Page:On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground.pdf/22

considered does not alter its albedo as a consequence of the altered temperature. In that case entirely different circumstances enter. If, for instance, an element of the surface which is not now snow-covered, in consequence of falling temperature becomes clothed with snow, we must in the last formula not only alter $\beta$ but also $\nu$. In this case we must remember that $\alpha$ is very small compared to $\beta$. For $\alpha$ we will choose the value $0.40$ in accordance with Langley's^[1] estimate. Certainly a great part of this value depends upon the diffusely reflected part of the sun's heat, which is absorbed by the earth's atmosphere, and therefore should not be included in $\alpha$, as we have defined it above. On the other hand, the sun may in general stand a little lower than in Langley's measurements, which were executed with a relatively high sun, and in consequence of this $\alpha$ may be a little greater, so that these circumstances may compensate each other. For $\beta$ we will choose the value $0.70$, which corresponds when ${\text{K}}=1$ and ${\text{W}}=0.3$ (a little below the freezing-point) with the factor $1.66$ (see p. 253). In this case we find the relation between ${\text{T}}$ (uncovered) and ${\text{T}}_{1}$ (snow-covered surface) to be ${\text{T}}^{4}:{{\text{T}}_{1}}^{4}={\frac {{\text{A}}(1+1-0.40)+{\text{M}}}{\gamma (1+1-0.70)}}:{\frac {{\text{A}}(1+0.50-0.20)+{\text{M}}}{\gamma (1+0.50-0.35)}}={\frac {1.60+\phi }{1.30}}:{\frac {1.30+\phi }{1.15}}$, if ${\text{M}}=\phi {\text{A}}$. We have to bear in mind that the mean ${\text{M}}$ for the whole earth is zero, for the equatorial regions negative and for the polar regions positive. For a mean latitude ${\text{M}}=0$, and in this case ${\text{T}}_{1}$ becomes $267.3$ if ${\text{T}}=273$, that is the temperature decreases in consequence of the snow-covering by 5°.7 C.^[2] The decrease of temperature from this cause will be valid until $\phi =1$, i. e. till the heat delivered by convection to the air exceeds the whole radiation of the sun. This can only occur in the winter and in polar regions.

But this is a secondary phenomenon. The chief effect that we examine is the direct influence of an alteration of $\beta$ upon the temperature ${\text{T}}$ of the earth's surface. If we start from a value ${\text{T}}=273$ and $\beta =0.70$, we find the alteration ($t$) in the

1. Langley, "Temperature of the Moon," p. 189. On p. 197 he estimates $\alpha$ to be only $0.33$.
2. According to the correction introduced in the sequel for the different heights of the absorbing and radiating layers of the atmosphere, the number 5°.7 is reduced to 4°.0. But as about half the sky is cloud-covered, the effect will be only half as great as for cloudless sky, i. e. the mean effect will be a lowering of about 2° C.