Predator-PreyAnalogsfor
Saturn’sNon-LinearRingDynamicsLWEsposito,SMadhusudhanan,MRehnberg,ZBrown,NAlbers
LASP- UniversityofColoradoJEColwell
UniversityofCentralFlorida
FinalCassiniSymposiumBoulderAugust2018
Transientringstructuresappearwheretheringsareperturbedstrongly
• EquinoxobjectsattheMimas2:1resonance• Strawbetweendensitywavecrests• Excessvarianceincreasesbetweenwavecrests• Gapedgeswhenamoonpassesby• SolitarywaveswheretheJanusresonancefallsonEpimetheusdensitywaveevery8years
AggregatesformatouterBringedge
‘Straw’seenbetweendensitywavecrestsmustforminlessthan10hours
SaturnOrbitInsertion2004
StrawfromCassiniGrandFinale
Janus2:1DW
Particle statistics show larger structures between density wave crests
Gap % Coverage TrendsTrend: gaps cover more linear area in troughs for Janus 4:3, Mimas 5:3(t significances: 1e-6,8e-12)
Trend: gaps cover more linear area in troughs for Janus 5:4 and 6:5(t significances: 5e-16,1e-15)Trend: gaps coverage is
greater in troughs for Prometheus 28:27(t significance: 5e-13)
DaphnisEdgeWakeshowsdownstreameffect
ClumpsForm
RingEdgeShearsandSeparates
Anothernon-linearphenomenon:AsolitonexcitedbyJanus-Epimetheus swap,every8years
Solitary wave propagating through A ring, seen every 8 years
Howtoexplainthisdynamicstructure?
• Solitarywavesandthelargeamplitude,rapidlygrowingtransientstructuresindicatenon-linearphenomena
• N-bodysimulationsaretooslow,anddon’tincludeallthephysics• Useasimplermodelwithanecologicalanalogy:Predator–Prey• Trackthemassandvelocitydispersion:Relativevelocityisstirredupbyclumps,velocitydisruptsclumps
• Forcethesystembythemoon’sgravitydrivingthesurfacemassdensityandthevelocitydispersion
• Allowfordiskinstability,usingToomre’sdispersionformula• Usenumericalsimulationresultsforoutcomesofstochasticcollisions(Hyodo&Ohtsuki;Leinhardt&Stewart)
Predator-PreyEquationsforRingClumping(Espositoetal 2012)
M=∫n(m)m2dm/<M>;Vrel2=∫n(m)Vrel2 dm/N
dM/dt=M/Tacc – Vrel2/vth2 M/Tcoll[accretion][fragmentation/erosion]
dVrel2/dt=-(1-ε2)Vrel2/Tcoll +(M/M0)2 Vesc2/Tstir[dissipation] [gravitationalstirring]
- A0 cos(ωt)[forcingbystreamline crowding]
Theaggregatemass,Misthe‘prey’;ThedispersionVrel2 isthe‘predator’:Itfeedsoffofthemassbygravstirring;Thepredatorreducesthepreybyerosion
PredatorPreyModelwithLogisticGrowth
20 0
2
2
2 2 22
' 1
:
2 '
0 0
( )(1 )' '
s sL
orbS
S
rel
S DI S th
DI
rel rel escB B
S S S S
Basic equationTT
VdM M S M M Sdt T R T T R VM whenT
dV V V Mdt T T
MM
πτ
ρ ρρ ρ
ω
τ τε
τ
τ τ
τ
=
⎡ ⎤= + −⎢ ⎥⎣ ⎦
= ≥
⎡ ⎤= − − + +⎢ ⎥
⎣ ⎦
⎛ ⎞= −⎜ ⎟
⎝ ⎠
Motivation:WhentheirMass reaches alimitingvalue,theaggregatescannotgrowfurtherbyringparticle sweepup,sincewehaveonlyafinitenumberofringparticles tosticktothegrowingaggregates.This ismodelled byaddingalogisticgrowthtermlimiting theopticaldepthofsmalleraggregates.
Thus,theclosertheaggregatemassMtoM_limit,theslowerthegrowthrate.
Note:Afterthe limitingmass isreached,theaggregateschange inmassonlyduetostochasticcollisions whichyieldaccretionsanddisruptions.
