AGB - Asymptotic Giant Branch wykład IV Model atmosphers of AGB stars Ryszard Szczerba Centrum Astronomiczne im. M. Kopernika, Toruń (56) ext. 27

„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 Atmospheres 2.Dynamics - pulsations - dust formation

The modelling of AGB Stars atmospheres – Basic equations 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

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 System of rate equations

Equation of radiative transfer (time-independent) The modelling of AGB Stars atmospheres – approximations: static case System of rate equations (statistical equilibrium) 2.Momentum: Hydr. equilibrium with rad. pressure 3.Energy: radiative equilibrium

The modelling of AGB Stars atmospheres – approximations: static case – further simplifications LTE for gas: Saha equation (ion. st.) Boltzman equation (l.p.) 3.Energy: radiative equilibrium

The modelling of AGB Stars atmospheres – heating and cooling b.f. – blocking factor  = 0.4 =>  T=500K;T=3000K

The modelling of AGB Stars atmospheres – heating and cooling: temperature behaviour. R i – R at which cont. is formed (T eff ). Inserting into radiative equilibrium Equation: For  = const. (grey atm.) R i /R=1/1.5 =>  T=-500K

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)

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

AGB Stars: Physics and characteristic conditions – Scale Height due to turbulence Hydrostatic equilibrium Scale Height: Turbulent pressure  t ~ 0.5 Turbulence can extend atmosphere by about 50%

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? For HCN mass is 27 times larger and ~1.5 km/s From Doppler shift:  ~ /c =

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 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)

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).

AGB Stars: grids of static models.

The newest models: Gustafsson et al. (2005)

AGB Stars: grids of static models.

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.

The modelling of AGB Stars atmospheres – Pulsations

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).

The modelling of AGB Stars atmospheres – Pulsations 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.

AGB Stars: dust formation The grain will grow until there is condensable material around and the condensation rate (R cond ) > the evaporation rate (R evap ). a – grain size; n – „monomers” number N-number of „monomers” in grain Dynamical equilibrium defines: equilibrium degree of condenstaion Homogenous grain growth

AGB Stars: dust formation Too simlified picture: different „monomers” can stick to the grain, grain drift was neglected, sticking probability is not 1, T dust may not be equal to T gas. Note that grain with a=0.01  m contains ~10 8 atoms!!! A growth rate. a 1 – monomer radius

AGB Stars: dust formation Assumptions: enough of condensable material; T=const => R growth =const; for t=0 (reduced) a~0 C is a monomer in C-rich stars. Monomer size a 1 = [cm], a/a 1 = ; R cond >>R evap For v=10 km/s  r= cm density drops by fact. 2 t puls ~1 year

AGB Stars: pulsations and dynamical models (Hoefner)

AGB Stars: dust formation Assumptions: enough of condensable material; T=const => R growth =const; for t=0 (reduced) a~0 C is a monomer in C-rich stars. Monomer size a 1 = [cm], a/a 1 = ; R cond >>R evap For v=10 km/s  r= cm density drops by fact. 2 t puls ~1 year

AGB Stars: Physics and characteristic conditions – Pulsations and dust formation 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 Assumptions: dust and gas a coupled (no drift). We demand that the radiative acceleration is large enough to overcome gravity  bulk =2 g/cm 3 ; = cm -1 M=1M o, L= L o =>  d /  > Overestimated since other forces also counteract gravity

AGB Stars: model atmospheres for pulsating stars Hoefner et al. (1997) Bowen (1988) Sedlmayer Hoefner