11institutetext:  Laboratoire de physique des 2 infinis Irène Joliot-Curie,
CNRS Université Paris-Saclay, Bât. 100, Faculté des sciences, F-91405 Orsay
Cedex, France

#  Blending from binarity in microlensing searches toward the Large Magellanic
Cloud

T. Blaineau 11 M. Moniez 11<EMAIL_ADDRESS>
(Received 22/03/2023, accepted 03/08/2023)

###### Abstract

Context.  Studies of gravitational microlensing effects require the estimation
of their detection efficiency as soon as one wants to quantify the massive
compact objects along the line of sight of source targets. This is
particularly important for setting limits on the contribution of massive
compact objects to the Galactic halo. These estimates of detection efficiency
must not only account for the blending effects of accidentally superimposed
sources in crowded fields, but also for possible mixing of light from stars
belonging to multiple gravitationally bound stellar systems.

Aims.  Until now, only blending due to accidental alignment of stars had been
studied, in particular as a result of high-resolution space images. In this
paper, we address the impact of unresolved binary sources that are physically
gravitationally bound and not accidentally aligned, in the case of
microlensing detection efficiencies toward the Large Magellanic Cloud (LMC).

Methods.  We used the Gaia catalog of nearby stars to constrain the local
binarity rate, which we extrapolated to the distance of the LMC. Then we
estimated an upper limit to the impact of this binarity on the detection
efficiency of microlensing effects, as a function of lens mass.

Results.  We find that a maximum of 6.2% of microlensing events on LMC sources
due to halo lenses heavier than $30M_{\odot}$ could be affected as a result of
the sources belonging to unresolved binary systems. This number is the maximum
fraction of events for which the source is a binary system separated by about
one angular Einstein radius or more in a configuration where light-curve
distortion could affect the efficiency of some detection algorithms. For
events caused by lighter lenses on LMC sources, our study shows that the
chances of blending effects by binary systems is likely to be higher and
should be studied in more detail to improve the accuracy of efficiency
calculations.

###### Key Words.:

Gravitational lensing: micro - Cosmology: dark matter - surveys - stars:
binaries - Galaxy: halo - Galaxy: kinematics and dynamics

††offprints: M. Moniez,

##  1 Introduction

Objects cataloged in dense fields are frequently composed of several blended
sources. Not accounting for this may distort the statistical conclusions of
the microlensing searches because of its impact on the shape of the light-
curve (Stefano & Esin 1995), which can have repercussions on detection
efficiency (Woźniak & Paczyński 1997). Some of the consequences of blending on
microlensing have been studied by comparing ground-based images with high-
resolution deep space images, obtained notably with the Hubble Space Telescope
(HST) (HST archive 2002). These space images allow one to quantify the impact
of accidental alignments of sources in the catalogs of the ground-based
surveys, due to the high density of the field (Alibert, Y. et al. 2005;
Tisserand et al. 2007; Wyrzykowski et al. 2010). However, in the direction of
the Magellanic Clouds, a specific component of the blending remains poorly
understood, which is a result of the mixing of light from multiple
gravitational bound stars being too close together to be resolved, even by
space telescopes, at a distance larger than $50\,kpc$. Their existence is an
additional cause of blending, distinct from the mixing caused by coincidental
alignments. In this paper we study the possible consequences, currently poorly
known, of the binarity of the Large Magellanic Cloud (LMC) stars on the
detectability of the gravitational microlensing effects they may experience.
In particular, we study the case of the detection efficiency of microlensing
effects due to high-mass ($>30M_{\odot}$) Galactic halo objects, which have
been recently searched in the LMC direction and excluded as a significant
component of the hidden mass of the Galaxy (Blaineau et al. 2022). In section
2, we provide an overview of the fundamentals of the gravitational
microlensing effect. In Section 3 we introduce the blending effects and their
impact on the detection efficiency. We introduce the case of multiple sources
and distinguish between three blending regimes. In section 4, we present our
statistical analysis tools and show that we cannot extract statistical
information on the LMC binary systems from HST images because of the
separation limit. In section 5, we describe our methodology to estimate upper
values of the binarity rate in the local volume (at a distance of less than
$500\,pc$). We show how the distribution of distances between the components
of star pairs in a complete Gaia catalog population allows us to quantify the
rate of widely separated double systems in the Galactic plane. We explain how
we extrapolated the local binarity rate down to $50\,au$ separations for stars
closer than $500\,pc$ to the Sun in section 6. In section 7, we detail how we
estimated a conservative upper limit of the impact of binarity on the
microlensing detection efficiency toward the LMC as a function of the
projected lens Einstein radius. We discuss the validity domain and limitations
of our study, and address the question of the dependence of binarity rates on
the stellar type in Section 8. We conclude and summarize our results in
Section 9, and propose some prospects for future microlensing surveys.

##  2 Overview of microlensing

###  2.1 Description of a microlensing event

The gravitational microlensing effect (Paczynski 1986), which was first
discovered in 1993 (Alcock et al. 1993; Aubourg et al. 1993; Udalski et al.
1993), occurs when a massive compact object (the lens) passes close enough to
the line of sight of a background source and temporarily magnifies its
brightness. Reviews of the microlensing formalism can be found in Schneider et
al. (2006) and Rahvar (2015). When a single point lens of mass $M_{L}$ located
at distance $D_{L}$ deflects the light from a point source located at distance
$D_{S}$, a situation hereafter referred to as Point-Source Point-Lens (PSPL)
event, an observer receives light from two images not separated in the
telescopes. The total magnification $A(t)$ of the apparent luminosity of the
source at time $t$ is given by the following (Paczynski 1986):

$A(t)=\frac{u(t)^{2}+2}{u(t)\sqrt{u(t)^{2}+4}}\ ,$ (1)

where $u(t)$ is the distance of the lens to the undeflected line of sight,
divided by the Einstein radius $r_{\mathrm{E}}$,

$r_{\mathrm{E}}\\!=\\!\\!\sqrt{\frac{4GM_{L}}{c^{2}}D_{S}x(1\\!-\\!x)}\\!\simeq\\!4.5\,\mathrm{au}\times\left[\frac{M_{L}}{M_{\odot}}\right]^{\frac{1}{2}}\\!\left[\frac{D_{S}}{10\,kpc}\right]^{\frac{1}{2}}\\!\\!\frac{\left[x(1\\!-\\!x)\right]^{\frac{1}{2}}}{0.5}\\!.$
(2)

Here $G$ is the Newtonian gravitational constant, and

$x=D_{L}/D_{S}.$ (3)

If the point lens has a constant relative transverse velocity $v_{T}$ with
respect to the point source (PSPL rectilinear event) and $u(t)$ reaches its
minimum value $u_{0}$ (impact parameter) at instant $t_{0}$, then

$u(t)=\sqrt{u_{0}^{2}+(t-t_{0})^{2}/t_{\mathrm{E}}^{2}},$ (4)

where $t_{\mathrm{E}}=r_{\mathrm{E}}/v_{T}$ is the lensing timescale,

$t_{\mathrm{E}}\sim 79\
\mathrm{days}\times\left[\frac{v_{T}}{100\,km/s}\right]^{-1}\left[\frac{M_{L}}{M_{\odot}}\right]^{\frac{1}{2}}\left[\frac{D_{S}}{10\,kpc}\right]^{\frac{1}{2}}\frac{[x(1-x)]^{\frac{1}{2}}}{0.5}.$
(5)

In the case of the study of a population of stellar sources within the LMC,
and assuming that independently of their mass the lenses are distributed in a
spherical, isotropic, and isothermal halo with the density distribution
described in Griest (1991) and with the parameters used in Blaineau et al.
(2022) (hereafter called the standard Galactic halo), the average duration of
the events due to the lenses of a given mass $M_{L}$ is $\langle
t_{E}\rangle\sim 63\,days\times\sqrt{M_{L}/M_{\odot}}$.

