intermezzo: stellar atmospheres in practice · 2014. 1. 27. · usm stellar atmospheres in practice...

USM

Intermezzo: Stellar Atmospheres in practice

A tour de modeling and analysis of stellar atmospheres throughout the HRD

except for

white dwarfs

117

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Stellar Atmospheres in practice

Some different types of stars…

r ~10 - 20RSun

M ~ 8-100MSun

Hot luminous stars:

Massive,

main-sequence (MS)

or evolved, ~10 Rsun.

Strong, fast stellar

winds

Cool, luminous stars

(RSG, AGB):

Massive or low/interme-

diate mass, evolved,

several 100 (!) Rsun.

Strong, slow stellar winds

Solar-type stars:

Low-mass, on or near MS,

hot surrounding coronae,

weak stellar wind

(e.g. solar wind)

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Different regimes

require different

key input physics

and assumptions

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality

•Velocity fields and

outflows

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Spectroscopy and Photometry

ALSO:

Analysis

of different

WAVELENGTH

BANDS

is different

(X-ray, UV,

optical, infra-

red…)

Depends on where in

atmosphere light

escapes from

Question: Why is this

“formation depth”

different for different

wavebands and

diagnostics?

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Spectroscopy and Photometry (see part 1)

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UV “P-Cygni”

lines formed in

rapidly accel-

erating, hot

stellar winds

(quasi-)

Continuum

formed in

(quasi-)

hydrostatic

photosphere

Spectroscopy (see part 1)

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Lines and

continuum in

the optical

around 5200 Å,

in cool solar-

type stars,

formed in the

photosphere

Spectroscopy

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X-rays from

hot stars,

formed in

shocks in

stellar wind

X-rays from

cool stars,

formed in

hot corona

Spectroscopy

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Stellar Winds

– 2nd part of

course!

KEY QUESTION:

What provides the

force able to

overcome gravity?

125

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

4 810 ...10 /M M yr

1410 /M M yr

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KEY QUESTION: What

provides the force able to

overcome gravity?

Pressure gradient

in hot coronae of

solar-type stars

Radiation force:

Dust scattering

(in pulsation-levitated

material)

in cool AGB stars

(Höffner and colleagues)

Same mechanism In cool

RSGs?

126

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

4 810 ...10 /M M yr

1410 /M M yr

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KEY QUESTION: What

provides the force able to

overcome gravity?

Radiation force:

line scattering in hot,

luminous stars

done here at USM,

more to follow in part 2

4 810 ...10 /M M yr

1410 /M M yr

Question: How do you think the high mass loss of stars with high luminosities

affects the evolution of the star and its surroundings? 127

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

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Stellar Winds from evolved hot and cool

stars control late evolution, and feed the

ISM with nuclear processed material

from introductory slides …

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In the following,

we focus on stellar

photospheres

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•OBSER-

VATIONS!!!

130

From part 1

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131

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

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LTE or NLTE? (see part 1)

HOT STARS: Complete model atmosphere and synthetic

spectrum must be calculated in NLTE

COOL STARS: Standard to neglect NLTE-effects on atmospheric

structure, might be included when calculating line

spectra for individual “trace” elements (typically used

for chemical abundance determinations)

BUT: See work by Phoenix-team (Hauschildt et al.)

ALSO: RSGs still somewhat open question NLTE calculations for various applications

(including Supernovae remnants) within the

expertise of USM

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133

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

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•Effects of numerous -- literally millions -- of (primarily metal) spectral lines

upon the atmospheric structure and flux distribution

•Q: Why is this tricky business?

Spectral line blocking/blanketing

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•Effects of numerous -- literally millions -- of (primarily metal) spectral lines

upon the atmospheric structure and flux distribution

•Q: Why is this tricky business?

- Lots of atomic data required (thus atomic physics and/or experiments)

- LTE or NLTE?

- What lines are relevant? (i.e., what ionization stages? Are there molecules present?)

Techniques: Opacity Distribution Functions

Opacity-Sampling

Direct line by line calculations


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Back-warming (and surface-cooling)


“Blanket” warms deep

layers

“Blanket” typically cools

uppermost layers

Heat (photons) enters

atmosphere from

sub-photospheric layers

Numerous absorption lines

“block” (E)UV radiation flux

Total flux conservation

demands these photons be

emitted elsewhere

redistributed to

optical/infra-red

Lines act as “blanket”, whereby

back-scattered line photons

are (partly) thermalized and

thus heat up deeper layers

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Back-warming and flux redistribution


…occur in stars of all spectral types

Back warming in cool stars

(from Gustafsson et al. 2008) UV to optical flux redistribution in hot stars

(from Repolust, Puls & Hererro 2004)

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Spectral line blocking/blanketing F

rom

Puls

et

al. 2

008

138

Back-warming and flux redistribution

…occur in stars of all spectral types

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Back-warming ‒ effect on effective temperature


RECALL: Teff -- or total flux (plane-

parallel) -- fundamental input

parameter in model atmosphere!

4

B effF T

From Gustafsson et al. 2008:

Estimate effect by assuming

a blanketed model with Teff

such that the deeper layers

correspond to an unblanketed

model with effective

temperature T’eff > Teff

Question: Why

does the line

blocking fraction

increase for very

cool stars?

Teff in cool stars derived,

e.g., by optical photometry

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Previous slide were LTE models. In hot

stars, everything has to be done in

NLTE…

Question: Why is optical

photometry generally NOT

well suited to derive Teff in

hot stars?

140


RECALL: Teff -- or total flux (plane-

parallel) -- fundamental input

parameter in model atmosphere!

4

B effF T

Back-warming ‒ effect on effective temperature

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Instead, He ionization-balance is typically used

(or N for the very hottest stars, or, e.g., Si for B-stars)

Simultaneous fits to observed HeI and HeII lines

-– from Repolust, Puls, Hererro (2004)

Back-warming shifts ionization balance toward more completely

ionized Helium in blanketed models, thus fitting the same observed

spectrum requires lower Teff than in unblanketed models

• black – blanketed Teff=45 kK

• red – unblanketed Teff=45 kK

• blue – unblanketed Teff= 50 kK

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Result: In hot O-stars with

Teff~40,000 K, back-

warming can lower the

derived Teff as compared to

unblanketed models by

several thousand degrees!

(~ 10 %)

New Teff scale for O-dwarf stars. Solid line – unblanketed models.

Dashed – blanketed calibration, dots – observed blanketed values

(from Puls et al. 2008)

142


Instead, He ionization-balance is typically used

(or N for the very hottest stars, or, e.g., Si for B-stars)

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143

•LTE or NLTE

•Spectral line

blocking/blanketing

•(sub-) Surface

convection

•Geometry and

dimensionality


outflows

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Surface Convection

144

from part 1

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Surface Convection

• H/He recombines in

atmospheres of cool stars

Provides MUCH opacity

Convective Energy

transport

OBSERVATIONS:

“Sub-surface”

convection in layers

T~160,000 K (due to

iron-opacity peak)

currently discussed

also in hot stars

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Traditionally accounted for by rudimentary

“mixing-length theory (see part 1) in

1-D atmosphere codes

BUT:

• Solar observations show very dynamic structure

• Granulation and lateral inhomogeneity

Need for full 3-D radiation-hydrodynamics

simulations in which convective motions occur

spontaneously if required conditions fulfilled

(all physics of convection ‘naturally’ included)

Surface Convection

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Solar-type stars:

Photospheric extent << stellar radius

Small granulation patterns

Surface Convection

147 from part 1

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From Wolfgang Hayek

Solar-type stars:

Atmospheric extent << stellar radius

Small granulation patterns

Box-in-a-star

Simulations

(cmp. plane-parallel approximation)

Surface Convection

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From Wolfgang Hayek

Approach

(teams by Nordlund, Steffen):

Solve radiation-hydrodynamical

conservation equations of

mass, momentum, and energy

(closed by equation of state).

