Rayleigh Scattering: Optical Depth



The average scattering cross section per particle is given by

sigma(lambda) = (128 pi5 alpha2)/(3 lambda4) · [ (6 + 3 delta)/(6 - 7 delta) ]
alpha = (n - 1)/(2 pi N0) approx (n2 - 1)/(4 pi N0),   for n near unity

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), delta 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:

tauray(lambda) = integ N(z) · sigma(lambda) dz approx sigma(lambda) · integ N(z) · dz = Z · N0 · sigma(lambda)

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/lambda2),   lambda in µm

(n-1)2 = A2 · (1 + 2 B/lambda2 + B2/lambda4)

tauray = Z a0/lambda4 · (1 + a1/lambda2 + a2/lambda4)   where,

a0 = (32 pi3)/(3 N02) · [(6 + 3 delta)/(6 - 7 delta) ] · sum qi · A2

a1 = sum qi · 2 B,   and   a2 = sum qi · B2

tauray(H2) = Z · 2.19 · 10-4/lambda4 · ( 1 + 0.0157248/lambda2 + 0.0001978/lambda4 ),   lambda 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 lambda8 term.

sigmaray(H2) = (8.14 · 10-13/lambda4 + 1.28 · 10-6/lambda6 + 1.61/lambda8) cm2,   lambda in Å

The optical depth per Km-amagat of hydrogen, denoted by tau1(H2), is then

tau1(H2) = 2.687 · 1024 · sigmaray(H2) = 2.687 (8.14 · 1011/lambda4 + 1.28 · 1018/lambda6 + 1.61 · 1024/lambda8),   lambda in Å

tau1(H2) = 2.687 (8.14 · 10-21 f4 + 1.28 · 10-30 f6 + 1.61 · 10-40 f8),   f = 108/lambda   and   lambda in Å

And the total optical depth for a mixture of gases is given as

tauray = tau1(H2) · sum 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/lambda2),   lambda in µm

ray = (n-1)2/(nH2 - 1)2



from Allen (pg. 92)
raynA
(10-5)
B
(10-3)
alpha
(10-24)
deltaa0
µm4/Km
air4.44591.000291828.715.673.400.03110.7E-4
H21.00001.000138413.587.521.610.022.35E-4
He0.06411.00003503.482.30
O23.86341.00027226.635.070.054
N24.60351.00029729.067.73.440.03010.9E-4
H2O3.36901.000254
CO210.56111.000449843.96.40.09
CO5.82471.00033432.78.1
NH37.34271.00037537.012.0
NO4.60351.00029728.97.4
CH410.15091.000441


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):0.10000.20000.26400.30000.40000.5000
tau(P0):209.576.952.051.190.360.14
P(tau=1):4.8145.7493.2851.02834.77094.5


Jupiter P0 = 1000.00, g=2425.3, µ=2.22, Z=37.45/0.9
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):299.828.622.471.420.420.17
P(tau=1):3.3116.1405.2706.32390.86030.4


Saturn P0 = 1000.00, g= 1000, µ = 2.14, Z=100.5/0.96
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):754.3121.686.213.561.050.42
P(tau=1):1.346.1161.0280.7950.32396.9


Titan P0 = 1500.00, g= 136, µ= 28.0, Z=58.85
wave(µm):0.10000.20000.26400.30000.40000.5000
tau(P0):5378.65152.5943.5824.987.372.92
P(tau=1):0.39.834.460.1203.6513.9


The gas continuum absorption contains a term which is supposed to represent the effect of Raman scattering due to the nu1 vibration of H2 at 4161 cm-1.
Ref.: Belton et al. (1971). Atm. of Uranus. ApJ. 164, 191-209

tauraman = 0.0208 · tauray