# Prediction of Astrometric-Microlensing Events from Gaia eDR3 Proper Motions
111The results of the predicted microlensing events is available in electronic
form via the GAVO Data Center https://dc.g-vo.org/amlensing/q3/q/form 222The
source code for this study is made publicly available
https://github.com/jkluter/amlensing

Jonas Klüter  Department of Physics and Astronomy, Louisiana State University,
202 Nicholson Hall, Baton Rouge, LA 70803 USA  Zentrum für Astronomie der
Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14,
69120 Heidelberg, Germany  Ulrich Bastian  Zentrum für Astronomie der
Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14,
69120 Heidelberg, Germany  Markus Demleitner  Zentrum für Astronomie der
Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14,
69120 Heidelberg, Germany  Joachim Wambsganss  Zentrum für Astronomie der
Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14,
69120 Heidelberg, Germany  International Space Science Institute, Hallerstr.
6, 3012 Bern, Switzerland

(Received 13 Sep. 2021; Accepted 17 Dec. 2021)

###### Abstract

Astrometric microlensing is a unique tool to measure stellar masses. It allows
us to determine the mass of the lensing star with an accuracy of a few per
cent. In this paper, we update, extend, and refine our predictions of
astrometric-microlensing events based on Gaia’s early Data release 3 (eDR3).
We selected about 500.000 high-proper-motion stars from Gaia eDR3 with
$\mu_{tot}>100\,\mathrm{mas/yr}$ and searched for background sources close to
their paths. We applied various selection criteria and cuts in order to
exclude spurious sources and co-moving stars. By forecasting the future
positions of lens and source we determined epoch of and angular separation at
closest approach, and determined an expected positional shift and
magnification. Using Gaia eDR3, we predict 1758 new microlensing events with
expected shifts larger than $0.1\,\mathrm{mas}$ between the epochs J2010.5 and
mid J2066.0. Further we provide more precise information on the angular
separation at closest approach for 3084 previously predicted events. This
helps to select better targets for observations, especially for events which
occur within the next decade. Our search lead to the new prediction of an
interesting astrometric-microlensing event by the white dwarf Gaia
eDR3-4053455379420641152. In 2025 it will pass by a $G=20.25\,\mathrm{mag}$
star, which will lead to a positional shift of the major image of
$\delta\theta_{+}=1.2^{+2.0}_{-0.5}\,\mathrm{mas}$. Since the background
source is only $\Delta G=2.45\,\mathrm{mag}$ fainter than the lens, also the
shift of the combined center of light will be measurable, especially using a
near infrared filter, where the background star is brighter than the lens
($\Delta Ks=-1.1\,\mathrm{mag}$).

Astrometry — Proper motions— Catalogues — Gravitational lensing: Astrometric-
microlensing effect — Stellar properties: Stellar masses — White dwarf stars —

††journal: AJ††software:  astropy (The Astropy Collaboration et al., 2013),
astroquery, (Ginsburg et al., 2019), matplotlib (Hunter, 2007), numpy (Harris
et al., 2020), TOPCAT (Taylor, 2005),

##  1 Introduction

The mass of a star is one of its most important parameters. It defines its
luminosity, temperature, surface gravity, appearance, and evolutionary path.
Testing evolutionary and stellar models requires accurate and direct
measurements of fundamental stellar parameters. Direct masses are usually
derived from double-lined spectroscopic and eclipsing binaries. However, for
most of the isolated stars, masses can only be derived indirectly, typically
by using the mass-luminosity or mass-radius relations. For the determination
of such relations, a set of accurately known masses is required. These are
mainly derived from binary stars (Andersen, 1991; Torres et al., 2010).
However, binary stars and isolated stars may evolve differently. Therefore it
is not known how well these empirical relations describe the masses of single
stars. For a better understanding of the mass-luminosity relations, direct
mass measurements of single stars are important. Besides asteroseismology,
which itself is strongly model dependent, gravitational microlensing is the
only available tool. Further, the direct determined mass of white dwarfs
provides a unique test sample for comparison with theoretical mass-radius
relations and evolutionary cooling tracks of white dwarfs, a first such
measurement was achieved by Sahu et al. (2017).

As a sub-area of gravitational lensing, microlensing describes the time-
dependent positional deflection (astrometric microlensing) and magnification
(photometric microlensing) of a background source (BGS) due to an intervening
star (“lens”) (Paczyński, 1986a, b). Using microlensing, it is possible to
determine the mass of the lens star with uncertainties in the order of a few
per cent, either by detecting finite-source effects in photometric
microlensing events and measuring the microlens parallaxes (Gould, 1992) or by
observing the positional deflection of astrometric-microlensing events
(Paczyński, 1991, 1995). While the usage of photometric microlensing is
slightly dependent on empirical relations to determine the radius of the BGS,
astrometric microlensing is completely model independent and depends directly
on the mass of the lens and its distance. In the last few years, this was used
by Sahu et al. (2017) and Zurlo et al. (2018) to determine the masses of Stein
2051b and Proxima Centauri, respectively. In comparison to photometric
microlensing, astrometric microlensing has two additional advantages: It can
be observed at larger angular separations between the background source and
the foreground lens which results furthermore in a longer time-scale for the
event of the order of months to years (Dominik & Sahu, 2000). The second
advantage is the possibility to confidentially predict astrometric-
microlensing events for stars with known proper motions (Refsdal, 1964;
Paczyński, 1995). Further, these precise predictions are also needed for the
mass determination to derive the unlensed separation.

In 2000, Salim & Gould (2000) systematically searched for astrometric-
microlensing events for the first time, followed by several studies based on
non Gaia proper motions (Proft et al., 2011; Lépine & DiStefano, 2012; Harding
et al., 2018). Nowadays the most precise predictions make use of astrometric
data from the Gaia satellite (Gaia Collaboration et al., 2016). Using Gaia
DR1, the first data release, McGill et al. (2018) predicted one event caused
by a white dwarf in 2019. With the second Gaia data release (Gaia DR2, Gaia
Collaboration et al., 2018) about 5700 events were predicted, either by using
solely Gaia DR2 (Klüter et al., 2018a; Bramich, 2018; Bramich & Nielsen, 2018;
Klüter et al., 2018b; McGill et al., 2019b) or by combining it with additional
catalogues (Nielsen & Bramich, 2018; McGill et al., 2019a), such as Pan-STARRS
(Chambers et al., 2016)333Panoramic Survey Telescope and Rapid Response
System,
https://panstarrs.stsci.edu, or VVV, (Minniti et al., 2010)444VISTA Variables
in the Via Lactea https://vvvsurvey.org, respectively. Additionally, Mustill
et al. (2018) searched for astrometric-microlensing events with impact
parameters of the order of one Einstein radius. Such events will lead to a
potentially measurable photometric magnification of the source star. They
found 30 such events that will occur until the year 2032 which have a $>$10%
probability to have an impact parameter smaller than the Einstein radius.
However, the astrometric precision of Gaia DR2 (and also of eDR3) is not
sufficient to securely predict photometric microlensing events.

In December 2020, Gaia published its early Data Release 3 (Gaia eDR3, Gaia
Collaboration et al., 2020a), which contains updated and more precise
astrometric parameters of about two billion stars. McGill et al. (2020) showed
that events predicted from Gaia DR2 which occur in the coming few years are
contaminated by spurious background stars and by pairs of co-moving stars.
Gaia eDR3 includes fewer spurious sources, and a better exclusion of co-moving
stars is possible due to the larger fraction of stars with available proper
motions. Further, the increase in the astrometric precision allows more
accurate predictions.