3-D radiative transfer included to

calculate net radiative

heating/cooling qrad in energy

equation, typically assuming LTE

and a very simplified treatment of

line-blanketing

Surface Convection

149 (= 0 in case of radiative equilibrium)

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http://www.astro.uu.se/~bf/ From Berndt Freytag’s homepage:

Surface Convection

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From Stein & Nordlund (1998)

Surface Convection

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Surface Convection

Question: This does not look much like the traditional 1-D models we’ve discussed

during the previous lecture! – Do you think we should throw them in the garbage?

Some key features:

Slow, broad upward motions, and

faster, thinner downward

motions

Non-thermal velocity fields

Overshooting from zone where

convection is efficient according

to stability criteria (see part 1)

Energy balance in upper layers not

only controlled by radiative

heating/cooling, but also by

cooling from adiabatic

expansion

See Stein & Nordlund (1998);

Collet et al. (2006), etc

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Surface Convection

In many (though not all) cases, AVERAGE properties

still quite OK:

Convection in energy balance approximated by

“mixing-length theory”

Non-thermal velocity fields due to convective motions

included by means of so-called “micro-” and

“macro-turbulence”

BUT quantitatively we always need to ask:

To what extent can average properties be modeled by

traditional 1-D codes?

Unfortunately, a general answer very difficult to give,

need to be considered case by case

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Surface Convection

Metal-poor red giant, simulation by Remo Collet,

figure from talk by M. Bergemann

For example:

In metal-poor cool stars spectral lines are scarce

(Question: Why?),

and energy balance in upper photosphere controlled to

a higher degree by adiabatic expansion of convectively

overshot material.

In classical 1-D models though, these layers are

convectively stable, and energy balance controlled only

by radiation (radiative equilibrium, see part1).

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Surface Convection

3-D radiation-hydro models successful in reproducing many solar features

(see overview in Asplund et al. 2009), e.g:

Center-to-limb intensity variation

Line profiles and their shifts and variations (without micro/macroturbulence)

Observed granulation patterns

Fro

m ta

lk b

y H

ayek

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Surface Convection

affects chemical abundance

(determined by means of line profile

fitting to observations)

One MAJOR result:

Effects on line formation has led to a

downward revision of the CNO solar

abundances and the solar metallicity,

and thus to a revision of the

standard cosmic chemical

abundance scale

Fig. from Asplund et al. (2009) – “The Chemical Composition of the Sun”

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Surface Convection

Also potentially critical for Galactic archeology…

…which traces the chemical evolution of the Universe by analyzing

VERY old, metal-poor Globular Cluster stars –- relics from the early

epochs (e.g. Anna Frebel and collaborators)

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Surface Convection

•Giant Convection Cells in the

low-gravity, extended atmo-

spheres of Red Supergiants

•Question: Why extended?

2

2 2

4

/ (with the

isothermal speed of sound)

/ / 0.5...0.6

/ 10 !

(see part 1)

RSG sun RSG sun

RSG sun

H a g a

a a T T

g g

Out to Jupiter…

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Surface Convection

Supergiants (or models including a stellar wind):

Atmospheric extent > stellar radius:

Box-in-a-star Star-in-a-box

(1D: Plane-parallel Spherical symmetry,

see part 1)

Star to model: Betelgeuse

Mass: 5 solar masses

Radius: 600 Rsun

Luminosity: 41400 Lsun

Grid: Cartesian cubical grid with 1713 points

Edge length of box 1674 solar radii

Model by Berndt Freytag, note the HUGE convective cells visible in the emergent intensity map!!

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Surface Convection

Star to model: Betelgeuse

Mass: 5 solar masses

Radius: 600 Rsun

Luminosity: 41400 Lsun

Grid: Cartesian cubical grid with 1713 points

Edge length of box 1674 solar radii

Movie time span: 7.5 years

http://www.astro.uu.se/~bf/movie/dst35gm04n26/

movie.html

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Surface Convection

Extremely challenging,

models still in their infancies.

LOTS of exciting physics to explore, like

Pulsations

Convection

Numerical radiation-hydrodynamics

Role of magnetic fields

Stellar wind mechanisms

Also, to what extent can main effects be

captured by 1-D models?

For quantitative applications like….

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Question: Why are RSGs ideal for

extragalactic observational stellar astrophysics

using new generations of extremely large infra-red telescopes?

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important codes and their features ….

163


Codes FASTWIND

CMFGEN

PoWR

WM-basic TLUSTY

Detail/Surface

Phoenix MARCS

Atlas

CO5BOLD

STAGGER

geometry 1-D

spherical

1-D

spherical

1-D

plane-parallel

1-D/3-D

spherical/

plane-parallel

1-D

plane-parallel

(MARCS also

spherical)

3-D

Cartesian

LTE/NLTE NLTE NLTE NLTE NLTE/LTE LTE LTE simplified

dynamics quasi-static

photosphere +

prescribed

supersonic outflow

time-independent

hydrodynamics

hydrostatic hydrostatic or

allowing for

supersonic

outflows

hydrostatic hydrodynamic

stellar wind yes yes no yes no no

major application hot stars with winds hot stars with

dense winds,

ion. fluxes, SNRs

hot stars with

negligible winds

cool stars, brown

dwarfs, SNRs

cool stars cool stars

comments CMFGEN also for

SNRs; FASTWIND

using approx. line-

blocking

line-transfer in

Sobolev approx.

(see part 2)

Detail/Surface

with LTE-

blanketing

convection via

mixing-length

theory

convection via

mixing-length

theory

very long execution

times, but model

grids start to

emerge

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And then there are, e.g.,

•Luminous Blue Variables (LBVs)

like Eta Carina,

•Wolf-Rayet Stars (WRs)

•Planetary Nebulae (and their

Central Stars)

•Be-stars with disks

•Brown Dwarfs

•Pre main-sequence T-Tauri and

Herbig stars

…and many other interesting

objects

Stellar astronomy alive and

kicking! Very rich in both

Physics

Observational applications

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Chap. 8 ‒ Stellar winds: an overview

ubiquitous phenomenon solar type stars (incl. the sun) red supergiants/AGB-stars

("normal" + Mira Variables) hot stars (OBA supergiants,

Luminous Blue Variables, OB-dwarfs, Central Stars of PN, sdO, sdB, Wolf-Rayet stars) T-Tauri stars and many more

165

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comet Halley,

with „kink“ in

tail

comet Hale-Bopp

with dust and

plasma tail (blue)

• comet tails directed away from the sun

• Kepler: influence of solar radiation pressure (-> radiation driven winds)

• Ionic tail: emits own radiation, sometimes different direction

• Hoffmeister (1943, subsequently Biermann): solar particle radiation

different direction, since v (particle) comparable to v (comet)

The solar wind – a suspicion

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• Eugene Parker (1958): theoretical(!) investigation of coronal equilibrium:

high temperature leads to (solar) wind (more detailed later on)

• confirmed by

• Soviet measurements (Lunik2/3) with “ion-traps” (1959)

• Explorer 10 (1961)

• Mariner II (1962): measurement of fast and slow flows

(27 day cycle -> co-rotating, related “coronal holes” and sun spots)

The solar wind – the discovery

167

USM The solar wind – Ulysses ...

