PX282 - C7 - mean free path

\frac{I_{λ}}{I_{λ, 0}} = \exp (- \int_{0}^{z} κ_{λ} ρ d z)

using this to define the mean free path for a photon, $l$ , which is the characteristic distance a photon travels:

\begin{matrix} l = \frac{1}{κ_{λ} ρ} \\ ⟹ \frac{I_{λ}}{I_{λ, 0}} = e^{- 1} \end{matrix}

taking a typical stellar opacity, $κ \sim 0.1 m^{2} k g^{- 1}$ , and density, $ρ \sim 1.5 \times 10^{5} k g m^{- 3}$ (core), $\sim 1.4 \times 10^{3} k g m^{- 3}$ (average), $\sim 10^{- 6} k g m^{- 3}$ (photosphere), gives the mean free paths:

example

\frac{I}{I_{0}} = e^{- κ ρ z} = 0.3 ⟹ k ρ z \approx 1.2

considering the density, $ρ (O_{3}) = 0.3$ ppm, and assuming it is all in the stratosphere $(0 - 50 k m)$
the mass in the air column is $≃ 6.4 \times 10^{- 3} k g m^{- 2}$
the average density of ozone $≃ 1.3 \times 10^{- 7} k g m^{- 3}$
therefore, the absorption coefficient is:

κ \approx 185 m^{2} k g^{- 1}