Line-by-line Absorption ####################### This section describes the physical process of absorption lines of different molecules absorbing and emitting spectral radiance in the atmosphere. These are the types of line-by-line absorption considered here: - :ref:`lbl-plain`, where each absorption line is considered separately. - Without Zeeman effect - With Zeeman effect - :ref:`lbl-ecs`, where the absorption lines of similar energies of a molecule are mixed together. .. _lbl-plain: Line-by-line Absorption Overview ******************************** 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 :ref:`lbl-line-strength` operator, and :math:`F` is the :ref:`lbl-line-shape` operator. Both :math:`S` and :math:`F` change slightly if Zeeman effect is considered. The main way that Zeeman effect changes the calculations is via the polarization it introduces to the propagation matrix summation. Without Zeeman effect ===================== The contribution to :ref:`propagation matrix ` from all non Zeeman-split absorption lines is simply .. math:: K_{A, lbl} = \mathrm{Re} \sum_i \alpha_{i}, and from this the full matrix is .. math:: \mathbf{K}_{lbl} = \left[ \begin{array}{llll} K_{A, lbl}&0&0&0\\ 0&K_{A, lbl}&0&0\\0&0&K_{A, lbl}&0\\0&0&0&K_{A, lbl} \end{array} \right], where :math:`i` is the pseudo-index of the absorption line and ``lbl`` is the pseudo-index of the plain line-by-line absorption for the sake of :ref:`summing up absorption `. .. note:: Plain line-by-line absorption only contribute towards the diagonal of the :ref:`propagation matrix `. With Zeeman effect ================== When Zeeman effect is considered, there are effectively 3 separate kinds of polarized absorption added to the :ref:`propagation matrix ` .. math:: K_{\sigma_\pm, z} &= \sum_i \alpha_{i, \sigma_\pm} \\ K_{\pi, z} &= \sum_i \alpha_{i, \pi} and from this, the full matrix contribution is .. math:: \mathbf{K}_{z} = \sum_\pm \left( \mathrm{Re} K_{\sigma_\pm,z} \left[ \begin{array}{rrrr} 1 + \cos^2\theta_m & \sin^2\theta_m\cos 2\eta_m & \sin^2\theta_m\sin 2\eta_m & \mp 2\cos\theta_m \\ \sin^2\theta_m\cos 2\eta_m & 1 + \cos^2\theta_m & 0 & 0 \\ \sin^2\theta_m\sin 2\eta_m & 0 & 1 + \cos^2\theta_m & 0 \\ \mp 2\cos\theta_m & 0 & 0 & 1 + \cos^2\theta_m \end{array} \right] + \mathrm{Im} K_{\sigma_\pm,z} \left[ \begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & \mp 4\cos\theta_m & 2\sin^2\theta_m\sin 2\eta_m \\ 0 & \pm 4\cos\theta_m & 0 & - 2 \sin^2\theta_m\cos 2\eta_m \\ 0 & - 2\sin^2\theta_m\sin 2\eta_m & 2 \sin^2\theta_m\cos 2\eta_m & 0 \end{array} \right] \right) + \\ \mathrm{Re} K_{\pi,z} \left[ \begin{array}{rrrr} \sin^2\theta_m & - \sin^2\theta_m\cos 2\eta_m & - \sin^2\theta_m\sin 2\eta_m & 0 \\ - \sin^2\theta_m\cos 2\eta_m & \sin^2\theta_m & 0 & 0 \\ - \sin^2\theta_m\sin 2\eta_m & 0 & \sin^2\theta_m & 0 \\ 0 & 0 & 0 & \sin^2\theta_m \end{array} \right] + \mathrm{Im} K_{\pi,z} \left[ \begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 2\sin^2\theta_m\sin 2\eta_m \\ 0 & 0 & 0 & 2 \sin^2\theta_m\cos 2\eta_m \\ 0 & 2\sin^2\theta_m\sin 2\eta_m & - 2 \sin^2\theta_m\cos 2\eta_m & 0 \end{array} \right], where the somewhat weird :math:`\pm`-sum is over the sigma components. Here the angles :math:`\theta_m` and :math:`\eta_m` are the angles with regards to the magnetic field. Given a spherical coordinate observation system with zenith angle :math:`\theta_z` and aziumuth angle :math:`\eta_a` and a local magnetic field with upwards facing strength :math:`B_w`, eastward facing strength :math:`B_u` and northward facing strength :math:`B_v`, these angles are given by .. math:: \theta_m = \arccos\left(\frac{B_v \cos\eta_a \sin\theta_z + B_u \sin\eta_a \sin\theta_z + B_w \cos\theta_z}{ \sqrt{B_w^2 + B_u^2 + B_v^2} } \right) \\ \eta_m = \mathrm{atan2}\left(B_u \cos\eta_a - B_v \sin\eta_a,\; B_w \cos\eta_a\cos\theta_z + B_u\sin\eta_a\cos\theta_z - B_w\sin\theta_z \right) .. _lbl-line-shape: Line Shapes =========== Line shapes should distribute absorption as a function of frequency. By convention, the line shape is normalized to have an integral of 1. 