**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** <I_{λ}> 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*+*dλ*.

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

**Radiative flux**: Net energy having a wavelength between *λ* and *λ*+*dλ* 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 (*F _{λ}* = 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** P_{rad} 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.* κ _{λ}* is the

**opacity**, and depends on the

*composition of the gas phase*, the

*density*, and the

*temperature*.

**Optical depth**, *τ _{λ}*: 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 τ_{λ}) to position 1 (τ_{λ,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.

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