Basic Model Ingredients

From Chapters 1 and 2 we know that low- to intermediate-mass stars (1 to 8 M⊙) are found to evolve along the Asymptotic Giant Branch (AGB). These stars are surrounded by large, extended dust shells and are characterized by pronounced time variations. This is particularly true for the main constituents of the AGB – Miras and Long-Period Variables (LPVs) – the light curves of which exhibit a more or less well-defined periodicity, in this way showing a kind of an oscillating behavior. This is assumed to be caused by pulsations of the deeper layers driven by kind of a κ mechanism (see e.g., Section 1.4.4). By these internal pulsations, hydrodynamic waves are generated that travel outward into a medium with decreasing density and temperature, causing the waves to increase in amplitude and finally grow and steepen to shock waves.

These shock fronts moving outward through the atmosphere have a significant bearing on the actual local thermal and chemical state of the shell, producing either favorable or unfavourable conditions for grain nucleation and growth, respectively. By means of these processes, a complex interplay between the internal pulsation, the dynamics of the circumstellar shell, and dust formation and growth is induced, the nonlinear treatment of which allows a reliable understanding not only of the detailed shell dynamics and its particular wind characteristics but also of its detailed spectral appearance, as illustrated in Figure 16.1, where the causal interplay of the various processes that govern the local and global dynamic shell structure is sketched.

The absorption and scattering properties of dust grains strongly depend on the ratio of the particle size to the wavelength of the wave interacting with a grain. Generally it is assumed that most of the grains in circumstellar dust shells are much smaller in size than 1 μ m but that some fraction of the grains has sizes up to a few microns (cf., e.g., Jura 1996). This assumption is based on (1) the observation that circumstellar grains seem not to strongly scatter radiation in the visual and infrared wavelength regions, but that scattering becomes important in the ultraviolet (UV) region (cf. Kruszewski et al. 1968; Serkowski and Shawl 2001) and (2) the theory of scattering of electromagnetic radiation by small particles that shows particles to strongly scatter radiation only if their size is comparable with or larger than the wavelength of radiation. The typical wavelengths of the radiation emitted by stars and their circumstellar dust shells considerably exceed the size of most of the grains, and one can determine the extinction properties of the dust in the limit case of small particles. Only if one is interested for some reason in the UV part of the spectrum has one to consider the case of grains bigger than the wavelength, but even then the particles are not very big compared with the wavelengths. In the theory of circumstellar dust shells there is fortunately no need to consider really big grains, and one avoids the problems encountered by calculating their extinction properties.

Basic Ingredients for Reliable Shell Modeling

Any attempt at a reliable modeling of a circumstellar dust shell has to rely on the mutual coupling of matter and radiation, where with respect to the matter one has to distinguish between gas and dust, making such an object, in principle, a coupled three-component system: gas, dust, and radiation, which requires an adequate treatment of each of the components and a consistent three-component approach to the system as a whole. In Part II, various approximations were outlined that were appropriate at different levels of approach for treating the gas, the dust, and the radiation complex, respectively.

In those cases where the gas and the dust components are dynamically tightly coupled (see Section 3.5.6), a one-fluid description for the matter is justified with regard to the hydrodynamic part of the problem. The same holds for the thermodynamical description, if the collisional coupling between the various material components is efficient enough to establish local thermodynamic equilibrium (LTE) such that there results a common temperature shared by all material components. Thus, in many circumstellar dust shells, conditions prevail where a one-component description of both the dynamics and the energetics of the matter is appropriate.

Although, in a historical view (see Section 1.1), R Coronae Borealis stars – usually addressed as R CrB stars – were the first kind of objects observed exhibiting pronounced occultations that seemed to appear episodically in their light curves (Figure 18.1). Despite the considerable endeavor in particular in the last century to physically and chemically reveal this conspicuous stellar behavior, there is still no final explanation of these spectacular occultations based on a consistent modeling satisfying all observational and theoretical aspects.

This unsatisfactory situation also concerns the conclusive physical explanation of the occultation process itself, as, in a more general sense, classification of the real evolutionary state of R CrB stars, all exhibiting a dramatic atmospheric hydrogen deficiency (εH,R CrB/εH,⊙ ≤ 10−3) accompanied by an unusual large carbon abundance (εC,R CrB/εC,⊙ ≥ 3 to 10), constitutes characteristic features of this small yet important class of spectacular objects. R CrB stars in total seem to populate an extremely narrow stellar mass range between 0.8 and 0.9 M⊙ yet exhibit a broad distribution of their effective temperature, ranging from 3,500 K, which indicates a spectral type R (e.g., S Aps), up to values around 15,400 K, which fits to spectral type early B (e.g., MV Sgr). In this range, the standard star R CrB with Teff ≃ 7,000 K occupies some intermediate position corresponding to a spectral class F.

