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

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of Fox and Wood (1985) [31] are qualitatively similar to those of Huguet and Lafon. However,
they extend further in the wake of the shock and describe a wider variety of initial
conditions. Here, the shocked system returns to equilibrium about 10 6 m behind the shock
front. To estimate the effect of shocks on 11 µm intensities from the star, it is necessary
to determine if shocks passing through the stellar atmosphere are transparent at 11 µm, or
if they contribute to the flux. The photosphere of the Höfner model reaches 11 µm optical
depth unity at radii no less than 11 mas (from Figure 3.17.) At this radius, the density (in
the pre-shocked state) is about 10 18 m −3 or 2 × 10 −12 g/cm 3 , the pre-shocked temperature
is about 2500 K and the velocity discontinuity at the shock front is about 30 km/s. This
should serve as an upper limit to the shock strength present in Miras.
A shock propogating at 30 km/s into a medium having a temperature of 3000 K
and a density of 10 −11 g/cm 3 was used to estimate the maximum effect that a (non-LTE)
shocked region might have on 11 µm radiation. It should be noted that the pre-shocked
density is 5 times higher than the upper limit mentioned above (creating 25 times more freefree
opacity per unit distance than might be expected.) The model considered is from Fox
and Wood (1985) [31], Figure 8. The kinetic temperature, density, and ionization fraction
of the gas after passage of the shock is plotted in Figure 3.20. The shock initially heats up
the gas to very high (∼30,000 K) kinetic temperatures. Within about 1 ms after the shock
front has passed, the hydrogen gas behind it starts to ionize, absorbing most of the thermal
energy of the shock, and lowering the temperature. As the temperature cools, the gas
contracts, and the density increases. Eventually, variables relax to equilibrium values. The
11 µm opacity was calculated from this model including only free-free hydrogen processes.
The dominant opacity source is from free electrons scattering off free protons (H ff .) The
opacity is also shown in Figure 3.20 along with the integrated opacity, or optical depth of
the post-shock region. The 11 µm optical depth of the entire shocked region in this case is
about 0.004. Since this was an upper limit to the shocks we might observe in a Mira, we
can conclude that the thin non-equilibrium shocked region has negligible optical depth at
11 µm and ignoring it in Section 3.4.3 was justified. This is not to say that the shocks can
be ignored completely. The outcome of the passage of a shock is a significant increase in
temperature and density. These increases are accounted for in the Höfner model, however,
and their effect on 11 µm opacity may be adequately treated in LTE.