Stellar%20Winds

1

• Stellar Winds
• D. John Hillier
• University of Pittsburgh

2

• Stellar atmospheres
• The premier book on radiative transfer and
  non-LTE model atmospheres.

• Solution of transfer equation in CMF.
• Many of the transfer codes in use today, for
  solving radiative transfer in moving media, are
  based in whole, or in part, on a series of papers
  by Mihalas, Kunasz, and Hummer (1975-1977).
• All papers utilizing CMFGEN results utilize this
  work.

• Opacity Project
• With Mike Seaton, and David Hummer, one of the
  founding fathers. The opacity project, and
  related projects, have been immensely beneficial
  in supplying a vast quantity of atomic data to
  the astrophysical community.

• Letters of reference
• Dimitri was kind enough to write letters of
  reference for me, and since I eventually became a
  full professor, he must have been reasonably
  kind.

• Foundations of Radiation Hydrodynamics
• Modeling of SN time dependent radiative
  transfer.
• (Light bed-time reading).

3

• Fundamental Parameters
• Idealized models
• Masses
• Abundances
• Effective Temperatures
• Surface Gravities
• Luminosities
• Radii
• The real world
• Rotation
• Magnetic fields
• Binarity
• Stellar, galactic and cosmic evolution
• Mass-loss rates
• Wind momenta and energy
• Ionizing Fluxes
• Metal enrichment

Current Values ??? Initial Values
Current Values ??? Future Values
4

• The Complications
• Photosphere
• Microturbulence Macroturbulence
• Pulsations instabilities near Eddington limit
• Winds
• Inhomogenous
• Clumping interclump medium
• Non-spherical?
• Origin of X-rays
• Hydrodynamics
• Smooth? Clumped? Decoupling?
• Time dependent models -- Do static solutions
  exist?
• Observational issues
• Distances
• Reddening (E(B-V) and R)
• Echelle spectrographs

5
From Przybilla, Nieva, Butler, 2008, ApJ, 688,
L103
AGS/GS 0.74 0.72 0.67 0.58
6
Revised Effective temperature scale of O stars

• Blanketing increases ionization in photosphere.
  Thus a lower Teff is need to fit the observed
  ionization at a given spectral type.
• ?????? Cooler Teff ST relation.
• Reduction by 1500 to 4000 K for dwarfs.
• (Martins et al. 2002 Crowther et al. 2002
  Herrero et al. 2002 Bianchi Garcia 2002
  Garcia Bianchi 2002 Repolust et al.
  2004Martins et al. 2005)

Vacca et al.
o H, He models ? Blanketed models
LC V
Martins, Schaerer Hillier 2002, AA, 382, 999
7
Effect of Blanketing on Energy Distribution
Key H, He H, He, CNO H, He, CNO, Fe, etc
HD93250 L 1.3 x 106 L?, Teff
45,700 K, R 18.3 R?, log g 4.0 M
5.6 x 107 M?/yr , V8 3000 km s1
8
Effective temperature scale of O stars
Martins, Schaerer Hillier (2005, AA 436, 1049)
Grid of CMFGEN models Theoretical calibration of
O star parameters (solid lines) --- new Teff
scales for LC V, III and I Decrease of effective
temperatures as high as 8000 K for supergiants
(20 ) Individual stars analyses from Herrero et
al. (2002), Repolust et al. (2004), Martins et
al. (2005)
9
Martins, Schaerer, Hillier (2005, AA 436,
1049)
He I ionizing photons.
H I ionizing photons.
10

• Luminous Blue Variables (LBVs)
• Variable, strong line emission
• Episodic as well as continuous mass loss dM/dt
  106 to 103 M?/yr
• Teff lt 30000 K N(H) (1 to 5) x N(He) Large
  radii (70 Rsun)
• Wolf-Rayet Stars
• Teff gt 30,000K Radii 1 to 30 Rsun H/He 0 to
  solar dM/dt 105 to 105 M?/yr
• WN Stars He and N emission
• Products of CNO cycle revealed at
  surface
• WC Stars Strong He and C emission
• Products of He burning
  revealed by extensive mass loss at
  surface.
• Evolutionary connection to O Stars
• For M gt 60M?
• O ? Of ? WNLabs ? WN7 (WNE) ? WC ? SN
• For M lt 60M?
• O ? LBV or RSG ? WN8 ? (WNE) ? WC ? SN
• Langer (1994) proposed
• O ? Of ? WN(H rich) ? LBV or RSG ? WN(H
  poor) ? WC ? SN

