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O-C. Atic Correction,Resulting from the Effect of Causes Not Allowed for In VOL. 16, 1930 STA TISTICS: W. J. L UYTEN 257 ON SOME STATISTICAL PROPERTIES OF DOUBLE STARS IN SPACE. H1. ON THE MEAN PERIOD OF DOUBLE STARS IN SPA CE By WILLBM J. LUYTSN HARVARD COLLAGS OBS1RVATORY, CAMBRIDGE, MASSACHUSETTS Communicated February 15, 1930 In the previous paper a formula was derived for the computation of the period of a binary star when only the parallax, and the angular separation at a given moment are known. Before actually using such a formula, it is always well to test it on a group of stars for which the period, to be estimated from the formula, is already known with reasonable accuracy. For the data required in this computation, the writer is under obligation to Dr. W. H. Van den Bos who kindly supplied the results of the latest known measures of these double stars-in many cases his own unpublished measures. Table 1 contains the data needed for the actual comparison for 15 binary stars of known parallax and period, arranged in order of increasing period, and designated by the name under which these stars are known to double-star observers. The second, third, and fourth columns con- tain the angular separation, according to the latest known measure, the adopted parallax, and the mass of the system. The fifth column gives the logarithm of the period, calculated by means of the formula derived in the previous paper, which formula is again repeated at the foot of table 1. The sixth and seventh columns show the period derived by double star calculators, and its logarithm, while the eighth column gives the differences O-C. According to formula (1) the standard deviation to be expected in any individual logarithm of period is 0.34 (as a minimum), hence that in the mean of 15 periods should be 0.09. From table 1 we notice that the mean value of log P as calculated from the formula is. 1.83 i 0.09, while that of the observed periods is 1.89. The difference O-C = +0.06, only two-thirds of the standard deviation, is small enough to inspire confidence in formula (1). It may well be explained by accidental errors; yet the possibility should not be excluded that formula (1) requires a small system- atic correction, resulting from the effect of causes not allowed for in its derivation, such as observational selection and discovery chance. The influence of the latter, however, would be expected to act in the opposite sense. From the internal discordances of the values of O-C in the last column, a standard deviation of 0.27 would be indicated for an individual estimate made by means of the formula. Since this value is smaller than Downloaded by guest on October 1, 2021 258 STA TISTICS: W. J. L UYTEN PROC. N. A. S. that derived from the theory, it again inspires confidence in the formula, and we may assume that its present small size is accidental. Formula (1) may now be judged to be reasonably reliable, and we may thus proceed with its application to the problem of the estimation of the mean period of double stars in space. For the definitions used in judging what is a double star, and what is, or are the periods in a multiple system, reference must be made to the previous paper. It appears to the writer TABLE 1 ST" d M ss wGP P(O) woP(O) O-C j 733 0!72 0!100 1.0 1.43 26.7 1.43 0.00 r Her (22084) 1.2 0.112 2.1 1.51 34.5 1.54 +0.03 a CMi 3k 0.307 1.6 1.54 40.2 1.60 +0.06 Mlb 4 AB 1.26 0.140 1.1 1.40 42.2 1.62 +0.22 LHer BC 0.74 0.107 0.90 1.43 43.0 1.63 +0.20 Krii60 AB 1.36 0.258 0.45 1.06 44.3 1.65 +0.59 a CMa AB 10.6 0.366 3.4 2.07 50.0 1.70 -0.37 t UMa AB-CD 1.61 0.145 1.4 1.63 59.9 1.78 +0.15 a Cen AB 7.02 0.757 2.04 1.44 80.1 1.90 +0.46 70 Oph (2 2272) 6.56 0.194 1.7 2.32 87.7 1.94 -0.38 Brsb 13 3.60 0.140 0.8 2.32 130?