The dynamical behaviour of the envelopes of four asymptotic-giant-branch stars of composition (X, Z)=(0.68, 0.02) and luminosity log L/L⊙ = 3.41, 3.60, 3.85 and 4.14 has been studied. At log L/L⊙ = 3.41, the envelope pulsates in the first-overtone mode and exhibits properties similar to those of the Mira variables. The more luminous models were all found to pulsate in the fundamental mode while simultaneously undergoing violent envelope relaxation oscillations. A significant amount of mass loss occurred from the model with luminosity log L/L⊙ = 3.60. It is suggested that a redgiant star undergoing envelope relaxation oscillations would be recognised as a symbiotic star. In the two most luminous models, a distinct outward-moving shell containing almost all the hydrogen-rich material in the star is formed. A suggestion is made that a further increase in luminosity would completely eject this shell to form a planetary nebula.

The Magellanic Clouds are well known as being very suitable for observing the various stages of stellar evolution. During the last few years, I have been studying the RR Lyrae variable stars in each of the two Clouds. Some first results were reported at IAU Colloquium No. 21 in 1972 (Graham, 1973). Here, I would like to update these results on the basis of more recent data and to comment on some of the characteristics of the field RR Lyrae stars in each system. Periods and light curves are now available for 63 RR Lyrae stars in a 1° x 1.3° field centered on the cluster NGC 1783 in the Large Magellanic Cloud (LMC) and for 62 stars in a 1° x 1.3° field centered on the cluster NGC 121 in the Small Magellanic Cloud (SMC). Both ab and c type variables are represented and, viewed individually, the Cloud RR Lyraes are identical in characteristics to those known in our Galaxy. Studied as groups, however, there are small but significant differences between the RR Lyrae stars in each system. The following four specific features seem to be emerging from the study.

In spite of the rather good agreement between the theory of stellar evolution and the observations, there exist some difficulties when one compares closely the sequences of open star clusters and the theoretical isochrones. Several, if not all, of the old open star clusters seem to be concerned, especially those which are accurately measured, namely Praesepe, NGC 2360, 752, 3680 and M67. The problem concerns the gap occuring in the HR diagram at the end of the phase of hydrogen burning in the core; it corresponds to the phase of hydrogen exhaustion (or of overall contraction). The sequence of M67 has been studied by Racine (1971) and Torres-Peimbert (1971). The well apparent gap is located farther from the zero-age main sequence than indicated by the models and the hook towards a larger Teff predicted during this phase is not observed. Differences in chemical composition may not be held responsible for these anomalies. From Torres-Peimbert's models, it may be assumed that neither solar type, nor super metal rich composition are able to reduce the discrepancies. As a further illustration, let us mention the case of NGC 752. In Table I, the main features related to the gap are examined: the disagreement, like in M67, essentially concern features 1 and 2. The observations are based on a recent study of Grenon and Mermillod (1973) and on Bell's data (1972). Bell has also mentioned the existence of discrepancies. As in M67, the gap is too far from the zero-age main sequence and does not present any sudden turning towards a larger Teff.

A series of stellar models were evolved, all with a total mass of 0.65 M⊙, an initial carbon-oxygen core of mass 0.60 M⊙, an intermediate helium mantle and an outer hydrogen-rich envelope with mass varying from case to case. Although the most hydrogen-rich cases resulted in red giants, cases with ≲ 0.01 M⊙ in the hydrogen envelope evolved at high surface temperature. The early stages of development of these models are similar to observed central stars of planetary nebulae. The later stages (when the nebula should be very thin optically) still have a high luminosity; the relevance to ‘ultraviolet stars’ will be discussed.

A summary of the main results and problems concerning explosive and eruptive stars is given. The way in which the observations are related to the internal structure problems is briefly explained. Suggestions for further work are made: (1) dynamo theory, supersonic turbulence, stability of the envelope of T Tau stars; (2) rate of exchange of matter in close binary as the final clue to the thermal runaway theory of novae.

Those novae known to be binaries generally have orbital periods of the order of hours, exceptions being the atypical recurrent novae T CrB and RS Oph, which have giant companions and probably much longer periods. Since the orbital period in a semi-detached system relates to the mechanism of current mass exchange and also to the extent to which the primary evolved before mass exchange took place at an earlier stage, it is of interest to see if any classical novae are in more widely separated systems.

