In the calculation of linear nonradial oscillation modes in composite polytropes with a small density discontinuity, a discontinuity mode may occur. This mode consists of a wave propagating along the discontinuity interface with a large amplitude that declines exponentially away from the interface. The period P of this mode is well-estimated (to within 10%) by

where Δρ/<ρ> is the fractional density discontinuity and k is the horizontal wavenumber (c.f. Gabriel and Scuflaire 1980, in Nonradial and Nonlinear Stellar Pulsation, eds. H. Hill and W. Dziembowski, Springer-Verlag). For a 12 solar-mass polytrope with a radius of 4.27 R⊙, a 3% density discontinuity at fractional radius 0.15 produces a discontinuity mode with a period of 7.329 hours. As the density discontinuity increases the period P decreases, resulting in avoided crossings with the normal g-mode spectrum. Between these avoided crossings, the discontinuity mode has an unusually large amplitude at the location of the discontinuity.

The surface brightness version of the Baade-Wesselink method, has been applied to 7 field RR Lyrae stars with metallicity ranging from [Fe/H]= −0.2 to −1.5. V magnitudes, V-R and V-I colors and CORAVEL radial velocities were used, and the analysis was performed over a restricted phase range in order to avoid the complications caused by the pulsating atmospheres. The combination with previous results of the B-W method, which used comparable criteria (Jones, Carney, & Latham, 1988, preprint; Jameson, Fernley, & Longmore 1987, in press M.N.R.A.S; Cohen & Gordon 1987, Ap.J., 318, 215) leads to the following relation between the absolute luminosity and metallicity:

Mv = (1.0 ± 0.05) + (0.17 ± 0.05) [Fe/H]

This relation is in very good agreement with the preliminary results found by Liu and Janes (this volume). The application of the above relation to the RR Lyraes in M31 and the Magellanic Clouds leads to distance moduli of (m–M)o= 24.21 ± 0.20 for M31, (m–M)o = 18.26 ± 0.20 for the LMC, and (m–M)o =18.85 ± 0.20 for the SMC, and the distance to the galactic center turns out to be approximately 7.2 kpc. From the absolute magnitude of the RR Lyraes and adopting a constant visual magnitude difference between the RR Lyraes and the turn-off ΔV = 3.55 (Buonanno 1986, Mem.S.A.It., 57, 333), we estimate ages of 18.8 and 15.7 Gyr for globular clusters of metallicity [Fe/H] = −2.2 (e.g. M92) and [Fe/H] = −0.8 (e.g. 47 Tuc) respectively, using the age-turnoff luminosity relation derived by Sandage (1982, Ap.J., 252, 553) or 20.9 and 16.9 Gyr using Buonanno’s relation.

Numerous possible mechanisms for the cause of B star radial and nonradial pulsations have been investigated in the last 30 years, but none have been clearly shown that they actually work. I briefly review these ideas, and then focus on a new idea about the driving due to time-dependent convection. I suggest that the outward going photons are absorbed by CNONe ions so that they capture the photon momentum. Previous investigations by many have shown that the acceleration of these ions can be two orders of magnitude larger than the local gravity. Levitation and concentration of these elements can occur between temperatures of 100,000K and 600,000K, the former temperature set by the complete ionization of the competing photon capturing helium, and the latter by the existence of simpler and less absorbing excited electron structures of the CNONe elements. Then I suggest that the concentration of these elements gives such a large local photon opacity that a thin convection zone exists down to maybe 10−5 of the mass of a typical B star. However, there would be no overlap with the outer surface convection zone that exists down to 50,000K in the outer 10−9 of the mass. The long time scale of the convection, or even its periodic turning on and off, gives a source of time-lagged luminosity which can be shown in some cases to drive mechanical pulsations in one or more modes. This envelope pulsation mechanism could apply equally to radial and low degree nonradial B star pulsations. It could also explain why not all B stars pulsate unless this unconventional composition structure occurs, disappears, and maybe reoccurs at some evolution stage.