168

USM … surveying the polar regions

169

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polar wind:

fast and thin

equatorial wind:

slow and dense

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fast wind:

over coronal holes

(dark corona, “open”

field lines, e.g., in

polar regions)

coronal X-ray

emission

very high

temperatures

(Yohkoh Mission)

The solar wind – coronal holes

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The sun radius = 695,990 km = 109 terrestrial radii

mass = 1.989 1030 kg = 333,000 terrestrial masses

luminosity = 3.85 1033 erg/s = 3.85 1020 MW 1018 nuclear power plants

effective temperature = 5770 ºK

central temperature = 15,600,000 ºK

life time approx. 10 109 years

age = 4.57 109 years

distance sun earth approx. 150 106 km 400 times earth-moon

The solar wind temperature when leaving the corona: approx.1 106 K

average speed approx. 400-500 km/s (travel time sun-earth approx. 4 days)

particle density close to earth: approx. 6 cm-3 temperature close to earth: 105 K

mass-loss rate: approx 1012 g/s (1 Megaton/s) 10-14 solar masses/year

one Great-Salt-Lake-mass/day one Baltic-sea-mass/year

no consequence for solar evolution, since only 0.01% of total mass lost over total life time

The sun and its wind: mean properties

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Stellar winds – hydrodynamic description

vel. field grav. radiative (part of) accel.

accel. accel. by pressure gradient

positive for v > a inwards outwards outwards negative for v < a

equation of continuity:

conservation of mass

equation of motion:

from conservation of momentum

2

2 2 2

2 2

With mass-loss rate , radius , density and velocity

= 4 r ,

and with isothermal sound-speed

21

rad

M r v

M v

a

a dv GM a dav g

v dr r r dr

Need mechanism which accelerates material beyond escape velocity:

pressure driven winds

radiation driven winds

remember equation of motion (conservation of momentum + stationarity, cf. Chap. 6, p. 79)

Note: red giant winds still not understood,

only scaling relations available (“Reimers-formula”)

1 (in spherical symmetry)

extdv dpv g

dr dr

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vel. field grav. radiative pressure

accel. accel.

2 2 2

2 2

21

rad

a dv GM a dav g

v dr r r dr

The solar wind as a proto-type for pressure driven winds

present in stars which have an (extremely) hot corona (T 106 K)

with grad

≈ 0 and T ≈ const, the rhs of the equation of motion changes sign at

and obtain four possible solutions for v/vc ("c" = critical point)

only one (the "transonic") solution compatible with observations

pressure driven winds as described here rely on the presence of a hot corona

(large value of a!)

Mass-loss rate

has to be heated (dissipation of acoustic and magneto-hydrodynamic waves)

not completely understood so far

6

2

sun

; with a (T=1.5 10 K) 160 km/s,2

we find for the sun 3.9

c

c

GMr

a

r R

14

sun10 M / yr, terminal velocity v 500 km/sM

Pressure driven winds

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USM Radiation driven winds

accelerated by radiation pressure:

cool stars (AGB): major contribution from dust absorption;

coupling to “gas” by viscous drag force (gas – grain collisions)

hot stars: major contribution from metal line absorption;

coupling to bulk matter (H/He) by Coulomb collisions

6 510 ...10 M / yr, v 2,000 km/ssunM

610 M / yr, v 20 km/ssunM

2 2 2

2 2

important only inlowermost wind

21

rad

a dv GM a dav

v dr r r dg

r

pressure terms only of secondary order

(a ≈ 20 km/s for hot stars,

≈ 3 km/s for cool stars)

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dusty

winds

• o Ceti = Mira (the marvellous star)

• first detected periodic variable

(David Fabricius, 1596)

• brightness variations by 5.5 mag

(from 3.5 to 9),

corresponding to a factor of 160

11 months 11 months

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dust: approx. 1% of ISM, 70% of this fraction formed

in the winds of AGB-stars (cool, low-mass supergiants)

Red supergiants are located in dust-forming “window”

transition from gaseous phase to solid state possible only in

narrow range of temperature and density:

gas density must be high enough and temperature low

enough to allow for the chemical reactions:

sufficient number of dust forming molecules required

the dust particles formed have to be thermally stable

Material on this and following pages from

Chr. Helling, Sterne und Weltraum,

Feb/March 2002

dwarfs

giants

red supergiants

browns dwarfs

density ρ

Cool supergiants: The dust-factories of our Universe

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• decrease of density and temperature

• more and more complex structures are forming

• dust: macroscopic, solid state body,

approx. 10-7 m (1000 Angstrom), 109 atoms

first steps of a linear reaction chain, forming the seed of (TiO2)N

Growth of dust in matter outflow

ions, atoms

seed formation

molecules

dust growth

terrestrial, macroscopic rutile crystal

(TiO2, yellowish)

178

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• star emits photons

• photons absorbed by dust

• momentum transfer accelerates dust

• gas accelerated by viscous drag force

due to gas-dust collisions

acceleration

proportional to number of photons, i.e.,

proportional to stellar luminosity L

mass-loss rate L

dust driven winds at tip of AGB responsible

for ejection of envelope

Planetary Nebulae

winds from massive red supergiants still

not explained, but probably similar mechanism

Dust-driven winds: the principle

179

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• star (“surface”) pulsates,

• sound waves are created,

• steepen into shocks;

• matter is compressed,

• dust is formed

• and accelerated by

radiation pressure

dust shells are blown away,

following the pulsational cycle

periodic darkening of

stellar disc

brightness variations

snapshot of a time-dependent hydro-simulation of a

carbon-rich circumstellar envelope of an AGB-star.

Model parameters similar to next slide. deg

ree

of

conden

sati

on

log d

ensi

ty

vel

oci

ty

distance r (Rstar)

180

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velocity

dark colors: dust shells

simulation of a

dust-driven wind

(working group E. Sedlmayr,

TU Berlin)

T = 2600 K, L = 104 Lsun,

M = 1 Msun, v = 2 km/s

Earth Mars

Jupiter Saturn

Neptun

shock front shock front shock front

181

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The sun

Red

AGB-stars

Blue

supergiants

mass [M] 1 1 ... 3 10...100

luminosity [L] 1 104 105...106

stellar radius [R] 1 400 10...200

effective temperature [K] 5570 2500 104…5·104

wind temperature [K] 106 1000 8000...40000

mass loss rate [M /yr] 10-14 10-6 ...10-4 10-6 ... few 10-5

terminal velocity [km/s] 500 30 200...3000

life time [yr] 1010 105 107

total mass loss [M] 10-4 0.5 up to 90%

of total mass

Stars and their winds – typical parameters

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Bubble Nebula

(NGC 7635)

in Cassiopeia

wind-blown

bubble around

BD+602522

(O6.5IIIf)

