36 RADIATION 
simply by plotting experimental absorption values 
against the square roots of m, or by the usual repre- 
sentation of In (1 — A), we can determine whether we 
are dealing with a continuous distribution of the absorp- 
tion coefficient, or with a resolution into individual 
lines. It is important to realize that this method is 
applicable even if the apparatus is not sufficiently 
sensitive to resolve the individual lines. Strong [48] 
applied this method to entire bands whose separate 
lines may have very different values for ko. 
Radiation Diagrams 
The knowledge of the absorption of a spectral line 
gives us one basis for the calculation of atmospheric 
heat radiation. The second basis is the distribution of 
the values for ko over the various wave lengths. For 
water vapor, which is most effective in the troposphere, 
the available measurements have been tabulated by 
Elsasser [14] and Méller [34]. Callendar and Cwilong 
give analogous figures. The absorption shows very great 
differences. In the rotation spectrum, ky is of the order 
of 10* em* g™; in the rotation-oscillation spectrum at 
6 uw it is about 10° em? g+; whereas it decreases in the 
“window” of the water vapor to 1 em? g or below. In 
spite of these large differences we can never neglect one 
spectral range as compared to another, because the in- 
tensive radiation in the range of nee values of ko is 
readily absorbed even by very thin layers, whereas the 
low radiation intensities at small values of ko are scarcely 
absorbed. However, thick and distant layers can parti- 
cipate in the emission of this radiation. Therefore we 
need the cumulative effect of all wave lengths for com- 
parison with measurements. Miigge and Moller [37] 
were the first to use a graphical method in which the 
integration over all wave lengths is computed in ad- 
vance and represented in diagrammatic form. Elsasser 
[14] developed a similar diagram, which is the same in 
principle, but which differs somewhat in arrangement. 
According to the foregoing discussion, the radiation 
at wave length ) of a thin layer of gas of temperature 
T is given by 
dS, = 7&(X, T) dm, (11) 
OL (kym) 
am 
where kp varies from line to line. Even larger spectral 
ranges that comprise a number of lines can be com- 
bined as long as ko varies less than k, in the range of 
one line, that is, less than 100:1. Then the total radia- 
tion of a layer element dm is obtained by summation 
over all wave lengths, 
dS = xdm 2 GQ; P) ena, 
A, (12) 
1. (Note added July, 1950.) Callendar [11] used an empirical 
absorption law L(w) = w/(w + wo) in which wy is a constant 
characterizing the degree of absorption; this law is easily 
applicable in theoretical investigations and covers the observa- 
tions well. Using all experimental data of water vapor, Cwilong 
[12] recently deduced an empirical absorption function, but he 
gives numerical values only and not an analytic expression. 
The values of the function are available for narrow frequency 
intervals of the whole long-wave water-vapor spectrum. 
and the radiation of a layer-m of finite thickness upon a 
surface element situated in the one boundary surface 
of the layer m becomes 
S=zf ame Sa, ey, An. (13) 
This is the radiation of an atmospheric layer upon a 
unit area, for instance the radiation of the entire atmos- 
phere upon the unit area of the sensitive surface of a 
measuring instrument placed on the ground. The inte- 
gration over m is difficult at first, because the tempera- 
ture T is in general not constant but a function of m, 
that is, a function of the radiating mass situated be- 
tween an altitude above ground and the surface of the 
earth. If we consider an isothermal atmosphere of 
temperature 7, then 
S7,= rf dm &(A, nyc (kom) 
= X7,(m) (13a) 
will be a function X of m. If we now plot the absorption 
1.0 8 -6 4 of (0) 
0) OS) all 2 Bi) ! 2 0 
‘ oO .O5 .| 2 5 | A 9) 
@4o) Os all “2 He) I 2 5 © 
© Ol OS di a 23) | 2 5 10 © 
(0) -2 -4 -6 8 1.0 
Fic. 1.—Absorption functions. The linear scale at the top 
gives the transmitted radiation. The linear scale at the bottom 
gives the absorptive or emissive power. Numbers on the func- 
tion seales from A to D are the absorbing mass m. 
(A) 1 —e~*™ for k = 1.66. Parallel radiation. 
(B) 1 —2 H3(m). Diffuse radiation. 
(C)) IKE = (kom) for ky) = 6.5. Diffuse radiation of a 
spectral line with a = 5.5. 
(D) L4. 12 (kom) for ko = 20. Diffuse radiation of a spec- 
tral line with a = 12. 
The values of & and ky are chosen so that for the same mass 
m the absorption will be 0.5 in each case. 
function X as the abscissa and provide it with a scale of 
m, as in Fig. 1, we can read at the scale division m, the 
radiation intensity emitted by the isothermal layer of 
temperature 7’). However, X also indicates the amount 
absorbed by this layer when an infinite surface of 
temperature 7) transmits black-body radiation through 
this layer. If ko does not vanish for any wave length, 
an infinitely thick layer (m = ) will have total 
absorption. Accordingly, the radiation emitted by an 
infinitely thick layer of gas is equal to the black-body 
radiation: the point m = © of the scale lies at X = 
oT 0. 
It can be seen immediately that the radiation of a 
layer element dm of temperature 7 is also given by the 
differential of X: 
Us 2 sep) a. 
om 
In order to find the radiation of a layer element of a 
temperature other than 7», a new evaluation of the 