There is interest in the minimum mass that can exhibit solar-like nonradial p-mode oscillations. Inspection of models for the solar case shows that there is approximately equal pulsational driving and damping due to the standard radiative effects. It is widely believed that coupling with surface convective elements occasionally tips the balance by strongly driving modes for a few days with the timescale of the convection at the top of the convection zone. Then both the convection and the radiative damping decrease the mode amplitude at the observed decay rate, (which is close to that calculated by Kidman and Cox (1984)) until the next convective reexcitation. Population I composition stellar models at 0.30, 0.45, and 0.75 solar mass have been constructed to see if the near equality of radiative driving and damping exists in them also. If so, then stars down to late spectral class could possibly display solar-like oscillations. We find that the convection timescale decreases to less than 100 seconds at 0.30 solar mass, but the near equality of radiative driving and damping occurs only down to about 0.5 solar mass. Thus there is the possibility that 100 to 300 second oscillations can be observed through most of the K spectral class, if the magnetic activity often seen in these stars does not overwhelm these small amplitude nonradial pulsations.

Two precision solar models have been constructed with the Iben evolution program, one with no diffusion of the internal atomic nuclei, and another that includes the effects of gravitational settling, thermal diffusion, and concentration gradient diffusion on the element abundances. The opacity at the bottom of the convection zone was increased 15-20 percent (within its theoretical uncertainty) to allow a few microhertz agreement with the observed solar p-mode frequencies. The original helium mass fractions of the mixtures were 0.291 and 0.289 for the no-diffusion and diffusion models, respectively. The diffusion model evolved to a surface Y=0.256 at the solar age, and the original Z value of 0.0200 decreased to 0.0179. Agreement of asymptotic theory 1=0 and 2 p-mode frequency separations and those directly calculated with nonadiabatic theory is good, and there is good agreement with observations also. Calculation of g-mode solutions shows that they do not have equal period spacings until high radial order. Nonadiabatic solutions for these g-modes modes enable us to predict their relative surface visibility. For 1=1-5, the lowest order modes seem to be more detectable assuming that they all have the same kinetic energy. Our high helium results in high central temperatures that give over 9 SNUs from the B and 1.5 SNUs from the Be reactions, but a lower more conventional helium abundance would reduce this B neutrino flux by maybe two SNUs. Models with iron condensed-out deeper than the convection zone, and with the presumed WIMPs to cool the central regions and reduce the SNU flux, agree less well with the p-mode frequency separations.

After the discovery of pulsations in white dwarfs, predictions were made that these DA and the hotter DB stars should be pulsating in radial modes with periods of a few seconds or less. The mechanisms are the normal kappa and gamma effects that periodically block the flow of radiative luminosity and the blocking effect of the frozen-in convection at the bottom of the convection zone. Blue edges of the instability strips are between 12,000K and 13,000K for the DA and between 32,000K and 33,000K for the DB variables. Extensive observations, however, have shown that these stars pulsate only in the few-hundred-second nonradial modes and not in any few-second radial modes. We have added the time dependent convection model of Cox, Brownlee, and Eilers (1966) to our pulsation analyses to further investigate the white dwarf radial modes. Since the time scale of the convection is usually short compared to the radial pulsation periods, convection is able to carry luminosity rapidly enough to nullify the kappa and gamma effects periodic radiation blocking. We find that most, and maybe all, radial pulsations for 0.6 solar mass carbon-oxygen white dwarfs with thin hydrogen or helium surface layers are stabilized for both these DA and DB classes, now finally in agreement with observations.

IUE spectra have been used to investigate the companion of the classical Cepheid S Sge. At the phase of the IUE observations, 45 Dra (F7Ib) is a good match to S Sge, both in (B–V)o and from 2200 to 3200 Å. However, from 1700 to 1900 Å, the S Sge spectrum has excess flux, as compared to 45 Dra. The flux remaining after subtracting 45 Dra from S Sge is a good match to an A5V to A7V star. The magnitude difference between the Cepheid and the companion is consistent with the absolute magnitude and colors of an A5V star. This corresponds to a mass of 1.9 solar masses (Popper, 1981, Ann. Rev. Astr. Ap., 18, 115). A spectral type range from A3V to A7V corresponds to an uncertainty in the mass of ±0.2 solar masses. However, the mass function from the orbit and an evolutionary mass for the Cepheid require a companion mass of 2.7 solar masses or greater. We infer that the companion is itself a binary.