In this paper we update our previous search for astrometric-microlensing
events in Klüter et al. (2018b, hereafter K18b) by using Gaia eDR3. First, in
Section 2, we shortly explain astrometric and photometric microlensing. In
Section 3 we explain our search for astrometric-microlensing events, starting
with the selection of potential lens and source stars in 3.1 and 3.2,
respectively, and continuing with the determination of approximate masses and
Einstein radii in 3.3, the forecast of their paths and the detection of close
encounters in 3.4, and the determination of the expected effects and selection
of observable events in 3.5. In Section 4, we present the predicted
astrometric-microlensing events and compare this list with previous studies.
Finally, we draw conclusions in Section 5

##  2 Basics of microlensing

Microlensing describes the time-dependent photometric magnification and
positional change of a background source by a point-like foreground mass
(“lens”) passing by. While the lens is passing the source, two images of the
source are created: a bright major image ($+$) close to the unlensed source
position, and a faint minor image ($-$) close to the lens position. In case of
perfect alignment, the background source appears as a so-called Einstein ring.
Its radius is given by (Chwolson, 1924; Einstein, 1936; Paczyński, 1986a):

$\theta_{\rm E}=\sqrt{\frac{4GM_{\rm L}}{c^{2}}\frac{D_{\rm S}-D_{\rm
L}}{D_{\rm S}\cdot D_{\rm L}}}=2.854\,\mathrm{mas}\sqrt{\frac{M_{\rm
L}}{\,\mathrm{M_{\odot}}}\frac{\varpi_{\rm L}-\varpi_{\rm
S}}{1\,\mathrm{mas}}}$ (1)

where $G$ is the gravitational constant, $c$ is the speed of light, $M_{L}$ is
the mass of the lens, $D_{\rm L}$, $D_{\rm S}$ are the distances of lens and
source from the observer, and $\varpi_{\rm L}$, $\varpi_{\rm S}$ are the
parallaxes of lens and source, respectively. This Einstein radius is often
used as an angular scale for the microlensing event, e.g. to define the
(unlensed) scaled angular separation on the sky
$\boldsymbol{u}=\boldsymbol{\theta}/\theta_{\rm E}$, where
$\boldsymbol{\theta}$ is the two-dimensional unlensed angular separation.

###  2.1 Photometric microlensing

The photometric magnification only depends on this unit-less angular
separation $\boldsymbol{u}$. The total magnification of the two images is
given by (Paczyński, 1986a):

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

where $u=\lvert\boldsymbol{u}\rvert$. The apparent magnification of the
background source can be reduced due to blending with light from additional
unresolved light sources. For the here predicted events, the most important
additional light source is the lens itself. With a flux ratio $f_{\rm LS}$
between lens and source, it can be expressed as (Dominik & Sahu, 2000):

$A_{\rm lum}=\frac{f_{\rm LS}+A}{f_{\rm LS}+1},$ (3)

and in units of magnitude, it is given by

$\Delta m=2.5\cdot\log_{10}\left({\frac{f_{LS}+A}{f_{LS}+1}}\right).$ (4)

Due to a strong decline ($A\simeq 1+2/u^{4}$, Paczyński, 1996), a
magnification is only observable when the impact parameter is on the order of
the Einstein radius. This also means that a photometric microlensing event
usually has a short time scale in the order of days to weeks. This can be
expressed by the Einstein time (Gould, 1992):

$t_{E}=\frac{\theta_{E}}{\mu_{rel}},$ (5)

where $\mu_{rel}$ is the absolute value of the relative proper motion between
lens and source.

###  2.2 Astrometric microlensing

In astrometric microlensing, the signal of interest is the change of the
positions of the two images. The positional shift of the major image with
respect to the unlensed position of the source is given by:

$\delta\boldsymbol{\theta_{+}}=\frac{\sqrt{(u^{2}+4)}-u}{2}\cdot\frac{\boldsymbol{u}}{u}\cdot{\theta_{\rm
E}}.$ (6)

In the unresolved case, this deflection is reduced due to blending. First, due
to the minor image, which leads to a shift of the centre of light (in respect
to the unlensed position of the source),

$\delta\boldsymbol{\theta_{c}}=\frac{\boldsymbol{u}}{u^{2}+2}\cdot{\theta_{\rm
E}}.$ (7)

and second due to a likely blending with a luminous lens (Hog et al., 1995;
Miyamoto & Yoshii, 1995; Walker, 1995):

$\delta\boldsymbol{\theta_{c,\,lum}}=\frac{\boldsymbol{u}\cdot\theta_{\rm
E}}{1+f_{\rm LS}}\,\frac{1+f_{\rm LS}(u^{2}+3-u\sqrt{u^{2}+4})}{u^{2}+2+f_{\rm
LS}u\sqrt{u^{2}+4}}.$ (8)

This also changes the reference point to the center of light of the lens and
the unlensed source.

For large impact parameters ($u\gg\sqrt{2}$), this simplifies to (Dominik &
Sahu, 2000):

$\delta\boldsymbol{\theta_{c,\,lum}}\simeq\frac{\delta\boldsymbol{\theta_{c}}}{1+f_{ls}}.$
(9)

Please note that $\boldsymbol{\delta\theta_{c}}$ and
$\boldsymbol{\delta\theta_{c,\,lum}}$ have different reference points, so only
the amount of the positional shifts are connected via equation 9.

For $u\gg 5$, the center of light is dominated by the major image. Therefore,
the shift of the major images can be approximated by the shift of the center
of light:

$\delta\theta_{+}\simeq\delta\theta_{c}\simeq\frac{\theta_{\rm
E}}{u}=\frac{\theta_{\rm E}^{2}}{\theta}\propto\frac{M_{\rm L}(\varpi_{\rm
L}-\varpi_{\rm S})}{\theta},$ (10)

where $\theta=\lvert\boldsymbol{\theta}\rvert$. This direct dependency on the
mass shows the strength of astrometric microlensing. Further, due to the
weaker dependence on the impact parameter, which is $1/u$ rather than
$1/u^{4}$ for photometric microlensing, astrometric microlensing can be
observed at larger impact parameters, which also results in longer time scales
(Paczyński, 1996; Miralda-Escude, 1996). It can be described by (Honma, 2001)

$t_{aml}=2\cdot
t_{E}\sqrt{\left(\frac{\theta_{E}}{\theta_{min}}\right)^{2}-u_{min}^{2}},$
(11)

where $\theta_{min}$ is the astrometric precision threshold of the used
instrument. This assumes a rectilinear lens-source trajectory, and
$\theta_{E}/\theta_{min}\gg 5$, that means we can observe the effect in a
regime where Equation (10) is valid. With high-precision instruments this can
be in the order of $\theta_{min}=0.1\,\mathrm{mas}$, which may lead to time-
scales of many months or even a few years. Further, the weaker $u$ dependence
also allows more confident predictions of astrometric-microlensing events.

On the other hand, the long time scales can lead to small variation within a
given time, which can be hard to measure. To indicate how fast the position
changes due to microlensing, we estimate two time scales
$t_{0.1\,\mathrm{mas}}$ and $t_{50\%}$. These indicates how long it takes
until $\delta\boldsymbol{\theta}$ change by $0.1\,\mathrm{mas}$ (i.e.
$\lvert\delta\boldsymbol{\theta}_{+}(T_{CA})-\delta\boldsymbol{\theta}_{+}(T_{CA}-t_{0.1\,\mathrm{mas}})\rvert=0.1\,\mathrm{mas}$,
where $T_{CA}$ is the epoch of the closets approach) and
$0.5\cdot\,\delta\theta_{+}$ from the maximum deflection, respectively. We
note that $t_{50\%}$ is longer than $t_{0.1\,\mathrm{mas}}$ for events with an
shift larger than $0.2\,\mathrm{mas}$ and vice versa if the maximum shifts is
smaller than $0.2\,\mathrm{mas}$. Figure 1 shows the positional shift due to
astrometric microlensing. $t_{aml}$ corresponds the duration outside of the
dashed circle ($\delta\theta_{+}>0.1\,\mathrm{mas}$, orange and green part)
and $t_{0.1\,\mathrm{mas}}$ corresponds to the duration between entering the
solid circle ($0.1\,\mathrm{mas}$ around the point of the maximum positional
shift) and reaching the maximum positional shift (green part). We estimates
those time scales numerically using a one-week grid. If we encounter a
duration below 2 weeks, we switch to a 1 day grid. Later, typically effects
events with very high proper motions compared to their impact parameters
squared ($\mu_{rel}/u_{min}^{2}>\sim 5\,\mathrm{mas/yr}$).