###  2.2 Optical depth, event rate, and detection efficiency

The microlensing optical depth $\tau$ is defined as the probability that the
line of sight to a source is within one $r_{E}$ of a lens (corresponding to
$A_{max}>1.34$ according to Eq. (1)). The event rate is the number of
microlensing events per star per unit time with $u_{0}<1$. This rate is
related to the optical depth by the following relation:
$\Gamma=(2/\pi)\tau/\langle t_{E}\rangle$ (Griest 1991). The event rate is
estimated from the duration distribution of the observed events, and it must
take into account the observational average detection efficiency that depends
a priori on the time sampling, the photometric accuracy, the source
population, the background noise, and the search algorithm. The probability of
detecting a given event depends on the parameters of this event ($u_{0}$,
$t_{0}$, $t_{E}$, and source magnitude), and the efficiency $\epsilon(t_{E})$
is defined as the ratio between the following two numbers: The numerator is
the sum of the detection probabilities of all events with a measured duration
$t_{E}$ and a measured $u_{0}<1$ (regardless of their other parameters), and
the denominator is the total number of expected events with true duration
$t_{E}$, true $u_{0}<1$, and true $t_{0}$ within the survey duration. This
efficiency is usually calculated by a Monte-Carlo simulation of microlensing
events, combined with observed light-curves of stable stars to produce
realistic light-curves of microlensing events, and then subjected to the same
analysis as the real data.

Basic Monte-Carlo efficiency estimates assume that the events are of the PSPL
rectilinear type, that is that the magnification curve is given by equations
(1) and (4). Concerning the studies toward the LMC, it has been established
that the contribution of the events that could escape detection due to lens
binarity does not exceed $10\%$ (Mróz et al. 2019); the impact of the parallax
effects (due to the orbital motion of the Earth around the Sun), which distort
the magnification curve, is also negligible toward the LMC (Blaineau & Moniez
2020). The subject of this paper is to explore the impact of blending (see
Fig. 1) on the detection efficiency toward LMC, taking into account not only
the blending of accidentally aligned sources already discussed in several
works (see for example Tisserand et al. (2007) and Wyrzykowski et al. (2011)),
but also the – non-accidental – contribution of multiple gravitationally bound
stellar systems that cannot be directly estimated from LMC spatial images.

Figure 1:  Images of the same (dense) region of the LMC taken with the EROS
camera (left) and the WFPC2 HST camera. The circles are centered on the EROS2
cataloged sources. It is clearly visible that several sources resolved in the
HST image can merge into a single EROS cataloged object. We note that even in
the HST image, the contribution of hypothetical binary systems is not
resolved, and it exceeds the expectation of a random spatial distribution (see
Sect. 5).

##  3 Blending effects in microlensing searches: The unknown contribution of
binary stars

In the case where a star, whose flux $\phi_{1}$ is amplified by a microlensing
effect by a factor $A(t)$, is mixed with the light of one or more other stars
whose summed flux is $\phi_{2}$, the evolution of the total flux of the source
is the following:

$\phi(t)=\phi_{1}A(t)+\phi_{2}.$ (6)

By defining the blend fraction as the ratio
$f_{b}=\frac{\phi_{1}}{\phi_{1}+\phi_{2}}$,

$\phi(t)=(f_{b}A(t)+(1-f_{b}))(\phi_{1}+\phi_{2}).$ (7)

If ignored – that is using the PSPL approximation –, blending may affect the
efficiency of a microlensing search in several ways. On the one hand, the fact
that a cataloged object is made up of light from several unresolved stars must
be taken into account when estimating the effective number of stars monitored.
A microlensing event can indeed occur on each of the stars composing the
cataloged source, increasing the number of stars actually susceptible to
lensing compared to the number of sources counted in the catalog. On the other
hand, the light contribution of other stars modifies the shape of the observed
light-curve from a microlensing event, in a way that changes with the
passband. The efficiency of the search algorithms may then be affected by the
following effects:

The shape of the light-curve can no longer be exactly fitted by that of a
simple (PSPL) event, which could cause events to be missed with a low-
tolerance algorithm if blending is ignored. In particular, the apparent or
effective magnification $A_{\text{eff}}$ of the object as observed from the
ground is the following:

$A_{\text{eff}}(t)=\frac{\phi(t)}{\phi_{1}+\phi_{2}}=f_{b}(A(t)-1)+1,$ (8)

then

$\frac{A_{\text{eff}}(t)-1}{A(t)-1}=f_{b}<1,$ (9)

showing that the apparent magnification of the observed object is
systematically lower than the real magnification $A(t)$ of the lensed star at
all times (Stefano & Esin 1995). Therefore, the measured (or apparent) impact
parameter $u_{0}$ is larger than the physical (true) impact parameter.

As soon as the blend is composed of stars of different colors, the apparent
magnification is no longer achromatic: This could reduce the efficiency of a
search algorithm that would only consider achromatic events.

The apparent Einstein duration $t_{E}$ estimated by fitting the light-curve of
a blended event with the curve of a PSPL event is systematically shorter than
its real value. Since the efficiency of the analysis depends on the measured
duration of the event, this means that there is also a modification of the
expected number of events as a function of the duration.

All of these effects may impact the number of expected events in a nontrivial
way and can potentially modify it, so it is necessary to properly take this
blending effect into account for the interpretation of microlensing effect
searches, especially when evaluating the optical depths. Mróz et al. (2017)
have established that a complete fit of the flux light-curve – that is to say
beyond PSPL approximation –, which takes a blend fraction into account as in
Eq. (7), is able to restitute the correct parameters with no bias. However,
the increase in the number of potential sources and the reduced sophistication
of detection algorithms mean that detection efficiency can be affected by
blending.

As previously mentioned, the confusion of several stars in a single cataloged
object has several possible origins: The superposition may be due to an
accidental alignment (accidental blending), depending on the field crowding,
or it may be due to the multiplicity of the stellar systems (mainly binary
blending). The estimation of accidental blending, as it is classically
obtained, makes the assumption of a locally uniform density distribution of
stars; on the other hand, the impact of superposition due to multiple systems,
the binary blending, cannot be estimated in the same way.

