Rayleigh scattering - Revision history

Milcareck at 17:17, 15 June 2026

2026-06-15T17:17:47Z

Milcareck at 17:14, 15 June 2026

2026-06-15T17:14:21Z

Milcareck at 14:22, 21 January 2026

2026-01-21T14:22:40Z

Milcareck at 11:03, 13 January 2026

2026-01-13T11:03:21Z

Milcareck at 10:31, 13 January 2026

2026-01-13T10:31:58Z

Milcareck at 10:25, 13 January 2026

2026-01-13T10:25:28Z

Milcareck at 10:23, 13 January 2026

2026-01-13T10:23:39Z

Milcareck at 10:20, 13 January 2026

2026-01-13T10:20:34Z

Milcareck at 10:16, 13 January 2026

2026-01-13T10:16:00Z

Milcareck at 18:45, 12 January 2026

2026-01-12T18:45:42Z

Show changes

@@ Line 226: / Line 226: @@
 The mass extinction coefficient is related to the molar cross section of scattering by
-<math>k_{\nu,i} = \sigma_{\nu,i} \frac{N_A}{\bar{M}}</math>
+<math>k_{\nu} = \sigma_{\nu} \frac{N_A}{\bar{M}} =  \frac{\sigma_{\nu}}{m}</math>
 where <math>N_A</math> is Avogadro constant and <math>\bar{M}</math> is the average molecular mass of the atmosphere.
@@ Line 239: / Line 239: @@
 <math>\tau_{\nu} = \int_{z_1}^{z_2} \sigma_{\nu} N dz</math>
 <math>\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\rho dz</math>
@@ Line 250: / Line 252: @@
 <math>\tau_{\nu} = \int_{p_2}^{p_1} k_{\nu}\frac{dp}{g}</math>
-A coefficient <math>scalep</math> is added to convert <math>Pa</math> to <math>mbar</math> used in radiative transfer.
+A coefficient <math>scalep</math> is added to convert <math>mbar</math>, used in radiative transfer, to <math>Pa</math>.
 == Rayleigh scattering in Generic PCM ==

@@ Line 14: / Line 14: @@
 The cross section is expected to vary little with <math>T,P</math> in the case of a gas, but due to the dependence of <math>N</math> on <math>T,P</math> and the variation of the refractive index with <math>T,P</math>, if any dependence is omitted, the cross section can be significantly erroneous by several orders of magnitude. The H<sub>2</sub>O molecule is the only one for which the variation of the refractive index with <math>T</math> and <math>P</math> has been measured or estimated. Thus, the variation of the cross section with <math>T</math> and <math>P</math> can be applied to this molecule using the formula introduced below.
-Rayleigh scattering, Brillouin scattering (not applicable in atmospheres) and Raman scattering are not taken into account in the calculation of this cross section. Only Rayleigh scattering (also known as the Cabannes peak) is taken into account.
+Brillouin scattering (not applicable in atmospheres) and Raman scattering are not taken into account in the calculation of this cross section. Only Rayleigh scattering (also known as the Cabannes peak) is taken into account.
 The refractive indices and King factors of each molecule studied in the Generic PCM are described below.
@@ Line 244: / Line 244: @@
 However, according to the hydrostatic balance
-<math>dp = \rho g dz</math>
+<math>dp = -\rho g dz</math>
 Then, we have
-<math>\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\frac{dp}{g}</math>
+<math>\tau_{\nu} = \int_{p_2}^{p_1} k_{\nu}\frac{dp}{g}</math>
 A coefficient <math>scalep</math> is added to convert <math>Pa</math> to <math>mbar</math> used in radiative transfer.

@@ Line 253: / Line 253: @@
 == Rayleigh scattering in Generic PCM ==
 The Rayleigh scattering routine is determined for each spectral band defined in the correlated-k routine and at each pressure level. The routine follows the path below:
@@ Line 258: / Line 260: @@
 * The refractive index <math>ng</math> and the King factor <math>Fk</math> are set.
 * The extinction coefficient is calculated in two stages. <math>Tauvari</math>, the part dependent on the wavenumber, is calculated, then <math>Tauconsti</math>, the part dependent on pressure and temperature (if it exists). Otherwise, the reference pressure <math>P0</math> and temperature <math>T0</math> of the refractive index measurements are used for <math>Tauconsti</math>.
-* The product <math>Tauvari \times Tauconsti</math> is calculated at the end to give the extinction coefficient <math>Tauvar</math> .
+* The product <math>Tauvari \times Tauconsti</math> is calculated at the end to give the extinction coefficient <math>Tauvar</math> . The figure opposite shows this variable in function of wavenumber for each molecule used in the Generic PCM.
 * After this loop, the extinction coefficient is then weighted by the black body function for each spectral band as <math> \text{TAURAY(P,NW)} = \frac{\int_{\nu' \in \text{band}} \text{TAUVAR} (P,\nu') B_{\nu} \, \mathrm{d}\nu'}{\int B_{\nu} \, \mathrm{d}\nu'} </math> .
 * Finally, the optical depth is calculated in optcv.F90, where the thickness of the layer <math>dP</math> is added (<math>g</math> was already added on <math>Tauconsti</math>).
 Each unit in this routine follows the International System of Units. HOWEVER, for all other routines in the radiative part of the model, the pressure is in mbar/hPa! That is why we add a <math>scalep</math> factor at the end for <math>Tauray</math>.

