Double periodicity of light variations is shown in different ways in stars of different types. So the light curves of μ Cephei of RV Tauri-type giant stars are composed of more or less fast waves superposed on slow ones, such that between the duration of the short and long cycles there is a correlation.

Multiperiodicity has been recognized as an important aspect of β Canis Majoris-type variability ever since the intensive observations of the “classical” members of this group by Struve and his coworkers in the early 1950’s. About half the stars observed by Struve exhibited a radial-velocity variation whose period was several hours, but whose amplitude was modulated with a period of the order of days. This quickly became known as the “beat phenomenon,” and was interpreted as an interference between two nearly equal short periods.

The present state of our knowledge concerning the excitation mechanisms of β Cephei pulsation is reviewed. First we shall discuss whether or not oscillations of β Cephei stars can be explained by the conventional pulsation theory. After showing that the explanation by the conventional theory might be impossible, we shall discuss other specific excitation mechanisms suggested so far.

The peculiar A stars were first noted as peculiar because they exhibit anomalously strong lines of one or more of the elements Si, Cr, Mn, Sr, and Eu. These diverse objects are all called Ap stars because their assigned spectral types fall predominantly in the range B9-F0. Recent studies have suggested, however, that the Ap stars should be subdivided into two groups. The stars characterized by enhanced lines of Mn, and usually Hg as well, differ from the remaining Ap stars in composition and binary frequency; there are no confirmed spectrum, photometric, or magnetic variations in any Hg-Mn stars; and the upper limit on the magnetic field strengths of the Hg-Mn stars is about 200 gauss. Because of these differences, many observers (e.g. Sargent and Searle 1967; Preston 1971b; Wolff and Wolff 1974) have suggested that the Hg-Mn stars are fundamentally different in their properties, and possibly in their origin and evolution, from the SiCrEuSr stars. We shall be concerned today only with this latter group of stars, and in the review that follows only members of the SiCrEuSr class of stars will be referred to as Ap stars.

This paper will be concerned with those variables lying on the cool side of the cepheid strip in the region of the HR diagram occupied by the red giant and supergiant stars. The blue edge of the red variable region for old disk population stars has been found by Eggen (1973b) to be at (R-I)K˜ 0.9 (log Teff˜ 3.53). According to Eggen (1973a, 1973b), all giant stars cooler than this vary, although the amplitude is very small (ΔV~0.2 mag) near the blue edge. In the halo population the blue edge is considerably hotter at (R-I)K˜ 0.5 (log Teff˜ 3.62). On the other hand, some small amplitude red variables (eg. R Dor, AK Hya, IQ Her) have been found at the low temperatures (RK-RK > 1.6) where large amplitude red (Mira) variables exist. From these results, it appears that the position of the blue edge deoends on some unknown stellar parameter(s). Auman (1971) has suggested that the helium abundance may be involved.

The scope of this review will be to summarize our knowledge of Cepheids at the present time concentrating primarily on the observational results obtained but at the same time relating these to the relevant results of pulsation and evolution theory. Of the many previous reviews of pulsating stars we would mention especially Payne-Gaposchkin and Gaposchkin (1938), Ledoux and Walraven (1958), Christy (1966a) and Cox (1974) as providing an excellent background to the theoretical and observational aspects of the subject. Because of these reviews we will concentrate on the more recent results obtained in connection with Cepheids. At the end a special section is devoted to the double-mode Cepheids as they are particularly relevant to this Colloquium.

In this review of the situation with regard to the multiperiodic Cepheid variables, our subject matter is divided into four parts. The first discusses general causes of pulsation of Cepheids and other variable stars, and their locations on the H-R diagram. For this section we draw upon the work during the past 10-15 years of J. P. Cox, Baker, Kippenhahn, A. N. Cox, King, Christy, Castor, Stobie, Stellingwerf, Davey, Iben, and Tuggle, mostly with the small amplitude linear nonadiabatic radial pulsation theory. In the second section we review the linear adiabatic and nonadiabatic theory calculation of radial pulsation periods and their application to the problem of masses of double-mode Cepheids. Contributions discussed are by Cogan, J. P. Cox, King, Stellingwerf, Petersen, Hansen, and Ross. Periodic solutions, and their stability, of the nonlinear radial pulsation equations for Cepheids and RR Lyrae stars are considered in the third section. This research has been done by Stellingwerf with previous development of methods by Baker and von Sengbusch and current work by A. N. Cox and Davey at Los Alamos. In the last section we give the latest results on nonlinear, nonperiodic, radial pulsations for Cepheids and RR Lyrae stars. This work has been done by Stellingwerf, King, A. N. Cox, J. P. Cox, and Davey.

