Effect of Clumping on WCD Inhibition

Stan Owocki

Bartol Research Institute

University of Delaware

• Bjorkman & Cassinelli (1992) proposed kinematic Wind Compressed Disk (WCD) paradigm for Be disks.• General idea confirmed by dynamical CAK models with only central line-forces (OCB 1994).• But WCD inhibited in CAK models with nonradial line-force.

• What is effect of instability-generated clumping on WCD inhbition??• Here apply new “3-Ray” SSF method for 2D instability simulations with nonlocal line-force.• Key problem: artificial structure from grid staircase along oblate surface BC• Results still tentative.

Outline

WCD Inhibition bynon-radial line-forces

Vrot (km/s) = 200 250 300 350 400 450

Wind Compressed Disks

}“S-350”

Vector Line-Force

fasterpolarwind

slower equatorial wind

r

rFl

ux

r g line ~ dΩ

Ω*

∫ r n I*

r n⋅[∇(

r n⋅ r v)]α

dvn/dn

Max

[dv n

/dn]

Net poleward line force from:

(1) Stellar oblateness => poleward tilt in radiative flux

(2) Pole-equator aymmetry in velocity gradient

N

Line-Driven Instability from Perturbed Profile Doppler Shift

Instability growth rate:

Ω ~ g/vth ~ vv’/vth ~ v/L >> v/H ~ vv’/v

=> # e-folds in wind ~ v/vth ~ 100 !!!

For ≤ L = vth/(dv/dr), perturbation “optically thin”

u=v/vth

0.0 0.5 1.0

0

500

1000

1500

Clumped density

-15

-14

-13

-12

-11

-10

CAK

log

Den

sity

(g/

cm3 )

Radius (R*)

1D Simulation of Small-Scale Line-Driven Instability

Local vs. Nonlocal Line-Force

gκ ~κ dp0

R*

∫ I* dx−∞

∞

∫ ϕ (x−vz /vth)e−t(x,p, z)

t(x, p, z) ≡ dz'κρϕ (x−vz' / vth)z*

∞

∫

≈τ z dx'x−vz / vth

∞

∫ ϕ (x' )

τz ≡κρvth

dvz / dz

Sobolev approximation

Nonlocal ray optical depth

LocalSobolev optical depth

3-Ray Grid for 2D Rad-Hydro

Diagram: N = 9 ; = 10o

Actual code: N =157 ; = 0.01 radPole =0o

Equator =90o

I+

Io

I-

g ~ I

+ - I-

Co-Rotating Interaction Region Models

localCAKmodel

nonlocalsmoothmodel

nonlocalstructuredmodel

c.

log(Density)

b. a.

-1000 1000cm/s2

-20 1000km/s

-75 75km/s

-16.7 -14cm-3

CAK withonly RadialForces

CAK withnonRadialForces

SSF withnonRadialForces

logDensity

RadialVelocity

LatitudinalVelocity

LatitudinalLine-Force

CAK withNonradial

Forces

-1.e-9 1.e-9Msun/yr

0 5.e-9Msun/yr

SSF withNorradial

Forces

RadialMass Flux

CAK withonly Radial

Forces

LatitudinalMass Flux

“Grid Staircasing” along Oblate Lower Boundary

• A vexing problem in WCD simulations has been the tendency for wind to form artificial stream structure.

• This stems from irregular “staircase” of spherical grid along the lower boundardary defined at the oblate stellar surface.

• The problem worsens with increased number of latitudinal zone.

• It is particularly acute in the 3-ray grid models here, which have N, = 157.

Mdot in CAK (g =0) model of WCD using 3-ray SSF grid

Preliminary Result Summary

• WCD model– assumes radial driving

• Poleward line-force in CAK models– results from stellar oblateness &

asymmetric velocity gradient

– reverses equatorward flow

– inhibits WCD

• Nonlocal 2D SSF models– global asymmetries disrupted

– poleward force mixed, weaker

– little net flow to pole or equator

– WCD still weak or absent

• Future work:– improve oblate lbc

– 2D rad-hydro with short characteristics

– mass-ejection models of disk formation