Mercado laboral y su relación con la gestión humana

The early double star observers, from William Herschel in the late 1700s to Robert Aitken about a hundred years later, focused on discovering and cataloging thousands of visual double stars. Aitken’s Double Star Catalog (Aitken & Doolittle 1932) lists over 3,000 double stars, over 30% of which have angular separations<0′′.

5, and over half have separations<1′′

, confirming the improved capabilities, and increasing the confidence in a physical relationship due to proximity. Given the nascent nature of these surveys, however, there is little discussion of orbital elements or multiplicity statistics until a few years later.

With the evidence for a physical association among double stars mounting because of the large number of close doubles discovered, attention turned toward treating them as bound systems whose orbital properties could be studied. Hertzsprung (1922) developed an

empirical method of estimating the orbital period of visual binaries by deriving a relationship between the orbital period and the ratio of the radius vector to yearly orbital motion, based on his studies of 13 pairs within 10 pc of the Sun with known orbits. He also noted that of the 15,000 known double stars at that time, only 50 had reliable orbits and an additional 1,000 showed hints of orbital motion. A few years later, Luyten (1927) used matching proper motion measurements of the components of double stars to argue for a physical association. Luyten (1930a) noted that the results of Hertzsprung (1922) were incomplete and arbitrary, and developed a more rigorous statistical method for the estimation of orbital period based on Kepler’s Third Law. He measured separation at one epoch, estimated masses from measured luminosities, and statistically estimated the orbital eccentricity and inclination. He also provided observational support for this method by testing it on 15 binaries with known orbits (Luyten 1930b). In perhaps the earliest statistical analysis of a volume-limited sample, Luyten (1930b) presented a tally of 47 visual doubles, 15 of which had reliable orbits, and 5 spectroscopic binaries for the 10 pc sample, which included 105 stars (Luyten & Shapley 1930). Estimating the periods of the visual doubles without orbits, Luyten (1930b) developed a period distribution of the complete sample of known binaries within 10 pc, concluding that the logP distribution (with period in years) was unimodal with a mean of 2.5 and a dispersion of 1.7, noting that any undetected binaries were likely to have periods larger than the derived mean.

Kuiper (1935a,b) proposed a variety of problems that could be studied with double stars, and addressed some of them, deriving a companion fraction of 80%, i.e., for the 465 primaries

studied, he estimated a total of 372 companions. These results were based on an observed companion fraction of 33%, which was adjusted to account for the incompleteness of the survey for large ∆mag pairs. Further, he noted a roughly Gaussian distribution of the semi- major axis, with a peak at about 20 AU. In a follow up work, Kuiper (1942) reported raw multiplicity statistics for a sample of 254 stars with parallax≥0′′.

095. He presented multiplic- ity by spectral type and by total mass. For A-K stars, the Single:Double:Triple:Quadruple (S:D:T:Q) numbers were 44:23:5:1, which yields a companion fraction, i.e., the total number of companions divided by the total number of primaries, of 49%. This is significantly higher than his earlier work’s results of 33%, but no incompleteness analysis was performed in the later effort. The semi-major axis distribution was confirmed to be Gaussian, but the peak had moved out to 50 AU. Kuiper noted the incompleteness of the survey and suggested that a sample selected from a larger volume of space along with the completion of spectroscopic and visual surveys would provide more reliable results. Heintz (1969) studied a sample of stars within 20 pc and presented a companion fraction of 1.0 to 1.1, i.e., 100 systems would contain 200–210 stars. Based on spectroscopic and visual binaries, his work identified 30 single, 47 binary, and 23 multiple systems, with an asymmetric distribution of semi-major axis peaked at 45 AU. However, Heintz also noted that the statistics were limited by selection effects, primarily the discovery probability and confirmation of physical pairs.

Focusing on spectroscopic binaries, Jaschek & Jaschek (1957) found that roughly 16% of a sample of about 600 F-K stars were spectroscopic binaries, which were identified as pairs with known orbits, those noted as spectroscopic binaries in earlier efforts, or those with

mean velocity differences between observatories of more than 20 km s−1_{. This threshold was}

independent of spectral type or mass, and, to account for the dependence of velocity semi- amplitude on mass, they applied a correction factor and reported a corrected spectroscopic binary fraction of about 30%, noting that it was roughly constant along the main sequence, a conclusion earlier noted by Kuiper for visual binaries. Petrie (1960) studied the probable error distributions of radial velocities and estimated that∼ 51% of F-M stars showed radial velocity variations, higher than previous estimates, but once again confirmed a roughly flat distribution across the main sequence. However, Petrie’s work assumed a Gaussian distribution for the observational errors of constant velocity stars, which was later shown to be incorrect (Kirillova & Pavlovskaya 1963). Jaschek & G´omez (1970) reviewed these prior efforts and presented an updated percentage of variable radial velocity stars as about 45% for F-M stars based on 350 stars, once again noting similar results across all spectral types. Abt & Levy (1976) attempted a comprehensive systematic effort for the multiplicity statistics of solar-type stars based on a sample of 135 F3–G2 IV or V bright field stars (V < 5.5 mag) and about 20 radial velocity measurements for each star. Their results were S:D:T:Q = 42:46:9:2 with a median period of 14 years, considerably shorter than the estimates of 320 years obtained by Luyten (1930b) or 79 years by Kuiper (1935a,b). Based on an incompleteness analysis aimed at accounting for missed binaries, they derived a companion fraction of 1.4 companions for each primary, and concluded that two-thirds of solar-type stars have stellar companions and the remaining one-third have substellar companions. However, their results have been called into question based on selection effects and the validity of the

binaries reported. As analyzed by Branch (1976), their magnitude limited sample suffers from a serious selection effect favoring binaries with bright companions, because unresolved binaries are intrinsically brighter than single stars and will be counted out to a larger volume of space (the Malmquist bias). Further, Morbey & Griffin (1987) pointed out that 24 of the 25 new binaries reported by Abt & Levy are not statistically supported by their data and showed that 21 of these are likely not binaries at all. Another important conclusion of Abt & Levy (1976) was the bimodal distribution of mass-ratio, which they use as evidence of two formation mechanisms of binaries – fission for close systems and cloud fragmentation for wider systems. However, several authors have pointed out that these results are dominated by selection effects and that the true distribution of mass-ratio is unimodal with frequencies increasing towards lower mass-ratios (Scarfe 1986; Trimble 1987, 1990).

Zinnecker (1984) reviewed binary statistics and the distribution of mass-ratios, discussing the various selection effects that hampered such studies and suggested that none of the binary formations mechanisms (fission, fragmentation, capture, and disintegration of small clusters) could be ruled out. Halbwachs (1986) studied the multiplicity statistics of the stars in the 4th edition of the Yale Catalog of Bright Stars (Hoffleit & Jaschek 1982) and its supplement (Hoffleit et al. 1983). Accounting for selection biases, he reported that the sample of F7–M 2591 dwarfs contained 52% single stars, 36% binaries, and 12% multiple systems. Based on an estimation of missed binaries, he concluded that at most 23% of stars are truly single. Halbwachs (1987) analyzed the mass-ratio (q) distribution among spectroscopic binaries in the 7th Catalogue of Spectroscopic Binaries (Batten et al. 1978), and concluded that there

was no peak at q ∼ 1 as suggested by Abt & Levy (1976) but rather a possible peak near

q ∼ 0.4. Further, he concluded that there is no difference in the mass-ratio distribution between close-period spectroscopic systems and the long-period visual binaries, suggesting that all binaries may be formed by a single mechanism.