Considering the case of two stars blended into a single catalog object, we
distinguish three blending regimes depending on their angular separation
$\delta$:

  * •

As long as $\delta$ is large compared to both angular Einstein radii of the
lens configurations111If the two stars are not at the same distance $D_{L}$,
then the Einstein rings for microlensing by the same given lens are
different., we can consider the two stars as independent in the context of the
calculation of the expected number of events: We refer to this mixing regime
as classic or ordinary since it is the one that occurs almost exclusively in
the case of accidental mixing, and we then observe the addition of a constant
flux to an amplified flux. Previous studies have shown that the impact of this
blending regime is small ($<10\%$) and positive on the detection efficiency
(Wyrzykowski et al. (2011); Blaineau et al. (2022) and references included).

  * •

If the angular separation $\delta$ between the two stars is small compared to
the angular Einstein radii of both lens configurations, then both components
undergo the same apparent magnification simultaneously, and everything happens
as if a single star with a flux equal to the sum of the two stellar fluxes
undergoes a PSPL microlensing effect, as in the absence of blending. According
to simple geometrical considerations, this regime has a negligible probability
of occurring for accidental blending, but not in the case of blending due to
binarity. In the latter case, the Einstein radii are identical for the
microlensing of both stars. Ignoring this blending regime has no consequence
on the optical depth measurement.

Figure 2:  Lens paths and associated light-curves.
(top) Three different trajectories of a deflector (red dashed) in front of the
two stars A and B blended into a single cataloged source; on the left, stars A
and B share $80\%$ and $20\%$ of the total luminous flux, respectively, and on
the right both stars have the same luminosity. The black outline corresponds
to the positions where the deflector must be for the source system to undergo
a total magnification of 1.34. The coordinates are in units of Einstein
radius, which is the same for both stars (located at the same distance).
(bottom) Magnification curves of the cataloged source system corresponding to
the trajectories above, which travelled from left to right. The black line
corresponds to a magnification of 1.34. We note that $t_{origin}$ is the time
when the deflector is closest to the star A (and thus does not always coincide
with the time of maximum apparent amplification). Curves 3 and 6 are very
close to microlensing curves with ordinary blending because the deflector
passes far from star B.

  * •

If the angular separation $\delta$ between the two stars is on the order of
the angular Einstein radii of both lens configurations, then that would be an
intermediate regime, where the light-curve can neither be described by a PSPL
microlensing effect nor by the addition of a constant flux to an amplified
flux. In this case, we observe the superposition of two events, with different
magnifications, and maxima shifted in time. Depending on the geometrical
configuration, two clearly separated peaks or a single asymmetric peak may
appear. Several examples of magnification curves are shown in Fig. 2, with the
corresponding geometrical configurations of sources and deflectors. As with
the previous regime, this situation can occur only in the case of blending due
to binarity, and the Einstein radii are identical for the microlensing of both
stars. Only this regime is likely to cause complications when estimating the
average detection efficiency of microlensing effects. If there were no
physical reason for binary systems to exist, this configuration would be
extremely rare and could be neglected since the probability of such a small
angular distance between uncorrelated objects, even in the densest fields, is
negligible. But when we are dealing with multiple gravitationally bound
systems, this situation can happen more frequently.

The blending has been studied with the help of HST data, in two different
ways; in the galactic plane direction, some authors (for example Rahal et al.
(2009)) have used the HST images directly to associate objects detected on the
ground with stars detected in HST, enabling the content of objects to be
studied statistically. The advantage of this approach is that it benefits from
the true spatial distribution of objects, taking any nonuniformity into
account, and it is limited only by the resolution of HST. The other way of
using HST data is indirect, and is based on extracting the luminosity function
of stars up to 3 magnitudes beyond the detection limit of ground-based
surveys. Ground-based telescope images were simulated by injecting stars
according to this luminosity function, and the star content of reconstructed
objects was then statistically estimated. Authors using the latter technique
assumed that the spatial distribution of stars is uniform and neglected the
fact that the existence of bound systems results in an excess probability of
small separation between components compared to a random distribution
(Alibert, Y. et al. 2005; Smith et al. 2007; Tisserand et al. 2007;
Wyrzykowski et al. 2010). As far as the LMC is concerned, neither approach can
take the case of binary systems into account because at the LMC distance
($49.59\pm 0.09\pm 0.54\,kpc$ (Pietrzyński et al. 2019)) no excess of star
pairs can be detected with respect to a random uniform distribution of stars
in the HST images, as we show in the next section. This means that even the
resolution of the space telescopes is insufficient to separate two components
of a binary stellar system located at $\sim 50\,kpc$.

To quantify the impact of multiple sources on microlensing toward the LMC, we
need a way to count them, based on other data than LMC spatial images. The
following section shows why the limitations of these space-based observations
toward the LMC have led us to study a population of the solar environment
using Gaia data (Gaia Collaboration et al. 2016).

##  4 Angular and physical distance separation between LMC stars detected by
HST

The spatial distribution of LMC stars is not expected to be uniform if it
includes multiple gravitationally bound systems. This is why we work with the
distribution of the number of pairs of stars as a function of their separation
(angular or spatial) and the two-point angular correlation function. The
correlation function is calculated with the Landy-Szalay estimator (Landy &
Szalay 1993), which compares the number of pairs in the data with the number
of pairs in a simulation of a uniform spatial distribution in the same spatial
domain. The other way to study multiple systems is to count the number of
pairs as a function of their angular separation $\delta$. We expect this
distribution to be the sum of a contribution due to fortuitous alignments and
an excess at the smallest values of $\delta$ if there are multiple
gravitationally bound systems in the set of the pairs. From simple geometrical
considerations, the first contribution is a distribution which increases
linearly with $\delta$ as long as $\delta$ is small enough so that edge
effects do not limit the catalog used. The second contribution, when it
exists, is expected for smaller values of $\delta$.

Figure 3:  Number of pairs and two-point correlation function of stars in a
HST-crowded LMC field image.
(top) Number of pairs of stars per $1.2^{\prime\prime}$ interval as a function
of their angular separation. The dotted line gives the predicted distribution
of angular separations in the case of a spatially uniform distribution of
stars with the same average star density. (bottom) Measured two-point angular
correlation function of the detected stars. The lower scale is the angular
separation $\delta$ and the upper scale gives the corresponding transverse
separation in the LMC.

Figure 3 shows the distribution of the number of pairs (top) and the two-point
correlation function (bottom), as a function of the angular separation
$\delta$ of the stars detected by the SExtractor algorithm (Bertin & Arnouts
1996) on an image of a crowded field of the LMC. This image has been obtained
by coadding the images taken with the f555w and f814w filters by the Wide
Field Planetary Camera 2 (WFPC2) of HST222HST images
hst_08676_08_wfpc2_f555w_wf and hst_08676_08_wfpc2_f814w_wf, at
$(RA,Dec)=(05h06m35.18s,-69\degree 20^{\prime}45.6^{\prime\prime})$.(HST
archive 2002). We notice that the correlation function remains zero down to
the HST resolution limit (about $0.5^{\prime\prime}$), decreases below
$0.5^{\prime\prime}$, and is $-1$ below $0.25^{\prime\prime}$, which indicates
that the algorithm can never resolve two stars separated by less than
$0.25^{\prime\prime}$. Since there is no positive correlation for small
separations, this shows that we also do not detect an excess of pairs in the
HST data over a random distribution beyond a separation of
$0.5^{\prime\prime}$, which corresponds to a transverse separation of
$27500\,au$ in the LMC. This situation is not surprising since gravitationally
bound systems at such distances are very rare (Dhital et al. 2010; Duchêne &
Kraus 2013). The LMC is therefore too far away for HST to resolve multiple
systems that would cause a significant excess of close pairs compared to a
random spatial distribution.

