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Misaligned Planets, Disks and Rings
Dong Lai Cornell University
CITA, 4/4/2016
Misaligned Planets, Disks and Rings
(1) Binary-Disk-StarInterac4on!Misalignedprotoplanetarydisk
(2) Secularperturba4onfrombinaryonplanet!MisalignedHotJupiters
(3) Extended,misalignedring/diskaroundplanets/browndwarfs
MisalignedPlanets
ESO
Alsonote:Misalignedmul4-planetsystems:Kepler-56(2planets10.5&21days,40-55deg;Huber+2013);othercandidates(?):Hirano+2014
How to produce misaligned planets (HJs)?
--“Primordial”misalignmentbetweenspinanddisk
--“Lidov-Kozaimigra;on”inducedbyabinarycompanion--Planet-Planetinterac;ons:dynamicalscaVering,secularchaos,coplanarhigh-emigra4on(Rasio&Ford96;ChaVerjeeetal.08;Juric&Tremaine08;Nagasawaetal.08;Wu&Lithwick11;Beauge&Nesvorny2012;Petrovich15,16)
Observa;onalConstraints:seeHuang&Wu2016;Petrovich2016,etc.
Ideas for Producing Primordial Misalignments between Stellar Spin and Protoplanetary Disk
-- Star formation in turbulent medium (Bate et al. 2010; Fielding et al 2014 ) -- Magnetic Star – Disk Interaction (Lai et al 2011, Foucart & Lai 2011) -- Perturbation of Binary on Disk (Batygin 2012; Batygin & Adams 2014; Spalding et al 2015; Lai 2014; Zanazzi & DL)
Star-Disk-Binary Interactions
LbLdS
disk
Mb
DL2014
Companion makes disk precess
Disk behaves like a rigid body (bending waves, viscous stress, self-gravity) Foucart & Lai 2015; Caveat ? (J. Zanazzi & Lai, in prep)
⌦
pd
' �5⇥ 10
�6
✓Mb
M?
◆ ⇣ rout
50 AU
⌘3/2
⇣ ab
300 AU
⌘�3
⇥ cos ✓db
✓2⇡
yr
◆
Disk makes the star precess
Due to gravitational torque from disk on rotating (oblate) star
⌦
ps
' �5⇥ 10
�5
✓Md
0.1M?
◆ ✓¯
⌦?
0.1
◆ ✓rin
4R?
◆�2 ⇣ rout
50 AU
⌘�1
⇥ cos ✓sd
✓2⇡
yr
◆
where
Two limiting cases:
|⌦ps|� |⌦pd| : =) ✓sd ' constant(1)
|⌦ps|⌧ |⌦pd| : =) ✓sb ' constant(2)
⌦
pd
' �5⇥ 10
�6
✓Mb
M?
◆ ⇣ rout
50 AU
⌘3/2
⇣ ab
300 AU
⌘�3
⇥ cos ✓db
✓2⇡
yr
◆
⌦
ps
' �5⇥ 10
�5
✓Md
0.1M?
◆ ✓¯
⌦?
0.1
◆ ✓rin
4R?
◆�2 ⇣ rout
50 AU
⌘�1
⇥ cos ✓sd
✓2⇡
yr
◆
Ini4allyMddecreasesover~1Myr
|⌦ps|� |⌦pd|
✓sd = 5�✓db = 5�Initial:
DL2014
In the frame rotating at rate ⌦pdˆLb
Resonance ⌦ps = ⌦pd
⌦eLe
dSdt
= ⌦psLd ⇥ S
dSdt
!
