Models for M and C giant stars differ from those for hotter stars by having complicated state equations and opacities dominated by lines from diatomic and polyatomic molecules. In addition many cool giants have atmospheres which are extended, and which cannot therefore be adequately modelled using a plane-parallel approximation. Mira variable stars have atmospheres which are even more extended due to the regular passage of shock waves through their atmosphere. In this review we will discuss recent modelling of such atmospheres and show comparisons with observations of variable and non-variable M stars.

Tests of the model atmospheres of red giant stars by photometric observations and angular diameter measurements (including limb-darkening) revealed that the present models reasonably represent the thermal structure of the continuum-forming region of red giant atmospheres. On the other hand, tests by line spectra (including line-blanketed fluxes) revealed that classical model atmospheres of red giant stars cannot yet satisfactorily represent the line-forming region. Recent progress in observations also provide some tests of improved models incorporating non-LTE or sphericity effects. More importantly, recent observations offer the means to test some new approaches such as: hydrodynamical simulation of photospheric convection, modeling atmospheres with thermal bifurcation, and incorporating the effects of outer atmosphere having inhomogeneous structure consisting of warm chromosphere and cool molecular dissociation zone.

Low resolution spectroscopic and photometric studies of Magellanic Cloud AGB stars have shown that the M → S → C sequence is the result of the third dredge-up of 12C and s-process elements in AGB stars of low mass. In this paper, data on the chemical compositions of normal Galactic M → S → C stars are reviewed and shown to be broadly consistent with expectations for the third dredge-up on the AGB.

Tc is detected in many AGB stars providing unambiguous proof that recent nuclear s-processing and mixing (the third dredge-up) has taken place. During this evolutionary episode the atmospheres of AGB stars are progressively enhanced with helium burning products (primari ly l2C) and s-process elements as they evolve from M->MS->S->SC->C stars. The increase in s-process elements can be traced most easily by the presence and increasing strength of the Tc I lines accompanying this progression. We also find that the third dredge-up phase is accompanied by an increase in the amplitude of light variation since no non-variable or low amplitude variable M, MS, SC, or S (with one exception) have Tc lines. M star Mira variables show Tc if P > 300d (low mass Pop I stars). No Pop II star is known to have Tc. Nor do supergiants show Tc I lines. The significant fraction of MS, S and C stars that do not show Tc, are surmised to be cooler analogues to the Ba II stars, i.e. binaries. The source to provide the neutrons for the s-process is most likely the 13C(a, n)16O reaction since most of the stars in which we observe Tc are thought to have masses less than 3 solar masses.

We have measured the relative radial velocity of Arcturus using the HF absorption cell technique on 43 occasions from 1981 through 1985. The range of our velocities is 500 m s-1, which is much larger than our estimated internal errors (typically 10 m s-1). This confirms the radial velocity variability of Arcturus that has been previously reported by our group and others based on shorter observational time spans.

A model atmosphere abundance analysis has been performed for a large number of G and K giants in a limited set of lines. The study was aimed at measuring the frequency of mild barium stars among GK giants in order to provide a test for the evolutionary status of the mild barium stars.

The spectrogram of FG Sge obtained by the 6-m telescope in 1986 is analyzed. The iron peak elements are underabundant relative to the sun, and the heavy elements (Z>39) have abundances up to 3 dex (1, 000 times) higher than in the Sun.

High dispersion spectra of the high velocity star HD 39853 show that it is a slightly metal-poor ([Fe/H]=−0.5±0.1) giant of the old disk population, with exceptionally strong lithium lines. The abundance of Li, derived by synthetic spectra of the 6103.6 and 8126.4 Å lines, is log N(Li)=2.8±0.2. Abundances for the other observed elements are typical for mildly metal-poor giants: oxygen is slightly overabundant ([O/Fe]=+0.25±0.15); the C/N ratio is large (10±2); the 12C/13C ratio is small (6.6±1.2); and light elements (Na, Mg, Al, Si, Ca and Ti) are enhanced with respect to Fe by about a factor of 2.5. Observed s-elements (Zr and La) are not overabundant. Finally, no variation in the radial velocity of the star were detected at a level of 1 Km s-1.

Hydrogen abundances (or H/He ratios) are hard to determine in stars cooler than the Sun because the Balmer lines, when visible at all, are formed largely in the chromosphere, while the bands of CH and NH are often strongly saturated and badly blended with atomic lines. A few stars (the hydrogen-deficient carbon or HdC stars) are known to be extremely hydrogen-deficient, as their G bands of CH are absent despite an overabundance of carbon. A means of detecting less extreme cases of hydrogen deficiency would improve our understanding of red giant evolution. Minor variations in hydrogen content may be expected as the result of the mixing of processed material to the surface, and more radical changes might result from a star's shedding its entire hydrogen-rich envelope, say in the course of binary-star evolution.

