§1: Introduction:
Beta Orionis, Rigel, is a very impressive star any way you look at it. The brightest star in Orion, one of the brightest and most striking constellations in the sky, Rigel ranks seventh in visual magnitude of any star seen from Earth. Physically, the star is huge and impressive being 74,000 times the luminosity, 62 times the diameter, and approximately 20 times the mass of our own sun. This B supergiant lies 228 parsecs away and has a surface temperature of 11,380 degrees Kelvin. All in all, a giant among stars.
Like many B supergiants, Rigel is known to have emission features, characteristic of an extended atmosphere. While all stars (presumably) have some sort of boundary layer where the pressure and density of their material asymptotically approaches zero, in B stars the boundary layer tends to be much denser than in stars such as the sun. Also in contrast to the sun, stars such as Rigel may have atmospheres cooler than their photospheres. This leads to observable spectral effects. Light from the hot stellar photosphere suffers line absorption in the thick, cooler atmosphere. Furthermore, the atmosphere itself is hot and emits radiation. If the optical depth is high enough both absorption of photospheric light and emission may contribute significantly to the observed spectrum.
Early type stars with emission components can be divided into five categories. All are surrounded by some sort of cool, dense extended atmosphere which produces the lines. Of stars comprise roughly 15% of O type stars and all have spectral class O8 and earlier. Wolf-Rayet stars are not of concern either because their spectral are dominated by broad emission bands rather than the mixed lines seen in Rigel.
Be stars are B-type stars with emission features and make up about 15% of that population. They typically rotate rapidly which appears as a double emission line bracketing the hydrogen absorption line. The emission occurs most strongly in the lowest orders of the Balmer series but varies between individual stars. Good examples of Be star spectra can be found in Underhill (1960). Be stars have been observed to vary in spectral profile, but the timescale is 5-10 years and is not periodic.
Shell stars are similar in nature to Be stars. The rapid rotation of such a star forms a shell of gas extending from the equatorial region. A good estimate of the mass of the shell is 10-9 Mo (Underhill, 1960). Angular momentum is conserved in the shell, so the rotation slows with increasing radius. Bulk radial velocity can be detected in some shell stars. In the case of 48 Libra, the velocity was seen to change up to 70 km/sec with a period of about 10 years. However, the cause of these variations is not known.
Finally, some O and B-type Supergiants are seen to have emission characteristics. These can be explained by a spherically symmetrical expanding atmosphere of moderate density. Typical of this is the archetypal star P-Cygni with its sharp emission peaks and short wavelength-displaced absorption cores. This is the group which Rigel is thought to be a member of.

§2.1: A Simple Theory:
It is the nature of stellar atmospheres to move radially outward from the surface of the star. This is necessary because hydrostatic equilibrium does not hold in the coronal region and must be replaced with a hydrodynamic equilibrium. Consequently, the atmosphere moves outward and the star loses mass. If the material causing line absorption of photospheric light is moving with respect to the photosphere, we can expect that absorption to be doppler-shifted with respect to the stationary absorption line. If the same moving gas is producing line radiation, we can expect the emission line to be doppler shifted in the opposite sense. This is discussed in more detail below.
We can model a massive star suffering from mass loss (such as Rigel) as a star (producing its usual bevy of absorption lines) surrounded by a thin, radially moving shell of gas (Figure 1) . By dividing this model up into eight sections as shown, the spectral contributions can be analyzed and then summed to create a composite spectrum.
Area A is the most complex featuring both standard absorption from the star itself plus a blueshifted absorption from the expanding mass shell. It also contributes a blue-shifted emission component. Similarly, section B contributes blue-shifted emission, but since it is not eclipsing the stellar disk, there is no absorption. Section D is similar to B except it is receding, thus a redshifted emission line. Section C is emitting, but, since it has no radial velocity with respect to the observer, there is no doppler shift. Area E would produce the strongest redshift, however, it is obscured by the star itself and contributes nothing.
By summing these contributions (after assuming reasonable velocity and coefficients of intensity) we get Figure X; a classic P-Cygni profile. Notice the fact that the minimum of the absorption component has been blue-shifted by an amount proportional to the radial velocity. This fact may prove observationally valuable.

