We show that the theoretical white dwarf luminosity function depends very much on the assumed phase diagram for the carbon-oxygen dense plasma. Since it is still very uncertain, we compare the two possible extreme cases of complete miscibility and complete separation of carbon and oxygen in solid phase. In the latter case we find that the paucity of low luminosity — log(L/L⊙) ≤ −4.5 — white dwarfs can be explained by the formation of an oxygen core, which releases a large amount of gravitational energy and slows down the cooling rate.

Massive stars (MZAMS ≳ 30 M⊙) develop during their observable hydrostatic evolutionary phases — i.e. central H- and He-burning — three different large scale convective zones, which are: 1) The H-burning convective core, 2) the intermediate convective shell (ICZ) above the hydrogen shell source, which forms at time of hydrogen exhaustion, and 3) the He-burning convective core. The spatial extent of these convective regions, wherein the chemical structure is rapidly homogenised, can be predicted from theory only with a large uncertainty. Different assumptions on the efficiency of these mixing processes in stellar evolution calculations lead to quite different evolutionary pictures for massive stars, especially regarding their Wolf-Rayet (WR) phases. On the other side, many observational data concerning WR stars became available in recent years. For this reason, we attempt to perform a comparison of theoretical evolutionary sequences with observed properties of WR stars in order to derive restrictions on the efficiency of the three mixing processes mentioned above.

A grid of homogeneous stellar models for Wolf-Rayet stars in the mass range from 1 to 60 M⊙ has been computed. For each stellar mass, stars with eight different sets of chemical compositions — from pure helium stars with (Y,C,O)=(1,0,0) to extreme helium poor stars with (Y,C,O)=(0.02,0.11,0.87) — have been calculated in order to investigate the dependence of the stellar structure on the stellar mass and chemical composition. Modern input physics adapted to the exotic chemical composition of the Wolf-Rayet stars has been incorporated in the models, including effects of partial recombination of helium, carbon, and oxygen, and detailed opacity tables for 20 different combinations of the (Y,C,O)-abundances. Furthermore, we estimated the effect of the intense, partly optically thick Wolf-Rayet winds on their apparent effective temperature, using the formalism of de Loore et al. (1982, IAU-Symp. 99, 53) with the parameters: = 3 . 10−5M⊙yr−1, β = 2, and v∞ = 2000 km s−1.

Studies of chemical composition of the atmospheres of F-K supergiants have revealed the existence of two specific peculiarities, deficit of carbon and excess of nitrogen. Anomalous abundances of C and N in yellow supergiants is explained by mixing of surface material with CNO-processed material of stellar interiors (the first dredge-up). Somewhat unexpectedly, F-K supergiants were found to show one more general chemical peculiarity: overabundance of sodium. Fig.1 shows the available values of [Na/H]=log[N(Na)/N(H)]*−log[N(Na)/N(H)]⊙ as summarized by Boyarchuk and Lyubimkov( 1983). Different symbols refer to data of various authors. Fig.1 shows that sodium overabundance increases with the decrease of surface gravity. For small log g it reaches [Na/H] ~ ~ 1. Recently Sasselov(1986) has suggested that this correlation is essentially a consequence of a relation between [Na/H] and a mass of supergiant.

We investigated overstable convective modes of differentially rotating massive main sequence stars. It is examined that the overstable convective modes in a rapidly rotating convective core may excite envelope non-radial oscillations by the resonance coupling between them. Let us denote by q the ratio of the angular velocity of the rapidly rotating core to that of the envelope. Then, it is found that as the ratio q increases, the order of the non-radial g modes resonantly coupled with the convective mode becomes lower, so that the excited g modes come to have shorter periods in an inertial frame. It is also found that when the separation of the frequencies of the consecutive envelope g modes is wider than the resonance width, there occurs the alternation between the strong and the weak resonances, i.e., the resonantly coupled mode becomes overstable or neutral according to the strength of the resonance.