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• The He Ionization Balance.
• Ionization of He (as He2) is maintained by
  photoionizations out of the n2 state of He. To
  first order, the population of the n2 state is
  maintained by the radiation field in the He II
  2-1 resonance transition at 304Å.
• Banketing reduces the radiation shortward of 350,
  hence decreasing the He ionization. To maintain
  the ionization to the observed value, as set by
  the observed ratio of He II to He I line
  strengths, a higher Teff (and luminosity) is
  required.

Energy
He
Continuum
3
2
Pumping by UV continuum radiation field.
304Å
1
He
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Mokiem, de Koter, Evans, et al. 2007, AA, 465,
1003 28 O early B stars (LMC) ? dwarfs ?
girants ? supergiants
LMC, SMC, Galaxy
14
Massey, Zangari, Morrell et al. 2008 (ApJ, in
press) Sk-70?69 O5.5 V((f)) LMC Teff40500 K,
Log L/L? 5.35 , dM/dt 0.8 x 106 M?/yr
15
Massey, Zangari, Morrell et al. 2008 (ApJ, in
press)
SMC o dwarfs ? giants ? sgs
LMC o dwarfs ? giants ? sgs
Massey et al. 2004, 2005
16
Massey, Zangari, Morrell et al. 2008 (ApJ, in
press)
17
Najarro et al. (2006, 456, 1159) Fe IV lines
affects transfer in He I resonance lines
Alters 2p 1Po populations. Affects strength of
HeI singlet transitions.
f, f/2 f/5, f/10
f, N(Fe) f/2, N(Fe) f, N(Fe)/2
VD13.5 km/s VD10.0 km/s
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Modified Wind-Momentum (?) Luminosity Relation
(Kudritzki et al. 1995)
Vink, de Koter, Lamers (2000) Theoretical
relation (Monte Carlo, Sobolev approximation,
Multiple Scattering, Homogeneous, Momentum
deposition.
19
Marcolino, Bouret, Martins et al. 2009 (AA, in
press)
dM/dt 1.0 x 10-6 5.0 10-7 2.3 10-7 1.2
10-7 6.3 10-8 3.5 10-8 6.0 10-10
20
Mokiem et al. (2007, AA, 473, 603) Solid
FLAMES Grey CMFGEN Dmom dM/dt V? ?R? -0.37
correction (2.3 H? in emission)
Galaxy
LMC
SMC
21

• Line Formation in Winds.
• Recombination.
• H, He emission lines Most C lines in WC stars
• Dielectronic recombination
• Famous N III 4640 triplet in Of stars (Mihalas et
  al. 1972).
• C III 2296Å, 6740, many C II lines in WC stars.
• Resonance Scattering
• P Cygni profiles in UV spectra of O stars.
• Na D lines in LBVs
• Collisional Excitation
• C IV 1548, 1551 doublet in WC stars.
• N IV 1486 in WN stars, C III 1909 in WC stars.
• Continuum fluorescence
• C IV 5805, some N IV lines in WN Stars.
• Many metal lines (Mg II, Si II, Fe II) in P Cygni
  type stars.
• Bowen fluroescence (Raman scattering)

22

• Resonance Scattering

Photon (e.g., from continuum) trapped in
transition. Photon scatters until it escapes, or
until the upper level is collisionally
de-excited. In a stellar wind, it gives rise to
the classic P Cygni profile.
23
Formation of P Cygni Line Profiles
?
24
Sample P Cygni Profiles
25
Symmetric Emission Line
?
EB tells us that I S(t1). Thus if S(Line)
B(continuum) we get symmetric emission profile.
There is no line contribution from directly in
front of star, and no line contribution from
directly behind the star.
26

• Continuum Fluorescence

UV transition (e.g., 2s 2S 3p 2Po at 312Å)
intercepts continuum radiation. Transition is
optically thick, and radiation scatters many
times in UV transition. However there is a
significant probability that upper level (e.g.
3p 2Po) level will decay to alternative level
(i.e., 3s 2S) giving rise to emission at a longer
wavelength (e.g., 5801, 5812Å doublet) A(3p
2Po,2s 2S) /A(3p 2Po,3s 2S) 150. The process
is important for the formation of Fe II lines in
LBVs, and QSOs.
27

