Stellar Astrophysics/Energy Transport

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Within a star there are two main methods of energy transport: radiation and convection. Both of these take place within our Sun, but occur in different regions of the star. Modelling this transfer is important understand how stars function and evolve over time.

Radiative Transfer

Radiative transfer is the transfer of energy via photons, and is the dominant method of transport within the core of the Sun and in the solar atmosphere. Photons passing through a region of gas can be absorbed while other photons may be emitted from the same region, so to model radiative transfer one must consider both the absorption and emission of radiation.

Parallel ray equation

In this section, we assume that the light is emitted in parallel rays to the observer. The more general non-parallel case will be considered later.

Absorption

Imagine there is a column of gas with a side area of $A$ and length $dL$. Assume that the gas has a number density of $n$ absorbing particles per unit volume, with each particle having an effective cross-sectional area $\sigma _{\lambda }$ when interacting with light of a particular wavelength.

From this, the proportion of area A that blocks the light is:$$d\tau _{\lambda }={\frac {dVn\sigma _{\lambda }}{A}}={\frac {AdLn\sigma _{\lambda }}{A}}=n\sigma _{\lambda }dL.$$This dimensionless value is called optical depth, and is proportional to the amount of light absorbed. The number density can be rewritten in terms of the mass density $\rho$ and average molecular mass ${\bar {m}}$ as: $n={\frac {\rho }{\bar {m}}}$, which allows the above expression to be written as $\rho {\frac {\sigma _{\lambda }}{\bar {m}}}dL$. $\kappa _{\lambda }^{m}={\frac {\sigma _{\lambda }}{\bar {m}}}$ is also known as the mass absorption coefficient, or the opacity per unit mass.

From the definition of optical depth, it can be implied that the reduction in light intensity caused by absorption is:

$$dI_{\lambda }=-I_{\lambda }d\tau _{\lambda }$$

Therefore, if only absorbtion is taking place, the intensity of the light leaving the column can be determined by integrating over the length of the column:

$$\int _{I_{\lambda }^{\text{in}}}^{I_{\lambda }^{\text{out}}}(I_{\lambda })^{-1}d\tau _{\lambda }=-\int _{0}^{\tau _{\lambda }}1d\tau _{\lambda }$$$$[\ln(I_{\lambda })]_{I_{\lambda }^{\text{in}}}^{I_{\lambda }^{\text{out}}}=-\tau _{\lambda }$$$$\therefore I_{\lambda }^{\text{out}}=I_{\lambda }^{\text{in}}e^{-\tau _{\lambda }}$$