When we observe astronomical source, along the eye path cloud matter may absorb the radiation , scatter or emit further radiation.
Absorption
Particle number density n, cross section of one particle is $\sigma_\nu$ , beam of area dA propagates a distance ds into the cloud , the totla absorbing cross section is $\sigma_\nu n ds$ , $I_\nu$ is 单色辐射强度。
$\alpha_\nu = n \sigma_\nu$ is absorption coefficient, defined as the fractional loss of intensity per unit length, with dimensions m^-1. It follows that the PHOTON MEAN FREE PATH $l_\nu = 1/ \alpha_\nu$
Emission
Two ways of an excited atom return to ground state:
i) atom emits energy spontaneously
ii) stimulated into emission by the presence of electromagnetic radiation
Stimulated emission is proportional to I_nu, as was the amount of absorption. Therefore for simplicity it can be considered to be negative absorption.
Define a spontaneous emission coefficient j_nu which is the energy emitted per unit time per unit volume per unit solid angle per unit frequency. $W m^{-3}sr^{-1}Hz^{-1}$
$dE = j_\nu dV d\Omega d_\nu dt$
Crossing a length ds a beam's specific intensity is increased by spontaneous emission by
$dI_\nu = j_\nu ds$
The optical deptth
Opaque or optically thick means on average a photon cannot pass through the medium without absorption. Conversaely is optically thin. These properties are functions of wavelengh, for example , a pen of glass is optically thin in the optical , but optically thick in the infrared.
Define the optical depth tau_nu
$\tau_\nu = \int \alpha_\nu ds$
1) Optically thinck if $\tau_\nu > 1$
2) before absorption is when optical depth is 1
3) In an optically thick homogenous medium the number of steps taken for a photon to diffuse out sim tau^2 (explanation)
Example
$\tau$ through 20m of water sim 1, then we know that
$\tau$ through 2m of water sim 0.1, means you can see to the bottom of a swimming pool
$\tau$ through 200m of water sim 10, means you cannot see to the bottom of the sea
The Radiative Transfer Equation
$\frac{dI_\nu}{ds } = -\alpha_\nu I_\nu + j_\nu$
can also be written as
$\frac{dI_\nu}{d\tau_\nu} = - I_\nu + S_\nu$
where define source function $S_\nu = j_\nu /\alpha_\nu$