rot
= ⌦ps
Ld
⇥ S� ⌦pd
Lb
⇥ S
=⇣⌦
ps
Ld
� ⌦pd
Lb
⌘⇥ S
GeometricInterpreta;onLai2014
✓sd = 5�
Initial: ✓db = 30�
Complica;ons:
Accre;onandmagne;cinterac;on
Magnetic Star - Disk Interaction: Basic Picture
Magne4cstar
Nacc ' Mp
GM?rin
Nmag ⇠ µ2/r3inMagnetic (misalignment) torque:
Accretion torque:
Accre;on&Magne;cTorques(Order-of-Mag):
ALaboratoryExperiment
J
ALaboratoryExperiment
Nacc ' Mp
GM?rin
Nmag ⇠ µ2/r3inMagnetic (misalignment) torque:
rin ⇠✓
µ4
GM?M2
◆1/7
For ! Nacc ⇠ Nmag
Accretion torque:
Accre;on&Magne;cTorques(Order-of-Mag):
Note:Massive(>1.3Sun)TTauristarshaveweakerdipolefieldsthanlessmassivestars.Consequenceonspin-orbitmisalignment?(cf.Batygin&Spalding2015)
Star-Disk-Binary Interactions
Gravitational interactions… Now include Accretion and Magnetic Torques
DL2014
No accretion/magnetic
Accretion/magnetic damps SL-angle
Accretion/magnetic increases SL-angle
Summary(#1)Star-disk-binaryinterac;ons
• Withabinarycompanion,spin-diskmisalignmentis“easily”generated• Thekeyis“resonancecrossing”• Accre4on/magne4ctorquesaffectit,butnotdiminishtheeffect!“primordial”misalignmentofstellarspinanddisk
Toorobust??Diskwarping,planet-diskcoupling(J.Zanazzi&DL,inprep)
Formation of Hot Jupiters: Lidov-Kozai Migration driven by a binary
Holmanetal.97;Wu&Murray03;Fabrycky&Tremaine07;Naozetal.12;Petrovich15,….
Lidov-Kozai Oscillations
planetorbitalaxis
binaryaxis
• Eccentricityandinclina4onoscilla4onsinducedifi>40degrees.
• Ifilarge(85-90degrees),getextremelylargeeccentrici4es(e>0.99)
Lidov-Kozai Oscillations: Octupole Effecte.g.Fordetal2000;Naozetal11;Katzetal11;Lithwick&Naoz11;Lietal14;Antognini15
Octupole!--Verylargeeevenformodesti--Lpcanflipwrtouterbinary
IncludingShort-RangeForces!--emaxisreduced/limited--flipisdelayed/suppressed
B.Liu,D.Munoz&DL2015
Lidov-Kozai Oscillations: Octupole Effect + Short-Range Forces
BinLiu,D.Munoz,DL2015
elimWindowofoctupole
Lidov-Kozai Oscillations: Octupole Effect + Short-Range Forces
WindowofOctupole Munoz,DL&Liu16
✏oct
=✓
a
ab
◆eb
1� e2
b
Hot Jupiter formation
" Planetformsat~afewAU" Companionstarperiodicallypumpsplanetintohigh-eorbit(Lidov-Kozai)" Tidaldissipa4oninplanetduringhigh-ephasescausesorbitaldecay!Combinedeffectscanresultinplanetsin~fewdaysorbit
Holmanetal.97;Wu&Murray03;Fabrycky&Tremaine07;Naozetal.12,Katzetal.12;Petrovich15
Q: What is happening to the stellar spin axis ?
Chaotic Dynamics of Stellar Spin Driven by Planets Undergoing Lidov-Kozai Oscillations
With
Natalia Storch (Ph.D.15# Caltech) Kassandra Anderson
Storch, Anderson & DL 2014, Science Storch & DL 2015 MNRAS (Theory I) Anderson, Storch, DL 2016 (Pop Study) Storch, DL & Anderson2016 (Theory II) Diego Munoz, DL & Liu 2016
Precession of Stellar Spin
• Starisspinning(3-30days)#oblate#willprecess
Ωs
Mp
Spin Dynamics
• Stellar spin axis S wants to precess around planet orbital axis L. LB
S
L
θsl θlb
θsb
Ωps
OuterbinaryaxisPlanetorbitalaxisStellarspinaxis
Spin Dynamics
• Stellar spin axis S wants to precess around planet orbital axis L.
• But L itself is moving: – Nodal precession (L precesses
around binary axis Lb) – Nutation (cyclic changes in
inclination of L relative to Lb)
LB
S
L
θsl θlb
θsb
Ωpl
Ωps
OuterbinaryaxisPlanetorbitalaxisStellarspinaxis
Spin Dynamics
• Q: Can S keep up with L? • Answer depends on
LB
S
L
θsl θlb
θsb
Ωpl
Ωps
OuterbinaryaxisPlanetorbitalaxisStellarspinaxis
⌦ps vs ⌦pl
If|Ωps|>>|Ωpl|:YES(“adiaba4c”)
OuterbinaryaxisPlanetorbitalaxisStellarspinaxis
θsl=constant,i.e.ini4alspin-orbitmisalignmentismaintainedforall4me
N.Storch
If|Ωps|<<|Ωpl|:NO(“non-adiaba4c”)
OuterbinaryaxisPlanetorbitalaxisStellarspinaxis
N.Storch
Toanswer,needtosolveorbitalevolu4onequa4onstogetherwithspinprecessionequa4on….
