Rayleigh Scattering: Optical Depth
The average scattering cross section per particle is given by
where, n is the index of refraction, N0 is the number of molecules per unit volume at standard temperature and pressure ( i.e., conditions of index of refraction), is the depolarization factor.
The optical depth can be related to the ``thickness'' of the atmosphere. If Z is the thickness in Km-amagats then:
ray() = N(z) · () dz () · N(z) · dz = Z · N0 · ()
The index of refraction has a wavelength dependence. This is usually represented by two constants, A and B, as follows:
(n-1) = A · (1 + B/2), in µm
(n-1)2 = A2 · (1 + 2 B/2 + B2/4)
ray = Z a0/4 · (1 + a1/2 + a2/4) where,
a0 = (32 3)/(3 N02) · [(6 + 3 )/(6 - 7 ) ] · qi · A2
a1 = qi · 2 B, and a2 = qi · B2
ray(H2) = Z · 2.19 · 10-4/4 · ( 1 + 0.0157248/2 + 0.0001978/4 ), in µm
The wavelength dependence in the VIAMP code is taken from Dalgarno and Williams (ApJ, 1962). This equation is similar to the equation above, however, it differs in the 8 term.
ray(H2) = (8.14 · 10-13/4 + 1.28 · 10-6/6 + 1.61/8) cm2, in Å
The optical depth per Km-amagat of hydrogen, denoted by 1(H2), is then
1(H2) = 2.687 · 1024 · ray(H2) = 2.687 (8.14 · 1011/4 + 1.28 · 1018/6 + 1.61 · 1024/8), in Å
1(H2) = 2.687 (8.14 · 10-21 f4 + 1.28 · 10-30 f6 + 1.61 · 10-40 f8), f = 108/ and in Å
And the total optical depth for a mixture of gases is given as
ray = 1(H2) · Zi · [ (ni - 1)2 / (nH2 - 1)2 ]
Note that this formulation always uses the wavelength dependence of hydrogen, even if other gases are used.
(n-1) = A · (1 + B/2), in µm
ray = (n-1)2/(nH2 - 1)2
ray | n | A (10-5) | B (10-3) | (10-24) | a0 µm4/Km | ||
air | 4.4459 | 1.0002918 | 28.71 | 5.67 | 3.40 | 0.031 | 10.7E-4 |
H2 | 1.0000 | 1.0001384 | 13.58 | 7.52 | 1.61 | 0.02 | 2.35E-4 |
He | 0.0641 | 1.0000350 | 3.48 | 2.30 | |||
O2 | 3.8634 | 1.000272 | 26.63 | 5.07 | 0.054 | ||
N2 | 4.6035 | 1.000297 | 29.06 | 7.7 | 3.44 | 0.030 | 10.9E-4 |
H2O | 3.3690 | 1.000254 | |||||
CO2 | 10.5611 | 1.0004498 | 43.9 | 6.4 | 0.09 | ||
CO | 5.8247 | 1.000334 | 32.7 | 8.1 | |||
NH3 | 7.3427 | 1.000375 | 37.0 | 12.0 | |||
NO | 4.6035 | 1.000297 | 28.9 | 7.4 | |||
CH4 | 10.1509 | 1.000441 |
The depth of penetration of Rayleigh scattering is computed for various objects using the data above. A summary of the characteristics of the plots in Figure 9 is given below (g is gravity in cm/s2, µ = molecular weight in gm/mole, Z = KM-amagats per Bar).
wave(µm): | 0.1000 | 0.2000 | 0.2640 | 0.3000 | 0.4000 | 0.5000 |
(P0): | 209.57 | 6.95 | 2.05 | 1.19 | 0.36 | 0.14 |
P(=1): | 4.8 | 145.7 | 493.2 | 851.0 | 2834.7 | 7094.5 |
wave(µm): | 0.1000 | 0.2000 | 0.2640 | 0.3000 | 0.4000 | 0.5000 |
(P0): | 299.82 | 8.62 | 2.47 | 1.42 | 0.42 | 0.17 |
P(=1): | 3.3 | 116.1 | 405.2 | 706.3 | 2390.8 | 6030.4 |
wave(µm): | 0.1000 | 0.2000 | 0.2640 | 0.3000 | 0.4000 | 0.5000 |
(P0): | 754.31 | 21.68 | 6.21 | 3.56 | 1.05 | 0.42 |
P(=1): | 1.3 | 46.1 | 161.0 | 280.7 | 950.3 | 2396.9 |
wave(µm): | 0.1000 | 0.2000 | 0.2640 | 0.3000 | 0.4000 | 0.5000 |
(P0): | 5378.65 | 152.59 | 43.58 | 24.98 | 7.37 | 2.92 |
P(=1): | 0.3 | 9.8 | 34.4 | 60.1 | 203.6 | 513.9 |
The gas continuum absorption contains a term which is supposed
to represent the effect of Raman scattering due to the 1 vibration
of H2 at 4161 cm-1.
Ref.: Belton et al. (1971). Atm. of
Uranus. ApJ. 164, 191-209