Figure 1: Geometric illustration for the different time scales. The wobbling
line shows the position of the major image with respect to the unlensed
position of the source at the same epoch. During the event the the major
images moves along this line in a clockwise or anticlockwise direction
(clockwise for the shown event). The wobbling motion is caused by the
parallactic motion of the lens. The black dashed and solid circle are centered
on the origin, and the position of the major image at the epoch of the closest
approach. Both have an radius of $0.1\,\mathrm{mas}$. $t_{aml}$ describes the
duration where the major image is outside of the dashed circle (orange +
green, part of the line). $t_{0.1\,\mathrm{mas}}$ describes duration between
entering the solid circle an reaching the center (green part of the line)
Similar to $t_{0.1\,\mathrm{mas}}$, $t_{50\%}$ descries how long takes if
circle would have a radius of $0.5\delta\theta_{+}$.

##  3 Prediction of microlensing events555The source code for this analysis is
publicly available at
https://github.com/jkluter/amlensing.

In principle, our search follows the method we described in K18b or in Proft
et al. (2011). However, we made major adjustments in the selection of
potential BGS and high-proper-motion stars (HPMS). The selection processed is
shown in Fig 2.

Figure 2: Illustration of the selection process. The search for astrometric-
microlensing events is divided into four major parts. The selection of good
high proper-motion stars (blue), the selection of good background stars
(green), the exclusion of co-moving stars (red), and the determination of the
closest approach and estimation of the expected microlensing effect (black).
The search for close by BGS is performed on the sample of $\sim 500,000$ HPMS
and the selection of good HPMS and BGS is performed independently from each
other. The illustration lists the different selection criteria, with the
number of excluded HPMS, BGS, Candidates or events. In each process, stars can
be excluded due to multiple criteria. The small rectangular boxes indicate
where the major computation is performed

###  3.1 List of high-proper-motion stars

Figure 3: Proper motions ($\mu_{tot}$) and parallaxes ($\varpi$) for all used
HPMS(Blue), excluded HPMS(red), and BGS (green). The high number of HPMS
excluded with small parallaxes, are due to the selection of HPMS with
$\varpi/\sigma_{\varpi}>5$. The BGS and HPMS shows a clear separation, the few
BGS $\mu_{tot}>100\,\mathrm{mas/yr}$ are passed by HPMS with even higher
proper motions. Figure 4: Number of photometric observations by Gaia ($n_{\rm
obs}$) and significance of the G flux ($G_{\rm flux}/\sigma_{G_{\rm flux}}$)
for all potential high-proper-motion stars. The black squares indicate the
HPMS rejected in the GCNS. The HPMS with low fidelity values also cluster in
the bottom left region. The excluded HPMS are plotted in red, and the HPMS
which passes this criterion are plotted in blue.

We started with the selection of HPMS as potential lenses. Compared with our
previous study, we extended our search to HPMS with absolute proper motions
$\mu_{tot}=\sqrt{\mu_{\alpha^{*}}^{2}+\mu_{\delta}^{2}}>100\,\mathrm{mas/yr}$.
The limit of $100\,\mathrm{mas/yr}$ still allows a clear separation between
background stars and co-moving stars (see Fig. 3). In Gaia eDR3, about
$500,000$ sources fulfill this criterion. As mentioned by Fabricius et al.
(2020) and Lindegren et al. (2020), the Gaia eDR3 contains a small fraction of
erroneous data. For example, about 1000 HPMS show a significantly negative
parallax. To exclude such sources, and to ensure a good astrometric solution
in Gaia eDR3 we applied the following quality cuts. We first excluded all HPMS
with a re-normalized unit weight error (ruwe) above 2. This deleted about
$33,000$ HPMS. Further, we excluded HPMS with the significance of the parallax
$(\varpi/\sigma_{\varpi})$ less than 5. About $30,000$ do not pass this
criterion. While the ruwe limit excludes mostly bright HPMS
($G<18\,\mathrm{mag}$), faint HPMS ($G>18\,\mathrm{mag}$) are mostly excluded
by the parallax criterion. Finally, we excluded 413 sources, since Gaia eDR3
does not provide a G magnitude for them.

In order to validate our selection, we cross-matched our list of HPMS with the
Gaia Catalogue of Nearby Stars (GCNS, Gaia Collaboration et al., 2020b). We
found ${\sim}144,000$ common stars in the GCNS proper and ${\sim}50,000$
common stars in the GCNS rejected catalogue. The latter are almost exclusively
rejected due to parallaxes between $8\,\mathrm{mas}$ and $10\,\mathrm{mas}$.
Only 95 common sources are rejected due to a low GCNS probability.

In K18b we also found that suspicious data cluster in the number of
photometric observations $(n_{obs})$ \- G-flux significance
$(G_{flux}/\sigma_{G_{flux}})$ space. This is also true for most of the 95
HPMS rejected by the GCNS (see Fig. 4). Consequently, we excluded
${\sim}9,500$ HPMS with

$G_{flux}/\sigma_{G_{flux}}\cdot n_{obs}^{1.5}<3\cdot 10^{5}.$ (12)

This limit is indicated by the line in Fig. 4.

Using Gaia DR2, in K18b we had further found two populations of erroneous data
with typical tangential velocities of $(v_{tan}\simeq 4\,\mathrm{km/s})$ and
$(v_{tan}\simeq 11\,\mathrm{km/s})$, respectively. In Gaia eDR3, we could not
detect these, nor any other unexpected population (see Fig. 3). Consequently,
we did not apply a filter on the tangential velocity. About $440,000$ HPMS
pass all 5 criteria. For these, we searched for background sources close to
their expected paths on the sky.

Rybizki et al. (2021) determined a classifier in order to indicate sources
with spurious astrometric data in Gaia eDR3. Using a machine-learning
algorithm, they determined a fidelity value between 0 and 1 for each source
with a 5-parameter solution in Gaia eDR3. A value of 1.0 means a perfectly
trustable solution and the lowest possible value of 0.0 indicates a lot of
issues in the astrometric solution. Further, they propose a cut at 0.5 to
differentiate between “good” and spurious sources. Only 126 of all HPMS which
passed our five criteria have a fidelity value below 0.75 and none of those
result in a predicted event, even without any cuts on the fidelity value. On
the other hand, a large fraction of the excluded HPMS ($97\%$) have a fidelity
value above 0.75. However, to ensure a good prediction, we kept the above-
described criteria.

###  3.2 Potential background stars

Figure 5: Illustration of the rectangular box used in our search for
background stars. The thick solid blue line indicates the direction of the
lens proper motion (red star), and the origin is set to the J2016.0 position
of the lens. The blue dashed line indicates the real motion, which includes
the parallax (only five years are shown, and not to scale). The real size is
defined by the position of the lens in J2010.5 and J2066.0 and a half width of
$w=7^{\prime\prime}$. When the J2016.0 position of a background star (yellow
star) is within our rectangle, it is considered as candidate. To account for
the proper motion of the background source, the widening shape (dotted black
line) would be more physically accurate. Plot taken from K18b. Figure 6:
Goodness of fit for Gaia’s astrometry, versus number of good astrometric
observations. BGS with a ruwe less than 2 are plotted in green. These shows a
sharp edge which we used to determine a limit for stars which do not have a
5-parameter solution. Such sources which pass this criterion are shown in
blue, and the excluded background sources are plotted in red.

As in K18b, for each HPMS we defined a rectangular box on the sky, using the
J2010.5 and J2066.0 positions and a half-width of $w=7\,\mathrm{mas}$ (see
Fig. 5) to search for suitable background sources. In the following the pair
of a foreground HPMS and background star (BGS) is called a “candidate”, where
the BGS parameters are labelled with the prefix “$\mathrm{Sou}\\_$”.