§2.2: A More Detailed Theory:
The simple theory is adequate for explaining the qualitative behavior of a system. But for qualitative analysis, a more detailed model is in order. We must take into account the spherical symetry of the system, the temperature, velocity, thickness, density, and radius of the shell.
To start with, we can concider the contributions to the spectrum from each microscopic peice dV of the shell moving in a radial direction. By integrating the contributions from all visible points on the shell, a spectrum is obtained. This should give the emission from the system as a function of wavelength (half of the spectrum).
Next the absorption characteristics can be added in by adding the absorption line function from the star to that provided by the shell where it overlaps the stellar disk.
The source function for the emission (shell only) is simply the integral over the sphere of the source function of each element. If we assume a simple thermal gaussian profile from each volume element, then

where v is the radial expansion velocity, and the angular size of the stellar disk is
.
Completing the integral yeilds the solution
.
By changing the limits of integration (and assuming that the atmosphere is cooler than the photosphere) we can get a function for the spectrum of the absorption line
.

Do we need to take "self-absorption" into account from the shell? In other words, will the forward (observer-ward) sections of the emission lobe shell absorb the light from the rear sections? If we assume that the shell atoms have only a few energy levels and thus a small number of widely spaced transitions (a valid approximation for hydrogen and helium) then the emitted light from the back side of the shell will be doppler shifted to a different frequency out of the absorption range of the front side of the shell. This is not the case, however, for materials in front of the stellar disk (the absorption lobe. While the radiation falling on them is doppler shifted, it is continuum radiation from the photosphere and will certainly fall into the absorption range.
In terms of trying to actually apply this theory to the observed situation in an attempt to interpret the data, we discover a great many problems. There are many unknown variables, namely Xdelta nu, Xdelta, dR, C1, and C2 which are not defined and yet produce dramatic changes in the functions observed.
Xdelta nu is dependent on temperature which is impossible to determine with observations of only one spectral line. Observations of several lines in the Balmer series, for instance, would give relative excitative abundances, from which, given a density, a temperature could be determined. C1 and C2 are unknown constants involving optical depth and emmision rates. These will depend on density, wavelength, and temperature at least. dR is actually an artefact of our model only and probably has no real meaning in the case of a real extended atmosphere star. Finally, Xdelta is unknown though Burbidge and Burbidge (1956) suggest that the radius of a Be star may be only one tenth that of its atmosphere.
With these limitations in mind, it is probably best not trying to work out expected spectral profiles without a much more detailed model.
Such a model was attempted by Castor and Lamers (1979) and may prove to be useful in a more detailed analysis.

§3.1: Observations:
Three sets of data were used for this study. The first includes over fifty spectra from January through April, 1988. These were obtained by Dr. John Gaustad and a number of students using the 24" Polar Heliostat telescope and a high dispersion spectrograph at Swarthmore equipped with a Reticon detector. Unfortunately, the data exists only in printed form (in the form of spectral curves) now and is thus very difficult to gather quantitative information from. Some quantitative data was gathered by measuring (with a ruler) the data, but its accuracy is inferior to digital data.
More recent data were obtained between November 20 and December 12, 1994, and between January 25 and April 7, 1995, with the same instrument equipped with an SU200 thermoelectrically-cooled, 16-bit CCD camera. This data was saved in digital form with IPLab (v 2.3.1 for Macintosh, reduced and analyzed on a Power Macintosh computer using IPLab and Kaleidagraph (v 3.0). Visualization work was done with the help of MatLab (v 4.2d).
The actual observations were carried out as follows: The star was centered on the spectrograph slit (width varied between 450 µm and 150 µm) where it could be observed through a small telescope mounted off axis to the main beam path (see Figure X). A hydrogen comparison lamp located on a table nearby fed light into an optical fiber which terminated at the slit. This served the double purpose of illuminating the slit for lignment work and providing calibration hydrogen lines of zero velocity. The light from the slit (both stellar and comparison) was then passed through an interference filter (centered on 6576.5Å with width 97.1Å and transmission 50%) and passed into the spectrometer. The filter was found to occasionally produce interference in the image plane by Fabry-Perot internal reflection. Thus it was often placed at a slightly oblique angle to the image plane. A collimating mirror focussed the light at an extremely oblique angle onto an eschelle diffraction grating with line spacing of 316 lines/mm. The 9th order diffraction was selected by a parabolic camera mirror which reflected light to the CCD. The detector is a Photometrics SU200 512x512, 23 µm/pixel chip cooled thermoelectrically to -45 degrees Celsius. Total dispersion at 6563Å was .0515 Å/pixel or about 3 km/sec of radial velocity.

Generally two or three exposures were taken with a total integration time of 24 minutes although a few nights were longer or shorter due to weather and technical difficulty. This procedure gave significantly higher signal-noise ratios than a single integration of equivalent length would have.

