SCIENCE- GOSSIP. 



35' 



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OBSERVATIONS ON VARIABLE STARS. 



To the Editor O^SCIENCE-GOSSTI'. 



Sir, — It lias occurred to me that the fullest advan- 

 tage has not been taken of all the data which 

 observation of variable stars has given to us. I pro- 

 pose to show, taking the well-known variable Algol 

 as an example, that several more interesting facts 

 may be determined. The following is a summary of 

 what we already know about this star. Algol and 

 his satellite revolve in circular or approximately 

 circular orbits, for the variations of speed indicated 

 by the spectroscope are precisely those which would 

 be shown by a body revolving at a uniform rale ; 

 these orbits are of course in the same plane. The 

 line of sight from the earth to Algol also lies in this 

 plane, for Professor Pickering has found that the 

 method of diminution of light during eclipse is 

 exactly such as would be caused by a spherical body 

 passing in front of Algol. The eclipse is therefore 

 annular, not partial, and the conclusion just given 

 follows at once. The complete period of revolution 

 of either Algol or his satellite is very approximately 

 4,089 minutes, that of greatest eclipse is 20 m. , and 

 the total period of eclipse is 560 m. Algol varies 

 from the second magnitude at maximum to the fourth 

 at minimum ; i.e. at the time of greatest eclipse we 



I 

 receive from it , „ of the light we receive nor- 



mally. Now it follows from the "law of inverse 

 squares" that when an opaque sphere passes in front 

 of a large luminous one, that s, being surface of 

 luminous sphere, s 2 that of opaque one, I), distance 

 of former, D„ that of latter, l, light of former un- 

 eclipsed, u when at greatest eclipse, then 



s, (I.,- 1..) . l>_.- 



.". if A and 8 represent the diameters of Algol and 

 his satellite respectively 



,/L, 



D, I.. - I.. ' D„' 



But the difference between the distances of Algol ar.d 

 his satellite is so minute, compared with that of either 

 from the earth, that the ratio of D, to D., may, with- 

 out any appreciable error, be taken equal to unity. 



a _ ^i,,(i 



■ u). 



Taking I., : [.._, = (2-512)-, this gives that, corrected 

 to six places of decimals, the diameter of Algol is 

 1-090100 times that of his satellite. There is, how- 

 ever, another entirely different method of finding 

 this ratio which I will now explain. A little con- 

 sideration will 1 show that the distance across the 

 " line of sight " from the Earth to Algol which he 



and his satellite traverse during the total period of 

 eclipse is A + B, and similarly that passed over during 

 the 20 m. of greatest eclipse is A - B ; also, since 

 they are moving uniformly in similar orbits, the 

 distances which either separately passes over in these 

 two periods are in the ratio of A + B to A — B. The 

 distance across the line of sight which either of 

 them traverses during one of these periods is the 

 chord of its orbit on which stands the arc described 

 during the time. For, taking the mean line of sight 

 to be that from the earth to Algol at exactly the 

 middle of the eclipse, the arc described by either the 

 latter or his companion during eclipse is symmetrical 

 with regard to this line, and therefore the chord on 

 which it stands is perpendicular to it. Let the chord 

 described by either Algol or his satellite (it makes no 

 difference which we take) during the whole eclipse be 

 <.',, during time of maximum eclipse c... Let the 

 angles subtended at the centre of motion by these 

 chords be 0, and 6., respectively. 



Then c, : C._, :: sin 6, : sin 8.,. But 6, in degrees is 



360° xarc described during eclipse 

 circumference of orbit 



_36o 1 x period of eclipse _^ 360° y 560 _ o..r 



period of revolution 



4089 



.,. .. , „ 360° x 20 „ , , 



Similarly ft, = J — = i°45'65. 



40S9 



. A + 11 _ C, _ sin 0, _ sin 49- 18-9' _ 75S3 

 " 'a — B c\ sin flj, sin i° 45-65' 308 



.•,-= 1-084674 corrected to si\ places of decimals. 



B 



This result agrees very well with thai arrived at 

 by the previous perfectly independent process, [1 

 will be seen that the whole difference is only -005426. 

 which, if Algol's diameter was 1,000,000 miles, or 

 considerably greater than that of our sun, would only 

 make the comparatively very small difference of 

 4,6Si miles in estimating that of his satellite. Il 

 the line of sight was slightly tilted out of the plane 

 of Algol's orbit, not sufficient to make the eclipse 

 partial, but causing it to be unsynmicliically annular 

 as seen from the earth, the distances across the line 

 of sight passed over during the whole eclipse and 

 maximum eclipse would then lie 'ess than A + Il and 

 A - B ; consequently the result arrived at by the second 

 method would be rather too large. Since, however. 

 the latter is actually slightly the smaller of the two. 

 what difference there is cannot be accounted for in 

 this manner, and hence the orbit is not lilted. Thus 

 we reach the rather important conclusion that Algol's 

 orbital velocity, as shown by the spectroscope, ol 

 26-5 miles per second, is his true velocity ; for if the 

 orbit was tilted the latter would have :'. component 

 perpendicular to the line of sight which would not 

 affect the spectroscope. I think the small dis- 

 crepancy can be best accounted for by the very 

 likely supposition that the true variation during 

 eclipse is not really precisely two theoretical mag- 

 nitudes ; or, again, the orbits may not be exact/) 1 

 circular. 



If the fluctuations in light of a large number of 

 variable stars were compared with their periods of 

 eclipse in the above manner, some interesting results 

 might be obtained. When the two results were 

 approximately. equal, we should know that the star in 

 question revolved in a circular orbit in a common 

 plane with the line of sight. When they were 

 gr.eatlv different, it would show that either the orbit 