Hydrodynamic calculations of nonlinear pulsations were done for models of helium stars with mass 1 M⊙, luminosity from 3220 L⊙ to 12820 L⊙ and effective temperature from 6000K to 8000K. The models with L > 8000L⊙ were found to pulsate in the fundamental mode with large amplitude (ΔR/R ~ 1), whereas less luminous models (L < 8000L⊙) revealed small amplitude oscillations (ΔR/R ~ 0.2) in the first overtone. In the luminosity range considered the blue edge of the instability strip corresponds to an effective temperature of 7500K, that is to the upper limit of the effective temperatures of R CrB stars. Application of the period-luminosity relation to the variables R CrB and RY Sgr gives their luminosities to be of 9000L⊙.

The hydrodynamic calculations of nonlinear self-excited radial pulsations were done for the models of W Vir stars with mass 0.6 M⊙ and luminosities from 794L⊙ to 3981L⊙. The periods of the models are longer than 10 days. The pulsations are shown to be accompanied by periodic shocks that change the density distribution in the pulsating stellar atmosphere. At radii less than 5 Rph, where Rph is the radius of the photosphere, the mean dynamic scale height is nearly five times the static scale height. In this region of the atmosphere the mean radii of outer mass zones do not change perceptably. On the other hand, at radii larger than 5 Rph the scale height is of the order of Rph so that the outermost layers ultimately expand with a velocity exceeding the local escape velocity. The mass flux in the atmosphere increases with decreasing mass to radius ratio and mass loss rate in W Vir type variables is in the range from 2 X 10−6M⊙·yr−1 to 10−5M⊙·yr−1.

A summary of basic parameters of classical Cepheid variable stars is being prepared. It will contain stars from the fourth edition of the General Catalogue of Variable Stars in the categories CEP, CEP(B), DCEP, and DCEPS, excluding those which belong to Population II according to the criteria of Harris (1985, A.J.,90, 756). V and B intensity means, as well as velocity means are derived from Fourier fits using the N. Simon program MINFIT. In addition to the data contained in the catalogue by Fernie and Hube (1968, A.J., 73, 492), it will provide a comparison of reddenings from a number of sources, binary status, and period change information. We anticipate a hardcopy version, and also a version on floppy disks. The computer version uses dBaselll, and will include programs, for instance, to calculate phases using changing periods.

A method of calculating nonlinear stellar pulsations including nonstationary radiative transfer in a grey spherical atmosphere is described. With the help of this method eleven type II supergiant radiative models were constructed with masses of 0.6M⊙, luminosities ranging from 128L⊙ to 3123L⊙ and periods in the range from 1.123 to 46 days. A stable limit cycle was found to be accessible only by models with an effective temperature between 5700K and 6165K. The model with Te = 6165K is stable, whereas the models cooler than 5700K show nonregular behavior. Transition from strictly periodic to nonregular pulsation arises when M/R ≲ 0.018, due to high amplitudes, δr/r ≈ 1, and strong shocks in the atmosphere. The radiative transfer effects lead to some decay in the radial amplitude, as well as to a more significant decrease, about 0.6 magnitudes, in the light variation. A photometric comparison between the light curves of the models calculated with and without transfer and the observed light curve of the variable star No. 154 in M3 shows that the results predicted by the transfer model are in much better agreement with obervational data.

A simple one-zone model for nonlinear stellar pulsations is outlined and applied to the study of period doubling observed in some W Virginis and RV Tauri stars. The model reveals a number of period doubling bifurcations as the parameters are varied, similar to those found by Buchler & Kovacs in their series of hydrodynamic models. In the vicinity of a stable limit cycle, despite its large number of degrees of freedom, the model develops an essentially one-dimensional Poincaré’s return map, determining the modulation of the amplitude. The analysis of these maps confirmed that period doubling has its origin in a strong nonlinear increase of total energy losses per period as radial amplitude, δr/r, increases. An additional study of nearly periodic hydrodynamic models with P > 15 days, calculated including radiative transfer effects, shows that the rate of energy dissipation per period by shocks in the atmospheres increases rapidly with δr/r, whereas the excitation rate, δo, remains rather stable. This permitted us to construct an analytic return map for maxima of the total kinetic energy which clearly demonstrates the mechanism of successive period doubling as δo, the sole bifurcation parameter, grows monotonically.

