> [!key-idea] > Good Sources: > - http://spiff.rit.edu/classes/phys440/lectures/optd/optd.html > - https://www.astro.uvic.ca/~tatum/stellatm/atm5.pdf > > You'll notice many of the things below are just describing the same thing in different ways. In the end, all of them tie back to this core relationship. > $\alpha = \frac{1}{l} = \rho\kappa = n \sigma = \frac{\mathrm{d} \tau}{\mathrm{d} z}$ ## Kirchoff's Law of Radiation **Kirchoff's Law** makes three statements: 1. A perfect black body in thermodynamic equilibrium is the perfect insulator (absorbs all light that strikes it) and radiator (emits energy according to [[Blackbody Radiation#Stefan-Boltzmann Law]], universal for all perfect black bodies). 2. For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity. (see [[Radiative Transfer Equation|RTE]]) $\frac{\mathrm{d} I_{\nu}}{\mathrm{d} z} = j_{\nu} - \alpha_{\nu} I_{\nu} = 0 \hspace{1cm} \Rightarrow \hspace{1cm} j_{\nu} = \alpha_{\nu} I_{\nu} = \alpha_{\nu} B_{\nu}(T)$ 3. The emissivity of a source cannot exceed one (because the absorptivity cannot exceed one by conservation of energy), and thus, it is not possible to thermally radiate more energy than a black body, at equilibrium. ## Mean Free Path $l = \frac{1}{n \sigma} \hspace{1cm} \text{where} \hspace{1cm} \begin{align} n &\equiv \text{number (volume) density} \\ \sigma &\equiv \text{scattering cross sectional area} \\ \end{align}$ The **Mean Free Path** is the scale length for a scattering process. For a material of size $L$, the probability of a particle scattering while traveling some distance ($\mathrm{d} x$)... ![[Mean_free_path.png|align:center]] $\mathcal{P} = \int \left(\frac{\text{area of scatterers}}{\text{area of target}}\right) = \int\frac{\sigma \left( n \times L^{2} \; \mathrm{d} x\right)}{L^{2}} = \int n \sigma \; \mathrm{d} x$ For incident flux ($F$) on the differential length of material, we have... $\mathrm{d} F = - F n \sigma \, \mathrm{d} x$ ...which we can solve to get **Lambert-Beer Law**, which relates the incident flux from a source ($F_{0}$) to the [[#Optical Depth]] ($\tau$). $\int \frac{\mathrm{d} F}{F} = - n \sigma \int \mathrm{d} x \hspace{1cm} \Rightarrow \hspace{1cm} F = F_{0} e^{-x/l} = F_{0} e^{-\tau}$^lambert-beer-law > [!note] > We use [[Flux|flux]] instead of [[Intensity|intensity]] as one might see on wikipedia. This is done to avoid confusion with the prevalent astronomy usage of these terms. ## Optical Thickness The **Optical Thickness** is a loosely defined notion of "how difficult it is for light to get through a unit length of this material". Thickness is an appropriate word because when thinking about distance in terms of [[#Optical depth|optical depth]]. A "thick" material is one which has a very short mean free path, and thus a large optical depth. the converse is true for a "thin" material. - **Optically thick**: Many [[#Mean free path|mean free paths]] per unit length - **Optically thin**: Few [[#Mean free path|mean free paths]] per unit length ## Extinction Coefficient $\alpha = \frac{1}{l} \hspace{1cm} \text{where} \hspace{1cm} l \equiv \text{mean free path}$ The **Extinction Coefficient** is the inverse of the [[#Mean Free Path|mean free path]], and has dimensions of inverse length. It is a combination of the effects of absorption and scattering. A high extinction corresponds to short mean free path, and thus, light has a "harder time" moving through the material. ## Opacity (also known as the "absorption coefficient"?) **Rosseland Mean Opacity** ($\kappa$) is the frequency-independent