• Low Temperature Dielectronic Recombination (LTDR)

Occurs when a doubly excited state has an energy
just above the ionization limit (e.g., C III 2p
4d 1Fo lies above the C2 ionization limit).
Subject to certain selection rules, the state may
spontaneously decay to the ion plus a free
electron (e.g., C III 2p 4d 1Fo ? C IV 2s 1S
e). If the autoionization probability is high
(e.g. often 1012 to 1014!), state is in LTE with
respect to the ion ground state (e.g. C IV
2s). If a transition can occur to a lower state,
with a high probability (e.g., 6 x 109), we have
an efficient recombination mechanism. This will
also provide a selective excitation mechanism
(i.e., enhance some lines only).
Energy
C3(2s)
28
Photoionization Cross-section
LTDR can be treated assuming the autoionizing
levels are bound states, or by treating them as a
resonance in the photoionization cross-section.
The first peak in the top figure correspond to
the dielectronic recombination line at 412Å.To
avoid aliasing in radiative transfer codes, the
photoionization cross-sections are often
smoothed. NB HTDR (High temperature DR) refers
to the general dielectronic process, and doesnt
rely on the existence of specific states close to
the ionization edge. For CIII, HTDR refers to
dielectronic recombination through states of the
from 2pnl with n large.
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C III
30

• Formation of CIII 5696

In WC stars the ratio of C III 5696 to C IV 5805
is used as one of the principal classification
ratios. In WC4 stars C III 5696 is absent, while
its strength increases rapidly with later
spectral types. Interestingly, other C III lines
do not show the same dramatic variation --- other
C III lines are still present in WC4 stars. The
behavior of C III 5696 can be understood on the
basis of the atomic structure of the C III atom.
In WC4 stars the 2s3d 1D state primarily decays
to the 2s2p 1Po state. The emission in 5696 (2s3d
1D to 2s3p 1Po) increases in strength as the
primary decay route, at 574Å, becomes optically
thick. Further, the population in the 2s3p 1Po
state is efficiently drained by transitions to
2p2 1D and 2p2 1S. Thus 5696 can remain optically
thin, even when it is strongly in emission.
5696Å
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• Constant Velocity Surfaces

In stellar winds and SN, emission lines are (can
be) formed over a large volume of the wind.
Because of the velocity field, emission lines are
broadened. Emission at each frequency in the line
comes from a surface of constant velocity Vobs,
such that Vobs V(r) cos ?. In general the
surfaces are curved, but for V r (t-to) they
are planes.
32
Illustration of the origin of CIV 1548 in a WN 5
star. In the bottom figure the grey scale plot
illustrates how the profile originates as a
function of impact parameter and frequency. The
line is not a pure P Cygni profile (as evident by
the strength of the absorption relative to the
absorption) it also has collisional component.
From Dessart Hillier (2004). (Treated as a
single line.)
33

• Stellar Wind Diagnostics.
• P Cygni profiles
• Mass loss rates / clumping /velocity law /
  abundances / ionization state
• Emission lines
• Mass loss rates / velocity law / abundances /
  ionization state
• Electron scattering wings
• Mass loss rates / clumping
• Optical/Infra red continuum
• Mass loss rates / velocity law / clumping
• Radio continuum
• Mass loss rates / clumping
• Variability
• Wind structure / clumping

34
Stellar Wind Diagnostics.

• X-ray fluxes
• Shocked gas, wind instabilities, abundances
• X-ray profiles
• Profile shapes ? mass-loss rates, constraints on
  porosity
• Mass loss from binary system
• V444 Cygni, Period change ? dM/dt
• Colliding winds in binary system.
• dM/dt, V8 etc
• Wind blown bubbles
• Mass loss and kinetic energy integrated over
  lifetime of bubble
• Direct imaging
• Eta Car image outer wind due to presence of
  coronagraph
• Interferometry

