breaching the eddington limit in the most massive, most luminous stars stan owocki bartol research...
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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
Wind-Blown Bubbles in ISM
Some key scalings:
WR wind bubble NGC 2359 Superbubble in the
Large Magellanic Cloud
Henize 70: LMC SuperBubble
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
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
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
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
Eta Car’s Extreme PropertiesPresent day:
1840-60 Giant Eruption:
=> Mass loss is energy or “photon-tiring” limited
Porosity
• Same amount of material
• More light gets through
• Less interaction between matter and light
Incident light
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
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
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
Summary Themes
• Continuum vs. Line driving
• Prolate vs. Oblate mass loss
• Porous vs. Smooth medium