We have investigated the atmospheric abundances of upper AGB stars of the SMC, searching in particular for evidence of the hypothetical envelope burning process. To this end we have computed synthetic spectra with varying C, N, and 0 abundances selected by considering the effect of the processes of the 3rd dredge up and envelope burning of a degree sufficient to prevent C star formation. The synthetic spectra (covering 0.5 μm to 2.5 μm) were analysed for observable effects of these two atmospheric enrichment processes. By analysis of band strengths of TiO, CO and CN we have found that substantial envelope burning is detectable for stars with Teff ≥ 3000 K but not below, due to the temperature dependence of CN bands. The synthetic spectra were compared to near-infrared and infrared observations of a small sample of SMC upper AGB stars thought to be prime candidates for the occurrence of envelope burning. This comparison indicated that envelope burning, of the extent considered here, is not occurring in these stars but rather the spectra are consistent with mild C enhancements produced by the 3rd dredge up alone.

In the atmospheres of cool oxygen-rich stars, water vapor is the most important molecular absorber. Yet no laboratory or theoretical spectrum exists for water vapor of sufficient detail to be useful for radiative transfer calculations in a stellar atmosphere. Because of the complexities of its bent triatomic structure, this deficiency is not likely to the be resolved soon. We have, therefore, undertaken the preparation of a synthetic spectrum for water vapor.

We have carried out abundance analyses of four low-mass supergiant variable stars (the RV Tauri or RV Tau-like variables AI Cmi, RU Cen, and U Mon, and the Type II Cepheid Kappa Pav) and two Population I Cepheids (CO Aur and V378 Cen). We used model atmospheres in which hydrostatic equilibrium, plane-parallel geometry, and local thermodynamic equilibrium (LTE) were assumed. Discussion of the results, and of published analyses of additional low-mass variables, leads to the following conclusions. (1) The Population I Cepheids show normal, solar elemental abundance ratios (except for the CNO elements, which have been altered by hydrogen burning), lending some support to the validity of the above assumptions for analyses of luminous variable stars. (2) The low-mass variables show metallicities ranging from solar down to [Fe/H] values typical of thick-disk and, in a few cases, of halo stars. (3) Most low-mass variables show a systematic underabundance of the heavy s- and r-process elements. In a few cases this may indicate that the stars were initially of extremely low metal content, and are now hydrogen deficient. However, most of the variables do not appear to belong to the halo population, nor do they show other abundance patterns seen in halo stars. The origin of these underabundances, and their apparent confinement to luminous variables, are difficult to understand in the context of nuclear processing. (4) The heavy-element underabundances correlate with second ionization potential in a manner suggesting that they are non-LTE phenomenan arising from overionization by Lyman-continuum photons. Why a similar effect is not seen in Population I Cepheids is unclear, but may be related to their generally weaker hydrogen emission. (5) Several low-mass variables, including RU Cen and V553 Cen, show carbon enhancements and solar s-process abundances. Relative to the majority of the Type II variables, these stars are s-process enhanced, and we argue that they are related to the Ba II and CH stars.

TX Psc is one of the very few carbon stars for which spectroscopic data from the blue toward the near infrared can be found in the literature. We present calculations of the first synthetic spectra that are in agreement with all the major spectral features in the observed spectrum, including the H∼ peak and the 3 μm band. The spectra are calculated on the basis of available observational estimates of the basic stellar parameters and model atmospheres that include absorption from HCN, C2H2 and C3 in the opacity calculation.

The recent discovery (Jorissen and Mayor, 1988) that not only Ba II but also S stars appear to belong predominantly to binary systems with periods of several hundred days raises the question of the similarity between these peculiar red giants and symbiotic stars. That question has been addressed by looking whether symbiotics have enhanced s-element lines in their spectra. In most cases, definite conclusions are hampered by the composite nature of such spectra. Nevertheless, in the case of the “yellow symbiotic” BD -21.3873, the spectral classification criteria provided by Keenan and Wilson (1977) allow to assign without ambiguity the type GIII-II to BD -21., 3873, the symbiotic nature of that star being apparent only through weak Fe II emission lines. Some heavy element lines (such as Sm II λ4220.7, λ4221.1 and λ4566.3) are clearly enhanced. Comparison with a spectrum of a G3Ib supergiant ensures that luminosity is not responsible for that effect. Moreover, some of these spectral features are also enhanced in Barium stars, as seen on a comparison spectrum of a Barium Star.

A flux curve has been computed using a preliminary, spherically symmetric model for a 3000 K, oxygen rich, giant star. The model was computed using the opacity sampling method with an improved frequency set, improved molecular equilibrium data and an improved set of opacities. In addition, a continuum flux curve is computed using the same model and only continuum opacity sources. The relative and to some extent the absolute blanketing used to compute both the model and the flux curve derived from the model may be illustrated by dividing the normal flux curve by the continuum flux curve. This same procedure is used to illustrate the blanketing in an observed star by dividing the observed flux curve by the continuum flux curve. When this is done, the blanketing in an observed flux curve may be compared with the blanketing in a model. When this comparison is made, it is obvious that the treatment of blanketing in the “new” flux curve, computed using the spherically symmetric model and using new parameters, is superior to the flux curves based on earlier models. This is especially true in the regions of the fundamental, the first overtone and the second overtone of Co. Also, the new water vapor opacity is much improved. The new water vapor opacity is based on actual measurements of high temperature water vapor. Correct representation of water vapor opacity is extremely important for oxygen rich stars because it forms a psuedo continuum because of its many lines. The TiO opacity does not fit the observations well. When the spherically symmetrical model flux curve is compared directly to an observed flux curve, the new flux curve gives a better fit than do flux curves computed from previous models. There is still (at least for the non-Mira stars) a serious flux excess in the model flux curves at 1.6 microns in the region of the H minus b-f and f-f crossover. However, this excess is not as great for the spherically symmetric model as it is for earlier plane parallel models. It is not determined if this improvement is due to spherical symmetry or due to the new model parameters.