35
Variability ? Clumping
WR136(WN6)
WR138(WN5OB)
WR134(WN6)
Mean profile has been subtracted.
From Lèpine Moffat, 1999, ApJ, 514, 909
36
Electron Scattering Wings
f1.0 f0.1
HD165763 (WC5)
?(Å)
Hillier Miller (ApJ, 1999)
37
Polarization
Limit on asphericity of WC Stars Many WC stars
show no evidence for intrinsic polarization (e.g.
Moffat et al.). Analysis of HD 165763 suggests
that it probably departs by lt 10 from spherical
(Kurosawa, et al. AA, 1999).
Aspherical clumpy winds in LBVs Ag Car
variable in polarization, with random
polarization position angle. R127 has variability
at constant position angle (Davies et al. 2005)
38

• Owocki (Zeta Pup model)

e.g., Owocki, Castor, Rybicki, 1988, ApJ, 335, 914
39
2D Instability Simulations Dessart Owocki
(2005, AA, 437, 657)
Lateral Scale Transverse Sobolev length (r/v)
1 degree Lateral scale set by grid
resolution? Diffuse radiation field crucial.
40

• Clumping
• Specification
• Size density distribution.
• Velocity profile and distribution.
• Nature of the interclump medium.
• Amount of mechanical energy deposited, and
  where.
• Effects
• Allow lower mass-loss rates
• Enhanced emission (density squared effects)
• Porosity (spatial velocity)
• Different lines and continuua can have distinct
  responses, but many are (surprisingly) similar
  (e.g., ? ? or ?2).
• Potential problems(e.g., Williams 1992, ApJ,
  392, 99)
• Degeneracy
• Optically thick clumps.
• Clumps can have their own ionization structure.
• Shielding
• Non-spherical (driven rotation, pulsations,
  inhomogeneities)

41

• Volume Filling Factor Approach
• Assume medium is clumped, the clumps are uniform,
  and the fractional volume of the clumps is f. If
  the clumps are small compared to the photon
  mean-free path, then
• ?(clump) ?(smooth) / f
• ?(eff) f ?(clump)
• ?(eff) f ?(clump)
• Under these assumptions, calculation is exact.
• The real world
• Approximation excellent for continua, but poor
  (?) for lines, since scale length of clumps is of
  order the Sobolev length.
• Question
• What is the accuracy when assumptions are not
  met?
• How?
• Many diagnostics
• Alternative techniques

Support Variability Observation (e.g.,
Lepine Moffat, 2008) 2D simulations (e.g.,
Dessart Owocki 2005)
42
A Tale of 2 Stars (ApJ, 2003)

• AV83 AV69
• O7 Iaf OC7.5 III((f))
• V 13.58 13.35
• E(B-V) 0.16 0.16
• L( L? ) 3.5 x 105 3.5 x 105
• R(R? ) 18.5 18.6
• Teff (K) 32,800 33,900
• Log g 3.25 3.50
• M/f1/2(M?/yr) 2.3 x 10-6 0.92 x 10-6
• f 0.1 1?
• V?(km/s) 940 1800
• ? 2 1?
• N(He)/N(H) 0.2 0.1
• X(C)/X(C)? 0.08 0.1
• X(N)/X(N)? 1.8 0.02?
• X(O)/X(O)? 0.06 0.2

?
43
AV83(O7Iaf) AV69(OC7.5 III((f)) L3.5 x
105L?, Teff 34000 K
44
N(P)/N(P)? 0.23 0.028
N(S)/N(S)? 0.27 0.13
N(P)/N(P)? 0.23 0.078
N(S)/N(S)? 0.27 0.13
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46
HD 190429A O4If (solid f1.0, 6.0 x
106)(dashed f0.04, 1.8 x 106) Teff39,000K,
Log g 3.6 (Bouret et al, 2005, AA)
47
OVI in Zeta Pup
Produced by Auger ionization O IV h?X ? O VI
2e
Zsargo et al. (2008, ApJ, 685L149)
48

Study of UV resonance lines in SMC/LMC OB stars.
Massa, Fullerton, Sonneborn, Hutchings (ApJ, 2003)
49

Massa, Fullerton, Sonneborn, Hutchings (ApJ, 2003)
50
Puls, Markova, Scuderi et al (2006, AA, 454, 625)
H?/mm/radio study evidence the clumping
decreases in radio region.
f1
Variable f
51
Marcolino, Bouret, Martins et al 2008 (AA,
submitted) HD 66799 O 8-9V, L 105 L?, dM/dt
1.2 x 109 M?/yr (100!)
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• Macro Clumping
• Oskinova et al. (2007, AA, 476, 1331)
• An approximate formulation for handling
  optically thick clumps.
• Apply model to Zeta Pup
• Reasonable fit to observations
• Has significant spatial porosity.
• Is porosity important for explaining X-ray line
  profiles.
• Mass loss rates
• Lower than smooth wind results.
• Higher than that obtained using volume filling
  factor approach.

