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:

Line-by-line Absorption Overview, where each absorption line is considered separately.
- Without Zeeman effect
- With Zeeman effect
Line-mixing using Error-corrected Sudden, where the absorption lines of similar energies of a molecule are mixed together.

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:

\[\alpha = S(\cdots) F(\cdots),\]

where \(\alpha\) is the absorption coefficient, \(S\) is the Line Strength operator, and \(F\) is the Line Shapes operator.

Both \(S\) and \(F\) 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 propagation matrix from all non Zeeman-split absorption lines is simply

\[K_{A, lbl} = \mathrm{Re} \sum_i \alpha_{i},\]

and from this the full matrix is

\[\begin{split}\mathbf{K}_{lbl} = \left[ \begin{array}{llll} K_{A, lbl}&0&0&0\\ 0&K_{A, lbl}&0&0\\0&0&K_{A, lbl}&0\\0&0&0&K_{A, lbl} \end{array} \right],\end{split}\]

where \(i\) 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 summing up absorption.

Note

Plain line-by-line absorption only contribute towards the diagonal of the propagation matrix.

With Zeeman effect

When Zeeman effect is considered, there are effectively 3 separate kinds of polarized absorption added to the propagation matrix

\[\begin{split}K_{\sigma_\pm, z} &= \sum_i \alpha_{i, \sigma_\pm} \\ K_{\pi, z} &= \sum_i \alpha_{i, \pi}\end{split}\]

and from this, the full matrix contribution is

\[\begin{split}\mathbf{K}_{z} = \sum_\pm \left( \mathrm{Re} K_{\sigma_\pm,z} \left[ \begin{array}{rrrr} 1 + \cos^2\theta_m & \sin^2\theta_m\cos 2\eta_m & \sin^2\theta_m\sin 2\eta_m & \mp 2\cos\theta_m \\ \sin^2\theta_m\cos 2\eta_m & 1 + \cos^2\theta_m & 0 & 0 \\ \sin^2\theta_m\sin 2\eta_m & 0 & 1 + \cos^2\theta_m & 0 \\ \mp 2\cos\theta_m & 0 & 0 & 1 + \cos^2\theta_m \end{array} \right] + \mathrm{Im} K_{\sigma_\pm,z} \left[ \begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & \mp 4\cos\theta_m & 2\sin^2\theta_m\sin 2\eta_m \\ 0 & \pm 4\cos\theta_m & 0 & - 2 \sin^2\theta_m\cos 2\eta_m \\ 0 & - 2\sin^2\theta_m\sin 2\eta_m & 2 \sin^2\theta_m\cos 2\eta_m & 0 \end{array} \right] \right) + \\ \mathrm{Re} K_{\pi,z} \left[ \begin{array}{rrrr} \sin^2\theta_m & - \sin^2\theta_m\cos 2\eta_m & - \sin^2\theta_m\sin 2\eta_m & 0 \\ - \sin^2\theta_m\cos 2\eta_m & \sin^2\theta_m & 0 & 0 \\ - \sin^2\theta_m\sin 2\eta_m & 0 & \sin^2\theta_m & 0 \\ 0 & 0 & 0 & \sin^2\theta_m \end{array} \right] + \mathrm{Im} K_{\pi,z} \left[ \begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 2\sin^2\theta_m\sin 2\eta_m \\ 0 & 0 & 0 & 2 \sin^2\theta_m\cos 2\eta_m \\ 0 & 2\sin^2\theta_m\sin 2\eta_m & - 2 \sin^2\theta_m\cos 2\eta_m & 0 \end{array} \right],\end{split}\]

where the somewhat weird \(\pm\)-sum is over the sigma components. Here the angles \(\theta_m\) and \(\eta_m\) are the angles with regards to the magnetic field.

Given a spherical coordinate observation system with zenith angle \(\theta_z\) and aziumuth angle \(\eta_a\) and a local magnetic field with upwards facing strength \(B_w\), eastward facing strength \(B_u\) and northward facing strength \(B_v\), these angles are given by

\[\begin{split}\theta_m = \arccos\left(\frac{B_v \cos\eta_a \sin\theta_z + B_u \sin\eta_a \sin\theta_z + B_w \cos\theta_z}{ \sqrt{B_w^2 + B_u^2 + B_v^2} } \right) \\ \eta_m = \mathrm{atan2}\left(B_u \cos\eta_a - B_v \sin\eta_a,\; B_w \cos\eta_a\cos\theta_z + B_u\sin\eta_a\cos\theta_z - B_w\sin\theta_z \right)\end{split}\]

