Stellar Atmospheres

Radiation emitted by a star: Can be measured in terms of the specific intensity.

It corresponds to the amount of energy of a given wavelength per unit time dt crossing a unit area dA into a solid angle dΩ (think of the solid angle as a 2D angle in 3D space).

For a blackbody:

The mean intensity <Iλ> is the result of averaging the specific intensity over the solid angle (i.e. integrating it).

For an isotropic radiation field (i.e. the same in all directions):

Energy density: energy per unit volume with a wavelength between λ and λ+.

The total energy density u is obtained by integrating over all wavelengths.

Radiative flux: Net energy having a wavelength between λ and λ+ passing each second through a unit area in the direction perpendicular to the surface (we are talking about the surface of the star, or about an imaginary -“mathematical”- surface).

For an isotropic radiation field there is no energy transfer (Fλ = 0).

Radiation pressure: pressure caused by a beam of light, and in particular by the momentum of the photons, p = E/c.

The total radiation pressure Prad is obtained by integrating the previous expression over the wavelength λ.

For a blackbody, the radiation pressure is simply u/3.

Stars are not blackbodies. The spectra of a star show absorption lines. In particular, the absorption lines produced by the metallic elements of a star are called line blanketing. This is a phenomenon taking place in the atmosphere of a star.

Stars are not in thermodynamic equilibrium either, since there is a net outward flow of energy. The temperature of a star varies with the position, thus there are fluxes of gas travelling across the star between zones of different temperature. We can assume, however, zones of local thermodynamic equilibrium (LTE), if the mean free path of the particles (the distances travelled by particles and photons between collisions) is small in comparison with the distances over which the temperature changes significantly.

In such conditions, we discuss the absorption of photons of a beam of light as it travels through the stellar atmosphere.

As the beam travels and photons are absorbed by stellar gas, the change in intensity of the beam is:

thus the intensity decreases with the distance (obviously).

ds is the distance traveled. ρ is the density of the medium. κλ is the opacity, and depends on the composition of the gas phase, the density, and the temperature.

Optical depth, τλ: Is a measure of the transparency of a medium over a certain distance ds.

The optical depth is not a distance, it is a dimensionless unit and increases monotonically with the path’s length ds. For a beam of light travelling from position 0 (with τλ) to position 1 (τλ,1 = 0):

The relation between optical depth and intensity results:

The optical depth can be though of as the number of mean free paths from the original position to the surface, as measured along the ray’s path.