Stellar Magnetospheres part deux: Magnetic Hot Stars

Stellar Magnetospherespart deux:

Magnetic Hot Stars

Stan Owocki

Key concepts from lec. 1• MagRe# --> inf => “ideal” => frozen flux

• breaks down at small scales: reconnection

• Lorentz force ~ mag. pressure + tension

• plasma ~ Pgas/Pmag ~ (a/VA)2

• press. scale ht. H << R (except in corona)

• wind mag. conf. ~B2R2/Mdot*V

Wind Magnetic Confinement

for OB SG, to get*~ 1, need B* ~ 100 G

€

(r) ≡B2 /8π

ρv 2 /2

Ratio of magnetic to kinetic energy density:

e.g, for dipole field, q=3; ~ 1/r 4

€

=B2r2

M•

v=

B*2R*

2

M•

v∞

(r /R*)2−2q

(1− R* /r)β

*

Alfven Radius: ( RA) 1

For dipole (q=3): RA = *1/4 R*

€

=VA

2

v 2

Hot Stars vs. Sun

• Luminous Radiative Envelope (not conv.)• Radiatively Driven Wind (line scatt.)

– Twind ~= Teff – Mdot ~ 10-7 MO/yr >> Mdot,sun

• Detected B-fields often steady, dipole• Rotation can be rapid Vrot ~< Vcrit

NOT a (direct) solar Coronal analog!

• Light transports energy (& information)

• But it also has momentum, p=E/c

• Usually neglected, because c is so high

• But becomes significant for very bright stars,

• Key question: how big is force vs. gravity??

• e=ge/g < 1 (near but below “Edd. limit”)

• “CAK” wind model: lines = glines/g > 1

Light’s Momentum

• Force depends on gradient of Pressure

• grad Prad comes from opacity

• Deep interior (>>1), Prad nearly isotropic

–Prad + Pgas act together => H=a2/g(1-)

• But near surface ( ~1) radiation “beamed”

• wind is not “Radiation Pressure Corona” H ~< R

Aside: Radiation Pressure vs. Gas Pressure

Radiative force

€

rg rad = dν

0

∞

∫ κ νˆ n Iν /c

~

e.g., compare electron scattering force vs. gravity

gel

ggrav

eL4GMc

r

L4 r2c

Th

e

GM2

• For sun, O ~ 2 x 10-5

• But for hot-stars with L~ 106 LO ; M=10-50 MO

.. .

if gray

€

=F /c

Optically Thick Line-Absorption in an Accelerating Stellar Wind

gthick~gthin

τ~

1ρ

dvdr

€

≡ vth

dv /dr

LsobFor strong,

optically thick

lines:

€

~vth

v∞

R*

<< R*

CAK force for power-law line ensemble

€

glines ~gthin

τ α~

L

r2

1

ρ

dv

dr

⎛

⎝ ⎜

⎞

⎠ ⎟

α

CAK wind in zero-gas-pressure limit

€

vdv

dr= −

GM

r2−

a2

ρ

dρ

dr+ glines

€

′w = −1+ C ′ w α

€

′w ≡r2vdv/dr

GM

€

C ~1˙ M α

€

gthin

τ α~

L

r2

1

ρ

dv

dr

⎛

⎝ ⎜

⎞

⎠ ⎟

α

Graphical solution

€

˙ M = ˙ M CAK€

˙ M < ˙ M CAK

€

˙ M > ˙ M CAK

Magnetohydrodynamic (MHD) Equations

€

Dρ

Dt+ ρ∇ • v = 0

Mass Momentum

mag Induction Divergence free B€

P = ρa2 = (γ −1)e

€

∂e

∂t+∇ ⋅ev = −P∇ ⋅v + HM − n2Λ(T)

€

Dv

Dt= −∇p +

1

4π∇ × B( ) × B + ρ (glines − ggrav )

€

∂B

∂t=∇ × v × B( )

€

∇• B = 0

Energy Ideal Gas E.O.S.

Magnetohydrodynamic (MHD) Equations

€

Dρ

Dt+ ρ∇ • v = 0

Mass Momentum

mag Induction Divergence free B€

P = ρa2

€

T = const.€

Dv

Dt= −∇p +

1

4π∇ × B( ) × B + ρ (glines − ggrav )

€

∂B

∂t=∇ × v × B( )

€

∇• B = 0

Energy Ideal Gas E.O.S.

MHD Simulation of Wind Channeling

Confinement parameter

A. ud Doula PhD thesis 2002

IsothermalNo Rotation

*=1/3




€

* =1*=1




*= 3




*=10




€

* =10

RA~ *1/4 R*

*=10



€

* =10

RA~ *1/4 R*

*=1000 !!

295 G ; * = 1 Final state of isothermal models

1650 G ; * = 32 930 G ; * = 10520 G ; * = 3.2

165 G ; * = 0.3293 G ; * = 0.1

1 Ori C

MHD simulation for 1 Ori C (O7 V)

• Bobs ~ 1100 G + Prot = 14 days

• Mdot~ 10-7 Msun/yr

• *~= 14

• Magnetic Confined Wind Shock (MCWS)

• kev X-rays => match Chandra obs.