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

\[F = \frac{1 + G_{lm} - iY_{lm}}{\sqrt{\pi}G_D} w(z),\]

where

\[z = \frac{\nu - \nu_0 - \Delta\nu_{lm} - \Delta_nu_Z - \Delta\nu_{P,0} + iG_{P,0}}{G_D},\]

where

Parameter	Description
\(\nu\)	The sampling frequency
\(\nu_0\)	The line center frequency
\(G_D\)	The scaled Doppler broadening half-width half-maximum
\(\Delta\nu_Z\)	The Zeeman shift
\(G_{P,0}\)	The pressure broadening - half width half maximum in the Lorentz profile
\(\Delta\nu_{P,0}\)	The pressure shift
\(Y_{lm}\)	The 1st order Line-mixing parameter
\(G_{lm}\)	The 2nd-order strength modifying line mixing parameter
\(\Delta\nu_{lm}\)	The 2nd-order line-mixing shift
\(w(z)\)	The Faddeeva function.

For more information about how \(G_{P,0}\), \(\Delta\nu_{P,0}\), \(Y_{lm}\), \(G_{lm}\), and \(\Delta\nu_{lm}\) are computed see Line Shape Parameters. The scaled Doppler broadening half width half maximum is given by

\[G_D = \sqrt{\frac{2000 R T}{mc^2}} \nu_0,\]

where

Parameter	Description
\(R\)	The ideal gas constant in Joules per mole per Kelvin,
\(T\)	The temperature in Kelvin,
\(m\)	The mass of the molecule in grams per mole, and
\(c\)	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 Zaghloul and Ali [6].

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

\[\Delta\nu_Z = \frac{e} {4 \pi m_e} \left(M_l g_{l,z} - M_u g_{u,z} \right),\]

where \(e\) is the elementary charge, \(m_e\) is the mass of an electron, \(M_l\) and \(M_u\) are the projection of the lower and upper states, respectively, of the angular momentum quantum number on the magnetic field, and \(g_{l,z}\) and \(g_{u,z}\) are the lower and upper state Landé g-factors, respectively. The latter are generally computed ahead of time, e.g., as by Larsson and Lankhaar [2], Larsson et al. [3].

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.

Line Shape Parameters

The line shape parameters supported by ARTS are

Parameter	Description	Pressure Dependency
\(G_{P,0}\)	Pressure broadening half width half maximum, collision-independent.	\(P\)
\(G_{P,2}\)	Pressure broadening half width half maximum, collision-dependent.	\(P\)
\(\Delta\nu_{P,0}\)	Pressure shift, collision-independent.	\(P\)
\(\Delta\nu_{P,2}\)	Pressure shift, collision-dependent.	\(P\)
\(\nu_{VC}\)	Velocity changing frequency.	\(P\)
\(\eta\)	Correlation parameter.	\(-\)
\(Y_{lm}\)	1st order Line-mixing parameter.	\(P\)
\(G_{lm}\)	2nd-order strength modifying line mixing parameter.	\(P^2\)
\(\Delta\nu_{lm}\)	2nd-order line-mixing shift.	\(P^2\)

These parameters are all computed species-by-species before being volume-mixing ratio weighted and summed up. In equation form:

\[L = \frac{\sum_i x_i L_i}{\sum_i x_i},\]

where \(L\) is a placeholder for any of the line shape parameters, and \(x\) is the volume-mixing ratio, and \(i\) 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 \(L_i\) 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:

Name	Equation	Description
`T0`	\(L_i(T) = X_0\)	Constant regardless of temperature
`T1`	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{X_1}\)	Simple power law
`T2`	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} \left[1 + X_2 \log\left(\frac{T_0}{T}\right)\right]\)	Power law with compensation.
`T3`	\(L_i(T) = X_0 + X_1 \left(T - T_0\right)\)	Linear in temperature
`T4`	\(L_i(T) = \left[X_0 + X_1 \left(\frac{T_0}{T} - 1\right)\right] \left(\frac{T_0}{T}\right)^{X_2}\)	Power law with compensation. Used for line mixing.
`T5`	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right)^{\frac{1}{4} + \frac{3}{2}X_1}\)	Power law with offset.
`AER`	\(L_i(200) = X_0\), \(L_i(250) = X_1\), \(L_i(296) = X_2\), \(L_i(340) = X_3\). Linear interpolation inbetween.	Inspired by the way AER deals with linemixing.
`DPL`	\(L_i(T) = X_0 \left(\frac{T_0}{T}\right) ^ {X_1} + X_2 \left(\frac{T_0}{T}\right) ^ {X_3}\)	Double power law.
`POLY`	\(L_i(T) = X_0 + X_1 T + X_2 T ^ 2 + X_3 T ^ 3 + \cdots\)	Polynomial in temperature. Used internal in ARTS when training our own linemixing.

here, \(X_0\) … \(X_N\) are all model supplied constants whereas \(T\) is the temperature in Kelvin and \(T_0\) is the reference temperature of the model parameters.

