The question of whether or not the period changes observed in some classical cepheids are due to evolution across the HR diagram remains controversial. On the one hand, it is known that the stars are indeed evolving across the diagram and are therefore changing their radii and must therefore be changing their periods. On the other hand, only a fraction of cepheids appear to have period changes, and some of these seem to be in the form of sudden jumps. Is it then that the evolutionary changes are too small to be detected, and the observed changes due to something else? This note explores the question.

In spite of the fact that Cepheid variables pulsate quite regularly their pulsation period remains constant only in the first approximation. The pulsation period is subject to variations because of stellar evolution. The calculations made by Hofmeister (1965) predicted that the evolutionary period changes of classical Cepheids should be observed on a time scale of several decades or longer. No detailed analysis of the observed period changes has been made since Hofmeister's fundamental theoretical work.

We have observed in the ultraviolet the hot companions of the Cepheids SV Per, RW Cam, SY Nor and KN Cen. The study of the absolute and relative intensities reveals that all, except the companion for KN Cen are evolved stars which should fit on almost the same mass track as the Cepheid. We find however that with generally accepted reddening values the companions of at least SV Per and RW Cam are too faint. Either the Cepheid loops are more luminous than presently calculated or the reddening is larger than presently accepted.

The variability of HR 7308 was discovered by Breger (1969), and Percy et al (1979) found this star to be a new cepheid. The correct period and the variation of amplitude were first recognized by Burki and Mayor (1980 a, b). Further studies of this star were published by Percy and Evans (1980), van Genderen (1981), Breger (1981), Fernie (1981), Burki et al (1982) and Henriksson (1983). Here the up-to-date radial velocity observations made with the spectrophotometer CORAVEL are presented.

The O-C curves — widely used for investigating the period behaviour of variable stars — are generally constructed under the assumption that the period changes are linear in time. If it is indeed so, the main trend of the diagram should outline a positive or negative parabola and β values are determined to characterize the curvature of the parabolae, i.e. the rate of change of the period. This rate, however, even in the case of ω Cen does not indicate exact parabola (Belserene, 1973).

The effects of nonlinear, nonlocal, diffusive convection have been included in the structure equations of pulsational hydrodynamic analyses of a series of RR Lyrae models. We find a well-defined red edge of the unstable region that depends on mode of pulsation. It is shown that this result has strong implications for the nonlinear behavior of RR Lyrae stars.

Eleven double-mode (dm) RR Lyrae stars, with periods midway between the periods for the c-type and ab-type RR Lyrae stars, have been identified in the Draco dwarf galaxy by reanalyzing the photometry of Baade and Swope (1961) for 35 stars. The stars are V11, 72, 75, 83, 112, 138, 143, 156, 165, 169 and 190, three of which previously were noted as dm RR Lyraes by Goranskij (1982). The methods of Stellingwerf (1978) and Stobie (1970) were used to find the periods. The period ratios, periods and amplitudes suggest that the stars radially pulsate simultaneously in the fundamental and first overtone modes. The beat masses, estimated from the P1/P0 vs. P0 diagram (Petersen 1973), using as calibration the King Ia (Y=0.299, Z=0.001) models of Cox, Hodson and Clancy (1983, hereafter CHC), are Mbeat/Mo=0.65 for nine stars, = 0.60 for V75 and = 0.55 for V165. If the mass loss rate prior to arriving on the horizontal branch is proportional to the metal abundance (Stobie 1971, CHC), V75 and V165 should be more metal rich than the other nine dm RR Lyraes. In the period-amplitude (P-A) diagram, at a given amplitude, V165 has a smaller period shift relative to the standard M3 line than do the higher mass stars. Subsequently, the correlations of Preston (1959) and Sandage (1982) suggest that it is more metal-rich than the other nine dm RR Lyrae stars in Draco. Furthermore, reanalysis of the P-A and period-mean magnitude relations of all the RR Lyrae stars in Draco shows evidence for a widespread range in the metal abundances. Stars with large period shifts relative to the M3 relation are found to be more luminous than stars with smaller period shifts. The frequency of variable amplitude (ie. Blazhko effect) RR Lyrae stars is greatest for the ab-type stars with short periods. The brightnesses of the highest maxima appear to fit the P-A relation, and the amplitudes of the lowest maxima are variable, with V123 being an extreme example. Figure 3 of Szeidl (1975) shows the analogous situation for M3 RR Lyrae stars with the Blazhko effect. These findings suggest that the amplitude variability is related to the mode-switching activity of the dm RR Lyrae stars.

