agb - asymptotic giant branch wykład iv model atmosphers of agb stars ryszard szczerba centrum...

AGB - AGB - Asymptotic Giant BranchAsymptotic Giant Branch

wykład IVwykład IVModel atmosphers of AGB starsModel atmosphers of AGB stars

Ryszard Szczerba

Centrum Astronomiczne im. M. Kopernika, Toruń

[email protected]

(56) 62 19 249 ext. 27

http://www.ncac.torun.pl/~szczerba/

„„Asymptotic Giant Branch”Asymptotic Giant Branch”

Harm Habing, Hans Olofsson (Eds.)

A&A Library, 2004 Springer-Verlag

AGB Stars: the atmospheres• From microscopic properties of gas and

radiation in AGB stellar atmospheres

• To the macroscopic properties and overall structure of the atmospheres.

1. The modelling of AGB Star Atmospheres2. Dynamics - pulsations - dust formation

The modelling of AGB Stars atmospheres – Basic equations

v

vvvvvvv

gas

vvgas

SdvSJdvFc

u

ugpu

ueueut

dvFc

gpuut

u

ut

QfluxQdensityt

;)(41

)(

2

1

2

1

1)(

0)(

sinkssources)()(

00

22

0

A conservation Eq.

1.Mass:

2.Momentum: (Eq. of motion). RS= Hydr. equilibrium + rad. pressure

3.Energy: e-internal energy; work: -against pressure, -by gravity, -by rad. forces, -difference between heating and cooling

),,(),,(),,(),,(1

tnrItnrtnrtnrIstc vvvv

These 3 Eqs. of mass, momentum and energy conservation must be solved together with:

Equation of radiative transfer

The modelling of AGB Stars atmospheres – Basic equations

ijijijij

ijijijii CRPPnPnun

dt

dn

;)()(

System of rate equations

),(),(),(),(1 2

rIrrrIrr vvvv

Equation of radiative transfer (time-independent)

The modelling of AGB Stars atmospheres – approximations: static case

ijijijij

ijijij CRPPnPn

;0)(

System of rate equations (statistical equilibrium)

dvSJ

dvFc

gp

vvv

vvgas

)(0

10

0

0

2.Momentum: Hydr. equilibrium with rad. pressure

3.Energy: radiative equilibrium

The modelling of AGB Stars atmospheres – approximations: static case – further simplifications

dvSJ

JBS

vvv

vvvvv

vv

v

vv

)(0

;

0

LTE for gas: Saha equation (ion. st.) Boltzman equation (l.p.)

3.Energy: radiative equilibrium

linesvcontvvvvvv ldvJdvB ,.,

00

;

The modelling of AGB Stars atmospheres – heating and cooling

b.f. – blocking factor

= 0.4 =>

T=500K;T=3000K

14

1

1

11

)1()(

)1()(

..;

)(

4

44

4

,.,

4

.,

T

T

T

T

TTT

TTF

fbl

TFTBF

linesvcontvv

vv

contvv

The modelling of AGB Stars atmospheres – heating and cooling: temperature behaviour.

Ri – R at which cont. is formed (Teff).

Inserting into radiative equilibrium Equation:

For = const.

(grey atm.)

Ri/R=1/1.5 =>T=-500K

5.0

0

2

2

.,

84.0

0)(2

1)(

)(2

1

R

RTT

dvR

RTBTB

R

RTBJ

ieff

ieffvvv

ieffvv

contvv

dvJdvB vvvv

00

The modelling of AGB Stars atmospheres – heating and cooling: atmosphere extension

Atmosphere extension is caused by increase in opacity.