Line Strength

Local thermodynamic equilibrium

For local thermodynamic equilibrium (LTE), the line strength is given by

\[S_{LTE} = \rho \frac{c^2\nu}{8\pi} \left[1 - \exp\left(-\frac{h\nu}{kT}\right)\right] \frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \frac{A_{lu}}{\nu_0^3},\]

where \(\rho\) is the number density of the absorbing species,

\[\rho = \mathrm{VMR}\frac{P}{kT},\]

where VMR is the volume-mixing ratio of the absorbing species.

Non-local thermodynamic equilibrium

For non-LTE, the line strength is given by

\[S_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left(r_l \frac{g_u}{g_l} - r_u\right) \frac{A_{lu}} {\nu_0^3},\]

and the added emissions are given by

\[K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{r_u\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left(r_l \frac{g_u}{g_l} - r_u\right) \right\} \frac{ A_{lu}}{\nu_0^3},\]

where \(r_l\) and \(r_u\) are the ratios of the populations of the lower and upper states, respectively. Note that \(K_{LTE} = 0\), as it represents “additional” emission due to non-LTE conditions. Also note that \(K_{NLTE}\) 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 \(r_l\) and \(r_u\) would have in LTE according to the Boltzmann distribution:

\[r_l = \frac{g_l\exp\left(-\frac{E_l}{kT}\right)}{Q(T)}\]

and

\[r_u = \frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)}\]

Putting this into the ratio-expression for \(S_{NLTE}\) with the following simplification steps:

Expansion:

\[\left(r_l \frac{g_u}{g_l} - r_u\right) = \frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right].\]

Extract lower state energies:

\[\frac{g_u}{Q(T)}\left[\exp\left(-\frac{E_l}{kT}\right) - \exp\left(-\frac{E_u}{kT}\right)\right] \frac{\exp\left(-\frac{E_l}{kT}\right)}{\exp\left(-\frac{E_l}{kT}\right)} \rightarrow \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)},\]

where this last step is possible because we estimate that \(E_u-E_l = h\nu_0\). Note how the expression for \(K_{NLTE}\) is 0 under LTE conditions. As it should be. This is seen by putting the above RHS and the expression for \(r_u\) into the expression for \(K_{NLTE}\):

\[K_{NLTE} = \rho \frac{c^2\nu}{8\pi} \left\{\frac{g_u\exp\left(-\frac{E_u}{kT}\right)}{Q(T)}\left[ 1 - \exp\left(\frac{h\nu_0}{kT}\right)\right] - \left[1 - \exp\left(-\frac{h\nu_0}{kT}\right)\right]\frac{g_u\exp\left(-\frac{E_l}{kT}\right)}{Q(T)} \right\} \frac{ A_{lu}}{\nu_0^3} = 0.\]

The ratio between LTE and non-LTE line strength remaining is:

\[\frac{S_{NLTE}}{S_{LTE}} = \frac{1 - \exp\left(-\frac{h\nu_0}{kT}\right)}{1 - \exp\left(-\frac{h\nu}{kT}\right)}.\]

It is clear that the non-LTE expression is the one that is incorrect here. The energy of the emitted photon is not \(h\nu_0\) but \(h\nu\), and as such the actual energy of the transition is \(E'_u-E'_l = h\nu\), 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 ₂, for example, this introduces a factor of

\[\begin{split}S_z = f(\Delta M) \left( \begin{array}{ccc} J_l & 1 & J_u \\ M_l & \Delta M & - M_u \end{array} \right)^2,\end{split}\]

where \(\Delta M \in \left[-1,\;0,\;1\right]\) is the change in quantum number for angular rotational momentum projection along the magnetic field for \(\sigma_-\), \(\pi\), and \(\sigma_+\), respectively, \(f(\Delta M)\) is the 0.75 for \(\sigma_\pm\) and 1.5 for \(\pi\), and \(J_l\) and \(J_u\) are the lower and upper total angular rotational momentum quantum number. The \((:::)\) construct is the Wigner 3-j symbol. It can be computed using software such as that by Johansson and Forssén [1].

Line-mixing using Error-corrected Sudden

TBD