On Some Statistical Properties of Double Stars in Space. II. On the Mean Period of Double Stars in Space

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VOL. 16, 1930

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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H. Van den Bos who kindly supplied the results of the latest known measures of these double stars-in many cases his own unpublished measures.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 contain 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 systematic 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

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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 ST"

j 733 r Her (22084) a CMi Mlb 4 AB LHer BC Krii60 AB a CMa AB t UMa AB-CD a Cen AB 70 Oph (2 2272) Brsb 13 t Boo A5 (p Uri) o Eri BC v Cas

TABLE 1 M ss

d

0!72 1.2 3k 1.26 0.74 1.36 10.6 1.61 7.02 6.56 3.60 3.01 9.22 4.52 8.14

0!100 0.112 0.307 0.140

0.107 0.258 0.366 0.145 0.757

0.194 0.140 0.173 0.165 0.200 0.180

1.0 2.1 1.6 1.1 0.90 0.45 3.4 1.4 2.04 1.7 0.8 1.0 2.32 0.64

1.4

wGP 1.43

1.51 1.54

1.40 1.43 1.06 2.07 1.63 1.44 2.32 2.32 2.01 2.58 2.27 2.56

P(O)

woP(O)

26.7 34.5 40.2 42.2 43.0 44.3 50.0 59.9 80.1 87.7 130?* 151 219 248 508

1.43 1.54 1.60 1.62 1.63 1.65 1.70 1.78 1.90 1.94 2.11 2.18 2.34 2.39 2.71

O-C

0.00

+0.03 +0.06 +0.22

+0.20 +0.59 -0.37

+0.15 +0.46 -0.38 -0.21

+0.17 -0.22 +0.12 +0.15

+0.06 1.89 1.83 *0.07 ^0.09 * 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. (1) Log P = 1.460 log d -0.487 log M +0.168 * 0.34

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 parsecs. 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

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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

a Tri t UMa CD x Dra v Boo t UMa AB

0.0272 0.027 * 0.769 1.36 1.82

LOG

P

-1.57 -1.56 -0.11 +0.13 +0.26

TABLE 3 NAME

d

T

d(A. U.)

ABS. MAG.

MASS

LOG P

0.77 2.68 02 547 AB 9.3, 9.4 4V8 0!100 .48 1.11 5.28 0.100 3300 O2 547 AB-C 330 9.3, 9.4, 10.2 Bo 187 0.281 139 10.4, 13.0 0.51 3.44 39 20 Millb. 377 2.3 0.115 0.51 2.31 10. 13. 5.2, 7.4 K Tuc AB (h 3423) 1.50 2.51 46 5.1 0.11? 111.03 1.58 1.2 0.11? 7.8, 8.8 Lac 353 CD* (I 27) K Tuc-Lac 353 AB-CD 319.4 0.11? 2875 * 5.2, 7.4, 7.8, 8.8 2.53 5.02 0.91 4.60 0.142 1060 150 6.6, 12.2 Lpz II 961 1.8 5.13 5.2, 5.5 310 0.1? 3100 ' r Ret 0.200 415 6.0 + mass 0.64 1.38 3.92 o2 Eri A-BCt 83 -y Lep (HV 50) 1.58 4.17 0.149 639 4.7, 7.3 95 Z 1280 0.86 2.70 5.2 0.100 52 8.7, 9.1 0.84 3.24 9.1, 9.1 19.4 0.163 119 2 1321 0.99 2.62 48 5.6, 14 Hu 1128 5.0 0.104 0.95 3.01 87 7.3, 9.7 7.8 0.095 Q2 539 AC 1.00 1.91 15.7 8.0, 8.0 Brs 5 (P = 342y??) 1.57 0.100 15.5 + mass 2.04 2.20 5.77 a Cen AB-Ctt 6740 0.760 8860 0.94 2.95 78.8 7.1, 10.2 14.2 0.181 Sh 190 0.76 4.35 8.8 10.5 Chri 2448 63.7 0.095 670 0.56 5.41 512 0.160 3210 10.2, 12.4 W-Ott 5811 1.36 2.14 24.8 6.5, 6.5 4.3 0.174 Sh 243 AB 1.89 5.32 0.174 4200 6.5, 6.5, 7.8 730 36 Oph-30 Sco AB-C 11.1 + mass 1.1 1.40 3.51 0.144 215 31 Mlb 4 AB-C 3.7 + mass 0.9 2.20 3.64 0.105 305 32 IA Her A-BC 0.55 2.83 Z 2398 54.4 11.1, 11.7 0.294 16 0.78 2.51 7.2, 13.2 0.243 37 9 h 5173 0.95 2.87 70 8.0, 8.7 21 0.300 61 Cyg (Z 2758) * The period of 1 27 may well be 100 years. t A period of about 8600 years is estimated from circular orbit. tt The difference in parallax has not been taken into account since this is statistically accounted for in formula (1). =

