The Equation of Transfer in the Interstellar Medium

The equation of radiative transfer is

 (1/ρ) dI_ν/ds = j_ν/4π - κ_ν I_ν.

Using Kirchoff's Law,

j_ν/κ_ν = 4πB_ν(T),

we can write the transfer equation

(1/ρ) dI_ν/ds = κ_ν[B_ν(T) - I_ν].

In the interstellar medium, because of low T,  B_ν(T) << I_ν , so, the transfer equation is approximately,

dI_ν/ds = - κ_νρ I_ν ,  or,   dI_ν/I_ν = - κ_νρ ds .

The solution is

I_ν = I_ν0 e^-τν,

where τ_ν is the so-called optical depth, defined,

τ_ν = ò₀  κ_νρ ds .

We note that the intensity reduction due to attenuation, measured in magnitudes, is

Δm = 2.5 log (I₀/I) = 2.5 log (e^τν) = (2.5 log e) τ_ν = 1.085736205 τ_ν.