The Frequency and Mass-Ratio Distribution of Binaries in Clusters II: radial segregation in the nearby dissolving open clusters Hyades and Praesepe

We have determined the mass functions, mass-ratio distribution functions and fractions of binary stars with mass ratios above particular thresholds for radially-separated populations of stars in the nearby open clusters Hyades and Praesepe. Radial mass segregation is detected, with the populations of stars within the tidal radii having much flatter mass functions than those outside the tidal radii. Within the tidal radii, the frequency of binary stars with mass ratio q>0.5 is 50 - 75 per cent higher for Hyades and 5 - 30 per cent higher for Praesepe. We also, for the first time, detect mass-ratio radial segregation. Of the binaries for which q>0.5, ~ 80 per cent of the inner Hyades population also have q>0.75, while for the extra-tidal population, the ratio is ~ 50 per cent. For Praesepe, ~ 67 per cent of the inner sample have q>0.75, and 35 - 45 per cent of the outer sample


INTRODUCTION
Open star clusters are important targets for studying how stellar populations form and disperse in the Galaxy.Most field stars are thought to form in clusters or looser associations, and we ought to be able to reconcile the properties of open cluster stellar populations with those of the Galactic disk.
Properties of binary star systems offer powerful constraints on star formation and early cluster evolution (Kraus & Hillenbrand 2012).Binary stars in dense clusters provide a heat reservoir, slowing or preventing runaway core collapse through close gravitational encounters with single stars that (after possible exchange of partners) effectively provide kinetic energy to the single star through an increase in negative binding energy (hardening) of the binary (Elson et al. 1987;Hut et al. 1992;Hurley et al. 2005).Binaries can also form in cluster cores through 2-body capture interactions.In the less-dense environments of open clusters, these effects will be less significant, but may still play an important part in controlling the mass distribution of ejected (evaporated) stars.In turn, we would like to know whether binaries of different mass ratios are subject to different rates of disruption and capture-formation, and more-generally how the dynamical processing in cluster cores affects the mass-ratio distribution of binaries.
Binary stars have been observed to be more centrally concentrated than single stars in globular (Albrow et al. 2001) and open clusters (Childs et al. 2023), and generally this is in accord with expectations from N-body simulations (Geller et al. 2013).
Previous observational studies of individual clusters have found binary mass-ratio distribution functions that are flat (Torres et al. 2021) ★ E-mail:Michael.Albrow@canterbury.ac.nz or gently rising (Reggiani & Meyer 2013) or falling (Patience et al. 2002).The recent large open-cluster study of Cordoni et al. (2023) found that while the mean behaviour of the mass ratio distribution for  > 0.6 is fairly flat, individual clusters can display distribution functions that either rise or fall.In contrast, Malofeeva et al. (2023), finds a strongly decreasing distribution with  for Pleiades, Alpha Per, Praesepe, and NGC 1039.We note that different authors have used quite different methods for these analyses.Theoretical models may not reproduce the observed distributions -e.g. the N-body model of Geller et al. (2013), while successful in reproducing many properties of NGC 188, fails to explain the observed -distribution for main-sequence stars.
There is some discrepancy over the binary mass-ratio distribution for field stars.From radial velocity studies, Duquennoy & Mayor (1991) found a distribution that peaks around q = 0.25 and declines towards higher mass ratios, while Fisher et al. (2005) and Raghavan et al. (2010) find a flat distribution with a sharp upturn towards higher mass ratios,  ≳ 0.8.El-Badry et al. (2019) confirmed this high- peak, but showed that it only exists for closely-separated binaries,  ≲ 600 au.Bate (2009) and Moeckel & Bate (2010) provide simulations of an open cluster that results from the collapse of a 500  ⊙ molecular cloud.The resulting cluster contains 1253 stars (191  ⊙ in stars) that initially (0.3 Myr) have a binary mass-ratio distribution that rises slowly towards  = 1.After 10 Myr of dynamical processing, the distribution for binaries with primary mass > 0.5  ⊙ has flattened a little, but remains unchanged for systems with lower-mass primaries.
In contrast, recent simulations by Guszejnov et al. (2023) of cluster formation from a 2×10 4  ⊙ primordial cloud produce a binary mass ratio distribution that, after ∼ 10 Myr, is strongly peaked at  ≈ 0.2, Praesepe, located at a distance of 185 pc, is a little more tightly bound than Hyades, with a core radius of 1.6 pc (Gao 2019) which projects to 0.5 deg, and a half-mass radius of 4.8 pc (Röser et al. 2019).It also is associated with vast tidal tails that stretch across tens of degrees on the sky (Röser et al. 2019).
In Paper 1 of this series (Albrow & Ulusele 2022), we introduced a new probablistic method to measure the binary frequency and massratio distribution function for stellar populations.In this paper, we apply the method to radially segregated samples of stars from Hyades and Praesepe.