Massive stars determine energy (kinetic and radiation)

and momentum budget of surrounding ISM

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Chap. 9 – Line-driven winds: the standard model

accelerated by radiation pressure in lines momentum transfer from accelerated species (ions)

to bulk matter (H/He) via Coulomb collisions

7 5

sun10 ...10 M / yr, v 200 ... 3,000 km/sM

Prerequesites for radiative driving large number of photons => high luminosity

=> supergiants or hot dwarfs line driving:

large number of lines close to flux maximum (typically some 104...105 lines relevant) with high interaction probability (=> mass-loss dependent on metal abundances)

2 4

* effL R T

pioneering investigations by Lucy & Solomon, 1970, ApJ 159 Castor, Abbott & Klein, 1975, ApJ 195 (CAK) reviews by Kudritzki & Puls, 2000, ARAA 38 Puls et al. 2008 A&Arv 16, issue 3

line driven winds important for chemical evolution of (spiral) Galaxies, in particular for starbursts transfer of momentum (=> induces star formation, hot

stars mostly in associations), energy and nuclear processed material to surrounding environment dramatic impact on stellar evolution of massive stars

(mass-loss rate vs. life time!)

184

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9.1 Radiative line driving and line-statistics

Observational findings:

massive star have outflows, at least quasi-stationary

only small, in NO WAY dominant variability of global

quantities

have to be explained

diagnostic tools have to be developed

predictions have to be given

(M, v )

M, v , v(r)

‒ Morton & Underhill 1977

‒ Howarth (p.c.)

Δt several years

vmax≈2,500

km/s

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9.1.1 Equation of motion in the standard model

2

ext

2

s

H

s

(with 0, 1-D spherically symmetric)

mass-loss rate

v isothermal sound sp

4 ( )v( ) const =

v 1v ( )

( )

(equation

eed, mean molecu

of state)

lar

v

r r r M

d dpa r

dr r dr

kTp NkT

m

t

2 2 2

exts s s

2

2

s

ext true cont line

Rad Rad2

Thomson

Rad

grav

v 2v vv v 1

v r

v

( ) (1 ) ( ) ( )

weight

(assumption here: known)

is Eddington fa(

)

corrects for radiati

c

v

o ctor, nst(

e

)

dda

dr dr

T

GMa r g r g r

r

g r

g r

acceleration due to Thomson scattering

ext

ext

continuity equation

momentum equatio

(use cont

Hydro-equation

inuity equatio

n

equation of mot

( ) 0

( ) ( )

1( )

n

s

n

)

io

t

pt

pt

v

v vv a

v v v a

186

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a) scattering of continuum light in resonance lines

b) momentum transfer from metal ions (fraction 10-3)

to bulk plasma (H/He) via Coulomb collisions

(see Springmann & Pauldrach 1992)

velocity drift of ions w.r.t. H/He is compensated by

frictional force as long as vD/vth < 1

(linear regime, “Stokes” law)

θ

γin

γout

metal

ion

in

in

out

itot all lines

rad

cos 1

isotropic reemissionc

cos 0

t m t m

hP

PP

g

radial in out

in in out out( cos cos )

absorption reemission

P P P

h

c

9.1.2 Principle idea of line acceleration

187

USM

from Springmann & Pauldrach (1992, A&A 262)

see also Owocki & Puls (2002, ApJ 568)

fric

th

v v( ) is reduced mass

v (prot)

i j

ij ij ij ij ijR G x x A A

ion

ion ion Rad 2

bulk bulk 2

approximate description (supersonic regime)

by linear diffusion equation

drift velocity

bulk H/He,

relaxa

v v

v v

tio

n t

ib

bi

d GM wg

dr r

d GM

dr

w

w

r

1

ion tot tot

ion bulk

ion bu

tot

Rad Rad Rad

k

ion

l

ime between collisions

in order to obtain one-component fluid,

v vv v

tot bulk ion, is metallicity

for and

1 1 1 1

lo w /or

ib bi

d d

dr

w g g gZ

M

V

dr

Z

drift large runalow way Z

e.g., winds of A-dwarfs, Babel et al. 1995, A&A 301

188

USM

s

2 2 2

lines s s

Rad2 2

line

Rad2

2 line

Rad

s

2

s

e

supersonic approx., v > v , pressure forces negligi

v 2v vvv 1 (1 )

v r

v v (1 )

v4 (1 ) 4

4 (

4

1 )(

ble

v v )

R

dd GMg

dr dr r

d GMg

dr r

dM GM r g

dr

GMM s

r

d

s

r

TH

line

e Rad

wind

TH

line

e TH e Rad/ , ,

1

4

v

sR

PLs s g

cG

dr g dm

M m t

PLM

c t

THmomentum loss

of radi

Now so-called S(ingle) S(cattering) L(imit), SS

ation field

assume that each photon is sca

L

v 1

ttere

/

d onc

PM c

L c L t

in

( )1!

e somewhere

in the wind, with

( )number of photons per time an

d is

Pc c L hd

L t L h

c

d dh

c

hP

L

S

TH

SL

"performance number" or wind efficiency

v 1 1

/

M

L c

momentum rate

needed to support

wind against gravity

9.1.3 The single scattering limit/multi-line scattering

189

USM

Wind efficiencies for Galactic OBA supergiants. The

actual efficiency might be smaller, due to neglected

wind clumping.

From Markova & Puls 2008

NOTE: Wolf-Rayet stars have much larger wind-

efficiencies (η = O(10)), due to higher Mdot

(and also Γ and τ are larger).

→ Single-scattering not sufficient to provide enough

radiative acceleration

190

USM Multi-line scattering

Friend & Castor (1983)

Abbott & Lucy (1985)

→ Monte Carlo Method

Puls (1987) not very efficient in OB-star winds

Lucy & Abbott (1993) explain large wind-efficiencies of WR winds due

to multi-line scattering in stratified ionization

equilibrium

Springmann (1994)

Gayley et al. (1995)

from Abbott & Lucy (1985) Throughout following slides WR case not considered

assume that each line can be treated separately, i.e.,

no interaction between different lines

don’t misinterpret this assumption (“single-line

approximation”) with SSL!!!

η(SL) > η(SSL) !!!

tot

lines i

P / linei

P

191

USM The photon-tiring limit

22 2

esc

wind esc

wind *

What is the maximum mass-loss rate that can be accelerated???

mechanical luminosity in wind at infinity is

vv v 2 with

maximum m

v

ass loss, if

2 2 2

GM GML M M

R R

L L

*

max 2 2

esc

ma

max

es

x

2

max

es

c

c

( ) 0,

2

v v

v

star becomes invisible

typical values: v 2000...3000 km/s 0.01 , v / v 1/ 3

2

/ vv 1

200

v

L

LM

M c

c

L c

v(r)

1-R/r

tir

* *

tir 2 6

esc

(Owocki & Gayley 1997) is maximum mass-loss rate when wind just escapes

the gravitational potential, w v 0

20.032

v yr 10

ith

M

M ML L RM

L R M

e

0.0012yr

M R

R

192

USM 9.1.4 Calculation of the line force

1

line line line

Rad

0 1

li

Rad

crucial point of the problem

absorbed emitted

(in single-line approximation)

4 1( , ) ( , ) ( , )

