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

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of its three parameters. The contours represent the fractional change in size and they are
plotted against τ gas and the intensity ratio for three discrete values of R gas . Immediately,
we notice that a gas shell may increase or decrease the apparent size of a star depending
on its location, depth, and temperature. It is even possible for a single shell of gas to do
both! (A shell of gas at 1.2R ∗ with intensity ratio, 0.04, will appear to enlarge the star
for transitions with optical depth greater than about 2, or shrink it for weaker transitions.)
Comparing Figure 5.5 with Figure 5.2, we see that the effect a spectral feature might have
on a diameter measurement is not necessarily correlated with the size of the feature. For
instance, a gas shell of radius 2R ∗ , optical depth 0.30, and intensity ratio 0.12, would not
be evident in the spectrum because it has a line depth of 1.00. 3 However, the same gas shell
would increase the apparent size of the star by almost 15%. Hence, estimating the effect
a spectral feature will have on the measured diameter involves making some assumptions
concerning the location and temperature of the gas source.
The most frequently used bandpass with the ISI is centered around the P20 transition
in the CO 2 laser at a wavelength of 11.149 µm with a half-width about 0.001 µm(or
wavenumber 896.94 ± 0.08 cm −1 ). The accuracy of the bulk of our diameter measurements
depends on the transparency of the stellar atmosphere in this bandpass. The strongest H 2 O
line in the P20 bandpass for o Cet is at 897.03 cm −1 . Assuming the best fitting density,
temperature, and radii for each of the two shells from above, this line has an optical depth
of 0.00475 for the inner (absorption) shell, and 0.00412 for the outer (emission) shell. The
model implies absorption and emission lines of depths 0.997 and 1.012, respectively. (This
amount is within the noise level of the spectra from Figure 5.1 and is consistent with not
being observed.) The model also implies a change in apparent size of the star of -0.10% and
+0.23% on each line, respectively. If we assume each line has a width of 0.03 cm −1 and the
ISI bandwidth is 0.167 cm −1 , averaged across our bandpass, the integrated effect would be
only 0.023% increase in size. For R Leo, a slightly stronger line with optical depth 0.0121
is within the P20 bandpass. 4
This line would reduce the diameter of R Leo by 0.29% on
the absorption line or 0.05% when averaged over our bandpass. Although the H 2 O lines
in α Ori and α Her were not strong enough to be fit with a gas shell model, the strongest
water line in their case was the same one as from R Leo, and we anticipate no effect due to
3 Actually, the edges of the spectral feature would have lower optical depths while the other two parameters
were constant. So, the star would exhibit about 3% emission in the spectra on both sides of the spectral
line.
4 R Leo and o Cet have different redshifts, so their spectral lines are at somewhat different wavelengths.