DATA
All of the photometric data used in this paper comes from Gaia data release 3 (DR3 or EDR3) (Riello et al. 2021).For Hyades, we adopted the selection of stars from the Gaia Catalog of Nearby Stars (GCNS) (Gaia Collaboration et al. 2021), a compilation of Gaia sources within 100 pc from the Sun.Stars belonging to Hyades were selected in the GCNS based on their distances, proper motions and radial velocities, resulting in 920 member stars.
To compute distances, and hence absolute magnitudes we use the simple  (pc) = 1000/(mas) relation without priors.The median distances of the cluster core members are 47.4 pc (Hyades) and 184.8 pc (Praesepe).The distribution of cluster member stars across the sky is shown in Fig. 1.

AGE AND METALLICITY
The age of these clusters has been determined from CMD fitting by various authors.These estimates tend to depend on the adopted metallicity and particularly the adopted stellar rotation velocity and degree of convective core overshoot, with a higher rotation or overshoot allowing stars to turn off from the main sequence at a higher luminosity for a given age.
Praesepe is believed to be co-eval with the Hyades and share the same metallicity (Brandt & Huang 2015;Gossage et al. 2018).
For our modelling (described below), we adopt isochrones from the MESA Isochrones and Stellar Tracks project1 (Dotter 2016;Choi et al. 2016;Paxton et al. 2011Paxton et al. , 2013Paxton et al. , 2015)).From our own fitting of these isochrones to the main sequence and turnoff regions of the CMD's, we have adopted an age of 630 Myr and metallicity of [Fe/H] = 0.25 for both clusters.The models in our analysis to follow, which only use the isochrones as a tracer of main sequence mass as a function of   , are almost invariant to the adopted age and metallicity since we restrict ourselves to stars well below the MS turnoff.It is known (Brandner et al. 2023) that the MESA isochrones provide a poor fit (ie.are under-luminous) for single stars with masses, 0.25 ≲  ≲ 0.85 M ⊙ , so we adjust the   magnitude of the isochrones to match the MS ridge line.
We do an initial coarse filtering of obvious outlying data points blue-ward of the MS ridge line and red-ward of the equal-mass binary main sequence, and truncate using the binary mass-ratio track at the top and bottom of the main sequence.To avoid confusion between binary stars and single stars leaving the main sequence, we make an upper-MS cut to our data at   = 3.We make the lower cut at   = 9, where the CMD main sequences are still very narrow and obvious.
To investigate possible segregation effects within each cluster, we divide each set of stars into samples located inside and outside of a sphere with radius equal to the tidal radius.For Hyades, this is estimated as   = 10 pc (Perryman et al. 1998), and for Praesepe we adopt   = 10.8 pc (Röser et al. 2019).At the median distances of the clusters, these project as 12 deg and 3.3 deg respectively.Colourmagnitude diagrams for the final selections of stars for each cluster and region are shown in Fig. 2.

