Modeling Disks of sgB[e] Stars

Jon E. Bjorkman

Ritter Observatory

Dusty Hot Star Winds

• Hot stars with dust:– B[e] – WR– Novae and Supernovae

• Wind must cool below condensation temperature

• Dust forms at large distances• Problem: density must be large

enough that reaction rates are faster than flow times

Zickgraf, et al. 1986

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Tcond =1500 K << T* = 25000 K

Eta Carinae

Morse & Davidson 1996

General Wind Kinematics

• Radial Momentum Equation

• Radial Motion

v

dvdr

= -GM

r2+

vf2

r+ Fext

vr ~

Vesc (radiation-driven)

= a (Keplerian)

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ÓÔÔ

General Wind Kinematics

• Azimuthal Motion

vf µ

Vrotr- 1 (vr ? vf ; angular momentum-conserving)

Vrotr (magnetically dominated; solid body rotation)

Vcritr- 1/ 2 (vr = vf ; Keplerian)

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Ì

ÔÔÔÔÔ

Ó

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Bi-Stability & Disks

Lamers & Pauldrach 1991

• Ionization shift at low latitudes

– Higher mass loss

– Lower terminal speed

Rotationally Induced Bi-Stability

Pelupessy et al. 2000

Terminal Speed Mass Loss

req

rpole

: 10

v• ,eq

v• ,pole

: 1/ 3

Need additional factor

Rotating Stellar Winds

Bjorkman & Cassinelli, 1993

Low Density,High V∞

High Density, Low V∞

Ionization Structure

Krauss & Lamers 2003

WCD Inhibition

Radial Force Only Non-radial Force Effects

Owocki, Cranmer, & Gayley 1996

WCDs and Be Stars

• Non-radial line forces (prevent disk formation)• Outflow speed too large (~400 km/s)• Density too small (to explain IR excess)

– Disk “leaks”• Material falls back onto star

• Material flows outward through disk

• Must put material into orbit• Must remove radiative acceleration

Magnetic Channeling

Owocki & ud-Doula 2003Cassinelli et al. 2002

Asymmetric Mass Ejections

• Kroll’s gravity filter:– Point “explosion”

– Material thrown backward falls onto star

– Material thrown forward goes into orbit

Stellar Bright Spot Model Owocki 2003Spot + Line-Force Cutoff

Keplerian (Orbiting) Disks

• Fluid Equations

• Vertical scale height

(Keplerian orbit)

(Scale height)

(Hydrostatic)

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(vϖ << vφ;v z = 0)

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fϖ

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fz

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Fgrav

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T = 15000K

P = a2r

Dq = 6∞ H / v = 0.1

Viscosity in Keplerian Disks

• Viscosity

• Diffusion Timescale

n = aaH

tn = v 2 / n

=Vcrit

aa2v R

(eddy viscosity)

Lynden-Bell & Pringle 1974

t +dt t

Viscous Decretion Disk

• Lee, Saio, Osaki 1991

Disk Temperature

sT 4 = I nmdWÚ: (R / v )3

Flared Reprocessing DiskFlat Reprocessing Disk

T : v - 3/ 4 T : v - 1/ 2

Disk Winds

Model for HD 87643

Oudmaijer et al. 1998

Power Law Approximations

• Keplerian Decretion Disk

• Flaring

b = 98

a = 198

(flat passive disk; T µ r - 3/ 4)

b = 54

a = 114

(flared passive disk; T µ r - 1/ 2)

b = 32

a = 72

(isothermal disk; T = const)

r = r0(R* / v )a exp -z

H(v )

Ê

ËÁÁÁ

ˆ

¯˜̃˜̃

2È

Î

ÍÍÍ

˘

˚

˙˙˙

H = H0(v / R*)b

Monte Carlo Radiation Transfer

• Divide stellar luminosity into equal energy packets

• Pick random starting location and direction• Transport packet to random interaction location

• Randomly scatter or absorb photon packet• When photon escapes, place in observation bin

(frequency and direction)

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Eγ = LΔt / Nγ

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τ =−lnξ (ξ is a random number)

REPEAT 106-109 times

MC Radiative Equilibrium

• Sum energy absorbed by each cell

• Radiative equilibrium gives temperature

• When photon is absorbed, reemit at new frequency, depending on T

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Eabs = Eemit

nabsEγ = 4πmiκ PB(Ti )

T Tauri Envelope Absorption

T Tauri Disk Temperature

Whitney, Indebetouw, Bjorkman, & Wood 2004

T Tauri Disk Temperature

Snow LineWater Ice

Methane Ice

Effect of Disk on Temperature

• Inner edge of disk– heats up to optically thin radiative equilibrium

temperature

• At large radii– outer disk is shielded by inner disk

– temperatures lowered at disk mid-plane

• Does not solve dust formation problem; requires– high density at condensation radius

– additional opacity interior to condensation radius

Model of sgB[e] Star

• Reaction network

• Timescales

• Condensation Condition

Porter 2003(Gail & Sedlmayr 1988)

Dtdust = 1/ (n1ai )

Dtflow = Rdustcon / v•

Dust Formation

SED

Porter 2003

Bi-stabilityViscous Decretion Bi-stability

Viscous Decretion

NLTE Monte Carlo RT• Gas opacity depends on:

– temperature– degree of ionization – level populations

• During Monte Carlo simulation:– sample radiative rates

• Radiative Equilibrium– Whenever photon is absorbed, re-emit it

• After Monte Carlo simulation:– solve rate equations– update level populations and gas temperature– update disk density (solve hydrostatic equilibrium)

determined by radiation field

sgB[e] Density (pure H model)

Bi-stability Viscous Decretion

Gas (Electron) Temperature

Bi-stability

Viscous Decretion

Dust Temperature

Bi-stability Viscous Decretion

Mid-Plane Temp

Bi-stability Viscous Decretion

Rdust = 400 R* Rdust = 1300 R*

Density

Bi-stability Viscous Decretion

sgB[e] Model SED

Bi-stability Viscous Decretion

(m) (m)

IR Spectroscopy

Roche, Aitken, & Smith 1993

Dust Properties

Wood, Wolff, Bjorkman, & Whitney 2001

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amax = 1μm

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amax = 3μm

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amax = 1mm

Large Dust Grains

YSO (GM Aur) SED

• Inner Disk Hole = 4 AU

Rice et al. 2003

Line-Blanketed Disk Opacity

Bjorkman, Bjorkman, & Wood 2000

Conclusions• Bi-Stability:

– Pros:• Provides better shielding for dust formation

– Cons:• Requires small condensation radius

• Viscous Decretion– Pros:

• Slow outflow enables much larger condensation radius• Disk wind may produce low velocity outflow

– Cons:• Dust optical depth is much too small

• Generally,– need to increase disk outflow rate

(without increasing free-free excess)– Or provide more shielding to decrease condensation radius