In this paper we review the development of the concept of the stellar rotation parameter commonly known as v sin i. We emphasize that the interpretation of the parameter in terms of physical characteristics of the star always depends on comparison with a model that is intended to represent the physical properties of the star. To that end we will trace the development of such models along with the observational means of determining the parameter. Emphasis will be place on the traditional methods involving stellar spectroscopy, but some attention will be place on indirect methods involving direct measurement of the rotation period and recent interferometric determination of stellar oblatness. In addition we will comment on recent techniques involving the simultaneous measurement of many spectral lines and synthetic spectra to improve the accuracy of rotational half-widths.

The natural desire for simplicity of such models has often resulted in erroneous values for stellar parameters. This is particularly the case for the most rapidly rotating stars generally of early spectral type, but may also be present in some giants and supergiants where rapid rotation is difficult to detect. Finally, we will comment on the possibilities of improving the quality of both the measurement and interpretation of this important stellar rotation parameter.

Performance of the cross-correlation function (CCF) is compared with the linear, algebraic equations, least-squares deconvolution directly giving the broadening function (BF). The many disadvantages of the CCF do not outweigh its simplicity so that the BF approach is strongly advocated.

The UV spectra predicted by LTE line-blanketed model atmosphere calculations are used to obtain a preliminary calibration of FWHM vs. v for UV lines, which is then used to determine the values of v sin i for four O-type stars in the Small Magellanic Cloud.

We study the effects of a differential rotation upon the determination of the v sin i parameter. The effects are studied for several values of the ratio t = kinetic energy/gravitational potential energy, which include energy ratios higher than permitted for critical rigid rotation and using an internal conservative rotation law that allows for a latitudinal differential rotation in the stellar surface. Two effects are outstanding: when differential rotation is dependent on the stellar latitude the v sin i parameter does not necessarily correspond to the equatorial rotation velocity; the line width is a double valued function of v sin i and it is dependent on t and the aspect angle i.

Fast rotation is expected to flatten the star and to produce non uniform temperature and density distributions (i.e. gravitational darkening). While the flattening mostly increases the absolute flux level of the energy distribution, gravitational darkening makes an equator-on star apparently cooler than a star seen through the pole. Both effects (Collins et al. 1991) influence the colours and the location of the star in the HR diagram but also, in a more subtle way, its spectral line profiles. More particularly, in early B type stars, gravitational darkening tends to privilege at the poles the formation of the ions with the highest ionization potentials and directly affects line formation. Consequently, most spectral line shapes - and especially the weakest ones - become aspect angle dependent which in several cases may play a role in the fundamental parameter determination procedures or even in the determination of stellar chemical abundances.

We present a modification of our code for calculation of NLTE model atmospheres in planar and spherical geometry to the case of rotating atmospheres. This modification enables self-consistent inclusion of the effect of rotation on the atmospheric structure.

Accuracies reached in space astrometry now permit the accurate determination of astrometric radial velocities, without any use of spectroscopy. Knowing this true stellar motion, spectral shifts intrinsic to stellar atmospheres can be identified, for instance gravitational redshifts and those caused by velocity fields on stellar surfaces. The astrometric accuracy is independent of any spectral complexity, such as the smeared-out line profiles of rapidly rotating stars. Besides a better determination of stellar velocities, this permits more precise studies of atmospheric dynamics, such as possible modifications of stellar surface convection (granulation) by rotation-induced forces, as well as a potential for observing meridional flows across stellar surfaces.

Observational v sin i line-width measurements for OBA stars are summarized, and the relationship of these measurements to ve sin i, the true projected equatorial rotation velocity, reviewed. In general, ve sin i ≥ v sin i for Be Stars, while ve sin i ≤ v sin i for O-type stars. The implications are that (i) it is probable that Be Stars are closer to their critical rotation velocities than is generally supposed (rendering plausible a range of new physical mechanisms for the Be phenomenon); and (ii) some non-rotational macroscopic line broadening (‘turbulence’) occurs in the atmospheres of O stars (and BA supergiants). Physical interpretations of this ‘turbulence’ are considered.

As a first step to study the influence that rotation has in the evolution of the most massive stars, we obtained the projected rotational velocity (v sin i) of a sample of OB stars from clusters and associations. We found that most of our stars present v sin i lower than 200 km/s.

We present preliminary results of a long-term spectroscopic monitoring of a magnitude-limited (V < 7.5) sample of OB-supergiants (07.5-B9) aimed at establishing the incidence of co-rotating, large-scale wind structures. In the optical, this can be achieved by detecting rotationally modulated variability in Hα. Dramatic line-profile variations operating on a daily (and in some cases on a hourly) timescale are observed. Firm conclusions regarding the origin of the variability must, however, await a detailed period analysis. There is no clear evidence for a causal link between photospheric and wind activities.

The stellar winds of OB stars are perturbed by Corotating Interaction Regions (CIRs). We present a 3D radiative transfer model which predicts that variations should occur in the infrared and millimetre continuum.

Line driven wind in late B stars causes significant heating of layers just above the photosphere. This large heating is a source of a thermal wind, which together with fast rotation opens the possibility of a formation of a disk. We refer to such disk as to a radiation induced disk. Disk inhibition by nonradial line force has only little effect in this case, since absorption in spectral lines is not the main wind driving mechanism.

We have re-determined in an homogeneous way v sin i values for all the B-type stars of the Bright Star Catalogue with southern declinations.

Projected rotational velocities for 67 B, A, and early-F stars have been determined. Some of these stars are early-type, radial-velocity standard candidates. The spectra of seven stars show metallic lines with composite profiles that consist of a narrow component near the center of a broad component, suggesting that they may be shell stars or binaries.

The different velocity distributions (v sin i, and v) are determined for single, non peculiar, A-type main-sequence stars, using homogeneous data. These distributions, together with synthetic corrections on stellar magnitudes due to the rotation, are used to quantify the global effect of rotation on a simulated A-type main-sequence in the H-R diagram. The resulting effect is lower than (0.03, 0.006) in (B – V, MV) for 95% of the stars.

Open clusters offer us the means to study stellar properties in samples with well-defined ages and initial chemical composition. Measurements of the projected rotational velocities for large numbers of cluster stars are becoming practical with the advent of multi-object spectroscopy, and such data make it possible to study the time evolution of the rotational distributions of massive stars. Investigations of these rotational distributions provide information on the angular momentum distribution in newborn clusters, the rotational spin-down due to wind loss, magnetic fields, and binary companions, and the incidence of spin-up due to mass transfer in close binaries. Here we review some of the recent observational studies and highlight some preliminary results from our work.

We constructed line profile models for young OB stars in M17 using TLUSTY and SYNSPEC, including the effects of rotational flattening and gravitational darkening. The values of v sin i for 8 member stars (with spectral classifications O4.5 V to O9 V) range from 7 km s–1 to 168 km s–1. The mean value of v sin i (102 km s–1) is lower than expected for very young clusters.