Profiles of emission lines in Be stars

The broadening functions resulting from a gaseous ring in circular motion around a star according to Kepler's law have been studied. When the distribution in the gaseous ring has a circular symmetry in the equatorial plane, the broadening profile is related to the surface density distribution along the radius by an integral equation which can be transformed into the Abel integral equation and solved analytically. Profiles corresponding to gaseous rings with a uniform density distribution but different widths are used to illustrate the general properties of the profile broadened by the gaseous ring in circular motion. When the distribution in the gaseous ring lacks circular symmetry, the profile becomes not only asymmetric with respect to the center of the line but also time dependent. The variation of profiles has a time scale measured in terms of hours or days depending upon the sixe of the ring, and may explain what has been observed by Hutchings and others in some Be stars. It has also been shown that such a variation is damped by the differential rate of revolution of emitting atoms in accordance with Kepler's law of motion. The time scale of damping is determined by the radii of both the outer and inner boundary circles of the gaseous ring. It is expected that for reasonable sixes the damping will obliterate, observationally, the time variation in terms of years. That this kind of variation has often been observed only shows the existence of a continual process, such as frequent ejections of matter from the central star, that creates the uneven distribution along the circular arc from time to time. The emission profile has also been studied for cases in which the circular motion does not follow Kepler's law. It is shown that a change in the law of motion which gives the velocity as a function of radius is equivalent to a proper modification of the density distribution in the Keplerian case, provided that the velocity varies monotonically with the radius. Consequently, the integral equation can likewise be transformed to the Abel form. Finally, some unresolved problems concerning the formation of gaseous rings are briefly discussed.