From hydrostatic equilibrium and definition of:

Radiatiative forces also affect the structure

Similary to the extension caused by H- the extension is caused by formation of new molecules (like TiO) (pulsations, turbulence)

dvFc

g

d

pd

drdgp

vv

0

1

;

The modelling of AGB Stars atmospheres – heating and cooling: atmosphere extension

AGB Stars: Physics and characteristic conditions – Scale Height due to turbulence

;

)(;;

2*

2

M

TR

mG

kH

ppppr

MGggp

H

turbradgas

Hydrostatic equilibrium

Scale Height:

Turbulent pressure

t ~ 0.5

Turbulence can extend atmosphere by about 50%

HHskmc

m

Tk

MG

RH

p

tst

ttH

t

ttt

5.1/5

22

*

2

AGB Stars: grid of static models• Model atmospheres are used for comparison between computed and observed spectra.

Stellar parameters, abundances. Tests of nucleosynthesis; chemical evolution of galaxies.

• There are 2 problems (even for static case with LTE for gas and radiative equilibrium):Completness of molecular data.Treatment of absorption (resolution).

•How many points do we need to resolve an AGB spectrum?

skmK

T

m

Tk

HH /

30008

8 21

21

For HCN mass is 27 times larger and <v>~1.5 km/s

From Doppler shift:

~<v>/c =5 10-6.

AGB Stars: methods to reduce number of frequencies.

• ODF – Divide spectrum into spectral intervals and transform opacity within each interval into Opacity Distribution Function (e.g. Gustafsson et al. 1975).About 500 intervals are needed to perform integrations in:Equation of motion

Radiative equilibrium dvSJ

dvFc

gp

vvv

vvgas

)(0

10

0

0

ODF method assumes that opacity does not depend on => not applicable in case of AGB stars!!!

AGB Stars: methods to reduce number of frequencies.

• OS – Opacity Sampling (e.g. Ekberg et al. 1986). A few thousand (randomly distributed over frequency) points are needed to get T with accuracy of about 50 K.Almost imposible to improve accuracy (err~1/sqrt(n))The most popular and (relatively) easy to generalize to (non-LTE):

i.e. To solve system of rate equations (statistical equilibrium)

ijijijij

ijijij CRPPnPn

;0)(

AGB Stars: grids of static models.

•Pioneering grid of static LTE models for AGB stars (M-type) – Tsuji (1978).•Other grids for M-stars: Brown et al. (1989)•For C-stars: Qerci et al. (1975), Kurucz (1979), Johnson (1982), Jorgensen et al. (1992), Plez et al. (1992)

•Presently, there are two groups announcing spherically symmetric, static models computed with OS: (Hauschild et al. 2002) and group which uses MARCS code (Edvardsson, Eriksson, Gustafsson, Jorgensen, Plez).


The newest models: Gustafsson et al. (2005)

AGB Stars: static models vs observations

For non-Miras agreement is reasonable: Serote Ross et al. (1996), Alvarez & Plez (1998)

AGB Stars: static models vs observations

For C-stars agreement in the ISO range has been achieved: Jorgensen (2000)

The modelling of AGB Stars atmospheres – Pulsations

•The subsonic motions induced by pulsating

interior generates sound waves.

•In the AGB atmospheres there is strong T and gradient.

•Parts of waves which are deeper (in hotter gas)

move faster => shock waves are generated.

•The strongly decreasing T and in the stellar

atmosphere enhance steepening of the shock waves.

Hs m

Tkc


•Movment of matter which has been hit by shock

wave can be approximated by „balistic” solution

(with variable „g”)

•However, in contrast with pure balistic solution,

the trajectories will not be symmetric (the radiation

pressure is changing).

0

2

0

max

0max 2;

r

GMu

u

u

r

rresc

esc


Hoefner et al. (2003)

AGB Stars: dust formation

•Extended atmospheres of AGB stars are so cool and dense enough that dust can form.

•Kinetic description of grain formation => typical time scales.

KINETIC PICTURE

•With decreasing T increase amount of (complex) molecules.•Formation of cluster is possible (sticking and chemical reactions). This depends on thermodynamical conditions.•At some point when cluster of „critical size” is fomed it is more favorable energetically to add more molecules to it (nucleation theory – derived from chemical considerations).