MODEL
Our probablistic generative mixture model for the CMD is described in Albrow & Ulusele (2022).The data for each cluster is the set  = {  } for stars , occupying locations   = ( −  ,   )   on the CMD with uncertainty covariances S  .We employ a scaling for each data covariance, S  → ℎ 2 S  , where ℎ = ℎ 0 + ℎ 1 ( , −  ,0 ), with  ,0 set as the central magnitude of the data.
Briefly, the likelihood is described by These are used to compute the likelihoods for each   given its (covariant) uncertainty, and ultimately the total likelihood for a given .
The model has undergone several modifications or extensions since its initial construction that we detail below.

Mass function
The power-law form adopted for the mass function in Albrow & Ulusele (2022) was found to be inadequate for some clusters.This may be because their stellar populations have undergone significant dynamical shaping since their formation, for instance from preferential ejection of lower-mass stars.We now adopt a more general form, ( |, ,  0 ,  0 ,  1 ) i.e. a power law plus a linear function in  that is normalised by the scale factor   = (max( min ,  0 )) − .The final two terms truncate the function at (fixed) lower and upper bounds,  min and  max , and the tanh term allows the power law to roll-over at masses  ≲  0 .
A value of  0 ≪  min allows a sharp cut to the power law with no roll-over.

Mass-ratio distribution function
In Albrow & Ulusele (2022), we adopted a polynomial form for the binary-star mass-ratio distribution function.We have explored using several alternative forms, including single and two-sided power laws, quadratics, and piecewise linear.We note that other papers, e.g.Li et al. (2013Li et al. ( , 2020)); Li & Shao (2022), have adopted strict power laws for the mass-ratio distribution.Generally we have found that it is difficult to set non-informative priors on these analytic parametric models, and that the forms of the analytic models impose undesirable constraints on the distribution.
We have finally adopted two different models for the distribution function.Our first model is linear sum of six shifted Legendre basis functions (five parameters).The inclusion of more components compared with Albrow & Ulusele (2022) allows more flexibility, for instance the ability to curve upwards or downwards at either end of the distribution without affecting the other end.
Our second model is a histogram with   equal-width bins for the distribution, i.e.

Binary fraction
We have introduced a new parameter,  B , that allows the binary star fraction to vary with mass along the main sequence by  B () =  B,0 +  B ( max − ).
We emphasise that  B is a consequence of parameters of the model, and should not be interpreted literally.Since binaries with  ≲ 0.4 barely deviate from the main sequence, such stars can contribute to either or both of the single or binary-star likelihood functions, depending on the adopted model basis.We regard  = 0.5 as a safe lower limit for , beyond which binaries are well-distinguished from single stars in the Gaia photometric data sets in this paper.

Priors
The complete model has up to ten free parameters due to the mass function, binary and outlier fractions, and data error scaling,  = (log 10 ,  0 , ,  0 ,  1 ,  B ,  B ,  O , log ℎ 0 , ℎ 1 ), plus either six or   parameters for the mass-ratio distribution function.
Following Bayes theorem, the probability distribution for , where (|) is the likelihood function, () is the prior for  and  is the evidence (marginal likelihood), a constant for a given  and model.
To compute the posterior probability distribution for , we must define a prior distribution, ().We have adopted sensible uniform priors for most parameters, however setting the priors for the massratio distribution function parameters requires some thought.
As explained in 4.2, we now include six coefficients,  0 −  5 for the shifted-legendre polynomial model.The first of these,  0 = 1, due to the normalisation condition.We draw each of the remaining coefficients,  1 −  5 , from normal distributions,   ∼ N (0, ). is treated as a hyperparameter, which we draw from a reasonablyuninformative gamma distribution distribution, Γ(2, 3).This model strikes a natural balance between smaller values of  (which result in samples from N (0, ) that have a higher prior probability) and higher values of  that may be favoured by the likelihood.It thus chooses the smoothest and flattest functional forms that are consistent with the data, For the histogram distribution, we would like to give equal prior weight to all histogram bins within the constraint, Naively we might assign some prior permissible range to the first chosen bin, 0 ≤  1 <  max , with the range for the second bin then being 0 ≤  2 <   −  1 etc.However, this procedure results in a selection-order bias, with a declining prior range for each subsequent bin.
The correct equal-weighted prior for  = {  } is the Dirichlet distribution, Dir( ) (see for instance Leja et al. (2017)).This is often described as the prior for each sub-length of a fixed-length string cut into multiple pieces.If   samples,   , are drawn from the Beta distribution, then are Dirichlet-distributed.
For our nested sampling procedure, we require the prior transform, which is the function that maps the range [0, 1] onto the prior.This function is the inverse of the cumulative distribution, This inverse is also known as the quantile function or pixel percent function, defined as (   ) such that CDF() =   .By direct integration and normalisation, we find that After some algebra, this implies that . (10)

