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Astron. Astrophys. 325, 709-713 (1997)

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2. Method and model assumptions

I use an Eulerian one-dimensional, spherically symmetric, time-dependent radiation hydrodynamic code based upon the method of characteristics (Cuntz & Ulmschneider 1988). The code is suitable to study the propagation of shocks, which are treated as discontinuities. Boundary conditions for incoming and outgoing shocks are solved. Ionization of hydrogen is explicitly taken into account assuming a 3-level atom. It has been considered in a fully consistent manner both in the equations of thermodynamics and hydrodynamics. Noninstantaneous effects of hydrogen ionization have been omitted. The Lyman- [FORMULA] transition is computed by using first order escape probabilities as described by Hartmann & Avrett (1984). At the outer boundary of the atmosphere, the Lyman- [FORMULA] optical depth is assumed as [FORMULA] motivated by the Hartmann & Avrett model. Radiation damping is considered in the effectively thin plasma approximation based on a Cox and Tucker type law as given by Judge & Neff (1990). Judge (1990) presented theoretical arguments that the effectively thin plasma approximation should work well for chromospheric layers of the inactive, low-gravity star studied here.

I assume the following stellar parameters: [FORMULA] = 3750 K, [FORMULA] = [FORMULA], and log [FORMULA] = 0.0 (cgs), which are the same as used by Basri et al. (1981). The Balmer continuum brightness temperature is assumed as [FORMULA]. The initial atmosphere of the model extends up to 1.15 [FORMULA], which corresponds to about 16 pressure scale heights in the hydrodynamic models. The atmospheric extent has been chosen to encompass the scale length where the major part of the mechanical energy dissipation of the shocks occurs. For my study I calculate the propagation of shock waves with an initial wave amplitude of 0.273 Mach corresponding to an energy flux of [FORMULA] ergs cm-2  s-1. This wave energy flux is commensurate with measurements of chromospheric emission losses (Judge & Stencel 1991), which are believed to be directly related to the mechanical energy input. The shocks at a mass column density of log m = 1.08 (cgs), which corresponds to a position in the middle photosphere. The initial atmosphere is assumed to be isothermal with [FORMULA]  K. The reason for choosing shock waves over sinusoidal waves at the inner atmospheric boundary is due to findings of de Jager et al. (1991), who argued in favor of shock waves while trying to constrain photospheric motion fields of super- and hypergiant stars considering microturbulence data and wave travel restrictions.

Regarding the wave periods of the models, the following considerations need to be taken into account: Bohn (1984) found that the maximum acoustic energy flux occurs at the driving period [FORMULA], where [FORMULA] is the acoustic cutoff period, [FORMULA] is the sound speed at the top of the convection zone, [FORMULA] is the ratio of the specific heats and [FORMULA] is the stellar surface gravity. Considering log [FORMULA] = 0.0 given above, [FORMULA] can be estimated to [FORMULA] s. As the [FORMULA]  Ori surface gravity is uncertain by at least a factor of 5, this value should not be considered reliable. In addition, the parameter [FORMULA], which denotes the position of the energy maximum in the acoustic frequency spectrum, is also largely unconstrained due to uncertainties in the calculation of the initial acoustic energy spectrum from the model of stellar convection (see e.g. Ulmschneider et al. 1996). These uncertainties are particularly relevant for stars of low gravity. Reasonable estimates of [FORMULA] range from 0.2 to 0.5. Fortunately, the results for the acoustic shock wave models are not very sensitive to the shape of the frequency spectra anyhow, which reduces the relevance of above-mentioned uncertainties enormously.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998