•Any cluster larger than this „critical size” is a seed nucleus for a dust grain.


•The grain will grow until there is condensable material around and the condensation rate (Rcond) > the evaporation rate (Revap).

0

1)(4

48

22

222

dt

dN

R

RnaRRa

dt

dN

Ram

Tknana

cond

evapevapcond

condH

a – grain size; n – „monomers” number

N-number of „monomers” in grain

Dynamical equilibrium defines: equilibrium degree of condenstaion

Homogenous grain growth


•Too simlified picture: different „monomers” can stick to the grain, grain drift was neglected, sticking probability is not 1, Tdust may not be equal to Tgas.•Note that grain with a=0.01 m contains ~108 atoms!!!

growth

growth

cond

evapgrowth

cond

evapevapcond

Rdt

ad

a

aaNR

dt

dN

N

aa

R

RnaR

R

RnaRRa

dt

dN

3

1

;

;1

1)(4

1

32

331

21

22

A growth rate.

a1 – monomer radius


][102

104.1;106.2)(

)(

4.1)(

)(&10607.6

)(

)(

33

3

1

3

1

3

1

7

3

144

4

1

1

1

0

st

cm

gmn

Hn

Cn

On

Cn

Hn

On

Ra

aRat

tRdtRa

Rdt

ad

growth

HH

growthgrowthgrowth

growthgrowth

t

growth

growth

growth

Assumptions: enough of condensable material;

T=const => Rgrowth=const;

for t=0 (reduced) a~0

C is a monomer in C-rich stars. Monomer size

a1=1.29 10-8 [cm],

a/a1=5 102 ; Rcond>>Revap

For v=10 km/s r=2 1013 cm density drops by fact. 2

tpuls~1 year

AGB Stars: pulsations and dynamical models (Hoefner)


][102

104.1;106.2)(

)(

4.1)(

)(&10607.6

)(

)(

33

3

1

3

1

3

1

7

3

144

4

1

1

1

0

st

cm

gmn

On

Cn

On

Cn

Hn

On

Ra

aRat

tRdtRa

Rdt

ad

growth

HH

growthgrowthgrowth

growthgrowth

t

growth

growth

growth

Assumptions: enough of condensable material;

T=const => Rgrowth=const;

for t=0 (reduced) a~0

C is a monomer in C-rich stars. Monomer size

a1=1.29 10-8 [cm], a/a1=5

102 ; Rcond>>Revap

For v=10 km/s r=2 1013 cm density drops by fact. 2

tpuls~1 year

AGB Stars: Physics and characteristic conditions – Pulsations and dust formation

0

3

2

0

)()(;

),(),(;)(),()(

daanaQa

QconstQa

aQaaCdaanaC

oext

extoext

extextext

•Timescale for grain growth are comparable to timescale of pulsations and to timescale of dynamical changes in the outer atmosphere – non-equilibrium process. •Dust, if formed (low T and „large density”), interact with radiation much more efficiently than molecules.• Radiation pressure on dust my overcame gravitation.• •How much dust is needed to drive a stellar wind?

AGB Stars: dust – radiation interaction

L

M

Q

Gc

L

MGc

Q

daana

dF

dF

L

MGc

r

MG

rc

L

oext

bulkd

bulk

oext

d

bulkd

3

16

44

3

)(3

4

)(

)()(

44

3

22

Assumptions: dust and gas a coupled (no drift).

We demand that the radiative acceleration is large enough to overcome gravity

bulk=2 g/cm3; <Qext>=5 103 cm-1

M=1Mo, L=5 103 Lo => d/

> 1.4 10-3

Overestimated since other forces also counteract gravity

AGB Stars: model atmospheres for pulsating

stars

Hoefner et al. (1997)

Bowen (1988)

Sedlmayer

Hoefner

agb - asymptotic giant branch wykład iv model atmosphers of agb stars ryszard szczerba centrum...

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