© ESO, 2010

1. Introduction

The understanding of star formation can benefit from data of binary stars, since these systems carry information which is lost when single stars form (Larson 2001). However, formation processes of binary and multiple stars are far from being completely understood. Possible processes include simultaneous condensation from a common primordial nebula, fission, and after-condensation capture. Numerical simulations involving now available high-performance computing have provided some new insights on this question (Tohline & Durisen 2001). Alternative approaches can use models describing those processes taking as a fundamental parameter the mass ratio, q = m₂/m₁ (m₁ and m₂ being the primary and second masses of the binary components), on the grounds that some values of q are only possible from certain processes, and that different physical processes can favor higher or lower values of the mass ratio. The form of the mass ratio distribution, f(q), has been the object of extensive discussion on whether f(q) decreases or increases with increasing q, if it has a bimodal behavior, or if f(q) depends on the orbital period. Model-fitting approaches have produced results in every direction. Decreasing functions, of the form q^−7/3, were suggested by Jaschek & Ferrer (1972); this function has some theoretical background, because it is the original mass function of Salpeter (1955), as pointed out by Jaschek (1969). Besides, this paper by Jaschek and Ferrer introduced the parameter q₀, the minimal mass ratio that ensures that both stars are simultaneously at the main sequence, thus avoiding conditions where evolutionary processes lead to mass exchange and loss of information on primordial component masses. Concerning mass ratio distributions, similar results were published more recently by Heacox (1998). However, more results have been published favoring other f(q)’s. Hogeveen (1990; 1992a,b) finds that f(q) can be a power law, either decreasing or increasing, depending on whether the sample (from Batten et al. 1978 Eighth Catalogue of Spectroscopic Binaries – SB) was formed by single-line (SBI) or double-line (SBII) systems. This calls attention to the extent to which selection effects can influence results. Increasing f(q)s, with peaks near high q’s were suggested by Lucy & Ricco (1979), among others, for close SBs; however, most studies with similar results (Garmany et al. 1980; Abt & Levy 1985; Levato et al. 1987; Fekel et al. 1988) are based on samples composed of close systems, where contact or mass exchange lead to loss of information on the original mass of the components. Bimodal distributions have been suggested by Trimble (1974), and as well by Abt & Levy (1976); finally, both qⁿ and bimodal distributions were the results of model-fittings by Dabrowski & Beardsley (1977) and Trimble & Ostriker (1978). Also based on Batten’s Eighth Catalogue is the study by Fisher et al. (2005) concerning a sample of local field stars, which indicated a q distribution of field binaries with many q values near unity, and thus dominated by double-lined systems. The above results indicate that confusion does exist, a problem that can be approached by the use of carefully selected samples where biases are minimized. The recent release of a large database on spectroscopic binary stars has allowed more significant model-fitting tests. The choice of spectroscopic systems is convenient because of the availability of useful parameters, especially radial velocities, which in visual binaries lack the necessary accuracy and amplitude. Using a selected sample of spectroscopic binaries as a reference, the scope of this work is threefold: to look for the more probable mass ratio distributions, to test Initial Mass Functions acting during binary formation, and to derive values for the parameter q₀; to reach these objectives, we performed Monte Carlo simulations involving assumptions on mass ratio distributions, on initial mass functions, and testing several q₀ values.

2. Data and basic methodology

Following former compilations of spectroscopic binary observational parameters, for example, that by Batten et al. (1978), we selected the observational sample from the online version of the Ninth Catalogue of Spectroscopic Binary Orbits (Pourbaix et al. 2004), which provides information on 2385 systems. Observational biases are certainly present, and several criteria were applied to compile a sample with a bias as small as possible, as follows:

• a)

  An important bias in the Ninth Catalogue is that the percentageof eclipsing binaries is statistically increased. This happensbecause variability is easily detected, and systems with deepereclipses are even more easily discovered, because they havecomponents of similar brightness and, therefore, with highermass ratios. This bias in favor of eclipsing SBs is stronger infainter systems. To minimize this bias, the sample included onlybinaries brighter thanm_V = 7.0, on the grounds that those objects belonging to the bright star catalogue (BSC) and its Supplement (Warren & Hoffleit 1987), which list systems brighter than m_V = 7.0, have been extensively observed and few spectroscopic systems, if any, remain undetected.

