The masses of spectroscopic binaries

The Masses ot Spectroscopic Binaries ARTHUR

BEER

T h e Observatories, C a m b r i d g e SUMMARY T h e p r o b l e m s involved in t h e d e t e r m i n a t i o n of m a s s e s of t h e c o m p o n e n t s of spectroscopic binaries are discussed. A t t e n t i o n is p a i d b o t h to m e t h o d s w h i c h yield i n d i v i d u a l m a s s e s , a n d to s t a t i s t i c a l m e t h o d s applicable w h e r e i n d i v i d u a l m a s s e s c a n n o t be calculated b e c a u s e only one s p e c t r u m is o b s e r v e d or b e c a u s e t h e orbital inclination is u n k n o w n . Compilations a n d r e d u c t i o n s h a v e been m a d e of s o m e 1200 s t a r s (the m a j o r i t y of w h i c h h a v e n o t p r e v i o u s l y b e e n considered for m a s s - d e t e r m i n a t i o n , since t h e d a t a were insufficient to f u r n i s h i n d i v i d u a l values), a n d a p r o g r a m m e is o u t l i n e d for t h e i r s t a t i s t i c a l use in t h e i n v e s t i g a t i o n of their relation w i t h l u m i n o s i t y , radius, a n d o t h e r stellar p a r a m e t e r s .

1. I N T R O D U C T I O N

I~ this note the writer returns to a theme which was a favourite of his some twentyfive years ago (BEER, 1927), in order to review the present "vista" in this field. "The only direct method we have of determining the mass of a celestial body is to measure its effect upon the motion of another body. It follows that the binary stars are the only ones whose masses we can determine directly. Since a knowledge of stellar masses is fundamental in all studies of the dynamics of the stellar system, the methods by which we calculate the absolute and relative masses of the components in the visual and spectroscopic binaries merit careful attention." These words (AITKEN, 1918) apply to-day even more than at the time when they were first written, nearly four decades ago. The enormous development of theoretical astrophysics has come to supplement the demands of stellar dynamics for exact stellar masses, and the studies of the stellar interior and of stellar atmospheres hinge essentially on this parameter which is perhaps the most important in determining the phenomena we encounter in stellar behaviour. Both the major groups of binaries the visual and the spectroscopic systems, provide determinations of stellar masses. The characteristics of these groups have recently been reviewed by VAN BIESBROECK (1951), VAN DEN BOS (1955), and HYNEK (1951). References to the observational material of spectroscopic binaries were given by C A M P B E L L , C U R T I S , M O O R E , and •EUBAUER in the Lick Observatory Bulletins Nos. 79, 181, 355, 483, 521 (1905-48). It appears established that both groups are part of the same "family", with probably a common origin. Observational selection plays an important part in both groups. Amongst the observed binaries, those for which the plane of the orbit has a large inclination (measured from the tangential plane perpendicular to the line of sight) will be in the majority, as also will those with short periods, since these are more easily discovered by spectroscopy. Difference in spectral type between the components, large parallax of the system, and some difference in magnitude between the components, all appear to help the visual discovery. Different selection effects occur in the various methods of observation according as the telescope is used visually, with a spectrograph, photographic plate, or photocell; and depending as well on the methods used to measure the spectra, the use made of the micrometer, ]387

1:188

T h e m a s s e s of spectroscopic binaries

microphotometer, etc. All these complex effects have to be singled out before a really successful attack can be undertaken. In particular, we shall have to distinguish between the well-known classes of spectroscopic systems; namely, binaries showing two spectra separated by their Doppler displacements, stars showing composite spectra (combinations of two different spectral types) but without measurable displacements, pairs with such a faint secondary component that only the spectrum of the primary is recorded, and finally systems exhibiting dynamical interaction, caused, for instance, by a common gaseous envelope, perhaps with mutual transfer of matter. In this last class, as has become apparent in cases of close eclipsing binaries, the spectroscopic elements are falsified: e.g. photometric observations give circular orbits, while the physical disturbances simulate a spectroscopic ellipticity. Reference should be made to the recent book by STRUVE (1950C) which gives a full account of the fundamental work on these lines. Statistical relationships have been investigated by many authors, for example by AITKEN (1918), BEER (1927), and by KUIrER (1935a, 1935b). Since we shall here only be concerned with certain aspects of the mass-problem, some headings will be sufficient to indicate the scope of these other statistical researches. Detailed analyses have been made, for instance, of the distribution of the mutual distances of the components; of the periods and their relation to distances, mass, eccentricity, and longitude of p eriastron; of the absolute magnitudes of the components and their differences, and of the dependence on spectral type. Analyses have also been carried out of the general distribution over the sky; and the attempt has been made to evaluate statistically the probable total number of double stars (when components of multiple systems are counted separately). It appears from these studies that as many as 80 per cent of the stars may be binaries. The frequency of separations, as illustrated b y KUIPER (1935a, 1935b), shows the intrinsic continuity of all binaries from stars practically in contact up to the widest visual pairs. The median value of about 20 astronomical units is, to quote KUIPER (1951), "remarkably close to the distance of the major planets from the Sun. It almost looks as though the solar system is a degenerate double star in which the second mass did not condense into a single star but was spread out and formed the planets and comets". The study of the period-eccentricity relation also strengthens the view that all binaries really belong to one family, and so does the recognition of the similarity of mass-distribution among the components in all groups. Other items, which must for the present purpose be by-passed, concern features of axial rotation in relation to the masses of spectroscopic binaries, as well as cosmogonical aspects of these problems. As to the latter, reference is made to Section 16 of this book, in particular to AMBARTSUMIAN'Sarticle on p. 1708. The group intermediate between spectroscopic and visual binaries is of particular interest. Special attention has been paid to it, amongst others by HYNEK (1938; 1951, p. 472), who summarizes the matter by saying that KUIPER'S Gaussian distribution curve of the mutual distances indicates that these "spectrum binaries" should be about as numerous as ordinary spectroscopic binaries, and should have separations from 1 to 6 astronomical units and a range of periods between about ½ to 6 years. Thus duplicity will be difficult to detect visually, and the radial velocity variations will generally be too small for their spectroscopic discovery. A substantial

