Unresolved Binaries and Galactic Clusters' Mass Estimates

Haffner & Heckmann (1937) provided one of the first indications that star clusters harbor a large number of unresolved binary stars. Maeder (1974) showed what position binary stars have in the color–magnitude diagram (CMD) as a function of their mass ratio q = M₂/M₁ (where M₂ is the mass of the secondary while M₁ is the mass of the primary component). Hurley & Tout (1998) demonstrated that the secondary sequences routinely seen above the main sequence (MS) in clusters' CMDs are actually made of binaries with wide mass ratio ranges (and not merely by equal mass binaries). A summary of the results on the binary stars content of star clusters is presented by Duchêne & Kraus (2013).

The binary fraction α in Galactic globular clusters is relatively small and usually does not exceed ∼10% (Milone et al. 2012), with only rare exceptions. For instance, Li et al. (2017) found a much larger binary fraction for just three globular clusters (α = 0.6–0.8). Open clusters (OCl), on the other hand, host a more significant fraction of binaries with α ≥ 30% (Bonifazi et al. 1990; Khalaj & Baumgardt 2013; Sarro et al. 2014; Sheikhi et al. 2016; Li et al. 2017). This percentage is, however, smaller than the one among field stars in the solar vicinity (Duquennoy et al. 1991). It has also been noted that the binary percentage increases at increasing a primary mass. This fact is often linked to the dynamical evolution of clusters (Kaczmarek et al. 2011; Dorval et al. 2017). Nevertheless, it does not seem to be universal since, for instance, Patience et al. (2002) found an increase of the companion-star fraction toward smaller masses in α Persei and Praesepe.

A fundamental quantity is the mass ratio q distribution. Unfortunately, a consensus is still lacking. According to Duquennoy & Mayor (1991), the distribution of masses of the secondary in the field does not show a maximum close to unity. Instead, this distribution is continuously increased toward the low-mass end. Fisher et al. (2005), however, found a q distribution peaking near q = 1 for field stars. The same peak was found by Maxted et al. (2008) for the low-mass spectroscopic binaries in the young clusters around σ Ori and λ Ori. Raghavan et al. (2010) support this point of view, showing that the mass ratio distribution shows a preference for like-mass pairs, which occur more frequently in relatively close pairs. Reggiani & Meyer (2013) argue for a universal form of the q distribution both for solar-type and for M-dwarfs in the general Galactic field:

$$\begin{eqnarray}&&{dN}/{dq}\sim {q}^{\beta }\end{eqnarray} \tag{ 1 }$$

with the β = 0.25 ± 0.29 (flat within the errors ). Also, Milone et al. (2012) claim that in the interval q ∈ [0.5, 1.0] the distribution of q is nearly flat, with few possible deviations among Galactic globular clusters. Kouwenhoven et al. (2009) introduces two different q distributions: a power law (1) for q ∈ [q₀, 1] and different β values, and a Gaussian one

$$\begin{eqnarray}&&{dN}/{dq}\sim \exp \left[-{\left(q-{\mu }_{q}\right)}^{2}/2{\sigma }_{q}^{2}\right]\end{eqnarray} \tag{ 2 }$$

for q ∈ (0,1] with μ_q = 0.23 and ${\sigma }_{q}^{2}=0.42$. According to Patience et al. (2002), the q distribution depends on the stellar mass interval: the higher-mass systems reveal a decreasing mass ratio distribution, and the lower-mass systems reveal a deficit of low mass ratio companions (see Figure 8 in Patience et al. 2002). As a result, the combined samples show the deficiency of q > 0.85. However, a flat distribution is not ruled out (see Figure 6 in Patience et al. 2002).

The q distribution keeps a memory of the primordial binaries' properties. Some numerical experiments were carried out along this line (Kroupa 2011; Geller et al. 2013; Parker & Reggiani 2013). Geller et al. (2013) performed N-body simulations of the old open cluster NGC 188 and showed that the distribution of orbital parameters for short-period (P < 1000^d) solar-type binaries would not be changed significantly for several Gyr of evolution. This fact means that observations of the present-day binaries even in the oldest open clusters can bring essential information on the primordial binary population. On the other hand, Parker & Reggiani (2013) showed that while the overall binary fraction decreases, the shape of the q distribution remains unaltered during the evolution. The presence of unresolved binaries in star clusters affects any estimate of their mass, both photometric (via star counts) and dynamical (via velocity dispersion and the virial theorem). In the latter case, if the sample of stars selected for velocity dispersion calculation (through radial velocities) contains spectroscopic binaries, one can indeed artificially inflate the velocity dispersion, and hence increase the mass. This point has been recently underlined by Kouwenhoven & de Grijs (2009), Bianchini et al. (2016), and by Seleznev et al. (2017).