2

g d d r I r rc

g

1

ne i

lines i line 1

2 ( , ) ( , )

id d r I r

c

• two quantities to be known

force/line in response to χν

distribution of lines with χν and ν

The force per line • super-simplified

• simplified: Sobolev approximation

• “exact”:

comoving frame, special cases

observer’s frame, instability

193

USM Super-simplified theory

velocity One line with transition freq. ν0

radius shell with Δr

Δm = 4πr2ρΔr

v2

v1

νin = νobs ν2 ν1 λobs

photosphere at R*

} { Δv

{

Δνobs

0

* obs 0

0

CMF obs 0

interaction with line at , when comoving frame frequency of photon

starting at with is equal

(finite profile width neglected, interaction p

to

Doppl

robability

v(e

= 1)

( r)

:

R

r

c

obs 0

0 1 1

obs 0

0 2 2

0

obs

v ( )

v (

scat

radial photons, =1,

tering at larger v

shift,

requires

as

'bluer' photons

su ed)

)

m

v

r

c

c

c

r

obs obs obs

1 2 1 ob

obs

obs

Rad

ob

0

2

s

Rad

s

2 2

0

Number of photons in interval , per unit time

( )

( v)

1

=

c

v 1 v 1

4

N L

t h

h Lg

h m

d

dr r

Pg

t m

L

c

c

r r

194

v(r)

v(r)

r

r

r

r

1

2

1

dv( )

dr

2

dv( )

dr

0

1

0

0

shell of matter with spatial extent r,

dvand velocity v

dr

absorption of photons at

photons must start at higher (stellar)

frequencies, are "seen" at

in frame of m

r

in frame of matter

atter because of Doppler-effect.

Let be frequency band contributing to

acceleration of matter in

dvThe larger ,

dr

the larger

the more photons can be absorbed

the larger the acceleration

r

rad

dv g

dr

(assuming that each photon is absorbed,

i.e., acceleration from optically thick lines)

0 1

0 obs 1 0

0

0 obs 0

1 1

1 0 0

v dv ( )

c

v

c

dv (dv/dr)

c c

r

line with transition

frequency 0 at begin

and end of shell

2 2

2 0 0 1

dv (dv/dr)

c c

r

Why grad dv/dr ?

195

USM

rad

0

0

rad 2 2

Assumption was: is scattered

independent of cross-sections, occu

each p

ption

v 1

numbers etc.

4

(o

ne l

ine

o

hoto

nly dependent on hydro-struc

n

The

t

n

a

t

)

:

Lg

c r

g

r

interaction probability = 1 e , with optical depth

ure and flux distribution

happens if interaction probability < 1?

1 prob = 1

1 prob =

What

Now: divis

ion in two clas

ses

rad rad

optically thick lines, prob = 1 (saturation, independent of )

optically t

(optic

hin

all

lines

y thin

line)

prob =

=

1

(optically thic k line)

1

g g

196

USM

Calculating the optical depth:

The Sobolev-approximation (SA)

Note: ‘first’ interaction at highest CMF-freq., ‘blue’ edge

‘last’ interaction (final reemission) at ‘red’ edge

TRICK of Sobolev approximation (Sobolev, 1960; developed

around 1945)

• in the resonance zone (width ~ 2 times 3 vth), assume ‘macro’-

quantities such as opacity, source-function and density to be

constant or perform Taylor expansion

• account at least for v and dv/dr

• then, all integrals of radiative transfer can be performed

analytically and are exact within the assumptions

The validity of the SA can be checked by comparing the scale-

length of the macro-quantities with the co-called Sobolev length,

which is the scale length associated to the line-profile:

From dv/dr LS = vth, we find LS = [d(v/vth ) / dr]-1

Note: always required: v > vth ≈ vsound/√m; m mass of absorbing ion

0 th

0

0 th

CMFν CMF

( ) profile function obs 0

( )r

c

n vobs

*

obs

*

L obs

L

0

L

res.zone

0 obs 0

0

obs 0

( ) const in resonance zon

first assumption:

2nd

v( )

e at

v( )

( ) ( ) ( )

v

assumptio

( )( ) (

n

'

)

:

R

R

rr dr

c

rr dr

c

r r

r

c

dr

general definition

0

obs

0

obs

0

0

S L 0 0

L 0

0

S L 0 0

replace spatial by frequenntial integra

( v) const in res

( v)'

( )( ) ( ') ' ( ') '

( v) ( v)

( )

( v

onance zone

o)

l!

p

r

r

r

dd dr

c dr

rcr

d

d

d dd d

dr dr

r

d

dr

r

obs 00 0

tical depth in Sobolev theory,

v( ) from :with

cr

r

197

USM

Rad

Within Sobolev theory, all radiation field related

quantities can be calculated, e.g.,

and

After a number of manipulation, one finds

(see, e

int

( ) , ( )

4 ( )( ) ( )

elligen

)

t

.(

J J d H H d

rg r H r

c r

S1

Rad cS

2

c

S L L0 0

2 2

.g., Rybicki & Hummer 1978, ApJ 219)

with cone-angle core inten

1 exp( ( , ))4 ( ) 1( )

( ) 2 ( , )

1 , ( ),

( ) ( )and ( , )

( v) v / (1

si

)v /

ty

rrg d I

c r r

RI

r

r rr

d d dr r

dr

S 1

Rad cS

S

S

For r >>R (i.e., 1), this is the same as derived from

super-simplified theory (incl. interaction p

1 exp( (1, ))4 ( ) 1( )

( ) (1, ) 2

1 exp( ( ))4 ( )

( ) (

robability),

)

rrg d I

c r r

rrH

c r r

S

2 2

L 0

02 2 S

S L

S

a 2

0

0

R d 2

v1 exp( ( ))

4 ( )

( ) 16( )

1 optically thick lines, 1

1 v

4 optically thin lines, 1

(

v 11 exp( ( ))

)and ( )

v /

4

dr

Lr dr

c

L dg r

r rr

dL

r c dr

r

r

d d

c

r

dr

r

To calculate the total line acceleration, we

have to sum over all contributing lines! 198

USM

Line acceleration from a line ensemble

tot i j

Rad Rad Rad

thick thin

i i2 2

thick thin

Li i Li i

i

e t

i

i

h

1 i

optical

( ) ( ) ( )

1 v 1 v

depth of

( )

:

= 4

pre v/ v

cisely /

:v

i

k k

k r

d dr d

g r g r g r

d dL L

d

r c dr

r

dr

ks

i

i

i

i i i

i

is line strength cross section,

lower occup. number of line transition

line in Sobolev theory

roughly const

( )

( )

Which line str

ant

eng

in wi

th corresponds to

nd!!!

'bo

nk

r

n

r

k

i

tot

Rad 1 i i i2

1

1

2

1 1i i<

optically thick optically thin

rder' 1?

1 ( )

4

v/1

v

/

g r k L L k

k d drk

d dr

r c k k k k

depends on hydrostruct. depends on line-strength

199

USM

... are present

... and needed!

line transitions in FeV

i

i

tot i

Rad Rad

all lines

ithin i

Rad i i

thick i i

Rad ν i ν i 1

(line-s

,

, trength

/

)

v

g g

g L k k

d drg L L k

Millions of lines ....

