Astro 6570 Physics of the Planets Planetary Rings
Roche limit Synchronous orbit
Equivalent masses of rings and satellites
The Uranian Rings
Stellar occulta@on profiles of the Uranian rings (1985): P. Nicholson (unpubl.)
Neptune s Ring System Radial Brightness Profile M. Showalter (1989)
Desiderata: Par@cle composi@on? Visible/IR spectrum, radio/radar reflec@vity, radio absorp@on coeff., densi@es Par@cle size distribu@on? Op@cal phase func@on, thermal spectrum, radio/radar scayering proper@es, thermal @me constant, surface density Velocity dispersion? Ring thickness (direct observa@ons), opposi@on effect, damping of density/bending waves, apsidal phase shi\s (eccentric ringlets) Origin of ring structures? Large scale varia@ons (A, B, C rings) Small scale structure Gaps & ringlets Wave trains Origin of rings themselves?
P. Nicholson (unpubl.)
Clark 1980
Saturn s Rings L. Esposito (1984)
Par@cle Sizes in Saturn s Rings Zebker et al (1985) Voyager Radio Science
ScaYering X- Sec@ons
Surface mass densi@es derived from density waves P. Nicholson (unpubl.)
Planetary Ring Dynamics (Based on notes by P- Y. Longaref) Confinement of rings to the planet s equatorial plane: orbital p precession This occurs in the differential precession timescale:.1 t pr ~ 21 2 ( 4 J! R p $ + * 2 # & '-! a $ * " a % - " # /a% & ), After t >> t pr, the inclined ring is a thick swarm of particles, symmetric about the equatorial plane. The half-thickness of the swarm is H! a sin i where i = mean inclination of an orbit!! a i when i << 0 $ " # 2 % &
The rms "vertical" velocity of the particles is! z 2 2 =! orb sin 2 i sin 2 "!where " = orbital longitude. = 1 2 #2 a 2 i 2 (# = mean motion) i.e.,! z 2 1 2 = 1 2 #ai " #H Eventually inelas@c collisions between the ring par@cles will damp out the ver@cal velocity components, conserving the total ( = z) angular momentum of the ring. Deno@ng the residual random (non- circular) par@cle velocity by c, the equilibrium ring half thickness is given by H! c!
Op@cal Depth and Surface Density If the mean particle radius is R, and average particle volume density is "n" particles/vol., then the NORMAL OPTICAL DEPTH is! = n "R 2 ( ) 2H ( ) and the SURFACE MASS DENSITY of the ring is ( ) $n 2H # = 4 3 "R3 = 4 3 ( ) ( ) $R! where $ = particle density gcm%3 Example: Saturn's A ring 1 ' & = GM * 2 s ) ( a 3,! 2-10 %4 sec %1 + c ~ 0.1 cm/s. H ~ 5m # ~ 40 g cm %2 / 0. R ~ 60 cm, if $! 1! ~ 0.5 12 3 n ~ 10 %7 cm %3 and mean inter-particle separation ~ n %1 3! 200 cm! 3R!
Collision Frequency & Ring Viscosity The number of particle collisions a particle has in time!t is!n c ~ 2n ("R 2 )c!t so the collision rate is given by # c =!N c!t i.e., # C! c$ H! %$ ~ 2"R 2 nc 2R c c!t The mean collision time, # c &1 ~ P orb 2"$.
The effective KINEMATIC VISCOSITY of the ring is!! " c l 2 = #$l 2 where l is the collisional mean free path: For $ " 1, " c " # and l! c " c = c #$. For $ # 1, " c # # and the particle executes many orbits between successive collisions % l! c # (i.e., deviation from circular orbit)! H. For arbitrary $, a more detailed analysis gives l! c #( 1+ $ ) 1 2 2 The viscosity is then!! which goes to the above limits as $ & 0 or $ & '. c 2 $ ( ) # 1+ $ 2 Note: this simple picture ignores non-local viscosity, due to the finite size of ring particles in comparison with l, and the effects of particle wakes or agglomerations due to self-gravity.