##  5 Constraining the binarity rate in the LMC

Since the LMC is too far away, we studied a stellar population close to us,
using data from the Gaia mission, and then we extrapolated the results to the
distance of the LMC. First, we defined the binarity rate we used and describe
our estimation method based on star pair counts.

###  5.1 Methodology

We wished to know the proportion of LMC objects in EROS- or MACHO-type
catalogs that are binary systems as a function of their transverse separation
$a_{t}$. As we have just seen, the LMC is too far away even for space missions
to resolve such systems. We therefore studied a population of nearby stars
cataloged by Gaia, for which the resolution of systems is possible as soon as
$a_{t}>500\,au$. In the following, object refers to an element of the Gaia
catalog (which can be made of a single star or a more complex system) and
system refers to a cluster of objects that we consider to be a gravitationally
bound system (typically a resolved binary system). We then counted the binary
stars in the local space, in a situation where they are well separated. We
defined $n^{*}_{tot}$ as the total number of objects in the catalog, and
$n^{*}_{bin}(a_{t})$ as the number of objects belonging to a resolved binary
system (i.e., made of two cataloged objects), with a transverse separation
$>a_{t}$. From these simple counts of the resolved double star systems, we
could numerically compute the following function:

$f_{bin}(a_{t})=\frac{1}{n^{*}_{tot}}\frac{dn^{*}_{bin}(a_{t})}{da_{t}}.$ (10)

This function is the differential probability that a Gaia cataloged object
belongs to a double system separated by a projected distance $a_{t}$. This is
a function that we directly derived from the Gaia catalog, as explained in the
next sections. We then defined $F_{bin}(a_{t})$, the integrated system
binarity rate, as the ratio between the number of binary systems (unlike
objects) with a transverse separation $>a_{t}$, to the total number of
systems:

$F_{bin}(a_{t})=\frac{n^{*}_{bin}(a_{t})/2}{(n^{*}_{tot}-n^{*}_{bin})+n^{*}_{bin}/2}=\frac{n^{*}_{bin}(a_{t})/n^{*}_{tot}}{2-n^{*}_{bin}/n^{*}_{tot}}.$
(11)

We note that the total number of systems at the denominator corresponds to the
sum of the single star systems plus all the resolved binary systems – not only
the systems separated by more than $a_{t}$–333As no binary system is resolved
in the LMC, we need to use this definition of a rate relative to the systems,
instead of a rate relative to the – unknown – underlying number of stars..
From the definition of $f_{bin}(a_{t})$, we obtained

$\frac{n^{*}_{bin}(a_{t})}{n^{*}_{tot}}=\int_{a_{t}}^{\infty}f_{bin}(a)da,$
(12)

which is the fraction of objects belonging to a binary system with a
transverse separation $>a_{t}$. The number $n^{*}_{bin}$ is the total number
of objects belonging to a binary system resolved by Gaia (depending on Gaia’s
resolution power). This number was estimated from

$\frac{n^{*}_{bin}}{n^{*}_{tot}}=\int_{a_{t}^{resol}}^{\infty}f_{bin}(a_{t})da_{t},$
(13)

where $a_{t}^{resol}$ is the separation limit of Gaia in the catalog we used.
As Gaia’s resolution is not a step function, we started integrating from a
very small separation $a_{t}^{resol}=50\,au$, below which we were sure Gaia
could not separate two components, but to which we extrapolated our
measurements, as shown in Section 6. This conservative choice then gave us a
lower bound on $n^{*}_{bin}$, and thus an upper bound on $F_{bin}$ in equation
(11). This choice is not critical since we subsequently show that the integral
(13) is much smaller than two, and it therefore has a minor impact on the
denominator of Eq. (11).

###  5.2 Binarity of stars closer than $500\,pc$ in the Gaia-EDR3 catalog

We have used the Gaia-EDR3 catalog (Gaia Collaboration et al. 2020) for a
preliminary assessment of the impact of binarity in microlensing studies. This
exploratory work does not aim to extract binarity rates, but rather upper
limits on the number of multiple source systems that can give rise to complex
gravitational microlensing effects. Our immediate goal is therefore to
estimate the maximum fraction of binary systems separated by transverse (or
projected) distances $a_{t}$ comparable to or larger than the Einstein radius
of a possible lens projected in the LMC.

In our definitions of the binarity rate, we have neglected the contribution of
stellar systems made of more than two stars, since we find only $1\%$ of such
systems with an angular size smaller than $10^{\prime\prime}$ (corresponding
to $a_{t}<6000\,au$ at a $600\,pc$ distance) in the selection of stars defined
in the next paragraph. We therefore neglected this type of system in our
treatment of blending effects. The estimate of the transverse (or projected)
physical distance $a_{t}$ of two stars in the Gaia catalog was deduced from
the angular distance $\delta$ between the two components and their annual
parallax measurements $\pi$, assuming that they both have an intrinsic uniform
and rectilinear motion444 Since we are trying to quantify binary systems, we
must remember that the rotation of the stars of a system around the center of
gravity alters the uniform rectilinear motion assumed in the estimation of
parallaxes. As we are only interested in binary systems separated by more than
$50\,au$, with orbital periods on the order of a few centuries, we can
consider that the variation of orbital velocity on the apparent trajectory of
the stars on the sky has only a negligible impact on the estimation of
parallaxes in Gaia.  (Lindegren et al. 2012, 2021), by
$a_{t}=1\,au\times\delta/\pi$. With the parallax differences between the
components being too imprecise compared to the angular accuracies, we could
not carry out a study in three dimensions, which would include the
longitudinal distance (along the line of sight). As we explain in the
following analysis, parallaxes were just used to convert angular distances
into transverse distances and also to reduce the risks of association of very
distant objects, accidentally located on the same line of sight, by
considering only pairs of stars with compatible parallaxes.

We selected the stars of the Gaia-EDR3 catalog (Gaia Collaboration et al.
2020), whose apparent magnitude is $3<g<18$, a domain in which the catalog is
complete, and thus not biased (Fabricius et al. 2020). Then we limited our
sample to stars closer than $600\,pc$ (corresponding to a parallax
$\pi>1.66\,mas$). At this distance, a separation of $1^{\prime\prime}$
corresponds to $600\,au$. We also required that the parallax accuracy be
better than $20\%$. The bias (Luri et al. 2018) induced by this selection can
be neglected because we rejected only $2\%$ of the stars that are the most
distant of our sample. Finally, we have restricted our study to Galactic
latitudes higher than $20\degree$ to avoid the very crowded and inhomogeneous
areas of the Galactic plane (figure 4).