• b)

  Mass exchange may influence the evolution of close systems, so to ensure a sample that conserves the original information of both components, the sample included only systems with periods longer than four days, corresponding to orbits with semi-major axis of about 7.4 million kilometers. This value is about 40% higher than the largest O stars in the sample, and accordingly, systems with possible contact between the components were not included in the compilation. This criterion leads to the exclusion of certain pairs of multiple systems (HD 25 204, HD 36 695, HD 71 663, HD 157 482, HD 201 433), even if these systems would have remained in the sample due to other criteria.

• c)

  Another condition imposed to avoid mass exchange was the exclusion of all systems with non-main sequence primaries. These last two criteria are complementary, producing a sample with non-interacting stars even for the eventual systems which, being formed by capture, do not have coeval components.

The final, homogeneous sample, had 249 systems, with primary spectral types distributed as follows: 29% OB, 55% AF, 16% GK. The list of stars is given in Table 2 at the end of this paper. This sample is significantly greater than those used in most of the papers cited in Sect. 1, and includes single (144 objects) and double-lined (105) systems.

A special mention has to be made concerning the paper by Fisher et al. (2005), because this paper, in a first analysis, has partially similar scopes and sources of data. However, an important objective for us was to derive values of q₀, a parameter which was not approached in their paper because it is seldom studied. Approaches to the other scopes are heavily influenced by the sample used, and here, it is true that the present sample of 249 systems is smaller than that of Fisher and collaborators, which has 371 objects; these samples, however, have some important differences. The larger sample was compiled with data from Batten’s Eighth Catalogue and from private communications; it used all values of periods, thus allowing the inclusion of contact systems and evolved stars, where mass exchanges can occur, processes which modify the original mas ratios at system formation. Unfortunately, the paper by Fisher et al. (2005) does not provide the list of systems forming the sample used for their study, so a direct comparison is not possible. However, from their criteria of d ≤ 100 pc and M_V ≤ 4, it can be deduced that for later stars, G and K dwarfs are excluded, because of the luminosity criterion. On the earlier side, an examination of the Bright Star Catalogue suggests that there are significant differences between Fisher’s and our sample. For example, from the BSC it is evident that the criterion for distances smaller than 100 pc only allows the inclusion, in Fisher’s sample, of about eight dwarfs of spectral type B3 or hotter, not necessarily primaries of spectroscopic binaries. On the other hand, GK dwarfs make up 16% of primaries of our sample, which besides includes 31 B3, or earlier, dwarf primaries. The same reasoning extends to all B dwarfs. Consequently, even if the exact content of Fisher’s list is not known, a clear perception arises that the ensemble of these differences is very significant and results in very different samples, for the spectral types and luminosity classes, and also for the volume of space studied.

Mass ratios can be directly derived from the radial velocities measured from SBIIs; however, with the present number of 105 systems, they are not enough to provide reliable information about the form of a mass ratio distribution. Besides, in doing this, the resulting sample presents a bias towards higher mass ratios, a perception already expressed at the modelings performed by Tout (1991). This author, and also Hogeveen (1992a), use different arguments to show that an inverse bias is present if only SBI systems are used. Indeed, both cite and use a former result, summarized in Fig. 1 of Staniucha (1979). The database used to produce this result carries the biases which are avoided in our sample. Also, given a fixed sample, the relative proportion of SBI to SBII will change steadily in favor of SBII, as time passes and observational techniques improve. Modeling this shifting subsamples separately would give unreliable results. So, the question is how to avoid those biases, and at the same time, how to put the observational information from systems with only one spectrum into a useful form. The so-called “mass function”, f(m), is a well known expression that addresses this problem. Not to be confused with the Initial Mass Function, the f(m) for spectroscopic binaries is derived from Kepler’s third law, and has been presented and used in Halbwachs (1987) or in Duquennoy & Mayor (1991) among others. It is given by $\mathit{f}\mathrm{\left(}\mathit{m}\mathrm{\right)}\mathrm{=}\frac{{\mathit{m}}_{\mathrm{2}}^{\mathrm{3}}{\sin }^{\mathrm{3}}\mathit{i}}{\mathrm{(}{\mathit{m}}_{\mathrm{1}}\mathrm{+}{\mathit{m}}_{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}}$(1)with m₁ and m₂ defined as above, and i the inclination of the plane of the system with respect to the line of sight. The mass function can also be expressed in terms of observational quantities, in the form $\mathit{f}\mathrm{\left(}\mathit{m}\mathrm{\right)}\mathrm{=}\mathrm{1.0385}\mathrm{\times }{\mathrm{10}}^{-7}{{\mathit{k}}_{\mathrm{1}}}^{\mathrm{3}}\mathit{P}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{e}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{3}\mathit{/}\mathrm{2}}$(2)where k₁ is the radial velocity of the primary component, in km s^-1, P is the period in days, and e is the system’s orbital eccentricity. This allows the use of SBI in exactly the same way as for SBII, as was pointed by Trimbe (1990), with the advantage that only information from the primary is needed. In a lengthy analysis, and after stating that it is impossible to determine an overall mass-ratio distribution by adding the observed q distributions of SBI and SBII systems, Hogeveen (1992b) finally suggests that the observed q distribution of SBII systems is compatible with the assumption that all binary stars in the solar neighborhood have mass ratios distributed according to the intrinsic q distribution found from SBI systems. The purpose of this study is to perform a comparison of observational information from Eq. (2) with theoretical predictions from Eq. (1); here, a problem arises because to use Eq. (1) one has to know m₂, a piece of information that is absent from Eq. (2).

