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

Michael D. Albrow

1School of Physical and Chemical Sciences, University of Canterbury, Private
Bag 4800, Christchurch, New Zealand
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

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 $q>0.5$, $\sim$80 per cent of the inner
Hyades population also have $q>0.75$, while for the extra-tidal population,
the ratio is $\sim$50 per cent. For Praesepe, $\sim$67 per cent of the inner
sample have $q>0.75$, and $35-45$ per cent of the outer sample.

###### keywords:

stars : binaries – open clusters and associations – methods : statistical –
methods : data analysis

††pubyear: 2024††pagerange: The Frequency and Mass-Ratio Distribution of
Binaries in Clusters II: radial segregation in the nearby dissolving open
clusters Hyades and Praesepe. –A

##  1 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) 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 $q>0.6$ 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 $q$ 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 $q$-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, $q\gtrsim 0.8$.
El-Badry et al. (2019) confirmed this high-$q$ peak, but showed that it only
exists for closely-separated binaries, $s\lesssim 600$ au.

Bate (2009) and Moeckel & Bate (2010) provide simulations of an open cluster
that results from the collapse of a 500 $M_{\odot}$ molecular cloud. The
resulting cluster contains 1253 stars (191 $M_{\odot}$ in stars) that
initially (0.3 Myr) have a binary mass-ratio distribution that rises slowly
towards $q=1$. After 10 Myr of dynamical processing, the distribution for
binaries with primary mass > 0.5 $M_{\odot}$ 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 $2\times 10^{4}\,M_{\odot}$ primordial cloud produce a binary
mass ratio distribution that, after $\sim$ 10 Myr, is strongly peaked at
$q\approx 0.2$, consistent with random sampling from the IMF. The distribution
is almost flat for $q>0.4$, but has a deficit of equal-mass binaries.

In this paper, we investigate the nearby coeval open clusters, Hyades (Mel 25,
Col 50) and Praesepe (M 44, Mel 88, NGC 2632). At a mean distance of 47 pc,
Hyades is the closest open star cluster to Earth. It has a core radius of 2.7
pc (Perryman et al., 1998), mapping to an angular size of $\sim 4$ deg, and a
half-mass radius of 5.75 pc, Evans & Oh (2022)). It also contains member stars
that are dispersed across some 90 deg of the sky (Meingast & Alves, 2019;
Jerabkova et al., 2021). Oh & Evans (2020) have found that Hyades is rapidly
disintegrating, estimating a current mass loss rate of at least $0.26\,{\rm
M_{\odot}\,Myr^{-1}}$, with a further 30 Myr until final dissolution. The
estimated initial mass of the cluster $\gtrsim 750\,{\rm M_{\odot}}$.

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 mass-ratio
distribution function for stellar populations. In this paper, we apply the
method to radially segregated samples of stars from Hyades and Praesepe.

Figure 1: Aitoff projection, in galactic coordinates, of the Hyades (blue) and
Praesepe (red) cluster members.

##  2 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.

For Praesepe, we adopt the selection of Röser et al. (2019) from Gaia DR2
(1394 stars). These were cross-matched to DR3 using the table
gaiadr3.dr2.neighbourhood provided by Gaia.

To compute distances, and hence absolute magnitudes we use the simple $d(\rm
pc)=1000/\varpi({\rm mas})$ 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.

##  3 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.

For Hyades, estimated ages include $710$ Myr (Brandner et al., 2023), $650$
Myr (Lebreton et al., 2001), $800$ Myr (Brandt & Huang, 2015), and 600 - 800
Myr (Gossage et al., 2018). Lodieu (2020) have determined an age of $650\pm
70$ Myr spectroscopically, from lithium depletion in L dwarfs. Metallicity
estimates are in the range [Fe/H] $\approx$ 0.0 - 0.25 (Brandner et al.,
2023).

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 project111http://waps.cfa.harvard.edu/MIST/
(Dotter, 2016; Choi et al., 2016; Paxton et al., 2011; Paxton et al., 2013,
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 $M_{G}$, 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, $0.25\lesssim M\lesssim
0.85\,{\rm M}_{\odot}$, so we adjust the $M_{G}$ 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 $M_{G}=3$. We make the
lower cut at $M_{G}=9$, 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 $r_{t}=10$
pc (Perryman et al., 1998), and for Praesepe we adopt $r_{t}=10.8$ pc (Röser
et al., 2019). At the median distances of the clusters, these project as 12
deg and 3.3 deg respectively. Colour-magnitude diagrams for the final
selections of stars for each cluster and region are shown in Fig. 2.

