Breaching the Eddington Limitin the Most Massive, Most Luminous Stars

Stan Owocki

Bartol Research Institute

University of Delaware

Collaborators:Nir Shaviv Hebrew U.Ken Gayley U. IowaRich Townsend U. DelawareAsif ud-Doula U. DelawareLuc Dessart U. Arizona

Massive Stars in “Pillars of Creation”

Massive Stars in the Whirlpool Galaxy

Wind-Blown Bubbles in ISM

Some key scalings:

WR wind bubble NGC 2359 Superbubble in the

Large Magellanic Cloud

Henize 70: LMC SuperBubble

Pistol Nebula

Eta Carinae

Massive Stars are Cosmic Engines

• Help trigger star formation

• Power HII regions

• Explode as SuperNovae

• GRBs from Rotating core collapse Hypernovae

• “First Stars” reionized Universe after BB

& Cosmic Beacons

Q: So, what limits the mass & luminosity of stars?

A: The force of light!

– light has momentum, p=E/c

– leads to “Radiation Pressure”

– radiation force from gradient of Prad

– gradient is from opacity of matter

– opacity from both Continuum or Lines

Continuum opacity fromFree Electron Scattering

Thompson Cross Section

th

e-

Th= 8/3 re2

= 2/3 barn= 0.66 x 10-24 cm2

Eddington limitEddington limit

The Eddington limit is when the radiative force on free electronsjust balances the gravitational force.

SUPER-Eddington: radiative force exceeds the force of gravity

Gravitational Force Radiative Force

Radiative force

€

grad = dν0

∞

∫ κ ν Fν /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

Mass-Luminosity Relation

€

dPgas

dr= −ρg

€

T ~M

R=>

€

ρT

R~

ρM

R2=>

€

L ~R4T 4

κM=>

€

T 4

κM /R2~

L

R2=>

Hydrostatic equilibrium (<<1):

€

dPrad

dτ=

F

c

Radiative diffusion:

€

L ~M 3

κ

€

~ M 2

€

(1− Γ)4

Mass-Luminosity Relation

Key point

• Stars with M ~ 100 Msun have L ~ 106 Lsun

=> near Eddington limit!

• Provides natural explanation why we don’t

see stars much more luminous (& massive)

than this.

Line-Driven Stellar Winds

• Stars near but below the Edd. limit have

“stellar winds”

• Driven by line scattering of light by

electrons bound to metal ions

• This has some key differences from free

electron scattering...

Q~ ~ 1015 Hz * 10-8 s ~ 107

Q ~ Z Q ~ 10-4 107 ~ 103

Line Scattering: Bound Electron Resonance

lines~QTh

glines~103g el

LLthin} iflines

~103

el 1

for high Quality Line Resonance,

cross section >> electron scattering

Optically Thick Line-Absorption in an Accelerating Stellar Wind

gthick~gthin

τ~

1ρ

dvdr

€

≡ρ vth

dv /dr

LsobFor strong,

optically thick

lines:

€

~vth

v∞

R*

<< R*

CAK model of steady-state wind

inertia gravity CAK line-force

Solve for:

˙ M ≈Lc2

Q−⎛

⎝⎜

⎞

⎠⎟

α

−Mass loss rate

˙ M v∞ ∝ L1

αWind-Momentum

Luminosity Law

α ≈0.6€

v(r) ≈ v∞(1− R∗ /r)1/ 2

Velocity law

~vesc

Equation of motion:

€

v ′ v ≈ −GM(1− Γ)

r2+

Q L

r2

r2v ′ v ˙ M Q

⎛

⎝ ⎜

⎞

⎠ ⎟α α< 1

CAK ensemble ofthick & thin lines

Summary: Key CAK Scaling Results

€

˙ m ~F1/α

g1/α −1

€

~F 2

g

e.g., for α1/2

€

~ g

Mass Flux:

Wind Speed:

€

V∞ ~ Vesc

How is such a wind affected by (rapid) stellar rotation?

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

Hydrodynamical Simulations of Wind Compressed Disks

But caution: These assume purely radial driving of wind

WCD Inhibition

• Net poleward line-force inhibits WCD

• Stellar oblateness => poleward tilt in radiative flux

r

Flux

N

Gravity Darkening

increasing stellar rotation

Non-radial line-force + gravity dark.:

Effect of gravity darkening on line-driven mass flux

€

~F 2(θ)

geff (θ)

€

α =1/2e.g., for

w/o gravity darkening, if F()=const.

€

~1

geff (θ)

€

˙ m (θ) highest at equator

€

~ F(θ)w/ gravity darkening, if F()~geff()

€

˙ m (θ) highest at pole

€

˙ m (θ) ~F(θ)1/α

geff (θ)1/α −1

Recall:

Effect of rotation on flow speed

€

V∞(θ) ~ Veff (θ) ~ geff (θ)

€

ω ≡Ω /Ωcrit

*€

ω =1

€

ω =0.9

€

geff (θ) ~ 1−ω2 Sin 2θ

Smith et al. 2002

Eta Car’s Extreme PropertiesPresent day:

1840-60 Giant Eruption:

=> Mass loss is energy or “photon-tiring” limited

Super-Eddington Continuum-Driven Winds

moderated by “porosity”

Porosity

• Same amount of material

• More light gets through

• Less interaction between matter and light

Incident light

Monte Carlo

Monte Carlo results for eff. opacity vs. density in a porous medium

Log(average density)

~1/ρLog(eff. opacity)

blobsopt. thin

blobsopt. thick

“critical densityρc

Pure-abs. model for “blob opacity”

€

eff ≈ l 2 [1− e−τ b ]

€

l

€

b ≡ κρ bl = κρl / f

€

eff ≡σ eff

mb

= κ1− e−τ b

τ b

€

≈ b

; τ b >>1

€

l

€

l

Key Point

€

eff ~1

ρFor optically thick blobs:

€

b = κρ cl / f ≡1

Blobs become opt. thick for densities above critical density ρc, defined by:

€

ρc =f

κl

volume filling factor

G. DindermanSky & Tel.

Shaviv 2001

Power-law porosity

Power-law porosity

€

M•

= 4πR*2ρ Sa

At sonic point:

€

≈L*

acΓ−1+1/α

€

eff (rS ) = Γρ c

ρ S

⎛

⎝ ⎜

⎞

⎠ ⎟

α

≡1

€

M•

CAK ≈L*

c 2QΓ( )

−1+1/α

Results for Power-law porosity model

Effect of gravity darkening on porosity-moderated mass flux

€

˙ m ≡˙ M

4πR2

€

~ F(θ)F(θ)

geff (θ)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1+1/α

€

˙ m (θ)

€

~ F(θ)w/ gravity darkening, if F()~geff()

€

˙ m (θ) highest at pole

€

v∞(θ) ~ vgeff (θ) ~ geff (θ) highest at pole

Eta Carinae

Summary Themes

• Continuum vs. Line driving

• Prolate vs. Oblate mass loss

• Porous vs. Smooth medium

Future Work• Radiation hydro simulations of porous driving

• Cause of L > LEdd?

– Interior vs. envelope; energy source

• Cause of rapid rotation

– Angular momentum loss/gain

• Implications for:

– Collapse of rotating core, Gamma Ray Bursts

– Low-metalicity mass loss, First Stars