Figure 4:  Projection perpendicular to the Galactic plane of the spatial
distribution of the stars of the Gaia catalog, and representation of the
limits of the shells used in our study. The representation is centered on the
Sun.  Figure 5:  Distribution of the number of stars in Gaia EDR3 as a
function of their distance and absolute magnitude (left scale). The right
scale gives the apparent magnitude that these stars would have in the LMC. The
Gaia catalog is complete between the thick red curves. The red lines are the
iso-apparent-magnitude lines in Gaia in the (distance, absolute magnitude)
plane. We restricted the study to stars within the absolute magnitude range
delimited by the thick horizontal black lines ($-0.5<G<9.5$), which ensures
the homogeneity of the stellar type over the volume we considered; the
vertical solid lines correspond to the distance limits of our sample. The
vertical dashed black lines delineate the distance domains of the shells we
consider in our analysis (between $50$ and $500\,pc$).

Figure 5 shows the distribution of the absolute magnitude $G$ of the stars of
the Gaia catalog closer than $600\,pc$, as a function of their distance,
estimated by their parallax. This catalog is complete between the lines of
equal apparent magnitude marked by the thick red curves ($g=3$ and $g=18$). We
defined five shells (Fig. 4 and 5), delimited by the spheres of radii (50,
140, 230, 320, 410, and $500\,pc$), and studied the pairs of stars in each
shell separately (Table 1). These five shells define a fiducial volume that is
not affected by the $600\,pc$ cutoff.

|  $a_{t}$ for $\delta=2^{\prime\prime}$ | number of stars
---|---|---
shell limits | separation | in shell | effective
$[50-140]\,pc$ | $[100-280]\,au$ | 160409 | 158000
$[140-230]\,pc$ | $[280-460]\,au$ | 481496 | 465000
$[230-320]\,pc$ | $[460-640]\,au$ | 867259 | 807000
$[320-410]\,pc$ | $[640-820]\,au$ | 1259719 | 1109000
$[410-500]\,pc$ | $[820-1000]\,au$ | 1634717 | 1313000
Table 1: Limits and contents of the shells (see text).

These subdivisions allowed us to verify that the separation distributions of
the binaries are indeed functions of the physical distances $a_{t}$,
independently of the angular distances $\delta$, as long as they are greater
than the separation power of Gaia. To ensure that the distributions studied in
the five shells considered are all complete, we only studied the stellar
population whose absolute magnitude is in the range $-0.5<G<9.5$ (between the
thick black lines in Figure 5). Finally, we only considered pairs of stars
with a magnitude difference of less than $2.5$, beyond which we could neglect
the impact of the less luminous component on the light-curve of a microlensing
effect.

Figure 6:  Number of star pairs per $0.5^{\prime\prime}$ interval as a
function of their angular separation. The dotted line gives the predicted
distribution in the case of a spatially uniform distribution with the same
average star density. Only stars within $600\,pc$ of the Sun and with a
Galactic latitude larger than $60\degree$ are considered here (1,233,947
stars). We measured about 29,000 pairs in excess of accidental pairs.

Figure 6 shows the angular separation distribution $\delta$ for the pairs of
stars closer than $600\,pc$. We first observed a clear excess of pairs for
$\delta<20^{\prime\prime}$, in agreement with Zavada & Píška (2020), which
demonstrates the existence of Gaia-detectable bound systems within $600\,pc$,
in contrast to the situation in Fig. 3 which showed the inability of HST to
separate bound systems in the LMC. Second, we noted that similar to HST, our
algorithm failed to separate stars that are too close in angular distance, in
this case around $0.5^{\prime\prime}$. The question of the separation limit,
as it also relates to Gaia’s scanning law, is beyond the scope of this paper
(see Blaineau (2021) for more details). For our study, it is sufficient to
know that we decided to focus only on pairs separated by at least
$2^{\prime\prime}$ in order not to be limited by the resolution of Gaia. We
have shown that the results of the next sections vary by less than $2\%$ (half
of the estimated uncertainty) if we change this minimal separation in the
range $[1.5^{\prime\prime}-2.5^{\prime\prime}]$ (Blaineau 2021).

To estimate the binarity rate of stars in a shell, we first built the
distribution of the type shown in Figure 6. We then calculated the number of
pairs remaining after subtracting the component due to accidental alignments,
as a function of the separation. This expected random component is a linear
function, fitted between $60^{\prime\prime}$555Two stars at $\sim 450\,pc$
separated by $60^{\prime\prime}$ would be separated by only
$0.5^{\prime\prime}$ if they were in the LMC. This directly illustrates the
fact that an excess count of pairs in the LMC is only expected for separations
much smaller than the resolution limit of HST. and $120^{\prime\prime}$
(dashed line in Figure 6). Once the angular separation was converted to
physical transverse separation $a_{t}$ (by using the measured parallax $\pi$),
from the excess in each channel of $a_{t}$ we could derive an estimate of the
differential stellar binarity rate $f_{bin}(a_{t})$ defined by expression
(10), plotted in Figure 7 for each shell. We would like to remind readers that
this quantity represents the differential fraction of stars in pairs in excess
of the expected number of accidental pairs separated by $a_{t}$, per unit of
$a_{t}$.

Figure 7:  Differential rate of binaries $f_{bin}(a_{t})$ measured in the five
shells as a function of transverse separation $a_{t}$. The red dashed line
gives the means for all shells. Error bars are not shown to avoid overloading
the figure, but the scatter of each series gives an indication.

The figure shows that the rates found do not depend on the shell considered,
which allowed us to use the average distribution, weighted by the effective
number of stars in each shell. These effective numbers are the numbers of
stars found within the intervals of parallax of each shell, which were
statistically corrected for misattributions of pairs in the shells, due to
uncertainties in the measured parallaxes 666 These uncertainties result in
uncertainties on the distance of the stars; a pair can then be either
misattributed if both components are effectively in another shell, or it can
be missed if only one of the components is in another shell.  (see also Table
1). It should be noted that the closest shells are obviously those that
allowed us to estimate the binarity rate at the smallest separations $a_{t}$,
but at the expense of the statistics limited by the small volume of the shell.
For shells at larger distances, the statistics is larger, but our angular
separation limit of $2^{\prime\prime}$ prevents estimates at the smallest
physical separations.

##  6 Estimation of the local stellar binarity rate

In this section, we explain how we fit and extrapolated the differential
binarity rate down to $50\,au$ transverse separations. Following Duquennoy &
Mayor (1991) and Raghavan et al. (2010), we considered a log-normal model for
the distribution of semi-major axes $a$ of binary systems. Since we measured
projected separations $a_{t}$ and not semi-major axes, this distribution was
modified as follows, by considering that the orbits have no preferred
inclination:

$f_{bin}^{DM}(a_{t})=A\int^{+\infty}_{a_{t}}\frac{a_{t}}{r^{2}\sigma\sqrt{2\pi(r^{2}-a_{t}^{2})}}\exp{-\frac{\left(\ln{r/r_{\text{mode}}}-\sigma^{2}\right)^{2}}{2\sigma^{2}}}dr,$
(14)

where $A$ and $\sigma$ were fitted to our data once $r_{mode}$ was chosen,
corresponding to the maximum of the distribution.

