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