This difficulty can be negotiated by dividing both Eqs. (1) and (2) by m₁. This gives, for the theoretical expression, $\frac{\mathit{f}\mathrm{\left(}\mathit{m}\mathrm{\right)}}{{\mathit{m}}_{\mathrm{1}}}\mathrm{=}\frac{{\mathit{q}}^{\mathrm{3}}{\sin }^{\mathrm{3}}\mathit{i}}{\mathrm{(}\mathrm{1}\mathrm{+}\mathit{q}{\mathrm{)}}^{\mathrm{2}}}\mathrm{\equiv }{\mathit{Y}}_{\mathrm{th}}$(3)and for the observational equation $\frac{\mathit{f}\mathrm{\left(}\mathit{m}\mathrm{\right)}}{{\mathit{m}}_{\mathrm{1}}}\mathrm{=}\frac{\mathrm{1.0385}\mathrm{\times }{\mathrm{10}}^{-7}{{\mathit{k}}_{\mathrm{1}}}^{\mathrm{3}}\mathit{P}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{e}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{3}\mathit{/}\mathrm{2}}}{{\mathit{m}}_{\mathrm{1}}}\mathrm{\equiv }{\mathit{Y}}_{\mathrm{obs}}\mathit{.}$(4)The modeling of Y_th, as just defined, will be compared with observations, expressed by Y_obs. To obtain Y_obs, the masses of the primary stars are needed. With the spectral types (main sequence) of the primaries being known for all 249 objects in the sample, the masses of the primaries can be obtained from a spectral type-mass table, like that from Lang (1991), where missing spectral types had their masses determined by iteration. With data on periods and eccentricity from the Ninth Catalogue, it was possible to derive the mass function, and Y_obs. The histogram for Y_obs is shown in Fig. 1. To investigate a possible dependency of the period on the mass function, the observational sample was divided into two groups: periods longer or shorter than 40 days, a value that approximately divides the sample into two equal parts. The normalized traces for these sub-samples are also shown in Fig. 1.

3. Simulations from mass ratio functions

The function Y_th depends on the form of the distribution function of mass ratios, f(q). The modeling of Y_th, which will be compared to Y_obs, thus totally depends on the expression of f(q). This is done by assuming several test f(q)’s, which are each used to generate a Y_th. With respect to the sini dependency in Eq. (3), inclinations of systems are supposed to be random, as widely accepted (Halbwachs 1987). The choice of test f(q)’s must cover most of the possible slopes or forms. Table 1 lists the f(q)’s used in simulations.