ARTHUR BEEI'~

1389

difference of spectral type of the two components, however, will lead to classification as a star with a composite spectrum. It is known that only about 3 per cent of the known spectroscopic binaries, and only about 1 per cent of the visual systems with separations below 1", have composite spectra. A continued systematic investigation of all such spectra should thus lead to promising results in view of the recognized cosmogonic importance of this group. The following references, covering about the past twenty-five years (and, on the whole, not discussed elsewhere in this paper), deal with the determination of star masses, visually and spectroscopically, and the relations amongst them and other connected problems: SHAJ]~T (1928, 1930), PITMAN (1929), JACKSON (1932), VAN MAANEN (1933), BLEKSLEY (1934, 1935), LOSSEVA (1936, 1938), BARNES (1937), LUNDMARK (1940), MASANI (1950), LEONE (1952), IRWIN (1952), KURZEMNIECE])AUBE (1955), KOPAL (1955). The mass-ratios of binary-components have been examined in this period mainly by STRUVE (1927, 1948a, 1954), S~AJN (1929, 1937), KREIKEN (1930), ZESSEWITSCH (1930), HUFFER (1934), PITMAN (1935), GABOVITS (1938), COLACEVICH (1938), VAN DE KAMP (1940), WILSON (1941), LEONE (1949), BLANCO (1952). In addition, the masses and statistics of eclipsing variables and of single-line, i.e. one-spectrum-binaries, are referred to in papers by GAPOSCHKIN(1932, 1938, 1940, 1951), HOLMBERG (1934), PLUMMER (1938), PARENAGO and MASSEVIC]t (1950), PLAUT (1950, 1953), FRACASTORO (1953). 2. THE TWO-SPECTRA STARS Determination of masses of visual binaries requires the knowledge of the parallax ~r of the system, according to Kepler's third law: ....(1)

/zl ~ - 1 ~ 2 = a3/rr3P2,

where /~1 and/~e are the masses of the two binary components, a the semi-major axis of the relative orbit (expressed in astronomical units), and P the period in years; we note that an error of ± 10 per cent in the parallax causes an error of ± 30 per cent in the mass. On the other hand, spectroscopically determined masses are free from this restriction (which confines the knowledge of reliable "visual" masses essentially to late spectral types). However, another complicating factor interferes with the determination of the spectroscopic masses: the orbital inclination i, which appears in the expression for the total mass /~1 -~- /~2 - -

1.0385 sina i . l07 (KI ~- K2)aP(1 -- e~)~.

.

.

.

.

(2)

(where K 1 and K 2 denote the semi-amplitudes of the radial-velocity curve in kilometres, P the period in days, and e the eccentricity); or in the corresponding equations for the relative masses of the components, when both spectra have been measured separately: #1,2 sina i --~ 1.0385. 10-7 . ( K 1 ~ K 2 ) 2 K 1 , 2 P ( 1 - - e~) ~I. . . . . (3) The importance of mass determinations from visual binaries is dealt with elsewhere in this book by VAN DEN BOS and VAN DE KAMP (1956) (Vol. 2, p. 1035 and p. 1040), and the following survey is therefore mainly concerned with the spectroscopic binaries. (Other aspects in this field are discussed in this volume (Sections 12 and 14) by FRACASTORO (p. 1198), JOHNSON (p. 1407), OVENDEN (p. 1193), STRUVE (p. 1371), TCHENG MAO-LIN and BLOCH (p. 1412), and WOOD (p. 1171).)

13.9U

'l'h*~ m a s s e s of spe(,tros('opi(, binal'.~s

The increase of the observational material since the writer's 1927 survey has been enormous: at that time a total of 1003 stars were known to be spectroscopically double, and for only 303 of them were the orbital elements available. These figures have now been doubled. Stars with variable radial velocity are listed in the recent catalogue by R. E. W~LSO.< (1953); we have orbits for 615 of these, according to a recent compilation (BEER, 1954). Parallel to this great observational advance, which has widened the numerical basis of these investigations, corresponding progress has been made in the methods of interpretation.

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For an actual knowledge of the individual masses and the total mass of the spectroscopic system, we are obviously restricted to the cases of stars with known inclination i, which constitute only about 25 per cent of the total number. These are essentially eclipsing binaries, supplemented by a few data from interferometric results and also from micrometric measurements for the longest periods. For the majority of cases the only recourse is to the use of a statistical mean inclination, in order to arrive at some general results for certain groups of stars, within which we may justifiably assume statistical randomness of the/-values. To CAMPBELL'S (1910) theoretical random value of 57?3, SCHLESINGER and BAKER (1910) added the assumption of a discovery probability proportional to sin i, and arrived at sin a i = 2/3. In a detailed monograph, BERTHOMIEU (1948) took up the whole question again and proposed a sub-division of the mean inclinations i0 of spectroscopic binaries into three classes S, E, and (E ~- S), namely spectroscopic binaries without eclipse, giving i o' ---- 55?9; those with eclipse giving i o " = 80?8;

ARTHUR B E E R

1391

and the combined material with i0" ~ 63°3. The corresponding mean values of sin s i 0 are 0.563, 0.942, and 0.675, respectively; PLAUT (1953, p. 25) gives 0.939 for the group E, considering all eclipsing binaries brighter than 8m5 at maximum. Fig. 1 contains all two-spectra spectroscopic binaries available to-day (BEER, ]954) and represents the logarithms of the masses of the components (log ,ul, 2) plotted against the spectral type as given by the observer. The inclination-factor has been eliminated: (a) by individually known values of i, determined photometrically or, in thirty-eight cases, visually or otherwise; or (b) by using 0.942 for eclipsing binaries with unknown photometric orbits; or (c) with 0.563 for the remaining non-eclipsing stars, which are added for statistical comparison. The diagram contains all available components, except for two stars, log /~1 ~-- Jr 1.44 and log /~2---- Jr 1.25, which could not be entered in the diagram, the spectra being WC7 and WN6, respectively; two other stars (08, + 2.13 and B9, .-7 2-30) fall outside the top margin of the diagram. The primary components (mass #1) of the above-mentioned cases (a) and (b) are marked with dots (o), the secondaries (mass #2) with crosses ( × ) ; the cases (c), which are usable only statistically, are marked with open circles (0) and the diagram does not distinguish b e t w e e n / ~ and/~2- In addition Fig. 1 contains, drawn as a solid line, a curve adopted by PLAUT (1953, p. 23) for the mean/Zl-Values , together with a dotted line based on the log ~-spectrum relation given by RUSSELL and MOORE (1940, p. 89, Table 28). To single out the data of the eclipsing variables within our catalogue, use was made of the references of the spectroscopic observers, of the catalogue by PRAGER (1941), of the General Catalogue of Variable Stars by KUKARKIN and PARENAOO (1948) and its Supplements (1948-53), and of the lists given by PLAUT (1950, 1953), PARENAOO and MASSEVICH (1950), PILOWSKI (1954), KOPAL (1954); particular mention should be made of the very helpful tables of CHUDOVICHEV (1952) and the invaluable Finding List by WOOD (1953). If we group the 343 available main-sequence masses together within thirty spectral sub-divisions, and take the means, we obtain Table 1.