If|Ωps|~|Ωpl|:“trans-adiaba4c”
If|Ωps|~|Ωpl|:“trans-adiaba4c”
OuterbinaryaxisPlanetorbitalaxisStellarspinaxisQ:Isitreallychao4c?
N.Storch
If|Ωps|~|Ωpl|:“trans-adiaba4c”
Storch,Anderson&DL14
Lyapunov4me~6Myrs
LB
SL
θsl θlb
θsb
Ωpl
Ωps
Complication & Richness Ωps&Ωplarestrongfunc4onsofeccentricity(and4me)
✏ =⌦pl0
⌦ps0
Key parameter:
/ a9/2
Mp⌦s
Thera4ooforbitalprecessionfrequencytospinprecessionfrequencyatzeroeccentricity
Bifurcation Diagram
Valuesrecordedateccentricitymaxima,for1500LK-cycles
Periodicislandsintheoceanofchaos
Quasi-chao4cregionsinthe“calm”sea
Storch,Anderson&DL14;Storch&DL15
Whatabout;daldissipa;on?
Lidov-Kozai + Tidal Dissipation
af
✓fsl
A tiny spread in initial conditions can lead to a large spread in the final spin-orbit misalignment
Memory of Chaos
Initial orbital inclination θlb=85± 0.05°, with spindown
Storch,Anderson&DL14
Distribu;onsofthefinalspin-orbitangle
Parameters:ab=200AUM*=Mb=1MsunStellarspin-downcalibratedsuchthatspinperiod=27daysat5Gyr(Skumanichlaw)Uniformdistribu4onofini4alsemi-majoraxes(a=1.5–3.5AU)
Anderson,Storch&DL16
Solar-typestar FStar(1.4Msun)
Anderson,Storch&DL16
Forma;onofHotJupitersinStellarBinariesAnderson,Storch&DL2016
GStars
Forma;oninofHotJupitersinStellarBinariesAnderson,Storch&DL2016
FStars
TheoryofSpinChaos/Evolu;on
InHamiltoniansystem,Chaosarisesfromoverlappingresonances(Chirikovcriterion;1979)
Ωps&Ωplarestrongfunc4onsofeandt(withperiodPe)Whatresonances??
LB
SL
θslθlb
θsb
Ωpl
Ωps
Storch&DL15Storch,DL&Anderson16
Spinprecession Orbit’snodalprecession Orbit’snuta4on
HamiltonianPerturba;onTheory
!
!
Spin-OrbitResonances
Averagespinprecessionfrequency
Integermul4pleofmeanLidov-Kozaifrequency
=
¯
⌦ps = �↵ cos ✓sl = N2⇡
Pe
IndividualResonance
BoundaryofChao;cZoneAnaly4caltheory:Fromoutmostoverlappingresonances
NumericalBifurca;onDiagramStar4ngfromzerospin-orbitangle
Storch&DL2015
Numerical
Theory
TidalDissipa;on:PathstoFinalMisalignmentsStorch,DL,Anderson2016
TidalDissipa;on:PathstoFinalMisalignmentsCHAOTIC
TidalDissipa;on:PathstoFinalMisalignments“Regular”
Adiaba;cResonanceAdvec;on
N.Storch
Bifurca;on!BimodalDistribu;onMisalignment
N.Storch
Condi;onforLKMigra;on,Frac;ons...Forgivenini4alaofplanet,emaxmustbesufficientlylargesothata(1–emax)issmallenoughforefficient4daldissipa4on
emax
= emax
(a, ab, M?, Mb, Mp, Rp, · · · ; i0
)
Condi;onforLKMigra;on,Frac;ons...Forgivenini4alaofplanet,emaxmustbesufficientlylargesothata(1–emax)issmallenoughforefficient4daldissipa4on
emax
= emax
(a, ab, M?, Mb, Mp, Rp, · · · ; i0
)
(1) elim(a, ab, M?, Mb, Mp, Rp, · · · ) large enough
Anderson,Storch&DL16
Condi;onforLKMigra;on,Frac;ons...Forgivenini4alaofplanet,emaxmustbesufficientlylargesothata(1–emax)issmallenoughforefficient4daldissipa4on
emax
= emax
(a, ab, M?, Mb, Mp, Rp, · · · ; i0
)
(1)(2)i0largeenough(Octupolewindow)
elim(a, ab, M?, Mb, Mp, Rp, · · · ) large enough
Analy4ccalcua4onsofmigra4on/disrup4onfrac4onsforalltypesofplanets(Munoz,DL&Liu2016)
Summary(#2)Chao;cStellarSpinandForma;onofHotJupiters
• Dynamicsofstellarspinisimportantfortheforma4onofhotJupiters,e.g.affectstheobservedspin-orbitmisalignments(dependenceonplanetmass,stellarrota4on/historyetc)
• Spindynamicscanbechao4c• Spindynamics/evolu4oncanbeunderstoodfromresonancetheory• Migra4onfrac4onscanbecalculatedanaly4cally
Extended&InclindedDisks/RingsaroundPlanets/BrownDwarfs
J.J.Zanazzi&DL2016
Lightcurveof1SWASPJ1407(~16Myr,0.9Msun)Aseries“eclipses”las4ng56daysaroundApril2007:
bigeclipse(3.5mag)withsymmetricpairsofsmaller(~mag)eclipses
Mamajeketal.2012vanWerkhovenetal.2014Kenworthyetal.2015
!Largering/disksystemaroundunseensubstellarcompanion(?)