Concerning the scattering of OB stars in the HR diagram (Humphry 1980), the effects of overshooting of convective core (Maeder 1984), mass loss (cf. chiosi and Maeder 1986), and generous stability criterion of semi-convection (Stothers and Chin 1976) have been discussed. Here, we will note the dredge up effect is caused by the sound waves emitted from a convective core.

The sound mode of nonradial oscillation, with the spherical harmonics Ylm(θ, ϕ) and the frequency ω, can exist in the propagation zone, where the bottom boundary locates at the position of , and the upper boundary does near the photosphere. Here, Ll is called as the Lamb frequency, and cs is the sound velocity.

A review is given of rate of mass-loss values in the upper part of the Hertzsprung-Russell diagram. Near the luminosity limit of stellar existance = −10−4 M⊙ yr−1. Episodical mass loss in bright variable super- and hypergiants does not significantly increase this value. For Wolf-Rayet stars the rate of mass loss is larger by a factor 140 than for non-evolved stars with the same Teff and L; for C stars this factor is ten. This can be explained qualitatively. Rotation appears hardly to influence the rate of mass loss except for vrot-values close to the break-up velocity. This is in accordance with theory. We suggest the existence of a Red Supergiant Branch; along that branch mass loss is virtually independent of luminosity. Stellar winds along the upper limit of stellar existence are mainly due: to radiation pressure for hot supergiants (≳ 10 000 K); to turbulent pressure for cool supergiants (3000-10 000 K), and to dust-driven and pulsation-driven winds for cooler stars. The turbulent pressure may originate in largescale stochastic motions as observed in Alpha Cyg. Episodical mass loss, as observed in P Cyg, HR 8752 and other Very Luminous Variables may be due to occasional violent stochastic motions, resulting in a shock-driven episodical mass-loss component.

Modern quantitative spectroscopy of hot stars has two aspects: the analysis of photospheric lines and stellar wind lines. The first one is meanwhile established as an almost classical tool to determine stellar parameters. NLTE model atmosphere and line formation calculations yield Teff, log g and abundances with high precision (see recent reviews by Husfeld (this meeting), Kudritzki (1987), Kudritzki and Hummer (1986), Kudritzki, (1985)). The second aspect, however, the quantitative analysis of stellar wind lines is still at its very beginning. For long time the stellar wind lines have been used to determine mass-loss rates and terminal velocities v∞ only. While these studies were pioneering and of enormous importance, it was also clear that very approximate calculations were done with respect to NLTE ionization and excitation and the radiative transfer in stellar winds. Thus, stellar wind lines could be used only in a more qualitative comparative sense, with no theory behind, which allowed the determination of precise and reliable numbers.

From the analysis of the galactic counterparts of the HS variables, η Car, AG Car and P Cyg we conclude that their atmospheres are near critical physical conditions so that small changes of the mass outflow could produce a large variation of the atmospheric structure and nova-like explosions at nearly constant Mbol.

Eta Carinae is a well-known example of a star with a massive circumstellar nebulous shell. The shell is regarded as a remnant of a great outburst of the star in the last century. η Car parameters remain to be a matter of scientific debates: most often investigators revise stellar temperature T, mass loss rate , the velocity of outflow V. We have worked out several numerical evolutionary models of η Car envelope (the homunculus) paying attention to the dependence on the mentioned parameters. The dependence on the parameters of grain evolution was also considered. For a representative model we fixed: V=500km/s; M=10−1 M⊙/y during the first 15 years of the outburst and slowly decreasing in time, T=20000K. A coupled two-components hydrodynamical and radiation model transfer problem has been solved.

Spectral studies of super- and hypergiants show that the (outward directed) turbulent acceleration approaches the value of the gravitational acceleration for the most luminous stars, which makes their atmospheres unstable.