It is now well known that Nature can make Carbon stars at lower luminosities than can (human) theorists. A number of workers, stimulated by this challenge, have been attracted to the problem. In this paper I review recent evolutionary models of relatively low mass AGB stars, with emphasis placed on the mixing of carbon to the stellar surface. In particular I discuss some recent improvements in the physics used to construct stellar models. These topics include: breathing pulses of the convective core found during core helium exhaustion; the effects of carbon recombination; the occurrence of semiconvection in the region between the two nuclear burning shells; and the importance of mass loss. Recent calculations have successfully produced models of low luminosity Carbon stars. The strengths and weaknesses of these models will be contrasted.

After a brief description of the developments of the theory of s-process nucleosynthesis, the difficulties recently encountered in envisaging reliable astrophysical conditions for obtaining a solar-system distribution of s-isotopes are discussed. In particular, while the reaction 22Ne(α, n)25Mg may account for the nucleosynthesis of the weak s-component in massive stars, it fails to reproduce the main s-component in intermediate mass stars. The efficiency of the alternative reaction 13C(α, n)160 occurring in low mass stars during recurring thermal instabilities of the He shell is then analyzed. It is shown that, contrary to previous expectations, the 13C source well reproduces the main component, provided that realistic physical conditions are assumed for the temporal behaviour of the pulse and the effect of the light n-absorbers (especially 12C) is properly taken into account. The results satisfactorily compare with the constraints of the classical s-analysis. Key observational evidences also appear to be in agreement with this scenario.

There are several types of Peculiar Red Giants (PRG's) which have enhanced s-process elements and/or carbon, but which cannot be explained by mixing during helium shell-flashing in the late stages of Asymptotic Giant Branch (AGB) evolution. These are the Ball, CH, sgCH, and the hotter R-type carbon stars. All these PRG's have absolute magnitudes which range down to zero and fainter. The Ball stars (Bidelman & Keenan 1951) are G-K giants whose spectra have strong CN and CH bands, and in extreme cases bands of C2 become noticeable. In addition they have strong lines of s-process elements such as Ball and SrII. The CH stars (Keenan 1942) are Population II equivalents of the Ball stars, and show similar spectral features, but with weaker metal lines, and usually stronger carbon bands. Bond (1974) introduced a new class of late F and G stars named the CH subgiants, which he suggests are fainter than the classical CH and Ball stars. They appear to be of mixed population (Luck & Bond 1982), and they are probably subgiants which will eventually evolve up the giant branch to become classical CH and Ball stars. The R-type carbon stars, at least the hotter (R0-R4) ones, differ from many of the other peculiar red giants in that they do not show enhanced s-process element abundances. The R stars have strong carbon bands, but relative to the N-type carbon stars they have less blue and ultraviolet absorption in their spectra. They range in absolute magnitude from near zero, similar to the Ball stars, up to several magnitudes brighter than this.

After a brief commentary on the place of “peculiar red giants” in the overall scheme of stellar evolution, an outline is given of the various possibilities for post asymptotic giant branch (AGB) evolution. The behavior of a post-AGB model star is crucially dependent on where in a thermal pulse cycle the mass of the hydrogen-rich envelope is reduced to such an extent that departure from the AGB must follow on a thermal time scale. If departure from the AGB occurs while the model is still burning hydrogen, post-AGB behavior depends on the mass of the helium buffer zone (- zone containing predominantly helium which has been processed through the hydrogen-burning shell following the last thermal pulse on the AGB). If departure occurs at an arbitrary time during the hydrogen-burning phase, then: (1) in - 25% of all cases, the post-AGB model will experience a final helium shell flash, and, in consequence of additional mass loss, may become a non-DA white dwarf; (2) in - 60% of all cases, the model will cease burning hydrogen when the mass in its hydrogen-rich envelope is reduced to ∼ 10-4Mʘ and will evolve into a DA white dwarf; and (3) in - 15% of all cases, the model will experience a final hydrogen shell flash, but the outcome with regard to spectroscopic type is unclear. If departure from the AGB occurs while the model is burning helium, the result is either the same as in option (3) just described, or mass loss during the post-AGB helium-burning phase may turn the star into a non-DA white dwarf.

Binarity seems to be a feature shared by various classes of Peculiar Red Giants (PRG). This observational fact has led to the general agreement that those stars result from a mass transfer originating from an asymptotic giant branch companion star.