July 6, 1883.] 



♦ KNOWLEDGE ♦ 



LAWS OF BRIGHTNESS. 



y. 

 By Richard A. Proctor. 



THE law obtained at page 372 relates to smooth spheres, 

 or spheres having a mat surface (German, matt, dfiiul, 

 as applied to surfaces), unpolished. It is useful to know 

 the law, as it affords a means of determining how far the 

 observed changes of brightness of those planets which pre- 

 sent phases accord with the changes which would result if 

 the surfaces of the planets were smooth. 



To begin with the moon : — 



Now here, before proceeding to consider the moon's 

 total brightness at her phases, we find ourselves confronted 

 by a circumstance which in no sen.se accords with what 

 was shown in the preceding part of this paper, as to the 

 brightness of a smooth sphere illuminated directly. We 

 see that the disc of the full moon does not present that 

 shading off" towards the edge or limb which theory re- 

 quires in the case of a smooth sphere. Setting aside the 

 remarkable variety of brightness seen in different parts 

 of the moon's disc, we find no perceptible tendency to 

 diminution of brightness towards the moon's limb. ]f 

 we look at the moon through a tinted glass suitably 

 graduated, we find that the outline remains distinctly 

 visible as long as any part of the moon (except, of course, 

 the exceptionally bright spots) can be seen. 



It is obvious that we must look for the explanation of 

 this circumstance in the unevenness of the moon's surface. 

 We can easily see, for instance, that if the moon's surface 

 were covered all over with steep conical hills, something 

 like the observed result would follow. For let ", b, c, 

 Fig. II, be steep conical hills on a globe, supposed to be 

 illuminated from above ; the light rays meet the surface at 



Fig. 11. 



a small angle, and the illumination is therefore small ; 

 hence, supposing these hills in the centre of the disc (that 

 is, the globe " full "), they would appear as three dark 

 spots, as ii', V, c! ; and if the whole surface of the globe 

 were covered with such hills the whole of the middle part 

 of the disc would be dark, comparing its light with that 

 from the middle of a smooth sphere of the same substance 

 and under equal illumination. Such hills on the edge of 

 the disc, as at d, c, /"(and supposed to be illuminated in the 

 same direction that the globe is viewed) would not appear 

 uniformly bright, for at their edges they would be under 

 very oblicjuc illumination, while along a central streak the 

 illumination would be nearly square (the hills being steep) ; 

 but it is easily seen that the total illumination of a hill- 

 covered region so placed might be (according to the slope of 

 the hills) rather greater than, or very little less than, the 

 illumination of the spots, «', h\ c . 



It appears to me, however, that it is on the whole more 

 satisfactory ta regard the general unevenness of the moon's 

 surface as due rather to crateriform elevations than to 

 conical hills. (It must be noticed that we are not con- 

 cerned here with the features revealed by the telescope ; 

 since the nature of the illumination of difl'erent parts of 

 the full moon must be mainly due to irregularities not 

 discernible by the telescope.) Now, if we take a crater 

 having such a section as is shown in Fig. 12, we see that 

 under vertical illumination we should have the base C D 

 fully illuminated, and the slopes B A, E V somewhat 



Fig. 12. 



obliquely illuminated ; while under the oblique illumina- 

 tion from the right, indicated by the transverse lines, we 

 have B C and E F (the only parts under illumination), 

 Ijoth illuminated nearly squarely. Of course, the latter 

 remarks only relate to one cross-section of the crater ; but 

 it is easily seen that the oblique illumination may give the 

 same apparent illumination for the crater (regarded as a 

 whole, and viewed in the same direction that the light 

 falls) that direct illumination gives. If C D be of less 

 reflective power than ABC and D E F, of course the 

 oblique illumination would have a further advantage, the 

 direction of the line of light being such that the dark 

 bottom of the crater would be concealed behind the wall 

 D E F. 



I forbear from entering further, however, into such 

 considerations as these, simply because the varieties and 

 combinations of slope, position, tint, itc, are endless. It 

 suffices here that the brightness of the moon's disc near the 

 edge is explicable in a general way by the circumstance 

 that the moon's surface is uneven. 



It will be obvious that the total light received from the 

 moon must necessarily be affected by a condition of her 

 surface which renders the seeming illumination of difl'erent 

 parts of her disc so different from that which would result 

 were she smooth.* It is not easy to determine to what 



* Some very strange assertions are made in a section of the 

 " Encyc. Brit." devoted to the question of the moon's brightness. 

 They illu.strate what I liave said respecting the errors cnmmonly 

 made on the subject of brightness, and I therefore quote the passage 

 at leiigtli. The writer of the passage lias given certain arguments 

 (based on Bouguer's and Leslie's inexact estimates of the moon's 

 total brightness) which appear to show that the moon is self-Iurai- 

 nons. He proceeds as follows; — "Although these arguments go 

 far to support the ancient opinion of the native light of the moon, 

 they are not entirely conclusive, and indeed cannot bo easily recon- 

 ciled with some of the phenomena. If the nioou shines in virtue 

 of her native light, rays will be emitted in all directions from every 

 point of lier surface; whence, since a visual angle of a given mag- 

 nitude includes a much larger portion of a spherical surface near 

 the extremities of its apparent disc than towards iho centre, and as 

 the number of rays is proportional to the suiface from which they 

 ])rocecd, it follows that the intensity of the moon's light ought 

 to be greater near the border than at the centre of her disc." 

 (The writer was clearly unacquainted with the experimental 

 law of the emission of light from self-luminous bodies). 

 " Tlie reason why this is not the case with regard to the 

 sun" (ho should have said, the reason why a variation of the 

 opposite kind exists) " is that a greater proportion of the rays are 

 ai)sorbed in passing through a greater extent of the solar atmo- 

 sphere; but the moon, having no atmosphere, ought to be sensibly 

 most brilliant near the circumference of her orb. The contrary is, 

 however, the case ; her light is greatest at the centre and less 

 intense towards the circumference (I), exactly as it rught to be on 