Advances in theoretical modeling of rapidly expanding atmospheres in the past few years made it possible to determine the stellar parameters of the Wolf-Rayet stars. This progress is mainly due to the improvement of the models with respect to their spatial extension: The new generation of models treat spherically-symmetric expanding atmospheres, i.e. the models are one-dimensional. Older models describe the wind by only one representative point. The older models are in fact ‘core-halo’ approximations. They have been introduced by Castor and van Blerkom (1970), and were extensively employed in the past (cf. e.g. Willis and Wilson, 1978; Smith and Willis, 1982). First results from new one-dimensional model calculations are published by Hillier (1984), Schmutz (1984), Hamann (1985), Hillier (1986), and Schmutz et al. (1987a); more detailed results are presented by Schmutz and Hamann (1986), Hamann and Schmutz (1987), Hillier (1987a,b), Wessolowski et al. (1987), Hillier (1987c) and Hamann et al. (1987). These results demonstrate that the step from zero- to one-dimensional calculations is essential. The important point is that the complicated interrelation between NLTE-level populations and radiation field is treated adequately (Schmutz and Hamann, 1986; Hillier, 1987). For this interrelation it is crucial to model consistently not only the line-formation region, but also the layers where the continuum is emitted. In fact, it is the core-halo approximation that causes the one-point models to fail in certain aspects.

Temperatures, mass-loss rates and luminosities of 30 galactic Wolf-Rayet stars (24 WN, 6 WC) are derived by fitting the observed equivalent widths of He I λ5876 and He II λ5412 and the absolute visual magnitude. A three-dimensional grid (T*-R*-) of model calculations provides the theoretical values.

The results are summarized in Figure 1. The different symbols denote the spectral classification of the individual stars, whereas four groups are distinguished: WNEA, WNE-B, WNL and WC stars. The size of the symbols indicates the mass-loss rates. The uncertainties of the results are estimated to be 0.1 dex in T*, 0.4 dex in and 0.5 dex in L. For the ll stars in common with the sample of Abbott et al. (1986) we find our mass-loss rates to be compatible with their radio flux if the correct ionization equilibrium in the radio emitting region is applied (Schmutz and Hamann, 1986). The model calculations show that for all but the WN2 and WN3 stars helium recombines to He+ before the ions enter the radio-emitting region.

A model atmosphere code that accounts for the special physical conditions in Wolf-Rayet atmospheres (Hamann and Schmutz, 1986; Wessolowski et al., 1987) is used to analyse the spectrum of the Wolf-Rayet star HD193077 (WN5+abs). The stellar parameters are determined such that the profiles of the helium lines He I λλ4471, 5876, He II λ5412, and the absolute visual magnitude are reproduced.

In order to estimate the systematic errors introduced by the model assumptions, we perform some test calculations. Instead of the velocity-law exponent of ß=1, another fit is obtained with ß=0.5 (Fig. 1). This fit yields an effective temperature of about 3000K higher than with ß=1. In order to simulate the effect of the suspected hydrogen content, a further fit (Fig. 2) is made when one free electron per helium atom is added artificially. This has only marginal influence on the derived temperature (+100K). Thus, we conclude that our model assumptions may introduce a systematic error of the order of 5000K.

The helium spectrum of the WN5 star HD 50896 (EZ Canis Majoris, WR6) is studied. Our aim is to establish a technique which allows the determination of the parameters of a Wolf-Rayet star from a systematic analysis of its spectral lines. Since the method of “iteration with approximate Lambda operators” became available for application to expanding atmospheres (Hamann, 1986, 1987), we are now able to compare observed spectra to realistic model calculations (Hamann and Schmutz, 1987; Wessolowski et al., 1987).

We consider the non-LTE spectral formation in a spherically expanding atmosphere. The velocity field v(r) is specified in its supersonic part by the usual analytical law with the parameters v∞ (final velocity) and the exponent ß=1. The temperature structure is derived from the assumption of radiative equilibrium, but only approximately evaluated for the grey LTE case. The atmosphere is assumed to consist of pure helium. The model atom has a total of 28 energy levels, among these 17 levels of He I. The line radiation transfer is treated in the “comoving frame”.