This communication concerns the star RR Telescopii, which has all the characteristics of a slow nova – a range in amplitude greater than seven magnitudes, a spectral type at maximum of F, and a decline through a nebular stage of increasing ionization level (e.g. Thackeray, 1955). RR Tel was seen as a variable before outburst, but little is known about this variable apart from its period of 387 days, although doubts have been expressed about its being a red variable (Payne-Gaposchkin, 1957). Dr Thackeray has Radcliffe spectrograms of RR Tel from soon after maximum to the present. On some of the more recent of these, bands of TiO have become visible, presumably as the hot star has faded, and Dr Thackeray and I interpret these as meaning that the original variable is still there and is an M giant. Thus RR Tel is in a binary system containing a giant M star and a hot companion and such a system cannot have an orbital period of hours like the classical novae mentioned above.

In 1972, Dr Glass and I examined RR Tel in the infrared between 1.2 and 20 µ. The energy distribution does not resemble the cool star that might be expected, but is exactly like that of free-free radiation over the whole wavelength range. The puzzle is that the infrared is two orders of magnitude stronger than we would predict from the optical spectrum for free-free radiation.

The high speed photometric observations of dwarf novae during outbursts that led to the discovery (Warner and Robinson, 1972) of rapid pulsations have been extended to other objects. Pulsations are usually seen near outburst maximum; an exception is VW Hyi which showed oscillations on December 25, 1972, when it was well down the declining part of its light curve.

The periods of the oscillations are found to vary; in most cases a rapid change from one discrete mode to another is observed. The dwarf nova stays in one oscillation mode for about an hour, and changes mode over a time of about 10 min. In the case of the VW Hyi observation, the period changed continuously from 28 s to 34 s over a time of 4 h, during which the apparent brightness decreased by 40%.

The existence of many closely spaced oscillation modes suggests that the dwarf novae are oscillating in non-radial modes. This is supported by observations of the 71-s oscillations of DQ Her: during eclipse a phase shift is seen indicating an l = 2, m = 2 mode of oscillation. In UX UMa, the 29-s oscillations show an opposite phase shift from that in DQ Her and indicate an l = 2, m = −2 mode.

A period-luminosity relationship can be deduced for the oscillations in dwarf novae, and this is in good agreement with recent calculations on quadrupole oscillations of hot white dwarfs made by Osaki and Hansen (1973).

Pulsations have so far been detected in the following objects: Z Cam (17 s), SY Cnc (24.6), CN Ori (25), KT Per (27), UX UMa (29), VW Hyi (28–34), CD −42° 14462 (29 and 30), AH Her (31).

I should like to follow up Dr Schatzman's talk with a few words on models of Nova nebulae. Geometrical models for novae Aql 1921 and Her 1934 have been derived by Mustel and Boyarchuk (1970), for novae Del 1967, Vul 1968 and Ser 1970 by Hutchings (1972), for Aql 1921 by Weaver (1974) and for Del 1967 by Malakpur (1973). Very different techniques and lines of reasoning have been used by all these people, but the models derived all have very strong similarities which should lead us to take them seriously in considering the nature of the nova mechanism.

It appears that nova remnants are characterized by two polar cones or blobs of similar but opposite velocity and similar density, moving along the axis of rotation of the system. In addition, one or more rings of matter may exist symmetrically about the equatorial plane, with a recession velocity and density which, in general, are different from the polar matter. The polar blobs appear to be associated with the matter giving rise to the ‘principal’ spectrum in the bright phases of the nova.

Three possible main causes have been suggested for the formation of such remnants, all connected with the initial outburst: (1) Weaver has proposed that an initially spherically symmetrical expansion is impeded and deflected by matter circulating in an accretion ring around the degenerate star. (2) Schatzman has suggested that the presence of the ring may favour an initial outburst preferentially in the polar regions. (3) Hutchings has suggested that physical interference of an initially spherical expansion may occur by a secondary component whose size is a large fraction of the separation of the two stars.

I feel that these points need to be considered by theoreticians in proposing detailed mechanisms of the nova outburst.

