524 



NATURE 



[August i, 1907 



exactly with a yearly motion of the Sun through space 

 equal to four times the distance of the Sun from the Earth. 



Thus the Sun's yearly motion being four times the Sun's 

 distance, the parallactic motion of stars in which this 

 motion is unforeshortened must be four times their parallax. 

 How this number varies with the amount of foreshortening 

 is of course readily calculated. The point is that from the 

 mean parallactic motion of a group of stars we are now 

 enabled to derive at once its mean parallax. 



This research has been carried out by Kapteyn for stars 

 of different magnitudes. It leads to the result that the 

 parallax of stars differing ]ivc magnitudes does not differ 

 in the proportion of one to ten, as would follow from the 

 supposition of equal luminosity of stars throughout the 

 universe, but only in the proportion of about one to five." 



The same method cannot be applied to groups of stars 

 of different proper inotions, and it is only by a somewhat 

 indirect proof, and by calling in the aid of such trust- 

 worthy results of direct parallax determination as we 

 possess, that the variation of parallax with proper motion 

 could be satisfactorily dealt with. 



The Mean Parallaxes of Stars of Different Magnitude 

 and Proper Motion. 



.As a final result Kapteyn derived an empirical formula 

 giving the average parallax for stars of different spectral 

 types, and of any given magnitude and proper motion. 

 This formula was published at Groningen in 1901." Within 

 the past few months the results of researches on stellar 

 parallax, made under the direction of Dr. Elkin, at the 

 Astronomical Observatory of Yale University, during the 

 past thirteen years,' have been published, and they afford 

 a most crucial and entirely independent check on the 

 soundness of Kapteyn 's conclusions. 



In considering the comparison between the more or less 

 theoretical results of Kapteyn and the practical determin- 

 ations of Yale, we have to remember that Kapteyn 's tables 

 refer only to the means of groups of a large number of 

 stars having on the average a specified magnitude and 

 proper motion, whilst the latter are direct determinations 

 affected by the accidental errors of the separate determin- 

 ations and by such uncertainty as attaches to the unknown 

 parallaxes of the comparison stars — parallaxes which we 

 have supplied from Kaptcyn's general tables. 



The Yale results consist of the determination of the 

 parallax of 17;, stars, of which only ten had been previously 

 known to Kapteyn and had been utilised bv him. Dividing 

 these results into groups we get the following com- 

 parison ; — 



Comparison Groups arranged in order of Proper Motion. 



Groups arranged in order of Magnitude. 



Yale ' Kafteyn 



Vale— 

 Kapteyn 



5-6 



6-7 

 -■6 

 S'3 



.' i?^""""- •^''^< A'-A V(/r«, No. 34S7, table iii. ; and .4st. Joiirn., p. 566. 

 - Publications A^lron. Laboratory, Uroningen, No. 8, p. 24. 

 3 Trans. Asiron. Observatory of Yale Univ., vol. ii., part i. 



us. rtsiron. ".^Dservatory ol Vale U 



NO. 1 070, VOI. 76] 



These results agree in a surprisingly satisfactory w-a> 

 having regard to the comparatively small number of star- 

 in each group and the great range of parallax which w 

 know to exist amongst individual stars having the sani' 

 magnitude and proper motion. In the iiiean perhaps th' 

 tabular parallaxes are in a minute degree too large, but 

 we have unquestionable proof from this comparison thai 

 our knowledge of stellar distances now rests on a solid 

 foundation. 



the Dislribtdion of Varieties of Luminosity of Stars. 



But, besides the mean parallax of stars of a particul.n 

 magnitude and proper motion, it is essential that \\ 

 should know approximately what percentage of the star^ 

 of such a group have twice, three times, &c., the mean 

 parallax of the group, and what percentage only one- 

 half, one-third of that parallax, and so on. In principle, 

 at least, this frequency-law may be obtained by ineans 

 of the directly determined parallaxes. For the stars of 

 which we have trustworthy determinations we can com- 

 pare these true parallaxes with the mean parallax of stars 

 having corresponding magnitude and proper motion, and 

 this comparison will lead to a knowledge of the frequency- 

 law required. It is true that, owing to the scarcity of 

 material at present available, the determination of the 

 frequency-law is not so strong as may be desirable, but 

 further improvement is simply a question of time and thi- 

 augmentation of parallax-determination. 



Adopting provisionally the frequency-law found in thi> 

 way by Kapteyn,' wo can localise all the stars in space 

 down to about the ninth magnitude. 



Take, for example, the stars of magnitude 5-5 to 6-5. 

 There are about 4800 of these stars in the whole sky. 

 According to .Auwers-Bradley, about q.j per cent, of these 

 stars, or some 460 in all, have proper motions betw-een 

 o"o4 and o"o5. Now-, according to Kapteyn's empiric 

 formula, the satisfactory agreement of which with the Yale 

 results has just been shown, the mean parallax of such 

 stars is almost exactly o"oi. Further, according to his 

 frequency-law-, 29 per cent, of the stars have parallaxes 

 between the mean value and double the mean value ; 

 6 per cent, have parallaxes between twice and three 

 times the mean value; iJ per cent, between three and 

 four times the mean value. Therefore of our 460 stars 

 133 will have parallaxes between o"-oi and o"-02, twenty- 

 eight between o"-02 and o"-03, seven between o"-03 and 

 o"o4, and so on. 



Localising in the same way the stars of the sixth 

 magnitude having other proper motions, and then treat- 

 ing the stars of the first magnitude, second magnitude, 

 third magnitude, and so on to the ninth magnitude 

 in the same way, we finally locate all these stars in 

 space. ^ • 



It is true we have not localised the individual stars, 

 but we know approximately and within certain limits of 

 magnitude the number of stars at each distance from 

 the Sun. 



Thus the apparent brightness and the distance being 

 know-n we have the means of determining the light-energy 

 or absolule himinositv of the stars, provided it can be 

 assumed that liglil does not suffer any extinction in its 

 passage through interstellar space. 



On this assumption Kapteyn w-as led to the following 

 results, viz., that within a sphere the radius of which 

 is 560 light-years (a distance which corresponds with 

 that of the average star of the ninth magnitude) there 

 will be found : — 