Table 1 Spectrum

,Io

(Sp.)

Mean mass (3)

05.6 O 8.4 B 0.0 B 1.0 B 2.0 B 3-0 B 4.0 B 5.0 B 6.6 B 8-0 B 9.0 A 0.0 A 1-7 A 3.0 A 4-8

33.9 21.8 17.3 13.6 11.5 11.3 9.7 7.0 7.7 7.8 5-1 2.0 2.5 2.5 2.4

Number

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Number

2.1 1'6 2.1 2'1 1.7 2.4 1"9 1.9 1.4 1.6 1.3 1.0 1.5 1'4 0.7

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13!)2

masses of spc('tvo,~copi(, tfinaries

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massive stars of the giant sequence were exch~ded from the whole material, as well as seven peculiar stars (O8-A0) with exceptionally large masses. Graphically smoothing out Table 1, and taking account of the weight of the points as given by n, we might summarize the results for the masses of the twospectra stars by Table 2. Table 2

Spectrum fi(©) 06

Spectrumi fi(())

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.2.2 1.9 1.7 1.5 1.3 1.1 0"8

I n a d d i t i o n , we h a v e d e t e r m i n e d for t h e last c o l u m n o f T a b l e 3 (see p. 1399), for c o m p a r i s o n w i t h t h e results for t h e o n e - s p e c t r u m stars, t h e m a s s - v a l u e s d e r i v e d f r o m T a b l e 2 for t h e a p p r o p r i a t e m e a n s p e c t r a used in this s u b s e q u e n t table. I'OO

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Fig. 2. Spectral distribution of the spectroscopic mass-ratios I f we n o w p l o t t h e 207 available i n d i v i d u a l m a s s - r a t i o s ~ - - # 2 / # 1 (defining < 1) a g a i n s t t h e s p e c t r a l t y p e o f t h e p r i m a r y c o m p o n e n t for all t w o - s p e c t r a stars (eclipsing a n d non-eclipsing) we arrive at Fig. 2. F e a t u r e s i l l u s t r a t e d b y this figure t h a t t h e r e are r e l a t i v e l y few stars w i t h ~, b e t w e e n 0.6 a n d 0.8; for t h e B stars

ARTHUR BEER

1393

covers the wide range from 0.2 to 1.0; most of the F stars, on the other hand, lie between 0.8 and 1.0. Obviously, selection-effects will influence this situation. The next two diagrams, Figs. 3 and 4, illustrate the relations between the two components of those fifty-six pairs for which the observers explicitly indicated differences in spectral type. In Fig. 3 the ordinates are the mass-ratios, and the lines in the direction of the abscissae point from the primary spectrum (i.e. the spectrum belonging to the more massive star) towards that of the secondary. It can be seen that in forty cases the secondary is of a later, and only in sixteen cases of an earlier type. I.OC D

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Fig. 3. Component spectra and mass-ratio Differences in spectral type are more frequent for small ~-values (when #3 ~ ~1, with #3 often belonging to the less massive late dwarfs). Fig. 4, in which eclipsing stars are given as dots and the other two-spectra stars as open circles, plots spectrum I against spectrum II and thus supplements the indications of Fig. 3 ; altogether 120 systems are now available, since this diagram is not restricted to the knowledge of ~ and also contains all systems having components of the same type. In an increasing number of cases spectrophotometric analysis has led to a knowledge of the magnitude differences AM between the two components in particular through the fundamental work by PETRIE (1934, 1939, 1948, 1950a, 1950b, 1950c, 1950d, 1952, 1955; see also PETRIE and MAUNSELL, 1950). These AM-values can be plotted against the mass-ratios, using an empirical mass-luminosity relation. From RUSSELL and MOORE'S (1940, p. 112) expression M z , ~ = 5.23--9.54 log/~l, 2 for the absolute bolometric magnitudes, we obtain for AM = M s - - M 1 the curve AM---- 9.54 log ~, which can be compared with the observed (AM, c¢)-points. A

13!)4

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diagram reproduced by HYNEK (1951, } } l 462) relates to fifty-five systems, based onl PETRIE; it is apparent that the positi<)ns of the l)oi~lts rel)resenting small massratios, i.e. large differences ill mass, deviate fr<)m the theoretical curve in the sense that at least one of the two components is "overluminous" for its mass. Our Fig. 5 contains the whole material used in the present paper, comprising seventyone systems, for both components of which absolute magnitudes have been calculated by the various methods mentioned below; this is sixteen systems fewer than were available for Fig. 7 below (for eleven we have no reliable ~ and for five no AM). Mo

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The diagram covers the range ~ ~ 0.30 and AM ~ 4m; only two stars fall outside this range, (0.36, 4:~7) and (0.58, 7m7). The theoretical curve AM _~ 9.54 log :¢ has been added for comparison. I t has been possible to extend the derivation of absolute magnitudes to eighty-six eclipsing stars, forty-five of which were previously used by PLAUT (1953). For seventeen of the other forty-one stars, photometrically determined radii r (expressed in rG) have become accessible through papers by GAPOSC~KIN, HAFFNER, KOPAL, LOHMANN, NEKRASOVA, OOSTERI-IOFF, PEARCE, and STRVVE. For eight stars individual luminosities became available, partly with the help of a parallax and a photometric value for L J L r For the remaining sixteen stars, with unknown photometric radii, direct or indirect parallaxes were used to derive the total visual absolute magnitudes MI,~ (viE) of the systems. We then have: Mbol ~ Jr 41.40 -- 5 log r 10 log T and Mvis --~ Mboi -- BC, where the effective temperatures T and the -