Spreading Timescale This viscosity, combined with the Keplerian shear in the ring, leads to a radial transport of angular momentum outward in the rings, and to radial spreading of a narrow ring: Consider 2 segments of the ring, each of width!a, in contact over an area A = 2HL. The effective differential velocity at the boundary!"! a! # = a d# da!a, so the velocity shear is d" da =a d# da.
The tangential force exerted on the inner segment is # f T =! a d" & ) $ % da ' ( A where ) is the mass density of the ring per unit volume. This leads to a radial ANGULAR MOMENTUM FLUX, per unit arc length L, given by F L = *a f T L = *2 d" )!a2 H da d" = *+!a 2 since + = 2H ). da For a keplerian " a ( ) ~ a * 3 2, F L! + 3 2 +!a " This flux transports angular momentum outwards in the ring, removing it from particles at the inner edge and adding it to those at the outer edge, the ring spreads radially.
Consider a narrow ring of width 2!a and surface mass density ". The angular momentum of each 1 2 of the ring, per unit length, is L = " #a 2!a, and the change in L associated with the ring spreading from 0 $!a is!l ~ 1 2 L %!a ( & ' a ) * ~ 1 " #a!a 2 The characteristic spreading time is thus ( ) 2 1, since #a 2! a 2. + sp =!L F L ( "!a ) 2, % =!a & ' # ( ) * 2 - ċ 1 $equivalent to a particle's random walk over a distance!a 0!a = 100 km = 10 7 cm Example : Uranus' / ring: 1 23 + " 1, c ~ 0.1 cm/s 4, ~ 25 cm 2 s.1 4 + sp " 4 5 10 12 sec ~ 10 5 yr
Collisional Equilibrium What determines c, the velocity dispersion? Collisions convert ordered keplerian shear veloci@es into random veloci@es Inelas@c collisions remove energy from the random veloci@es è heat. i.e., Kepler shear è random veloci@es è heat
(a) Work done by viscous force!w!t ( ) + f T # "# = $a ~ a " 1 2! "f T # +!#! 1 3 f T a! $ = " 1 3 f T a d$ da!a ' = " %& a d$ * ( ) da +, ' = "& a d$ * ( ) da +, 2 2 A!a ~& $ 2 for Kepler shear " per unit ring mass
(b) Energy dissipa@on rate!e collision! m % 1 2 "# r & ' ( ) 2 $ 1 2 # 2 r ( ) * where "= coefficient of restitution +!E!t! $ 1 2! $ 1 2, c ( 1$ "2 )c 2 m (m= particle mass) ( 1$ "2 )c 2 " per unit mass Balancing (a) and (b) -. c 2 /0! /0 2 ( 1$ "2 )c 2 1+ 0 1 " 2 ~ 1$. where. ~ 0( 1) 1+ 0 2 Since "( c), this is an implicit relation between c and 0.
Comments The equilibrium value of c depends on the elasticity (via!) of the ring particles, which must be measured experimentally, and on ". Detailed calculations give # ~ 0.6 $! min ~ 0.6, and numerical simulations have confirmed the above! " ( ) relation for hard spheres. The equilibrium is stable if d! < 0; this is the case for most (all?) materials. dc If! <! min, dissipation wins and c % 0 $ ring collapses to a dense (mono) layer where the effective l & R $ ' ~ (R 2 ". ) # ( 3 R 2 "! ("c ( 2 1*! ) 2 + # ) c! - -, 1*! 2 $ H ~ c ( ~ R ( ) We have neglected the following: - realistic particle size distribution - energy associated with particle spins - inter-particle gravitational interactions. 0 0 / 1 2 (R ~ (R ~ 0.01 cm/sec for R! 50 cm - non-keplerian shear associated with satellite perturbations (e.g., at resonances) - non-isotropic velocity dispersions: c r 1 c z - collisional destruction and/or accretion of ring particles
Angular momentum transport & shepherding We have seen above that viscosity (i.e., collisions) leads to an outward flux of angular mementum, F L in a Keplerian disk. This can also be seen from simple angular momentum/energy considerations: two equal-mass ringlets interact conserving L but dissipating E: an amount of angular momentum,! L is transferred from the inner ringlet to the outer. the overall change in mechanical energy is given by the net work done on the two ringlets:!e = "# in!l + # out!l $ = d# d#!a " ( da da "!a ' & )) % (!L = 2 d#!a!l da 3#!a!L = " a Since!E must be 0 or negative, we must have! L > 0 i.e., momentum is transferred outward.