New BV photographic photometry for 56 RR Lyrae variables in M15, combined with original measures by Sandage, Katem and Sandage (1981), yield data of improved precision which enable a rediscussion of the RR Lyrae characteristics and Oosterhoff problem.

FG Sge is a unique stellar object possessing a number of unusual properties which are as follows: (1) FG Sge is a core of a planetary nebula, (2) at least since 1894 this star has crossed the HR diagram from the left to the right, (3) the atmosphere of FG Sge was enriched in s-elements, (4) FG Sge is a pulsating variable the period of which progressively increases from 15 days in 1962 to 110 days in 1980.

During the seventeenth century, in the wake of the discovery of the solar differential rotation, some scientists argued that stellar variability was a direct consequence of axial rotation, the spinning body showing alternately its bright (unspotted) and dark (spotted) hemispheres to the observer (Brunet 1931). Although this idea did not withstand the passage of time, it is nevertheless an interesting one because it is clearly indicative of the kind of fascination stellar rotation has aroused since its inception. And yet, at this writing there is no longer any doubt that spherically symmetric models do explain the major observed properties of stars. Moreover, if one excepts the very early and very late moments of a star's lifetime, the effects of rotation on stellar structure are apparently dynamically unimportant (e.g., Tassoul 1978, hereafter T.R.S.; Moss and Smith 1981, and references therein). What is the purpose, then, to discuss the role of rotation on the main-sequence and post-main-sequence phases of stellar evolution?

The different mechanisms by which mixing can take place in stellar interiors are considered : the classical Rayleigh-Benard instability with penetrative convection and over-shooting, semi-convection, gravitationnal and radiative settling, turbulent mixing. The latter mechanism is thoroughly described, from the driving force of turbulent mixing to its influence on stellar structure, stellar evolution and the analysis of the corresponding observationnal data.

Turbulent mixing has to be considered each time the building up of a concentration gradient takes place, either by gravitationnal or radiative settling or by nuclear reactions. Turbulent mixing, as a first approximation, can be described by an isotropic diffusion coefficient. The process is then governed by a diffusion equation. The behaviour of the solution of the diffusion equation needs some explanation in order to be well understood.

A number of examples concerning surface abundances of chemical elements are given (3He, 7Li, Be, 12C, 13C, 14N), as well as a discussion of the solar neutrinos problem.

The building up of a µ-barrier, which stops the turbulence allows stellar evolution towards the giant branch and explains nitrogen abundance at the surface of giants of the first ascending branch.

Turbulent mixing is also of some importance for the transfer of angular momentum and has to be taken into account for explaining the abundance of the elements in Wolf-Rayet stars.

The principal question studied in Mestel (1953) was: under what conditions will the E-S circulation through the radiative envelope ensure that a Cowling-type main sequence star stays effectively homogeneous? The E-S velocities v are of order (L/Mg)(Ω2r/g), so that the circulation time tcirc is ⋍ t(ΩKelvin-Helmholtz)/(Ω2 r/g), where the bar indicates a mean value. The circulation advects nuclear-processed material from the convective core, and so sets up horizontal variations δµ in mean molecular weight µ. The condition of hydrostatic support requires corresponding horizontal temperature variations; the consequent local breakdown in radiative equilibrium yields a “µ-current” (analogous to the E-S “Ω-current”) with velocities vµ estimated roughly as (L/Mg) (δµ/µ). The total velocity vμ is the linear superposition v∽Ω + v∽µ, each constructed using first-order perturbation theory. The µ-distribution that fixes v∽µ is itself determined by the circulation:

The method used by Tassoul and Tassoul assumes a “turbulent viscosity” which is important in determining the dynamics, but is unimportant in heat transport. This approximation is inconsistent, the argument is as follows:

The rotation is expanded about uniform rotation in a series.