-

ARTHUR BEER

1395

bolometric corrections BC are tabulated by KUIrER (193S, pp. 446, 453). Mainly by consultation of the light curve, approximate intensity ratios were assigned to the two components, wherever required. In six cases it was necessary to perform the bOO

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splitting into M 1 and M 2 with a statistical value of the AM of the components, derived from the distribution of all the observed AM-values. This was done by plotting a frequency curve for all the eighty completely determined systems: more -I0 o

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than 80 per cent of these have AM < lm4, and 0'.~7 was taken as representative for the two-spectra systems. Since 93 per cent have AM < 2m2, the latter value was taken as the lower limit for the "transition" of a two-spectra into a one-spectrum star. Fig. 6 contains 172 points (Mvis, Sp), i.e. all of our eighty-seven systems except for the two Wolf-Rayet components (WC7 at + 2m3, WN6 at -- 3 .ml). Primary stars

1396

Th(~ masses of sp(~(.t,ros(.~)t)i(, bilt~ri(,s

are marked by dots (e) and secondaries by circles (o). The points which deviate most below the main trend belong to t.he B-stars AO Mou ([ and II), AQ Cas (I and II), and 31 Cyg (II). To conclude, we now represent in Fig. 7 the quantities Mbol against log ~/~. The material, taken from that used in Fig. 2, comprises all the eclipsing stars in this figure and twelve additional systems not contained in Fig. 2 because of their missing e-values. Thus all available eighty-seven systems for which it has been possible to calculate, from their orbital inclinations, individual masses and absolute visual magnitudes M 1 and M e of both components, are now used. To these values of Mvis the bolometric corrections have been added according to KVIPER (1938, pp. 446,

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453), as also used by PLAUT (1953); the values given by the latter for the spectral type of the secondaries, Sp~ (comp), have been accepted in our diagram as the appropriate ones, wherever they are available, as these take the individual photometric characteristics more into account. The primaries are marked with dots ($), and the secondaries with circles (0) ; the circle which deviates most in the upward direction belongs to AQ Cas, that below to R X Cas. RUSSELL and MOORE (1940, p. 89, eq. 94) derive for the brighter component log #b1/a ---- -- 0.0400 (Mb -- 5-20), and find that this empirical relation "represents the course of the mass-luminosity relation over the whole range for which direct observational data are available, within little more than the casual error of the individual normals with which it is compared. The observational range covers 15m in M b or 17 m in the true absolute magnitude, and 0-72 in log ~/~b, corresponding to a factor of 140 in the mass". This straight-line relation has been inserted into Fig. 7, and the comparison with to-day's material is indeed of great interest. Originally, this curve was based on all reliable individual data relating to visual and spectroscopic binaries. Some of the outstanding discrepancies which we notice have previously been discussed by STRUVE (1948a). For instance, XZ Sgr indicates

A R T H UI~, ]'~ E EI'~

1397

a luminosity difference between its A- and G-components of about 2'!'5; the massfunction is 0.004; i = 90 ° a n d / t 1 = 3 give a mass-ratio which leads to #2 = 0.35. The ( A M - ~)-relation, mentioned above, would therefore require a magnitude difference of about 9 m, which is in excess of the observed one by 6m5 (i.e. the secondary is more than 200 times more luminous than predicted by the curve). Other deviating cases are, according to STRUVE,the W Ursae Majoris stars with an average mass-ratio of 0.54, which would require an average AM of 2:~53; spectroscopic and photometric evidence, however, points towards very similar luminosities of the two components. Furthermore, the fainter components of many Algol-stars are much more luminous than would correspond to the mass-luminosity relation. Here the situation has been extensively discussed, and very interesting results arrived at, by STRUVE (1954)*. Recently, infra-red studies of Algol-star companions have also proved promising; see, e.g., BEER and KOPAL (1954). PETRIE (1939, 1948) pointed to large discrepancies among the ordinary giants, and so did HY~EK (1948). Here again, spectroscopic binaries provide fundamental material to test and limit the validity of the massluminosity law. We also refer to another paper by STRUVE (1948b) and to his George Darwin Lecture (1949) on spectroscopic binaries; here he also emphasizes early relevant researches by F. E. BAXANDALL, who was F. J. M. STRATTON'S collaborator in Cambridge. To complete this section, we finally list some further interesting work, not yet mentioned, devoted to the exploration of the mass-luminosity relation, its discrepancies and related problems, carried out during the past twenty-five years or so. This comprises the papers by REDMAN (1927, 1928), MCLAUOHLIN(1927), LUNDMARK and LUYTEN (1928), PETRIE (1934--56, see p. 1046), ELLSWORTH (1934, 1938), COLACEViCI~ (1936), DURAND (1936, 1938a, 1938b, 1938c, 1938d), PARENAOO (1937, 1939), L w o w (1939), LU~CDMARK(1940), BAIZE (1943), BERTHOMIEU(1943), LOHMANN (1948), KVIPER (1948), RUDKJ6BINO (1950), PILOWSKI (1950, 1951a, 1951b, 1951c, 1952, 1953), MASSEVITCH and PARENAGO (1950), GRATTON (1950), KOSIREV (1951), SHAKOVA (1952), SAHAnE (1952), McNAMARA (1953). 3. THE O~¢E-SPECTRU~ STARS We now turn to the stars with only one observable spectrum ; ill these cases we must exclusively rely upon the mass-function f which can be written :

f = 1.0385.10-TK13p(1 -- e2)I.

.

.

.

.

(4)

or

tt2 sin 3 i ~3 sin 3 i f -- (ttj ÷/~2) ~ -- (tq -4- #2) "(1 ÷ ~ ) 3 .

.

.

.

.