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

##  4 Model

Our probablistic generative mixture model for the CMD is described in Albrow &
Ulusele (2022). The data for each cluster is the set
${\bm{D}}=\\{{\bm{D}_{k}}\\}$ for stars $k$, occupying locations
${\bm{D}_{k}}=(G-G_{RP},M_{G})_{k}^{T}$ on the CMD with uncertainty
covariances $\textbf{{S}}_{k}$. We employ a scaling for each data covariance,
$\textbf{{S}}_{k}\rightarrow h^{2}\textbf{{S}}_{k}$, where
$h=h_{0}+h_{1}(M_{G,k}\,-M_{G,0})$, with $M_{G,0}$ set as the central
magnitude of the data.

Briefly, the likelihood is described by

$P({\bm{D}_{k}}|{\bm{\theta}})=(1-f_{\rm B}-f_{\rm O})P_{\rm
S}({\bm{D}_{k}}|{\bm{\theta}})+f_{\rm B}P_{\rm
B}({\bm{D}_{k}}|{\bm{\theta}})+f_{\rm O}P_{\rm
O}({\bm{D}_{k}}|{\bm{\theta}}),$ (1)

where $P_{\rm S}({\bm{D}}|{\bm{\theta}}),P_{\rm
B}({\bm{D}}|{\bm{\theta}}),P_{\rm O}({\bm{D}}|{\bm{\theta}})$ are respectively
the likelihood functions for single stars, binaries and outliers
(contaminants), $f_{\rm B}$ and $f_{\rm O}$ are the fractions of binary stars
and outliers, and ${\bm{\theta}}$ is the vector of model parameters. The
likelihood function for single stars depends on a parameterised mass function.
For binary stars, the likelihood depends additionally on a parameterised mass-
ratio distribution function. The mass and mass-ratio distribution functions
are each represented as linear combinations of Gaussian basis functions,
which, in combination, map onto bivariate Gaussians on the CMD. These are used
to compute the likelihoods for each ${\bm{D}_{k}}$ given its (covariant)
uncertainty, and ultimately the total likelihood for a given ${\bm{\theta}}$.

The model has undergone several modifications or extensions since its initial
construction that we detail below.

###  4.1 Mass function

---
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).

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,

$\begin{split}\frac{dP(M|\gamma,k,M_{\rm
0},c_{0},c_{1})}{dM}=C_{M}\times\left([(c_{0}+c_{1}(M-M_{\rm
min})]s_{c}+M^{-\gamma}\right)\times\\\ \tanh(-k(M-M_{0}))\times H(M-M_{\rm
min})\times H(M_{\rm max}-M),\end{split}$ (2)

i.e. a power law plus a linear function in $M$ that is normalised by the scale
factor $s_{c}=\left(\max(M_{\rm min},M_{0})\right)^{-\gamma}$. The final two
terms truncate the function at (fixed) lower and upper bounds, $M_{\rm min}$
and $M_{\rm max}$, and the $\tanh$ term allows the power law to roll-over at
masses $M\lesssim M_{0}$. A value of $M_{0}\ll M_{\rm min}$ allows a sharp cut
to the power law with no roll-over.

###  4.2 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. (2013); Li 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 $N_{q}$ equal-width bins for the
distribution, i.e.

$\frac{dP(q|p_{1},...,p_{N_{q}})}{dq}=p_{i},\quad q_{i}<q\leq q_{i+1}.$ (3)

There are $N_{q}$ parameters, $p_{1}\,...\,p_{N_{q}}$, for the model, with a
normalisation constraint, $\Sigma_{i}p_{i}=N_{q}$.

###  4.3 Binary fraction

We have introduced a new parameter, $\dot{f}_{\rm B}$, that allows the binary
star fraction to vary with mass along the main sequence by $f_{\rm
B}(M)=f_{{\rm B},0}+\dot{f}_{\rm B}(M_{\rm max}-M)$.