* 2.11 -0.21 t Boo 3.01 0.173 1.0 2.01 151 2.18 +0.17 A5 (p Uri) 9.22 0.165 2.32 2.58 219 2.34 -0.22 o EriBC 4.52 0.200 0.64 2.27 248 2.39 +0.12 v Cas 8.14 0.180 1.4 2.56 508 2.71 +0.15 1.83 1.89 +0.06 ^0.09 *0.07 * According to the orbit derived by Van den Bos (B. A. N., 2, 29, and 3, 154) the period of Brsb 13 = 41G Arae is 100.9 years. From his own recent measures, however, Dr. Van den Bos concludes that this period is much too short, and estimates the real period at something like 130 years. Log P = 1.460 log d -0.487 log M +0.168 * 0.34 (1) that the solution of the problem may best be attempted by calculating the periods of all double stars nearer than a given distance-ten parsecs in the present investigation. The material has been taken from the list of stars in Harvard Annals, Vol. 85, No. 5. To these have been added a Trianguli, since the re-determination of its parallax at the McCormick Observatory now leaves little doubt about its being nearer than ten par- secs. Among southern stars, the new Yale parallax of p Eridani places this binary well within our limits, while, for the sake of completeness, t Reticuli has also been included, although the parallax of this star is known with but scant measure of certainty. It is possible that another system of parallaxes than the one used here would place 02547, 2 1280, Brsb 5 and ,B 733 = 85 Pegasi further away than 10 parsecs; while it is almost certain that 02;539 AC, and Chri 2448 will fall outside those Downloaded by guest on October 1, 2021 VoL. 16, 1930 STA TISTICS: W. J. L UYTEN 259 limits. In an investigation such as the present, the inclusion of these stars does no harm, provided that no other double star has been omitted nearer than the furthest of those in the present list. This is believed and hoped to be the case, insofar as contemporary parallax information TABLE 2 STAR PRIUOD LOG P a Tri 0.0272 -1.57 t UMa CD 0.027 * -1.56 x Dra 0.769 -0.11 v Boo 1.36 +0.13 t UMa AB 1.82 +0.26 TABLE 3 NAME d T d(A. U.) ABS. MAG. MASS LOG P 02 547 AB 4V8 0!100 .48 9.3, 9.4 0.77 2.68 O2 547 AB-C 330 0.100 3300 9.3, 9.4, 10.2 1.11 5.28 Bo 187 39 0.281 139 10.4, 13.0 0.51 3.44 Millb. 377 2.3 0.115 20 10. 13. 0.51 2.31 K Tuc AB (h 3423) 5.1 0.11? 46 = 5.2, 7.4 1.50 2.51 Lac 353 CD* (I 27) 1.2 0.11? 11- 7.8, 8.8 1.03 1.58 K Tuc-Lac 353 AB-CD 319.4 0.11? 2875 * 5.2, 7.4, 7.8, 8.8 2.53 5.02 Lpz II 961 150 0.142 1060 6.6, 12.2 0.91 4.60 r Ret 310 0.1? 3100 ' 5.2, 5.5 1.8 5.13 o2 Eri A-BCt 83 0.200 415 6.0 + mass 0.64 1.38 3.92 -y Lep (HV 50) 95 0.149 639 4.7, 7.3 1.58 4.17 Z 1280 5.2 0.100 52 8.7, 9.1 0.86 2.70 2 1321 19.4 0.163 119 9.1, 9.1 0.84 3.24 Hu 1128 5.0 0.104 48 5.6, 14 0.99 2.62 Q2 539 AC 7.8 0.095 87 7.3, 9.7 0.95 3.01 Brs 5 (P = 342y??) 1.57 0.100 15.7 8.0, 8.0 1.00 1.91 a Cen AB-Ctt 6740 0.760 8860 15.5 + mass 2.04 2.20 5.77 Sh 190 14.2 0.181 78.8 7.1, 10.2 0.94 2.95 Chri 2448 63.7 0.095 670 8.8 10.5 0.76 4.35 W-Ott 5811 512 0.160 3210 10.2, 12.4 0.56 5.41 Sh 243 AB 4.3 0.174 24.8 6.5, 6.5 1.36 2.14 36 Oph-30 Sco AB-C 730 0.174 4200 6.5, 6.5, 7.8 1.89 5.32 Mlb 4 AB-C 31 0.144 215 11.1 + mass 1.1 1.40 3.51 IA Her A-BC 32 0.105 305 3.7 + mass 0.9 2.20 3.64 Z 2398 16 0.294 54.4 11.1, 11.7 0.55 2.83 h 5173 9 0.243 37 7.2, 13.2 0.78 2.51 61 Cyg (Z 2758) 21 0.300 70 8.0, 8.7 0.95 2.87 * The period of 1 27 may well be 100 years.
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