(5)

The one-spectrum stars are by far the largest group: among the 615 stars with known orbits, there are 408 one-spectrum stars and 207 two-spectra stars. The circumstances which prevent the secondary from being seen can be illustrated by a "threshold diagram", plotting the spectra of both components as abscissae and their * To quote J. L. GREENSTEIN (1955) in his D r a f t R e p o r t of Commission 29 (p. 219) for tile 1955 Dublin Meeting of tile i n t e r n a t i o n a l Astronomical Union: " T h e sub-giant components of Algol-type variables proved to be entirely different f r o m normal subgiants, or those in wide visual binaries. I t is possible t h a t they h a v e lost appreciable mass in evolving, due to close p r o x i m i t y o f a denser b i n a r y " .

1398

The masses of Sl)(~ctros('opi(' bim~l'i(.s

absolute magnitudes as ordinates; threshohl curves of the components allow the general behaviour of different coml)onent-combinations to be predicted ( H Y N E K , 1951, p. 465). In the one-spectrum stars we can obtain a first general estimate of the masses of the invisible components by using mean mass-values of the main components, together with a statistical mean inclination and a mean mass-flmction (taking only values exceeding 0.0002); the result is ~ = 0.19. Now, according to the data in the preceding section, the mean mass-ratio of all two-spectra stars is 0.77; this corresponds to a magnitude difference of I ml. The above value 0.19 for the one-spectrum stars, however, corresponds on the mass-luminosity curve to a magnitude-difference of 6m.8. Thus, on the average, secondary components are not only "just invisible" but are in fact far below the "threshold of detection" (they will be mainly red dwarfs, whose average mass turns out to be much smaller than that of the Sun). The number of secondary spectra which can be detected may be significantly increased by the development of a proposal described briefly in Vol. 1 of this book, p. 475, etc., and in more detail in FELLGETT (1955). In the previously mentioned paper (BEER, 1927, p. 111) an attempt was made to derive the mass ratio of the one-spectrum stars, ~ I, for various stellar groups, by modifying the function f as follows. Let f I be its mean value for all one-spectrum stars within a given range of spectral type, and f ~ the corresponding value for all two-spectra stars in this group ; we then form the ratio f #f ii, and evaluate ~ I as follows :

. ., f i i l 4 _ ~

• 1 - - "f

u

ii

.... ((2)

For the 149 values available ill 1927 the author obtained, for the three spectralranges then used, the following mean values of ~ I: A, 0.415; F, 0.466; G, 0.508; and as the total mean 0.463. Repeating this calculation with the new material, and after individual selection, KURZEMNIECE (1954) was able to use 440 systems (out of 476 orbits) ; 294 of these belonged to the previously mentioned group S and 146 to E. The rejected cases comprised all stars with f < 0.001, as well as all unreliable, and all extremely large f-values. The calculated data for ~ I can then also be used for the determination (from the f 1-values of given spectral groups) of the mean masses of the primary component in the one-spectrum systems.

~tl(I) =

fI ,i

(1 + ~ I) ~ (~ i)~ sina il

. . . . (7)

Investigating separately the groups E and S, it could be seen t h a t there is no significant difference between the masses of these groups ; they differ only through their geometrical arrangement with reference to the observer, and not through any deeper physical cause. In what follows both groups have therefore been cmnbined. Obviously ~ I, f, and (/~1 q- #~) will be affected simultaneously, but, due to observational selection, in a different manner from spectral class to spectral class. Furthermore, it is pointed out t h a t eclipsing variables appear to be more or less

AaTHUa BEER

1399

mfiformly distributed among the spectroscopic binaries, a result which is also supported by some recent theoretical considerations by SAAKIAN (1952). The following Table 3 gives the mean masses of the principal components for nine spectral groups, both for the one-spectrum and the two-spectra binaries, as derived from the two preceding equations. Table 3 Spectrum

.T]

n

fT~

O-B2 B3-B7 B8-A5 A6-F5 FS-F9 dF6-dG5 dG6-dK5 sgG0-sgK0 gF6-gK5

1.305 0.103 0.056 O.065 0.073 0.052 0.034 0.047 0.114

18 38 86 39 6 17 6 7 44

1.724 0.701 0.237 0.256 0.158 0.159 0.079 0.261 0.239

~-~ 24 26 48 37 3 13 6 3 4

0.73 0.66 0.78 0.88 0.89 0.77 0.64 0.71 0.84

0.62 0.27 0.37 0.42 0.58 0.43 0-42 0'30 0.55

Spectrum

BB0'2

4.0 A1.5 0.8 F 7.4 dG 0.4 d K 1.4 sgG 6.7 gG 8.3

pT~) 20'3 9-9 2'6 2'2 1.4 1'3 1.1 2'4 3"3

20.6 12.6 2.9 2.6 2.0 2.0 1.4 7.0 2.9

16.5 9.5 2.5 1.8 1.6 1.5 1.0 ---

The spectra F 6 - K 5 are here subdivided into the main sequence stars F6-F9, the dwarf-groups dF6-dG5 ("subdwarfs"), and d G 6 - - d K S ; the giants gF6-gK5 are separated, and also the main subgiant group sgG0-sgK0 (while a few more sg-stars are contained in the group F6-F9). The last column, fi, gives for comparison the results obtained for the two-spectra stars, derived in the preceding section (see p. 1389 and Tables 1 and 2). The/~1 (II)-values in the various groups show r.m.s.-errors between 10 and 52 per cent, the #1 (I) between 21 and 58 per cent. It can be seen from Table 3 that f I is always less than f II, and ~-~ < ~ II. This was to be expected: f depends on the total mass, the mass ratio and the inclination; since the companion in a one-spectrum system is invisible because it is too faint, its mass will be smaller than that of the companion of a two-spectra binary of the same type, and, therefore, I <

:¢ I i

and

( # 1 ~ - # ~ ) z < ( # 1 ~ - # 2 ) IL

Furthermore, it is also plausible that we should have i I < i IZ, because the discovery of small amplitudes is favoured in the one-spectrum stars (1) of the S class above, i.e. the non-eclipsing stars, discovered by spectroscopic means only: in the two-spectra stars the spectral lines of both components show the variations and displacements with greater difficulty than do the single unblended lines in the onespectrum stars. A smaller amplitude, however, means that on the average the binary's orbit has a smaller inclination. The observed mean i-value of the twospectra stars will thus exceed those of the one-spectrum systems; in other words, there will be more systems with a small i among the group I than among H. In Table 4 both groups I and IX have been combined. Table 4