We propose the examination of trailed photographs as a medium of search for fast variable stars. Diurnal rotation and seeing permit access to the period range from 0.1s to 10s for a resolution of 1″, exposures of one minute and a plate field of 30′, while slower or faster periods can be studied by mechanically trailing at the appropriate rate. This permits examination of a wide period band over a field of perhaps 0.1 square degree in a minute's time, while photoelectric devices now available permit only star by star examination. Thus we can contemplate examination of an appreciable fraction of the sky with moderate telescope time required.

The problem of the adiabatic non-radial oscillations of spherical stars is reviewed and results recalled for a variety of models. The anomalous behaviour of the eigenfunctions for highly condensed models is related to the apparent mobile singularities depending on the eigenvalues. Tables of Q-values are provided to facilitate possible applications to variable stars.

In the case of the gravity modes, the existence of multiple spectra some stable (g+ modes) some unstable (g− modes) if superadiabatic and subadiabatic regions alternate is discussed and the interest of further investigations underlined.

As far as vibrational stability (effects of the non adiabatic terms) is concerned, a general expression is given for the ‘damping coefficient’. The attention is drawn to the possibility for g+ modes of becoming vibrationally unstable under the effect of various factors and in various models, including the Sun where this was advocated as a possibility of relieving the neutrinos difficulty.

Finally the present status of the most obvious candidates among variable stars for non-radial oscillations, the β Canis Majoris stars and the rapid blue variables (white dwarfs) is briefly reviewed.

Physical arguments are provided which suggest the following:

(a) The Sun rotates rapidly internally, with a period near one day. The arguments are based upon a low ‘effective’ plasma dynamic viscosity associated with a negative ‘effective’ magnetic density, ϱM = −B2/4πν2. This low (near zero) viscosity allows several calculations of the angular velocity of the solar core to be made. A fluid dynamical argument based upon the inviscous Navier-Stokes relation shows that for objects seated in a non-expanding magnetohydrodynamic fluid, the usual Kepler law should be replaced by T2 ∝ r4.

(b) The rapid rotation suggests that the solar magnetic field is deeply buried and, furthermore, that the Sun violates the Ferraro theorem. This violation results from a radial electric field necessary to support the solar plasma. This radial field requires the Sun to be charged with a charge q = + 2 × 1011 esu. Thus the Sun's differential rotation may be viewed as arising from B']θ=0 in the rest frame of the fluid where Er ≠ 0 or alternatively as a polar spin down by the solar wind flow. Thús in the fluid frame, the solar activity cycle may be viewed as an Alfvén wave, with T= Ω/vA = 20 yr. The velocity of material along the field is such that (G−1 cm−2 s−1), governed by the collapse of mass in the solar core; (4H + 4e → 1He + + 2e) results in a reduction of gas pressure unless ~7 × 1014 gm s−1 of material continually collapse so as to conserve the particle number in the solar interior.

(c) This requires the Sun to emit gravity waves with an energy comparable to its luminosity. The virial of the nuclear reactions in the core governs whether the energy goes into gravity waves or heat. A variable positron-electron core allows this.

It has been shown (Markey and Tayler, 1973; Tayler, 1973; Wright, 1973) that a wide range of simple magnetic field configurations in stars are unstable. Although the ultimate effect of the instabilities is unclear, it seems likely that they would lead to enhanced destruction of magnetic flux, so that magnetic field decay would be much more rapid than previously supposed. Instability is almost certain in a non-rotating star containing either a purely toroidal field or a purely poloidal field, which has closed field lines inside the star. In both cases the instability resembles the well known instabilities of cylindrical and toroidal current channels, modified by the constraint that motion must be almost entirely along surfaces of constant gravitational potential.

If both toroidal and poloidal fields are present, the problem is more complicated. In a toroidal plasma with a helical field, the worst instabilities are also helical but it is impossible for a helical disturbance to be parallel to a surface of constant gravitational potential everywhere. As a result, the admixture of toroidal and poloidal fields has a stabilizing influence, but it is not at present clear whether the majority of such configurations are completely stable.

The effect of rotation has not yet been studied but it will certainly be important if the rotation period is less than the time taken for an Alfvén wave to cross the region of interest. This is true in most stars unless the internal magnetic field is very much stronger than any observed field.

