Mean observed rotational velocities for single, normal, main-sequence stars are reviewed and compared with the mean observed v sin i’s for giant and supergiant stars, Be stars, peculiar A-type and metallic-line stars, and Population II objects.

Research into the effect of rotation on the internal structure of stars has a long and detailed history. The names of McLaurin, Jacobi, Kelvin and Jeans are associated with detailed work on the structure and stability of rotating liquid masses assuming uniform rotation. Much of this work is summarized in Jeans’ (1929) Astronomy and Cosmogony and more recent summaries are given in Lyttleton (1953) and Lebovitz (1967).

In the following a method is presented for computing the internal structure of nonspherical stars assuming that the force per gram causing the deviation from spherical symmetry is conservative. The method has the advantage that in a normal (spherical) stellar structure code only slight changes have to be made in order to obtain nonspherical stellar models. The method can be applied as well to rotating stars as to stars distorted by tidal effects. Although it is similar to that of Faulkner et al. (1968) in the case of purely rotating stars, it is not necessary to use the division into two zones, where either slow rotation or negligible contribution to the gravitional potential is assumed.

The equations for a rotating convective spherical shell are solved in the Herring approximation as an initial value problem. The main results are

• (1) The most unstable modes (those that maximize the heat flux) correspond to convective cells stretching from pole to pole.

• (2) The calculations of the Reynolds stresses show transport of angular momentum towards the equator. That is, differential rotation sets in with equatorial acceleration.

• (3) The convective heat transport is maximum at the equator. This would give rise to an equator-pole flux difference.

• (4) If convection is non-axisymmetric (as in the most unstable modes) there are no time independent solutions. The time dependence is oscillatory and of the form ωt + mφ.

The perturbation method has been applied to the problem of the oscillations of a gaseous star rotating around a fixed z-axis according to a general law of the type

Mean rotational velocities are presented for stars on and above the main sequence from spectral type A5 to F9. The velocities are shown to be inconsistent with the hypothesis of stellar rotation in completely uncoupled shells for stars that have expanded in radius by less than a factor of four. They do, however, support the hypothesis of solid-body rotation for these stars. The data for the giant stars are discussed, but no firm conclusions about the mode of rotation in these stars can be drawn.

The evidence in favor of the hypothesis that Be stars owe their emission properties to material rotationally ejected from the equator is reviewed. The evolutionary state of Be Stars is then discussed with reference to evolutionary sequences of stellar models. It is concluded that (i) Be stars are not confined to the secondary contraction phase as previously proposed (ii) evolution probably proceeds with uniform rotation at least outside the initial convective core. Mechanisms for transporting angular momentum are briefly discussed.

To compute the evolution of a rotating star, the following approximations were used in order to obtain some of the main effects of rotation with simple models of spherical symmetry:

It has become clear in recent years through the work of Collins (1963, 1965, 1966), Collins and Harrington (1966), Hardorp and Strittmatter (1968a, b) and Roxburgh and Strittmatter (1965) how seriously the spectrum of a star may be affected by rotation. However, before atmospheric models for rotating stars can be computed, the results of corresponding interior models have to be available. For polytropic configurations a great many rotating interior models have been constructed (Chandrasekhar, 1933; James, 1964; Stoeckly, 1965; Monaghan and Roxburgh, 1965; and Anand, 1968), but for rotating main-sequence stars few computations exist (Sweet and Roy, 1953; Roxburgh et al, 1965; Ostriker and Mark, 1968; Faulkner et al., 1968). In fact for B- and Be-type stars for which the most rapid rotation along the main sequence is observed, not a single interior model has yet been computed except in this conference reported by Kippenhahn and Thomas (1970). It is this gap that the present paper attempts to close.

The braking of stellar rotation in the wholly convective phase in the pre-main sequence is numerically discussed. The structure of stars in that phase is expressed by a rotating polytrope with an index of 1.5 and the Schatzman-type mechanism is used as the means of loss of angular momentum. The magnetic energy is assumed to change with evolution as H02/8π(R/R0)s, where H0 and R0 are initial magnetic field and radius, and s is a free parameter. The changes of angular momentum, rotational velocity, etc. with contraction are calculated from the initial state, which is taken to be the state when the stars flared up to the Helmholtz-Kelvin contraction. It is shown that the exponent s must be in the range from – 1 to – 3 so that the stars with adequate strength of the initial magnetic field may lose almost all of their angular momenta in a suitable rate if they are initially in the state of rotational instability.