200

USM

Logarithmic plot of line-strength distribution function for an O-type wind at 40,000 K and corresponding power-law fit (see Puls et al. 2000, A&AS 141)

2( ), 0.6...0.7

dN kk

dk

20

( , ) ( )dN k N f d k dk

+ 2nd empirical finding:

valid in each frequential

subinterval

Note curvature

of distr. function

2 / 3

The line distribution function

pioneering work by Castor, Abbott & Klein (CAK, 1975): • from glance at CIII atom in LTE, they suggested that ALL line-strengths follow a power-law distribution

first realistic line-strength distribution function by Kudritzki et al. (1988)

NOW: couple of Ml (Mega lines), 150 ionization stages (H – Zn), NLTE

201

USM Force/line + line-strength distribution

1

max 1

max

1

tot

Rad 1 i i i2 2

0

12 2

0 0

0 2

12 2

11 1

1 1i i

1

1

<

1( )

4

1 ( ) ( , ) ( ) ( , )

4

( ) ( )

4

k

k k

k

k

k ka

g r k L L kr c

k L dN k L kdN kr c

N L f dk k dk

r c

k k k k

1

tot

Rad 1

02

1

2

2

2

0

1

1

1

(1 )

1

final result

( ) ( )4 v

const ( )

4 r

c v ; onst/

v

(1 )

k

k

k

g

d dr

k

N

k k

L f ddk r

M d

dk

r c

r

very ‘strange’ acceleration,

non-linear in dv/dr

202

USM The force-multiplier concept

neglected so far

– non-radial photons (μ ≈ 1 justified only for r >> R)

– ionization effects (have assumed that ni/ρ = const throughout wind)

line-force expressed in terms of Thomson acceleration

Rad

grav

e th E E E

CAK CAK CAK 1

CAK

( ) with "force-multiplier"

v dv( ) ( , , )

dv/d W d W W

, , "force-multiplier parameter", with

t

gM t

g

s n n nM t k CF r v k t CF k k CF

r r

k

1

1

ionization parameter,

O(0.1) under O-star conditions

optical depth in Sobolev-approx., if line-strength identical with

strength

t k

e

11 -3

E

of Thomson-scattering (= ) [correctly normalized]

electron density in units of 10 cm

0.5(1 - ) dilution factor of radiation field

CF "finite con

s

n

W

e angle correction factor", correction for non-radial photons

CAK 1975

Abbott 1982 Pauldrach, Puls & Kudritzki 1986

203

USM

if everything has been correctly normalized.

for O-stars, kCAK is of order 0.1

kCAK can be interpreted as the fraction of photospheric flux which would be blocked if ALL lines were optically thick, divided by α.

a different parameterization has been suggested by Gayley (1995). Both parameterizations are consistent though.

for line-driving in hot, pure H/He winds (first stars) one can show that α + δ = 1, i.e., δ ≈ 0.33.

for all subtleties and further discussion, see Puls et al. 2000, A&ASS 141.

0 th

CAK

( ) ( )dv

,L c (1 )

o

L fN

k

204

USM

9.2 Theoretical predictions for line-driven winds

first hydro-solution developed by CAK 1975, ApJ 195,

improved for non-radial photons and ionization effects

by Pauldrach, Puls & Kudritzki 1986, A&A 164 and Friend

& Abbott 1986, ApJ 311

2 2 2

exts s s

2

ext true cont line

Rad Rad

line

Rad 12

2

1

2

equation of motion

(

v( , v, , ) if all subtleti

)

ves included

dr

v 2v vvv 1 ( )

v r

( ) (1 ) ( ) ( )

had

Al

/

(4 )

v

/

Lg r k

r

r d

dda r

dr dr

GMa r g r g r

r

f

drk f

df r M

M

2 2 2

2s s s

2 2 2

v 2v vv vv 1 (1 ) v

v r 4

l together

dd GM f L M dr

dr r dr r dr

for ‘normal’ winds

• non-linear differential equation

• has ‘singular point’ in analogy to solar wind

• vcrit >>vs (100… 200 km/s)

• solution: iteration of singular point location/velocity,

integration inwards and outwards

205

USM 9.2.1 Approximate solution

finally …

(see also Kudritzki et al., 1989, A&A 219)

• supersonic → pressure terms vanish

• radially streaming photons → f (4π)α → const

for unique solution, derivatives have to be EQUAL!

Mdot too small Mdot OK

2

2

2

2

const

vwith (1 ),

v const vv (1 ) ( v )

graphical solu (Cassinelli et al. 1979, ARAA 17,

is consta

tio

nt

Kudri

n

v

y A L M y

dA GM

d GM L dM r

d

y

y r

r r r

r

d

d

r

tzki et al. 1989)

1

1

at critical point

equation of motion

and

in equation of motion at critical poin

equality of derivativ

const

1 const

1,

e

1

const

t

s

i.e.

c

c

c

c

y A L M y

L M y

y A y

y

M yL

!

1, (1 ) (1 )

(1 )1

c

c

y GM

y GM y

206

USM

Eddington factor, accounting for acceleration by

Thomson-scattering, diminishes effective gravity

Neff number of lines effectively driving the wind ( kCAK), dependent on metallicity and spectral type

1 1 1

1' ' '

eff

2

*

Integration between and R

scaling law for

0.5 for approx. solution, "CAK-velocity law"

, =0.8 (O-stars) ... 2

(1 )

v v (1 )

1

v( ) v 1

M N L M

dr GM

dr

rr

M

R

11

2

e c

2

s

(BA-SG), see next slide

scaling law 2 (1 )

v

, if all subtleties includev 2.25 d

fo

1

1

v

r GM

Rv

exponent of line-strength distribution function,

0 < < 1

large value: more optically thick lines

’ = ,with ionization parameter,

typical value for O-stars: ’ 0.6

Scaling relations for line-driven winds (without rotation)

207

USM

consistent solution

• inclusion of finite cone-angle and

(nE/W)δ term:

Pauldrach, Puls & Kudritzki (1986) and

Friend & Abbott (1986)

• major effect

y no longer constant,

steeper slope in subcritical,

flatter slope in supercritical wind

• critical point closer to photosphere

– lower Mdot, larger vinf

“Cooking recipe” by Kudritzki et al.

(1989, A&A 219)

• very fast calculation of Mdot, vinf for given

force-multiplier parameters

*

L

1

2

CAK velocity

v( ) v

From follo

N

ws from

law

the

1!!!

v /

OT

1

E

c

S

y y const

kconst

d d

r

r y

R

r

• this basically explains why resonance lines remain

optically thick also in the outer wind part

• generalized velocity law

o from consistent solution

o from ‘β-velocity law‘

* inv( ) v 1 , most cases =0.8...1.3 R

rr

… consistent solution

of complete equation

‒- β=0.8 velocity law

+ photospheric structure

(see Santolaya Rey,

Puls, & Herrero, 1997,

A&A, 488)

with same mass-loss rate

and terminal velocity as

in consistent solution

208

USM

use scaling relations for Mdot and v∞, calculate

modified wind-momentum rate

3 / 2 1 / '1 / 2 1 / ' 1 / '

* effv (1 )M R N L M

1 / 2

1 1 / '1 / ' 1 / '

eff 1 / 2

*

(1 )v (1 )

MM N L M

R

9.2.2 The wind-momentum luminosity relation (WLR)

209

USM

use scaling relations for Mdot and v∞, calculate

modified wind-momentum rate

(Kudritzki, Lennon & Puls 1995)

(at least) two applications (1) construct observed WLR, calibrate as a function of

spectral type and metallicity (Neff and α’ depend on both parameter)

– independent tool to measure extragalactic distances

from wind-properties, Teff and metallicity

(2) compare with theoretical WLR to test validity of radiation driven wind theory

1 / 2 1 / ' 1 / '

* eff

1 / 2

*

independent of and

2v since ( ' )

3

!!!!!