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}, where .. list-table:: :header-rows: 1 * - Parameter - Description * - :math:`\nu` - The sampling frequency * - :math:`\nu_0` - The line center frequency * - :math:`G_D` - The scaled Doppler broadening half-width half-maximum * - :math:`\Delta\nu_Z` - The Zeeman shift * - :math:`G_{P,0}` - The pressure broadening - half width half maximum in the Lorentz profile * - :math:`\Delta\nu_{P,0}` - The pressure shift * - :math:`Y_{lm}` - The 1st order Line-mixing parameter * - :math:`G_{lm}` - The 2nd-order strength modifying line mixing parameter * - :math:`\Delta\nu_{lm}` - The 2nd-order line-mixing shift * - :math:`w(z)` - The Faddeeva function. For more information about how :math:`G_{P,0}`, :math:`\Delta\nu_{P,0}`, :math:`Y_{lm}`, :math:`G_{lm}`, and :math:`\Delta\nu_{lm}` are computed see :ref:`lbl-line-shape-params`. The scaled Doppler broadening half width half maximum is given by .. math:: G_D = \sqrt{\frac{2000 R T}{mc^2}} \nu_0, where .. list-table:: :header-rows: 1 * - Parameter - Description * - :math:`R` - The ideal gas constant in Joules per mole per Kelvin, * - :math:`T` - The temperature in Kelvin, * - :math:`m` - The mass of the molecule in grams per mole, and * - :math:`c` - The speed of light in meters per second. The factor 2000 is to convert to SI units. The Faddeeva function is in ARTS computed using the MIT-licensed `Faddeeva package `_, which is based in large parts on the work by :cite:t:`zaghloul12:_algorithm916_acm`. The Zeeman line-shift is derived from the magnetic field strength and the magnetic quantum number of the transition. A linear Zeeman effect is assumed such that .. math:: \Delta\nu_Z = \frac{e} {4 \pi m_e} \left(M_l g_{l,z} - M_u g_{u,z} \right), where :math:`e` is the elementary charge, :math:`m_e` is the mass of an electron, :math:`M_l` and :math:`M_u` are the projection of the lower and upper states, respectively, of the angular momentum quantum number on the magnetic field, and :math:`g_{l,z}` and :math:`g_{u,z}` are the lower and upper state Landé g-factors, respectively. The latter are generally computed ahead of time, e.g., as by :cite:t:`larsson19:_updated_jqsrt,larsson20:_zeeman_jqsrt`. .. note:: It is important to not confuse the line-mixing parameters used here with full line mixing as described below. The line-mixing paramters here are still plain line-by-line absorption, but it is important that there are no cut lines and that the data for *all* line-paramters are derived toghether. .. _lbl-line-shape-params: Line Shape Parameters --------------------- The line shape parameters supported by ARTS are .. list-table:: :header-rows: 1 * - Parameter - Description - Pressure Dependency * - :math:`G_{P,0}` - Pressure broadening half width half maximum, collision-independent. - :math:`P` * - :math:`G_{P,2}` - Pressure broadening half width half maximum, collision-dependent. - :math:`P` * - :math:`\Delta\nu_{P,0}` - Pressure shift, collision-independent. - :math:`P` * - :math:`\Delta\nu_{P,2}` - Pressure shift, collision-dependent. - :math:`P` * - :math:`\nu_{VC}` - Velocity changing frequency. - :math:`P` * - :math:`\eta` - Correlation parameter. - :math:`-` * - :math:`Y_{lm}` - 