 2 (1 − tan 2 θ ) Tr D P J = d/R − 4(1 + ∆ φ tan θ ) d/R Changes of stability at Tr D P J = ± 2 8

Circular case: Periodic orbits and stability Introduction Scattering approach J/ ( r − d ) 2 = 0 . 29325 J/ ( r − d ) 2 = 0 . 29218 Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary p = − d − R cos α − v sin( α − θ ) 8

Occurrence of rings Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary 9

Occurrence of rings Introduction Scattering approach Occurrence of rings Rotating billiard Periodic orbits Occurrence of rings Structure in rings Summary 9

Strands and Arcs b) 0.02 Introduction Scattering approach 0.015 Occurrence of rings 0.01 Structure in rings Strands and Arcs 0.005 Phase-space volume M-M Resonaces Y 0 Dynamics -0.005 Summary -0.01 -0.015 -0.02 -0.3004 -0.3002 -0.3 -0.2998 -0.2996 X ε = 0 . 00165 10

Strands and Arcs b) 0.02 Introduction Scattering approach 0.015 Occurrence of rings 0.01 Structure in rings Strands and Arcs 0.005 Phase-space volume M-M Resonaces Y 0 Dynamics -0.005 Summary -0.01 -0.015 -0.02 -0.3004 -0.3002 -0.3 -0.2998 -0.2996 X ε = 0 . 00167 10

Strands and Arcs c) 0.02 Introduction Scattering approach 0.015 Occurrence of rings 0.01 Structure in rings Strands and Arcs 0.005 Phase-space volume M-M Resonaces Y 0 Dynamics -0.005 Summary -0.01 -0.015 -0.02 -0.3004 -0.3002 -0.3 -0.2998 -0.2996 X ε = 0 . 00168 10

Strands and Arcs d) 0.02 Introduction Scattering approach 0.015 Occurrence of rings 0.01 Structure in rings Strands and Arcs 0.005 Phase-space volume M-M Resonaces Y 0 Dynamics -0.005 Summary -0.01 -0.015 -0.02 -0.3004 -0.3002 -0.3 -0.2998 -0.2996 X ε = 0 . 001683 10

Phase-space volume of trapped regions ε = 0 1:1 1:6 1:5 1:4 1:3 1:2 700 a) 600 500 400 Ν(<∆ t>) 300 200 100 0 4.08 4.1 4.12 4.14 4.16 <∆ t> Stability resonance: e iα , cos( α ) = 2Tr D P J , α p : q / (2 π ) = p/q 11