1log ( v ) log ( , sp.type

'

)

M R N L

M

M R L const z

► stellar winds

contain info

about

stellar radius!!!

9.2.2 The wind-momentum luminosity relation (WLR)

210

USM

Modified wind momenta of Galactic O-, early B-, mid B- and A-supergiants as a function of luminosity, together

with specific WLR obtained from linear regression. (From Kudritzki & Puls, 2000, ARAA 38).

211

USM

Simple, however interesting argument (cf. Puls et al., 2000, A&ASS141)

Remember

for resonance lines k ~ f (lower level = ground state of ion)

cross sectio

2

n

2( )

, abs

e

ndN k efk k

dk m c

9.2.3 Why α 2/3?

inclusion of other (non hydrogenic) ions (particularly

from iron group elements) complicates situation

general trend: α decreases !

2 3

3

3

The hydrogen atom

for resonance lines, from Q

The m

.

32 1 1(1, ) 1

3 3

(summed over all contributing angular moment

ost simp

'Kramersfo

le case:

.Mrmul '

a)

a

f nn nn

C

1

3

max

m

max

max max

1 1

3 3max max

a

max 3

max

x

distrib

Number of lines until principal quantum number :

( ) 1

(

( )

= number of lines with f-values larger than a given

uti

) ( ) 1

on f

( )

o e

u c

n

n

n

N n n

N f

Cn

f

Cf n

n

n

n

f n C f

4

2

3 compare with

powerl

2

aw,

!!!3

tion

dN

dN

f

dk

fd

k

5p

4p

3p

2p

1s example: 4 resonance

lines until n=5

212

USM

number of effective lines

scales (roughly) with Z1-α

• more metallicity => more lines

consequence

both mass-loss and wind-momentum

should scale with

example for Z=0.2 (≈ SMC abundance)

• Mdot (40kK) factor of 0.45 decrease

• Mdot (10kK) factor of 0.09 decrease

Let Z be the (global) abundance relative to its solar value, i.e., solar comp. is Z =1

1

'

1.5

Z for , ' 2/3 (O-type winds)

... Z for , ' 0.4 (A-type winds)

Z

9.2.4 Predictions from line statistics

adapted from Puls et al., 2000, A&ASS 141

Teff = 40kK

slope=0.56

Teff = 10kK

slope=1.35

213

USM Predictions from line statistics

Differential importance of Fe-group and lighter elements

(CNO) • cf. Pauldrach 1987; Vink et al. 1999, 2001; Puls et al 2000; Kriticka 2005

• lines from Fe group elements dominate acceleration of lower wind

determine mass-loss rate Mdot

• lines from light elements (few dozens!) dominate acceleration of outer wind

determine terminal velocity v∞

From Kritcka, 2005

214

USM

Pauldrach (1987) and

Pauldrach et al. (1994/2001): “WM-basic”

consistent hydrodynamic solution, force-

multiplier from regression to NLTE line-

force

NLTE, since strong radiation field and low

densities

150 ions in total (≈ 2 MegaLines),

reduced computational effort due to

Sobolev line transfer

since 2001, line-blocking/blanketing and

multi-line effects included

9.2.5 Theoretical wind-models

From Pauldrach et al (1994)

(see also Pauldrach et al. 2001 for inclusion of line-

blocking/blanketing) 215

USM

Vink et al. (2000/2001)

• Monte-Carlo approach following Abbott & Lucy (1985):

• derive (iterate) Mdot from global energy conservation

• occupation numbers: NLTE, with Sobolev line transfer

• advantage: precise treatment of multi-line scattering

• disadvantage: only scattering processes can be considered,

no line-blocking/blanketing in NLTE

Krticka & Kubat (2000/2001/2004), Krticka 2006

• similar approach as Pauldrach et al., but

– disadvantage: no line-blocking, no multi-line effects

– advantage: more component description (metal ions + H/He)

– allows to investigate de-coupling in stationary wind-models

Kudritzki (2002, based on Kudritzki et al. 1989)

• “cooking recipe” coupled with approx. NLTE, very fast

– allows for depth-dependent force-multiplier parameters

2 2

esc

esc

1 1

input:

calculate via Monte-Carlo:

calculate new estimate: from , update occupation numb

1(v

ers

v ) ( ) ( )

, calculate

iterate until c

2

v , v , , ( ),

( )

( )

onverges

( )

i

i

i i i

i

M L R L

L R M

L

M L L

M

216

USM Validity of WLR concept

Theoretical wind-momentum rates as a function of luminosity, as calculated by Vink et al. (2000). Though

multi-line effects are included, the WLR concept (derived from simplified arguments) holds!

models for different luminosity classes!

217

USM Consistency of different codes

Results from WM-Basic

From Puls et al. 2003 (IAU Symp. 212)

218

USM

9.2.6 Predictions from hydrodynamical models

OB stars:

• Vink et al. (2000): “Mass-loss recipe” for solar abundances

in agreement with independent models

– by Kudritzki (2002), with

– by Puls et al. (2003), using WM-Basic (A. Pauldrach and co-workers)

– by Krticka & Kubat (2004)

• Vink et al. (2001):

• Krticka (2006):

0.69

0.64

for O-stars,

for B-supergiants

M z

M z

0.67

0.06

for O-stars

v

M z

z

0.12v z

219

USM Summary Chap. 9

radiative line acceleration:

scaling relations for line-driven winds

wind-momentum luminosity relation (WLR)

• mass-dependency vanishes or weak, since 1/x= α´≈0.6 (for OB-stars)

• offset D (and, to a lesser extent, slope x) depend on spectral type and metallicity

predictions from theoretical models • metallicity dependence

esc

1 11-

' ´eff

v v

v( ) v (1 )

M L M

Rr

r

1

2log v ( / ) log /M R R x L L D

rad

Doppler-effect

dv dv for optically thick lines, for ensemble of lines

d d

!

gr r

220

USM

Determine atmospheric parameters from observed spectrum

Required

Teff

, log g, R, YHe

, Mdot, v, + metal abundances)

(R stellar radius at R

= 2/3)

also necessary

vrad

(radial velocity)

v sin i (projected rotational velocity)

Given

reduced optical spectra (eventually +UV, +IR, +X-ray)

,resolution of observed spectrum

Visual brightness V

distance d (from cluster/association membership), partly rather insecure

NLTE-code(s), "model grid"

1. Rectify spectrum, i.e. divide by continuum (experience required)

2. Shift observed spectrum to lab wavelengths (use narrow stellar

lines as reference):

rad

lab obs rad1 , assumed as positive if object moves away from observer

c

vv

Alternative set of parameters

L, M, R or

L, M, Teff

or

Teff

. log g, R …

interrelations

Useful scaling relations

If L, M, R in solar units, then

2 4

* eff

2

*

4B

L R T

GMg

R

0.5

7

* 2

eff

4

2

*

11

esc *

4 15

e eff

He He

e

He

He

3.327 10

log log 2.74 10

(1 ) 1.392 10

/ 1.8913 10

10.4

1 4

with number of free electrons

per Helium atom

(e.g.,=2, if completely ionized)

