266 
refer to an atmosphere in which the density p varies 
with height h above the ground according to the re- 
lation 
hl hla 
’ 
—h/ = 
Pp = poe > & = Noe 
where H is a constant (called the scale height), and po 
denotes the density at the ground (h = 0). The alterna- 
tive form n = noe!” refers to the number of particles 
per cubic centimetre, n at height h, no at the ground. 
Such formulas do not apply to the actual atmosphere 
unless H is itself regarded as a function of h, but they 
are useful approximations over a range of height in 
which H does not vary greatly. The value of H is 
kT /mg, where k is Boltzmann’s constant (1.380 * 10-1 
ergs per degree C); 7’ denotes the absolute temperature, 
g the acceleration of gravity (981 cm sec), and m the 
mean mass of the particles composing the atmosphere. 
Also H = RT/Mg, where M (= Nm) is the mean molec- 
ular weight, and R (= KN) is called the gas constant; 
R = 8.314 X 10’ ergs per degree C per mole. 
It is of special interest to consider the absorption of 
light in such an atmosphere, because of the simplicity 
of the relations involved, and their approximation to 
the actual conditions in our atmosphere. 
It will be supposed that outside the atmosphere the 
intensity of the light considered is J, and that at 
height hf it is 7. The absorption will be supposed to take 
place with a definite absorption coefficient k,, so that 
in effect the light must be monochromatic, of a definite 
frequency v (or rather, within a narrow band of fre- 
quency v to vy + dy). Instead of 7 and J,, one might 
therefore alternatively consider the flux of photons of 
this frequency, Q per square centimetre per second at 
height h, and Q,, at h = © (outside the atmosphere): 
Q« Tor Q/Q,= 1/T, - 
The absorption may be due to a particular atmos- 
pheric constituent, and if so, n and p will refer not to 
the whole air but to this particular constituent only. 
If the atmosphere is so static that the different con- 
stituents are each distributed in accordance with their 
molecular weight, each will have its own value of H; 
for example, for O2 at 273° absolute (OC), H is about 
7.2 km, and for O it is twice as great; the only assump- 
tion here made, however, is that H can be regarded as 
independent of h. 
Let x denote the zenith angle of the sun, which is 
also the angle made by the beam of light with the 
vertical (the curvature of the level layers of the at- 
mosphere being neglected). The equation of absorption 
is dI = —k,nI sec x dh, of which the integral is 
I = 1, exp(—hkymHe"” sec x). 
The rate of absorption q per cubic centimetre at height 
h is —dI cos x/dh, which has the value kynJI. This has 
its maximum at the level 
Imax = H In (kymoH sec x) = ho + H In sec x, 
where In denotes the logarithm to the Napierian base 
é (= 2.718), and ho is the value of hmax for x = 0 (vertical 
sunlight); at this level n has the value nmax given by 
Mmax = (1/kyH) cos x, 
THE UPPER ATMOSPHERE 
and q has the value qmax given by 
Qmax = (I,, cos x)/H exp 1, 
where exp 1 = 2.718. 
Let 2 = (h — Mmax)/H, and q! = @/qmax; then the 
relations above lead to 
qd = el-z—e * 4 
A graph of this function is shown in Fig. 1. It repre- 
sents the proportionate distribution of absorption per 
0.8 1.0 
Fic. 1—The proportionate distribution of absorption of 
monochromatic radiation per unit volume of gas in an expo- 
nentially distributed atmosphere, given as a ratio (q’) of the 
absorption at any level, to that at the level of maximum ab- 
sorption; the level is indicated by z, reckoned from the height 
of maximum absorption, in units of the scale-height of the 
atmosphere. 
unit volume as a function of height z, reckoned up- 
wards or downwards from the level of maximum, in 
terms of H as the unit of height. The form of the curve 
is independent of the particular value of H and of the 
initial light-intensity J,,, as well as of the absorption 
coefficient k, Though the actual level of maximum 
absorption depends on k,, no, H, and x, the value of 
the maximum absorption depends on Jf,, H, and x, 
but not on ky or no. 
Below the level hmax the beam intensity and the 
absorption fall away very rapidly, the beam being 
quickly attenuated by the increasingly dense air. The 
decrease of q’ upwards is slower; at great heights q’ 
varies approximately as e!-*. The main part of the 
absorption lies in a layer of thickness about 3H 
(from z = —1 toz = 2). Throughout the day, gmax 
varies as cos x, and hmax is In see x above hy. When 
x = 60°, sec x = 2, In sec x = 0.7, and so tmx = 
ho + 0.7H. 
The height ho (= H In kynH) may be negative for 
sufficiently small values of k, and no, that is, if the 
absorption is weak; q then increases downwards to the 
ground, and much of the radiation penetrates to ground 
level. For oxygen (O2), for which no is of order 10* , 
and H of order 108 cm (7.2 km), the minimum value 
of k, for which hy will not be below ground, namely 
