Theory of Disk Accretion onto Magnetic (Compact)...
Transcript of Theory of Disk Accretion onto Magnetic (Compact)...
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Theory of Disk Accretion onto Magnetic (Compact) Stars
Dong Lai Cornell University
“Physics at the Magnetospheric Boundary”, Geneva, 6/25/2013
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Magnetosphere - Disk Interaction: Basic Picture
Accreting x-ray pulsars: Accreting ms pulsars (LMXBs): CV’s (Intermediate polars): Classical T Tauri stars:
B ∼ 1012G, rm ∼ 102R
B ∼ 107G, rm ∼ 10R
B ∼ 103G, rm ∼ (a few)R
B ∼ 108G, rm ∼ (a few)R
rm ∼
µ4
GMM2
1/7
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Ghosh & Lamb 1979
Arons, McKee, Pudritz (Arons 1986) Shu et al. 1994
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Lovelace et al. 1995 Matt & Pudritz 2005
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Simulations… Hayashi, Shibata & Matsumoto, Miller & Stone, Goodson, Winglee & Bohm, Fendt & Elastner, Matt et al, Romanova, Lovelace, Kulkarni, Long, Lii et al, Zanni & Ferreira, ……
Romanova et al. Zanni & Ferreira
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Issues: (uncertainties, possible applications…)
Magnetosphere boundary vs disk inner radius Magnetic linkage between star and disk: Width of interaction zone, steady vs cyclic behaviors Magnetosphere outflows Propeller effect Torque on the star: Spinup/spindown Misaligned dipole: Effect on disks Spin-disk misalignment
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Dipole Field Invaded by a Conducting Disk
η ∼ ν = αHcs
tdiff ∼H
2
η∼ 1
αΩ
Exact solution (Aly 1980)
Disk is a good conductor: Diffusion of B into disk:
Istomin’s talk
Bz(rin) B(rin)
rin
H
1/2
(MRI)
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Dipole Field Invaded by a Conducting Disk
η ∼ ν = αHcs
tdiff ∼H
2
η∼ 1
αΩ
Exact solution (Aly 1980)
Disk is a good conductor: Diffusion of B into disk:
Instabilities at inner edge (KH, RT, reconnection) Boundary layer
Istomin’s talk
Bz(rin) B(rin)
rin
H
1/2
(MRI)
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Magnetosphere Boundary Layer rm → rm + ∆rm
Magnetic torque on BL (per unit area)
rBzBφ+
2π= Σ
d(r2Ω)dt
= Σ vr∂(r2Ω)
∂r
Bφ+ = −ζBz
Note : For rc < rm :
rm ∼
µ4
GMM2
1/7
depends on ζ, ∆rm.
rm ∼
µ2
ΩM
1/5
Transition from ΩK to Ω
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rm = rin?
-- Disk-like structure could exist inside -- The rm expression assumes steady state. If mass accumulates at boundary episodic accretion e.g., when Mdot is small, rm larger than rc: rin could be << rm (Spruit & Taam 1993; D’Angelo & Spruit)
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Star-Disk Linkage (Width, Time-dependence…)
Linked fields are twisted by differential rotation… Field inflates, breaks the linkage
Aly; Lovelace et al.; Uzdensky,…
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Star-Disk Linkage (Width, Time-dependence…)
Linked fields are twisted by differential rotation… Field inflates, breaks the linkage
Aly; Lovelace et al.; Uzdensky,…
Bφ+
Bz
max
∼ 1Maximum twist:
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Star-Disk Linkage (Width, Time-dependence…)
Steady-state?
∂Bφ
∂t∼ Bz (Ω− Ωs)−
Bφ
tdiff= 0
Bφ
Bz
∼ |Ω− Ωs| tdiff
Steady-state linkage possible only very near corotation ∆r
rc∼ α for η = αHcs
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Stellar field penetrates the inner region of disk; Field lines linking star and disk are twisted --> toroidal field --> field inflation Reconnection of inflated fields restore linkage
Star-Disk Linkage (Width, Time-dependence…)
Quasi-cyclic behavior
Inevitable…
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Stellar field penetrates the inner region of disk; Field lines linking star and disk are twisted --> toroidal field --> field inflation Reconnection of inflated fields restore linkage
Star-Disk Linkage (Width, Time-dependence…)
Quasi-cyclic state
QUESTION: Connection with QPOs in LMXBs (and other systems) ?