In this review paper I concentrate exclusively on the multiple periodic RR Lyrae stars and I do not wish to discuss the general problems of cluster variables. There are some excellent review papers on them (e.g. Preston, 1964).

The majority of the RR Lyrae stars repeat their light curves from cycle to cycle with astonishing regularity. It was, however, discovered already in the early days of variable star research that some of them showed conspicuous changes in the height of maxima of their light curve and simultaneously the time of maxima could not be fitted in by a linear formula.

The subject “Multiperiodic RR Lyrae Stars” encompasses two entirely different types of phenomena: 1) long-term modulation of amplitude known as the Blazhko-effect and 2) simultaneous excitation of several radial modes. In the first category the observational material is now quite extensive, but very little theoretical analysis has been undertaken. In the second case the results of pulsation theory are applicable, but only one such object has been found. Much progress in our understanding of RR Lyrae stars has been made in recent years, however, and many results do have a bearing on the present problem. Several excellent reviews of current pulsation theory are available (see Iben 1971c, or Cox 1974a,b). I will therefore restrict my comments to those results bearing directly on choice of mode for RR Lyrae stars.

On reviewing the observational literature of the recent past on RRs (or dwarf Cepheid or AI Velorum) and δ Scuti stars, it appeared that most present controversy is centered on three specific questions:

1. (1) Are δ Scuti stars really periodic, or only quasi-periodic?

2. (2) Are tidal modulations responsible for the slow cyclic variations observed in many of these stars, or not?

3. (3) Are there any real physical differences between the members of these two groups, or are they actually the same kind of object?

I would like to restrict this review primarily to these questions; and since some of our recent observational results bear directly on questions CI) and (2), and indirectly on question (3), I hope I will be forgiven for describing briefly here some of our current relevant findings.

Theoretical problems in connection with δ Sct and RRs type variables are reviewed. All evidence shows that δ Sct stars are normal main-sequence or early post main-sequence stars. Results of linear stability analyses of δ Sct models agree well with observations, the presence of non-variables in the strip being understandable in terms of helium diffusion. Excitation of non-radial modes may also occur; comparison of observed multiple periodicities with the theoretically derived period patterns for radial and non-radial modes is discussed as a means to distinguish between these possibilities.

It is not known whether RRs variables are of the same nature as δ Sct stars or in a late evolutionary state with low mass. RRs variables pulsate in radial modes, and recent linear stability analyses seem to agree well with observations for both normal and low masses. The reason for the difference in amplitude between RRs and δ Sct stars is not known. Non-linear investigations may provide important information in the near future. The observed period ratios of RRs variables indicate for most of the double mode pulsators a slight deficiency in heavy elements and a normal mass; but more detailed investigations of this problem are needed.

The purpose of this paper is to contribute to answer the question : why among A stars are there variable and “constant” stars? To do this we will first review the observational properties of these stars, then discuss the theoretical models based on the interplay between microscopic diffusion and rotation, and finally discuss the “unsolved” questions !

The normal dwarf nova (DN) outburst is characterized by a rapid rise, a brief stay near maximum and a slower decline to minimum light; the whole sequence lasts in general for 4 to 6 days, but can take as long as 10 or 12 days in some stars. While not strictly periodic, the DN outbursts are quasi-periodic in the sense that the mean periods derived from intervals sufficient to contain at least 50 outbursts (i.e. a few years for the shortest period systems, up to 30 years for longer period systems) remain sensibly constant. It should be added, however, that in the case of U Gem there is evidence for an increase in its recurrence period which may indicate some evolutionary change: for the first 50 years of observation of U Gem the mean period was 96. 5 days and for the second 50 years it was 107.6 days (Mayall, 1957); during the past four years the mean interval has been 291 days (with only one outburst in 1974).

The current status of the theory of non-radial oscillations of white dwarfs and hydrogen shel1-burning degenerate stars, in the linear, quasi-adiabatic approximation, is reviewed. Relevant aspects of the thermal structure of such stars are summarized, and several problem areas requiring further investigation are identified. Among the most interesting and important of these are the question of the existence of g modes in white dwarfs; the nature of the excitation mechanisms for non-radial oscillations; the possibility of tidal excitation; the problem of mode-coupling between pulsations and convection; and the question of the effects of a solid, crystallizing core.

This paper will review reasonably thoroughly and comprehensively the many observational properties of the remarkable RS CVn-type binaries. In addition this paper will show that many of these same properties are observed in other types of binary systems.