June 1, 1895.] 



KNOWLEDGE. 



131 



one of the larger planets when in "opposition," we can 

 determine by the photometer its exact stellar magnitude. 

 We can also compute the apparent diameter of the planet 

 as seen from the sun, and thus ascertain the fraction 

 representing the area of its disc compared with the area of 

 the hemisphere illuminated by the sun. If the surface of 

 the planet were a perfect reflector of light, we could in 

 this way — knowing the distance of the sun and planet — 

 compute the brightness of the sun in terms of the apparent 

 brightness of the planet. But as no surface is a perfect 

 reflector, a correction must be made for the " albedo," or 

 reflecting power, of the planet in question. Let us see 

 what result this will give in the case of Mars, Jupiter, and 

 Saturn. 



The mean of a number of determinations of the diameter 

 of Mars, as seen at the earth's mean distance from the 

 sun, gives about 9-4". Eeducing this to the mean distance 

 of Mars from the sun, I find that the mean diameter of 

 Mars in opposition, as seen from the sun, is 6-17 '. This 

 gives the area of its disc 29-9 square seconds. Now, in 

 a hemisphere of the star sphere there are 20,G2G'5 x 

 12,900,000 = 267,319,440,000 square seconds, and hence 

 area of hemisphere = 8,940,450,000 times the area of the 

 disc of Mars as seen from the sun. Hence, if the surface 

 of Mars were a perfect reflector, the sun as seen from 

 Mars would be 8,940,450,000 times brighter than Mars 

 appears to us when in opposition. But Mars is not a 

 perfect reflector. Its albedo, or reflecting power, is, 

 according to Zollner, only 0-2672 (that of a perfect reflector 

 being 1) ; hence we must divide the above number by 

 0'2672, which gives 33,459,768,000 for the ratio of the 

 light of the sun to the reflected light of Mars. Now, as 

 the mean distance of Mars from the sun is 1-5237 (that of 

 the earth being 1), we must multiply the above result by 

 the square of 0-5237, or 0-2742, to obtain the light of the 

 sun as seen from the earth. This gives the light of the 

 sun as seen from the earth 9,174,668,385 times the light 

 of Mars when in opposition, a number which corresponds 

 to 24-9 stellar magnitudes. Now, Prof. Pickering finds 

 the stellar magnitude of Mars at mean opposition to be 

 — 2-25, that is 2J magnitudes brighter than a star of the 

 zero magnitude, or about Ij- magnitudes brighter than 

 Sirius ; hence we have the sun's stellar magnitude, as 

 deducedfrom the apparent brightness of Mars -(24-9 + 2-25) 

 = — 27-15, a considerably higher mean than that usually 

 adopted. 



Let us now see what value can be derived from the planet 

 Jupiter. The mean diameter of Jupiter as seen from the 

 sun (allowing for the ellipticity or compression of its disc) 

 may be taken — as the result of five determinations by 

 difterent astronomers— at 30-4". This gives an area of 

 disc = 1040-6 square seconds, and a ratio of the area of 

 the hemisphere to that of Jupiter's disc = 256,889,717. 

 Dividing this number by Jupiter's albedo, as found by 

 Zollner, viz. 0-6238, we obtain 411,814,230 for the ratio 

 of the light of the sun to the reflected light of Jupiter. 

 Now, as the mean distance of Jupiter from the sun is 

 5-2028 (that of the earth being 1), its distance from the 

 earth, when in opposition, is 4-2028, and we must multiply 

 the above number by the square of this, or 17-603, which 

 gives 7,273,874,744, a number which corresponds to 24-65 

 stellar magnitudes. Prof. Pickering finds the stellar magni- 

 tude of Jupiter in opposition to be —2-52, and addmg this 

 to 21-65 we obtain - 27-17 for the sun's stellar magnitude, 

 a result in close agreement with that found from Mars. 