Any treatment claiming a realistic description of a circumstellar dust shell has to take into account at least five physically and technically rather different complexes:

1. • Hydrodynamics

2. • Thermodynamics

3. • Radiative transfer

4. • Chemistry

5. • Dust condensation

By their combined action, these complexes determine the local physical behavior and global spectral appearance of the circumstellar dust shell. These fundamental complexes and their mutual causal interplay are outlined in Figure 3.1. The coupling indicated by arrows is rather tight, making any consistent realistic modeling an extremely nonlinear problem with regard to both a reliable physical and chemical description and the appropriate mathematical and numerical treatment.

Figure 3.1 displays the general situation for a typical circumstellar dust shell and thus provides a principal frame containing the main ingredients of any reliable approach, yet it does not show which level of description in each box and for each coupling of the different boxes has to be adopted for an appropriate and consistent theoretical and structural modeling of a specific situation. For this aim, the main complexes and their essential physical and chemical couplings, highlighted in this figure, will be outlined in detail in the following chapters.

General Scenario and Historical Background

In the year 1783, the astronomer Eduard Pigott discovered that a star in the constellation Coronae Borealis decreased rapidly in brightness within about a month. Finally, the star disappeared, but after some period of darkness, the light curve of this object recovered within about 500 days and then remained fairly constant for nearly a decade until another obscuration phenomenon repeated (Pigott 1797). Since that time, regular observations have confirmed this star – now named R Coronae Borealis (RCrB) – as an irregular variable representing a small, very specific class of objects all exhibiting this episodic brightness variations and other very similar and peculiar stellar characteristics, with the striking feature that the atmospheres of these stars are entirely dominated by the presence of helium and carbon but show nearly no hydrogen. Based on this finding, in 1934 E. Loreta suggested that the obscuration phenomenon of R CrB stars might be the result of an episodic avalanche of soot formation in the carbon-rich atmospheric environment (Loreta 1934), due to which the stellar brightness is dramatically reduced by extinction, an interpretation strongly supported by J. A. O'Keefe five years later by vapor-pressure arguments based on thermodynamic calculations (O'Keefe 1939).

Theoretical Description of Growth Processes

The basic process for the growth of a dust grain is the collision of a species from the gas phase with the surface of a dust grain. In some fraction of such collisions, translational energy of the gas-phase particle is consumed by exciting vibrational states of the solid, and the particle is captured into a bound state of a surface oscillator. The particle is then bound to the surface of the solid. This process is called adsorption; it is depicted schematically in Figure 12.1. If the particle is not adsorbed, it is scattered back into the gas phase. As a result of adsorption, the surface of any solid embedded in a gaseous environment always is covered with various kinds of particles from the gas phase (also called adatoms).

The freshly adsorbed particles initially are only weakly bound to the surface because adsorption occurs only rarely at a site where the incoming particle immediately forms a strong chemical bond with surface atoms of the solid. In most cases the particle is bound by weak van der Waals forces (at low temperatures) or electrostatic interactions between charged and electrostatically polarizable atoms or molecular groups.

The atoms at the surface of the solid are in a state of permanent vibration due to the thermal excitation of lattice vibrations. By interaction with the vibrating substrate, the adsorbed particles are excited to vibrations relative to the particles at the surface of the solid.

The formation of dust in circumstellar environments is essentially a pure chemical problem. The conditions, however, under which chemical reactions occur in such environments are quite different from the conditions under which chemical reactions proceed in the laboratory.

First, particle densities in circumstellar shells generally are lower than in the laboratory by about 10 decades. This means that chemical reactions in circumstellar shells proceed much more slowly than in the laboratory, often so slowly, that reaction conditions change considerably before a reaction has run to completion. Especially the cooling of matter in an expanding stellar outflow may increase reaction time scales dramatically if activation-energy barriers are involved. This necessitates in the case of circumstellar environments that we consider in detail how reactions proceed in time in an environment where reaction conditions continuously change during the course of the reaction. The low densities additionally disable the stabilization of a complex of colliding particles by transferring the bond energy of two particles to a third particle while undergoing a transition into a bound state. Such ternary reactions are often essential in reactions under laboratory conditions but are completely negligible under circumstellar conditions. Important reaction routes under laboratory conditions thus are inaccessible in circumstellar shells.