Modeling the Structure of Hot Star Disks

Jon E. Bjorkman

Ritter Observatory

Systems with Disks

• Infall + Rotation– Young Stellar Objects (T Tauri, Herbig Ae/Be)– Mass Transfer Binaries– Active Galactic Nuclei (Black Hole Accretion Disks)

• Outflow + Rotation– AGBs (bipolar planetary nebulae)– LBVs (e.g., Eta Carinae)– Oe/Be, B[e]

• Rapidly rotating (Vrot = 350 km s-1)• Hot stars (T = 20000K)• Ideal laboratory for studying disks

General Wind Considerations

• Radial Momentum Equation

• Radial Motion

• Be disk line profiles– Widths and symmetry =>

v

dv

dr= -

GM

r2+

vf2

r+ grad

vr ~

Vesc (radiation-driven)

= a (Keplerian)

ÏÌÔÔ

ÓÔÔ

vr £ a (Dachs, Hanuschik, …)

Disk probably is Keplerian

General Wind Considerations

• Azimuthal Motion

vf µ

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

r (magnetically dominated; solid body rotation)

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

Ï

Ì

ÔÔÔÔÔ

Ó

ÔÔÔÔÔ

Rotating (in/out) Flows

• Fluid Equations (cylindrical coords: )

• Equation of State

€

ϖ,z,φ

€

P = a2ρ

€

a =kT

μmH

(isothermal sound speed)

Keplerian Disks

• Fluid Equations

• Vertical scale height

(Keplerian orbit)

(Scale height)

(Hydrostatic)

€

(vϖ << vφ;v z = 0)

€

fϖ

€

fz

€

Fgrav

€

T = 15000K

€

H /ϖ = 0.04

€

Δθ =2.5°

Disk Variability

• Viscosity

• Viscous Diffusion Timescale

– too large, unless a ~ 0.1–1

n = aaH

tn = v 2 / n

=Vcrit

aa2v R

= 20yr 0.01

a

Ê

ËÁÁÁ

ˆ

¯˜̃˜̃

v

R

(eddy viscosity)

Viscous Decretion -Disks

vf =Vcrit R / v

vv =&M

2pv S

S =&MVcritR

1/ 2

3paa2v 3/ 2

Rmax

v- 1

È

Î

ÍÍÍ

˘

˚

˙˙˙

r =S

2pHe- 0.5(z/H )2

H = (a / vf )v

(surface density)

(scale height)

(Keplerian orbit)

(hydrostatic)

(continuity eq.)

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

Isothermal Keplerian Disk Density

= 1.5

= 3.5

€

ρ(ϖ ) = ρ0 (R* / ϖ )α exp −12

zH (ϖ ) ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

H (ϖ ) = H0 (ϖ / R* )β

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)

€

Eγ = L Δt / Nγ

€

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

• Problem: Don’t know T a priori• Solution: Change T each time a photon is

absorbed and correct previous frequency distribution

avoids iteration

€

Eabs = Eemit

nabsEγ = 4πmiκ PB(Ti )

Temperature Correction

Bjorkman & Wood 2001

Frequency Distribution:

€

dP

dν= jν (T + ΔT ) − jν (T )

=κ ν ΔTdBν

dT

Model of B[e] Star

Bjorkman 1998

Disk Temperature

Bjorkman 1998

B[e] SED

Bjorkman 1998

T Tauri Envelope Absorption

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

• Permits dust formation in outer disk

• Requires a different opacity source at smaller radii

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

Disk Temperature

Carciofi & Bjorkman 2004

Disk Density

Carciofi & Bjorkman 2004

NLTE Level Populations

Carciofi & Bjorkman 2004

SED and Polarization

Carciofi & Bjorkman 2004

IR Excess

Carciofi & Bjorkman 2004

LTE Line-Blanketed Polarization

Observed Observed

MC Simulation MC Simulation

Flux Polarization

Bjorkman & Carciofi 2003

Inner Disk:• NLTE Hydrogen • Flared Keplerian• h0 = 0.07, = 1.5

• R* < r < Rdust

Outer Disk:• Dust • Flared Keplerian• h0 = 0.017, = 1.25

• Rdust < r < 10000 R*

HAeBe Model

Summary• Viscous Timescale:

– 20 years ( = 0.01)– probably a bit too long (but may be larger)

• NLTE Modeling of Keplerian Disk– Fully self-consistent 3-D model

• determines radiative equilibrium temperature• vertical hydrostatic equilibrium• steady state disk surface density• Single parameter: (and inclination angle i)

– Reproduces Polarization and SED– Temperature

• Inner disk: falls as r -3/4 (like thin blackbody)• Outer disk: isothermal

&M

&M = (a / 0.01)¥ 10- 11Me yr- 1

Acknowledgments

• Rotating winds and bipolar nebulae– NASA NAGW-3248

• Ionization and temperature structure– NSF AST-9819928– NSF AST-0307686

• Geometry and evolution of low mass star formation– NASA NAG5-8794

• Collaborators: A. Carciofi, K.Wood, B.Whitney, K. Bjorkman, J.Cassinelli, A.Frank, M.Wolff

• UT Students: B. Abbott, I. Mihaylov, J. Thomas• REU Students: A. Moorhead, A. Gault

Be Star H Profile

Carciofi and Bjorkman 2003

i = 82º

Polarization vs IR Excess

Coté & Waters 1987

P ~ sin2i

Edge-on

Pole-on

Gault, Bjorkman & Bjorkman 2002

MC Polarization vs IR Excess

Disk Orientation: Inclination

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Quirrenbach et al. 1996

Interferometric sin2i

Pol

arim

etri

c s

in2 i