green leaves of plants. But the same -solar radiation, when
broken up into diffused sky light, which has no definite direction,
has fallen into equilibrium with a much lower temperature,
through loss of its reversibility. It has been remarked that the
temperatures of the planets can be roughly compared by means
of this principle, if their coefficients of absorption of the solar
radiation are assumed; that of Neptune comes out below
−200° C., if we suppose that it is not kept higher by a supply
of internal heat.
To obtain dynamical precision in this discussion an exact definition of the narrow beam such as is usually called a ray is essential. It can be specified as a, narrow filament of radiation, such as may be isolated within an infinitely thin, impermeable, bounding tube without thereby producing any disturbance of the motion. If either the tube or the surrounding radiation were not present to keep the beam in shape, it would spread sideways, as in optical diffraction. But the function of the tube is one of pure constraint; thus the change of energy content of a given length of the tube is represented by energy flowing into it at the end where the radiation enters, and leaving it at the other end, but with no leakage at the sides. The total radiation may be considered as made up of such filaments.
9. Temperature of the Sun.—The mean temperature of the radiating layers of the Sun may be estimated from Stefan's law, by computing the intensity of the radiation at his surface from that terrestrially observed, on the basis of the law of inverse squares; the result is about 6500° C. The application of Wien's law, which makes the wave-length of maximum energy vary inversely as the temperature, for the case of a perfectly radiating source, gives a. result 5 500° C. These numbers will naturally differ because (i) the Sun is not a perfect radiator, the constitution of his radiation in fact not following the law of that of a black body, (ii) the various radiating layers have different temperatures, (iii) the radiation may be in part due to chemical and electrical causes, and in so far would not be determined by the temperature alone. The fair agreement of these two estimates indicates, however, that the radiation is largely regulated by the temperature, that the layers from which the main part of it comes are at temperatures not very different, and that not very much of the complete radiation established in these layers and emitted from them is absorbed by the overlying layers.
10. Fluorescence.—When radiation of certain wave-lengths falls on a fluorescent body, it is largely absorbed, but in such manner as directly to excite other radiation of different type which is emitted in addition to the true temperature-radiation of the body. The distinction involved is that the latter radiation is spontaneously convertible with the heat of the absorbing body at its own temperature, without any external stimulus or compensation; it is, in fact, on the basis of this convertibility that the thermodynamic relations of the temperature-radiation have been established. According to the experimental law of Stokes, the wave-lengths of the fluorescent radiation are longer than those of the radiation which excites it. If the latter were directly transformed, in undiminished amount, into the fluorescent kind, this is what would be expected. For such a spontaneous change must involve loss of availability; and, beyond the wave-length of maximum energy in the spectrum, the temperature of a given density of radiation is greater the shorter its wave-length, as it is a function of that density and the wave-length alone such that greater radiation always corresponds to higher temperature. *But it would appear that the opposite should be the case for radiation of long wavelengths, lying on the other side of the maximum, in which the tendency would thus be for spontaneous change into shorter waves; this may perhaps be related to the fact that the lines of longer wave-lengths in spectra often come out brighter at lower temperatures, for they are then thrown on the other side of the maximum and cannot be thus degraded. The principle does not, however, have free play in the present case, even when the incident radiation is diffused and so has not the abnormally high temperature associated with a directed beam (§ 8), since part of it might be degraded into low-temperature heat, or there might be other compensation of chemical type for any abnormally high availability that might exist in the fluorescent radiation. It has been found that fluorescent radiation, showing a continuous or banded spectrum, can be excited in many gases and vapours; milky phosphorescence of considerable duration, and thus doubtless associated with chemical change, is produced in vacuum tubes, containing oxygen or other complexly constituted gases, by the electric discharge.
11. Entropy of a Ray.—If each definitely constituted beam of radiation has its own temperature and everything is reversible as above, a question arises as to the location of the process of averaging which enters into the idea of temperature. The answer can depend only on the fact, that although the beam is definite as to wave-length and intensity, yet it is far from being a simple wave-train, in that it is constituted of trains of limited lengths and various phases and polarizations, coming from the independent radiating molecules. When such a beam has once emerged, it travels without change, and can be reflected back intact to its source, and is in so far reversible; but when it has arrived there, the molecules of the source will have changed their positions, and it cannot be Wholly reabsorbed in the same manner as it was emitted. There must thus be some feature in the ultimate averaged constitution of the beam, emitted from a body in the definite steady state of internal motion determined by its temperature, which adapts it for spontaneous uncompensated re absorption into a body at its own (or a lower) temperature, but not at a higher one.
The question of the determination of the form of the function φ in § 6 would thus' appear to be closely connected with the other problems, hitherto imperfectly fathomed, relating to the statistics of kinetic molecular theory. A very interesting attack on the problem from this point of view has recently been made in various forms by Planck. It of course suffices to examine some simple type of radiating system, and the results will be of general validity. He considers an enclosure filled with radiation involving an entirely arbitrary succession of phases and polarizations along each ray, and also containing a system of fixed linear electric oscillators of the Hertzian type, which are taken' to represent the transforming action of radiating and absorbing matter. The radiation contained in the enclosure will be passed through these oscillators over and over again, now absorbed, now radiated, and each constituent will thus settle down in a unilateral or irreversible manner towards some definite intensity and composition. But it does not appear that a system of vibrators of this kind, each with its own period, can perform one of the main functions of a material absorber, namely, the transformation of the relative intensities of the various types of radiation in the enclosure to those corresponding to a common temperature. There would be equilibrium established only between the mean internal vibratory energy in the vibrators of each period and the density of radiation of that period; there is needed also some means of interchanging energy between vibrators of different periods, which probably involves doing away with their fixity, or else employing more complex vibrators and assuming a law of distribution of their internal energy. In the absence of any method of introducing this temperature equilibrium directly, Planck originally sought, in the case of each, independent constituent, for a function of its intensity of energy and its wave-length, restricted as to form by a certain assumed molecular relation, which has the property of continually increasing after the manner of entropy, during the progress of that constituent of the radiation in such a system towards its steady state. If the actual entropy S per unit volume could be thus determined, the relation of Clausius δS=δE/T would supply the connexion between the temperature and the density of radiant energy E. This procedure led him, in an indirect and tentative manner, to a relation d2S/dE2=−α/E, so that S=−αE logβ E, where α, B are functions of M an expression which conducts through Clausius's relation to E= (eβ)−1e−1/αT.