In the case of Saturn, we may take its mean apparent 

 diameter, as seen from the sun, at 16-8" (allowing for the 

 ellipticity of its disc). This gives an area of 221-67 square 

 seconds, and a ratio of the area of the hemisphere to that 



of Saturn's disc of 1,205,934,244. Dividing this number 

 by Saturn's albedo, 0-4981, as found by Zollner, we 

 obtain 2,421,068,548 for the ratio of the light of the 

 sun to the reflected light of Saturn. Now, as the mean 

 distance of Saturn from the sun is 9-5388, its distance 

 from the earth when in opposition is 8-5388, and we must 

 multiply by the square of this, or 72-9111, which gives 

 170,522,771,010, a number which corresponds to 28-11 

 stellar magnitudes. Now it has been found by photo- 

 metric measures that Saturn, when in opposition, and the 

 rings invisible, is about equal in brightness to Aldebaran, 

 which may be considered as a standard star of the first 

 magnitude. Hence we have the sun's stellar magni- 

 tude = -27-11, a result in close agreement with those 

 found from Mars and Jupiter. 



From the above calculations we see that, on the lowest 

 estimate, the sun's stellar magnitude is at least —27. If 

 this result be correct, the results arrived at in my paper 

 on " The Distance and Mass of the Binary Stars " will 

 require considerable modification. The magnitudes there 

 found for the sun, supposed placed at the distance of the binary 

 star, would have to be diminished by 1-5 magnitude. Thus, 

 in the case of 7 Leonis, the sun at the distance indicated 

 by the hypothetical parallax would shine as a star of 

 8-29 — 1-15 or 6-79 magnitude, denoting that y Leonis is 

 only sixty-six times brighter than the sun. This would 

 diminish the parallax of 0-58" to 0-29", and would increase 

 the mass of the system eight times, or to about ■^^'^\i. of 

 the sun's mass. Similar corrections would have to be 

 made in the case of the other binaries discussed in my 

 paper, the general result being to increase the mass of each 

 system eight times. For J Scorpii we should have a mass 

 of about ^,th of the sun's mass ; for i Leonis the mass 

 would be about \ ; for 35 Comfe about \, and for 

 T Cygni about i, results which are perhaps more probable 

 than those found in my previous paper. In the case of 

 the other binary stars, however, referred to in that paper, 

 the masses would still be small. For it Cephei we have 

 a mass of about ^^ ; for w Leonis about .7'^ ; for 

 j8 Delphini ^V ; for t Ophiuchi yV, and for J Aquarii 

 J^. An opposite effect would, of course, be produced in 

 the remarkable case of j)S Herculis. Here the sun would 

 shine as a star of 4-5 magnitude, if placed at the distance 

 indicated by the hypothetical parallax (0-107"), denoting 

 that the sun is about one hundred times brighter than the 

 star for equal masses. To reduce the sun to a 9-5 magni- 

 tude star, the parallax should be reduced to 001", and 

 this would increase the mass of the system to one 

 thousand two hundred and forty times the mass of the sun. 



We may compare the relative brightness of the sun 

 with that of binary stars having spectra of the solar type. 

 Suppose an imaginary planet to revolve round the sun 

 at x^o^li of the earth's distance from the sun ; the period 

 of revolution would be yo'^^th of a year, and the semi- 

 axis major of its orbit, as seen from the earth, would be 

 ?JL6_2_6_5_ 2062-65". Substituting these values of P and a 

 in Mr. Monck's formula (given in my catalogue of binary 

 star orbits), and taking the sun's stellar magnitude at -27, 

 I find that the relative brightness of the sun, compared 

 with that of £ Ursfe Majoris (which has a spectra of the 

 solar type), taken as unity, is 1-2748. Conversely, if we 

 assume the brightness of the sun to be equal to that of 

 f UrsK, I find that the sun's stellar magnitude would 

 be -26-73. 



Assuming the sun's stellar magnitude at — 27, I find 

 that the sun would be reduced to about the same brightness 

 as a Centauri if placed at the distance indicated by a 

 parallax of 0-76" found for that star. The spectrum of a, 

 Centauri is of the solar type. 