Angular momentum transport & shepherding The same argument applies to the gravita@onal interac@on of a ring and a satellite as long as energy is dissipated as a result of the interac@on (e.g., by damping of density waves driven in the ring), the satellite effec@vely repels the ring. This gives rise to the concept of satellites shepherding ringlets, and clearing gaps in wider rings. Let s consider two situa@ons: (1) a satellite (mass M s ) and a narrow ring of mass M r, or (2) a ring edge of surface mass density σ. The torque between the satellite and the ring for these two cases is given by*: ( )! 0.4 G2 M 2 s M r! 2 a s " a r T 1 ( ) 4 T 2 ( )! 8# ag 2 2 M s 27! 2 a s " a r ( ) 3 In equilibrium, T s must balance F L within the ringlet, or on either side of the gap: T s = 2$a F L = 3$a 2 # %! (In fact, F L may be reduced if the satellite perturbations modify the normal Kepler shear, d! da.) *This torque scales as M s 2 because it is effectively a "tidal" torque on the ring.
!" ~e i kr #iwt $ 2 = k 2 c 2 # 2% G& k +' 2 Gravita@onal Stability Dispersion relation for axisymmetric oscillations in a self-gravitating disk: In a keplerian disk '! (. In order to avoid a range of k's where the disk is unstable (i.e., $ 2 < 0) we need Q = (c % G& > 1. "Toomre (1966) This sets the minimum velocity dispersion for a stable disk: c > % G& 4% G)R"! ( 3( * = 4G)" - +,. / 0 (R ( 2 The critical wavelength, where instability first occurs, is 1 crit = 4% 2 G& = 4% c ( 2 Q( = 4% Q H Example : Saturn's B ring, a = 10 5 km, ( = 2 2 10 #4 s #1,"! 1 3 c min! 0.07cms #1 if R = 50cm and 1 crit = 30m if Q = 1.5 (a typical value?) Saturn's A and B rings are close to gravitational instability (i.e., Q! 1), leading to ubiquitous "self-gravity wakes" with a wavelength # 1 crit.
Mimas 5:3 density and bending waves.
Outer A ring density waves
Encke Gap Satellite Wakes
Evolu@onary Timescales: Collisional spreading: An ini@ally narrow ring spreads to width ΔR in random walk @me Uranus ε ring: " T s!!r % # $ l & ' 2 T coll "! (P!R % # $ H & ' "! ) 2 ( % $ # c 2 P ' &!R 2!! 1, P! 8 hr, "R! 60km H! 10m #T s ~ 2 $ 10 4 yr 2 Saturn s B ring:!, P similar; "R ~ 25 # 10 3 km H! 50m $T s ~ 10 8 yr Satellite torques: Torques due to shepherding, or due to density waves transfer angular momentum to or from satellites: Prometheus & Pandora dri\ outward: ~ 10 7 yrs A Ring dri\s inward: ~ 10 8 yrs ε Ring shepherd life@me: 6 10 8 yrs α, β Ring shepherd life@me: 5 10 7 yrs
Timescales (con@nued): Micrometeoroid impacts: Reduce specific angular momentum of rings: C Ring collapse: < 2 10 8 yr Contaminate icy ring material: Saturn s Rings darken: ~ 10 8 yr Gross erosion of ring par@cles & redistribu@on of mass: Saturn s Rings: 1 cm/10 5 yr Drag processes (small par@cles): Poyn@ng- Robertson (light) drag: Jovian Ring: ~ 10 5 yr Exospheric drag (neutral gas): Uranian rings: 10 8-10 9 yr Neptunian rings? Plasma drag: Jovian Ring: 10 2±1 yr Destruc@on of small ring par@cles: Ion spuyering Jovian Ring: 10 3±1 yr Micro meteorid impact Uranian dust bands: 10 4 yr