Neither of them give us any information, however, about the actual distribution of energy at any one temperature from which we may calculate that at any other temperature. For that, some relation must be found between the energy and the wave-length. Planck, by reasoning founded on the electromagnetic character of the waves, derived such a relation, but both his reasoning and his results are a little too complicated to be introduced here. His results have been confirmed in the most striking manner by experiments carried out by Rubens and Kurlbaum (Ann. der Physik, 4, p. 649, 1901). They measured the energy in a particular wave-length (.0051 cms., i.e. nearly 100 times the wave-length of red light) in the radiation of a full radiator from a temperature of 85° up to 1773° absolute, and their results are given in the following table:

Absolute Temperature. Observed Energy. Energy calculated from
Planck's Formula.
85 -20.6 -21.9
193 -11.8 -12
293 0 0
523 +31 +30.4
773 64.6 63.8
1023 98.1 97.2
1273 132 132
1523 164 160
1773 196 200

We have therefore the means of calculating both the total quantity and the kind of radiation given out by any full radiator at any temperature, and a number of very interesting problems may be solved by means of the results.

Efficiency in Lighting.—One very simple problem is concerned with efficiency in lighting. We see by reference to Fig. 16, that in the radiation from the electric arc very little of the energy is in the visible part of the spectrum even though the temperature in the arc is the highest yet obtained on the earth, whereas the energy in the visible part of the spectrum from a gas flame is almost wholly negligible. The problem of efficient lighting is to get as big a proportion as possible of the energy into the visible part of the spectrum, and therefore the higher the temperature the greater the efficiency. This is the reason of the greater efficiency of the incandescent gas mantle over the ordinary gas burner, for the introduction of the air into the gas allows the combustion to be much more complete, and therefore the temperature of the mantle becomes very much higher than that of the carbon particles in the ordinary flame. The modern metallic filament electric lamps have filaments made of metals whose melting point is extremely high, and they may therefore be raised to a much higher temperature than the older carbon filaments. The arc is even more efficient than the metallic filament lamps, because its temperature is higher still; and we must assume that the temperature of the sun is very much higher even than the arc, since its maximum of energy lies in the visible spectrum.

Temperature of the Sun.—The actual temperature of the sun may be calculated approximately by means of Stefan's fourth power law. We will first assume that the earth and the sun are both full radiators, and that the earth is a good conductor, so that its temperature is the same all over. The first assumption is very nearly true, and we will make a correction for the small error it introduces; and the second, although far from true, makes very little difference to the final result, for it is found that the values obtained on the opposite assumption that the earth is an absolute non-conductor differ by less than 2 per cent. from those calculated on the first assumption. We will further assume that the heat radiated out by the earth is exactly equal to the heat which it receives from the sun. This is scarcely an assumption, but rather an experimental fact, for experiment shows that heat is conducted from the interior of the earth to the exterior, and so is radiated, but at such a small rate that it is perfectly negligible compared with the rate at which the earth is receiving heat from the sun.

The sun occupies just about one 94,000th part of the hemisphere of the heavens or one 188,000th part of the whole sphere. If the whole sphere surrounding the earth were of sun brightness, the earth would be in an enclosure at the temperature of the sun, and would therefore be at that temperature itself. The sphere would be sending heat at 188,000 times the rate at which the sun is sending it, and the earth would be radiating it at 188,000 times its present rate. But the rate at which it radiates is proportional to the fourth power of its absolute temperature, and therefore its temperature would be the fourth root of 188,000 times its present temperature, i.e. 20.8 times. If the radiating or absorbing power of the earth's surface be taken as 9/10, which is somewhere near the mark, the calculation gives the number 21.5 instead of 20.8. The average temperature of the earth's surface is probably about 17° C. or 290° absolute, and therefore the temperature of the sun is 290 x 21.5, i.e. about 6200° absolute.

It is easy to see that if we had known the temperature of the sun and not of the earth, we could have calculated that of the earth by reversing the process.

By this means we can estimate the temperatures of the other planets, at any rate of those for which we may make the same assumptions as for the earth. Probably those planets which are very much larger than the earth are still radiating a considerable amount of heat of their own, and therefore to them the calculation will not apply; but the smaller planets Mercury, Venus and Mars have probably already radiated nearly all their own heat and are now radiating only such heat as they receive from the sun. The temperatures calculated in this way are—

Average
Absolute Temperature
Mercury . . . . . . . . . 467°
Venus . . . . . . . . . . 342°
Earth . . . . . . . . . . 290°
Mars . . . . . . . . . . 235°