0"77 0'76 0'8 0-78

62 95 -165

~{I

nI

0"46 --0.44

149 261

Author BEER, 1927 NEKRASOVA, 1938 STRUVE, 1950 KURZEMNIECE, 1954

1400

The masses of" spectros<,opic binaries

In early investigations SMART (1925), il~?Eg (1927), and SHAJN (1928), had suggested that the mean s-values increase if one proceeds in the spectral sequence towards the later types, t'[OLMBERG (1934), NEKRASOVA (193S), and BLANCO (1952) also mention such a dependence. The new material, however, as analysed by KURZEMNIECE (1954), suggests that such a relationship cannot be established with certainty. Fig. 8 based on Table 3, giving e for the one-spectruIn stars ([) and twospectra stars (~I) plotted against spectrum, illustrates this for the new detailed subdivisions and shows that, indeed, both ~ n and e I appear to be practically constant. I'0 o .

.

.

.

.c~

,0" .i~

~

'

og

l O.l

0'~

IIIIlllll

0

[l[lll[ll

Bo

I[1111,[I

Ao

I[llLIlil

11111[1[I

Fo

Go

i

go

Fig. 8. Mean mass-ratios against type for the one-spectrum and the two-spectra sta~'s

From the theoretical point of view, however, we should actually expect an increase of ~ for the later spectral types. This can be seen if we evaluate K from equation (2): K1:

212 -

~+i

(#1 d- #2) '~ sinai P-~(1 -- e2)-'~.

. (8) "

"

K is therefore approximately proportional to sin i, (#1 ÷/~2), and u. While the inclination is of course independent of the spectrum, the latter's correlation with the total mass of the system is well established, the mass decreasing towards later types (Fig. 1, etc.). For a given K, we should consequently expect, for the early-type stars with their larger mass, a smaller ~; and correspondingly for the later types, with their smaller total mass, and for the same given K, a larger ~. This reasoning, however, does not guarantee t h a t we shall really be able actually to observe such an increase of ~ with type, and in fact we seem to be unable to do so ; another selection effect has entered. It must be kept in mind that the ease with which the amplitude K can be observed in itself depends on the spectral type. Among stars of early type (which usually have diffuse lines) only those that happen to show sharply-defined displaced lines, will be singled out as spectroscopic binaries. This implies that the known eases of early-type binaries have the larger K-values as compared with those of the average late types ; and this corresponds to a relatively larger ~, according to equation (8), and this effect thus tends to compensate the decrease of ~ with advancing spectral class, which would follow from the general decrease of mass. This combination of two effects seems the most likely explanation of the apparent independence of ~ with respect to spectral type, shown in Fig. 8. The following Table 5 is derived from KURZEMN~EeE'S Table 1 and gives #-~ for spectral combinations of 425 systems, selected from the above-mentioned 440 (S -4- E) systems; /~1(I) and ttl(II) have been grouped together, weighing them with their respective errors.

ARTHUR BEER

1401

Table 5 Mean spectrum

Spectral range O-B2 B3-B7 B8-A5 A6-F5 F6-F9 dF6-dG5 dG6-dK5 sgG0-sgK0 gF6-gK5

~

Percentage

0.2 4.0 1.5

20.0 10.0 2.7

± ]6 ± 22 ± ]2

7.4 do04 d K 1-4 sgG 6.7 g G 8.3

1.5 13 1.1 2.6 3.1

17

~

42 64 134 76 9 30 12 10 48

"" ± 42 ± 39

The above arrangement of the late types has been introduced following PARENAGO and MASSEVICg (1950, 1951), who placed the sg-stars on the mass-luminosity diagram somewhat earlier than the main sequence. Furthermore, the mean mass of

+1.5 -/~ 0 X

x 0 0

I

~.

A:

+0.5

~o

X

•

o .~.~

X •

A

•

O

• •

"~

X

Oo

go

X Oo

O.O

•

x •

~•

• o

A eO

-0.5

B•

Ao

,Co

Go Sp

K•

It,l•

-

Fig. 9. T h e l o g a r i t h m of ~a1, t h e m a s s of t h e p r i m a r y , p l o t t e d a g a i n s t spectral type. T h e ( × ) r e p r e s e n t the/21 v a l u e s ta~ken f r o m T a b l e 5, a n d t h e o t h e r p o i n t s r e p r e s e n t different d e r i v a t i o n m e t h o d s (see text). T h e a g r e e m e n t of t h e ( × ) w i t h t h e s e o t h e r p o i n t s gives confidence in t h e v a l i d i t y of t h e m e t h o d on w h i c h Table 5 is b a s e d

the dwarf-stars given above must be larger than it would be for an undisturbed average: observational selection will have singled out the large masses, which have the larger K's and are therefore more easily discovered. The large value for the subgiants is uncertain, due to the inhomogeneity of the physical characteristics in this small group of stars. Fig. 9 summarizes all log / ~ l - V a l u e s , after KURZEMNIECE, plotted against spectral class, and gives: (a) observed mean masses from Table 5 (marked ×); (b) and (c)

141~2

T i m m a s s e s o f .~i)e(,troseopic. b i n a r i e s

values given by [ARENAGO an(l MASSEVI(~H (1951) (O); an(t by RUSSELLand MO()tcE (1940) (A); (d) masses derived fl'om visual 1)iaaries by l)h]~:Ni(;O and MASSEW(~H (1951) (O). Each point rei)resents the mean of a number of stars. The agreement between the points is on the whole satisfactory and indicates that the various methods are systematically consistent with each other. Recent work has not only provided new material but has also been concerned with convenient methods for the evaluation of secondary masses in one-spectrum

+2 -

/~2o

,ob

5O 40 30

,°f

!

o 0.5 0.2 o.I o.o5 0.02 o-oi 0.005 0-OO2

20

IO1

÷I-

I

('3,(~C31

I

0"0001 r,..