Figure 8:  Projected log-normal function $f_{bin}^{DM}$ for $r_{mode}=0.1\,au$
(dotted line), and power function $f_{bin}^{PL}$ (full line), fitted to the
differential fraction of binary, as a function of the projected separation.
The vertical red line corresponds to a projected separation of $200\,au$. The
inset shows that the fit of $f_{bin}^{DM}$ is not satisfactory at small
separations.

If we chose $r_{mode}=0.1\,au$, then we find $A=0.126$ and $\sigma=2.72$, but
the fit is unsatisfactory for $a_{t}<600\,au$, and this is the case regardless
of the parameter $r_{mode}$ (Figure 8). This poor agreement can probably be
attributed to the fact that this log-normal distribution was established for
solar-type stars, while we studied a different and more extended population.
Nevertheless, following this fitted model, we computed the binarity rate for
systems with $a_{t}>200\,au$ 777This length corresponds to the most likely
projected Einstein radius (at the LMC) of a $\sim 50M_{\odot}$ Galactic halo
lens magnifying light from a LMC source. using Eqs (13), then (12) and (11),
and found $F_{bin}(200\,au)=2.76\pm 0.03\%$. By varying the $r_{mode}$
parameter within $10^{-4}\,au<r_{mode}<10^{2}\,au$, we found values close to
each other, contained within $2.05\%<F_{bin}(200\,au)<2.8\%$. Similarly, we
found that $F_{bin}(100\,au)<3.6\%$ regardless of the $r_{mode}$ parameter.

We also fitted the following empirical power-law parameterization to our
measurements:

$f_{bin}^{PL}(a_{t})=(0.19\,au^{-1})\times(a_{t}/1\,au)^{-1.379},$ (15)

which better describes our data for small $a_{t}$ values (Figure 8). With this
alternative model for the distribution of projected separations of binaries,
we found that $F_{bin}(200\,au)=3.48\pm 0.03\%$, a value compatible with that
found with the previous model.

In section 5, we mentioned that the total probability of a Gaia object being a
member of a resolved binary system is less than the value given by expression
(13). The value we found by integrating Eq. (15) from $a_{t}=50\,au$ is
$0.114$, which is an upper limit of the fraction of Gaia objects belonging to
binaries resolved in our sample. This number is indeed small compared to two,
and its exact value does not impact the computation of $F_{bin}$ from Eq.
(11), especially since we are interested in the upper limit. It should be
further noted that nothing can be said from our data about the binarity rate
for $a_{t}<50\,au$ because extrapolation below this value is not constrained.

##  7 Extrapolation at the LMC; impact of binarity on microlensing detection

The previous study concerns a population of nearby Milky Way stars with
$-0.5<G<9.5$. In the LMC, they would have an apparent magnitude $18<g<28$,
that is they would be among the faintest stars in a classical catalog
searching for gravitational microlensing effects. This is a limitation of the
extrapolation of this work to observations toward the LMC, in addition to the
fact that we assume that this stellar population has the same binarity
statistical characteristics in the LMC as in the Milky Way disk.

We saw in section 3 that if the angular separation of the components of a
blend is much smaller than the angular Einstein radius – expressed by
$a_{t}<<R_{E}/x$, where $x$ is given by Eq. (3) –, then both sources undergo
roughly the same magnification and everything happens as if there were only
one source. Ignoring this blending has therefore no impact on the detection
efficiency of the ground surveys. As a consequence, only the pairs with
$a_{t}\sim R_{E}/x$ or $\geq R_{E}/x$ should be considered for a possible
impact on the statistics of microlensing effects. From the
$f_{bin}^{PL}(a_{t})$ function of the differential binarity rate and using Eq.
(11), we can estimate the maximum proportion of situations where binarity
induces a risk of failure to detect microlensing effects due to light-curve
distortion.

Figure 9:  Rates of binarity $F_{bin}$ as a function of the projected Einstein
radius of the lens $R_{E}/x$, obtained for different ways of integration: from
a component separation larger than $R_{E}/x$ (solid red line), or larger than
$R_{E}/2x$ (dashed red line), or by weighting contributions with a ramp
function (black line) (see text). Typical masses corresponding to the Einstein
radii are plotted on the upper abscissa for sources within the LMC. This rate
is an upper limit to the risk of not detecting microlensing effects due to
binarity-induced distortion of the light-curve.

Figure 9 shows this maximum proportion as a function of the Einstein radius
projected in the LMC, $R_{E}/x$, after integration over the distribution of
$a_{t}$ under three different assumptions: assuming that the blend effect is
significant as soon as $a_{t}>R_{E}/2x$ (and thus integrating the differential
distribution from $R_{E}/2x$ to infinity), or only when $a_{t}>R_{E}/x$, or
weighting the differential distribution between zero and one (ramping) when
$a_{t}$ varies from $0.1R_{E}/x$ to $1.75R_{E}/x$. The latter assumption was
derived from the study of Griest & Hu (1992), which discusses the distortions
expected by a microlensing curve for a composite source in detail. Figure 9
shows, for example, that under the most pessimistic assumption (a blending
effect taken into account as soon as $a_{t}>R_{E}/2x$), no more than $7\%$ of
the sources are binary systems with a luminosity difference of less than 2.5
magnitude between the components, a configuration that could affect the
identification of light-curves of microlensing events when $R_{E}/x>50\,au$.

##  8 Discussion

###  8.1 Field of application

One must first remember the limitations of this work. Only the population of
stars of absolute magnitude $-0.5<g<9.5$ could be thoroughly studied, and any
use for another population is an extrapolation, either for an identical
population though in another galaxy possibly with a different metallicity, or
for a stellar population in the Milky Way with more extensive types.

Figure 10:  Distribution of the projected Einstein radius
$R_{E}/x=R_{E}D_{S}/D_{L}$ for the microlensing events toward the LMC due to
$1M_{\odot}$ lenses within the standard Galactic halo (blue line). The
distribution expected for other lens masses $M_{L}$ can be obtained by scaling
the abscissa by the factor $\sqrt{M_{L}/M_{\odot}}$. The purple line shows the
cumulative distribution. The most probable value of $R_{E}/x$ is $28.6\,au$,
the mean is $44.0\,au$, and the median is $36.5\,au$.