When the cluster mass is evaluated through the luminosity function (LF) obtained via star counts, the mass estimate derived neglecting unresolved binaries would be smaller than the actual mass. This is straightforward to show, since the mass of a binary system is larger than the mass of a single star at the same magnitude due to the strong mass dependence of the stars' luminosity (approximately (L/L_⊙) ∼ (M/M_⊙)⁴ for the MS stars, see Figure 7 on page 209 in Carroll & Ostlie 2014).

If a single star and a binary system have the same magnitude, their luminosities are also equal L_s = L₁ + L₂, where suffix s marks the single star, while 1 and 2 are the primary and secondary, respectively. Therefore, ${M}_{s}^{4}={M}_{1}^{4}+{M}_{2}^{4}$. Instead, ${({M}_{1}+{M}_{2})}^{4}\,=\,{M}_{1}^{4}\,+\,{M}_{2}^{4}\,$+ $4{M}_{1}{M}_{2}^{3}\,+\,6{M}_{1}^{2}{M}_{2}^{2}\,+\,4{M}_{1}^{3}{M}_{2}\,$ $={M}_{s}^{4}+4{M}_{1}{M}_{2}^{3}+6{M}_{1}^{2}{M}_{2}^{2}+4{M}_{1}^{3}{M}_{2}$. Since all terms are positive, ${({M}_{1}+{M}_{2})}^{4}\gt {M}_{s}^{4}$, and ${M}_{1}+{M}_{2}\gt {M}_{s}$.

For example, the presence of unresolved binaries was taken into account by Khalaj & Baumgardt (2013) to estimate the Praesepe cluster mass. Khalaj & Baumgardt (2013) found a binary fraction of 35% ± 5% in Preasepe and used a correction (multiplicative) factor of 1.35 for the cluster mass estimate. Following them, the same correction was applied by Seleznev (2016a) to estimate of NGC 1502 stellar mass. Unfortunately, Khalaj & Baumgardt (2013) provided little information on how they obtained the multiplicative correction factor 1.35, which leaves room for further investigation.

To amend this, in this work, we present a novel approach and estimate the mass of five open clusters of different ages and metallicities, starting from their LF. In this case, one can provide an independent estimate of this correction factor and assess its dependence both on binary fraction α and on q distribution.

The layout of the paper is as follows. Section 2 is devoted to the description of our approach and the associated algorithms. Section 3 contains our results for NGC 1912, NGC 2099, NGC 6834, NGC 7142, and IC 2714. Section 4 is dedicated to a summary of our results, provides the paper conclusions, and discusses some future perspectives.

Two ingredients are needed to derive the correction factor for applying to the photometric mass because of the presence of unresolved binaries. The first one is the binary fraction. In this study, we adopt a binary fraction independent on magnitude. A larger binary fraction for brighter stars would not increase cluster mass significantly since bright, massive stars typically only number a few. We consider a binary fraction in the range of 10%–90%. The second one is the mass ratio q distribution. We explore four different distribution functions for q:

• 1.  
  A δ function with q = 1.
• 2.  
  A flat distribution function.
• 3.  
  A Gaussian distribution (2) as in Kouwenhoven et al. (2009).
• 4.  
  A Gaussian distribution (2) with mode shifted to q = 1 to reproduce Fisher et al. (2005), Maxted et al. (2008), and Raghavan et al. (2010) function.

The last distribution was taken with μ_q = 0.60 and ${\sigma }_{q}^{2}=0.42$, the latter is the same as in Kouwenhoven et al. (2009).

Kouwenhoven et al. (2009) summarized different methods of assignment of the mass values to the binary components, "pairing" methods, as they called them. Our task is different because at the beginning we have the stellar magnitude of the binary and require the mass of each component. The procedure described below could then be called "luminosity-limited pairing", following the terminology of Kouwenhoven et al. (2009).

We use a quadratic mass–luminosity relation following Eker et al. (2015):

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}L & = & -(0.705\pm 0.041){\left(\mathrm{log}M\right)}^{2}\\ & & +\ (4.655\pm 0.042)(\mathrm{log}M)-(0.025\pm 0.010)\end{array}\end{eqnarray} \tag{ 3 }$$

where L is the luminosity, and M is the stellar mass (in solar units). Relation (3) refers to single MS stars; consequently, we assume that all unresolved binaries have their components at the MS and have not experienced mass transfer. It is reasonable for stars below the MS turn-off. Then, only NGC 7142 with an age logarithm of 9.2 (see Table 1 below) could contain a detectable number of binary stars after this stage of evolution. Nevertheless, even for stars above the turn-off in NGC 7142, we could find only a few binary stars after the mass transfer. There could probably be five blue stragglers (see Figures 3 and 4 in Straižys et al. 2014) and one to two evolved (yellow) stragglers among the upper part of CMD.

