This review is concerned with recent theoretical investigations and numerical models of star formation with varions symmetries. Observations strongly support the fact that stars condense out of cool (≈10 K) and dense (103-104 atoms/cm3) interstellar clouds due to gravitational instability and collapse. Bright, young stellar objects (0- and B-stars are always found in the vicinity of coloud complexes.

In this paper we report briefly the results of an extensive set of 3-D hydrodynamic calculations that have been performed during the past two and one-half years to investigate the susceptibility of rotating clouds to gravitational fragmentation. Because of the immensity of parameter space and the expense of computations, we have chosen to restrict this investigation to strictly isothermal collapse sequences. Isothermality is, fortunately, a fairly accurate description of the optically thin phases of collapse of protostellar clouds: a one solar mass, Jeans unstable interstellar cloud, having a temperature of 10 K and density ~ 106 particles/cm3, for example, will collapse isothermally through about 6 orders of magnitude in density before heating up significantly (Gerola and Glassgold, 1978). The assumption also applies to protogalactic clouds (Silk, 1977). The results, therefore, should add specifically to our understanding of the dynamics of the earliest phases in a protostellar (or protogalactic) cloud’s evolution. A much more complete discussion of these calculations can be found in Bodenheimer, Tohline and Black (1980), hereafter referred to as BTB.

Numerical calculations of the collapse of adiabatic clouds from uniform density and rotation initial conditions show that when restricted to axisymmetry, the clouds form either near-equilibrium spheroids or rings. Rings form in the collapse of low thermal energy clouds and have β = T/|w|≳ 0.43. When the axisymmetric constraint is removed and an initial m=2 density variation is introduced, clouds either collapse to form near-equilibrium ellipsoids or else fragment into binary systems through a bar phase. Ellipsoids form in the collapse of high thermal energy clouds and have β ≲ 0.27. The results are consistent with the critical values of β for instabilities in Maclaurin spheroids, and suggest that protostellar clouds may undergo a dynamic fragmentation in the nonisothermal collapse regime.

The formation of a 1 M⊙ protostar in spherical symmetry has been followed in time dependent hydrodynamics with a detailed description of the equation of state and a careful treatment of radiative transport. The comparison of the dynamic evolution with observation is made in terms of the Hertzprung-Russell diagram. It is found that the evolution following from Larson’s initial condition produces quantitatively and qualitatively better agreement than that following from the initial condition of Hayashi, including the first appearance of the object as an infrared source. Of particular importance for a correct physical understanding of the formation process is an adequate description of the accretion shock. Pressure ionization and electron degeneracy effects are of increasing importance for the internal structure of lower mass objects.

Analytic stability analyses (Lyttleton, 1953; Chandresekhar, 1969) have shown that Maclaurin spheroids are dynamically unstable to non-axisymmetric perturbations if

Two-dimensional calculations of ionization-shockwave propagation into a curved molecular cloud are presented. Density enhancement occurs due to the combined effects of cloud curvature and radiation flow. The star formation process is expected to be enhanced near the edges of irregularly shaped molecular clouds.

Recent observational evidence on solar oscillations is reviewed; this evidence strongly favors the global interpretation for much of the observed spectrum. Implications of these observations for the study of the solar interior and atmosphere are discussed.

The solar granulation has been simulated by numerical solution of the multidimensional, time-dependent, nonlinear Navier-Stokes equations applied to the solar atmosphere. Granules may be explained as buoyantly rising bubbles created at the level where T = 8000 K, and which have collapsed into vortex rings. The calculation is in quantitative agreement with observations and has a number of implications for solar physics and convection theory.

The nonlocal character of the radiation field sinnificantly modifies the radiative damping of perturbations in the solar photosphere. Gravity waves are not usually considered to exist in the solar photosphere because the radiative damping time, when based on the Newtonian approximation, is too short. However, this restriction does not apply to low order gravity waves. In fact, with the inclusion of nonlocal effects, the radiative damping for low order gravity waves becomes negative for some region in the photosphere and thus acts as a driving mechanism for gravity waves there.

A fundamental problem in contemporary stellar atmospheres research concerns the cause of what the spectroscoplst calls “macroturbulence.” Even in so well studied a star as the Sun, it is unclear as to which of the many resolved velocity fields is most responsible for the broadening of the disk-integrated spectrum. There are several uncertainties attached to the identification of this primary velocity field. To cite one, Beckers (1980) indicates in a recent review that the two principal contributors to macroturbulence, convective granulation and the five-minute nonradial oscilltion pattern, each add only an r.m. s. velocity of 1/2 km s at τ5000 = 0.1. According to him, even when they are put together with related unresolved patterns [e.g. subgranulation and short period (<30) oscilltions] the sum of all known velocities seems to fall short of macroturbulence obtained from line broadening studies [~ 3 km s-1; radial-tangential model (Gray 1977, Smith 1978)]. The most recent models of the solar granulation field (Keil 1980) suggest somewhat higher velocities, e.g. 1.1 km s-1 at τ5000 = 0.1, when revised corrections for terrestrial seeing are taken into account. Nonetheless, such corrections must be added to both the convection and oscillation amplitudes, so it is still not clear whether one of these fields dominates the line formation.