Shell model Artificial but can be solved
exactly Useful for studying effects that will
arise in other clumped models. Alternative
technique.
54
Shell like density structure adopted for
transfer models (blue line). Red line shows
radiative instability model computed for Zeta Pup
by Owocki (private communication).

55

• For AV83 (O7 Iaf not shown), an shell clumping
  approach gives a similar spectrum to that
  obtained using the filling-factor approach, but
  detailed analysis required.
• For a WN5 star, the shell clump model gives a
  similar continuum, but generally weaker lines.
  The lines and continua behave differently because
  continuum photons generally have much greater
  mean free paths (hence shells are effectively
  thin) than do line photons (shells may be
  effectively thick). Since the lines are
  inconsistent with the continuum, clumped model is
  invalid ?? thinner clumps?

56

• WN5 Model

R?2.5R? f0.1 (V
100km/s) L 3x105 L? Mdot2.5
x 10-5 M?/yr Teff 85,000K
57

Volume filling factor approach
Shell model
58

• THE
• END

59

• Modeling with Clumping
• The answer you get depends on the nature of the
  clumps. To constrain the effects of clumping it
  is essential that we understand the clumping
  mechanism and hence the type of clumps.
• Disks Macroscopic clump.
• Bullets Can have lots of mass but cant be seen
  spectroscopically.
• Balls or pancakes --- effects, for example,
  porosity.
• Nature of interclump medium?

Credit NASA, ESAC.R. O'Dell (Vanderbilt
University), M. Meixner and P. McCullough.
Credit NSAA and The Hubble Heritage Team (STScI,
AURA)..
60

• Computational Issues
• Computing the spectrum (i.e., the formal
  solution) from a clumped model, whose populations
  are known, is feasible. The difficulty is
  computing the non-LTE populations.
• Possible exception Scattering resonance line
  arising from an ion which is the dominant
  ionization stage.
• Non-LTE modeling
• Homogeneous models
• t(2D) gt 100 t(1D)
• t(3D) gt 5000 t(1D)
• Inomogeneous models
• t(3D) gt 106 t(1D)
• Cant do full problem Need approximations.

61
Reliability
Abundances Accurate H/He CNO (50)
Fe (factor of 2?) Abundances not strong
function of model. Vinf 10 (turbulence
meaning?) Mass-loss rates Mdot/sqrt(f) --
some dependence on ? f (factor of 2)
f(r)? Luminosities 50? Factor of 2
(or more) increase in L when line blanketing
included?
62
Reliability
Radii Wind dependent. Biased by
assumptions. Hydrodynamics values more
consistent with evolution? Thick winds ?
difficult to constrain observationally. Effective
temperatures As for radii V(r) Thick winds ?
difficult to constrain observationally. Hyrodynam
ics? Stellar masses Binaries. Mass loss rates
(sensitive to ?)
63
Mass loss Rates from 1st Principals
Fe Opacity bumps (OPAL, Opacity Project) Nugis
Lamers (2002, AA, 389 162) Graefener Hamann
(2005, AA, 432, 633) Graefener Hamann (2008,
AA, 482, 945) ? Optically thick stellar
winds ? Critical point is sonic point ? Opacity
increases through critical point Require
T(sonic) 160,000 K (hot bump) 40,000 to
70,000 K (cool bump) ? No guarentee that Mdot
set by critical point can be driven to
infinity. ? Multiple critical points?
64
Momentum Equation
where
With asound speed we have
Continuum driven The Fe bump is caused by
million of lines (pseudo continuum).
g(rad) only weak dependence on dv/dr
65
HD165763 (WR111 WC5)
Graefener Hamann (2005, AA, 432, 633)
66
For WCE, WNE stars necessary to include very high
ionization stages of Fe. Graefener
Hamann (2005, AA, 432, 633) CAK alpha close to
0 (i.e., winds driven by thin lines.)
67
WR22 (WN70 system HD 92270)
Graefener Hamann (2008, AA, 482, 945)
68

• WR22 Dynamical Comparison
• Blue (VD100 km/s) Red (VD50 km/s).

