.. _Sec Absorption: Absorption ########## Absorption is the physical process that reduces the ``_ of a beam of light as it passes through a medium. It can be described be Beer's law: .. math:: \vec{I}(r) = \vec{I}(0) \exp(-\mathbf{K} z), where :math:`\vec{I}(r)` is the ``_ of the light at some distance :math:`r`, and :math:`\mathbf{K}` is the ``_ of the medium. Propagation Matrix ****************** The propagation matrix conceptually describes how ``_ propagates through a system. The propagation matrix is a square matrix with strict symmetries, and it has the form .. math:: \mathbf{K} = \left[ \begin{array}{rrrr} A & B & C & D \\ B & A & U & V \\ C &-U & A & W \\ D &-V &-W & A \end{array} \right], where :math:`A` describes the total power reduction, :math:`B` describes the difference in power reduction between horizontal and vertical linear polarizations, :math:`C` describes the difference in power reduction between plus 45 and minus 45 linear polarizations, :math:`D` describes the difference in power reduction between right and left circular polarizations, :math:`U` describes the phase delay between right and left circular polarizations, :math:`V` describes the phase delay between plus 45 and minus 45 linear polarizations, and :math:`W` describes the phase delay between horizontal and vertical linear polarizations. The unit of all of these is m :math:`^{-1}`. Spectral Radiance ***************** The unit of spectral radiance is W sr :math:`^{-1}` m :math:`^{-2}` Hz :math:`^{-1}`. Line-by-line Absorption *********************** This section describes the physical process of absorption lines of different molecules absorbing and emitting ``_ in the atmosphere. These are the types of line-by-line absorption considered here: #. `Plain line-by-line absorption `_, where each absorption line is considered separately. #. Zeeman splitting, where the otherwise single absorption line is split into multiple lines due to the presence of a magnetic field. #. Line-mixing by means of error-corrected sudden approximation, where the absorption lines of similar energies of a molecule are mixed together. Plain Line-by-line Absorption ============================= The absorption in plain line-by-line absorption is simply the sum of all absorption by each absorption line. The absorption of a single absorption line is described by the following equations: .. math:: \alpha = S(\cdots) F(\cdots), where :math:`\alpha` is the absorption coefficient, :math:`S` is the ``_ operator, and :math:`F` is the ``_ operator. Line Shape ========== Line shapes should distribute absorption as a function of frequency. By convention, the line shape is normalized to have an integral of 1. The following line shapes are available in ARTS 3. Voigt Line Shape ---------------- .. math:: F = \frac{1 + G_{lm} - iY_{lm}}{\sqrt{\pi}G_D} w(z), where .. math:: z = \frac{\nu - \nu_0 - \Delta\nu_{lm} - \Delta\nu_Z - \Delta\nu_{P,0} + iG_{P,0}}{G_D}, :math:`\nu` is the frequency, :math:`\nu_0` is the line center frequency, :math:`\Delta\nu_{lm}` is the line mixing shift, :math:`\Delta\nu_Z` is the Zeeman splitting, :math:`\Delta\nu_{P,0}` is the pressure shift, :math:`G_{P,0}` is the pressure broadening - half width half maximum in the Lorentz profile, :math:`G_{lm}` is the strength modifying line mixing parameter, :math:`Y_{lm}` is the phase-introducing line mixing parameter, and :math:`G_D` is the scaled Doppler broadening half-width half-maximum. :math:`\nu` is just a sampling frequency, it can be anything positive. :math:`\nu_0` is provided by the absorption line catalog. The :math:`\Delta\nu_{lm}`, :math:`\Delta\nu_{P,0}`, and :math:`G_{P,0}` are Line Shape Parameters. :math:`\Delta\nu_Z` is the Zeeman splitting, which depends on molecule and magnetic field strength as is described in Zeeman Effect. The scaled Doppler broadening half width half maximum is given by .. math:: G_D = \sqrt{\frac{2000 R T}{mc^2}} \nu_0, where :math:`R` is the ideal gas constant in Joules per mole per Kelvin, :math:`T` is the temperature in Kelvin, :math:`m` is the mass of the molecule in grams per mole, and :math:`c` is the speed of light in meters per second. The factor 2000 is to convert to SI units. Line Strength ============= Local thermodynamic equilibrium ------------------------------- For local thermodynamic equilibrium (LTE), the line strength is given by .. math:: S_{LTE} = \rho \frac{c^2\nu}{8\pi} \left[1 - \exp\left(-\frac{h\nu}{kT}\right)\right] \frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \frac{A_{lu}}{\nu_0^3} Non-local thermodynamic equilibrium ----------------------------------- For non-LTE, the line strength is given by .. math:: S_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left(r_l \frac{g_u}{g_l} - r_u\right) \frac{A_{lu}} {\nu_0^3}, and the added emissions are given by .. math:: K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{r_u\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left(r_l \frac{g_u}{g_l} - r_u\right) \right\} \frac{ A_{lu}}{\nu_0^3}, where :math:`r_l` and :math:`r_u` are the ratios of the populations of the lower and upper states, respectively. Note that :math:`K_{LTE} = 0`, as it represents "additional" emission due to non-LTE conditions. Also note that :math:`K_{NLTE}` may be negative. To ensure ourselves that this can be turned into the expression for LTE, we can rewrite the above for the expression that :math:`r_l` and :math:`r_u` would have in LTE according to the Boltzmann distribution: .. math:: r_l = \frac{g_l\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} and .. math:: r_u = \frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)} Putting this into the ratio-expression for :math:`S_{NLTE}` with the following simplification steps: Expansion: .. math:: \left(r_l \frac{g_u}{g_l} - r_u\right) = \frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right]. Extract lower state energies: .. math:: \frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right] \frac{\exp\left(-\frac{E_l}{kT}\right)}{\exp\left(-\frac{E_l}{kT}\right)} \rightarrow \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)}, where this last step is possible because we estimate that :math:`E_u-E_l = h\nu_0`. Note how the expression for :math:`K_{NLTE}` is 0 under LTE conditions. As it should be. This is seen by putting the above RHS and the expression for :math:`r_u` into the expression for :math:`K_{NLTE}`: .. math:: K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{\frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)}\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \right\} \frac{ A_{lu}}{\nu_0^3} = 0. The ratio between LTE and non-LTE line strength remaining is: .. math:: \frac{S_{NLTE}}{S_{LTE}} = \frac{1 - \exp\left(-\frac{h\nu_0}{kT}\right)}{1 - \exp\left(-\frac{h\nu}{kT}\right)}. It is clear that the non-LTE expression is the one that is incorrect here. The energy of the emitted photon is not :math:`h\nu_0` but :math:`h\nu`, and as such the actual energy of the transition is :math:`E'_u-E'_l = h\nu`, but this should be relatively close in cases where we actually care about non-LTE (which is low density, low collision atmospheres).