Until recently, the β Cep stars were regarded as a small, homogeneous group of pulsating stars, well separated from the Cepheid instability strip in the H-R diagram. Now, the picture has been complicated by the discovery of other types of variables in and around the β Cep instability strip. These include (i) variable B supergiants, which may connect the β Cep strip to the Cepheid strip through the variable A and F super-giants (Maeder, 1980), (ii) the line profile variable stars or 53 Per stars, which extend to mid to late B types, and which may connect through the hypothetical Maia sequence to the δ Set strip (Smith, 1977), (iii) the short-period variable Be stars (Percy et al, 1980) and (iv) the ultra-short-period variable B stars discovered by Jakate (1979b). This variety of variables may reflect the variety of physical processes and pulsation mechanisms which are important in the B type stars.

The best known β Cephei variable is a α Virginis (Spica) because it is a binary star (P = 4.01 days) and has a interferometer measurement of its angular diameter. In this study to see if the pulsations can be caused by mixing of hydrogen and helium in the semiconvective zone, the parameters we take are 11.5 M⊙, 26,300 K, 6.5 x 1037 erg/s with an envelope composition of Y = 0.28, Z = 0.02. These data agree within the errors of observations as reported recently by Odell (1980), and the theoretical radial fundamental mode period of 0.171 day compares well with the single observed pulsation period of 0.174 day.

An experimental Fairchild CCD-211 was placed in the 2.1-m coudé spectrograph at the Kitt Peak National Observatory and used to record an 87 Å spectral band centered on Hα. On the night of 1979 July 11 UT date, this system was used to observe continuously the extreme β Cephei variable BW Vulpeculae throughout one of its 4h 49m cycles with an average exposure of 13 minutes per observation. A total of 18 observations was secured. The results show dramatic profile variations of Hα, including features not previously reported, and extreme variations of the C II λλ6578, 6582 lines, including variations of equivalent width. The fast time-resolution capability and the photometric linearity of the CCD have permitted the detection of subtle effects that have been missed by photographic observations and has led directly to important new interpretations of the complex atmospheric pulsations in this star, including effects of altered opacity on the formation of spectral lines and the suggestion of a helium-ionization heat engine as a mechanism for driving atmospheric pulsations.

BW Vulpeculae (HD 199140) is a well known Beta Cenhei star, with several unusual characteristics. It possesses the largest amplitude in both light and radial velocity variations, of any star in this classification. The most outstanding feature of the observations is a standstill or bump, on the light curve, accompanied by doubled lines in the radial velocity observations. In this paper, we have attempted to explain these phenomena with two different models. Firstly, the resonance mechanism, from Simon and Schmidt, and secondly, a non-linear calculation.

The profile of the Hydrogen alpha spectral line in the large-amplitude Beta Cephei star BW Vulpeculae was modeled in terms of a linear, non-radially moving, and rotating stellar atmosphere. The pulsation and rotation parameters were derived from fitting the radial velocity amplitude, and the best fit was obtained for l=2, m=-2, i=90°, rotation velocity=120 km/sec, and pulsation amplitude=27 km/sec. When the line profiles are compared to this model, the fit is quite encouraging for most phases, but several discrepancies remain. At the time of writing, it is not clear whether tliese can be made to disappear by increasing the sophistication of the model, or whether a completely different physical model (e.g. radial) is necessary.

Assuming that frequencies observed in multiperiod β Cephei variables correspond to normal pulsation modes leads to the following conclusions: (1) a variety of low Z oscillations are excited in these stars, (2) at most one radial mode is seen in any of them.

Pulsation constants can be derived for two β Cephei stars without recourse to photometric calibrations. In both cases a 0.5 mag. downward revision of the Balmer line absolute magnitudes is indicated. Implications of this result for Stellingwerf’s opacity bump mechanism are briefly considered.

The text of the paper will appear in Proceedings of the Fifth European Regional Meeting of the IAU.

Almost every survey for β Cephei detection has been restricted to the small area of the HR diagram that contained the 15-20 first discovered. (Table 1). Of course, every new short period pulsating star found in such an a priori limited region has strengthened the impression that an instability box could be derived from the study of the statistics of β Cep variables. That conclusion could have led to the discovery of a β Cep pulsation mechanism related to a particular mass - age internal structure status. On the contrary if the phenomenon is due to a mechanism acting in a wider extension of the HR diagram (and so probably better related to envelope properties) one still has to explain why its detection seems to be easier in the B0-B1 zone. Moreover, even within the “instability box” some classical β Cep could be either in core hydrogen burning phase or others in shell hydrogen burning phase, i.e. with quite different internal structures.