Among all stars in the boxes we first excluded BGS with significantly negative
parallaxes
($\mathrm{Sou}\\_\varpi+3\cdot\mathrm{Sou}\\_\sigma_{\varpi}<0\,\mathrm{mas}$),
$G$ magnitudes fainter than $G=21.5$ and without G magnitudes in Gaia eDR3.
Second, we excluded all BGS with a ruwe above 2. This criterion can only be
applied for BGS with a five-parameter solution in Gaia eDR3. However, the ruwe
strongly correlates with the astrometric goodness-of-fit over the square root
of the number of good astrometric microlensing along-scan observations
($\mathrm{Sou}\\_GoF/\sqrt{\mathrm{Sou}\\_N}$, see Fig. 6). Hence, if a five-
parameter solution does not exists in Gaia eDR3, we only considered sources
with $\mathrm{Sou}\\_GoF/\sqrt{(}\mathrm{Sou}\\_N)$ ¡ 1.24, which is
equivalent to ${\rm\texttt{ruwe}}<2$.

Figure 7: Difference in the proper motion between HPMS and BGS. The left panel
shows the stars with a proper motion listed in Gaia eDR3, while in the right
panel we used the positional displacement between DR2 and DR3 as an estimate
for the proper motion. The excluded candidates are plotted in red. The
sparsely scattered candidates with large difference in the right panel are
excluded since the absolute value of the proper motions are similar.

In order to avoid physical binary stars as well as co-moving pairs of stars,
we apply various filters on the parallax and proper motion. We first consider
only candidates with the parallax of the source smaller than 0.9 times the
parallax of the lens. i.e:

$(\mathrm{Sou}\\_\varpi/\varpi<0.9)$ (13)

The main purpose of this cut is to avoid imaginary Einstein radii (i.e. the
”source star” being closer than the ”lens star”), since the proper motion is
more effective to distinguish between background and co-moving stars,
especially for distant stars $(\varpi<10\,\mathrm{mas})$. For the difference
in proper motion we applied the following two criteria:

$\displaystyle\lvert\mathrm{Sou}\\_\boldsymbol{\mu}-\boldsymbol{\mu}\rvert>0.7\cdot\mu_{tot}$
(14) $\displaystyle\mathrm{Sou}\\_\mu_{tot}<0.8\cdot\mu_{tot}$ (15)

The difference in proper motion between HPMS and BGS are shown in the left
panel of Fig. 7

####  3.2.1 Displacement between DR2 and eDR3 as proxy for missing proper
motions

Figure 8: Absolute value of the magnitude difference ($\Delta G$) versus
angular separation ($\Delta\phi$) for the cross match between Gaia eDR3 and
DR2, as listed in the DR2 neighbourhood catalogue. The grey dots in the
background show the distribution for a random sample. The green dots
($\Delta\phi>400\,\mathrm{mas}$) are interpreted as false matches, these are
ignored, while the good matches are shown in blue. The population of stars
with $\Delta\phi$ around $100\,\mathrm{mas}$ are interpreted as co-moving
stars, without a 5-parameter solution in Gaia DR2. The red dots, indicate
sources excluded due to a mismatch in the magnitudes. The orange lines
indicate the limits for our classification. We note hat including more false
matches lead to a stricter selection of candidates.

These criteria can only be used when a proper motion is given in Gaia eDR3.
However, a visual inspection (originally required due to reasons mentioned in
subsection 3.2.2) has shown that most bright background sources without a
five-parameter solution are within a small angular separation to an even
brighter star, which results in a limited performance of Gaia. An example is
shown in panel (a) of Fig. 19. For many of these supposed BGS, we observed
that the difference between the Gaia DR2 and Gaia eDR3 positions is similar to
the difference between the positions of the HPMS, and that they are therefore
co-moving stars. To distinguish between co-moving stars and true candidates,
we used this positional offset to determine an approximate proper motion.

Using the DR2 neighbourhood catalogue Torra et al. (2020), we searched for
matched sources. For ${\sim}185,000$ of our BGS, we found at least one entry
in this catalogue. To ensure a correct match we applied the following steps.
Some sources have multiple entries in this DR2-neighbourhood catalogue. For
those, we only considered the eDR3-DR2 pair with the smallest angular
distance. We then selected only matches where the angular distance
($\Delta\phi$) is $\Delta\phi<400\,\mathrm{mas}$ and the magnitude difference
is $\lvert\Delta
G\rvert<0.3\,\mathrm{mag}\cdot(\Delta\phi/1\,\mathrm{mas})^{0.2}$ (see Fig.
8). We also excluded 1061 sources where we found a Gaia DR2 source within an
angular distance less than $400\,\mathrm{mas}$, but with a magnitude
difference larger than the above limit. Typically, these are brighter in Gaia
DR2. We found that for several candidates a star can be identified at the
listed position, but much fainter than the listed $G$ magnitude in Gaia DR2.
Hence the reason for the disagreement might be an incorrect $G$ magnitude in
Gaia DR2.

Figure 9: Approximate proper motion from the position difference Gaia DR2
minus eDR3 versus the actual proper motion given in Gaia eDR3. For most BGS
these two are in a good agreement, especially if a 5-parameter solution exists
in Gaia DR2 (green dots). The few dozen candidates with large discrepancy are
most likely mismatches.

We calculated an estimate of the proper motion from the positional
displacement between Gaia eDR3 and Gaia DR2, that is

$\mathrm{Sou}\\_\boldsymbol{\Delta\phi}_{\times
2}=\left(\begin{pmatrix}\mathrm{Sou}\\_\alpha_{\mathrm{DR3}}\\\
\mathrm{Sou}\\_\delta_{\mathrm{DR3}}\end{pmatrix}-\begin{pmatrix}\mathrm{Sou}\\_\alpha_{\mathrm{DR2}}\\\
\mathrm{Sou}\\_\delta_{\mathrm{DR2}}\end{pmatrix}\right)\cdot\begin{pmatrix}\cos{\mathrm{Sou}\\_\delta_{\mathrm{DR3}}}\\\
1\end{pmatrix}/\Delta t$ (16)

where $\Delta t$ is the difference between the catalogue epochs of Gaia eDR3
and Gaia DR2 (0.5 years). We also computed a rough estimate for the parallax
as follows:

$\mathrm{Sou}\\_\varpi_{\mathrm{approx}}=4.74\frac{\mathrm{km/s}}{\,\mathrm{mas/yr}}\cdot\frac{\lvert\mathrm{Sou}\\_\Delta\phi_{\times
2}\rvert/1\mathrm{yr}}{v_{\mathrm{tan}}}\,\mathrm{mas},$ (17)

where we use a typical tangential velocity of $v_{\mathrm{tan}}=75km/s$.

If the proper motion of the BGS is not given in Gaia eDR3, we used
$\mathrm{Sou}\\_\boldsymbol{\Delta\phi_{\times 2}}$ and
$\mathrm{Sou}\\_\varpi_{\mathrm{approx}}$ as an estimate for the proper motion
and parallax of the BGS, with an assumed standard error of
$\mathrm{Sou}\\_\sigma_{\Delta\phi_{\times
2}\mathrm{,\,ra/dec}}=5\,\mathrm{mas}$ and
$\mathrm{Sou}\\_\sigma_{\varpi_{\mathrm{approx}}}=3\,\mathrm{mas}$,
respectively. These standard errors roughly reflect the half width of the
distributions of all BGS with a five-parameter solution, for the corresponding
parameter. Finally we applied the same criteria as those we had for BGS with
five-parameter solutions (i.e Equations (13) - (15)).

For candidates where neither Gaia eDR3 lists a proper motion nor a partner
could be found in Gaia DR2 we set the parallax and proper motion to zero.
These cases were also visually inspected. We excluded 52 out of the 279
inspected candidates since we could not identify the BGS in any Pan-STARRS,
DSS2666Digitised Sky Survey, http://archive.eso.org/dss/dss/, 2MASS (Skrutskie
et al., 2006)777Two Micron All Sky Survey ,
https://irsa.ipac.caltech.edu/Missions/2mass.html or VVV Image. We note that
the large majority of candidates were only excluded because the source is
blended in all of the considered surveys. We draw this conclusion from the
fact that two thirds are located at a declination less than $-30^{\circ}$ and
outside of the Pan-STARRS footprint, while only two fifths of all inspected
candidates are below $-30^{\circ}$, and most of the inspected candidates are
too faint to be identified in a DSS2 or 2MASS image (compare the images in the
right column in Fig. 19). Further, we note that we only inspected candidates
which passed all criteria and would lead to an measurable effect.

