## 2. Method and model assumptionsI 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- transition is computed by using first order escape probabilities as described by Hartmann & Avrett (1984). At the outer boundary of the atmosphere, the Lyman- optical depth is assumed as 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: =
3750 K, = , and log
= 0.0 (cgs), which are the same as used by Basri
et al. (1981). The Balmer continuum brightness temperature is assumed
as . The initial atmosphere of the model extends
up to 1.15 , 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
ergs cm 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 , where is the acoustic cutoff period, is the sound speed at the top of the convection zone, is the ratio of the specific heats and is the stellar surface gravity. Considering log = 0.0 given above, can be estimated to s. As the Ori surface gravity is uncertain by at least a factor of 5, this value should not be considered reliable. In addition, the parameter , 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 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. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |