# Gaia uncovers difference in B and Be star binarity at small scales: evidence
for mass transfer causing the Be phenomenon

Jonathan M. Dodd,1 René D. Oudmaijer,1 Isaac C. Radley,1 Miguel Vioque,2,3 and
Abigail J. Frost4,5
1School of Physics & Astronomy, University of Leeds, Woodhouse Lane, Leeds,
LS2 9JT, UK
2Joint ALMA Observatory, Alonso de Córdova 3107, Vitacura, Santiago, 763-0355,
Chile
3National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA
4Institute of Astronomy, KU Leuven, Celestijnlaan 200D, Leuven, 3001, Belgium
5European Southern Observatory (ESO), Alonso de Córdova 3107, Vitacura,
Santiago, 763-0355, Chile E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Be stars make up almost 20% of the B star population, and are rapidly rotating
stars surrounded by a disc; however the origin of this rotation remains
unclear. Mass transfer within close binaries provides the leading hypothesis,
with previous detections of stripped companions to Be stars supporting this.
Here, we exploit the exquisite astrometric precision of Gaia to carry out the
largest to date comparative study into the binarity of matched samples of
nearby B and Be stars from the Bright Star Catalogue. By utilising new “proper
motion anomaly” values, derived from Gaia DR2 and DR3 astrometric data
alongside previous values calculated using Hipparcos and Gaia data, and the
Gaia provided RUWE, we demonstrate that we can identify unresolved binaries
down to separations of 0.02$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$. Using
these measures, we find that the binary fractions of B and Be stars are
similar between 0.04 – 10$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$, but the
Be binary fraction is significantly lower than that of the B stars for
separations below 0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$. As the
separation range of these “missing” binaries is too large for mass transfer,
and stripped companions are not retrieved by these measures, we suggest the
companions migrate inwards via binary hardening within a triple system. This
confirms statistically for the first time the hypothesis that binary
interaction causes the Be phenomenon, with migration causing the dearth of Be
binaries between 0.02 - 0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$.
Furthermore, we suggest that triplicity plays a vital role in this migration,
and thus in the formation of Be stars as a whole.

###### keywords:

Astrometry and celestial mechanics: proper motions – stars: emission-line, Be
– binaries: close

††pubyear: 2023††pagerange: Gaia uncovers difference in B and Be star binarity
at small scales: evidence for mass transfer causing the Be phenomenon–A

##  1 Introduction

Be stars make up around $20\%$ of the total population of B-type stars
(Bodensteiner et al., 2020a) and are defined, in part, by the presence of
Balmer emission lines. These lines are thought to originate from a slowly
outflowing ionised circumstellar gas-disc (Porter & Rivinius, 2003; Rivinius
et al., 2013; Jones et al., 2022). While the dynamics of these discs have been
studied extensively (Wisniewski et al., 2010; Draper et al., 2014; Suffak et
al., 2022), the mechanism by which they form is undetermined. Consensus
suggests that the formation is related to the rapid rotation observed in
practically the entire Be population (Zorec et al., 2016, 2017), and which is
suggested to be at or near the critical velocities of such stars (Townsend et
al., 2004; Granada et al., 2013). Such rapid rotation could thus allow for
processes such as turbulence (Townsend et al., 2004) or non-radial pulsations
(Owocki & Cranmer, 2002; Semaan et al., 2018) to contribute significantly to
the formation of the decretion discs around Be stars due to decreased
effective gravity at the surface of such stars (Granada et al., 2013). This,
however, only leads to a further question - what exactly causes Be stars to
rapidly rotate?

Three mechanisms have been suggested as the origin of this characteristic
rotation; it could be a relic from the parent molecular cloud, the rotation
thus a result of inherited angular momentum (Bodenheimer, 1995); the envelope
of the star could be spun up via momentum transport from a contracting core
(Granada et al., 2013; Hastings et al., 2020); or, crucially for this study,
it could be a result of mass and angular momentum transfer via binary
interaction (Shao & Li, 2014; Bodensteiner et al., 2020a).

If binary interaction is the source of this rapid rotation, then one could
expect that the binary statistics of B and Be stars will differ. This line of
reasoning provided the basis for the study performed in Oudmaijer & Parr
(2010), which found the binary fraction of B and Be-type stars to be similar
between separations of $0.1$ and $8$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$
based on a sample of $36$ B and $37$ Be stars, and concluded that binarity, at
least at these scales, could not be responsible for all Be stars. However,
while a number of studies into the binarity of Be stars exist (Boubert & Evans
2018; Klement et al. 2019; Hastings et al. 2021 and El-Badry et al. 2022,
among others) and suggest the stripping of the companion to the Be star as the
cause (Han et al., 2002; Nazé et al., 2022; El-Badry & Quataert, 2021; El-
Badry et al., 2022), very few comparative studies into the binarity of both B
and Be-type stars exist. The next most recent points of comparison are Abt &
Levy (1978) and Abt & Cardona (1984), each from around 40 years ago, and which
similarly conclude that binarity cannot be responsible for the Be phenomenon
at the scales they probe. However, Abt & Levy (1978), which utilises radial
velocity variations to detect binarity, suffers from the effects of small
sample sizes, while Abt & Cardona (1984) is hampered by inhomogeneous biases
as it is, in effect, a review of the then available literature.

With the advent of high-precision astrometric all-sky surveys, we are now able
to mitigate these issues and investigate the multiplicity of large samples.
Recent studies have employed astrometric data from surveys such as Hipparcos
(Van Leeuwen, 2007) and Gaia (Gaia Collaboration et al., 2021) in order to
detect binarity in the overall stellar population, both in the case of wide
(El-Badry et al., 2021) and close (Kervella et al., 2019, 2022) binaries. The
time is thus ripe for such an investigation into the binarity of B and Be
stars.

Here we utilise the Proper Motion Anomaly (PMa), as introduced in Kervella et
al. (2019) (see also Brandt 2018), which is a measure of the difference
between long and short term proper motions of a point source, in order to
determine the fractions of B and Be type stars in visually unresolved binary
systems. Alongside previously determined PMa values from Kervella et al.
(2022), calculated using Hipparcos and Gaia DR3 data (a similar catalogue can
also be found in Brandt 2021), we use new values, calculated for the first
time in this work using a combination of Gaia DR2 and DR3 data (Gaia
Collaboration et al., 2018, 2022). Using these metrics we probe binary systems
with separations between $0.02$ and
$1.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and with magnitude
differences up to $4$\mathrm{m}\mathrm{a}\mathrm{g}$$.

Additionally we utilise the Gaia-provided Renormalised Unit Weight Error
(RUWE) to further constrain the binary fractions of these two populations
(Stassun & Torres, 2021). Thus we are able to bridge the gap in binary
separations between, and expand upon previous radial velocity (Abt & Levy,
1978) and adaptive optics (Oudmaijer & Parr, 2010) studies of these stars.
Additionally, by querying a sample over an order of magnitude larger than
present in the most recent similar study (Oudmaijer & Parr, 2010), this paper
provides the largest scale comparative study of B and Be star binarity to
date.

This paper is structured as follows: in Section 2 we detail the two binary
detection methods used (the PMa and the RUWE), and demonstrate their
sensitivities and limitations; in Section 3 we report the results of utilising
these methods on samples of B and Be stars; and in Section 4 we discuss the
implications of these newly reported binary fractions on models for the
formation of Be stars.

##  2 Methods

###  2.1 Binary Detections via Proper Motion Anomaly

For any binary system we can define two similar variables, the centre of mass,
which the two component stars orbit, and the photocentre, which describes the
location of the apparent point source in an unresolved system. In general, the
positions of these two points differ, thus the photocentre also orbits the
centre of mass - leading to a deviation from the linear proper motion expected
for single sources in the case of unresolved binary systems.

Figure 1: Cartoon illustrating the principle behind the Proper Motion Anomaly.
The large and small stars are the primary and secondary companion
respectively, the black dot is the centre of mass and the yellow dot the
photocentre. Two sets of observations are taken at $A$ and $B$, with a time
interval of $\Delta t$ between them. $\mu_{A}$ and $\mu_{B}$ are the short
term proper motions measured during their respective observations, and
$\mu_{AB}$ is the long term proper motion, determined from the change of
observed position over $\Delta t$. Arrows labelled $\mu_{PC}$ are the
instantaneous proper motions of the photocentre in the frame which the centre
of mass is stationary.

This deviation from the expected linear proper motion, referred to as the
Proper Motion Anomaly (PMa), $\Delta\mu$, can be detected through a comparison
of long and short term proper motion measurements ($\mu_{lt}$ and $\mu_{st}$
respectively), with the long term effectively tracing the motion of the centre
of mass and the short term tracing the sum of the centre of mass and relative
photocentre motions. An illustration of this principle can be found in Figure
1. The PMa can thus be expressed as:

$\Delta\mu=\mu_{st}-\mu_{lt}$ (1)

Thanks to current advances in the precision of proper motion measurements,
alongside high precision positional measurements allowing for long term proper
motions to be calculated, the PMa thus provides a powerful tool in the search
for unresolved binary candidates within large samples of stars.

This measure is sensitive to the period of a given binary system. Periods
significantly longer than the time elapsed over the long term proper motion
measurement, $\Delta t$ will have a deviation in proper motion too small to be
detected. Similarly, systems wherein a whole number of complete orbits have
occurred during the measurement of the long term proper motion will not be
detected, again as the deviation in long and short term proper motions will be
too small to detect.

Following this logic, peak sensitivity of the PMa should be achieved for
systems where an $\frac{2n+1}{2}$ orbital periods have elapsed between the two
observations, as the absolute change in orbital phase is at a maximum here.
However, in reality, due to the time-window smearing effects present over the
observation periods of the two constituent astrometric surveys (Kervella et
al., 2022) there is a general decrease in sensitivity towards smaller periods,
with peak sensitivity being achieved for periods of $P\simeq\Delta t$
(Kervella et al., 2022). This also explains why, although of much higher
astrometric precision, the DR2-DR3 derived PMa’s probe larger periods than the
Hipparcos-DR3 PMa values (as will be shown in the following sub-subsection).
Below we will probe the sensitivity limits of the various methods both
theoretically and empirically.

####  2.1.1 Separation Limits of the Proper Motion Anomaly

Whereas the sensitivity of the PMa to mass as a function of orbital period
(and thus linear separation) has previously been investigated (Kervella et
al., 2022), the sensitivity to angular separation remains unexamined. As
detailed in Appendix A, using an idealised theoretical model, consisting of a
face-on, circularly orbiting binary system, we arrive at a minimum detectable
separation of $0.01$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ for PMa values
calculated using Hipparcos and Gaia DR3 data. Additionally, when considering
the PMa calculated using Gaia DR2 and DR3 data, the theoretical minimum
detectable separation is $0.0025$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.
However, this computation does not take into account the effect of the time-
window smearing during the PM measurement - which leads to the DR2-DR3 PMa
losing out on the shorter separations.

It is thus useful to obtain an empirical value as a point of comparison. In
order to evaluate this sensitivity of the PMa to varying separations, we use a
sample of binary systems listed in the Washington Double Star (WDS, Mason et
al. 2022) catalogue. The WDS contains a compilation of multiple star systems
for which at least one measured separation is available within the literature.
The sources in the WDS were cross-matched with the list provided in Kervella
et al. (2022) using the xMatch service provided by CDS, Strasbourg (Pineau et
al., 2020). No restrictions on mass or spectral type were placed on the stars
within the binaries in order to generate the largest, general sample. Reported
higher order multiple systems were then removed for the sake of simplicity, as
were any systems with separations greater than
$10$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. While the PMa primarily probes
the unresolved regime of separations by nature, this
$10$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ limit was selected in order to
ensure that no unpredictable effects were materialising at higher separations.

For the resulting $11\,116$ unique binary systems, we obtained values of the
PMa from the Kervella et al. (2022) catalogue (which uses data reported in the
Hipparcos re-reduction (Van Leeuwen, 2007) and the Gaia DR3 catalogues, giving
$\Delta t\approx 25\,$\mathrm{y}\mathrm{r}\mathrm{s}$$) and calculated new
values, using data available in the Gaia Data Release 2 and Data Release 3
(Gaia Collaboration et al. 2018, 2021; $\Delta t\approx
0.5\,$\mathrm{y}\mathrm{r}\mathrm{s}$$). The methodology used in Kervella et
al. (2022) was utilised for the calculation of these new values, with
calibration performed via re-calculating Hipparcos-Gaia DR3 PMa’s replicating
the reported PMa values exactly and reported errors to within a precision of
$1\%$.

Figure 2: Median values of the Hipparcos - Gaia DR3 and Gaia DR2 - DR3 PMa
S/N’s and the RUWE, of systems reported as binary in the WDS (Mason et al.,
2022). Here values greater than 3 constitute a detection of binarity by either
PMa, and values greater than 1.4 constitute a detection by the RUWE - see
Section 2.2 for details.

The final sample contains $10\,283$ sources with reported PMa values; note
that not all values surpass the binary detection threshold of S/N$>$3\. The
majority of these systems had separations reported to a precision of
$0.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. $316$ systems have
separations reported as $0$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ in the
WDS Catalogue (i.e. their separations were below
$0.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$). For each of these systems,
higher precision separations were retrieved from supplemental WDS material
(Mason et al., 2022). Where orbital solutions had previously been reported,
the separation available at the Gaia DR3 epoch was used.

Figure 2 shows the median values of the signal-to-noise ratio for the
Hipparcos-Gaia DR3 PMa for the WDS sample at varying separations. These
systems have been placed into bins of width
$0.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ at separations greater than or
equal to $0.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. At separations below
$0.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ the systems have been placed
into 10 bins such that each bin has an equal number of systems within it
(approximately 30 per bin). For these bins the mean reported separation of the
systems within it is plotted.

Based on the data presented in Figure 2, it can be seen that for the
DR3-Hipparcos PMa, there is an increase of SNR towards lower reported
separations. On average, this PMa remains above the detection threshold of $3$
for all separations between $0.02$ and
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ (based on the median values).
As a result, $0.02-0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ appears to
be an empirical range of separations detectable via this PMa; at least half of
all WDS binary systems with separations within this range are detected (note
that we wouldn’t expect for all binaries within this range to be detected due
to projection effects). This is in agreement with the calculated theoretical
minimum detectable separation and the expectation that the upper limit occurs
somewhere around the spatial resolution limit of the surveys.

It is also to be noted that high PMa systems at separations greater than the
resolvable limit (i.e. with separations greater than $\sim
1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$) exist in non-negligible numbers.
These systems have reported separations much greater than the PMa can detect
(as it measures the motion of an essentially unresolved binary), and it is
thus likely that they are, in fact, as of yet undetected higher order multiple
systems.

Alongside the examination of the Kervella et al. (2022) reported Hipparcos-
Gaia DR3 PMa values, an equivalent consideration of Gaia DR2-DR3 PMa values
has been conducted. As there does not exist an equivalent catalogue as in
Kervella et al. (2022), these PMa’s were thus calculated via the same process,
but using data from the Gaia Data Releases 2 and 3.

The median values of these PMa SNR values at a given separation can also be
found in Figure 2. Again, a rise in the median PMa can be seen towards lower
separations, here beginning at separations of approximately
$1.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, increasing towards a peak at
around $0.6$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and remaining above
the detection threshold down to
$0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. While this lower limit is in
disagreement with the theoretical limit of
$0.0025$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ derived in Appendix A, the
theoretical considerations largely ignore the effect of smearing over the
observing timeframe. As the DR2-DR3 PMa is smeared across the entirety of the
long-term proper motions timeframe, this will mean that the lowest separations
are not probed, although the number of potential observable objects increases
significantly (as Gaia has observed orders or magnitude more objects than
Hipparcos). Additionally the increase in precision achieved since Hipparcos
has both resulted in the upper limit of separations detectable to be expanded
beyond the Gaia resolution limit, and increased the total fraction of systems
detected within the sensitive limit.

These results remain consistent when examining B and Be type binaries within
the WDS (Mason et al., 2022), with systems within the two sensitive ranges
being detected at significantly higher rates than those outside that range.

We thus show that the Hipparcos-Gaia DR3 PMa is sensitive to binary systems
with angular separations between
$0.02-0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and the Gaia DR2 - DR3
PMa is sensitive between separations of
$0.2-1.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

####  2.1.2 Magnitude Limits of the PMa

Figure 3: Median PMa of WDS systems, as a function of the difference in
magnitude of the two stars for the Hipparcos-Gaia DR3 and Gaia DR2-DR3 PMa’s
and the RUWE. Both plots have been cropped to a maximum difference in
magnitude of $8$\mathrm{m}\mathrm{a}\mathrm{g}$$, as the number of systems
with magnitude differences greater than this decreases to the point where
meaningful conclusions cannot be drawn. Detections are reported out to
differences of up to $12$\mathrm{m}\mathrm{a}\mathrm{g}$$ in all three cases.

In order to evaluate the limitations of the Proper Motion Anomaly, we must
also consider the sensitivity of this measure to the difference in magnitude
between the two components. Here systems contained within the WDS catalogue
are once again considered, now limiting the sample to those systems with
reported separations between $0$ and
$1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ in line with the more favourably
sensitive range determined in the previous section.

In the case of the Hipparcos-Gaia DR3 PMa, and as can be seen in Figure 3,
examining the difference in the WDS reported apparent magnitude reveals that
the PMa shows some preferential sensitivity within the $1.0$ to
$3.5$\mathrm{m}\mathrm{a}\mathrm{g}$$ range, peaking at around $\Delta
V_{mag}=3$\mathrm{m}\mathrm{a}\mathrm{g}$$.

In the case of the Gaia DR2-DR3 PMa, an equivalent examination reveals a
general increase in sensitivity across most magnitude differences available
via the WDS sample, with the preferentially detected region expanding to
systems with a difference in magnitude $0<\Delta V_{mag}\leq 4.5$, peaking at
around $\Delta V_{mag}=2.5$, although significant numbers of detections are
still made at greater differences in magnitude.

###  2.2 Binary Detections via the RUWE

The Renormalised Unit Weight Error, or the RUWE, is a reduced chi-squared
parameter that describes the ‘goodness-of-fit’ of a single body astrometric
model to an observed source. As such, it would follow that such a parameter is
sensitive to the presence of unresolved binary companions - with the
photocentre wobble caused by such a companion causing deviations from what is
expected of a single body, resulting in a worse fit astrometric solution.
Indeed, the RUWE parameter has been suggested to act as a binarity indicator
(Belokurov et al., 2020), with values significantly above the normalised value
of $1$ (i.e $\geq 1.4$ as per Stassun & Torres 2021) being regarded as
indicative of the source being ‘badly fit’ by the astrometric solution and
thus potentially binary.

Like the PMa, the RUWE varies in sensitivity according to the period of a
binary system, with the RUWE having some function of sensitivity to binarity
determined both by the period of the system in question and by the time
baseline over which the RUWE is determined (approximately six months in the
case of Gaia DR2 and DR3), with Penoyre et al. (2022) suggesting that for Gaia
DR3, RUWE is an effective measure of binarity for periods of around a month to
a decade.

However, due to sensitivity to other (non-multiplicity related) factors which
can inflate the RUWE (Belokurov et al., 2020), it is important to note that
there may be some fraction of false positive detections - for example if the
sample contains stars known to have circumstellar discs or extended regions of
emission (Oudmaijer et al., 2022). This is unlikely to hamper our study of B
and Be stars however, as the discs of Be stars are very small when compared to
the resolution achieved by Gaia DR3 (Quirrenbach et al., 1997; Wheelwright et
al., 2012).

####  2.2.1 Separation Limits of the RUWE

The separation limits of the RUWE were determined using the same process as
for the PMa above. Here, the detection threshold is given to be $1.4$ (as
suggested by Stassun & Torres 2021). Based on the data presented in Figure 2,
we find that the RUWE exhibits a sharp and marked increase above the detection
threshold at binary separations below the resolution limit of Gaia
($0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$). This increase peaks at
approximately $0.09$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and, on
average, remains above the threshold down to separations of
$0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. Thus, we find that the RUWE
is sensitive to those binary systems with separations ranging between
$0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ and
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

####  2.2.2 Magnitude Limits of the RUWE

As with the PMa, it is useful to quantify the sensitivity of the RUWE to the
magnitude differences of the binary systems probed. Figure 3 demonstrates, by
repeating the same process as used to determine the magnitude differences the
PMa is sensitive to, that it would seem that the RUWE is primarily sensitive
to systems with magnitude differences between $1.5$ and
$3.0$\mathrm{m}\mathrm{a}\mathrm{g}$$, peaking somewhere between $2.0$ and
$2.5$\mathrm{m}\mathrm{a}\mathrm{g}$$.

We have thus shown that, via these astrometric methods, we can consistently
detect visually unresolved binaries with separations as low as
$0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ using the RUWE, and
$0.02$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ using the Hipparcos-Gaia PMa
and with magnitude differences up to $4.5$\mathrm{m}\mathrm{a}\mathrm{g}$$,
depending on the measure used. Thus, thanks to the precision achieved by
modern astrometric surveys, we will be able to probe the origins of Be stars
with a statistical basis heretofore unavailable.

##  3 The Binarity of B and Be Stars

###  3.1 Sample Selection

Here we utilise the same initial sample of B and Be stars extracted from the
Bright Star Catalogue (BSC) 5th Revised Edition (Hoffleit & Warren, 1995;
Hoffleit & Saladyga, 1997) as used in Radley et al. (in preparation). This
sample was then cross-matched within
$5$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ against the Gaia Data Release 3
(Gaia Collaboration et al., 2021) data. Stars without proper motion
measurements in the Hipparcos or Gaia Data Releases were removed. Stars with
non-V luminosity classes were also removed in order to ensure that the
comparisons made between our two samples are between like sources that are
subject to the same selection effects. A summary of the spectral types of the
stars included in this sample can be found in Table 1.

Table 1: Populations of main sequence B and Be type stars within the sample, divided by spectral type. Spectral Type | B | Be
---|---|---
B0V | 12 | 4
B1V | 47 | 10
B2V | 104 | 31
B3V | 97 | 19
B4V | 31 | 12
B5V | 111 | 13
B6V | 43 | 8
B7V | 48 | 4
B8V | 156 | 13
B9V | 258 | 9
Total | 907 | 123

This sample provides an almost thirty-fold increase in the number of B-type
stars, and a four-fold increase in the number of Be-type stars compared to
similar comparative studies (Abt & Levy, 1978; Oudmaijer & Parr, 2010) that
probe a well defined separation range. Additionally, by virtue of being
selected from the same catalogue of the brightest stars in the sky, these
stars will be subject to the same selection effects, and have similar
brightnesses and parallaxes, thus ensuring their usefulness in comparing how B
and Be type stars differ in their astrometric properties. These stars are also
amongst the most nearby ones, thus allowing us to detect the physically
closest companions whilst remaining within the range of angular separations
detectable by our methods (see Sections 2.1.1 and 2.2.1), with the lower end
of our detectable range, $0.02^{\prime\prime}$ equating to
$5\,$\mathrm{a}\mathrm{u}$$ at the mean distance to the stars in these
samples.

###  3.2 Detection of Unresolved Binaries

We first consider the complete, unmodified samples of $907$ B and $123$ Be
stars, utilising a combination of PMa values found within the catalogue
contained within Kervella et al. (2022), calculated using Hipparcos and Gaia
DR3 data, and new PMa values calculated using data reported in Gaia Data
Releases 2 and 3, as well as the Gaia-provided RUWE. These measures allow us
to detect unresolved binary systems via differences in long and short term
proper motions (for the PMa; see Section 2.1) and via deviations from a single
body astrometric fit (in the case of the RUWE; see Section 2.2).

Using the Hipparcos-Gaia DR3 PMa, which probes the range 0.02 $-$ 0.7” (see
Section 2.1.1), equating to around $5\,-\,180\,$\mathrm{a}\mathrm{u}$$ at the
average distance of $260$\mathrm{p}\mathrm{c}$$, we arrive at binary fractions
of $42\pm 2\%$ and $28\pm 4\%$ for the B and Be samples
respectively111Uncertainties here are calculated using the binomial method.
Error bars are given at $1\sigma$.. This proves to be a greater than $3\sigma$
difference.

In contrast, the Gaia DR2 - DR3 PMa values reveal similar B and Be binary
fractions in the 0.2 $-$ 1.1” range (around
$50\,-\,290\,$\mathrm{a}\mathrm{u}$$ at $260$\mathrm{p}\mathrm{c}$$). In this
case, the fraction of B stars detected as being in a binary system is found to
be $27\pm 1\%$ and the equivalent for Be stars is found to be $29\pm 4\%$.

Finally, the fractions of B and Be stars detected via applying a detection
threshold of $RUWE>1.4$, tracing binaries with separations between 0.04 $-$
1.1” (10 $-$ 290 $\mathrm{a}\mathrm{u}$), were afound to be $27\pm 1\%$ and
$20\pm 4\%$ respectively. See Table 2 further down for a summary of these
binary fractions.

###  3.3 Correcting Biases

Although the global properties of the B and Be stars in the BSC are very
similar, the two samples exhibit slightly different properties across a number
of observables. For example, by virtue of being rarer, the Be stars are on
average further away than the B stars, with the average B star in our sample
sitting at a parallax of $5.26$\mathrm{m}\mathrm{a}\mathrm{s}$$ with a
standard deviation of $3.38$\mathrm{m}\mathrm{a}\mathrm{s}$$, whereas the
average Be star sits at a parallax of $3.81$\mathrm{m}\mathrm{a}\mathrm{s}$$
with a standard deviation of $2.42$\mathrm{m}\mathrm{a}\mathrm{s}$$. Likewise
the average apparent magnitudes of the B and Be populations are
$6.06$\mathrm{m}\mathrm{a}\mathrm{g}$$ (standard deviation of
$1.07$\mathrm{m}\mathrm{a}\mathrm{g}$$) and
$5.95$\mathrm{m}\mathrm{a}\mathrm{g}$$ (standard deviation of
$1.32$\mathrm{m}\mathrm{a}\mathrm{g}$$) respectively. In addition to this, the
two populations differ in the distribution of their spectral types (for
example, almost 30% of the B stars are B9V, while only 7% of the Be stars are
B9Ve). As such, the variations in the distributions of these properties must
be taken into account to ensure that the binary fractions are not affected by
them.

Thus, in order to remove the effects of these biases, a variety of sub-samples
of B stars were created. These included random samples, alongside “semi-
random” samples wherein the B population was binned according to a given
variable, mentioned above, and stars were randomly picked from each bin in
order to match the distribution of said variable in the Be sample. In each
case a Kolmogorov-Smirnov test was performed in order to confirm that the sub-
sample was likely drawn from the same distribution as the Be population. The
binary fraction of each sub-sample was then determined using the PMa and RUWE
detection thresholds detailed above.

A summary of these samples and their binary fractions can be found in Figure
4. Therein, the greater than $3\sigma$ dearth of Be binaries, when compared to
the number of B binaries, is still seen to be present at the tightest
separations probed by our methods.

Figure 4: Summary of the binary fractions of B and Be stars, as determined by
Hipparcos-Gaia DR3, Gaia DR2 - DR3 PMa values, Gaia DR3 RUWE values and
resolved binaries (see text). Also presented are the binary fractions of the
various sub-samples of the B type stars, taken in order to account for any
biasing effects. ‘Matched’ here indicates that a sub-sample of the B stars has
been created such that the distribution of a given variable in the sub-sample
is drawn from the same parent distribution as the Be population. Note that the
‘Wide Binaries’ plot is noisier due to smaller number statistics. The
Hipparcos - Gaia DR3 values, probing the smallest separations, show a
significantly lower Be binary frequency (in each case a greater than $3\sigma$
difference), while all other approaches return similar binary fractions to
within the errors. Based upon this we report a dearth of Be binaries with
separations between $0.02$ and
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and, as discussed in Section
3.5 we further constrain these missing binaries to between $0.02$ and
$0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

###  3.4 Wider Separation Binaries

While the focus of this study is on close binaries, we now consider the
presence of binary companions outside the
$0.02-0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ range. Oudmaijer & Parr
(2010), who probe separations of
$0.1-8$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, determined binary fractions
of B and Be stars at $29\pm 8$ and $30\pm 8\%$. This finding could imply that
the difference in the two binary fractions disappears towards larger
separations.

Exploiting the added information provided by Gaia, we performed a basic search
for spatially resolved B and Be binaries by cross-matching the sample depicted
in Fig. 4, and looking for sources within a radius of
10$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$. Any ‘duplicate’ sources that
were retrieved were then deemed to be probable companions to the primary if
they satisfied:

$\lvert\omega_{primary}-\omega_{duplicate}\rvert\leq\sigma_{primary,\omega}+\sigma_{duplicate,\omega}$
(2)

where $\omega$ is the parallax of either star, and $\sigma_{\omega}$ is the
uncertainty corresponding to either parallax.

Based on this basic test, $6\pm 1\%$ of B and $10\pm 3\%$ of Be V-type stars
are found to have potential resolved companions within a distance of
$10$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. These values remained
consistent even when randomly sampling or when generating semi-random samples
of B stars intended to match the spectral type, parallax or apparent magnitude
distributions of the Be stars. A summary of these results can also be found in
Figure 4. In the case of each of the distribution matching samples, a
Kolmogorov-Smirnov test was performed in order to verify that the distribution
of each given parameter in the B star sample is likely being drawn from the
same parent distribution as the Be star sample.

Presuming that each of these additional sources is indeed a binary companion,
poses an interesting question, that being why do Be stars have binary
companions at a lower rate than their B type equivalents at close (i.e.
$<0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ separations, but at a similar
rates at greater separations?

###  3.5 Further Constraints on Binary Separation

In this study we have determined, through the use of various astrometric
measures, between separations of
$0.02\,-\,0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, the binary fraction
of Be stars is significantly lower than that of their B star equivalents,
while they are similar at larger separations. Let us continue by further
identifying at which separation the B binary stars become more prevalent.

Table 2: Percentage of B and Be stars detected as binary at varying ranges of separations. By utilising combinations of the previously explored astrometric detection methods we are able to constrain the greater than $3\sigma$ death of Be binaries to between $0.02$ and $0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. Separation Range / $\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$ | B | Be | Detection Method
---|---|---|---
$0.02\leq x\leq 0.2$ | $28\pm 1\%$ | $17\pm 3\%$ | Detected by the Hipparcos-DR3
|  |  | PMa but not the DR2-DR3 PMa.
$0.02\leq x\leq 0.7$ | $42\pm 2\%$ | $28\pm 4\%$ | Detected by the Hipparcos-DR3
|  |  | PMa.
$0.2\leq x\leq 1.1$ | $27\pm 1\%$ | $29\pm 4\%$ | Detected by the Gaia DR2-DR3
|  |  | PMa.
$0.04\leq x\leq 0.7$ | $27\pm 1\%$ | $20\pm 4\%$ | Detected by the RUWE.
$0.1\leq x\leq 8$ | $29\pm 8\%$ | $30\pm 8\%$ | Detected by Oudmaijer & Parr (2010)
$0.7\leq x\leq 10$ | $6\pm 1\%$ | $10\pm 3\%$ | Resolved as binary by Gaia.

Table 2 reveals that by comparing the two PMa values and looking at those
systems detected by the Hipparcos-Gaia DR3 PMa but not by the Gaia DR2-DR3
PMa, we can constrain the lack of Be binaries to between $0.02$ and
$0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. We also note that at
separations greater than $0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, B
and Be stars have similar fractions of their populations in binary systems -
in agreement with the previous most recent comparative study of these stars
(Oudmaijer & Parr, 2010) 222Although we do not wish to make quantitative
comparisons between values reported by our different detection methods, a
larger binary fraction detected by the Hipparcos-Gaia DR3 PMa ($42\pm 2\%$)
than the Gaia DR2-DR3 PMa ($27\pm 1\%$) would be expected in the case of a
log-flat distribution of separations (as per Öpiks law; Öpik 1924). This is
because the separation range probed by the Hipparcos-Gaia DR3 PMa ($0.02$ to
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$) is almost twice the width in
log-space as that of the Gaia DR2-DR3 PMa ($0.2$ to
$1.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$). Conversely, by themselves
the values reported for the Be stars may indicate a distribution decreasing
towards to closer separations that we are sensitive to.. Additionally, we note
that the RUWE detects statistically similar rates of B and Be stars as being
in binary systems. Thus, by virtue of the RUWE’s sensitivity stretching
between $0.04$ and $0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, we are
able to confine the dearth of Be binaries even further - restricting them to
between $0.02$ and $0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

##  4 Discussion

In previous sections we have established, for the first time, ranges of
angular separation and magnitude difference within which the PMa and RUWE are
sensitive to binarity.