§3.2 Data Reduction and Presentation:
Reduction of the 1994-95 data was accomplished by subtracting readout noise and thermal noise (approximately 253±1 and 1±2 counts/pixel respectively) from each image. A program was written which collapsed the spectral data from three to two dimensions and then exported it to text format where it was pasted into a spreadsheet. When each image had been thus reduced, the spreadsheet columns were summed for the total spectrum. The comparison line from each image was similarly treated and appended to the nightly spreadsheets.
To compare different night's data, each spectrum was normalized to a continuum level of one and the spectrum-comparison line pair were shifted so comparison lines lay at the same pixel value. Finally, the doppler shift due to the orbital velocity of the earth was removed from each spectrum. This correction turned out to be a somewhat substantial 10 pixels in some cases. The result was a set of normalized, zero velocity spectra covering some 120 days.
Since the primary purpose of this investigation was to search for a periodicity in Rigel's emission behavior, I wanted to look at the data as a whole. Unfortunately the data was not contiguous for the entire period, observing having been interupted by bad weather, equipment failure, and absence. To solve this I interpolated over spots with no data using a simple linear interpolation between real data points. This approximation seems fairly valid because there is often little change between nights. It should be noted, however, that the interpolation scheme becomes less and less valid as the timespace between real observations grows larger. It does seem like a useful, if only qualitative tool. No interpolated data was used in determining the actual quantitative behavior.
Reducing the Reticon data from 1988 was accomplished by measuring to the best of my ability (about 1 mm) the heights and relative wavelength positions of various spectral features. This produced some surprisingly nice results.

§4: A Look at the Data:
If the complete set of data is examined a period shows. Though it is less obvious in the 1994 data (figure 4.1), the 1995 data (figure 4.2) shows a definite pattern. Notice the pattern of weak double emission followed by a strong red emission. Notice also how the apparent width of the absorption line changes over the cycle.
Figure 4.3a shows a little more concretely the periodicity. In it, the heights of the blue and the red emission features from each observation is represented. Notice that there appear to be three "peaks" where the red features become very intense.
If the whole data set is "wrapped" around upon itself with a period of 38 days, it looks like figure 4.3b. Here we can see that the

§5: Conclusions:
Rigel's spectral emission does indeed seem to be periodic in nature. This is shown first in figures X and X, the three-dimensional plots of the real and interpolated data. Measuring certain parameters, such as peak height and absorption line edge position, allows the data to be "wrapped" - superimposed with itself offset by a time lag. This technique, represented in figures 4.3b, 4.4b and 4.5b, shows that some of these parameters are roughly repetetive although there are minor variations. On top of this, the best period for the 1988 data appears to be 30 days while, seven years later, the period has shifted to about 38 days.

radial vs non-radial pulsation
What type of star is this anyway?

Acknowledgements:
The author would like to thank Makiko Sakai of Bryn Mawr College for helping out with the observations along with all the people who kept me from going insane in the night. John Gaustad's wisdom and experience helped me to understand what I was seeing and when I was being foolish. Wulff Heitz provided a great many facts and resources to help me understand the subject better. John Boccio provided the visualization software as well as lots of help using it.

References:
Bohm-Vitense, Erika, Stellar Astrophysics, Volume 2: Stellar Atmospheres
Leitherer, Claus, "H as a Tracer of Mass Loss from OB Stars", ApJ, 326, 356-367, 1988 March 1.
Underhill, Anne B. "Early Type Stars with Extended Atmospheres", in Stellar Atmospheres, Greenstein, Jesse L, (ed.), University of Chicago Press, 1960.
Castors, J.I. And Lamers, H.J.G.L.M., "An Atlas of Theoretical P Cygni Profiles", ApJSS, 39, 481-511, 1979 April.
Maeder, André, "Radial and Non-Radial Pulsation sin Wolf-Rayet Stars and Supergiants," in IAU Highlights of Astronomy, volume 7, 273-279, 1986.
Lasala, J. And Kurtz, M.J., "A Fast, Reliable Spectral Rectification Technique," ASP, p605, 1985, July.
Underhill, A. And Doazan, V. (Eds.), B Stars With and Without Emission Lines, Monograph Series on Nonthermal Phenomena in Stellar Atmospheres, NASA, 1982.
Roth, G.D. (trans. Augensen, H.J. and Heintz, W.D.), Compendium of Practical Astronomy, Volume 3, Springer-Verlag, Germany, 1994.
Saleh, B.A.E. And Teich, M.C., Fundamentals of Photonics, John Wiley and Sons, Inc., 1991.
Burbidge, E.M. and Burbidge, G.R., Vistas in Astronomy, ed. A. Beer (London and New York: Pergamon Press, 2, 1470, 1956.