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

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extension is reflected at wavelengths containing significant molecular absorption in which
the monochromatic radius can be more than twice as large as the continuum radius.
The variable star models of Höfner et al. (1998) [47] predict atmospheric structure
for a wide variety of luminosities, effective temperatures, piston velocity amplitudes, and
carbon to oxygen ratios. The latter parameter mostly affects the velocity at large radii
where molecular absorption from oxides is significant, and the dust which forms in greater
quantities in carbon-rich environments. As in the previous models, shocks are formed as the
density wave travels outward, and these shocks are characterized by a velocity discontinuity
of around 20 km/s. A temperature increase is associated with each shock front. The stellar
atmosphere is greatly extended in the Höfner model as well. Figure 3.12 illustrates the
structure of a 7000L ⊙ , 1M ⊙ star with an effective temperature of 2880 K, and a carbon
to oxygen ratio of 1.4, driven with a piston velocity of 4 km/s. Shown is the predicted
gas velocity, density, temperature, and degree of dust condensation as a function of radius
at different phases (reproduction of Figure 5 from Höfner et al. (1998) [47]). The density
is greater than the hydrostatic case for densities lower than about 10 −10.75 g/cm 3 . The
shock fronts are most pronounced as discontinuities in the velocity structure but are also
evident as increases in the density by an order of magnitude or so and as increases in the
temperature. These features can be seen expanding with increasing phase. Dust begins to
form around 1.9R ∗ . The prototypical mira, o Cet, is thought to have an overabundance of
oxygen relative to carbon. Höfner et al. (1998) [47] contains an oxygen-rich model which
has a structure that appears identical to the above model for radii less than 2R ∗ , and differs
at greater radii only in that dust does not form, and the velocity approaches about 10 km/s
at large radii. The O-rich model was not plotted with as great an accuracy in the paper,
however, and we will consider the carbon-rich model without dust further in Section 3.4.3.
There is experimental evidence as well that Mira variable atmospheres are extended
significantly. H 2 spectral lines were observed by Hinkle et al. (2000) [43] and H 2 column
densities higher than 5 × 10 24 cm −2 were inferred. Assuming a reasonable scale height and
radius, densities greater than ∼ 10 −11.5 gm/cm 3 are needed in the atmospheric layers cool
enough to form H 2 . The radio continuum measurements of Reid and Menten (1997) [84]
are consistent with H − ff and H 2 − ff opacity3 begin dominant in the atmospheres of Mira
variables and imply densities around 10 −11.8 gm/cm 3 at a radius about twice the optical
3 Here and elsewhere the notation X ff is used to describe the transitions X + +e − ⇐⇒ X + +e − . X − ff refers
to X + e − ⇐⇒ X + e − , and X bf refers to X ⇐⇒ X + + e − .