We use the cluster LF φ(J) to count the number of stars in different magnitude intervals. The LFs are evaluated statistically with the use of 2MASS database (Skrutskie et al. 2006); that is, we obtain the LFs for the cluster region ("cluster plus field") and an equal area nearby reference field ("field") and get the cluster LF as the difference between "cluster plus field" and "field". With this approach, we do not take into account a possible difference in the mass function between the cluster center and outskirts. This procedure has been described in detail in Seleznev (1998), Seleznev et al. (2000), Prisinzano et al. (2001), Seleznev (2016b), and Seleznev et al. (2017). The magnitude distribution is binned in intervals ΔJ, and in each of them, we count number of stars and then derive the number of binaries, using a binary fraction α:

$$\begin{eqnarray}&&N={\int }_{J}^{J+{\rm{\Delta }}J}\varphi (J){dJ},\quad {N}_{b}=\alpha {\int }_{J}^{J+{\rm{\Delta }}J}\varphi (J){dJ}.\end{eqnarray} \tag{ 4 }$$

We round star numbers to integers and tune the number of intervals to obtain each bin occupied by at least one star. Figure 1 illustrates the process and shows the LF of NGC 7142 obtained as in Seleznev (2016b). For each magnitude bin, the mean magnitude is considered for further calculations.

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Stellar magnitudes are converted into luminosities with the use of the isochrone tables (Bressan et al. 2012) as follows. An isochrone corresponding to the cluster age is first selected. We took the cluster ages from Loktin & Popova (2017), but then refined them by comparing with isochrones (Bressan et al. 2012) the cluster CMD for the probable cluster members selected from Gaia DR2 (Gaia Collaboration et al. 2016, 2018), filtering by parallaxes and proper motions. Then, absolute magnitude is obtained from cluster distance modulus and color excess. The set of adopted cluster data is listed in Table 1. The photometric distances are then compared with Gaia DR2 distances, derived from parallaxes. We find that for distances closer than about 1.5 kpc, photometric and Gaia distances agree exceptionally well. Beyond this distance, the figures provided by GAIA tend to be significantly larger than the photometric ones. We tentatively impute such differences to the actual GAIA release, which is probably not very precise for large distances. Future releases will surely alleviate these differences.

Table 1.  Star Cluster Characteristics

Cluster	$\mathrm{log}t$	$(m-M{)}_{0}^{1}$ a	$E(B-V)$ a	dPHOT	dGAIA	Mmin	Mmax
	t in years	(mag)	(mag)	(pc)	(pc)	(M⊙)	(M⊙)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
IC 2714	8.6	10.48	0.34	1250	1390	0.73	2.82
NGC 1912	8.3	10.29	0.25	1140	1150	0.68	3.60
NGC 2099	8.7	10.74	0.30	1410	1510	0.76	2.77
NGC 6834	7.9	11.59	0.71	2080	3570	1.07	5.12
NGC 7142	9.2	11.25	0.39	1780	2600	0.87	1.80

Note.

^aLoktin & Popova (2017).

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Then, the star mass and the luminosity value are extracted from the isochrone table corresponding to each cluster age. Table 1 contains the limits in the star masses covered by our LFs: column 7 contains the minimum mass (it corresponds to J = 16 mag with exception to NGC 6834, where the minimum mass corresponds to J = 15.9 mag; these magnitudes, in turn, correspond to the completeness limit of the 2MASS data) and column 8 contains the maximum mass. Stars with masses close to the upper mass limit have been evolved from the MS. Due to this reason, we use another isochrone table with an age of 4 · 10⁷ yr to determine the luminosity of the evolved stars at the MS stage with the same mass as evolved star mass. For each binary, the following system of equations holds:

$$\begin{eqnarray}&&\left\{\begin{array}{l}L={L}_{1}+{L}_{2}\\ \mathrm{log}{L}_{1}=-0.705{\left(\mathrm{log}{M}_{1}\right)}^{2}+4.655(\mathrm{log}{M}_{1})-0.025\\ \mathrm{log}{L}_{2}=-0.705{\left(\mathrm{log}{M}_{2}\right)}^{2}+4.655(\mathrm{log}{M}_{2})-0.025\\ q={M}_{2}/{M}_{1}\end{array}\right.\end{eqnarray} \tag{ 5 }$$

where L is luminosity of binary star, L₁ and L₂ are luminosities of the binary components, and M₁ and M₂ are masses of the primary and secondary components of the binary star, respectively. For each binary star, we extract mass ratio q from the component mass ratio distribution from Monte-Carlo simulations.