~.o-

-s -2

-O.OOOO1

O,5 0.4 0"2

-

O.I

(:).07

:).03

0 " 0 1 ~

0"0{

I

-2

I /I

2

L

I

If

I

L,f

I

3 4 5

L

7

O'L

2

3

4 S

I

I

]

I

I

I

I

I

I

L

I

I

7

1'0

2

]

4 5

7

I0

2

'] 4 S

I

-I

O Io9/~2 Fig. ]0.

The

I

+l

i

i

K

i

7 1~90

I

+2

Q(#I, #~) n o m o g r a m

systems. PARENAGO (1950), assuming t h a t the principal component belongs to the main sequence, derived #z-values from the mass-luminosity relation, and then obtained statistically useful values of #3 from equation (5); with ~ ~ #3/#1 and Q = f/sin 3 i, we have Q -- ~3/(1 ÷ ~)2. The solutions are facilitated by the nomogram reproduced in Fig. 10, which gives the relation between ~ (or #1 and #3) and Q. For a given observed mass-function f and a calculated or statistically estimated inclination i, the nomogram is entered with Q ; if the mass/~z of the primary is known by any means, the mass #3 of the secondary is then obtained directly. I t should be noticed t h a t if/~z is taken from a spectrum-mass relationship, a certain degree of logical circularity is introduced into mass-luminosity relations derived from it. I t is of interest to note that PARENAGO is able to list eighty-three one-spectrum

ARTH UI¢ BO'EI¢

1403

stars with their Q,/A1,/A2 and with the semi-major axis A of the relative orbit of both components, expressed in solar radii R ) (7. 105 km), i.e., A = a sin i/R•. For sixtytwo cases exact photometric values of i have been available, and for the remaining twenty-one stars it has been assumed that sin i ~-~ 1. For sixty-one stars of this list the photometrically determined values r 1 and r 2 for the radii of the stars are known, expressed in units of the semi-major axis; through multiplication by A, their radii R 1 and R 2 are obtained in solar units. Furthermore, using the photometric value of L1/L 2, the magnitude-difference Am of the two components is found as Am = 2-5 log (L1/L2). Adding this Am to the absolute magnitude M 1 of the main component (with the help of which/~1 was found above), we obtain M e. For these last-mentioned sixty-one systems, therefore, values of log M2, log #2, and log R 2 can be plotted in PARENAOO'S diagram, using as coordinates ~ = log (#/R) and ~ = log (M/#5). These coordinates have also been used in a detailed study of the mass-luminosity relation by PARENAGO and MASSEVICI~ (1950) (see also KOUROANOFF, 1950), which was to include all reliable visual double stars, eclipsing binaries, and spectroscopic values. Supplementary references to other and mainly earlier investigations of this fundamental relation have been given above (p. 1397). The new data were actually taken from the card-catalogues of the Moscow Astrophysical Institute (GAISH). In all, 144 points of the new diagrams represent stars on the main sequence, twenty-one are slightly above it, seven slightly below it, three are white dwarfs, twenty-five subdwarfs, twenty-six subgiants, and ten supergiants; six are giants taken from BERGLUND'S (1936) study of visual binaries; in addition, five mean values are used for slow visual pairs; and one point represents the mean mass for spectroscopic binaries of average type G5 and absolute magnitude 0.m0, obtained by a statistical method (BEER, 1927). Altogether, besides these lastmentioned six means, 242 individual stars were used. The main conclusion of these authors is that there is no unique mass-luminosity relation which is valid for all the various star groups. There appear to exist two straight sections in the main sequence, represented by L = 1.12. #a.92 for O to G4, and by L --~ 0.41 . #2.29 for G7 to M-types. Giants and supergiants are close to the upper section, white dwarfs far above (under-luminous for their mass) and subdwarfs and subgiants far below the main sequence. The mass-radius relation turns out to be an irregular curve, intersecting the line log # = log R at the three points 0.43, 0.15, and 0.00; these have the co-ordinates (B9, M~, = -- 0:~8), (F2, ~ 3~73), and (G4, -~ 4~77), respectively. Additional diagrams relate to certain functions of #, L, and R. In addition to the above-mentioned (~, ~?)-relation ($ is the logarithm of the potential energy of the star), they discuss, for example, log p ---- log/~ -- 3. log R (p is the logarithm of the mean density) and ~ = log L ~- 3.5 log R -- 6-5 log #. Such diagrams show well-marked principal sequences, and the aim was to establish for each of them a HERTZSPRUNGRUSSELL diagram of the form L ~-- 10~/~R ~ (where x can involve the mean molecular weight); various solutions have been proposed and their respective correlation coefficients listed (see also STRUVE, 1950a). The above-mentioned extensive tabulations by PLAUT (1953), comprising all eclipsing binaries photographically brighter than 8~.5 at maximum, are accompanied (on p. 22, etc.) by a discussion of recommended mass values, both for one-spectrum and for two-spectra systems, derived from four sources.