Magnetically Confined Wind-ShocksBabel & Montmerle 1997

Magnetic Ap-Bp stars

Magnetically Confined Wind Shock

~ 2 kev X-rays fit Chandra spectrum

for 1 Ori C

log T

but now with RadiativeCooling

still no rotation

-1 Ori C

QuickTime™ and a decompressor

are needed to see this picture.

Gagne et al. 2005, ApJ, 528, 986

Prot = 13 days

1 Ori C

~ 45o

i ~ 45o

X-ray Light Curve for 1Ori C

Magnetically Torqued Disk (MTD)Cassinelli et al. 2002

Alfven vs. Kepler Radius

when RA > RK :Magnetic spin-up => centrifugal support & ejection

Kepler co-rotation Radius, RK:

GM/ RK2 = V

2/RK

RK = w2/3 R*

w=Vrot/Vcrit Vcrit2 = GM/R*

= Vrot2RK/R*

2

Alfven radius: (RA)=1

RA = *1/4 R*

e.g, for dipole field, ~ 1/r 4

Keplerian spin-up vs. escape

w = Vrot/Vcrit0.2 0.4 0.6 0.8 1

2

4

6

8

10

sqrt(*) ~ B

v > v

escv < v

kep

MHD Sims of Field-Aligned Rotation *=10 ; vrot=250 km/s (w=1/2)

DensityV /Vrigid

RKRA RKRA

Field aligned rotation

€

* = 32

RK RA

Vrot=250 km/s = Vcrit/2

againisothermal, but now with

Magnetically Torqued Disk

Owocki & ud-Doula 2003Cassinelli et al. 2002

Magnetic Bp Stars

• Ori E (B2p V)– Prot = 1.2 days => vrot/vcrit ~ 1/2

– Bobs ~ 104 G => * ~ 107 !

– => VAlfven very large => Courant time very small

– => Direct MHD impractical

• Instead treat fields lines as “Rigid”

Strong field limit: B*,* --> ∞• Field lines act as rigid guides for wind outflow

– Torque up wind outflow

– Hold down disk material vs. centrifugal force

• Problem: VAlfven ->∞ => can’t do MHD

• But can model semi-analytically

– Rigidly Rotating Magnetosphere (RRM)

– gas accumulates at minima in grav+cent. potential

Effective Gravitational+Centrifugal Potential

Rigidly Rotating Magnetosphere

Townsend & Owocki (2005)

Accumulation Surface

Townsend & Owocki (2005)

RRM model

for Ori E

δδδ

B*~104 G

tilt ~ 55o

*~106 !

EM +B-field

δδδδδδ

polarimetry

photometryRRM model for Ori E

δδδ

B*~104 G

tilt ~ 55o

*~106 !

Townsend, Owocki & Groote (2005)

Ori ERRM Model H Observations

Sanz-Forcada et al. (2004)

MHD sim of centrifugal breakout

*= 620 vrot=vcrit/2

log()

*= 620 vrot=vcrit/2

log T

Centrifugally Driven Reconnection Flare

~ 4-5 kevX-rayflares,as seen in Ori E

Model grid for 2-Parameter study: Rotation & Magnetic Confinement

Isothermal, vrot=vcrit/2, *=100

RK RA

Isothermal, vrot=vcrit/2, *=1000

RK

RA~7R*

Radial disk mass: dme/dr

€

dme

dr≡ 2πr2 ρ sinθdθ

π / 2−Λθ / 2

π / 2+Λθ / 2

∫

Equatorial mass distribution dme/dr R

adiu

s (R

*)

Time (Msec)0

*=100

1

5

W=1/2

Equatorial mass distribution dme/dr R

adiu

s (R

*)

Time (Msec)0

*=100

1

5

W=0

Parameter study: rotation & magnetic confinement

Between RRM and full MHD: “Rigid Field Hydro-Dynamics”

RFHD

Solve time-dependent hydro along completely rigid individual field lines

The flow along an individual field line

Rigid Field - Hydro Model

Summary• Magnetic field + hot star wind:

– * >10: channel wind into shock (MCWS)• explains X-rays from 1 Ori C

– spin up wind past Kepler co-rotation radius– confine material against centrifugal

• Rigidly Rotating Magnetosphere (RRM)

• explains H-vs. rot. phase in Ori E

– mass build up => centrifugal breakout• reconnection => T >~ 108 K

• can explain hard X-ray flares in Ori E

• “centrifugally driven reconnection”

Current & Future Work• 3D MHD of MCWS

– lateral structure scale, tilted field case

• Comparison with observations– Chandra & XMM

• X-rays from shocks and flares

– Optical photometry, spectroscopy, polarimetry• Rotation Spin-Down??

– VLTI to resolve disk

• Application to other types of magnetospheres?

• RRM shock as site for Fermi acceleration??– Could produce up to TeV Gamma Rays!

Stellar Magnetospheres part deux: Magnetic Hot Stars

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