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

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changes and asymmetry in the velocity field of α Ori’s atmosphere is presented in Lobel and
Dupree (2001) [62]. So, it may be that hydrostatic radiative transfer modelling of α Ori is
insufficient at reproducing all of the observable features. Particularly, a hydrostatic theory
tends to under-estimate the extension occuring in a dynamic stellar atmosphere.
The long period variable models of Bowen (1998) [15], Beach et al. (1998) [7],
Bowen (1990) [16], and Bowen (1992) [17] provide a description of density stratification,
temperature, and gas velocities in the atmosphere surrounding a driven photosphere. High
temperature shock fronts, with temperatures as high as 10,000 K, are predicted to expand
outward radially. The velocity discontinuity of the shock front is a function of the stellar
mass, driving period, piston amplitude, and effective temperature, but typically reaches a
maximum value between 20 and 30 km/s. The other notable effect of the pulsation is an
increase in the density outside of the photosphere relative to hydrostatic predictions. A
plot of the density vs. radius (predicted by the “standard” Bowen model) for five different
driving amplitude values is shown in Figure 3.10. This is a reproduction of Figure 1
from Bowen (1988) [15]. The effect that the pulsation has on the density distribution is
clear. Increasing the piston velocity amplitude causes significant extension in the stellar
atmosphere. Consequently, the density (and optical depth) of outer atmospheric layers is
increased by several orders of magnitude. A piston amplitude of 3.5 km/s was chosen by
the author as the “standard” because of its correspondance with observable post-shock radiation.
The “standard” mass, effective temperature, and period were chosen to be 1.2M ⊙ ,
3000 K, and 350 days, respectively. At this amplitude, the density distribution is roughly
hydrostatic down to about 10 −13.5 g/cm 3 at which point it decreases much more gradually
than the hydrostatic model. We also see three spikes in each of the density plots at radii
corresponding to the location of three outgoing shock fronts. These spikes travel outward
and are damped as they expand.
The Mira models of Bessell et al. (1989) [9] and Bessell et al. (1996) [10] also show
significant atmospheric extension relative to its static “parent” star. A variable star with
period, 330 days, mass, 1M ⊙ , and “parent” effective temperature of 3020 K, is predicted to
have luminosity varying between 2200 L ⊙ and 5000 L ⊙ , and effective temperature between
2700 K and 3050 K. The Rosseland radius 2 of the model varies between 0.9 and 1.09 times
the parent stellar radius of 236R ⊙ . Shocks are also formed in the atmosphere of the Bessell
2 Radius at which τ Rosseland (r) = 1.