Graefener Hamann (2008, AA, 482, 945)
69

• Mass loss rate is a strong function of the
  Eddington Parameter (?)

Models computed for fixed L(2 x 106 L?)
Graefener Hamann (2008, AA, 482, 945)
70
HD 92740 (WN7) L 2 x 106 Lsun
Mdot 2.0 x 10?5 Msun/yr Teff 44,000
K Mass 87 Msun Vinf 1785 km/s
V(cl) 30 or 100 km/s ?
Species included H, He, C, N,O, Ne, Si,
S, Cl, Ar, Ca, Fe, Ni
71
Model with current parameters is a little too
hot. Low ionization features (e.g., NIII 4640,
HeI 5876) are too weak.
72
CAK ?
73
Pulsations
Strange Mode Instabilities Occurs when radiation
pressure is important. Opacity modified acoustic
waves. Glatzel Kiriakidis (1993, MNRAS, 263,
375) Glatzel Kaltsshmidt (2002, MNRAS, 337,
743) Glatzel (1994, MNRAS, 271, 61)
MOST observations HD 165763 (WC5, WR111) No
coherent fourier amplitudes greater than 50
parts/ million for periods lt 2.4 hr). Expected
periods 10 to 30 minutes. Moffat et al. (2008,
ApJ, 679, L45)
74
Pulsations
MOST observations of WR123 (WN8) Lefévre et al.,
Ap J, 2005, 634, L109 Stable 9.8 hour period. lt 2
mmag (10 d?1 lt f lt 1400 d?1) Complex power
spectra with amplitudes 5-20 mmag. WR103 (WC8)
shows similar complex power spectra.
Dorfi , Gautschy, Saio, 2006, AA, 453,
L35 Pulsation periods consistent with 10
hours. Pulsations damped/modified by wind Motions
outer layers not synchronized with the
pulsations Velocities of 100 km/s (up down)
75
Hydrodynamic Instabilities in Atmospheres Near
the Eddington Limit
Instabilities gt 0.5 to 0.8 Lead to
inhomogeneities Continuum driven winds Larger
mass-loss rates. Super Eddington
Luminosities. Eta Carinae Novae
Shaviv, 2001, MNRAS, 326, 126
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Inconsistent! Continuum matched but lines do not.
78

• Conclusions
• The Fe opacity bumps play a key role in
  initiating W-R winds. Other effects (e.g.,
  pulsations) may be needed for some W-R
  subclasses.
• Accuracy of current W-R models (particularly
  radii and effective temperatures) is limited by
  uncertainties in the wind hydrodynamics and
  clumping.
• Due to the LARGE number of parameters needed to
  parameterize clumping, need to use ALL available
  diagnostics (radio to X-ray spectra, variability)
  to provide constraints.
• Filling factor approach is useful, but results
  need verification by other means.
• Urgently needed
• Alternative approaches to handling clumping
• Handling of complex velocity fields ??gt
  additional diagnostics
• Additional theoretical insight into clumping
  structure, and its variation in the wind.
• Linking X-rays/structure.

79

• General Thoughts
• Hydrodynamics
• Smooth
• Clumped
• Time dependent
• Do static solutions exist?
• Stellar issues
• Pulsations
• Instabilities near Eddington limit
• Other issues
• Distances
• Reddening (E(B-V) and R)

80
Mokiem, de Koter, Evans, 2006, AA, 456,
1131 FLAMES Study of SMC O Early B Stars
Galactic objects(Penny1996, ApJ, 463, 737)
81
Mokiem, de Koter, Evans, 2006, AA, 456,
1131 FLAMES Study of SMC O Early B Stars
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83
HD 92740 (WN7) L 2 x 106 Lsun
Mdot 2.0 x 10?5 Msun/yr Teff 44,000
K Mass 87 Msun Vinf 1785 km/s
V(cl) 30 or 100 km/s ?
Species included H, He, C, N,O, Ne, Si,
S, Cl, Ar, Ca, Fe, Ni
84
Model with current parameters is a little too
hot. Low ionization features (e.g., NIII 4640,
HeI 5876) are too weak.
85