Kenworthy&Mamajek2015
Claimedbestfitmodel:Ringsize~0.6AU,withgaps(moons?),totalmass~1M_earthOrbitalperiod10-30years(eccentric)
Ques;on:Underwhatcondi;onscananextended,inclinedring/diskexistaroundaplanet/BD?
• Outerradiusofring
• Howtoproduce/maintaininclina4on?
rout
. 0.3rH rH = a
✓Mp
3M?
◆1/3
planet
M?S
ring
lorb
l(r)
Ring at radius r experiences two torques
From star :
From oblate planet:
Slorb
l(r)
T? ⇠GM?r2
a3
Tp ⇠G(J2MpR2
p)r3
rL ⇠✓
J2Mp
M?R2
pa3
◆1/5
⇠ (3J2R2pr
3H)1/5
T? = Tp at Laplace radius :
�
Slorb
l(r)
�
EquilibriumState:LaplaceSurface
Howtomakethedisksufficiently“rigid”tohaveappreciable
• Internalstresses(pressure,viscosity):Noteffec4veforthinrings
• Self-gravity:
�(rout
)?
Tsg ⇠ ⇡Gr⌃
T? ⇠GM?r2
a3 Tp ⇠G(J2MpR2
p)r3Compareto
Howtomakethedisksufficiently“rigid”tohaveappreciable
• Internalstresses(pressure,viscosity):Noteffec4veforthinrings
• Self-gravity:
�(rout
)?
Tsg ⇠ ⇡Gr⌃
T? ⇠GM?r2
a3 Tp ⇠G(J2MpR2
p)r3Compareto
Tsg
(rout
) ⇠ GMd
rout
& T?(rout
)
� ⌘ Md
M?
✓a
rout
◆3
=3Md
Mp
✓rH
rout
◆3
& 1
EquilibriumStatewithSelf-Gravity
T? + Tp + Tsg = 0
EquilibriumStatewithSelf-Gravity(modifiedLaplaceSurface)
EquilibriumStatewithSelf-Gravity(modifiedLaplaceSurface)
Whatifdisknotinequilibrium?Candisk“precess”coherently?
Slorb
l(r)
�
Timeevolu;onofdiskr2⌦
@ l(r, t)@t
= T? + Tp + Tsg
Initial l(r, 0) aligned with S (at � = 40�)
Inner disk (Tp & Tsg, T?): stays aligned with S
Outer disk (Tsg & Tp): evolves coherently if Tsg(r) & T?(r)() � & 1
nutation in �, precession (smaller �)/libration (larger �) in �
Timeevolu;onofdisk
Summary(#3)Extended,InclinedDisk/RingAroundPlanets/BDs
• Intriguing“eclipses”ofJ1407duetoextendeddisk/ring?• rHandrL• Ifrout<rL,diskisinclinedwrtorbit(provideSinclinedwrtorbit)• Ifrout>rL,requiresufficientdiskmassforself-gravitytomaintaindiskcoherenceandoutdiskinclina4on--generalizedLaplacesurface(withself-gravity)--precessing/libra4ng/nuta4ngself-gravita4ngdisk
Md
Mp& 1
3
✓rout
rH
◆3
Thanks