1. Introduction In this note, the formalism used to compute a simple stellar atmosphere model is described. The definition of a stellar atmosphere is strongly connected with the radiation field crossing the boundary layers of a star: the atmosphere is composed of those layers where photons interact with matter for the last time before leaving the star. The goal of a model is to calculate the radiation field at all frequencies, depths and directions consistent with the properties of matter (temperature, number densities, pressure, etc.) at all depths.

2. General description 2.1. General assumptions The atmosphere is modelled as a finite slab of gas, with spatial coordinate z ∈ [0, z ∗ ] and angular coordinate µ = cos θ ∈ [−1, 1]. We suppose azimuthal symmetry (Fig. 1). The atmosphere is static, in a steady state, in hydrostatic and radiative equilibrium.

/ -(0 (01)*32! 454 "

!" $#&% 67 6 8 9 '()*,+-.+ 8 :. =; < ?>

Fig. 1. Schematic representation of a plane-parallel atmosphere. The author is grateful to B. Rutily for fruitful discussion on this topic.

2

L. Chevallier

2.2. The chemical composition A stellar atmosphere is composed with many kinds of particles, namely atoms, ions, free electrons, possibly molecules or even dust grains. To simplify the description of the model, we suppose that the atmosphere is composed of a single chemical species (e.g., hydrogen, helium) whose atoms can be ionized several times when losing their electrons. Distinct energy states of this atom and associated ions are denoted by the subscript i ∈ [[1, N ]], where i = N refers to completely ionized atoms. The average number of particles in state i per unit volume (number density) is denoted by ni .

2.3. Unknown quantities The basic unknown quantities of this model are the specific intensity of the radiation field I(z, µ, ν), the complete set of number densities n i (z) for i ∈ [[1, N ]], the number density of free electrons ne (z) and the temperature T (z). The variables are the depth z ∈ [0, z ∗ ], the cosine of the inclination angle from the outer normal to the atmosphere µ ∈ [−1, 1], and the frequency ν ∈ ]0, +∞[.

2.4. Input parameters Input parameters of a model are: • the surface gravity g := GM/R2 , where M and R are the mass and the radius of the star respectively, and G is the gravitational constant, 4 • the effective temperature Teff , as defined by L = 4πR2 σTeff , where L is the luminosity of the star and σ is the Stefan-Boltzmann constant,

• the chemical composition (abundances of chemical species with respect to hydrogen), not introduced in this single species model. The gas pressure P0 on the boundary layer z = z ∗ is also required in models with P0 6= 0.

3. The equations 3.1 Radiative transfer equation This equation yields the specific intensity I(z, µ, ν) of the radiation field for given ni (z), ne (z) and T (z). It describes the distribution of photons in space (z), direction (µ) and frequency (ν) as a result of their interaction with matter. It is usually written in terms of the optical depth τ rather than the geometrical depth z. For a given frequency ν, the relation between z and τ is given by a bijection τν from [0, z ∗ ] to [0, τν∗ ], where τν∗ := τν (0) > 0 (see [1] for details). Then Iν (τ, µ) := I(τν−1 (τ ), µ, ν).

Stellar atmospheres modelling

3

The radiative transfer equation is, for all ν ∈ ]0, +∞[, τ ∈ ]0, τ ν∗ [ and µ ∈ [−1, 1], Z 1 ∂Iν 1 Iν (τ, µ0 ) dµ0 − Sν∗ (τ ), µ (τ, µ) = Iν (τ, µ) − 2 $ν (τ ) ∂τ −1 Iν (0, µ) = 0 µ 0

Sν∗ (τ ) is the primary source function and $ν (τ ) ∈ [0, 1] is the albedo at depth τ .[1] It is usual to introduce, instead of $ ν (τ ), the destruction probability of photons εν (τ ) = 1−$ν (τ ). The primary source function and the albedo depend on ni (z), ne (z) and T (z). We have supposed isotropic and monochromatic (i.e., with no change of frequency) scattering to simplify the presentation. Once the radiative transfer equation is solved for the variable τ , its solution is written in terms of the z-variable via the bijection τν−1 .

3.2. Chemical structure equations This system of N + 1 equations yields all number densities n i (z) for i ∈ [[1, N ]], together with the electron number density n e (z), for given I(z, µ, ν) and T (z). We introduce the total number density of particles n(z) :=

X

ni (z) + ne (z),

i∈[[1,N ]]

the volumic mass ρ(z) :=

X

mi ni (z) + me ne (z),

i∈[[1,N ]]

and the pressure P (z) := n(z)kT (z), assuming that the atmosphere is a perfect gas. mi and me are particles i and electron mass respectively, and k is the Boltzmann constant. The chemical structure equations are, for all z ∈ [0, z ∗ ], X X ni (z) Pi,j (z) = nj (z)Pj,i (z) (i ∈ [[1, N − 1]]), j∈[[1,N ]]\{i} j∈[[1,N ]]\{i} X qi ni (z) = ne (z), i∈[[1,N ]] dP (z) = −gρ(z), with boundary condition P (z ∗ ) = P . 0 dz

The first N −1 equations are the statistical equilibrium equations. They describe the change of state of particles i, due to their radiative and collisional interactions. Pi,j (z) is the rate of change from state i to state j. It

4

L. Chevallier

depends on I(z, µ, ν), ne (z) and T (z). The equation for i = N is unnecessary since it can be deduced from the first N − 1 equations. The second equation is the charge conservation equation. It means that the atmosphere is electrically neutral (qi is the electrical charge of particles i when expressed in unit of electron charge). The last equation is the hydrostatic equilibrium equation. It means that the atmosphere is static, since the pressure forces (left-hand side) are equal to the gravity forces (right-hand side). We have neglected radiation and turbulent pressure in this equation.

3.3. Radiative equilibrium equation This equation yields the temperature T (z) for given I(z, µ, ν), n i (z) and ne (z). For all z ∈ [0, z ∗ ], Fr (z) := 2π

Z

1 −1

Z

+∞ 0

4 I(z, µ, ν) µ dµdν = 4πσTeff ,

where Fr (z) is the integrated (on frequency) radiative flux at z. This equation involves the coefficients of the radiative transfer equation at z, which depend on T (z 0 ) at every point z 0 of the atmosphere. Making this dependence explicit in the radiative equilibrium equation leads to a non linear integral equation for T (z). This equation is very difficult to solve because of the highly non linear dependence of I(z, µ, ν) on the temperature. In a stellar atmosphere in radiative equilibrium, energy is carried out by radiation only (there is no convection) and the total flux is conserved 4 leaving the star. within the atmosphere. It is thus equal to the flux 4πσT eff

4. Starting a model As seen in Sec. 3, all quantities are coupled and the system of equations they satisfy has to be solved iteratively. To start the iterative process, a first idealized model is computed by assuming that the atmosphere is in local thermodynamic equilibrium.

5. Validation A model is declared valid if the calculated emerging monochromatic flux R1 Fr (0, ν) := 2π −1 I(z, µ, ν)µ dµ fits the observed one for all ν > 0. The model then defines the boundary conditions of a stellar interior model.

References 1. B. Rutily, Multiple scattering theory and integral equations, in this volume.