To compute the fraction of potentially complex events due to binarity as a
function of the lens mass, we combined the rates from Fig. 9 with the
projected Einstein radius $R_{E}/x$ distribution expected for that lens mass.
Figure 10 shows this generic distribution for lenses of $1M_{\odot}$ mass,
changing with $M_{L}$ by simply scaling the abscissa with
$\sqrt{M_{L}/M_{\odot}}$. This has been established by assuming that the
lenses are spatially distributed according to the standard Galactic halo
described in Blaineau et al. (2022). Table 2 shows, for a series of lens
masses, the maximum expected fractions of situations where the shape of the
microlensing events can be affected by binarity, splitted into three domains
of projected Einstein radius $R_{E}/x$. For each lens mass, the event fraction
for each domain was deduced from the properly scaled distribution of Fig. 10:
For $R_{E}/x<50\,au$, we conservatively assumed that $100\%$ of the events can
be affected by blending due to binarity; for $50\,au<R_{E}/x<1000\,au$, we
integrated the most pessimistic function of Fig. 9, weighted by the normalized
distribution of $R_{E}/x$ ; for $R_{E}/x>1000\,au$, we considered that no more
than $2\%$ of the events can be affected by blending due to binarity.

$M_{L}$ ($M_{\odot}$) | 1 | 3 | 10 | 30 | 100
---|---|---|---|---|---
$R_{E}/x<50\,au$ | 71% | 34% | 9% | 2% | 0.4%
max. contribution | 71% | 34% | 9% | 2% | 0.4%
$50<R_{E}/x<1000$ | 29% | 66% | 91% | 97% | 95%
max. contribution | 1.8% | 3.9% | 4.6% | 4.0% | 4.8%
$R_{E}/x>1000\,au$ | 0% | 0% | 0.2% | 0.8% | 5%
max. contribution | 0% | 0% | 0% | 0% | 0.1%
Total | 73% | 38% | 14% | 6.2% | 5.3%
Table 2: Fractions of events toward the LMC within the three domains of the
projected Einstein radius and their maximum contributions to the fraction of
events with significant blending due to binarity.

The totals given in the table are the maximum proportions of events whose
shape can be affected by binarity. This does not necessarily mean that such
events systematically escape detection (a lot depends on the selection
algorithm). Nevertheless, for the sake of caution, we will keep these
proportions as an upper limit for situations where the classic detection
efficiency calculation does not apply. In the absence of a specific
simulation, the status of these events on binaries in terms of the detection
efficiency is indeed poorly known, and this must be taken into account as a
systematic uncertainty for the measurement of optical depths and event rates.

Our study shows that the impact of source binarity can be neglected to first
order when searching for gravitational microlensing effects from lenses
heavier than $30M_{\odot}$. The maximum value of uncertainty of $6.2\%$ is
even smaller when the lenses are more massive (Fig. 10), showing that the
additional effect of binarity on the blending effects can be neglected for
heavy lenses, as was done in Blaineau et al. (2022). Figures 9 and 10 allows
for these numbers to be estimated in the case of even more massive lenses
toward the LMC. For an estimate corresponding to another Galactic halo model,
or to other targets than LMC, Fig. 10 needs to be rebuilt.

For events due to lighter lenses with a higher probability of projected
Einstein radii $R_{E}/x<50\,au$, it is currently not possible to draw reliable
conclusions on the impact of the blend due to the binarity of the LMC sources
with our technique without risky extrapolation. An alternative method for
estimating the differential binarity rate with a projected separation of less
than $50\,au$ is needed, coupled with a specific simulation to estimate the
detection efficiency of non-PSPL events.

Skowron et al. (2009) found that $0.5\%$ of the events observed by OGLE toward
the Galactic Center (not LMC) have two peaks because they are due to binary
systems (lens or source). This suggests that a larger proportion must suffer
lesser distortions (Di Stefano & Mao 1996), but this leads in particular to
them being assigned an incorrect duration, and this may affect the efficiency
calculation. It is not easy to extrapolate these observations to the LMC (but
see Griest & Hu (1992)), but nonetheless it seems to us that the question is
not strictly closed for microlensing toward the LMC caused by light halo
objects, even if it is likely that the effect we are talking about is
negligible.

###  8.2 Discussion on stellar populations

Since the majority of the stars observed in the LMC correspond to the
brightest stars in the Gaia data that we used (Fig. 5), we investigated the
variation of the binarity rate $F_{bin}(200\,au)$ with the magnitude of the
stars. We therefore reproduced our analysis by restricting our sample to two
absolute magnitude $g$ domains, between -0.5 and 4.5 mag ($3,090,000$ stars)
and between 4.5 and 9.5 mag ($612,000$ stars), always limiting the maximum
difference in magnitude between the two components to 2.5. The statistical
power here was significantly reduced for the population with $-0.5<g<4.5$, but
we nevertheless found a significantly lower binarity rate for bright star
pairs ($F_{bin}(200\,au)=1.42\pm 0.08\%$) than for faint star pairs
($F_{bin}(200\,au)=2.58\pm 0.04\%$). It is therefore likely that in LMC
catalogs such as the EROS2 or MACHO surveys, which are composed of rather
bright stars, we overestimated the binarity rate by assuming that the stellar
populations from our Gaia sample and in the LMC are similar.

(a) Accidental pairs

(b) Pairs $1000\,au<a_{t}<2500\,au$

(c) Pairs $a_{t}<1000\,au$

Figure 11:  Magnitude differences between components of pairs, for different
separation ranges. (a) Magnitude differences for pairs separated by more than
$30000\,au$ (only due to fortuitous alignments). (b) and (c) Same
distributions for pairs separated by $1000<a_{t}<2500\,au$ and
$a_{t}<1000\,au$ (blue), distributions for the fortuitous alignments (orange),
normalized according to the sample size, and the difference between the two
distributions (green; i.e., the distribution of the magnitude difference for
binary systems only). These histograms are sampled in 0.05 magnitude steps
with one entry per star. In panel (c) the number of fortuitous alignments is
negligible, and thus the orange distribution is almost zero and green and blue
distributions are superimposed.

We also investigated whether there are correlations between the magnitude
differences and the separation $a_{t}$ of binaries. Figure 11 shows in green
the distributions of magnitude differences $\Delta G$ between components for
binaries with $1000\,au<a_{t}<2500\,au$ (Fig. 11(b)) and with $a_{t}<1000\,au$
(Fig. 11(c)), obtained by subtracting the expected distributions for random
pairs (Fig. 11(a)), properly normalized, from the observed (blue)
distributions. The distribution for random pairs was deduced from that of the
difference $\Delta G$ of pairs of uncorrelated stars, separated by
$a_{t}>30000\,au$ – which turns out to be approximately uniform. It appears
that the closer the binary system is, the smaller the difference in magnitude
between components. It is tempting to explain this difference by the
intervention of a gravitational capture mechanism, which would favor the
formation of distant binaries, whose luminosity would be consequently less
correlated. These observations are corroborated by the distribution of mass
ratios as a function of the semi-major axis or period of the binary systems in
Moe & Di Stefano (2017).

Finally, we examined the case of red giants, which constitute an important
part of the catalogs of the historical microlensing surveys to the LMC.
Unfortunately, they represent too few stars in our Gaia sample to establish a
reliable binarity rate. However, we examined the magnitude distribution of
stars in pairs with separation $a_{t}<2500\,au$ containing at least one giant.
In this sample of pairs, we found that there are almost no giant binary
systems (less than $1\%$ of the sample), but only binaries consisting of a
giant and a main sequence star, with magnitude difference $\Delta G$ larger
than $2.5$ in $85\%$ of the cases. This fact reinforces our conclusion that we
probably overestimated the proportion of binaries in EROS2/MACHO-type catalogs
by extrapolating the binarity rate measured in Section 7.