The construction of the histograms for Y_th was made for each q distribution listed in Table 1, by generation of successive Y_th values using random numbers. The procedure is as follows: given any distribution f(q), $\frac{{\mathrm{\int }}_{{\mathit{q}}_{\mathrm{0}}}^{\mathit{q}}\mathit{f}\mathrm{\left(}\mathit{q}\mathrm{\right)}\mathrm{d}\mathit{q}}{{\mathrm{\int }}_{{\mathit{q}}_{\mathrm{0}}}^{\mathrm{1}}\mathit{f}\mathrm{\left(}\mathit{q}\mathrm{\right)}\mathrm{d}\mathit{q}}\mathrm{=}\mathit{x}\mathit{.}$(5)Here, x is a random number and the q₀ parameter is introduced, which is the minimal mass ratio that ensures that both stars are simultaneously at the main sequence, thus avoiding conditions where evolutionary processes lead to mass exchange and loss of information on primordial component masses; in this sense, q₀ is a critical parameter for this study, and finding its value is a result in itself. It can be determined from evolutionary lifetime models, following Iben (1964). A system in which the primary is too massive with respect to its secondary will never have both components simultaneously in the main sequence, because by the time the secondary arrives at the ZAMS, the primary will already have left it. The approximate limit of q₀ for coexistence as dwarfs would be around 0.17, as suggested by Giannone & Giannuzzi (1969) and Jaschek (1969). Solving the integrals (Eq. (5)) for each distribution leads to an expression for q that contains both q₀ and x. A similar procedure is performed for the sine component. Values of modeled Y_th (Eq. (3)) are generated in this way and successive executions lead to the construction of histograms, one for each adopted f(q) distribution, where a q₀ value is adopted. These modeled histograms have to be compared to the Y_obs histogram. The comparison takes into account that the observational sample contains non-negligible noise, because it is relatively small (249 objects), even if it is one of the largest compiled to date. So, simulations for each f(q) were, accordingly, generated up to 249 executions, producing histograms of Y_th which have the same size as the Y_obs histogram; the histogram for Y_th has a noise similar to the Y_obs noise and, because it has the same size, allows also a comparison on a normalized basis. Both distributions, Y_obs and Y_th, are compared point-to-point for 25 partitions in the domain of possible Y values, which go from 0 to 0.25. The modulus of the difference within any partition k is $\mathit{di}{\mathit{f}}_{\mathit{k}}\mathrm{=}\mathrm{\left|}{\mathit{Y}}_{\mathrm{obs}\mathit{,k}}\mathrm{-}{\mathit{Y}}_{\mathrm{th}\mathit{,k}}\mathrm{\right|}$(6)and the total difference between the histograms is $\mathit{D}\mathrm{=}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathrm{25}}\mathit{di}{\mathit{f}}_{\mathit{k}}\mathit{.}$(7)This comparison is performed repeatedly to generate a mean difference, which is given by $\overline{\mathit{D}}\mathrm{=}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathit{D}}_{\mathit{j}}\mathit{.}$(8)Here, N was set to 1000, a value sufficient for the convergence of $\overline{\mathit{D}}$. This procedure, which has made on a normalized basis, produces values of D within the interval (0, 1), where $\overline{\mathit{D}}\mathrm{=}\mathrm{1}$ means a model totally different from the observations. Values of D for selected test distributions from Table 1 are presented in Fig. 2, which also gives information on the minimal mass ratio, q₀.

4. Simulations from initial mass functions

Another modeling of the observed Y was performed by creating binary systems from the random combination, two by two, of stars generated following a formation law: the Initial Mass Function. The expression of the IMF, since its first formulation by Salpeter (1955), has been frequently modified. More recently, Kennicut (1998) suggested that observations are best fitted by a composite IMF, with a slope 1.4 for formation up to one solar mass, and a slope 2.35 for greater masses. However, the situation can be different when star-forming cores undergo fragmentation, a scenario studied by Goodwin & Kroupa (2005). In these cases, a binary can be formed after additional members are ejected from the system, leading to the formation of close binaries. This would fit the description of many spectroscopic binaries, in which case a slightly different IMF could apply. The fragmentation process can produce stars of different masses, the smaller ones are the first to be ejected. The two remaining masses may or may not obey a specific, even composite IMF. To test these possibilities we performed a simulation of the random combination of stars that are independently formed. Here, the variables are the minimum and maximum masses possible in the star-formation processes and the IMF slopes; many values around those suggested by Kennicut were tested. The minimal mass was set to 0.06 solar masses; the upper limit for stellar masses was set to 120 solar masses, a conservative approach to the value suggested by Figer (2005). The generation of masses m₁ and m₂ starts with the equation $\frac{{\mathrm{\int }}_{{\mathit{m}}_{\min }}^{\mathit{m}}{\mathit{m}}^{\mathrm{-}\mathit{t}}\mathrm{d}\mathit{m}}{{\mathrm{\int }}_{{\mathit{m}}_{\min }}^{{\mathit{m}}_{\max }}{\mathit{m}}^{\mathrm{-}\mathit{t}}\mathrm{d}\mathit{m}}\mathrm{=}\mathit{x}$(9)where t, the slope of the IMF is also a variable with values t₁ and t₂, allowing the existence of a composite IMF (Kennicut 1998). The generation of m₁ and m₂ is subject to conditions that, if t is t₁, masses greater than a critical value (here, one solar mass) are discarded; and if t is t₂, masses smaller than the critical value are discarded. Here, t₁ and t₂ are first defined; which one will be used comes from a random sorting; this t_i value is then used to generate m₁, and after a new t_i definition, the mass m₂ is generated. Therefore, in all systems created by this simulation, IMFs for primaries and secondaries can be different. The same consideration for q₀ already made in Sect. 3 applies. Repeated executions are performed, and applied in Eq. (3) together with the random sini, to produce a histogram of simulated Y_th, which can be compared to the observations. We did this following the same procedure described in the previous section; the number of modules of point-to-point subtractions, observational minus modeling, was also set to be 1000. We show some results for various slopes and q₀ in Fig. 3.