4%variation
( )( )01 , 0.97
1 sin syn
mm tω
Σ = Σ =+
Strengthregimesurfacemassdensityforcingphaseplots
2 2 2 2
2
2
0sG k v kω κ π
ω
= − Σ +
<
Diskinstabilitywhen:
Equilibriumdistributionofaggregatesfromstochasticcollisionsisapowerlaw
PowerLawindexforradiusdistribution: -1.0158
Conclusions• Ringstructureshowstransientclumpinginperturbedregions:Weconcludethatforcingbythemoontriggersaggregation.Thisalsoincreasestherelativevelocity,liberatingsmallparticles
• Thestructureformsrapidly,onorbitaltimescales,outofphasewiththemoon… anddownstreamofthemoon’swakeinitiation
• Wefindthatgrowthbysweep-upistooslowtoexplaintheexcessstructureobserved inbetweendensitywavecrests
• Gravitationaldiskinstability canactonorbitaltimescales;WeuseToomre’s stabilityparameterQtoestimatethegrowthrateforclumps
• Weachieverapidgrowthbymodulatingthesurfacemassdensity,decreasingthevelocitydispersionorbydecreasingtheshear
• Aggregatesfromstochasticcollisionshaveapower-lawsizedistribution
Takeawaymessage:Moonforcingdrivesaccretion,triggersdiskinstability,producingtransientclumpsdownstream:AcontinuingprocessofCosmicRecycling
Back-Up
M=0.97Vrel=0for0.01torb
WecanforcethePredator-Preymodelbysurfacemassdensityorbyvelocityvariations,whichgivesimilaroutcomes.
ClumpmassM,fromVrel=0for>0.3Torb
Massvariationsaroundthefixedpoints:0.7torb=64.4%0.6torb=60%0.5torb=51%0.4torb=48%0.3torb=31%
UsingNumericalsimulationsresults
• Whathappenswhenequalsizedobjectcolliderandomly?• CanNumericalsimulationsbeusedtobasethestatisticsofrandomevents?
UsingtheresultsofHyodoandOhtsuki:• Theoutcomesofthestochastic eventsarebasedontheratioofImpact/Escapevelocityandthedirectionofcollision.
• Thedirectionofcollision canberadial,azimuthal andvertical.Direction ischosenwithequalprobability.
Radial
Azimuthal
Vertical
Radial
Azimuthal
Vertical
Accretion
Disruption
RandomEventOutcomes:• Accretion:Greenregion:Thiseventdoublesthecurrent
mass.• Hitandrun:Blueregion:Thiseventdoesnotchangethe
mass.• disruption:Redregion:Thiseventhalvesthemass.
Note:Thissimulationconsiderspresenceofstrongtidalwaves.(DistancefromSaturn:140kkm)
HitandRun
HyodoandOhtsuki:140Kkmcasesimulation
Limitingmasscalculation:computedbasedoncellsize
40
4 6 2
12 9
60
3 30 0
500 10
5 10 10 100 /
5 10 5 10
1.0472 10
4.7746 10 M ~ 5 10
L
L
L
L
M m mM cm cm g cmM g or kgM kgM M
= × ×Σ
= × × ×
= × ×
= ×
= × ×
Inputparameters:tauS =1tauB =0.1;tauS/tauB=10.epsilon=0.1Coefficientofrestitutionrho0=0.25g/cm3.Uncompresseddensityofringparticleaggregates.m0=1.05x109 g,massofR0=10mspherewithrho0=0.25g/cm3.Referencemass.S=300cm,smallparticleradius,frommassdensityrho0=0.25g/cm3,andopticaldepthtauS=0.1.Vthresh(M0)=1cm/sec
EquilibriumdistributionofMassofaggregates:
ThePowerLawforindexradiusdistributionwasfoundtobe:-0.3386
FinalPlots(DistancefromSaturn140Kkm,presenceoftidalenvironment)
PowerLawIndex:-1.0158PowerLawIndex:-0.3386
Conclusion:• ThePredatorPreymodelcanincludetheoutcomesofrandomcollisions inthepresenceoftidalenvironmentbyusingtheresultsofnumericalsimulations.
• Thepower lawindexofmassdistributionwasfoundtobe:-0.3386• Thepower lawindexofradiusofaggregatesdistributionwasfoundtobe:-1.0158
• Thepower lawindexofthemassdistributionobtained fromthesimulationmatchwellwithresultsobtainedfromobservations. (moreexplanationmightbeneededforthispoint)
• TheMassdistribution canbecomputed fordifferentsettingsoftidalenvironment.• TheLongtermbehavioroftheringscanbestatisticallypredictedusingtheequilibriummassdistributionsusingPredatorPreymodel,whichcouldotherwisebeverytimeconsuming.
• Thoughthereisastrongpresenceoftidalenvironment(140kkm,thereisstillapossibilityoffindingaggregateswithhighmasses,thiscouldexplainthepresenceofStrawsinFring?(Notverysureaboutthispoint)
