Cranmer, S. R. 1996, ``Dynamical Models of Winds around Rotating Hot Stars: Pulsations, Waves, and Streams,'' at Spectroscopic Diagnostics of Small-Scale Structure in Stellar Atmospheres, CCP7 Workshop, University of Glasgow, 28-29 Mar 1996, published in the CCP7 Newsletter of the British Collaborative Computational Project on the Analysis of Astronomical Spectra, Number 23, May 1996.


Hot luminous stars (spectral types O, B, Wolf-Rayet) are observed to have strong stellar winds which exhibit variability on time scales ranging from hours to years. Many classes of these stars are also seen, via photospheric line-profile or photometric variability, to pulsate radially or nonradially. It has been suspected for some time that these oscillations can induce periodic modulations in the surrounding stellar wind and produce observational signatures in, e.g., ultraviolet P Cygni line profiles (see Fullerton and Kaper 1995 for a recent review). Here we begin to investigate the means by which stellar pulsations can propagate into an accelerating wind, and present preliminary comparisons between theoretical radiation-hydrodynamics models and observations of the wind-modulated structure in BW Vul.

Although pulsations represent a discrete spectrum of standing waves in a stellar interior, we are interested in their transformation into traveling waves in the photosphere and wind. The simple gravo-acoustic wave theory of Lamb (1932) for a static, isothermal, and gravitationally-stratified medium predicts three types of possible oscillations (see also Mihalas and Mihalas 1984):

  1. acoustic waves, which have wavelengths significantly shorter than the atmospheric density scale height H, and thus are not strongly affected by the stratification,

  2. internal gravity waves, which are primarily driven by buoyancy forces in a convectively-stable medium, but only exist for nonradial perturbations with wavelengths long compared to H,

  3. evanescent oscillations, which exist for wavelengths of the same order as the scale height, and vary exponentially with radius, instead of propagating sinusoidally.

Not coincidentally, most low-order and low-degree pulsation modes of stars are evanescent in the photosphere, and not much has been known about how these non-propagating oscillations can make their way out into the wind before damping to negligible amplitudes. We find, however, that the presence of an accelerating wind can provide the necessary impetus for evanescent modes to effectively ``tunnel'' their way out of the interior. First, in the subsonic, or near-static wind, the reference frame of the temporal oscillations is itself beginning to propagate, and this implies that a small degree of group velocity is imparted to the evanescent waves. Second, in the wind, the density no longer falls off exponentially, but much more slowly [as approx. r^(-2)], so the effective scale height grows much larger. Frequencies previously evanescent here no longer ``see'' as much of an underlying density gradient, and are free to propagate nearly acoustically. Indeed, Abbott (1980) studied the dispersion of waves in a radiatively-driven wind by ignoring all background gradients, and found no evanescent solutions whatsoever.

In addition to the standard analytic and linear (small-amplitude) methods of studying wave dispersion, we directly model the propagation of oscillations into a hot-star wind via radiation-hydrodynamical models. Using the VH-1 code described in Cranmer and Owocki (1996), we perturb the density, pressure, and horizontal velocity at the subsonic lower boundary of the wind models, and we apply the gravo-acoustic wave theory described above to specify the amplitudes and phases of these base variations. The radial velocity at the base is not specified ab initio, since the wind solution settles down to the unique mass flux and velocity law that can be driven by the line-driving radiative force. Note that we are currently constrained to impose these base perturbations at a point several scale heights above the photosphere (where the continuum optical depth is approx. 0.001 to 0.1), and the full problem of interior-to-photosphere-to-wind propagation is not yet solved self-consistently. However, of course, since most spectral diagnostics of low-order and low-degree pulsations are in fact in the photosphere, we can be reasonably assured that these evanescent modes have significant enough amplitudes at optical depths approximately equal to 1 to still survive in the subsonic wind.

Thus, models of small-amplitude base perturbations show that evanescence is indeed not a hindrance to producing wind variability correlated with stellar pulsations. The eventual amplitude of propagating waves in the wind, however, is smaller for evanescent modes than for acoustic modes. In hot-star winds, internal gravity modes disappear (and behave identically to evanescent modes) because of the presence of strong radiative cooling which rapidly damps out the temperature fluctuations required for buoyancy waves.

Preliminary models of strong (nonlinear) radial wind oscillations of the beta Cephei variable BW Vulpeculae show good agreement between observed and modeled base ``radial velocity curves'' and wind-contaminated UV profile variability of C IV 1550 (see Burger et al. 1982; Blomme and Hensbere 1985; Furenlid et al. 1987). Indeed, BW Vul pulsates with a period of 4.82 hours, which is significantly longer than its acoustic cutoff period (about 1-1.5 hours, depending on pulsational phase), and strong density shells and velocity-gradient ``kinks'' are able to propagate supersonically along with the wind each period. We are currently applying this general modeling technique to other systems, especially those which rotate rapidly and exhibit nonradial oscillations (e.g., zeta Puppis and HD 64760, extensively observed by the IUE MEGA project).


Abbott, D. C. 1980, Ap. J., 242, 1183.

Blomme, R., and Hensberge, H. 1985, Astron. Astrophys., 148, 97.

Burger, M., de Jager, C., van den Oord, G. H. J., and Sato, N. 1982, Astron. Astrophys., 107, 320.

Cranmer, S. R., and Owocki, S. P. 1996, Ap. J., 462, 469.

Fullerton, A. W., and Kaper, L. 1995, ``Variability in the Photospheres and Stellar Winds of Hot Stars: The `Photospheric Connection' At Last?'' in the Hot Star Newsletter, ed. P. Eenens, 15, 2.

Furenlid, I., Young, A., Meylan, T., Haag, C., and Crinklaw, G. 1987, Ap. J., 319, 264.

Lamb, H. 1932, Hydrodynamics (New York: Dover Publications).

Mihalas, D., and Mihalas, B. W. 1984, Foundations of Radiation Hydrodynamics, (New York: Oxford University Press).

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