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Van der Klis 2005
Quasi-Periodic Oscillations (QPOs) Power density spectrum of x-ray flux variations of accreting millisecond pulsars
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Stellar field penetrates the inner region of disk; Field lines linking star and disk are twisted --> toroidal field --> field inflation Reconnection of inflated fields restore linkage
Star-Disk Linkage (Width, Time-dependence…)
Quasi-cyclic state
QUESTION: Episodic outflow (X-wind?)… Connection with observations?
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Ejection from Magnetospheric Boundary
Romanova et al. 2009
Zanni & Ferreira 2013
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Jets from Accreting Neutron Stars
Cir X-1: Miller-Jones et al.2011
Jets observed in -- Atolls (Aql X-1), Z-sources (Sco X-1), AMXPs (SAX J1808) -- In soft x-ray (banana) states (Ser X-1, 4U1820-30) (cf. BHs: no jet in thermal state)
(Migliari et al. 2011)
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Stellar field penetrates the inner region of disk; Field lines linking star and disk are twisted --> toroidal field --> field inflation Reconnection of inflated fields restore linkage
Star-Disk Linkage (Width, Time-dependence…)
Quasi-cyclic state
QUESTION: On average, what is the width of the linked region? ∆r
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Torque on Star
dJ
dt Maccr
2mΩ(rm) + Tm
Tm −r2BzBφ+
rm
∆r = ζr2B2
z
rm
∆r for Bφ+ = −ζBz
|ζ| ∼ 1: ζ > 0 when r < rc and ζ < 0 when r > rc
Note: |Tm| ∼ Mr2
mΩ(rm) if ∆r ∼ width of BL
rc
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Propeller Effect
rmΩ > vesc(rm) or rm > 21/3rcWhen
accreted plasma in atmosphere may be ejected
dJ
dt Maccr
2mΩ(rm) + Tm − Mejectr
2mΩ
Torque on star:
Lii’s talk
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Propeller Effect
rmΩ > vesc(rm) or rm > 21/3rcWhen
accreted plasma in atmosphere may be ejected
dJ
dt Maccr
2mΩ(rm) + Tm − Mejectr
2mΩ
Torque on star:
Question: Propeller outflow/jet: Connection with observations? (cf. boundary layer ejection)
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Jets from Accreting Neutron Stars
Cir X-1: Miller-Jones et al.2011
Jets observed in -- Atolls (Aql X-1), Z-sources (Sco X-1), AMXPs (SAX J1808) -- In soft x-ray (banana) states (Ser X-1, 4U1820-30) (cf. BHs: no jet in thermal state)
(Migliari et al. 2011)
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Propeller Effect
rmΩ > vesc(rm) or rm > 21/3rcWhen
accreted plasma in atmosphere may be ejected
dJ
dt Maccr
2mΩ(rm) + Tm − Mejectr
2mΩ
Torque on star:
Question: Is there accretion onto star? Dead disk? (D’Angelo & Spruit)
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Propeller Effect
rmΩ > vesc(rm) or rm > 21/3rcWhen
accreted plasma in atmosphere may be ejected
dJ
dt Maccr
2mΩ(rm) + Tm − Mejectr
2mΩ
Torque on star:
Question: Equilibrium spin: T Tauri stars, millisecond pulsars,
Long-period pulsars ( magnetar field?) Can we understand spinup/spindown of X-ray pulsars?
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Spinup/Spindown of Accreting X-ray pulsars
Bildsten et al. 1997
Postnov’s talk
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1990.6
2008.2
Camero-Arranz et al. 2010,2012
4U1626-67 7.66s Transition lasted 150 days
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Chakrabarty et al.1997; Gonzalez-Galan et al 2012
GX 1+4
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So far, physics of aligned dipole…
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Misaligned Dipole
-- Funnel flow to polar caps -- Accretion through instabilities
Simulations by Romanova et al…
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Misaligned Dipole: Exciting Waves in Disks
Vertical force on disk:
m = 1, ω = Ω, 2Ω
Excitation of Bending waves in the disk
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Misaligned Dipole: Exciting Waves in Disks
Vertical force on disk:
m = 1, ω = Ω, 2Ω
Excitation of Bending waves in the disk
DL & Zhang 08
Perturbations most “visible” at Lindblad/Vertical Resonance
Ω(rL) =ω
2=
Ω
2,Ω
Lovelace’s talk Question: QPOs….
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Van der Klis 2005
kHz QPOs in Accreting Millisecond Pulsars
SAX J1808.4-3658: XTE J1807.4-294:
Beating of high-freq. QPO with perturbed fluid at L/VR ?
