LONG-WAVE RADIATION 35 
(where dw is the solid angle), and if the mass m has the 
same temperature 7’, then m radiates, according to 
Kirchhoff’s law, the same amount of energy as it ab- 
sorbs; that is, 1t emits 
B = &(, T) dw-A = &(d, T) dw (1 — e-4"). (3) 
A mass element dm of temperature 7’ therefore sends 
through the absorbing mass m the radiation 
dH = &(X, T) dw:dA = &(, T) dw-e-"kydm. (4) 
If the mass element has a different temperature 74, 
the radiation is simply 
G(A, T1) dwa-e"kydm (5) 
in which it is assumed that k) is not a function of the 
temperature of the mass m. 
Diffuse Radiation. The simple basic law represented 
by equations (8) and (4) holds essentially for all com- 
plications that appear in the atmosphere. First of all 
we must take into consideration the fact that the radia- 
tion is not parallel. Let us consider the radiation of a 
layer of air of infinite horizontal extent which is part 
of a uniformly stratified air mass and which is to con- 
tain the radiating mass dm over an area of one square 
centimeter. Then the radiation emitted from this mass 
will arrive on a receiving surface at all angles of inci- 
dence 0°< yg <90°. The integration of equation (4) 
over » can be reduced to known functions and the 
radiation of the layer element is 
dS = r& (A, T) 2Ho(kym) kydm. (6) 
The radiation of a layer of finite thickness with a tem- 
perature T is then 
S = x& (0, £) [1 — 20s (kym)]. (7) 
The functions H» and A; (called Hi) and Hi; by Elsasser) 
are tabulated [27], so that they can be used for exact 
calculations. The expression A? = [1 — 2H; (kym)] 
can be considered as an absorption function of diffuse 
radiation, and its differential is accordingly 
dA? = 2H.(kym) kydm. 
Within a large range the following approximations can 
be made: 
2H (x) & 1.66e+* and 2H,(x) Y e)6, 
which means that the laws for diffuse radiation between 
atmospheric layers are replaced by the laws for parallel 
radiation in which, however, the absorption coefficient 
is multiplied by 54. 
Radiation from a Spectral Line. A further modifica- 
tion of the simple absorption laws is required by the 
physical processes that take place when gas masses 
radiate. The absorption bands of the multimolecular 
gases are not continuous, but are resolved into numer- 
ous closely spaced absorption lines. The absorption 
coefficient is extremely high in the center of each of 
these lines, while it is smaller by about two orders of 
magnitude between two lines. Thus, even if we consider 
only a very small spectral band Ad which contains only 
a single line, we must take into consideration a varia- 
tion of k, at a ratio of 1:100. This is most easily done 
by determining once and for all the absorption function 
of a line. The form of a spectral line, that is, the law of 
the decrease of the absorption coefficient from the 
center towards the edges, is known as the dispersion 
form: 
a od /4 
= 1)? + 8/4 
(Instead of \, the frequency v = 1/\ cm™! is used; 
6 is also given in em.) The significant values in this 
law are (1) the absorption coefficient m the center ko, 
(2) the half-value width 6 of the line, and (3) the dis- 
tance between two adjacent lines Av. The ratio 6/Av 
determines to what fraction of ko the absorption coefhi- 
cient k decreases between two lines. Neither 6 nor Ap 
has the same value in one band, let alone in different 
bands, of the same spectrum. The distance between 
lines Ay varies irregularly, because of the overlapping 
of differmg laws for the various spectral lines. Only 
very few values have been determined for 6, because 
the measurements are extremely difficult to make. 
Therefore, it 1s usually assumed that 6, as well as Ay, 
is constant for the entire spectrum. Although this as- 
sumption is only an expedient, there is no possibility 
of a more exact evaluation at the present time. 
If we integrate the absorption laws for parallel or 
diffuse radiation over such a dispersion form of a 
spectral line, we arrive at new laws, that is, new absorp- 
tion functions, which can be expressed by 
Li (kom) 
ky (8) 
il +Ay/2 [ ( koms"/4 )] 
= — 2H; w (9 
Gd seas on NG ee Ey A eS 
for the radiating layer m, and by 
L (kom) 
1 +Ay/2 Ieom8?/4 
for the radiating column. As before, the radiation of a 
layer element is the first derivative of this function with 
OL (kom) 
respect to m, that is, dm for parallel or diffuse 
radiation. It is of no consequence whether this integral 
can be solved analytically or numerically. If it can be 
tabulated, it can be used for any further calculations. 
The two new equations, (9) and (10), contain as a 
parameter the ratio between the half-value width and 
the distance between lines, a = Ay/6. 
An approximate solution for L may be obtained, if 
we assume 62/4 in the denominator to be negligible 
compared to (vy — vo). In that case 
L (kom) & V kowm/o2. 
This indicates that the radiation of a gas layer of 
finite thickness m is proportional to the square root of 
m if the absorption occurs in individual lines. If, on 
the other hand, we were dealing with continuous absorp- 
tion, we would arrive at an exponential function. Thus, 