We have then used these measures to reveal a greater than $3\sigma$ dearth of
Be stars in binary systems at separations unresolved by Gaia, compared to B
stars, we then go to further constrain this absence of Be binaries to
separations between $0.02$ and
$0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

These results pose a conundrum, as one might naively assume that this implies
that the Be phenomenon more favourably occurs via single star pathways, rather
than the currently favoured binary pathways. Therefore, in order to continue
to pursue the binary pathway for Be formation, we must consider: what sort of
companion will remain undetected by these methods, whilst also remaining
consistent with such models of Be formation?

In this section we will posit some explanations for this difference in binary
fraction, alongside the implication for models regarding the formation of the
Be phenomenon.

###  4.1 Stripped Companions

One such solution invokes the presence of a companion whose envelope has been
stripped by the putative Be star. Accretion from such a companion would
provide a source for the initial spin-up mechanism that eventually results in
the formation of the Be stars’ characteristic disc (El-Badry et al., 2022;
Jones et al., 2022), while also allowing the companion to become low-mass, dim
or close enough to the host star (Frost et al., 2022) to be unlikely to be
detected via the PMa.

Such a model has been suggested to be the route through which $20-100\%$ of
field Be stars form (El-Badry & Quataert, 2021), and a significant number of
these stripped companions to Be stars have been reported in the literature
(El-Badry et al. 2022; Bodensteiner et al. 2020b; Frost et al. 2022; see Table
3). Additionally, no close main sequence companions to Be stars have been
confirmed (Bodensteiner et al., 2020a), which is consistent with what would be
expected in the case of a stripped companion origin for Be stars (although it
should be noted that this study only considers massive Be stars - i.e.
spectral types earlier than B1.5). As such, they may be responsible for
hydrogen-poor supernovae and neutron star mergers resulting in gravitational
wave sources.

The periods proposed to be required in order for stripping to begin by Han et
al. (2002) are of order $0.1-10$ days for Common Envelope Ejection and
$400-1500$ days for Roche Lobe Overflow. Using the average distance to the Be
stars in our sample ($\sim 260$\mathrm{p}\mathrm{c}$$), and assuming a
circular, Keplerian orbit, these stars would require a binary companion at a
separation of $0.02$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ (approximately
$5\,$\mathrm{a}\mathrm{u}$$) or closer in order for stripping to initiate.
While, the process of mass transfer would then act to widen the orbit of this
companion, the resultant stripped star would be low-mass and faint. We would
thus expect that the majority of stripped companions, if they indeed exist,
would be missed by both our PMa values and the RUWE as they will too close to
the host Be star, or too low mass and dim to be detected.

In support of this, PMa and RUWE values were obtained for $16$ Be stars
confirmed via X-ray observations to host a stripped companion (all of those
from Nazé et al. 2022). In order for the above suggestion to have any
validity, these stars, that are known to host stripped companions, should not
be detected as binaries using the measures detailed within this paper. A
summary of these objects and their Gaia data can be found in Table 3.

Table 3: Summary of Gaia DR3 - Hipparcos PMa and RUWE values for Be stars quoted in Nazé et al. (2022) as being known to host stripped companions. PMa and RUWE values marked with an asterisk (*) constitute binary detections. Name | Spectral Type | PMa | RUWE
---|---|---|---
$\varphi$ Per | B1.5Ve | $0.23$ | $1.804^{*}$
HD 29441 | B2.5Vne | $1.17$ | $1.125$
HD 41335 | B1.5IV-Ve | $8.13^{*}$ | $1.515^{*}$
HD 43544 | B3Ve | $0.66$ | $1.047$
HD 51354 | B3Ve | $0.79$ | $1.055$
HD 58978 | B0.5IVe | $2.52$ | $0.941$
HD 60855 | B2Ve | $1.52$ | $1.097$
HD 113120 | B2IVne | $6.93^{*}$ | $2.385^{*}$
$\kappa^{1}$ Aps | B2Vnpe | $1.48$ | $1.060$
HD 152478 | B3Vnpe | $1.82$ | $0.990$
HD 157832 | B1.5Ve | $1.60$ | $0.735$
28 Cyg | B3IVe | $1.55$ | $1.113$
HD 194335 | B2IIIe | $3.81^{*}$ | $1.041$
59 Cyg | B1Ve | $6.38^{*}$ | $2.724^{*}$
60 Cyg | B1Ve | $6.31^{*}$ | $0.142$
8 Lac A | B1IVe | $0.87$ | $1.480^{*}$

Seven of these 16 systems are detected by either method; 5 by the Hipparcos -
Gaia DR3 PMa, and 5 by the RUWE, with 3 objects detected by both. However, 5
of these 7 detected systems (the exceptions being HD 194335 and HD 200310)
have known additional companions (Mason et al., 2022) at separations within
the range the PMa’s and RUWE are sensitive to. In addition, these companions
are at separations significantly larger than required for the periods reported
by Nazé et al. (2022), and are therefore inconsistent with being the reported
stripped companion. An undetected, third, companion has also been suggested in
the case of HD 200310 to explain the eccentric orbit of the stripped companion
(Klement et al., 2022), making the system a potential triple. Thus, the true
fraction of detected stripped companions would seem to be either $6$ or $12\%$
depending on how one wishes to treat the case of HD 200310. Therefore, we
reach the conclusion that the stripped companions listed in Nazé et al. (2022)
are by and large not detected by the PMa or RUWE, whose detections turn out to
trace a third body in these systems instead. Thus, the potential for the Be
phenomenon being caused by the presence of stripped companions remains
consistent with our results.

###  4.2 Triples

Intriguingly, the separation range of the “missing” Be binaries (0.02 -
0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$; with
0.02$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$ corresponding to
5$\mathrm{a}\mathrm{u}$) is typically too large for mass transfer to have
occurred (typically less than $4$\mathrm{a}\mathrm{u}$$ is required per the
periods reported in Han et al. 2002). The natural next question to ask is
then, why would the Be stars lack binaries in this particular separation
range, while the stripping of a close companion should occur at smaller
separations?

A clue can be found in the fact that the systems listed in Table 3 show a high
incidence rate of higher order multiplicity, with $56\pm 18\%$ of this sample
of Be stars known to host a stripped companion having additional companions
(Mason et al., 2022) at separations larger than those reported in Nazé et al.
(2022). Additionally this number is also somewhat elevated with the expected
incidence rate of higher order multiplicity posited by Moe & Di Stefano (2017)
\- being around $40\%$.