Stellar rotation from the time of star formation to the main sequence stage is discussed. Also, the formation of the solar system and other planetary systems is discussed, with respect to the braking.

In discussing the status of the theory of rotating stellar atmospheres, it is necessary to draw upon the contributions of many well established aspects of astrophysics and to interconnect them in a cohesive pattern structured so as to provide insight into a rather specific problem – namely, the structure and characteristics of a surface of the star undergoing axial rotation. Many different connections are possible having varying degrees of emphasis and, of necessity, those given here represent only one such presentation. The discussion could be much simplified if it were not necessary to test the efficacy of the theoretical development by referring to observations. Unfortunately, such a comparison is necessary and the results are at the moment somewhat inconclusive. This unhappy situation arises from the retrospectively obvious fact that axial rotation does not play a dominant role in determining the directly observable properties of stars. Indeed, if rotation were a dominant factor, earlier attempts at describing stellar structure and evolution would have met with little success. However, it is becoming increasingly clear that the structure and final evolution of highly evolved stars are greatly influenced by the total angular momentum which they retain from their earlier history. In order to understand the angular momentum distribution present in the final state, it is necessary to understand the effects of stellar evolution on the total angular momentum and its distribution. But even before this step can be taken, one must first successfully describe the rotational structure of the main-sequence phase as it is this state which provides the initial conditions necessary for any further study. It is further appropriate that we attempt to describe this period of a star’s life as the largest body of observational material with which we must test our results exists for these stars. In addition, we should expect a study of the atmospheric structure to be the most fruitful as this is the region of the star which provides the final modification of the radiation we observe.

The effect of rapid uniform rotation on equivalent widths of spectral lines is studied in the case of a star seen pole-on. Contrary to the situation for the strong lines MgII 4481 and SiII 4128, the effect may be significant for the weak ones, especially those of the EuII-type. Because the mean atmospheric parameters of a rapidly rotating star seen pole-on are very similar to those of a non-rotating star of the same mass, this may cause significant errors in abundance determinations, especially for Eu-type elements. The aspect effect on the central intensity of two EuII lines of different strengths is also studied.

For 121 southern stars of spectral types between 05 and A3, including 35 supergiants, equivalent widths and values of ve sini have been measured for absorption lines on direct-intensity tracings of coudé spectrograms from Mount Stromlo, with original dispersion 150 microns per angstrom. For stars of a particular temperature and luminosity class, the strength of the hydrogen lines is weaker and the triplet helium lines stronger in fast rotators than in sharp-lined objects which are rotating more slowly. Full details are in print (Monthly Notices Roy. Astron. Soc. 144 (1969), 1): subsequent papers in press present detailed profiles of the Balmer lines for 23 slowly rotating stars and details of emission features shown by several fast rotators.

The investigation is being continued with spectrograms of northern B stars secured at the Dominion Astrophysical Observatory, supported in part by grant GP-13544 from the National Science Foundation.

Models of rotating stellar atmospheres have been constructed for masses appropriate to middle and late A stars, and some of the photometric properties of these models are discussed. A specific model for the rapidly rotating star Altair has been constructed matching the observed radius, luminosity, and projected rotational velocity. It is found that the energy distribution predicted for the model agrees well with that observed for Altair, after the latter has been corrected for line blanketing.

A statistical investigation has been made of rotation and macroturbulence in early type la and lab supergiants. The principal results are: (1) At all spectral types in the range B0-A5 both rotation and macroturbulence contribute to the observed line broadening. In the early B stars rotation is as important a broadening agent as large-scale mass motion. In the middle B and A stars turbulence dominates but there still is an appreciable contribution from rotation. (2) In spite of the fact that the stars observed seem to be losing mass, there is no strong evidence for significant angular momentum loss.

When studying the axial rotation of the components of binary systems, we ask the following fundamental question: How much is axial rotation affected by the other star? In particular, is there any synchronism between the periods of axial rotation and orbital revolution?

Thus in fact we are more interested here in the angular than in the linear velocity of rotation. It was pointed out by McNally (1965) that the angular velocity of rotation reaches its maximum near A5 and drops off rather rapidly on both sides so that the GOV and 05V stars have approximately the same average period of rotation. The accompanying Table I is an adaptation of McNally’s figures.