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

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from several models. The 10 µm feature stems from a resonance of the Si–O bond. We
see that the dust is much more opaque in the visible. In addition, there is a great deal of
scattering that occurs in the visible but not in the mid-infrared due to the characteristic
grain size of several hundred nanometers.
We can estimate the structure of the dust shell by assuming that the dust particles
are in thermodynamic equilibrium and are heated by the central star. In this case:
Radiation Absorbed = Radiation Emitted
L ∗ /4πr 2 = σT 4 (3.2)
Hence, if the temperature at R ∗ is T eff , then
T (r) = T eff (r/R ∗ ) −1/2 (3.3)
Suh (1999) [96] claims silicate dust condenses around 1000 K for the gas densities characteristic
of stellar atmospheres. For o Cet, with an effective temperature of 2500 K, we estimate
the inner radius of dust formation to be 6.25R ∗ . For α Ori, assuming T eff = 3300K, the
radius of dust formation is 10.89R ∗ . The estimated formation radius for o Cet is very
close to the value of 6R ∗ predicted by Lobel et al. (2000) [59] from modelling the spectral
energy distribution of Mira. However, it is much larger than the experimentally observed
inner dust shell radius of approximately 2R ∗ from Danchi et al. (1994) [22]. For α Ori,
Lobel et al. (2000) [61] estimate the inner radius of the dust shell to be ∼ 35R ∗ , whereas
Danchi et al. (1994) [22] measured a thin dust shell to exist at a radius about 40R ∗ surrounding
α Ori. Danchi et al. (1994) [22] also measured inner radii of dust formation for
R Leo, χ Cyg, and α Her, to be respectively, 1.3 − 2.3R ∗ , 15R ∗ , and 12.5R ∗ .
Much of the discrepancy between the above dust formation radii values and the
simple predictions stems from the variability of the stars considered. α Ori is thought
to exhibit non-regular variations and there is evidence suggesting that thin dust shells
form at various times and are observed expanding outward in concentric shells (Danchi
et al. (1994) [22]). For o Cet, dynamical models have shown that pulsationally driven
shocks propagating outward through the atmosphere of long period variables are followed
by an expansion which cools the gas sufficiently well to cause condensation of dust to occur
at much lower radii than would be possible in a static atmosphere (Bowen (1988) [15]). The
dynamic models of Höfner et al. (1998) [47] predict dust formation to occur at radii as low
as 2R ∗ .