542 



SCIElSrCE, 



[Vol. VI., No. 150. 



in comparison with which tbe heat coming from the 

 stars, and that radiated and reflected by the earth, 

 may be neglected without any sensible error. But 

 by the generally recognized principle that the rela- 

 tive radiating and absorbing powers of bodies are 

 equal, the ratio between radiation and absorption is 

 the same for all bodies at a given temperature; so that 

 it is not necessary to consider the radiating power of 

 the moon, but to simply satisfy the condition that the 

 moon, with a surface of maximum radiating power, 

 such as a lampblack surface, shall radiate heat as 

 fast as it is received from the sun. 



All bodies are so constituted that their absolute 

 radiating power is a function of the temperature, the 

 former increasing with the latter, but by no means 

 in proportion. If, therefore, we know the relation 

 between the temperature of a body and its rate of 

 radiating heat, and also know the rate with which it 

 is receiving heat from its surroundings, we can, by 

 means of the preceding condition, form an equation 

 of condition which determines the temperature. 



According to Pouillet's determination from the ex- 

 periments of Dulong and Petit, a square centimetre of 

 surface of maximum radiating power, and at the tem- 

 perature of 0°C., radiates 1.146 calories of heat per 

 minute ; and hence, by the law of Dulong and Petit, 

 the rate of radiating heat for any other temperature 



e, is 1.146u^, in which 



1.0077. The rate with 



which a square centimetre of surface normal to the 

 direction oi the sun's rays receives heat from the 

 sun is what is called the solar constant, usually de- 

 noted by A. Putting, therefore, s for the area of the 

 moon's surface in square centimetres, and a for that 

 of a great circle, the rate with which heat is radiated 

 from the moon's surface is expressed by 1.146 ^^s, 

 and the rate with which it is received from the sun, 

 by yla. Hence, by the conditions above, since s = 4a, 

 we get in the case of the moon in space, in which it 

 loses heat by radiation only, and receives it from the 

 sun only, the equation 



^ 4.584 

 for determining e where A is known. Since log /a is 

 exactly equal to 1-300, this may be put into the fol- 

 lowing convenient and practical form : — 



A 



=300 log = 30n (log A - 0.G612). 



4.584 



From this equation, deduced as a simple case from 

 a more general and mathematical treatment of the 

 subject in the * Temperature of the atmosphere 

 and earth's surface,' ' the v,'riter, with the assumed 

 value of ^ = 2.2, deduced tbe value of e = — 96° C. 

 But as there is some uncertainty with regard to the 

 value of this constant, since some of the solar rays 

 may be entirely absorbed before reaching the earth's 

 surface, and it is thought by some to be considerably 

 greater than this, we shall put it here equal to 2.5. 

 With this value we get e = — 79°. This must be un- 

 derstood to be the mean surface temperature of the 

 moon, or, more accurately, the temperature of a sur- 

 face uniformly heated which would radiate as much 

 heat as the surface of the moon, which, of course, 

 has very different temperatures on opposite sides at 

 any given time. 



The law of Dulong and Petit being an empirical one, 

 which satisfied the experiments from 0° to 300° only, 

 there is some uncertainty in extending it down to 

 —79"; but this is very small in comparison with what 



1 Professional papers of the signal service, No. xiii. 



it is in extending it in the other direction, up to the 

 temperature of the sun, as has been done by Pouillet 

 and others, in forming an equation for determining 

 its temperature. The uncertainty in the true value 

 of A, together with that in the extension of the law 

 down to so low a temperature, causes some uncer- 

 tainty in the mean temperature of the moon as thus 

 determined ; but this is not very great in a matter of 

 this sort, for it amounts to only 17° in an uncertainty 

 of one-eighth part in the value of A. 



But when we attempt to determine the tempera- 

 ture of the side of the full moon exposed to the sun 

 and earth, the uncertainty becomes very much 

 greater. In this case the heat is not only radiated 

 from the surface, but it is also conducted inward 

 from the surface heated far above the mean tempera- 

 ture of the moon, and stored away for the time. The 

 rate with which it is conducted in depends upon the 

 conductivity and capacity of the lunar soil for heat, 

 which are unknown to us ; and the problem would be 

 extremely complex if they were known. The tem- 

 perature of the moon's surface, in this case, can only 

 be determined for the two extreme hypotheses of in- 

 finitely great and infinitely small conductivities for 

 heat. Upon the first hypothesis, the heat received 

 and absorbed by the moon would be instantly dis- 

 tributed through the whole mass, and radiated 

 equally by all parts of the moon's surface, and the 

 temperature of the part exposed to the sun's rays 

 would be the mean temperature of the moon as ob- 

 tained above. Upon the other hypothesis, it would 

 not be conducted away at all from the surface receiv- 

 ing it, but, in case of a static temperature, it would 

 all have to be radiated away by the same surface re- 

 ceiving it. Hence, in this case, instead of the radi- 

 ating surface being four times as great as the surface, 

 or normal sectional area receiving it, it is only equal 

 to it for the part of the moon's surface upon which 

 the sun's rays fall perpendicularly, and we must 

 therefore have l.!l46|u.^ =■ A, or 



A 



= 300 log = 30O Gog ^ - 0.0592), 



1.146 



instead of the preceding similar expression. 



With the assumed value of A = 2.5, this gives 

 0= 101° for the temperature of the central part of 

 the moon's disk as viewed from the sun, and from 

 the earth at full moon. For other parts, the value 

 of A in the preceding expression must be multiplied 

 into the cosine of the angle of incidence of the sun's 

 rays upon the moon's surface, and thus this expres- 

 sion will give the temperature down as low as it is 

 safe to extend Dulong and Petit's law. The same re- 

 sults would be obtained sensibly with any ordinary 

 conduotivity for heat if the same side of the moon 

 were permanently exposed to the sun, for the tem- 

 perature gradient by which the heat would be con- 

 ducted inward would soon become so small, in this 

 case, that the rate by which heat would be conducted 

 inward would be insensible, as in the case in which 

 heat is conducted outward from the interior of the 

 earth. 



The result above, of 101°, which is a little above 

 the temperature of boiling water, must be regarded 

 simply as a limit beyond which, in a large range of 

 uncertainty , the temperature cannot go. The other 

 limit is — 79°. If we suppose the temperature of the 

 warmest part of the moon's disk to fall halfway be- 

 tween these extremes, it would be a very little above 

 a freezing temperature. William Ferrel. 



