La desterritorialización permanece

The effective temperatureT is equal to the radiative temperature Trad and the kinetic tem- peratureTkin which describes the distribution of the particle velocities. m denotes the mass of the particles.

Thermodynamic equilibrium is not in agreement with observations, but to simplify the model we can assume local thermodynamic equilibrium (LTE). This means locally a blob of gas in a certain layer is in thermodynamic equilibrium but from one layer of the star to the next higher layer, the temperature changes. The temperature depends on the depth in the atmosphere. In LTE we assume that the kinetic temperature of the plasma is almost equal to its radiative temperature. This can be done since the amount of energy lost on the surface of the star is small compared to energy content of the star.

3.2

Energy transport

Next we have to explain how energy transport through the stellar atmosphere takes place. We assume that no energy is produced in the atmosphere but only in the central part of the star. The atmosphere depends strongly on the amount of energy produced in the center, how it is transported, and how the photons interact with the material of the atmosphere. For energy transport, convection as well as radiative transport can occur. The interaction can be absorption and emission of photons by the chemical elements contained in the atmosphere, collisions between electrons, electrons and atoms, and atoms as well as with molecules.

We assume hydrostatic equilibrium between the gravitational force that contracts the star and the pressure of the gas, and the radiation that expands the star. The energy in our model is generated in the interior of the star. The interior of the star is assumed to be uniform. Energy transport from the core to the outer layers of the atmosphere not only takes place by convective motion, but also by radiation.

In the atmosphere energy is absorbed by the atoms and reemitted at another wavelength range. The effect was accounted for by introducing line opacities of the chemical atomic and molecular species contained in the star. The radiation (energy) which is escaping the star can be observed and this is the only way to learn more about the stellar interior. We assume that the total amount of energy emitted is produced in the central not observable region of the star and transported to the outside. In G-stars the energy is transported by radiation rather than by convection. The problem to be solved is the description of how the physical parameters of the material making up the star couples to the spectrum.

3.2.1 Radiative energy transport

We set up the coordinate system of our plane-parallel atmosphere the way that the atmosphere is aligned perpendicular to the x coordinate. Energy is transported by radiation in the directionx. The change of the specific intensityIν over an increment of path length dxis the

sum of losses and gains over that path length,

dIν =−κνρIνdx+jνρdx. (3.3)

κν andjν are the absorption and emission coefficients at a frequencyν, respectively,ρdenotes

the density of the gas.

3.2.2 Convective energy transport

Convective energy transport takes place in most stars. Especially in cool stars, the convection zone is deep and in the outer part of the atmosphere the energy transport is dominated by con- vection. Convection is a hydrodynamical process and resolving the hydrodynamical equations

32 3 STELLAR ATMOSPHERES MODELS

numerically takes a lot of computation power. We therefore introduce a phenomenological description of the problem.

On the surface of the star we assume that convection takes place as observed on the sun. This motion is described by the path an average gas cell takes on its way through the atmosphere. On the sun we observe large convective cells and also smaller convective motions which is called granulation and can be divided into super granulation and micro granulation. On the surface of an unresolved star these smaller and larger convection regions cannot be resolved. The overall dynamic motion is therefore described by the mixing length theory and it can be observed in Doppler broadening of spectral lines.

Convection only takes place if a rising gas cell is not able to dissipate its energy fast enough by radiative energy transport. If this happens the rising cell must have a sufficiently high opacity to prevent complete energy loss by radiation. It ends when the rising gas volume has disposed its energy completely to the surrounding medium or else it stops on the boundary of the atmosphere. In our model we do not account for convective overshooting. The flux Φ that is transported through the layers can be expressed by the physical parameters of the cell: ∆T is the temperature excess of the cell above the surrounding medium, ρ is the gas density in the cell,Cp the specific heat at constant pressure andv the cells upward velocity:

Φ =Cpvρ∆T. (3.4)

Convection is favored if the opacity of the gas is high. The mixing length formulation

A simplified way to think of convection is to describe the length of the path an average gas cell takes on its way through the atmosphere (often referred to as the mixing length theory).

Λ =αHp (3.5)

where Λ denotes the mixing length. From observations of the solar atmosphere, convection is standing in a linear relation to the the pressure scale height, Hp. The pressure scale height

is proportional to the height in which in the pressure drops by a factor 1/e. To put the pressures scale height into a relation with the mixing length, we introduce the factor α, the mixing length parameter. The main problem arises when scaling the mixing length parameter. The parameter mixing length Λ cannot even be observed directly on the surface of the sun. We have to make sure that the energy transport is not only performed by convection but also by radiation. For a discussion on the problems of scaling the mixing length parameter,α, see the PhD thesis of Jan Bernkopf (2001).

Conditions for convection - The Schwarzschild criterion

The convective cell as it rises through the atmosphere is buoyed additionally at each level as it continues to rise. Since the density in the outer layers drops and the cell is expected to stay in pressure equilibrium, it must expand. To figure out if the cell will continue to rise or if it will start to sink, we calculate the density change in the cell.

Assuming the convective cell behaves adiabatically, meaning no energy leakage occurs from the rising cell, we obtain the Schwarzschild criterion (1906):

P ρ−β =const. (3.6)

3.3. INTERACTION BETWEEN RADIATION AND MATERIAL 33