• Conclusions
• The Fe opacity bumps play a key role in
  initiating W-R winds. Other effects (e.g.,
  pulsations) may be needed for some W-R
  subclasses.
• Accuracy of current W-R models (particularly
  radii and effective temperatures) is limited by
  uncertainties in the wind hydrodynamics and
  clumping.
• Due to the LARGE number of parameters needed to
  parameterize clumping, need to use ALL available
  diagnostics (radio to X-ray spectra, variability)
  to provide constraints.
• Filling factor approach is useful, but results
  need verification by other means.
• Urgently needed
• Alternative approaches to handling clumping
• Handling of complex velocity fields ??gt
  additional diagnostics
• Additional theoretical insight into clumping
  structure, and its variation in the wind.
• Linking X-rays/structure.

86

• General Thoughts
• Key assumptions
• Beals model (hot core surrounded by dense
  out-flowing wind)
• Spherical (but some exceptions)
• Key requirements
• Atomic data
• Line and continuum cross-sections
• Collision rates
• Charge exchange
• Do details matter?
• Accurate wavelengths?
• Completeness of opacities?
• All species?
• Clumping
• Volume filling factor approach
• Macro-clumping

87

• Energy Balance
• In a static atmosphere (d/dt0) all three
  equations equivalent in the sense that they lead
  to the same (unique) temperature structure.
  However, mathematically and numerically, they are
  different.
• R.E. Eqn.
• Used in model atmosphere calculations, but may
  have numerical difficulties at large optical
  depths. Not used for stellar model calculations.
• Flux Conservation
• Used in interior calculations, and can be used
  when t gt 1. Not useful in outer layers, since
  energy removed from radiation field is small.
  Numerical errors will dominate the accuracy of
  flux calculation. Related to the R.E. Eqn. by the
  radiative transfer equation.
• Electron Energy Balance
• Mathematically equivalent to R.E. Eqn. However
  assumptions (e.g. LTE, super levels) and
  numerical errors and approximations mean that
  they are not equivalent in model atmosphere
  calculations. Related to the R.E. Eqn. by the
  equations of Statistical Equilibrium.

88

• Energy Equations

Radiative Equilibrium
Energy emitted by gas - Energy absorbed from
radiation field 0
Flux Conservation
Electron Energy Balance
Equations, as written, assume a static
atmosphere. Appropriate modifications can be made
so that they are valid in stellar winds or in SN.
89

• Electron Energy Balance

Recombination cooling
Photoionization heating
Collisional cooling
Collisional heating
Collisional ionization cooling
Collisional recomb. heating
Free-Free heating
Free-Free cooling
90
Recombination Lines

• Most important for the most abundant species H
  He. In WC stars most of the C lines form by
  recombination, or a recombination like process
  (e.g., LTDR). In the recombination process, an
  electron is captured into an excited state. This
  electron then cascades to the ground state,
  emitting one or more photons. In nebula, with
  their low densities, it is usual to consider 2
  cases
• Case A All lines treated as optically thin.
• Case B Lines to ground state, which will easily
  have the largest population, treated as optically
  thick. This means that any photon emitted in a
  Lyman transition scatters in the same transition
  until the electron cascades to another level, and
  emits a photon of lower energy. (Obvious
  exception is Ly a in hydrogen. Why?).
• In both cases it is necessary to consider
  consider collisions with electrons (and ions)
  which cause electrons in excited states to be
  shuffled into another excited state.
• For hydrogen, generally write the effective
  total recombination rate as nenHaA (only valid
  when Lyman continuum is optically thin) and
  nenHaB. The later is usually appropriate since
  recombinations to the ground state emit a photon
  which immediately reionizes another H atom. NB
  a is the recombination coefficient, and should
  not be confused with the photoionization
  cross-section used earlier.
• Tables of effective recombination coefficients
  (i.e., into an individual levels), line fluxes
  (usually expressed relative to some reference
  line such as Hß, and total recombination rates
  are available for H and He ions (e.g., Hummer and
  Storey, 1987, MNRAS, 224, 801).
• In winds, optical depths in transitions other
  than the Lyman lines are important, and must be
  taken into account.