1404

The masses of spectroscopic binaries

(A) By applying equation (3) to all two-spectra stars with known K 1 and K 2 (only a few peculiar systems with well-established disturbances by gaseous streams, shells, etc., having been omitted). (B) The mass-function f of the one-spectrum systems was used in an adaptation of PARENAGO'S (1950) method (see also PARENAGO and MASSEVICH, 1950; KOUROANOFF, 1950; STRUVE, 1950a, 1950b). In the expression (5) for f, the factor sin 3 i is known from the light curve, and the values o f / ~ are then inferred through the application of a plot of the kind presented here as Fig. 1. The mean curve (see p. 1391) can, of course, generally only be used for stars earlier than F0, if giants and dwarfs cannot be separated. For a star with a given observed spectral type, "individual" /~:values are read off from this curve; and we then obtain ~-values and therefore /~2 from f by solving, in principle, the equation ~3 _ A~2 _ 2A~ -- A ---- 0, if we call f . (#1 sin3 i) -1 = A. The graphical method for obtaining these data rapidly has been discussed above (p. 1402). (C) The second modification of PARENAGO'S method, as used by PLAUT, is suitable for all stars for which the absolute visual magnitude M h of the heavier star is available, either through its parallax or b y other methods. These M-values then lead through the mass-luminosity relation to the mass #1 (I~USSELL and MOORE, 1940, eq. 94). The calculation of/~2 then proceeds in the same way as above for (A). It is emphasized that this method has the advantage over (A) that here the purely statistical relation between #1 and spectral class is avoided. (D) The fourth method, mentioned by PLAUT, is a modification of (C), b y introducing PETRIE's M-values referred to on p. 1393. The accuracy of PETRIE'S method in the case of two-spectra systems is considerably greater than (B) and (C); for these stars, however, PLAUT has in fact used method (A). It is also noted that other methods have previously been discussed in the papers by HOLMBERG (1934) and GAPOSCHKIN (1940). The present situation as to the mass-radius diagram is reviewed by PILOWSKI (1954) on the basis of his previous work (1950-53). He plots all eclipsing variables having completely known and reliable photometric and spectroscopic elements (altogether forty-seven pairs), and states that the fine-structure of the early part of the main sequence as proposed b y EOOEN (1950a, 1950b, 1950c, 1950d) appears essentially verified. The writer is at present engaged in an attempt to extend the discussion of the important #-R diagram by supplementing the systems mentioned above which have been determined individually, with the very extensive statistical data available to-day. A few concluding remarks about this will summarize present possibilities. From what has been said above, the observational status of two-spectra stars can be divided into : (a) the well-determined bright eclipsing binaries, (b) the fainter and less completely known systems, and (c) eclipsing stars without an /-determination where in our discussion a mean value from all eclipsing stars has been used. There are further: (d) the non-eclipsing two-spectra stars, to be used with sin 3 i =-- 0.563. In one-spectrum stars we have: (e) the eclipsing binaries with known i; and (f) the remaining eclipsing binaries, to which we apply the mean /-values as used in (c). Classes (e) and (f) lead to expressions f(/~lq-~u2, ~). There are, finally, (g) those one-spectrum stars without i where we again apply the procedure used in (d). Then, for the one-spectrum systems, the combined use of the stars (a) and (b) furnishes the #1 (from known luminosities, as well as from the mass-spectrum relation). The Q-nomogram then gives the #:values. Furthermore, these can also be determined with the other method outlined previously (p. 1398), which furnishes spectral

ARTHUR I:~EEIt

1405

means of a I --~ ¢ (f I, f II, ~ II); these latter are then applied to the particular spectral types, leading via ~ I to # I and ~ II. The results of the various methods are averaged and, finally, more than 1200 single #-values can thus be investigated statistically. In conclusion, the value of the study of binaries lies, as HYNEK recently emphasized again, not only in the fact that they are the only available means of discovering stellar masses, but also in the circumstance that the existence and dynamical state of binary and multiple stars constitute an important datum which any comprehensive theory of stellar evolution must take into account. These two aspects of the study of binaries form a stringent test which no evolutionary theory has so far passed in a fully satisfactory manner. ]:~EFERENCES AITKEK, R . G .

.

.

.

.

.

.

.

.

.

1918

BAIZE, P . . . . . . . . . . . BARNES, C . . . . . . . . . . . BEER, A . . . . . . . . . . .

1943 1937 1927 1954

BEER, A. and KoPAL, Z . . . . . . . BEaGLUND, F . . . . . . . . . . BERT~OMIEU, H . . . . . . . . . BLANCO, V. M .

.

.

.

.

.

.

.

.

.

BLEKSLEY, A. E. H .

.

.

.

.

.

.

.

CAMPBELL, W. W . . . . . . . . . CHUDOWCHEV, N. I . . . . . . . . COLACEWCH, A . . . . . . . . .

DU~A~D, G .

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . FRACASTORO, M. G . . . . . . . GABOVITS, J . . . . . . . . . GAeOSCHKIN, S . . . . . . . .

. . . .

E~GEN, O .

.

.

.

.

ELLSWOI%TH, J .

.

.

.

.

.

.

.

.

.

.

.

.

FELLGETT, P .

GRATTON, L . . . . . . . . . . HOLMBERG, E . . . . . . . . . . HUFFER, R. C . . . . . . . . . . HYNEK, J. A . . . . . . . . . .

1954 1936 1943 1948 1951 1952 1934 1935 1910 1952 1936 1938 1936 1938a 1938b 1938c 1938d 1950a 1950b 1950c 1950d 1934 1938 1955 1953 1938 1932 1938 1940 1951 1950 1934 1934 1938 1948 1951

The Binary Stars,

2nd ed., 1935, (McGraw-Hill, New York.) C.R. Acad. Sci. (Paris), 216, 633. M.N., 97, 454.

p.

216.

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1406

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1930 1935a 1935b 1938 1948 1951 1.948

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1407

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Evidence For and Against Binary Hypotheses of the Blue-red Erupting Stars MARTII~ J O H N S O N

Physics Department, The University, Birmingham SUMMARY"

The peculiar and erupting stars which exhibit simultaneously a cool oxide spectrum and a hot helium spectrum have been variously regarded, either as binary pairs with blue and red components or as single stars with composite envelopes. The implications and obstacles in each of these hypotheses are here critically examined. A suggestion is made for avoiding one difficulty of the binary hypothesis, by a mechanism in which a prominence erupting into a gaseous ring becomes amplified by suppression, resulting in local instability.

1. N A T U R E OF THE P R O B L E M

spectrum dominated by titanium oxide normally denotes the low temperature (20000-3000 °) at which stability and excitation of that material would occur in thermodynamic equilibrium. Correspondingly, at the other extreme of visible stellar temperatures, the lines of ionized helium must imply at least 20,000 °30,000 ° in thermodynamic equilibrium. We shall discuss in this note some old and new explanations of those few stars whose spectrum combines both the coolest and the hottest, the oxide and the helium simultaneously imitating phenomena of such extremely differing temperatures that either two separate bodies or two regions in radical departure from thermodynamic equilibrium must be sought. It has commonly been supposed that these "combination" spectra imply binaries, one component of each pair being a normal hot B-star and the other a normal cool M-star; great distance from us or small distance between the components could conceal the binary character, and eclipses and orbits could be hidden by orientation and the motions of a common envelope. The present author (1951) calculated some time-constants of fluorescence around these stars in terms of such binary hypothesis; some authors employing the same hypothesis have written "companion star" in inverted commas recognizing the uncertainty; MENZEL, who once propounded the A STELLAR

4I