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Astron. Astrophys. 337, 149-177 (1998)

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1. Introduction

Before turning into planetary nebulae, low to intermediate mass stars (1-[FORMULA]) are found to evolve along the Asymptotic Giant Branch (AGB). In this phase, mass loss dominates over nuclear burning and finally terminates the AGB evolution by reducing the mass of the stellar envelope below some critical limit. High mass loss rates between 10-6 and 10[FORMULA] are very often detected, mostly by excess continuum radiation in the infrared, originating from thermal emission of circumstellar dust (e.g. Herman et al. 1986; Jura 1987; Bedijn 1987; Schutte & Tielens 1989), and by the presence of molecular rotation lines from different molecules seen in emission at submm, mm, and cm wavelengths (e.g. Knapp & Morris 1985; Netzer & Knapp 1987).

It is now generally accepted that the mechanism responsible for such high mass loss rates during the AGB evolution is based on the efficiency of radiation pressure on dust grains. Shock waves generated by Mira-type stellar pulsations are essential for accelerating the outflow from the stellar surface to the sonic point where the gas becomes cool enough at sufficiently high densities to allow heavy elements to condense as grains. The dust grains belong to one of two different types: (1) silicate-type grains, found around so called "oxygen stars" with an abundance ratio C/O [FORMULA] 1, and (2) carbon-based grains around "carbon stars" with C/O [FORMULA] 1. These dust particles efficiently scatter and absorb photons, extracting momentum and energy from the stellar radiation field. The acquired momentum is transferred to the gas by collisions between dust particles and gas molecules (see e.g. Gilman 1972; Salpeter 1974; Kwok 1975; Goldreich & Scoville 1976), while the absorbed energy is re-radiated at infrared wavelengths.

Up to now, hydrodynamical models of dust driven winds on the AGB do not generally include the "long-term" variations of the stellar parameters and mass loss rate (on stellar evolution time scales of the order of [FORMULA] to [FORMULA] years , as opposed to stellar radial pulsation time scales of several 100 days ), although it is well known that the stellar luminosity and (very likely) the mass loss rate undergo significant variations when so-called "thermal pulses" occur on the upper AGB (e.g. Iben & Renzini 1983).

In fact, recent observations have clearly revealed the existence of so called "detached shells" around a number of AGB stars, which has been taken as strong evidence that mass loss may be temporarily interrupted (e.g. Willems & de Jong 1988; Chan & Kwok 1988; Zijlstra et al. 1992; Olofsson et al. 1996; Izumiura et al. 1996, 1997). However, the exact evolutionary behavior of the mass loss remains unknown. Two general scenarios have been proposed that relate mass loss interruptions to the rapid luminosity variations occurring when AGB stars undergo a thermal pulse cycle. Both of them are based on the observation that all carbon stars with excess emission at [FORMULA] 60 µm are optically visible. In the first scenario (favored by e.g. Zuckerman 1993 or Olofsson et al. 1996) the far infrared excess is explained by a rather low steady mass loss, only interrupted by a short high mass loss peak, possibly caused by a thermal pulse. In the second one (e.g. van der Veen & Habing 1988; Egan et al. 1996) a quite long period of high mass loss followed by a phase of greatly reduced mass loss in the aftermath of a thermal pulse produces a detached shell and corresponding excess emission, provided that the mass loss minimum lasts long enough to allow for sufficient expansion and cooling of the dust shell before the high mass loss resumes and obscures the star again. In the former case it seems that the mass loss "pulse" does not last long enough to create a detached dust shell containing enough matter to account for the observed emission at long wavelengths (Egan et al. 1996), a conclusion supported by the present work.

From the theoretical side, it is presently not possible to derive mass loss rates along the AGB from first principles (except for some very cool, high luminosity carbon stars; see Dominik et al. 1990, Arndt et al. 1997). Existing stellar evolution calculations resort to semi-empirical mass loss prescriptions (Vassiliadis & Wood 1993; Blöcker 1995), which, due to the occurrence of thermal pulses, lead to considerable variations of the mass loss rate on time scales which are short compared to the flight time of a gas parcel through the circumstellar envelope. Hence, the situation is far from steady state and time-dependent hydrodynamics/radiative transfer calculations taking into account the "long-term" effects of stellar evolution are needed for a physically consistent interpretation of the observed spectral energy and surface brightness distributions of mass losing AGB stars. This is particularly important for understanding the formation and structure of multiple shells, revealing part of the the previous mass loss history. Ultimately, analysis of the observed properties of the circumstellar shells of a large number of AGB stars should allow to check the presently adopted mass loss laws and, if necessary, to derive empirical corrections.

A first brief report of time-dependent hydrodynamical wind calculations similar to those presented here was given by Vassiliadis & Wood (1992), who used a simple one-component hydrodynamics code (Wood 1979) ignoring the details of dust radiative transfer. The circumstellar density and velocity structures they find seem to be very similar to those obtained in the present work. However, Vassiliadis & Wood (1992) did not compute the emergent spectral energy distribution for a comparison with observations.

We have developed a new code which is suitable to treat the time-dependent two-component radiation hydrodynamics problem of dust driven stellar outflows in spherical symmetry. The code includes a detailed solution of frequency-dependent radiative transfer in the dust component and provides synthetic emergent spectra. It is designed to take into account the evolutionary changes of the stellar parameters and the resulting variable mass loss rate through a time-dependent inner boundary condition for the system of partial differential equations describing the model of the circumstellar shell. In a previous paper (Steffen et al. 1997a, henceforth Paper I), we have tested this code and have used it to study the hydrodynamical properties and spectral energy distributions of steady state solutions for a variety of different parameters. In the present work we apply the code to the time-dependent case, elaborating on preliminary investigations published before (Szczerba & Marten 1993; Schönberner et al. 1997, 1998; Steffen et al. 1997b; Steffen & Szczerba 1997).

In Sect. 2 we describe the modifications (relative to the equations given in Paper I) and additional assumptions incorporated into our code for the treatment of the time-dependent case. The input data taken from stellar evolution calculations with mass loss (Blöcker, 1995) are briefly discussed in Sect. 3, while the main results of our computations for one particular evolutionary track, but assuming different dust properties, are presented in Sect. 4. In Sect. 5, we compare the observed distribution of AGB objects in the IRAS two-color-diagram with the color evolution computed from our dynamical models, and show that the observed spectral energy distribution of the prominent carbon star S Scuti (Groenewegen & de Jong 1994; Olofsson et al. 1996) and the detached dust shell of the another well-known carbon star, Y CVn, (Izumiura et al. 1996) are explained in a natural way by our models, as is the rapid transition to the post-AGB phase. Finally, in Sect. 6, we summarize the main conclusions of this work and close with some remarks on possible future improvements of the AGB winds models.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998