The Size, Structure, and Variability of Late-Type Stars Measured ...

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due to random thermal velocities.
The procedure for calculating a spectral line shape from the dynamical Mira model
of Höfner is to calculate the intensity profile (as in Section 3.4.3) at wavelengths near
the natural spectral line wavelength. Specifically, some wavenumber close to the natural
wavenumber is chosen. The H 2 O transition will introduce opacity at this wavelength only
if the component of the velocity of the gas along the line of sight is such that its light
will be redshifted to the chosen wavenumber. At positions where the velocity satisfies this
condition, the extra opacity is added to the model and the integration over a blackbody
source function is carried out. The intensity profile resulting from the density, temperature,
and velocity structure given by the Höfner model at phase 0.3 near an H 2 O line is shown in
Figure 5.7. A dust shell forming at 2R ∗ was added for consistency with observations. The
H 2 O was assumed to exist everywhere with a mass density of 1.67 × 10 −5 times the local
density (given by the Höfner model). The specific spectral line illustrated here was chosen
to have a maximum line strength of 3 × 10 −22 cm at a temperature of 2000 K. Directly on
the spectral line (relative velocity of 0 km/s), only H 2 O gas not moving towards or away
from the observer is seen. Most of this gas is moving radially towards the star, but is located
on the sides of the star so as to have no velocity component along the line of sight. The
resulting intensity profile contains more flux at radii larger than the continuum radius. At
an observing frequency redshifted by 6 km/s relative to the natural frequency, only H 2 O
gas moving away from the observer is seen. Since the Höfner model predicts infall near the
photosphere at phase 0.3, H 2 O is seen directly in front of the star. Since this gas is at larger
radii than the photosphere, it is cooler and the intensity from the star is reduced relative
to the continuum.
The model spectral line shape is calculated by integrating the annular intensity
for a range of frequencies near the transition frequency. In the case above, the spectral line
will show emission at the natural frequency, and absorption at the redshifted frequency. Of
course, all frequencies will be redshifted an additional amount if the star has some motion
relative to the earth. The shape and size of a spectral line will depend on both the line
strength and its temperature dependance. Assuming a mass density of H 2 O of 1.67 × 10 −5
times the local density, the spectral line profiles of a variety of lines are shown in Figure 5.8.
The spectral lines were calculated for lines with strengths of 3 × 10 −23 , 3 × 10 −22 , and
3 × 10 −21 cm and having maximum opacities at temperatures of 1000 K, 2000 K, 3000 K,
and 4000 K. For low temperature lines, the cooler gas at larger radii has opacity and this