####  3.2.2 Selection on the $\Psi$ value

Figure 10: $\Psi$ vs. $G$ magnitude. Top: Showing the distribution for all
potential BGS. BGS which would not pass the criterion defined by McGill et al.
(2020) are shown in red. BGS which had passed all of our selection criteria
are plotted in yellow. Only a few dozen of those would not have passed the
criterion of McGill et al. (2020). A sample of randomly selected stars is
shown in grey and $90\%$ of those are below the orange line. We note that the
our sample has slightly larger $\Psi$ values compared to the random sample.
Bottom: In blue the BGS of all predicted events are shown. The ones excluded
after the visual inspection are plotted as red ($\delta>-30^{\circ}$) and
magenta ($\delta<-30^{\circ}$) squares. The excluded candidates below the
orange line were inspected since Gaia eDR3 does not provide proper motions and
they were not found in DR2. The grey dots indicate the random sample as for
the top panel, with the 90th percentile shown as orange line.

For sources brighter than $G=18\,\mathrm{mag}$, McGill et al. (2020) proposed
to use

$\Psi=\frac{\sigma\\_5d\\_max}{1.2\,\mathrm{mas}\cdot\gamma(G)}<1$ (18)

as an additional criterion, to exclude spurious data, where
$\sigma\\_5d\\_max$ is the square root of the largest singular value of the
$5\times 5$ covariance matrix of the astrometric parameters, and

$\gamma=\max\left[1,10^{0.2(G-18)}\right]$ (19)

The distribution of $\Psi$-values as function of $G$ magnitude is shown in
Fig. 10. In our raw sample (i.e. if none of our exclusion criteria is
applied), 2071 candidates would not fulfil this criterion. Out of those, 1545
violate either our criterion on the
$\mathrm{Sou}\\_GoF/\sqrt{\mathrm{Sou}\\_N}$ (1450) or show a bad match
between DR2 and DR3 (1468). The comparison with Gaia DR2 has shown that a
large fraction of those are most likely physical binary stars, which leads to
the selection described above. Further 98 candidate pairs are excluded due to
the displacement between Gaia eDR3 and DR2

Instead of a strict criterion, we visually inspected the Pan-STARRS, 2MASS and
DSS2 images for 562 of the candidates where the background stars show a $\Psi$
larger than the 90th percentile of the stars in a random sample, with similar
magnitudes (shown as orange line in the top panel of Fig. 10). This leads to
the exclusion of 210 candidates. Example of the visual inspection are shown in
Fig. 19. The bottom panel of Fig. 10 shows the $\Psi$-value for our predicted
events in blue, the excluded candidates are plotted as red or magenta squares.
We note that for most of the events a clear falsification was not possible due
to the same reasons as mentioned in Section 3.2.1.

###  3.3 Approximate Masses and Einstein Radii

Figure 11: Colour-magnitude diagram of all potential lenses (grey) with full
Gaia eDR3 photometry $G$, $G_{RP}$, and $G_{Bp}$. The blue dots indicate the
lenses of the predicted events. All stars above the green line are considered
as red giants and all sources below the red line as white dwarfs.

For each of the potential astrometric-microlensing events, we determined the
Einstein radii based on an assumed lens mass. We used this to determine an
expected deflection and magnification of each event. We distinguished between
main-sequence stars (MSs), white dwarfs (WDs) and red giants (RGs), using the
following limits (see Fig. 11):

$\displaystyle WD:\,\,\,$ $\displaystyle
G_{BP,abs}>4.0\cdot(G-G_{RP})^{2}+4.5\cdot(G-G_{RP})+7.4$ (20) $\displaystyle
RG:\,\,\,$ $\displaystyle
G_{BP,abs}<-50.0\cdot(G-G_{RP})^{2}+70.0\cdot(G-G_{RP})-20.0$ (21)

where $G_{BP,abs}$, is the absolute $G_{BP}$ magnitude determined using the
distance modulus. These limits were slightly adjusted compared to K18b. HPMS
without a $G_{BP}$ and $G_{RP}$ magnitude given in Gaia eDR3 are assumed to be
main-sequence stars.

For WDs and RGs we assumed typical masses of $M_{\rm WD}=(0.65\pm
0.15)M_{\odot}$ and $M_{\rm RG}=(1.0\pm 0.5)M_{\odot}$, respectively, and for
MSs we determined the mass based on the absolute $G$ magnitude ($G_{abs}$),
using the following equation based on the mass-luminosity function as
determined in K18b (Section 3.4):
for $G_{abs}<8.85:$

$\qquad\ln\left(\frac{M}{M_{\odot}}\right)=0.00786\,G_{abs}^{2}-0.290\,G_{abs}+1.18$
(22)

for $8.85<G_{abs}<15:$

$\qquad\ln\left(\frac{M}{M_{\odot}}\right)=-0.301\,G_{abs}+1.89,$ (23)

For MSs we assumed an uncertainty of $10\%$. Stars with $G_{abs}>15$ lie in
the Brown Dwarf (BD) regime, where the mass-luminosity function is not valid.
For these, we assumed a mass of

$M_{BD}=(0.07\pm 0.03)\,\mathrm{M_{\odot}}$ (24)

We note that this is only a rough estimate. For example it treats sub dwarfs
like MS Stars. However, measuring the mass is aim of the observations of the
astrometric microlensing event.

Using Eq. (1) we then determined the Einstein radii, using the masses
determined above and the Gaia eDR3 parallaxes of lens and source. For the
source parallaxes, we also used the prior derived from the DR2-eDR3
displacement, if only a two-parameter solution is provided by Gaia eDR3.

###  3.4 Forecasting the motion

For about $136\,000$ candidates, we searched for the closest approach between
source and lens. Using elementary geometry, we first determined an approximate
minimum angular separation $(d_{\rm CA,\,approx})$ and approximate epoch of
the closest approach ($T_{\rm CA,\,approx}$):

$d_{\rm
CA,\,approx}=\left\lvert\frac{\Delta\alpha^{\ast}\cdot\Delta\mu_{\delta}-\Delta\delta\cdot\Delta\mu_{\alpha^{\ast}}}{\sqrt{\Delta\mu_{\alpha^{\ast}}^{2}+\Delta\mu_{\delta}^{2}}}\right\rvert$
(25) $T_{\rm
CA,\,approx}=-\frac{\Delta\alpha^{\ast}\cdot\Delta\mu_{\alpha^{\ast}}+\Delta\delta\cdot\Delta\mu_{\delta}}{\Delta\mu_{\alpha^{\ast}}^{2}+\Delta\mu_{\delta}^{2}}+2016.0$
(26)

where $\Delta\alpha^{\ast}$, $\Delta\delta$, $\Delta\mu_{\alpha^{\ast}}$,
$\Delta\mu_{\delta}$ are the differences in positions and proper motions
between lens and source in right ascension and declination. Note that this
approximation neglects the periodic motion caused by the parallax difference.

For the next step, we only considered the ${\sim}16,000$ candidates where the
predicted shift of the major image is larger than $0.03\,\mathrm{mas}$ or the
approximate minimum angular separation is less than two times the parallax
difference. For those we searched for the exact closest approach. We
determined the position of lens and source in equatorial Cartesian coordinates
as a function of time ($t$) after the Gaia reference epoch $t_{0}=J2016.0$,

$\boldsymbol{x}(t)=\begin{pmatrix}\cos(\delta+\mu_{\delta}\,t)\cdot\cos(\alpha+\mu_{\alpha}\,t)\\\
\cos(\delta+\mu_{\delta}\,t)\cdot\sin(\alpha+\mu_{\alpha}\,t)\\\
\sin(\delta+\mu_{\delta}t)\end{pmatrix}\frac{1000\,\mathrm{mas}\cdot
pc}{\varpi}+\boldsymbol{E}(t).$ (27)

where $\boldsymbol{E}(t)$ is the geocentric location of the Sun at $t_{0}+t$,
in equatorial Cartesian coordinates. We note that
$\mu_{\alpha}=\mu_{\alpha^{\ast}}/\cos(\delta)$. This is retrieved from the
astropy packages (The Astropy Collaboration et al., 2013). Further, we
determined the angular separation between lens and source via:

$d(t)=2\cdot\arcsin\left(\left|\frac{\boldsymbol{x}(t)}{\lvert\boldsymbol{x}(t)\rvert}-\frac{\mathrm{Sou}\\_\boldsymbol{x}(t)}{\lvert\mathrm{Sou}\\_\boldsymbol{x}(t)\rvert}\right|\cdot
0.5\right).$ (28)

We then evaluated this function on a one-week grid within two years around the
previously determined approximate epoch. By comparing consecutive grid points,
we detected local minima and then applied a nested-intervals algorithm to
determine the precise epoch and minimum distance for each local minimum. As
edges ($t_{0}$, $t_{3}$) of the starting interval we use the two adjacent
neighbours of the previous found minimum minimum, and split it according to
the golden ration at $t_{1}=t_{0}+38.2\%\cdot(t_{3}-t_{0})$,
$t_{2}=61.8\%\cdot(t_{3}-t_{0})$ of the interval length. We then shrink the
interval such that in each step the epoch for the minimum of
$d(t_{1}),d(t_{2})$ stays in the center of our interval (the new intervals is
$t_{0}-t_{2}$ or $t_{1}-t_{3}$, respectively), until the width is less than
$3\cdot 10^{-10}yr\approx{10\mathrm{ms}}$, i.e much smaller than the expected
error. Finally, by comparing all detected local minima, we selected the global
minimum.

###  3.5 Computing the expected astrometric effects

Using equations (6) - (8) and (4), we computed the expected astrometric shifts
for the major image ($\delta\theta_{+}$), for the centre of light
($\delta\theta$), and for the centre of light including luminous-lens effects
($\delta\theta_{\mathrm{lum}}$), as well as the expected photometric
magnification. We determined the uncertainties of these predictions using a
Monte Carlo approach, where we picked 5000 realisation. The results
corresponds to the median and the upper and lower errors to the $15.87th-50th$
and $(84.13th-50th$ percentile, respectively. We do not include any co
variances between the different input parameters.

Finally, we only selected predicted events with an expected shift of the major
image larger than $\delta\theta_{+}=0.1\,\mathrm{mas}$ or with an expected
magnification larger than $\Delta m=1\,\,\mathrm{mmag}$.

Since several high-precision astrometric instruments like Gaia or JWST are or
will be located at the Sun-Earth Lagrange point L2, we repeated our
calculations with a $1\%$ larger parallax. These results are labelled in the
online table with “L2_” as a prefix. However, the epochs and expected
separations differ only very little.

##  4 Results: Predicted astrometric-microlensing events

Figure 12: Top: Expected maximum shifts for all astrometric-microlensing
events with a magnitude difference $\Delta G<6\,\mathrm{mag}$. The grey dots
indicate the events where the proper motion and parallax of the source is
unknown. The blue dots show the events with a five-parameter solution for the
background source, as well as the determined standard errors. On the x-axis
the epoch of closest approach $T_{\rm CA}$ is displayed. The apparent paucity
of events during the Gaia mission time is due to the angular resolution limit
of Gaia eDR3. Bottom: Number of predicted events per year. The red bars shows
the events with $\Delta G<3\,\mathrm{mag}$, the red + blue bars show the
events with $\Delta G<6\,\mathrm{mag}$, and the red + blue + grey shwos all
predicted events.

We report the prediction of 4842 astrometric-microlensing events based on Gaia
eDR3, caused by 3791 distinct High-Proper-Motion Stars. The relevant date on
the individual events are listed in the online table. This can be accessed
through the GAVO Data Center888German Astrophysical Virtual Observatory,
https://dc.g-vo.org/amlensing/q3/q/form, and is also available through the TAP
service999https://dc.g-vo.org/tap or through the Virtual Observatory (look for
“Astrometric-Microlensing Events Predicted from Gaia eDR3”) A description of
all the coloumns can be find in Table 2 \+ 3

The table contains updated parameters for 3084 events already predicted from
Gaia DR2 (see subsection 4.1). For those we provide updated and more precise
information on the separation and epoch of the closest approach. The 1758
newly predicted events are mainly due to the reduced lower limit for the
proper motion of the HPMS ($\mu_{tot}>100\,\mathrm{mas/yr}$ instead of
$\mu_{tot}>150\,\mathrm{mas/yr}$ in K18b; see subsection 4.2).

A total of 869 events predicted in K18b could not be re-discovered with Gaia
eDR3, mainly due to the stricter quality requirements. For about 200 of those
events, Gaia eDR3 led to strong evidence that lens and source are co-moving
stars, based either on the proper motion given in Gaia eDR3 or on the
positional displacement between Gaia eDR3 and Gaia DR2.

###  4.1 Updated results from Gaia DR2

In Klüter et al. (2018a) and K18b we predicted 3908 astrometric-microlensing
events, and Bramich & Nielsen (2018) and Bramich (2018) predicted 79 and 2509
further events, respectively. These partly overlap, so that a total of 5766
events had been predicted from Gaia DR2. In this section we investigate which
of the events are found within our new results, and why certain events do not
show up within our new sample. In the rest of this section the numbers are
given for the sample of the 3908 events which we predicted with Gaia DR2,
followed in parentheses by the number for the full sample of 5766 Gaia DR2
predictions .

####  4.1.1 Match between Gaia DR2 and eDR3

In order to match the source IDs between Gaia eDR3 and Gaia DR2, we again used
the DR2 neighbourhood catalogue provided with Gaia eDR3. If in Gaia eDR3
multiple sources are provided for a single Gaia DR2 source, we only considered
those with the smaller angular separation as a genuine match. Source pairs
with angular separations below $450\,\mathrm{mas}$ and with magnitude
differences below $0.5\,\mathrm{mag}$ are labelled as good matches, the others
as bad matches. For each of the HPMS we were able to detect a counterpart in
Gaia eDR3. Among those, 45 (45) were labelled as bad matches due to large
angular separations and 3 (3) due to a missing $G$ magnitude in Gaia eDR3. For
5 (6) of the BGS, the DR2 neighbourhood catalogue does not list any entry. For
3 out of the 6, the BGS is clearly visible in the Pan-STARRS images, and for
one of them a slight distortion of the PSF can be observed in VVV. For the
other 2, the images were strongly blended by the HPMS. Additionally, 114 (140)
BGS show a bad match between Gaia DR2 and Gaia eDR3 ($d>450mas:26\,(39)$,
$\Delta G>0.5mag:90\,(103)$; $G_{\text{DR3}}=None:18\,(31)$). In total, for
166 (193) events, it was not possible to find a good match for the BGS and
HPMS. Further, for 21 (33) of those, the BGS and HPMS are matched to the same
Gaia eDR3 source. This is also true for 2 events where both matches seem to be
good. Additionally, two BGS are matched to the same Gaia eDR3 source, but one
of those matches is classified to be bad. Finally, the lenses of 26 (860) and
81 (82) events, are not classified as HPMS in Gaia eDR3, due to their proper
motion being less than $100\,\mathrm{mas/yr}$ or because no proper motion is
given, respectively, and for 21 (806) events the background source is outside
of our search rectangle. The high numbers for the full sample are due to the
extended search by Bramich & Nielsen (2018) for events until the year 2100,
and no cuts on the proper motion. In conclusion, within the search windows
around individual HPMS we were able to find 3755 (3990) events among our
original 3908 (5766) candidates before the quality cuts.

####  4.1.2 Excluded events

This subsection gives the statistics of the various reasons for excluding some
of the predicted events. We note that an event could be excluded due to
multiple reasons.