Let be $x=\mathrm{log}{M}_{1}$, a = −0.705, b = 4.655, and c = −0.025. After some algebra, the luminosity reads:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}L & = & \mathrm{ln}10\cdot ({{ax}}^{2}+{bx}+c)\\ & & +\ \mathrm{ln}\left(1+{e}^{\mathrm{ln}10\cdot \left(a{\left(\mathrm{log}q\right)}^{2}+\mathrm{log}q(b+2{ax})\right)}\right).\end{array}\end{eqnarray} \tag{ 6 }$$

The goal is to define x, so that we build up a function f(x), which is equal to zero when a solution to the system (5) is found:

$$\begin{eqnarray}\begin{array}{rcl}f(x) & = & \mathrm{ln}10\cdot ({{ax}}^{2}+{bx}+c)\\ & & +\ \mathrm{ln}\left(1+{e}^{\mathrm{ln}10\cdot \left(a{\left(\mathrm{log}q\right)}^{2}+\mathrm{log}q(b+2{ax})\right)}\right)-\mathrm{ln}L.\end{array}\end{eqnarray} \tag{ 7 }$$

To solve this equation, we use the Newton–Raphson method as

$$\begin{eqnarray}&&{x}_{k+1}={x}_{k}-f({x}_{k})/f^{\prime} ({x}_{k}),\quad \mathrm{where}\quad f^{\prime} (x)={df}(x)/{dx}\end{eqnarray} \tag{ 8 }$$

until the difference $| {x}_{k+1}-{x}_{k}|$ reaches the requested accuracy.

The Newton–Raphson method converges only if certain conditions are met. First, one needs to choose initial trial values, which are not too far from the root. Therefore, we build a for-loop with intervals of the mass [0.08; 10] M_⊙, (or $x\in [-1.097,1]$) with a small increase. The loop ends when we find those x_i that give $f({x}_{i})\cdot f({x}_{i+1})\lt 0;$ this implies that the root is in the interval $x\in [{x}_{i},{x}_{i+1}]$. We then consider x_i as a starting point for iteration. Second, the function f(x) should be smooth in its domain; this is easy to prove, as f(x) is a combination of smooth functions.

The final x_k will be the solution of the equation and, in turn, M₁ = 10^x the value for the primary component mass from the system (5). Hence, we can define the secondary component mass M₂ from the fourth equation of the system (5), and, finally, the total mass of the binary star M₁ + M₂. The described procedure is repeated for all N_b stars to eventually derive the total mass of binaries in the interval $J\in [J;J+{\rm{\Delta }}J]$. When extended to all magnitude bins, the procedure yields the total mass of the cluster binaries M_b in these bins.

Finally, to define the mass of the cluster, we need to find the mass stored in single stars M_s (whose number is ${N}_{s}=N-{N}_{b}$ in each magnitude interval). For these stars, we use an isochrone table, where we determine the mass according to the magnitude and the cluster parameters from Table 1 (see a description of the procedure above). As a result, we obtain the cluster mass $M={M}_{b}+{M}_{s}$ in the considered magnitude interval.

Let us now define M_wob as the cluster mass obtained assuming that all stars are single. Then the ratio M/M_wob would naturally give the cluster mass increment due to unresolved binaries.

In this work, we start from the LF of five open clusters: IC 2714, NGC 1912, NGC 2099, NGC 6834, and NGC 7142 obtained by star counts with 2MASS as described above.

For each cluster, we repeated the procedure described in the previous section up to 30 times both for cluster LF and for boundaries of the LF confidence interval. This procedure allowed us to evaluate the scatter of the mass increment factors. We explored the whole parameter space made of binary fraction α and mass ratio q distribution to quantify the spread in the estimates of the cluster mass when unresolved binaries are taken into account.

We considered two cases of equal mass components. The first case is when we take into account binary systems only. In the second case, we also take into account the multiple (triple and quadruple) systems following Tokovinin (2014), who found for systems with multiplicity of 1:2:3:4:5 ("1" means single star) the relative abundance ratio of 54:33:8:4:1. It is worthwhile because at distances of ∼1 kpc a hierarchical triple of separation ∼100 au has an angular separation of about 0.1 arcsec, then a triple system or a "binary of binaries" could be missed, just like tight unresolved binaries.