@@ Line 255: / Line 255: @@
 The Rayleigh scattering routine is determined for each spectral band defined in the correlated-k routine and at each pressure level. The routine follows the path below:
-* Each spectral band is recut by the <math>Nfine</math> parameter. Rayleigh scattering calculations are performed on a loop on Nfine for each spectral band.
+* Each spectral band <math>NW</math> is recut by the <math>Nfine</math> parameter. Rayleigh scattering calculations are performed on a loop on Nfine for each spectral band.
 * The refractive index <math>ng</math> and the King factor <math>Fk</math> are set.
 * The extinction coefficient is calculated in two stages. <math>Tauvari</math>, the part dependent on the wavenumber, is calculated, then <math>Tauconsti</math>, the part dependent on pressure and temperature (if it exists). Otherwise, the reference pressure <math>P0</math> and temperature <math>T0</math> of the refractive index measurements are used for <math>Tauconsti</math>.
-* After this loop, the extinction coefficient is then weighted by the black body function for each spectral band.
+* The product <math>Tauvari \times Tauconsti</math> is calculated at the end to give the extinction coefficient <math>Tauvar</math> .
 * Finally, the optical depth is calculated in optcv.F90, where the thickness of the layer <math>dP</math> is added (<math>g</math> was already added on <math>Tauconsti</math>).

@@ Line 260: / Line 260: @@
 * After this loop, the extinction coefficient is then weighted by the black body function for each spectral band.
 * Finally, the optical depth is calculated in optcv.F90, where the thickness of the layer <math>dP</math> is added (<math>g</math> was already added on <math>Tauconsti</math>).
 If you want to add a new molecule, simply enter its refractive index and King factor and use the same <math>Tauvari</math> and <math>Tauconsti</math> formulas, as for the previous molecules.

@@ Line 7: / Line 7: @@
 where <math>\nu</math> is the wave number in <math>m^{-1}</math>, <math>N</math> is the molar volume in <math>molecule.m^{-3}</math>, <math>n_{\nu,i}</math> is the real refractive index, <math>\delta_{\nu,i}</math> is the depolarisation factor.
-The term <math>\left( \frac{6+3\delta_{\nu,i}}{6-7\delta_{\nu,i}}\right)</math> is also called the King factor and is denoted by <math>F_k (\nu,i)</math>. So,
+The term <math>\left( \frac{6+3\delta_{\nu,i}}{6-7\delta_{\nu,i}}\right)</math> is also called the King factor and is denoted by <math>F_k (\nu,i)</math>. So,:
 <math>\sigma_{\nu,i} = \frac{24\pi^3 \nu^4}{N^2 } \left(\frac{n_{\nu,i}^2 -1}{n_{\nu,i}^2 +2}\right)^2 F_k (\nu,i)</math>
@@ Line 94: / Line 94: @@
 <math>b_{\lambda,i, T, p} = \left( a_0 + a_1\bar{\rho} + a_2\bar{T} + a_3\bar{T}\bar{\lambda}^2 + \frac{a_4}{\bar{\lambda}^2} + \frac{a_5}{\bar{\lambda}^2 - \lambda_{UV}^2} + \frac{a_6}{\bar{\lambda}^2 - \lambda_{IR}^2} \right)</math>
-With:
+With
 <math>    \bar{\rho} = \frac{\rho}{\rho_0} </math>
 <math>    \bar{T} = \frac{T}{T_0} </math>
@@ Line 243: / Line 242: @@
 <math>\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\rho dz</math>
-However, according to the hydrostatic balance:
+However, according to the hydrostatic balance
 <math>dp = \rho g dz</math>
-Then, we have:
+Then, we have
 <math>\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\frac{dp}{g}</math>

@@ Line 261: / Line 261: @@
 * After this loop, the extinction coefficient is then weighted by the black body function for each spectral band.
 * Next, the optical depth is calculated in optcv.F90, where the thickness of the layer <math>dP</math> is added (<math>g</math> was already added on <math>Tauconsti</math>).
 == Rayleigh scattering in exo_k ==