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Stellar Spin – Disk Misalignment
Standard story: S//L
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There are magnetic torques which tend to make the inner disk -- warp -- precess on timescale >> dynamical time (rotation/orbital period)
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There are magnetic torques which tend to make the inner disk -- warp -- precess on timescale >> dynamical time (rotation/orbital period)
Consider general geometry. Two limiting cases…
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Perfect conducting disk:
Torque on disk (per unit area): Averaging over stellar rotation: Precessional
Torque
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Magnetically Threaded disk:
threads the disk
Torque on disk (per unit area): Averaging over stellar rotation:
Warping torque
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A Laboratory Experiment
J
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A Laboratory Experiment
DL, Foucart & Lin 2011
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Magnetically Threaded disk:
threads the disk
Torque on disk (per unit area): Averaging over stellar rotation:
Warping torque
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Recap: Magnetic precessional torque and warping torque on disk (per area)
(Instability)
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Low-Frequency (10-50 Hz) QPOs in LMXBs
Lense-Thirring prcession of warped disk ? (e.g. Stella & Vietri 1998)
Other systems: mHz QPOs in x-ray pulsars? Photometric variabilities of AA Tau? (Bouvier+)
νLT ∝ ν2K
Magnetic effects induce warp and contribute to precession (Shirakawa & DL 02; Pfeiffer & DL 04)
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Possible Connection to (Exo)Planetary Systems
ESO
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1 10 100
100
0
-100
Orbital Period [Days]
Spin
-Orb
it M
isal
ignm
ent [
Deg
rees
]
exoplanets.org | 3/10/2013
Many “hot Jupiters” have Misaligned S*-Lp
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S*-Lp misalignment in “Hot Jupiter” Systems The Importance of few-body Interactions 1. Kozai + Tide migration by a distant companion star/planet 2. Planet-planet Interactions (Strong scatterings, secular chaos etc)
Misaligned protostar - protoplanetary disk ? Alternative possibility:
e.g., -- Solar system: orbital plane misaligned from solar equator by 7 deg. -- Two multi-planet systems (KOI-1241) with common orbital plane misaligned with the stellar equator (J. Carter et al.)
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Recall: Magnetic toques from the star want to make the inner disk warp and precess…
But disk will want to resist it by internal stresses (viscosity or bending wave propagation)
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Steady-state Disk Warp:
Foucart & DL 2011
For protoplanetary disk/star parameters, the disk warp is small
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Back-reaction Torque on Star: What is happening to the stellar spin direction? (Is there secular change to the spin direction?)
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Back-reaction Torque on Star: What is happening to the stellar spin direction? (Is there secular change to the spin direction?)
Warping torque
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Back-reaction Torque on Star: What is happening to the stellar spin direction? (Is there secular change to the spin direction?)
Warping torque Back-reaction torque
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Evolution of the stellar spin
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Evolution of the stellar spin
“weak” warping torque
“strong” warping torque
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A hierarchy of time scales: (1) Orbital period of inner disk, spin period (days) ==> short… Averaged out already (2) Warp growth time and precession period of inner disk (3) Disk warp evolution time: e.g., due to viscosity (4) Timescale to change the spin (longest!)
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How to test this?
• Measuring spin-orbit angles for systems with 2 transiting planets e.g., Two multi-planet systems (KOI-1241) with common orbital plane misaligned with star
• Measuring the orientation of stellar spin and disk Young star and disk (with jets)? MS stars with debris disks?
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Watson et al 2011
Greaves et al. 1998
CSO and Spitzer (MIPS) image Backman et al 2009 Consistent with face-on (Stapelfeldt 2010)
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Summary Issues, uncertainties, possible applications
rm vs rin
Star-disk linkage: width, steady vs cyclic Magnetosphere boundary outflow Propeller effect/outflow Torque on the star: spinup/spindown Misaligned dipole: effect on infall, effect on disk Spin-disk misalignment: disk warp/precession, exoplanets Observations: Variabilities (QPOs), NS jets (2 different types?), spin equilibrium (infer B?), spinup/down observations, warp/prcession, exoplanets, protoplanetary disks…
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THANKS
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Magnetosphere Boundary Layer rm → rm + ∆rm
Transition from ΩK to Ωs
Magnetic torque on BL (per unit area)
rBzBφ+
2π= Σ
d(r2Ω)dt
= Σ vr∂(r2Ω)
∂r
Bφ+ = −ζBz
rm
ζ∆rm
H
2/7µ
4
GMM2
1/7
rm
ζ∆rm
H
1/5µ
2
ΩsM
1/5
Note : For rc < rm :
Mdot is local
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Simulations by Cornell group: M. Romanova, Lovelace, etc