We emphasise that $f_{\rm B}$ is a consequence of parameters of the model, and
should not be interpreted literally. Since binaries with $q\lesssim 0.4$
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 $q=0.5$ as a safe lower limit for $q$, beyond
which binaries are well-distinguished from single stars in the Gaia
photometric data sets in this paper.

###  4.4 Priors

Figure 4: Mass-ratio distribution 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 $N_{q}=10$ 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 $q$-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.

The complete model has up to ten free parameters due to the mass function,
binary and outlier fractions, and data error scaling,
${\bm{\theta}}=(\log_{10}k,M_{0},\gamma,c_{0},c_{1},f_{\rm B},\dot{f_{\rm
B}},f_{\rm O},\log h_{0},h_{1})$, plus either six or $N_{q}$ parameters for
the mass-ratio distribution function.

Following Bayes theorem, the probability distribution for ${\bm{\theta}}$,

$P({\bm{\theta}}|{\bm{D}})=\frac{P({\bm{D}}|{\bm{\theta}})P({\bm{\theta}})}{Z},$
(4)

where $P({\bm{D}}|{\bm{\theta}})$ is the likelihood function,
$P({\bm{\theta}})$ is the prior for ${\bm{\theta}}$ and $Z$ is the evidence
(marginal likelihood), a constant for a given ${\bm{D}}$ and model.

To compute the posterior probability distribution for ${\bm{\theta}}$, we must
define a prior distribution, $P({\bm{\theta}})$. We have adopted sensible
uniform priors for most parameters, however setting the priors for the mass-
ratio distribution function parameters requires some thought.

As explained in 4.2, we now include six coefficients, $a_{0}-a_{5}$ for the
shifted-legendre polynomial model. The first of these, $a_{0}=1$, due to the
normalisation condition. We draw each of the remaining coefficients,
$a_{1}-a_{5}$, from normal distributions, $a_{i}\sim\mathcal{N}(0,\sigma)$.
$\sigma$ is treated as a hyperparameter, which we draw from a reasonably-
uninformative gamma distribution distribution, $\Gamma(2,3)$. This model
strikes a natural balance between smaller values of $\sigma$ (which result in
samples from $\mathcal{N}(0,\sigma)$ that have a higher prior probability) and
higher values of $\sigma$ 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,

$\Sigma_{i=1}^{N_{q}}p_{i}=N_{q},\quad\quad\quad p_{i}\geq 0.$ (5)

Naively we might assign some prior permissible range to the first chosen bin,
$0\leq p_{1}<p_{\rm max}$, with the range for the second bin then being $0\leq
p_{2}<N_{q}-p_{1}$ 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 ${\bm{p}}=\\{p_{i}\\}$ is the Dirichlet
distribution, ${\rm Dir}({\bm{p}})$ (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 $N_{q}$ samples, $y_{i}$, are drawn from
the Beta distribution,

$B(y)=\frac{\Gamma(N_{q})}{\Gamma(N_{q}-1)}(1-y)^{N_{q}-2}=N_{q}(1-y)^{N_{q}-2},$
(6)

then

$p_{i}=\frac{y_{i}}{\Sigma_{i}y_{i}}$ (7)

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,

${\rm CDF}(p_{i})=\int_{0}^{p_{i}}B(x)dx.$ (8)

This inverse is also known as the quantile function or pixel percent function,
defined as $Q(p_{i})$ such that ${\rm CDF}(Q)=p_{i}$. By direct integration
and normalisation, we find that

${\rm CDF}(p_{i})=1-(1-p_{i})^{N_{q}-1}.$ (9)

After some algebra, this implies that

$Q(p_{i})=1-\left(1-\frac{N_{q}-1}{N_{q}}p_{i}\right)^{1/(N_{q}-1)}.$ (10)

##  5 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 $((\log
k,M_{0}),c_{0},c_{1},f_{B,1},\log_{10}h_{0},h_{1})$ frozen to constants $(\log
k,M_{0})=(4.1,0.0)$ (an abrupt rather than $\tanh$ cutoff for the bottom of
the mass function); $c_{0}=0$; $c_{1}=0$; $f_{B,1}=0$; $\log_{10}h_{0}=0$; and
$h_{1}=0$. We have also tested different values for $N_{q}$ in the range 4 to
10. Since all the different runs have different numbers of free parameters, we
use the ratios of the evidence, $Z$, (Bayes factors) to decide which model or
models are best. We follow the Kass & Raftery (1995) interpretation that
$\Delta\log_{10}Z>(0.5,1,2)$ 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 $f_{B}$ 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 $\log_{10}k$ and $M_{\rm
0}$) was found to be unnecessary since we made sharp cuts at $M_{G}=9$. The
evidence disfavoured using $c_{0}$ and $c_{1}$ 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 $N_{q}=10$ 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
$0.4<q<=1.0$. 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 $q$-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.

