Absorption

Absorption is the physical process that reduces the spectral radiance of a beam of light as it passes through a medium. It can be described be Beer’s law:

\[\vec{I}(r) = \vec{I}(0) \exp(-\mathbf{K} z),\]

where \(\vec{I}(r)\) is the spectral radiance of the light at some distance \(r\), and \(\mathbf{K}\) is the propagation matrix of the medium.

Propagation Matrix

The propagation matrix conceptually describes how spectral radiance propagates through a system. The propagation matrix is a square matrix with strict symmetries, and it has the form

\[\begin{split}\mathbf{K} = \left[ \begin{array}{rrrr} A & B & C & D \\ B & A & U & V \\ C &-U & A & W \\ D &-V &-W & A \end{array} \right],\end{split}\]

where \(A\) describes the total power reduction, \(B\) describes the difference in power reduction between horizontal and vertical linear polarizations, \(C\) describes the difference in power reduction between plus 45 and minus 45 linear polarizations, \(D\) describes the difference in power reduction between right and left circular polarizations, \(U\) describes the phase delay between right and left circular polarizations, \(V\) describes the phase delay between plus 45 and minus 45 linear polarizations, and \(W\) describes the phase delay between horizontal and vertical linear polarizations. The unit of all of these is m \(^{-1}\).

Spectral Radiance

The unit of spectral radiance is W sr \(^{-1}\) m \(^{-2}\) Hz \(^{-1}\).

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:

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:

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

where \(\alpha\) is the absorption coefficient, \(S\) is the line strength operator, and \(F\) is the line shape 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

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

where

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

\(\nu\) is the frequency, \(\nu_0\) is the line center frequency, \(\Delta\nu_{lm}\) is the line mixing shift, \(\Delta\nu_Z\) is the Zeeman splitting, \(\Delta\nu_{P,0}\) is the pressure shift, \(G_{P,0}\) is the pressure broadening - half width half maximum in the Lorentz profile, \(G_{lm}\) is the strength modifying line mixing parameter, \(Y_{lm}\) is the phase-introducing line mixing parameter, and \(G_D\) is the scaled Doppler broadening half-width half-maximum. \(\nu\) is just a sampling frequency, it can be anything positive. \(\nu_0\) is provided by the absorption line catalog. The \(\Delta\nu_{lm}\), \(\Delta\nu_{P,0}\), and \(G_{P,0}\) are `<Line Shape Parameters_>`_<Line Shape Parameters_>. \(\Delta\nu_Z\) is the Zeeman splitting, which depends on molecule and magnetic field strength as is described in `Zeeman effect <Zeeman Effect_>`_ <Zeeman Effect_>. The scaled Doppler broadening half width half maximum is given by

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

where \(R\) is the ideal gas constant in Joules per mole per Kelvin, \(T\) is the temperature in Kelvin, \(m\) is the mass of the molecule in grams per mole, and \(c\) 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

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

Non-local thermodynamic equilibrium

For non-LTE, the line strength is given by

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

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 actualy care about non-LTE (which is low density, low collision atmospheres).