Several authors (myself included!) have suggested that turbulent mixing takes place in some, if not all, stars, and in particular that such mixing can explain the low solar neutrino flux. This turbulence is thought to be caused by differential rotation produced by braking due to angular momentum loss in a stellar wind, and/or to the effect of meridional circulation currents in redistributing angular momentum. Whilst such instabilities may exist even in the presence of a stabilizing distribution of chemical composition, they do not necessarily cause mixing. To be effective in mixing, the energy available to the instability be it differential rotation or any other mechanism, has to be sufficient to lift the helium rich matter in the interior of the star to the outer regions. This requires where Erot is the kinetic energy in rotation, Eg the gravitational energy, τth the thermal time scale and τnuc the nuclear evolution time scale of the star.

Following Schatzman and Maeder (1981) we compute the evolution of the sun with partial mixing by hydrodynamic instabilities. Instead of simply assuming a turbulent diffusion coefficient which is a constant multiple of the viscosity, we incorporate some of the properties of hydrodynamic instabilities. This puts limits on the amount of diffusion that can be obtained, and makes it dependent on time and position in the star.

We consider stellar evolution in low mass stars (1–3 Mo) near the main sequence with the hypothesis that mild turbulence is present within the all star. Turbulent transport of the elements is modeled by diffusion equations where the diffusion coefficient is chosen to be D = R✶eν where ν is the kinematical viscosity and R✶e is a Reynolds number. We consider the effects of the growth of the gradient of the mean molecular weight on turbulence. The main consequences of diffusion on stellar evolution are (1) an increase of the life time near the main sequence and (2) a change of the radial distributions of chemical species (12C, 13C, 14N, 160) (figure 1). The inhibition of the turbulence, when the gradient of mean molecular weight reaches a certain critical value, allows the evolution towards the red giant branch. When stars evolve towards the giant branch, chemical species are dredged up to the surface. At this stage models with and without diffusion, predict substantially different surface abundances (in particular the 12C/13C and C/N ratios). Comparison between models and the available data on giants during the first dredge-up show that abundance anomalies can be explained if turbulent mixing is present during the main sequence phase (figure 2).

It is well known that light elements like lithium are good indicators of the hydrodynamical behaviour of the outerlayers of the stars. As they are nuclearly destroyed at low temperature, i.e. close to the surface, their surface abundances reflect the nature of the transport process at work between the photosphere and the nuclear destruction region.

It has been suggested by Schatzman (1977) that the empirical lithium content/age relationship found by Herbig (1965) could be explained by exchange of matter between the convective zone and deeper layers, where the rate of lithium burning is higher, by turbulent diffusion. The claim by Tassoul and Tassoul (1983) that such a turbulence is necessary to obtain a self-consistent description of stellar rotation and meridional circulation has given some momentum to Schatzman's proposal, as well as other successes of the turbulent diffusion hypothesis (Schatzman et al. 1981).

1, 2, Visiting astronomer, Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique of France, and the University of Hawaii.

The question which has been raised in many chapters of this book is about the existence of constraints on stellar evolution coming from related topics like cosmology or in the case of the present chapter the chemical evolution of the galaxies. As it will be seen in this contribution it seems wiser to consider that chemical evolution of galaxies is indeed related to the problem of stellar evolution discussed here but is not going to provide as many constraints on it as one would expect. The purpose of this presentation is therefore to outline the principal relations between these two fields and to discuss the impact of some recent works on them.