Table 1: Fraction of binary stars with mass ratio greater than a given $q$ for each cluster. The final rows, $FQ_{75}$ are the measured ratio of the number of binaries for which $q>0.75$ to the number for which $q>0.5$, the single-star mass-function power law index, $\gamma$, and the value of $FQ_{75}$ implied by random sampling of the mass function.  | Hyades | Praesepe
---|---|---
| $r<r_{t}$ | $r\geq r_{t}$ | $r<r_{t}$ | $r\geq r_{t}$
$q$ | legendre | histogram | legendre | histogram | legendre | histogram | legendre | histogram
0.5 | $0.125_{-0.028}^{+0.030}$ | $0.116_{-0.027}^{+0.029}$ | $0.08_{-0.03}^{+0.04}$ | $0.067_{-0.024}^{+0.030}$ | $0.128_{-0.016}^{+0.018}$ | $0.113_{-0.017}^{+0.020}$ | $0.119_{-0.022}^{+0.026}$ | $0.086_{-0.024}^{+0.026}$
0.6 | $0.111_{-0.026}^{+0.029}$ | $0.105_{-0.024}^{+0.029}$ | $0.061_{-0.023}^{+0.030}$ | $0.055_{-0.021}^{+0.027}$ | $0.100_{-0.016}^{+0.018}$ | $0.093_{-0.016}^{+0.018}$ | $0.076_{-0.017}^{+0.022}$ | $0.067_{-0.020}^{+0.023}$
0.7 | $0.104_{-0.025}^{+0.029}$ | $0.094_{-0.024}^{+0.027}$ | $0.046_{-0.019}^{+0.027}$ | $0.040_{-0.017}^{+0.023}$ | $0.090_{-0.017}^{+0.019}$ | $0.079_{-0.015}^{+0.017}$ | $0.047_{-0.016}^{+0.022}$ | $0.055_{-0.018}^{+0.022}$
0.8 | $0.092_{-0.023}^{+0.026}$ | $0.078_{-0.022}^{+0.027}$ | $0.034_{-0.015}^{+0.024}$ | $0.025_{-0.012}^{+0.018}$ | $0.080_{-0.015}^{+0.016}$ | $0.068_{-0.014}^{+0.016}$ | $0.034_{-0.015}^{+0.021}$ | $0.022_{-0.010}^{+0.015}$
0.9 | $0.057_{-0.015}^{+0.020}$ | $0.045_{-0.021}^{+0.025}$ | $0.020_{-0.010}^{+0.016}$ | $0.013_{-0.008}^{+0.014}$ | $0.044_{-0.008}^{+0.009}$ | $0.023_{-0.012}^{+0.014}$ | $0.021_{-0.009}^{+0.012}$ | $0.012_{-0.007}^{+0.011}$
$FQ_{75}$ | $0.81_{-0.11}^{+0.07}$ | $0.76_{-0.11}^{+0.10}$ | $0.51_{-0.14}^{+0.16}$ | $0.52_{-0.16}^{+0.16}$ | $0.68_{-0.08}^{+0.08}$ | $0.66_{-0.09}^{+0.09}$ | $0.34_{-0.12}^{+0.15}$ | $0.46_{-0.12}^{+0.12}$
$\gamma$ | $0.80_{-0.25}^{+0.25}$ | $0.82_{-0.24}^{+0.22}$ | $1.65_{-0.30}^{+0.30}$ | $1.64_{-0.30}^{+0.30}$ | $1.28_{-0.17}^{+0.17}$ | $1.30_{-0.17}^{+0.17}$ | $1.90_{-0.24}^{+0.25}$ | $1.94_{-0.23}^{+0.23}$
Implied $FQ_{75}$ | $0.35_{-0.02}^{+0.02}$ | $0.35_{-0.02}^{+0.02}$ | $0.28_{-0.02}^{+0.02}$ | $0.28_{-0.02}^{+0.02}$ | $0.31_{-0.01}^{+0.01}$ | $0.31_{-0.01}^{+0.01}$ | $0.27_{-0.02}^{+0.02}$ | $0.26_{-0.02}^{+0.02}$

We note that there is a certain amount of degeneracy for mass ratios $q>0.75$,
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 mass-ratio distributions to rise towards higher $q$
(that integrates over the degeneracy), we introduce a metric, $FQ_{75}$, that
we define as the ratio of the number of cluster binaries for which $q>0.75$ to
the number for which $q>0.5$. Values of $FQ_{75}$ are listed in Table 1.