91
At 104 K, aB is a approximately 2.6 x 1019 m3
s1 (or 2.6 x 1013 cm3 s1). For hydrogenic like
ions with charge z, a(z,T) za(z1,T/z2).
Diagram taken from http//wiki.hmet.net/index.php
/HII_Case_B_Recombination_Coefficients
92

• More Thoughts on Line Formation
• Above picture is idealized. Sometimes more than
  one mechanism contributes, and sometimes it is
  difficult to discern.
• e.g., In WN5 stars, C IV 1548, 1551 clearly has
  both a resonance scattering contribution, and a
  collisional excitation contribution.
• For absorption lines in O stars, it may not make
  sense to ask What is the formation mechanism?
  Levels may be close to LTE with many processes,
  radiative collisional, driving the state
  towards LTE. The strength of most absorption
  lines is closely coupled with the photospheric
  temperature gradient. Of course, for some lines
  some processes may markedly change the strength
  of the line.
• e.g., NIII 4640 can be a photospheric line
  driven into emission by dielectronic
  recombination --- it is not necessarily a wind
  line.

93

• Collisional Excitation

Electron is collisionally excited to a state
which can radiative decay. Important for levels
close to the ground state, since they are more
easily excited (i.e., hv/kT 1). The strong O
II, O III, N II, S II lines seen in nebula
spectra arise from collisional excitation. The
forbidden nature of the lines is not a
prerequisite --- it just happens the levels in
CNO that are low lying are forbidden to decay by
electric dipole transitions. At higher
densities, collisional de-excitation can compete
with radiative decays, and the lines weaken
relative to other lines which still have a
density2 dependence.
Energy
NIV
Critical density Density at which half the decays
from the excited state are radiative, and half
are through collisional de-excitation.
2s2p 3Po
1496.5Å
2s2 1S
94
New O Star Parameters
Martins, Schaerer Hillier (2005, AA 436,
1049)
Vacca et al.
Other stellar parameters modified by
line-blanketing (see also Martins et al. 2002)
Luminosities reduced by up to 0.3 dex (for
given ST) Ionising fluxes reduced 0.2 to
0.8 dex (for given ST)
95
Martins, Schaerer, Hillier (2005, AA 436,
1049)
Qo H ionizing photons Q1 He I ionizing
photons
? dwarfs ? giants ?supergiants
96
In WR stars, adding line-blanketing caused
inferred Teffs ( L) to increase. Principal Teff
diagnostic, HeI/HeII ratios, are set in wind, and
determined by UV radiation field at 300.
97

• Chemistry
• H burning (Main Sequence)
• 4H ? He4 energy
• Process occurs by CNO cycle
• C12 H1 ? N13 ? (106 years)
• N13 ??C13 e ???(14 min)
• C13 H1 ? N14 ? (3 x105 years)
• N14 H1 ? O15 ? (3 x108 years)
• O15 ??N15 e ???(82 secs)
• N15 H1 ? C12 He4 (104 years)
• The CNO nuclei act as catalysts Most of the CNO
  is converted to N.
• Helium Burning
• 3He4 ? C12 energy

98
Meynet Maeder (2000, AA, 361,101)
Evolution of 20 M? star as a function of initial
equatorial velocity.
Location in H-R diagram depends on rotation
velocity as well as mass!
99
Meynet Maeder (2003, AA, 404, 975)
Influence of rotation on surface abundance as a
function of remnant mass for a star with an
initial mass of 60M?.
100

• Clumping
• Specification
• Size density distribution.
• Velocity profile and distribution.
• Nature of the interclump medium.
• Amount of mechanical energy deposited, and
  where.
• Effects
• Allow lower mass-loss rates
• Enhanced emission (density squared effects)
• Porosity
• Different lines and continuua can have distinct
  responses, but many are (surprisingly) similar
  (e.g., ? ? or ?2).
• Potential problems(e.g., Williams 1992, ApJ,
  392, 99)
• Degeneracy
• Optically thick clumps.
• Clumps can have their own ionization structure.
• Shielding

101
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102
Mokiem, de Koter, Evans, et al. 2006, AA, 456,
1131 FLAMES Study of SMC 31 O Early B Stars
Galactic objects (Penny1996, ApJ, 463, 737)