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

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H 2 O on their measured diameters greater than about 0.1%
As of yet, we have not taken into account the possibility that a species other than
water may be present and may contain a spectral line within the ISI P20 bandwidth. It
appeared from Figure 5.1 that α Ori and α Her and possibly the Miras also had spectral
features which were not attributable to H 2 O. Although unlikely, it would be possible for
a spectral line from one of these species to be present in our bandpass. We will attempt
to put an upper limit on their effect by examining the spectra of the P20 bandpass. The
spectra from Figure 5.1 are repeated in Figure 5.6. with a higher magnification and the P20
bandpass explicitly shown. Due to the motion of the Earth, the spectra will be redshifted
relative to this observation at other times in the year. The two sets of vertical lines denote
the edges of the P20 bandpass as they would appear at the extrema of Earth’s motion
relative to the star. The spectra of α Ori appears very flat within the P20 bandpass. The
four other stars have fluctuations in their spectra which appear slightly larger than the
noise. These “possible absorption lines” have a depth no greater than ∼8% for any of the
stars. Since the observed absorption lines in o Cet and R Leo originated in a shell just
outside the photosphere at a temperature ∼1600 K, we will calculate the effect that an 8%
absorption line will have on a diameter measurement using these values. To reproduce a
line depth of 0.92, a 1R ∗ /1600 K gas shell would need an optical depth of about 0.10. In
this case, the uniform disk diameter for the star would be reduced by 1.31% on the line, or
0.23% after averaging over the bandpass. The magnitude of the effect is not too dependent
on the choice of parameters. If the gas shell is positioned instead at a radius of 1.10R ∗ ,
with the same temperature, the star then appears 5.5% larger on the spectral line. The
maximum radius that a 1600 K gas shell could be at and still produce an 8% absorption line
is 1.30R ∗ , although the gas would have to be opaque in order to do this. If the temperature
of the gas were increased to 2000 K with the shell at 1R ∗ , the diameter fit would be 1.06%
too small, and if the temperature were 1200 K with the same radius, the measurement
would be 1.45% too small. These changes would be lowered by a factor of ∼6, or to less
than 0.25%, if a single line of width 0.03 cm −1 were present within the ISI bandpass of
width 0.17 cm −1 . With a good degree of certainty, based on the spectra and theory, we can
safely estimate any measured size change due to the extra opacity of a spectral line in the
P20 bandpass to be less than ∼1% for all of the stars considered.