The presence of such a third companion can neatly explain the absence of
detected Be binaries at larger separations, while also allowing for any
companion stars to become close enough for mass transfer to occur; it is well
known that higher order multiplicity can result in the hardening of an inner
binary. Indeed, a third body increases the occurrences of migration and
eventual binary interactions significantly (Toonen et al., 2022; Kummer et
al., 2023).

Recent models (Preece et al., 2022) suggest that of the systems that
eventually form stripped companions - half form via binary pathways and half
form via triple pathways. This can take place by either mass transfer
occurring within the inner binary - with the outer tertiary star potentially
becoming unbound - or a merger of the inner binary and mass transfer occurring
between the two remaining stars (Preece et al., 2022).

Our results therefore imply that close binary interactions are responsible for
the formation of Be stars, which constitute $20\%$ of the B star population,
with a migratory effect causing the lack of Be binaries in the $0.02$ to
$0.04$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$. Moreover, we suggest that
triplicity must play a vital role in triggering this migration, and thus in
the formation of Be stars as a whole.

###  4.3 Potential Limitations of the Methods

We will now address the various potential limitations of the methods as
described in Sections 2 and 3.

####  4.3.1 Sample Selection

First, we note that the Be phenomenon is known to be transient, with the disc
known to disappear and reappear on short timescales, which can lead to the
potential misclassification of Be stars as B stars Rivinius et al. (2013).
Thus, contamination of the B star sample with misclassified Be stars could
effect the previously detected binary fractions for the B star sample.
However, as the Be sample should have low contamination (as observation of the
emission line features associated with the Be phenomenon is required for
classification as a Be star, even if these features later disappear), the
detected Be binary fractions should be representative of the Be population as
a whole - and thus representative of any misclassified Be stars. Therefore,
should there be a significant amount of Be contaminants in the B star sample,
then they will act to bring the true B binary fraction down - thus the true B
binary fraction would differ more significantly from the Be fraction.

We also note that, as this sample is selected from the BSC (Hoffleit &
Jaschek, 1982; Hoffleit & Warren, 1995; Hoffleit & Saladyga, 1997), these
stars are necessarily bright. Whilst Gaia astrometric fits become systematic
dominated at $G<13$ (Lindegren, 2020), the resultant increase in uncertainties
associated with this effect ensures that the PMa is still a viable metric - as
a signal-to-noise ratio is used to quantify binary detection. Additionally, in
the calculation of the new Gaia DR2 - DR3 PMa’s corrections to the respective
proper motions of bright stars as proscribed by Lindegren et al. (2018) and
Cantat-Gaudin & Brandt (2021) have been applied. Steps have also been taken to
ensure that the two samples are subject to the same selection effects, and
these are detailed in Section 3.3, thus any systematic effects should be
consistent between the two samples. To check this, all stars brighter than
$5$\mathrm{m}\mathrm{a}\mathrm{g}$$ were removed from the B and Be samples;
which resulted in zero change to the reported dearth of Be binaries within the
$0.02\,-\,0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ range.

####  4.3.2 Variability

While photometric variability has not been quantitatively considered here, we
anticipate that it is unlikely to effect the PMa unless the variability is
occurring both asymmetrically and on a scale similar to the sensitive
separation range, here we look at this in more detail.

First, consider a single star exhibiting symmetrical variation; in this case
the position of the photocentre does not deviate from the centre of mass -
thus leading to no proper motion anomaly. When we then introduce an unresolved
companion, the position of the photocentre therefore deviates from the
position of the centre of mass, however the magnitude of the deviation varies
with the variability cycle. Thus, while such variability shouldn’t produce a
false positive detection of binarity, it is possible for it to induce a false
negative. The likelihood of this false negative thus depend on two factors:
the magnitude of the variability, and the timescale on which it occurs (as
observations taken at the same point in the variability cycle will obviously
be unaffected, and variability with a period shorter than either individual
observing period will be smeared, diminishing the effect on the PMa).

In the case of asymmetric variability - i.e. that caused by phenomena such as
mass ejection, or variations in surface brightness in stars such as red super-
giants (Chiavassa et al., 2022), or asymmetry within a disc (Meilland et al.,
2007; Stee et al., 2013) \- deviation of the photocentre from the centre of
mass will occur. However, such asymmetry would have to be on a similar scale
to the PMa’s sensitive range to in order for such deviation to be construed as
a false positive detection of binarity.

In the specific case of this study, we anticipate little effect of the
photometric variability of Be stars (Labadie-Bartz et al., 2017; Labadie-Bartz
et al., 2022) on our PMa values. This is due to the observed periods of Be
variability being significantly shorter than the observing periods of Gaia
DR2, DR3 and Hipparcos ($668$, $1038$ and $1227$ days respectively), with the
longest period variability found by Labadie-Bartz et al. (2017) being
approximately $200$ days. Similarly, variability caused by mass loss onto the
Be disc is unlikely to significantly effect the PMa, as the discs of Be stars,
with sizes of order milli-arcseconds, are very small when compared to the
resolution achieved by Gaia DR3 (i.e.
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$; Wheelwright et al. (2012);
Quirrenbach et al. (1997)). Additionally, by similar logic, asymmetry of the
discs themselves (Meilland et al., 2007; Stee et al., 2013) is also expected
to have little effect on the PMa measurements of these stars. Likewise, the
stars are necessariliy smaller than their disc, they are expected to be too
small to have asymmetric surface brightness that would effect the PMa.

##  5 Conclusions

In this paper we have performed the largest scale comparative study of B and
Be-star binarity to date, utilising both the Hipparcos - Gaia DR3 PMa values
found in Kervella et al. (2022) and new Gaia DR2 - DR3 PMa values calculated
for the first time here, alongside the Gaia provided RUWE.

We evaluate the limits in which these methods are sensitive to binarity.
Particularly we determine for the first time the range of angular separations
they are capable of reliably detecting, through the use of WDS (Mason et al.,
2022) data. We thus find that the Hipparcos - Gaia DR3 and Gaia DR2 - DR3
PMa’s are sensitive to binaries with separations in the
$0.02\,-\,0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ and
$0.2\,-\,1.1$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$ ranges respectively.
Likewise the RUWE is found to be sensitive to separations between $0.04$ and
$0.7$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.

Through the use of these astrometric methods in combination with one another,
we report a greater than $3\sigma$ dearth of Be binaries compared to B stars
at separations between $0.02$ and
$0.2$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, with the fractions of B and
Be stars found to be in binary systems at these separations being $28\pm 1$
and $17\pm 3\%$ respectively.