RESULTS
We have used the dynamical nested sampler code DYNESTY (Higson et al. 2019;Speagle 2020) to sample the posterior parameter distribution for each cluster.The code has the ability to treat particular parameters, or combinations of parameters as constant, rather than free.As well as allowing all parameters to vary, we have made runs with combinations of ((log ,  0 ),  0 ,  1 ,  ,1 , log 10 ℎ 0 , ℎ 1 ) frozen to constants (log ,  0 ) = (4.1,0.0) (an abrupt rather than tanh cutoff for the bottom of the mass function);  0 = 0;  1 = 0;  ,1 = 0; log 10 ℎ 0 = 0; and ℎ 1 = 0. We have also tested different values for   in the range 4 to 10.Since all the different runs have different numbers of free parameters, we use the ratios of the evidence, , (Bayes factors) to decide which model or models are best.We follow the Kass & Raftery (1995) interpretation that Δ log 10  > (0.5, 1, 2) represents (substantial, strong, decisive) evidence for a proposition.
We find in all cases that allowing scaled data uncertainties log 10 ℎ is necessary.Allowing ℎ to vary along the main sequence (i.e. using the ℎ 1 parameter) is only necessary for the Hyades.Allowing   to vary along the main sequence is very mildly disfavoured in all cases.A turnover in the mass function at the low-mass end (parameters log 10  and  0 ) was found to be unnecessary since we made sharp cuts at   = 9.The evidence disfavoured using  0 and  1 for these data, so we also discard these parameters and use a pure power-law for the mass function.
In Figs. 3 and 4 we show the mass-functions and mass-ratio distribution functions.For both clusters, the mass functions for the samples of stars outside of the tidal radius are consistent with the canonical mass-functions of Salpeter (1955) and Chabrier (2003) (which is very similar to Kroupa (2001) in this mass range).Inside the tidal radii, the mass functions are much flatter.
We show results for the mass-ratio distribution functions using the Legendre-function model and also as histograms with   = 10 bins.In order to be unaffected by different amounts of blending in the models between single and binary stars close to the main sequence, we renormalise these functions over 0.4 <  <= 1.0.Our final models contain between 9 and 15 parameters, which is higher than the quantity of information we actually extract for interpretative purposes.Effectively we are marginalising (integrating) over some dimensions of the parameter space.We choose to show the results based on these different model basis functions to qualitatively illustrate the uncertainty in the distributions, and thus guard against over-interpretation of the fine details.
Assuming a power-law mass function, then for a given primary  star, if the secondary star is randomly selected from the mass function, the mass-ratio distribution will follow the same functional form as the mass function.This form (implied by the particular mass function for each sample) is also shown in Fig. 4. Additionally, we show the implied -distribution for randomly sampled stars from the Chabrier (2003) single-star galactic-disk mass function, that we obtained through Monte Carlo sampling.The samples of stars from the outer regions of each cluster are somewhat consistent with a randomly-chosen-secondary hypothesis.In contrast, the samples of stars from within the tidal radius have a much higher frequency of high-mass-ratio binaries.We note that there is a certain amount of degeneracy for mass ratios  > 0.75, as all such stars tend to lie along the equal-mass binary main sequence (see fig. 3 in Albrow & Ulusele (2022)).As a simple way of representing the tendency (or not) of the massratio distributions to rise towards higher  (that integrates over the degeneracy), we introduce a metric,  75 , that we define as the ratio of the number of cluster binaries for which  > 0.75 to the number for which  > 0.5.Values of  75 are listed in Table 1.
In Fig. 5 we show  75 plotted against the overall fraction of observed cluster stars that are binaries with  > 0.5.There is a loose relationship, where the populations of stars with higher binary frequencies also have higher fractions of binaries that have  > 0.75.Referring to Table 1,  75 is higher for all populations than implied by random sampling of their present-day single-star mass functions, and in most cases higher than implied by random sampling of the Chabrier mass function (for which  75 = 0.35).
In Fig. 6 we show the total fractions of observed cluster stars that are binaries with  greater than a given value, based on the Legendrefunction models.These fractions are obtained by integrating the distributions in Fig. 4 backwards from  = 1.Numerical values of these functions for particular  are given in Table 1 for both the Legendre and histogram bases.The numbers are robust and and more-or-less independent of the choice of model basis and the other choices of including or excluding various parameters from the model.
In Appendix A, Figs.A1 -A4 each show seven random realisations of the CMD for each cluster.These realisations have been generated by taking (1 −   ) random samples of the posterior maximum model, where  is the number of stars in the final observed CMD.They are shown next to the final observed CMDs for visual comparison.In all cases, the model CMD's give an excellent representation of the observations.These simulations show that any apparent gaps in the observed distributions of binary stars in the CMD's are likely stochastic not physical.
Table 1.Fraction of binary stars mass ratio greater than a given  for each cluster.The final rows,  75 are the measured ratio of the number of binaries for which  > 0.75 to the number for which  > 0.5, the single-star mass-function power law index, , and the value of  75 implied by random sampling of the mass function.