Wolf-Rayet (WR) stars are the descendants of massive stars that have lost their hydrogen rich envelope. Recently more accurate data on WR stars have become available: mass-loss rates (van der Hucht et al. 1986), radii and luminosities (Underhill 1983, Nussbaumer et al. 1982).

It may therefore be worthwhile to investigate if combinations of observed parameters shed some light on the structure of the extended stellar wind of WR stars.

In many WR stars the photosphere is situated in the stellar wind. We assume that the wind is stationary and isotropic. Further we assume a velocity law v(r)=v∞(1−Rs/r)β where v∞ is the terminal velocity of the wind in km/s, Rs is the radius where the wind acceleration starts and β > 0 is a free parameter. We can then easily compute the level R in the wind where the photosphere is located (de Loore et al. 1982): R is the solution of the equation 6.27 10−9 τat R v∞/ = fβ(Rs/R) where τat is the optical depth at the photosphere (2/3 or 1), (>0) is the mass loss rate in M⊙/yr and fβ > 1 is a slowly varying function (Doom 1987).

Of/WN stars have been introduced by walborn (1977). These luminous stars are particularly characterized by the appearance of both the characteristic of emission lines of HeII and NIII and strong HeI and [NII] emissions. These objects have become particularly interesting due to the discovery of their link to the S Dor variables (Stahl et al. 1983, Stahl 1986, Wolf 1987). They are supposed to represent key objects in the evolution of massive stars being the immediate progenitors of WN stars. We observed the Of/WN transition type stars R127, R84, and S61 of the LMC with high dispersion spectroscopy in the optical and satellite-UV range. R127 is at present undergoing an S Dor-type outburst. In 1986 R127 became the visually brightest star of the LMC until it was surpassed by SN1987A in Feb. 1987. The spectrum in the optical range during the extraordinary bright outburst phase is distinguished by numerous P Cygni type lines of singly ionized metals. Since the beginning of the present outburst the spectral type has gradually become later and the equivalent spectral type as derived from our latest CASPE spectrum taken in 1986, August 22 was middle A. The IUE-spectrum of R127 during maximum is characterized by copious lines of singly ionized metals formed in the wind. A detailed disucssion of the optical and satellite UV high dispersion spectrograms of R127 taken during its present extraordinary bright outburst phase is forthcoming in Astronomy and Astrophysics (Wolf et al. 1987a).

The galactic distribution of Wolf-Rayet stars is redetermined. In the solar neighbourhood within d < 2. 5 kpc the WN/WC number ratio is 0.54, indicating a larger influence of mass loss and convectlve overshooting than most present evolutionary models account for. The WR density gradient In the solar neighbourhood is constant. From a comparison of WR galactocentric distances and WR cluster and association membership, we find indications that WCE stars descend from WNL stars at R > 6.5 kpc, and that WCL stars descend from WNL stars at R < 8.5 kpc.

Be stars are well known to be rapid rotator and to show intermittent emission-line activity. Such an activity is now interpreted as an abrupt mass-ejection and formation of a cool disk in the equatorial region and then its gradual disappearance mass-loss is called as episodic mass-loss.

Rapid rotation and mass-loss connection in Be stars was suggested for the first time by struve(1931), which necessarily leads to the requirement of break-up velocity in Be stars. However, Vsin i statistics suggests almost all Be stars are well below the break-up velocity. The additional forces have been searched for so far; i.e., stellar wind, magnetic field, mass accretion, and so on. At the moment, none of them can succeed in explaining the episodic mass-loss in Be stars.

A set of equations of stellar wind in a new steady state in spherically symmetry is presented. This equations are available also for the deep interior of stars, whereas the usual equations can be applied only to the surface region. The new equations have a variable mass flux which becomes zero at the inner boundary of the mass flow. The velocity also reduces zerj), whereas it diverges at r=0 in the usual continuity equation 4∏r2ρ v=constant. In the surface region, the present equations approach the usual equations.