The atmospheres of many stars are variable to such a degree that they are not adequately represented by either static or steady-state models. In this study, a computer programme capable of calculating completely time-dependent model atmospheres of variable stars was developed. Specifically, the programme was developed to model the expanding atmospheres of cool, carbon-rich stars. The driving force for the expansion was radiation pressure acting on graphite grains which were assumed to form in the outer layers of the star. The details of the condensation and growth of the graphite grains as well as the resultant interactions between (a) the grains and the stellar radiation field and (b) the grains and the ambient gas were included in the calculations. The models were calculated by coupling a model atmosphere programme to the hydrodynamical equations needed to describe the flows of the gas and the grains, and to the equations which govern the nucleation and growth of the grains. The atmospheres were assumed to be spherically symmetrical and to have a grey opacity. In addition, the thermal time scale was assumed to be negligible compared to the dynamical time scale. The grains were assumed to be spheres of graphite and to form in accordance with the Lothe-Pound nucleation theory. The stellar parameters adopted for the models were M = 1.5 M(,(CIRCLE)),. L = 1.94 x 10('4)L(,(CIRCLE)), C/H = 1.22 x 10(' -3) and C/O = 1.76. The surface-free-energy adopted for the grains was 1000 ergs/cm('2). Two time-dependent models were calculated. Model 1 had T* = 2500 K and was followed for an elapsed time of 27.3 x 10('7) sec. For Model 2, T* = 2400 K and the elapsed time was 56.3 x 10('7) sec. The structure of both models was such that carbon vapour was found to be highly supersaturated in the outer layers of the atmospheres and extensive grain condensation occurred. The radiation pressure on these grains was sufficient to generate mass flows in both models. The calculated mass loss rate for Model 1 was 6.2 x 10('-9) M(,(CIRCLE))/yr, and for Model 2, 7.4 x 10('-8) M(,(CIRCLE))/yr. In Model 1, the mass flow approached a steady-state but in Model 2, a small amplitude pulsation was superimposed upon the outward flow. Initially, this pulsation was very irregular but after an elapsed time of 27 x 10('7) sec, the model had relaxed into a steady pulsation mode with a period of 6.48 x 10('7) sec and a velocity amplitude at the surface of 0.8 km/sec. This mode was followed for four periods during which the amplitude of the pulses remained constant. The driving force for this pulsation appears to be an opacity controlled feedback mechanism which operates between the grain-forming region of the model and the hydrogen dissociation zone. It was found that in Model 1, the opacity of the grains was too small for this mechanism to produce pulsations. In both models, grain nucleation was negligible at supersaturation levels less than 5. As a result, the grain-forming region was restricted to layers for which r > 1.82R* in Model 1, and r > 1.58R* in Model 2. In both cases, the low atmospheric densities in the grain-forming regions severely limited the growth of the grains following their formation so that the grains remained very small (a (TURNEQ) 5.5 x 10('-8) cm) and, even at the surface of the models, only 75% of the free carbon vapour was in the form of grains. The optical depth of the grains at (lamda) = 0.7(mu) was 5.0 x 10('-3) in Model 1, and 1.0 x 10('-1) in Model 2. The pulsation in Model 2 produced a variation in the optical depth of the grains of 1.8 x 10('-2).
- Pub Date:
- Physics: Astronomy and Astrophysics