The case for variable magnetic fields, in the range 1–15 kG, in three of the X-ray binary candidate stars, is discussed. We refer to electronic Zeeman polarimetry by Kemp and Wolstencroft (1973a and b) and by Angel et al. (1973). Our detections in Hβ showed: (1) definite magnetic fields, roughly constant over periods of 2 h, in θ2 Orionis and HD 77581; and (2) complex time behavior in HD 153919, with field reversals over periods of ~ 10 min, but with usually small values (≲ 2 kG) as averaged over times of 1–2 h. (Photographic Zeeman spectroscopy is therefore unlikely to show detectable fields, i.e. > 2 kG, in the latter star.) The essentially null results of Angel et al. (1973) in Hβ, in HD 77581 and HD 153919, indicate that the emission is non-magnetic; only the underlying absorption (which is far weaker than the emission in Hβ) shows Zeeman effects. A plot of the Zeeman measure for HD 77581, against the 8.95-day period, is shown: our points are suggestive of periodicity; and the points of Angel et al. (1973) which unfortunately have larger relative error bars, are consistent with the indicated variation and with the general scale of our results.

It is shown that when a magnetic field is present along the axis of rotation, the point of bifurcation, where the Jacobi ellipsoids branch off from the Maclaurin spheroids, occurs at a value of eccentricity higher than the value e = 0.81267 that obtains in the absence of a magnetic field. This is in contrast with the effect of a toroidal magnetic field which, as has been shown earlier, leaves the point of bifurcation unaffected.

This review deals with the local instabilities arising when the effects of rotation are taken into account in the evolution of a non-magnetic star.

The Rayleigh and shear instabilities will be examined under the conditions prevailing in radiative zones where the effect of density stratification, thermal diffusion, viscosity and varying chemical composition must be taken into account. The possible consequences on the evolution of a star are finally outlined.

The study of the linear and adiabatic oscillations of a gaseous star gives rise to an eigenvalue problem for the pulsation σ, if perturbations proportional to eiσt are considered. In the presence of a rotation, a tidal action or a magnetic field, the equations are not separable in spherical coordinates. To get approximate expressions for the influence of these factors on the non-radial oscillations of a star, the author and his collaborators J. Denis and M. Goossens have used a perturbation method (Smeyers and Denis, 1971; Denis, 1972; Goossens, 1972; Denis, 1973). Their procedure corresponds to a generalization of the method proposed by Simon (1969) to study the second order rotational perturbation of the radial oscillations of a star.

Two types of perturbations are taken into account: volume perturbations due to the local variations of the equilibrium quantities and to the presence of a supplementary force in the equation of motion (Coriolis force, Lorentz force); surface perturbations related to the distortion of the equilibrium configuration and to the change of the condition at the surface in the presence of a magnetic field. The resulting expressions are accurate up to the second order in the angular velocity in the case of a rotational perturbation, to the third order in the ratio of the mean radius of the primary to the distance of the secondary in the case of a tidal perturbation, and to the second order in the magnetic field in the case of a perturbing magnetic field. These expressions can in principle be applied to any mode.

Numerical results have been obtained for a homogeneous model and for a polytropic model n = 3. In particular, the splitting of the frequencies of the fundamental radial mode and of the f-mode belonging to l = 2 and m = 0 has been studied for the critical value of y, in the case of a component of a synchronously rotating binary system.

In the wake of recent theoretical work on contact systems (e.g. Whelan, 1972; Biermann and Thomas, 1972; Lucy, 1968), it is of importance to determine fundamental data from observations. This has been done recently by several groups in analysing light curves (Mochnacki and Doughty, 1972; Hutchings and Hill, 1973; Wilson and Devinney, 1973), and it is found that shapes, temperature differences and distributions, and mass-ratios, can be determined in many cases. However, where spectroscopic data are also available, the mass-ratios are not always in agreement. Using the photometric models, it is possible (Hutchings, 1973) to calculate the distortion of line profiles resulting (primarily) from the non-uniform brightness over the component stars in these systems. This distortion leads to the characteristically observed ‘square’ velocity curves for the systems (e.g. Binnendijk, 1967). Correction for the effect in most cases (a) resolves the mass-ratio discrepancy and (b) leads to better estimates for the masses. The faintness of most contact systems makes detailed spectroscopy difficult, but there appears to be a need for further work in the directions outlined here to improve the fundamental data available on them. These results should also be borne in mind in inspecting previous work on contact binaries.