Rayleigh Scattering: Optical Depth

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 ) ] · q_i · A²

a₁ = q_i · 2 B,   and   a₂ = q_i · 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) · Z_i · [ (n_i - 1)² / (n_H2 - 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/(n_H2 - 1)²

from Allen (pg. 92)

	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).

Earth P₀ = 1013.25, g= 981, µ=28.97, Z=7.88

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

Jupiter P₀ = 1000.00, g=2425.3, µ=2.22, Z=37.45/0.9

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

Saturn P₀ = 1000.00, g= 1000, µ = 2.14, Z=100.5/0.96

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

Titan P₀ = 1500.00, g= 136, µ= 28.0, Z=58.85

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