Figure 2 shows the dependence of the cluster mass increment on the binary fraction for the five clusters. Each panel corresponds to a cluster, and different colors are used to indicate the various q distributions. At first glance, one can easily see that the equal mass component model significantly deviates from the other models, which do not appear much different.

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Khalaj & Baumgardt (2013) found the cluster mass increment value of 1.35 for a binary fraction of 0.35. According to our study, the increment value should be between 1.10 and 1.15 for realistic q distribution (see Figure 2). However, taking into account the possible presence of the multiple (triple and quadruple) systems in the cluster would increase the value of the increment on the average 1.32 times for the case of equal components. Then the value of 1.35 for the cluster mass increment found by Khalaj & Baumgardt (2013) for the Praesepe cluster is reasonable. We fitted the dependencies of the increment on the binary fraction via linear regression and provide fitting formulae in Table 2. The columns of Table 2 are: the binary components mass ratio model, the cluster, the coefficients A and B of the linear regression y = A + Bα (where y is the cluster mass increment, and α is the binary fraction), the χ² of the fit, and the goodness-of-fit probability Q (Press et al. 1992). Coefficient A does not differ significantly from the unity in virtually all cases. The coefficients B for the clusters lie within the limits of the q distribution model (except for NGC 7142, the oldest one). This fact demonstrates that the shape of the LF does not affect the dependence of the cluster mass increment on the binary fraction α significantly.

The LFs used in the present work are limited in magnitude because of the completeness limit of 2MASS. Therefore, we miss stars with masses lower than the limit listed in the seventh column of Table 1. How can the missing low-mass stars affect our results? We consider the binary fraction α independent of the stellar magnitude. In such a case, the cluster mass increment should be independent of the magnitude (and the mass) limit. In order to make this suggestion more solid, we performed the following experiment. For NGC 2099, we calculated the mass increment for a set of limiting magnitudes J = 14, 15, 16 mag in the case of flat q distribution. It turned out that the mass increment slightly increases with the limiting magnitude. For instance, for α = 0.8 y = 1.322 ± 0.013 for J_lim = 14 mag, y = 1.328 ± 0.009 for J_lim = 15 mag, and y = 1.334 ± 0.006 for J_lim = 16 mag. If the binary fraction increases with the stellar magnitude, the cluster mass increment would most probably increase with the stellar magnitude. If the binary fraction decreases with the stellar magnitude, we would expect the cluster mass increment to be independent of the stellar magnitude or even to decrease with the stellar magnitude.

In any case, we underline that even applying the mass increment, one would not obtain the total mass of the cluster but only slightly improve a lower limit estimate of it.

In this work, we attempt to quantify the increase of the cluster mass estimate—obtained by star counts—produced by the presence of unresolved binaries. The results are illustrated in Figure 2 and summarized in Table 2.

The most relevant results of this study are

• 1.  
  The dependence of the cluster mass increment on binary fraction is linear in most cases.
• 2.  
  The dependence of the cluster mass increment on the binary fraction α does not vary significantly for the realistic q distributions considered here. We checked three realistic distributions: a Gaussian distribution (2) with μ_q = 0.23, a flat distribution, and a Gaussian distribution (2) with μ_q = 0.60. An inspection of Figure 2 and Table 2 shows that the closer the distribution mode to unity, the higher the expected cluster mass increment.
• 3.  
  The dependence of the cluster mass increment on the binary fraction α within the limits of a specific q distribution model does not differ substantially among the selected clusters (except for NGC 7142, the oldest one). Then we can safely conclude that the form of the LF does not affect this dependence considerably.
• 4.  
  For the particular case of a binary fraction α = 0.35, the cluster mass increment is confined between 1.10 and 1.15 (for realistic q distributions, see Figure 2). However, taking into account the possible presence of the multiple (triple and quadruple) systems in the cluster would increase the value of the increment (in the mean 1.32 times for the case of equal components). Then the value of 1.35 for the cluster mass increment for the Praesepe cluster obtained by Khalaj & Baumgardt (2013) is reasonable.

Our results will help to improve the estimate of the mass of clusters containing unresolved binary stars in the broad range of the binary ratios α and with different assumptions on the distribution of the binary component mass ratio q.

The work of A.F.S. and V.M.D. was partly supported by the Ministry of Education and Science (the basic part of the State assignment, RK No. AAAA-A17-117030310283-7). The work of A.F.S. and V.M.D. was supported also by the Act No. 211 of the Government of the Russian Federation, agreement No. 02.A03.21.0006.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.