Out of the 3755 (3990) events described in the previous subsection, 402 (404)
events were excluded because the HPMS did not pass all quality criteria. This
was mainly caused due to a ${\rm\texttt{ruwe}}>2$ (396 (398) events). For 7
(7) events the HPMS did not pass the $G_{flux}/\sigma_{G_{flux}}\cdot
n_{obs}^{1.5}>3\cdot 10^{5}$ criterion. We note that Rybizki et al. (2021)
determined a fidelity value less than 0.5 only for two of the excluded events.
Except for one which has a value of 0.87, they have a value larger than 0.99
(the fidelity value lies between $1.0={\rm Good}$ and $0.0={\rm Bad}$). Hence
the ${\rm\texttt{ruwe}}=2$ might be too strict a limit. Consequently, events
which are only excluded due to a high ruwe might be real events.

Based on the information of the BGS we excluded 283 (285) events. These are
mainly due to a high $GoF/n>1.24$ (197 (197) events) which is equivalent to
${\rm\texttt{ruwe}}>2$, or due to a strange magnitude difference between DR2
and DR3 (207 (209) events). Additionally, 18 (19) events were excluded due to
a significantly negative parallax, and 34 (35) events were excluded due to a
standard error in the position larger than $10\,\mathrm{mas}$. Furthermore we
found 198 (200) events where the motion of the background source in Gaia eDR3,
or the positional change between Gaia eDR3 and Gaia DR2 indicates that the
supposed lens and source star are co-moving stars. Finally, 26 (29) events
were excluded after visual inspection of the BGS with an atypically high
$\Psi$ value. In addition, it is possible that some of the events found in
Gaia eDR3 but with an expected shift less than $0.1\,\mathrm{mas}$ would not
have passed the visual inspection.

In the end 3039 (3264) astrometric-microlensing events passed all of our
criteria (and 3352 (3579) when using the fidelity value instead of the ruwe
and parallax error).

####  4.1.3 Re-detected Events

For 150 (180) of these 3039 (3264) re-detected events, the expected effects
are below our selection limit of $\delta\theta_{+}>0.1\,\mathrm{mas}$ or
$\delta m>1\,\mathrm{mmag}$. For 8 (20) events this is caused by a significant
difference in the mass of the lens. For the 8 events in K18b this is due to a
classification as MS instead of a WD. For the other 142 (150) events, the
differences between the values determined from Gaia DR2 and eDR3 are not
significant ($<3\sigma$).

Finally, our result contains 2888 (3083) events previously predicted. The
majority of the events, 2818 (2997) have comparable angular separations. Only
for 70 (86) events we found a significant difference ($>3\sigma$) between Gaia
DR2 and eDR3. These are caused by significant changes within the Gaia Data.
For the epoch of the closest approach, we also find a good agreement for most
of the events. However, 181 (193) show offsets between 0.5 and 1.0 years.
Beside the significant changes within the Gaia Data, this is due to two local
minima with roughly the same minimum separation, resulting in a switch from
one to the other as global minimum. This effect is not covered in our error
estimation. However, for most of those events a measurable shift is also
expected during the epoch predicted from Gaia DR2. By using Gaia eDR3 the
uncertainties for both epoch and position are smaller by roughly a factor of
2.

Except for 17 (78) events, the expected effects are in a good agreement, too.
Significant differences are mainly caused by large differences in the assumed
masses –– this concerns 10 (56) events. For 7 (22) events this is caused by
significant differences in the impact parameters. We do not observe a
reduction of the mean errors, since these are dominated by the uncertainties
of the mass.

###  4.2 New events from Gaia eDR3

In addition to the 2888 (3083) previous events, we predict 1756 new
astrometric-microlensing events. This is mainly due to the reduction of the
lower limit in the HPMS proper motion. However, we also found 563 new events
with a proper motion larger than $150\,\mathrm{mas/yr}$. In most of these
cases (487 events), Gaia DR2 did not contain the background source, and for 76
events, Gaia eDR3 indicates the lens as binary star, while Gaia DR2 only lists
one component.

###  4.3 Properties of the full sample

Our full sample of predicted astrometric-microlensing events based on Gaia
eDR3 contains 4842 events caused by 3791 distinguished HPMS. The number of
events for the different criteria are summaized in Table 1. As in K18b, these
are mainly located towards dense areas in the galactic disc, or towards the
Magellanic Clouds. Since astrometric microlensing favours close-by massive
stars as lenses, roughly half (2160) of the events have a magnitude difference
larger than 6. For 2682 events the magnitude difference is smaller than 6, and
for 1246 events the lens star is at most 3 magnitudes brighter. In the
following we give the corresponding numbers for each of these two subsamples
in parentheses, i.e. the total number of events is followed by (events with
$\Delta G<6$, events with $\Delta G<3$). Finally, we found 285 events where
the BGS is brighter than the lens. We note that bright lenses tend to have
either large masses or small distances, both lead to large (angular) Einstein
radii. Hence, a measurable shift is also expected at larger angular
separations, where the source might be detectable next to a bright star.

Out of the 4842 (2682, 1246) events, 473 (260, 111) have an expected shift
larger than $1\,\mathrm{mas}$, and for 532 (301, 135) the expected shift is
between $0.5\,\mathrm{mas}$ and $1\,\mathrm{mas}$. Especially the 260 events
with $\Delta G<6$ and $\delta\theta_{+}>1\,\mathrm{mas}$ are promising
targets. However, also the impact parameter and observability should be
considered in order to select targets for observation, which can differ
between Telescopes and instruments.

In the unresolved case, i.e. when only the shift of the centre of light can be
observed, we found 393 (393, 380) events with an expected effect larger than
$\delta\theta_{\rm lum}>0.1\,\mathrm{mas}$. This can only be observed for
events with low contrast between HPMS and BGS, since the blending by the HPMS
strongly reduces the expected shift. All of the 50 events with an expected
shift of the centre of light above $0.5\,\mathrm{mas}$ have a magnitude
difference below $\Delta G=3$. About $40\%$ of those (147 out of 393) are
caused by a WD. We note that the contrast between HPMS and BGS, and therefore
the expected shift of the centre of light depends on the flux ratio in the
selected filters.

The expected shifts and predicted epochs of minimal separation are shown in
the top panel of Fig. 12, and the number of events per year are shown in the
bottom panel. The number of events per year is 94 (52, 24). Until the year
${\sim}2030$ the number of predicted events is low. This is caused by Gaia’s
limited spatial resolution and performance for close pairs. For the next
decade (2021-2031) we found 685 (443, 218) events; these and the events during
the Gaia mission are shown in Fig. 13. The latter will become especially
interesting when Gaia shall publish its individual astrometric measurements,
planned for the Data Release 4 (expected in the year 2024) and for the final
data release after the extended mission (expected about 2029). Within the next
decade the number of events per year strongly increases with time, where
events caused by faster HPMS tend to be earlier. Within the next decade we
expect 76 (46, 17) events with an expected shift larger than
$1\,\mathrm{mas}$.

Figure 13: Top: Maximum shifts for all expected astrometric-microlensing
events between 2014 and 2032 and a magnitude difference $\Delta
G<6\,\mathrm{mag}$, with (blue) and without (grey) five-parameter solutions
for the background source in Gaia eDR3. Bottom: Number of events per year
(counted in half-year bins and multiplied by a factor of 2) for all events
(grey + blue + red), events with a magnitude difference $\Delta
G<6\,\mathrm{mag}$ (blue + red) and events with a magnitude difference $\Delta
G<3\,\mathrm{mag}$ (red).

