LONG-WAVE RADIATION 37 
summation over \ in (12) or (13) with 7 as a parameter 
becomes necessary. Let the ratio of the radiation of 
the two layer elements be y; then 
SD ON) = L4(kym)An 
YT, To, m) = a Ue 
SEO. 1) = L#(kym)AX 
nN 
(14) 
We are now able to determine the radiation of the 
layer element dm of temperature 7 from the product 
Sa = 9 UK, Uo m)=-Xr5(m) dm. (15) 
m 
From (15) follows the key equation of the radiation 
diagram 
Ge i i, Poy 7) Sra, (16) 
where 7’ may now be any function of m, that may be 
given by observation or assumption. If we now plot y 
on the ordinate against X on the abscissa, we obtain a 
graph with curves for every value of 7, in which y = 
1 signifies that T = 7, and T < T> gives values of y 
less than 1 (Fig. 2a). The radiation of an isothermal 
O] 
1072 CAL CM=2 MIN™! 
40° 40° 
-40° 
Be 
io! | 
10° 2 
Ww 
Fie. 2a—Radiation diagram according to Méller. The heavy 
lines refer to downcoming radiation of the atmosphere at the 
ground (1), the radiation at 7 km received from below (II), 
and the radiation at 7 km received from above (III). 
atmospheric layer is then given by the area bounded 
laterally by parallels to the y-axis through m = 0 and 
m = m, by the X-axis at the bottom, and the line 7 
at the top. The radiation of a nonisotherma atmos- 
pheric layer in which 7’ is a function of m can be found 
by plotting the temperature distribution 7(m) in the 
network of curves for m and 7’, and integrating. This is 
the basic principle of all radiation diagrams. Aside from 
the use of different absorption values, ko, Elsasser’s 
chart is an authalic transformation of this principle, 
in which the isotherms are made rectilinear, and, as 
a result, the lines of equal m-values become curved. 
The curvature is hardly noticeable, because the lines 
for m = 0 and m = ~, which, as the boundaries of the 
graph, remain straight lines, intersect at the point 
T = 0. Thus the diagram assumes triangular or trape- 
zoidal shape (Fig. 2b). Furthermore, abscissa and ordi- 
nate are interchanged by comparison with Méller’s 
diagram. 
ae | 
40° o° -40° 
Fic. 2b.—Radiation chart according to Elsasser. The heavy 
lines correspond to those in Fig. 2a. 
=e0° 
Even though the principle of the two radiation dia- 
grams is the same, there are certain numerical differ- 
ences. These are best illustrated by a comparison of 
the functions Xioc and X—ssc in the two charts. The 
values 77 = +40C and —80C are the highest and 
lowest temperatures shown. Table I shows that there 
Tasie I. Raprarion oF AN IsorHeRMAL LAYER WITH A 
WaTER-VApPorR ConTENT w IN PER Cant oF oJ” As 
Grtven By Exsasser (E) anp Méuier (M) 
w (g cm=*) 
Temperature j | 
UOC Sea | at. |) 5 vo he 
Ica eectl mica) oe | % | % | % 
+40C 10; 17.0} 26.8 41-1 || 57.8 | 75.9 | 91.9) ) 100-0 
| M 18 |) Wee | ate) |) GAS || Wats | 89.0 | 100.0 
_g0q (B | 16-6 | 38.1 | 58.0 | 76.1 | 88.8 | 96.6 | 100.0 
M 13, Il || BRS |) Glase! |) Wee 85.2 | 93.3 | 100.0 
1 
is good agreement with differences of less than 1 per 
cent in the middle range of amounts of water vapor 
between 3 X 10- and3 X 107! g cm. In the range of 
smaller or larger amounts of water vapor Moller’s 
values are about 2 per cent lower than Elsasser’s. (The 
first edition of Elsasser’s chart, as well as the earlier 
edition of the chart by Mtigge and Moller both showed 
radiation values 10 to 15 per cent lower in the middle 
range of water-vapor amounts. The agreement of the 
revisions made by both authors independently during 
the war is rather remarkable.) 
There is more of a difference in the evaluation by the 
two authors of the radiation of carbon dioxide. For the 
amount of CO: normally present in the atmosphere, 
there is almost complete absorption in the extraordi- 
narily intense band around 14.9 «4 even by only very 
thin layers. The weak extreme boundaries of the band 
extend to 12.5 » and 17.5 u, respectively. Elsasser now 