1st order Line-mixing parameter. - :math:`P` * - :math:`G_{lm}` - 2nd-order strength modifying line mixing parameter. - :math:`P^2` * - :math:`\Delta\nu_{lm}` - 2nd-order line-mixing shift. - :math:`P^2` These parameters are all computed species-by-species before being volume-mixing ratio weighted and summed up. In equation form: .. math:: L = \frac{\sum_i x_i L_i}{\sum_i x_i}, where :math:`L` is a placeholder for any of the line shape parameters, and :math:`x` is the volume-mixing ratio, and :math:`i` is a species index. The normalization is there to allow fewer than all species to contribute to the line shape parameters. The temperature dependencies of the individual :math:`L_i` are computed based on avaiable data. There is no general form avaiable, so instead the temperature dependencies are computed based on the data avaiable for each species as: .. list-table:: :header-rows: 1 * - Name - Equation - Description * - ``T0`` - :math:`L_i(T) = X_0` - Constant regardless of temperature * - ``T1`` - :math:`L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{X_1}` - Simple power law * - ``T2`` - :math:`L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} \left[1 + X_2 \log\left(\frac{T_0}{T}\right)\right]` - Power law with compensation. * - ``T3`` - :math:`L_i(T) = X_0 + X_1 \left(T - T_0\right)` - Linear in temperature * - ``T4`` - :math:`L_i(T) = \left[X_0 + X_1 \left(\frac{T_0}{T} - 1\right)\right] \left(\frac{T_0}{T}\right)^{X_2}` - Power law with compensation. Used for line mixing. * - ``T5`` - :math:`L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{\frac{1}{4} + \frac{3}{2}X_1}` - Power law with offset. * - ``AER`` - :math:`L_i(200) = X_0`, :math:`L_i(250) = X_1`, :math:`L_i(296) = X_2`, :math:`L_i(340) = X_3`. Linear interpolation inbetween. - Inspired by the way `AER `_ deals with linemixing. * - ``DPL`` - :math:`L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} + X_2 \left(\frac{T_0}{T}\right) ^ {X_3}` - Double power law. * - ``POLY`` - :math:`L_i(T) = X_0 + X_1 T + X_2 T ^ 2 + X_3 T ^ 3 + \cdots` - Polynomial in temperature. Used internal in ARTS when training our own linemixing. here, :math:`X_0` ... :math:`X_N` are all model supplied constants whereas :math:`T` is the temperature in Kelvin and :math:`T_0` is the reference temperature of the model parameters. .. _lbl-line-strength: Line Strength ============= .. _lbl-lte: 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}, where :math:`\rho` is the number density of the absorbing species, .. math:: \rho = \mathrm{VMR}\frac{P}{kT}, where VMR is the volume-mixing ratio of the absorbing species. .. _lbl-nlte: 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). Zeeman effect ------------- If Zeeman effect is considered, the emission and absorption terms above are modified by quantum number state distribution. For O :sub:`2`, for example, this introduces a factor of .. math:: S_z = f(\Delta M) \left( \begin{array}{ccc} J_l & 1 & J_u \\ M_l & \Delta M & - M_u \end{array} \right)^2, where :math:`\Delta M \in \left[-1,\;0,\;1\right]` is the change in quantum number for angular rotational momentum projection along the magnetic field for :math:`\sigma_-`, :math:`\pi`, and :math:`\sigma_+`, respectively, :math:`f(\Delta M)` is the 0.75 for :math:`\sigma_\pm` and 1.5 for :math:`\pi`, and :math:`J_l` and :math:`J_u` are the lower and upper total angular rotational momentum quantum number. The :math:`(:::)` construct is the Wigner 3-j symbol. It can be `computed using software `_ such as that by :cite:t:`johansson2016`. .. _lbl-ecs: Line-mixing using Error-corrected Sudden **************************************** TBD