LR

T

Mg

R

v R g

s T g

I Ys

Y

I

Chap. 10 Quantitative spectroscopy

The exemplary case of hot stars

221

USM

SiIV NIII

CIV/SiIV

H H

H

HeII

HeII HeII

HeI/HeII

HeI HeI

HeI/HeII

HeI

rectified

optical spectrum,

("blue" and "red")

corrected for vrad

,

of the late O-SG

19 Cep

___ Hydrogen

...... Helium I

- - - Helium II

in "red":

"strategic" lines to

derive atmospheric

parameters in hot

stars

H

HeI

HeII

222

USM

strong line

intermediate line weak line

cont line

contline line

( )remember equivalent width 1 ( ) ,

area of profile under continuum, dim[ ] = Angstrom or milliAngstrom, mÅ

corresponds to width of saturated profile ( ( ) 0) with sa

H HW d R d

H

W

R

me area

223

USM

Use weak metal lines

to derive v sin i:

Convolve theoretical line

with rotational profile.

Convolve finally with

instrumental profile

(~ Gauss) according

to spectral resolution

Convolution with rotational and instrumental

profile conserves equivalent width!!!

sin

i

to

observer

material

moves

away from

obs.

-> lower freq.

material

moves

towards obs.

-> higher freq.

Determine projected rotational speed v sin i

224

H - log g and Teff

Iso-contours of equiv. widths for H(from model grid), for solar Helium abundance and (very) thin winds

degeneracy of profiles:

(almost) identical lines for

Teff

=40,000 and log g=4.0

and

Teff

=25,000 and log g=3.2

wings of Balmer lines

(Stark-broadened)

react strongly on electron-

density (as a function of ) => perfect gravity indicator

to derive Teff

, log g and YHe

,

at least 3 lines have to be fitted in parallel

(if no wind is present):

Hdefines log g (for given Teff

) HeII/HeI define T

eff (for given log g)

absolute strength of He lines define YHe

usually, wind emission

has to be accounted for

(profiles shallower)

225

Measured equivalent widths

Balmer lines HeI HeII

H 1.99 4387 0.32 4200 0.25

H 4471 0.86 4541 0.31 4922 0.46 4686 0.27

Note: Hand HeII 4686 mass-loss indicators

Result: Teff

30,000 K, log g 3.0 ...3.2,

YHe 0.10 ... 0.15, log Q -12.8

Fit diagram for YHe

= 0.1 (best fit at log Q = -12.8) Fit diagram for YHe

= 0.15 (best fit at log Q = -12.8)

Note: Wind parameters can be cast into one quantity

For same values of Q (albeit different combinations of

Mdot, v

and R* ), profiles look almost identical!

Fit diagram constructed from model grid with

eff

He

20, 000 K < T < 50,000 K with T = 2,500 K

2.2 < log g < 4.5 with log g = 0.25

-14 < log Q < -11 with log Q = 0.3, Y 0.10, 0.15, 0.20

1.5

*

log( )

MQ

R v

Coarse fit - analysis of equivalent widths

226

USM

obs 0 0

obs 0

obs 0

max 0m min

0 0

v( )1 ; line frequency in CMF

v( ) > 0 : blue side

v( ) 0 :

Absorption/Reemission in

atomic = flu

id f

DOP

r

am

r

PLER-EFFECT

ed side

v1

c

!!!

NO E

e

T :

r

c

r

r

maxat ,

o o

P Cygni profile formation and v∞

227

Note: interpretation of vmax ≈ v∞ (wind) requires

large interaction probability ~ 1-exp(-τ), i.e.,

optical depth τ must be large at large radii and

low densities ????

USM

obs 0 0

obs 0

obs 0

m max 0 min

0 0

( )1 ; line frequency in CMF

( ) > 0 : blue side

( ) 0 :

Absorption/Reemission in

atomic = fluid

fr

DOPPLER-EFFECT!!!

NOTE

red side

c

:

m

1

a e

v r

c

v r

v r

v

maxat ,

o o

P Cygni profile formation and v∞

5

m

1530v 2.998 10 1 3, 480 km/s

1548

O3III(f*) (LMC)

228

NV Si IV CIV

Determination of

terminal velocity from

UV-P Cygni profiles

observation with IUE

(International Ultra-

violet Explorer) no longer active

229

USM

H H

H

HeII

HeII HeII

HeI/HeII

HeI

HeI

HeI

HeI/HEII

HeI

___ Hydrogen

...... Helium I

- - -Helium II

HeII

H

HeII

indicated lines

used for fits

derived parameters

Teff

= 31,000 K

log g = 3.17

log Q = -12.87

YHe

= 0.10

= 1.0

with v

= 2050 km/ s

we have 1.5

*log( / ) 7.9M R

Fine fit - detailed comparison of line profiles

230

USM

IF you believe in stellar evolution

use evolutionary tracks to derive M from (measured) Teff

and log g => R

transformation of conventional HRD into log Teff

- log g diagram required

problematic for evolved massive objects, "mass discrepancy":

spectroscopic masses (see below) and evolutionary masses not consistent,

inclusion of rotation into stellar models improves situation

IF you believe in radiation driven wind theory

use wind-momentum luminosity relation

IF you know the distance and have theoretical fluxes

(from model atmospheres), proceed as follows

filter

9 -1 -2 -1

0

0

2.5 log + const

spectral response of photometric system

absolute flux calibration

0 corresponds to 3.66 10 erg s cm Å at 5, 500 Å outside earth's atmosphere

V S d

S

V

isophotal

0

filter filter filter

9

2

* sun

filter

*

wavelength such that ( ) , 2895 for Johnson V-filter

const = 2.5 log(3.66 10 2895) 12.437

2.5 log

5log

+ const10 pc

R 29.

V

S d S d S d

R RM S d

* V

-1 -2 -1

theo

f

th

ilt r

o

e

e

if R in solar units, M the absolute visual brightness (dereddened!) and

V 2.5 log 4 with the Eddington flux in units of [erg

553 ( )

s cm Å ]

V

H S d H theoretic

M

a

V

l

Determination of stellar radius –

if it cannot be resolved

231

USM

d from parallaxes (if close) or cluster/ association/ galaxy membership (hot stars)

(note: clusters/ assoc. radially extended!)

For Galactic objects, use compilation by

Roberta Humphreys, 1978, ApJS 38, 309 and/or

Ian Howarth & Raman Prinja, 1989, ApJS 69, 527

Back to our example

HD 209975 (19 Cep): Mv

= -5.7

check: belongs to Cep OB2 Assoc., d 0.83 kpc V = 5.11, A

V = 1.17 => M

V = -5.65, OK

From our final model, we calculate Vtheo

= -29.08 => R = 17.4 Rsun

Finally, from the result of our fine fit, , we find

Finished, determine metal abundances if required,

next star ....

but end of lecture …

remember relation between MV

and V (distance modulus)

5(1 log ) , distance in pc, reddeningV V V

M V d A d A

1.5

*log( / ) 7.9M R

6

sun0.91 10 M / yrM

232

intermezzo: stellar atmospheres in practice · 2014. 1. 27. · usm stellar atmospheres in practice...

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