Figure 2 .
Figure 2. Colour-magnitude diagrams for the inner and outer regions of each cluster.The red lines are the magnitude-corrected MESA isochrones.

Figure 3 .
Figure 3. Mass function for the inner and outer regions of each cluster.The blue lines are random samples from the posterior distribution, and the red line indicates the posterior median probability model.Black lines are canonical mass functions from Salpeter (1955) (dash-dot) and Chabrier (2003) (dash).

Figure 4 .
Figure 4. Mass-ratio functions for the inner and outer binary-star populations of the two clusters (columns).The top row is the Legendre-function model, modified from Albrow & Ulusele (2022).The lower row shows the histogram models for   = 10 bins.As in Fig. 3, the blue lines are random samples from the posterior distribution, and the red line is the a-posteriori maximum probability model.Orange and yellow lines show respectively the implied -distributions for randomly sampled stars from the measured mass function and the Chabrier (2003) single-star galactic-disk mass function.The first column shows the prior distribution.

Figure 5 .
Figure 5. Ratio of binaries with  > 0.75 to the number for which  > 0.5 plotted against overall frequency of binaries with  > 0.5 for Hyades (red) and Praesepe (blue).Samples within the tidal radius are plotted with solid lines and those outside the tidal radius as dashed lines.The left plot has been computed from the legendre model for the mass-ratio distribution function, and the right plot has used the histogram model.

Figure 6 .
Figure 6.Implied fraction of binary stars with mass ratio  ′ >  for each cluster (red) together with its 1- and 2- uncertainties from the Legendrefunction model.

Figure A1 .
Figure A1.The top left panel shows final CMD for the Hyades inside a 10.0-parsec radius, with the colour-adjusted isochrone in green.The remaining seven panels show random realisations of the posterior maximum model.

Figure A2 .
Figure A2.The top left panel shows the final CMD for Hyades outside a 10.0-parsec radius with the colour-adjusted isochrone in green.The remaining seven panels show random realisations of the posterior maximum model.

Figure A3 .
Figure A3.The top left panel shows final CMD for Praesepe inside a 10.8-parsec radius with the colour-adjusted isochrone in green.The remaining seven panels show random realisations of the posterior maximum model.

Figure A4 .
Figure A4.The top left panel shows the final CMD for Praesepe outside a 10.8-parsec radius with the colour-adjusted isochrone in green.The remaining seven panels show random realisations of the posterior maximum model.