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

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depth are the contours plotted against τ gas and the intensity ratio for three discrete values
of R gas .) We see in Figure 5.2 that it is possible to have either absorption or emission if
the gas shell is located at large enough radii (assuming the gas is cooler than the star.)
If we assume some column density of H 2 O in the thin gas shell, we can calculate
the optical depth for each of the many observed lines from a list of molecular line strengths.
Hence, the choice of some column density, N H2 O, along with a temperature, T gas , and gas
shell radius, R gas , allows us to calculate the fractional line depths for all of the observed
lines.
(We have tacitly assumed a line width of 0.018 cm −1 and a stellar blackbody of
temperature 2500 K. 2 ) By minimizing the differences between these calculated values and
the observed line depths, we can find the best fitting simple gas shell corresponding to
the observed spectra. This process has been applied to the observed absorption lines in
R Leo and the observed absorption and emission lines separately in o Cet.
(Assuming
the two line components in o Cet have different velocities, they will be transparent at the
characteristic wavelength of the other component and can be analyzed separately.) The
results of the fitting to R Leo’s lines are shown in Figure 5.3. Plotted is the best fitting
theoretically calculated absorption line depths vs. the observed absorption line depths for
the 56 strongest measured lines. The best fitting gas shell for R Leo had a radius of 1R ∗
(just beyond the blackbody photosphere), a temperature of 1593 K, and a column density of
3.14×10 18 cm −2 . The standard deviation of the error of this fit is 0.06 (while the absorption
varies between 0.60 and 1.00) indicating a relatively good fit. However, it appears that the
calculated line depths are systematically deeper than the observed lines for the strong lines,
and shallower for the weak lines. This is not unexpected since the model of an infinitely
thin shell is unrealistic.
If the shell instead had some thickness, the strong lines would
saturate at the edges of the star at slightly larger radii than the weak lines, and the increase
in size of the source would result in a higher flux and a lessening of the absorption depth.
This matches, to first order, the deviation we observe. The best fitting H 2 O gas shell for
the absorption component of o Cet contains a column density of 1.90 × 10 18 cm −2 , has a
temperature of 1707 K, and is at a radius of 1R ∗ . The emission line component of o Cet’s
spectra is best fit by a gas shell of temperature, 1075 K, radius, 1.883R ∗ , and column density,
1.37 × 10 19 cm −2 . The detailed form of the H 2 O spectral lines in o Cet stems from a gas
2 It is actually the intensity ratio, B ν(T gas)/B ν(T ∗), which is fit, and the gas temperature is known only
in terms of the stellar effective temperature which we have supposed to be 2500 K for convenience and
reference.