Figure 5: Ratio of binaries with $q>0.75$ to the number for which $q>0.5$
plotted against overall frequency of binaries with $q>0.5$ 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.

In Fig. 5 we show $FQ_{75}$ plotted against the overall fraction of observed
cluster stars that are binaries with $q>0.5$. There is a loose relationship,
where the populations of stars with higher binary frequencies also have higher
fractions of binaries that have $q>0.75$. Referring to Table 1, $FQ_{75}$ 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 $FQ_{75}=0.35$).

---
Figure 6: Implied fraction of binary stars with mass ratio $q^{\prime}>q$ for
each cluster (red) together with its 1-$\sigma$ and 2-$\sigma$ uncertainties
from the Legendre-function model.

In Fig. 6 we show the total fractions of observed cluster stars that are
binaries with $q$ greater than a given value, based on the Legendre-function
models. These fractions are obtained by integrating the distributions in Fig.
4 backwards from $q=1$. Numerical values of these functions for particular $q$
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. 7 \- 10 each show seven random realisations of the CMD
for each cluster. These realisations have been generated by taking
$(1-f_{O})N$ random samples of the posterior maximum model, where $N$ 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.

##  6 Conclusions

From the observational results above we note the following:

  1. 1.

The inner parts of both clusters have mass functions that are flatter than a
canonical power-law IMF. Stars from outside the tidal radius that are
evaporating from the clusters have steeper mass functions, similar to field
stars in the Galactic disk (Chabrier, 2003). For both inner and outer samples,
the mass functions for Praesepe are a little steeper than for Hyades. These
observations are consistent with a scenario that the clusters formed with mass
functions flatter than that of Chabrier (2003) or Kroupa (2001). Both clusters
are undergoing segregated mass loss, with lower-mass stars being lost from the
clusters at a greater rate then higher-mass stars, thus steepening the mass
function for ejected stars and flattening it for retained stars.

  2. 2.

The overall frequency of binaries with mass ratio $q>0.5$ is 50 - 75 per cent
higher within the tidal radius of Hyades than outside the tidal radius. For
Praesepe, this frequency is 5 - 30 per cent higher within the tidal radius.

  3. 3.

There is a much higher relative frequency of higher-mass-ratio binaries within
the tidal radius of each cluster. The fact that low-$q$ binaries are less-
prevalent than high-$q$ binaries in the interior samples is perhaps a result
of dynamical processing, with a higher rate of three-body interactions in the
cluster interiors. Such interactions can harden binaries or exchange binary
companions, with both processes resulting in a kinetic energy transfer to
recoil of the binary and/or single star. The exchange process (sometimes via
the intermediate phase of a triple system) is more-likely to retain the two
highest-mass stars as a binary (Heggie, 1975) thus converting lower-$q$
binaries to higher-$q$ . The ejected least massive star may leave the cluster
core, accentuating the single-star mass segregation.

## Data and Code Availability

Gaia Data Release 3 (DR3) data, and the Gaia Catalogue of Nearby Stars are
publicly available via the Gaia archive, https://gea.esac.esa.int/archive/ ,
and the Centre de Données astronomiques de Strasbourg(CDS) catalogue service,
https://vizier.cds.unistra.fr/viz-bin/VizieR.

The CMDFITTER code used for this analysis is written in PYTHON and CUDA (via
the pyCUDA python library). CUDA is an extension to C/C++ that uses an NVIDIA
graphical processing unit to perform parallel calculations. The code is
available at https://github.com/MichaelDAlbrow/CMDFitter.

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##  Appendix A Realisations of the the models

Figure 7: The top left panel shows the 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
8: The top left panel shows the final CMD for the 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 9: The top
left panel shows the 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 10: 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.