##  9 Conclusions: The impact of binarity on microlensing surveys

We conclude from this study that the search detection efficiency of long
duration microlensing events due to lenses heavier than $30M_{\odot}$ toward
the LMC is not significantly affected by the source binarity. This result is
useful not only in the recent combined analysis of EROS and MACHO data
spanning several decades (Blaineau et al. 2022), but also in future research
with Rubin-LSST.

On the other hand, for lenses lighter than $30M_{\odot}$, as soon as the
projected Einstein radius is less than a few tens of $au$, the binarity rates
extrapolated here are higher and less reliable. The fraction of events that
may be affected by blending due to source binarity becomes less negligible,
and another study must be undertaken to estimate the impact of a binarity rate
that may be more significant and probably positive on the detection
efficiency.

Although it is not possible to estimate from the Gaia database the
differential binarity rate for $a_{t}<50\,au$ in the LMC, one can conversely
consider detecting binarity effects by measuring distortions due to blending
with respect to a simple (PSPL) microlensing effect. The fraction of cases
where a fit with blend (according to Eq. (7)) is significantly better than the
PSPL fit would allow us to extract constraints on the binarity rate. This
should be particularly relevant to Rubin-LSST observations if the photometric
accuracy reaches a few milli-magnitudes.

Our study of the impact of binarity rates on microlensing detection efficiency
toward the LMC is easily transferable, through some scaling and modelization
of the lens spatial distribution, to studies of microlensing within the
Galactic plane. In this case, the source population should better resemble the
one studied in this paper. Our last comment is that the use of a tolerant
prefiltering, not too sensitive to the precise shape of the magnification
curve, remains the safest technique to mitigate the possible effects of
distortion due to the binarity of the source on the detection efficiency.

###### Acknowledgements.

We thank Olivier Perdereau for his useful comments on the manuscript. This
work was supported by the Paris Île-de-France Region.

## References

  * Alcock et al. (1993) Alcock, C., Akerlof, C., Allsman, R., et al. 1993, Nature, 365
  * Alibert, Y. et al. (2005) Alibert, Y., Chabrier, G., & Massacrier, G. 2005, A&A, 442, 501
  * Aubourg et al. (1993) Aubourg, E., Bareyre, P., Bréhin, S., et al. 1993, Nature, 365, 623
  * Bertin & Arnouts (1996) Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393
  * Blaineau (2021) Blaineau, T. 2021, PhD thesis, Ecole doctorale n°576, Particules Hadrons Energie et Noyau: Instrumentation, Image, Cosmos et Simulation
  * Blaineau & Moniez (2020) Blaineau, T. & Moniez, M. 2020, A&A, 636, L9
  * Blaineau et al. (2022) Blaineau, T., Moniez, M., Afonso, C., et al. 2022, A&A, 664, A106
  * Dhital et al. (2010) Dhital, S., West, A. A., Stassun, K. G., & Bochanski, J. J. 2010, The Astronomical Journal, 139, 2566
  * Di Stefano & Mao (1996) Di Stefano, R. & Mao, S. 1996, ApJ, 457, 93
  * Duchêne & Kraus (2013) Duchêne, G. & Kraus, A. 2013, Annual Review of Astronomy and Astrophysics, 51, 269
  * Duquennoy & Mayor (1991) Duquennoy, A. & Mayor, M. 1991, A&A, 500, 337
  * Fabricius et al. (2020) Fabricius, C., Luri, X., Arenou, F., et al. 2020, arXiv e-prints, arXiv:2012.06242
  * Gaia Collaboration et al. (2020) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2020, arXiv e-prints, arXiv:2012.01533
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Griest (1991) Griest, K. 1991, ApJ, 366, 412
  * Griest & Hu (1992) Griest, K. & Hu, W. 1992, ApJ, 397, 362
  * HST archive (2002) HST archive. 2002, https://archive.stsci.edu/
  * Landy & Szalay (1993) Landy, S. D. & Szalay, A. S. 1993, ApJ, 412, 64
  * Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2
  * Lindegren et al. (2012) Lindegren, L., Lammers, U., Hobbs, D., et al. 2012, A&A, 538, A78
  * Luri et al. (2018) Luri, X., Brown, A. G. A., Sarro, L. M., et al. 2018, A&A, 616, A9
  * Moe & Di Stefano (2017) Moe, M. & Di Stefano, R. 2017, ApJS, 230, 15
  * Mróz et al. (2017) Mróz, P., Udalski, A., Skowron, J., et al. 2017, Nature, 548, 183
  * Mróz et al. (2019) Mróz, P., Udalski, A., Skowron, J., et al. 2019, ApJS, 244, 29
  * Paczynski (1986) Paczynski, B. 1986, ApJ, 304, 1
  * Pietrzyński et al. (2019) Pietrzyński, G., Graczyk, D., Gallenne, A., et al. 2019, Nature, 567, 200
  * Raghavan et al. (2010) Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1
  * Rahal et al. (2009) Rahal, Y. R., Afonso, C., Albert, J. N., et al. 2009, A&A, 500, 1027
  * Rahvar (2015) Rahvar, S. 2015, International Journal of Modern Physics D, 24, 1530020
  * Schneider et al. (2006) Schneider, P., Kochanek, C., & Wambsganss, J. 2006, Gravitational Lensing: Strong, Weak and Micro (Springer Berlin, Heidelberg)
  * Skowron et al. (2009) Skowron, J., Wyrzykowski, L., Mao, S., & Jaroszyński, M. 2009, Monthly Notices of the Royal Astronomical Society, 393, 999
  * Smith et al. (2007) Smith, M. C., Woźniak, P., Mao, S., & Sumi, T. 2007, Monthly Notices of the Royal Astronomical Society, 380, 805
  * Stefano & Esin (1995) Stefano, R. D. & Esin, A. A. 1995, The Astrophysical Journal, 448, L1
  * Tisserand et al. (2007) Tisserand, P., Le Guillou, L., Afonso, C., et al. 2007, A&A, 469, 387
  * Udalski et al. (1993) Udalski, A., Szymanski, M., Kaluzny, J., et al. 1993, Acta Astron., 43, 289
  * Woźniak & Paczyński (1997) Woźniak, P. & Paczyński, B. 1997, ApJ, 487, 55
  * Wyrzykowski et al. (2010) Wyrzykowski, L., Kozłowski, S., Skowron, J., et al. 2010, Monthly Notices of the Royal Astronomical Society, 413, 493
  * Wyrzykowski et al. (2011) Wyrzykowski, L., Skowron, J., Kozłowski, S., et al. 2011, MNRAS, 416, 2949
  * Zavada & Píška (2020) Zavada, P. & Píška, K. 2020, AJ, 159, 33