We posit that this apparent lack of Be binaries provides evidence for the
presence of stripped companions to the Be stars, with the mass-transfer
process by which they are created having been previously suggested as an
origin for the Be phenomenon. Such stars would go undetected by the measures
used within this paper, as they are too close, low-mass and dim to induce a
significant enough deviation of the photocentre from the centre of mass. We
also find that Be stars previously reported to have a stripped companion have
additional companions (i.e. triples and higher order multiples) at a rate
somewhat elevated from the expected rate for B stars as a whole. Therefore, we
suggest that migration of binary companions to Be stars to separations at
which mass-transfer can occur, triggered by a third companion, must play an
important role in the formation of Be stars.

## Acknowledgements

The STARRY project has received funding from the European Union’s Horizon 2020
research and innovation programme under MSCA ITN-EID grant agreement No
676036. This project has also received funding from the European Union’s
Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under
the Marie Skłodowska-Curie Grant Agreement No. 823734. 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.

## Data Availability

The catalogues of the proper motion anomalies and RUWE values of both the
sample of B and Be stars, as well as the WDS binaries are available form the
corresponding author on reasonable request, and will also be made available
via VizieR.

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##  Appendix A Derivation of the Theoretical Limits of the PMa

In order to obtain a theoretical, lower bound binary separation detectable via
the PMa, it is easiest to consider a simple binary system. Take, for example,
a system of two intermediate mass (i.e. $2-5$\mathrm{M}_{\odot}$$) main
sequence stars in a circular orbit. In such a system, where both stars are of
a known mass, the centre of mass is easily derivable. Likewise, by applying
the mass-luminosity relationship:

$\frac{L}{L_{\odot}}\simeq 1.4\frac{M}{M_{\odot}}^{3.5}$ (3)

for such stars, as given by Griffiths et al. (1988), the location of the
photocentre is similarly able to be determined.

The minimum theoretically detectable separation will occur for a system where
(a) the difference between the centre of mass and the photocentre is at a
maximum and, (b) where the change in absolute orbital phase is at a maximum
(i.e the observations are spaced half an orbit apart).

The latter of these criteria (b) is fairly easy to evaluate, provided we know
the masses of the two stars. However, we must first evaluate the former
criteria (a) in order to determine the mass ratio at which the largest offset
between the photocentre and centre of mass is achieved. This proves to be a
somewhat more involved process.

By normalising the separation between the two stars to $1$, it is possible to
write the photocentre and centre of mass equations as:

$a_{1,M/L}+a_{2,M/L}=1$ (4)

$M_{1}a_{1,M}=M_{2}a_{2,M}$ (5)

$L_{1}a_{1,L}=L_{2}a_{2,L}$ (6)

$\delta=\lvert a_{1,M}-a_{1,L}\rvert$ (7)

where $a_{1/2,M/L}$ is the separation of the primary ($1$) or secondary ($2$)
companion from the centre of mass ($M$) or photocentre ($L$) depending on the
subscript, $M$ is the mass of an individual star, $L$ is the corresponding
luminosity and $\delta$ is the offset between the centre of mass and the
photocentre.

From this set of equations, alongside the mass-luminosity relation given by
Equation 3, it is possible to rewrite Equations 5 and 6 in terms of
$a_{1,M/L}$ and the mass ratio of the two stars, $M_{1}/M_{2}$, giving the
expression of $\delta$ as:

$\delta=\lvert\frac{M_{2}}{M_{1}}(\frac{M_{2}}{M_{1}}^{3.5}+1)^{-1}-\frac{M_{2}}{M_{1}}(\frac{M_{2}}{M_{1}}+1)^{-1}\rvert$
(8)

By taking the derivative of this with respect to the mass ratio, and solving
for $d\delta/d(\frac{M_{1}}{M_{2}})=0$, it is possible to obtain the mass
ratio for which the maximum offset between the centre of mass and photocentre
is achieved. This is found to occur for a mass ratio, $M_{1}/M_{2}=0.47$ for
intermediate mass main sequence stars.

If we now assume a circular orbit with a stationary centre of mass, then the
photocentre motion is resultant entirely from the orbital motion of the
system. Thus, as the maximum change in proper motion occurs for observations
separated by $\frac{3n+1}{2}$ orbital periods, where $n$ is some integer
greater or equal to zero, in this case resulting in the change in proper
motion being twice the instantaneous proper motion. Thus, we can simply
express this change in proper motion, $\Delta\mu$ as:

$\Delta\mu=2\mu_{st}=\omega r\delta=\frac{2\pi r}{P}\delta$ (9)

where $\mu_{st}$ is the short term proper motion measurement, which here can
be reduced to the tangential velocity of the photocentre. $P$ is the period of
the orbit and $r$ is the binary separation.

While this approximation of the relation between the change in proper motion
and the period of a binary system may seem to imply that the proper motion
changes increasingly towards shorter period binaries, this approximation
ignores the effects of smearing on the actual observationaly data. In reality
this smearing across the observational timeframe causes a drop in sensitivity
at shorter periods (Kervella et al., 2022), with the maximum sensitivity being
achieved for periods similar to twice that of the time elapsed between the two
observations. Here, this resolves to a period of $\approx
50$\mathrm{y}\mathrm{r}\mathrm{s}$$ for the Hipparcos-Gaia DR3 PMa, and
$\approx 1$\mathrm{y}\mathrm{r}$$ in the case of the Gaia DR2-DR3 PMa.

Using this calculated PMa, in concert with standard errors for the three
surveys in question (in this case the Hipparcos re-reduction (Van Leeuwen,
2007) and the Gaia DR2 and 3 (Lindegren et al., 2018, 2021) although other
values can be used as and where appropriate), a lower limit of separation can
be obtained. This is achieved via propagating the standard errors through the
calculations required to determine the PMa (as detailed in Kervella et al.
(2022)) and using this to generate a signal-to-noise ratio. As we have an
expression for the change in proper motion that is a function of the
separation we can thus write:

$\frac{\Delta\mu}{N}=\frac{2\pi r}{NP}\delta$ (10)

where $N$ is the standard error of calculating the PMa. Thus by setting the
above to equal the Kervella et al. (2022) defined detection threshold of $3$,
and solving for $r$ we arrive at a theoretical minimum binary separation
detectable by the PMa. In the case of the PMa obtained from Hipparcos re-
reduction and Gaia DR3 data this lower limit is estimated to be
$0.01$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$, and in the case of the Gaia
DR2-DR3 PMa this is $0.0025$\mathrm{{}^{\prime}}\mathrm{{}^{\prime}}$$.
However the effect of the observational smearing is underestimated,
particularly in the case of the Gaia DR2-DR3 PMa, where the PM measurement
period covers the entirety of long term proper motion’s timeframe - likely
resulting in less sensitivity to small separations than predicted here.