Surface brightness solutions using the (V-R) color index have been carried out for 52 southern Cepheids and 63 northern Cepheids, with 14 stars in common. For the southern stars the data came from the study by Gieren (1986, M.N.R.A.S, 222, 251), with new observations, and for the northern stars, from Moffett and Barnes (1987, Ap.J., 323, 280). All stars were reduced using the same surface brightness relation and the same transformation from radial velocity to pulsational velocity. The 14 stars in common show that there is only a small systematic difference between the southern and northern data sets and reductions. The combined solution for the 101 Cepheids in the period range 3 to 45 days is log R = 1.108 + 0.743 log P with uncertainties of ±0.023 in the zero point and ±0.023 in the slope. This result is consistent with the individual results of the above references but has much smaller uncertainty.

Linear and nonlinear pulsational properties of convective stellar envelopes relevant for W Vir and RV Tau stars are surveyed. All models show the same trend to pass from regular to irregular behavior when a control parameter is changed (the effective temperature). The transition to irregular pulsation follows well known systematic routes to chaos (as in the radiative case). Some rich structures were found in special cases; they deserve further research. We show that the chaotic behavior is sustained even when convection is taken into account. The effect of the inclusion of time dependent convection shows up mostly as a shift of Kovacs and Buchler (Ap.J 1988) results in the parameters plane (L,Teff) towards more realistic models.

Irregular pulsations are commonly observed in many groups of pulsating stars. An interpretation in terms of superposition of modes of pulsation is sometimes barely convincing when too many modes have to be considered and when unexpected ratios of frequenceis are observed.

A survey for Long-Period Variables has been made of the Bar and southern regions of the Large Magellanic Cloud. This has been combined with the results of Reid, Glass and Catchpole (1988) to present a global survey of the LMC. Periods are used as age indicators, giving the star formation history of the LMC from 1 to 10 Gyr, which indicates a major burst of star formation occurred approximately 4 Gyr ago.

The nature of irregular light variation in giants and supergiants has long been a matter of dispute. Our lack of understanding has two main sources: (i) a paucity of long, continuous, high quality observations and thorough analysis of the available data and (ii) until very recently, a lack of systematic nonlinear hydrodynamic studies for these stars.

The direct fit of theoretical pulsation frequencies to the observations (i.e. stellar seismology) proved to be a very efficient tool in the study of solar oscillations. In the case of other multiperiodic variables, like δ Scuti stars, Ap stars and white dwarfs the method suffers from the disturbing abundance of possible nonradial modes. Colour and/or radial velocity (or line profile) measurements can narrow down the number of possibilities, but these kinds of data are not often available with the desired accuracy and sampling rate. Since pulsational frequencies are the most readily and accurately computed and measured quantities of pulsation, we address the question of the accurate fit of the nonradial pulsation frequencies to the observations in the case of δ Scuti stars.

We have obtained UBVRI and JK photometry and radial velocities for 13 field RR Lyrae stars with a large range of metallicity. Our sample includes eleven ab-type and two c-type variables. The surface brightness method is applied to these RR Lyrae stars to determine their mean absolute magnitudes <MV> as well as their distances and radii. We have used the Kurucz’s model atmospheres to provide metallicity- and gravity-dependent color-temperature transformations and bolometric corrections. Our results indicate a clear correlation between the luminosities of the RR Lyrae stars and their metallicities such that metal-poor RR Lyrae stars are more luminous.

The log (Period) – infrared (2.2 micron) magnitude relationship has been measured for 7 clusters (M3, M4, M5, M15, M107, ω Cen and NGC 5466). For the clusters where there are several observations for each of the variables, enabling a good mean magnitude to be derived, there is no evidence for scatter in the relation outside observational error. This conclusion applies also to ω Cen, even though the variables observed covered a range of metallicity. It is argued that very accurate relative distances can be obtained which are insensitive to reddening errors and the effects of metallicity. Any mass difference between variables in different clusters may still introduce uncertainties in the relative distances. Recent Baade-Wesselink results by Fernley et al. have been used as an absolute calibration, including a small residual correction for metallicity effects. On this scale, the mean distance modulus for 6 of the clusters (excluding NGC5466) is essentially the same as that derived by Buonanno, Corsi and Fusi-Pecci (preprint) using a combination of horizontal branch magnitudes and main sequence fitting. However, individual distance moduli differ by typically 0.09 magnitudes. There is no clear correlation of this residual with metallicity.