goes. The next applicants for the list appear to be -y Virginis, B.D. + 350 2436, Z 1819, and ,B 996. Concerning one star definitely within the limits there exists some uncertainty as to its duplicity, viz., ir3 Orinis.*

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The actual material at our disposal consists of the 15 stars listed in table 1, 5 spectroscopic binaries, about which information is given in table 2, and 27 other doubles for which the data are shown in table 3arranged in the same manner as the first five columns of table 1, except that now two extra columns have been inserted, giving the conversion of the observed angular separation into astronomical units and the absolute magnitudes of the components, which have now served for the calculation of the mass. The stars have been arranged in order of R. A. and are designated by their usual double-star names, but they may be* easily identified with those in H. A., Vol. 85, No. 5. Using the fifteen logarithms of periods from the seventh column of table 1, the five from table 2 and the twenty seven from the seventh column of table 3 we now have 47 values of log P for a distribution. These 47 values are given in table 4, arranged in order of numerical size, while the statistical description of their distribution is shown below. TABLE 4

-1.57 -1.56 -0.11 +0.13 +0.26 +1.43

1.54 1.58 1.60

1.70 1.78 1.90

2.14 2.18 2.31

2.51 2.62 2.68

2.87 2.95 3.01

3.64 3.92 4.17

5.13 5.28 5.32

1.62

1.91

2.34

2.70

3.24

4.35

5.41

1.63 1.65

1.94 2.11

2.39 2.51

2.71 2.83

3.44 3.51

4.60 5.02

5.77

Median 2.51; quartiles at 1.65 and 3.54 indicating a standard deviation of 1.41. Arithmetic mean 2.57 =' 0.23, standard deviation, calculated from the actual squares, 1.59; after allowing for dispersion in estimated values of log P from formula (1), o- = 1.57 0.16. The distribution is irregular, which is not surprising since there are only 47 items. The curious part, however, is that most of the unexpected peak between log P = 1.5 and 2.0 is due to actually determined periods from table 1, no less than five out of fifteen lying between 40 and 50 years. Under these conditions it should be considered as accidental that the agreement between the median and the mean is so close, although this does inspire some confidence in the statistical stability of the result. It was remarked before in Harvard Annals, 85, 1923 (90) that the median value of log P for double stars in space was probably not far from 2.5 to 2.8. The lower limit of these two has now been reached by the observations, and it remains to be seen to what extent these observations may be considered as unselectively representative of the conditions in space. Examination of the list of stars in H. A., Vol. 85, and comparison with the present list of binaries calls forth the following comments: (a) The known spectroscopic binaries are all brighter than the sixth apparent magnitude. Whether or not spectroscopic binaries or eclipsing

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binaries (such as a Geminorum C) exist among the fainter stars is at present not capable of determination by observation. (b) For the visual binaries the search may be taken as complete down to the limits of the double star Durchmusterung, say about 9.0 on the Harvard scale. Below this limit nothing is known with certainty, and the only star fainter than this limit in the present list is B. D. + 660 34. (c) As far as distant companions are concerned, our present knowledge is based entirely upon accidental discoveries (with the exception of Proxima Centauri, which was deliberately searched for by Innes). In view of these facts it is almost impossible to say what the probable conditions in space are, and thus, in the absence of any reliable information, we may sum up our ignorance by stating that the chances are as good that the unknown binaries are of longer periods than that they are of shorter periods than the mean derived here. Likewise, it appears reasonable that the actual standard deviation is not much larger than the one derived here. The final conclusion concerning the periods of binary stars in space may therefore be stated as follows: the median-mean logarithm of the periods (corresponding to the median-geometric mean of the actual periods) is probably in the neighborhood of 2.5, corresponding to a little more than 300 years, while the dispersion in these logarithms is probably not more than 1.70, in other words, half the binaries in space may be expected to have periods between the limits of 20 and 4000 years. * Cf. Astrophys. Journ., 48, 1918 (270).