5. Discussion and concluding remarks

An inspection of Fig. 1 does not provide evidence of a period dependency on the Y distribution and, therefore, the form of f(q) does not seems to be period-dependent. An analogous histogram is found in Fisher et al. (2005), with a different shape; indeed, since the criteria used to compile the samples are very different, a similarity is not expected. The question of the form of the mass ratio distribution in spectroscopic binaries as well as on the q₀ value can be addressed by an analysis of Fig. 2, where the histogram of the entire sample of 249 stars was compared with the various simulations; similar calculations using the subsamples for shorter and longer periods did not produce significantly different results. We analysed the evolution of D values because several q₀ are used in each q distribution. From Fig. 2 it is clear that those q distributions that are bimodal or increase with q, have higher D, regardless of which q₀ is used. Therefore, this family of functions is unsuitable for the present simulations. Another group of functions seems to provide better fits. A mass ratio distribution with uniform probability (function 1) gives good results. However, linear, decreasing q functions in the form f(q) = 1 − aq with a ≈ 0.5, produce the best fits to observations, and also give important information on the q₀ value, which is around 0.08. These functions are preferred over function 1 because they provide information on the constraint on q₀, while function 1 only returns low D values for unphysical (i.e. too low) values of q₀. Function 5 with a = 0.5 seems to be the best compromise between low D and a physical q₀. Consequently estimates of q₀ around 0.2 do not produce the best results. A q₀ from 0.08 to 0.1, meaning that one component can be more than ten times more massive than the other, puts limits on the primary masses, giving preference to the formation of systems with non-massive primaries, and where both components, with low masses and slower evolutionary rates, can coexist as main-sequence stars. An examination of the observational sample shows that 29% of the primaries are OB stars; even if this frequency is higher than the frequency of OB stars in the Galaxy, it must be stressed that the sample may carry a bias toward brighter younger stars because it was limited to stars brighter than m_V = 7.0. This strongly suggests that the majority of SBs is composed of low-mass stars, and that a q₀ limit around 0.1, allowing simultaneous main sequence components, is the best fit to observations.

Considering the initial mass functions for spectroscopic binaries, Fig. 3 shows that steeper slopes, around 2.35, are not the best fits to observations. Indeed, slopes of about 1.4 seem

to be more adequate, even when combined with steeper ones. With respect to q₀, differences between the three curves at the bottom of Fig. 3 are small, with slight advantage to the homogeneous distribution with slope 1.4, which has its minimum around q₀ = 0.08 to q₀ = 0.1, reinforcing the perception gained in the former paragraph. So, a reasonably good fit to observations is obtained from the pairing of stars that are independently formed; the standard IMF is not followed, which suggests an alternative process of binary formation, different from that of single field stars. The process of fragmentation, which is defined as break-up occurring during the dynamical collapse phase of protostellar clouds, leading to equal mass binaries and a dominance of high mass-ratios (Boss 1988), is not confirmed by the present study, because the best fit to observations comes from models with mass ratio distributions favoring low values of q. Processes of spectroscopic binary formation, as expected, are different from those of visual binaries.

References

All Tables

All Figures