efficiency at which a particular medium absorbs and scatters light, with dimensions of "cross section per mass". It can be defined as: $\alpha = \frac{1}{l} = \rho\kappa = n \sigma = \frac{\mathrm{d} \tau}{\mathrm{d} z}$ where - $\kappa \equiv$ Rosseland Mean Opacity $\in [0,1]$ ($0=\text{transparent}$, $1=\text{opaque}$) - $\rho \equiv$ Mass Density - $l \equiv$ [[#Mean Free Path]] - $n \equiv$ number (volume) density - $\sigma \equiv$ scattering cross section - $\alpha \equiv$ [[#Extinction Coefficient]] **Opacity** is usually frequency-dependent though, and thus, the Rosseland Mean Opacity is typically defined terms of the frequency-dependent opacity ($\kappa_{\nu}$). $\frac{1}{\kappa} = \frac{\int_{0}^{\infty} \frac{1}{\kappa_{\nu}} \frac{\partial B_{\nu}}{\partial T} \; \mathrm{d} \nu}{\int_{0}^{\infty} \frac{\partial B_{\nu}}{\partial T} \; \mathrm{d} \nu} \hspace{1cm} \text{where} \hspace{1cm} B_{\nu} \equiv \text{blackbody spectral density}$ > [!note] > In principle, $\kappa$, $\sigma$ and $\alpha$ are all frequency-dependent. ## Kramer's Opacity Law **Kramer's Opacity Law** directly correlates the density and temperature of the medium to the average opacity. It can be derived from the Rosseland Mean Opacity for [[Bremsstrahlung Radiation|bremsstrahlung radiation]]. (skipped) $ \bar{\kappa} \propto \rho T^{-7/2} \hspace{1cm} \text{where} \hspace{1cm} \begin{aligned} \bar{\kappa} &\equiv \text{average opacity} \\ \rho &\equiv \text{density of medium} \\ T &\equiv \text{temperature of medium} \\ \end{aligned} $ Here, we can see that the opacity higher (more opaque) at lower temperatures. ## Optical Depth The **Optical Depth** ($\tau \equiv \tau_{\nu}$) is a frequency-dependent measurement that relates a physical distance to the absorptivity of a material over that distance. It can be thought of as the number of [[#Mean Free Path|mean free paths]] transversed in the given medium. $\tau(z) = \int_{0}^{z} \alpha \; \mathrm{d} z = \int_{0}^{z} \kappa \rho \; \mathrm{d} z = \int_{0}^{z} \sigma n \; \mathrm{d} z \underbrace{= \; \sigma N}_{\text{uniform material}}$ where - $\alpha \equiv$ [[#Extinction Coefficient]] - $\kappa \equiv$ [[#Opacity]] - $\rho \equiv$ density of medium - $\sigma \equiv$ scattering cross section - $N \equiv$ column number density ($\rm{particles\,m^{-2}}$) For a uniform material, the optical depth is just the length of the material transversed / mean free path. *(Note: This is the measure you use when what you care about is "how much light is absorbed".)* People often define the optical depth using the Rosseland mean [[#Opacity|opacity]], whereby $\alpha = \kappa \rho$ (as above) is frequency-independent. For the **frequency-dependent** optical depth: $\tau_{\nu}(z) = \int_{0}^{z} \alpha_{\nu} \; \mathrm{d} z = \int_{0}^{z} \kappa_{\nu} \, \rho \; \mathrm{d} z$ > [!note]- From the Radiative Transfer Equation > If we only consider the absorption term in the [[Radiative Transfer Equation|RTE]], then we can define optical depth ($\tau_{\nu}(z)$) as the exponential index. > > $\frac{\mathrm{d} I_{\nu}}{\mathrm{d} z} = - \alpha_{\nu} I_{\nu} \hspace{1cm} \Rightarrow \hspace{1cm} I_{\nu}(z) = I_{\nu,0} \exp \underbrace{\left[ - \int_{0}^{z} \alpha_{\nu}(z') \; \mathrm{d} z' \right]}_{- \tau_{\nu}(z) \equiv \text{optical depth}} = = I_{\nu,0} e^{- \tau_{\nu}(z)}$ > $\Downarrow$ > $\text{optical depth} \equiv \quad \tau_{\nu}(z) = \int_{0}^{z} \alpha_{\nu}(z') \; \mathrm{d} z' \hspace{1cm} \Rightarrow \hspace{1cm} \mathrm{d} \tau_{\nu} = \alpha_{\nu} \; \mathrm{d} z$