####  4.3.1 Time scales

Typically, a shift larger than $0.1\,\mathrm{mas}$ from the unlensed position
is expected over a time range between $1.1\,\mathrm{yr}$ (15.87th percentile)
$7.7\,\mathrm{yr}$ (84.13th percentile) with a median of $3.2\,\mathrm{yr}$.
And it takes typically between $0.037\,\mathrm{yr}$ ($27\mathrm{weeks}$,
15.87th percentile) and $3.6\,\mathrm{yr}$ (84.13th percentile) to observe a
change in the positional shift by $0.1\,\mathrm{mas}$, with a median of
$0.53\,\mathrm{yr}$. Both distributions are shown in Figure 14. For 303 events
$t_{0.1\,\mathrm{mas}}$ is between $5\,\mathrm{yr}$ and $10yr$, and for 209
events $t_{0.1\,\mathrm{mas}}$ is longer than $10\,\mathrm{yr}$. The expected
shifts for those events are below $0.2\,\mathrm{mas}$ and
$0.15\,\mathrm{mas}$, respectively. Additionally, 39 events do not list an
$t_{0.1\,\mathrm{mas}}$, since they show large uncertainties, an the expected
shift is below $0.1\,\mathrm{mas}$ when the uncertainties are not included in
the analysis. Since all of the events are included in the Gaia eDR3, events
with large $t_{0.1\,\mathrm{mas}}$ can benefit from the existence a precise
J2016.0 position. However, those are hard to observed with an accuracy of
$0.1\,\mathrm{mas}$ anyway. For 263 of the 512 ”slow” events
($t_{0.1\,\mathrm{mas}}>5\,\mathrm{yr}$) a change in the position by
$50\%\cdot\delta\theta_{+}(\simeq 0.05-0.8\,\mathrm{mas})$ is between
$2\,\mathrm{yr}$ and $5\,\mathrm{yr}$, and for 221 shorter than
$2\,\mathrm{yr}$. we found a strong correlation between
$\log(t_{0.1\,\mathrm{mas}})$ and $\log(u^{2}/\mu_{rel})$ with a correlation
coefficient of 0.98, leading to:

$t_{0.1\,\mathrm{mas}}\simeq
0.11\,\mathrm{yr}\cdot\left(\frac{u_{min}^{2}\cdot
1\,\mathrm{mas/yr}}{\mu_{rel}}\right)^{1.16}$ (29)

Figure 14: Distribution for $t_{aml}$ (blue), $t_{0.1\,\mathrm{mas}}$(green)
$t_{50\%}$ (red). The trends of the distribution for $t_{aml}$ and
$t_{0.1\,\mathrm{mas}}$ continues towards longer timescales. Note the
logarithmic scale.

####  4.3.2 Events caused by White Dwarfs

Figure 15: Maximum shifts for all expected events caused by a WD, with (blue)
and without (grey) five-parameter solution for the background source in Gaia
eDR3.

For 625 (563, 409) events, the HPMS is classified as WD. In the next decade
112 (95, 75) events occur, out of which 9 (5, 4) have an expected angular
shift larger than $1\,\mathrm{mas}$. As median we find an occurrence rate of
12 (10, 8) events per year with expected shifts larger than
$0.1\,\mathrm{mas}$, and about 1 event per year with expected shift larger
than $1\,\mathrm{mas}$, respectively (cf. Figure 15). These events are of
special interest, for two reasons. First, WDs are relatively faint for a given
mass, i.e. the magnitude differences between lens and source are much smaller
than for main sequence star-lenses. Hence, they provide ideal targets with
large effects and only little blending. This is also shown by the larger
fraction of events with smaller magnitude differences ($66\%$ events with
$\Delta G<3$, compared to $25\%$ for the full sample). Further, relations
concerning the masses of WD are currently only poorly known, due to the lack
of WD with directly determined masses. This increases the importance of direct
mass measurements for white dwarfs.

Harding et al. (2018) estimates the event rate of astrometric microlensing
events caused by stellar remnants. For Gaia DR2 sources and white dwarfs as
lenses, they estimates $1.52\cdot 10^{-3}$ events per white dwarf per decade.
With about 3800 white dwarfs in our sample of good HPMS, we would expect about
20 microlensing events between 2030 and 2065.5 (where our sample shows a
constant behaviour). However, for the same time range, we found 146 events
caused by a white dwarfs, with an expected shift larger than
$0.3\,\mathrm{mas}$ (the detection limit in Harding et al. (2018)). This would
lead to an event rate of $1.1\cdot 10^{-2}$. The higher event rate is due to
our selection of HPMS only, which are more likely to cause astrometric
microlensing events than the average population of white dwarfs.

####  4.3.3 Events caused by binary stars

Our sample also contains pairs of events with identical BGS, but different
HPMS. In total, we found 24 such pairs, caused by 15 binary systems. A visual
inspection showed that all of these are true events. For 15 of the 48 events
the magnitude difference is below $6$. Additionally, for 421 events, the HPMS
is part of a binary system, but a measurable effect is not expected for the
second component. We note that the orbital motion of the binary system is not
included in our analysis, nor accounted for in the error estimation. It is
possible that additional events are caused by binary stars which have angular
separations larger than 30” (which we used as radius for a cross-match), or by
binaries which are not resolved in Gaia eDR3, or where the other component
does not satisfy our quality criteria. Due to the orbital motion of the binary
components, the true epoch and the separation of the closest approach might
differ from our predictions. Corresponding corrections can be included in our
analysis, once the binary motion is known. For some of the events, this will
come with the full Gaia Data Release 3 (i.e. in 2022).

####  4.3.4 Astrometric-microlensing events with photometric signature

Figure 16: Expected maximum magnification for all astrometric-microlensing
events with photometric signature, except for one (grey dot) all events have a
five-parameter solution for the background source in Gaia eDR3.

In K18b we predicted 127 astrometric-microlensing events which might lead to a
noticeable photometric signal. Using Gaia eDR3, we re-detect 51 of those
events. Further 44 events were re-detected, but with an expected magnification
below 0.1 mmag. Additionally, we found 88 events with an expected
magnification larger than $1\,\mathrm{mmag}$. 32 of those were predicted in
K18b but with an expected magnification below $1\,\mathrm{mmag}$ the other
newly predicted events are caused by stars with proper motions between
$100\,\mathrm{mas/yr}$ and $150\,\mathrm{mas/yr}$. In total we found 139
astrometric-microlensing events with potential photometric signatures.
However, all predicted magnifications have large uncertainties. These are
shown in Fig. 16. For 49 (49, 40) events we expected a magnification larger
than $10\,\mathrm{mmag}$, and for 9 (9, 9) events even larger than
$100\,\mathrm{mmag}$. We do not find any photometric events with high contrast
($\Delta G>6$), due to the blending of the lens. We note that our estimation
only considers blending caused by the lens star, and neglects any blends from
additional background sources.

####  4.3.5 Astrometric-microlensing event by Gaia eDR3-4053455379420641152

As an example, we here want to show a promising newly predicted astrometric-
microlensing event caused by Gaia eDR3-4053455379420641152 (VVV 176144893) in
2025 (hereafter G-2025). G-2025 is located towards the galactic center(${\rm
RA}_{J2016}=17^{\mathrm{h}}38^{\mathrm{m}}37^{\mathrm{s}}.04721$; ${\rm
DEC}_{J2016}=-34^{\circ}27^{\prime}35^{\prime\prime}.4946$) with a proper
motion of ($\mu_{\alpha}*$, $\mu_{\delta}$) =
($-316.1\,\mathrm{mas/yr},\,-388.6\,\mathrm{mas/yr})$, and a parallax of
$\varpi=25.5\,\mathrm{mas}$. Gaia provides a magnitude of
$G=17.8\,\mathrm{mag}$, $G_{BP}=18.2\,\mathrm{mag}$ and
$G_{RP}=16.9\,\mathrm{mag}$, respectively. Therefore, we classify the star as
a white dwarf. For those, we determined an approximated mass of
$m=0.65\,\mathrm{M_{\odot}}$. G-2025 is located within the VVV footprint, and
VVV lists a brightness of $KS=15.5\,\mathrm{mag}$. A stamp from VVV is shown
in Figure 17, where G-2025 is marked by the green square. G-2025 has a similar
proper motion ($\mu_{\alpha}^{*}=-326.3\,\mathrm{mas/yr}$,
$\mu_{\delta}=-391.1\,\mathrm{mas/yr}$) and parallax
($\varpi=25.9\,\mathrm{mas}$) as the close-by star Gaia
eDR3-4053455379465036800 (2MASS J17383723-3427304; VVV 176850871, marked by
the black cross in Fig. 17). This is also true for the proper motion provided
by VIRAC (Smith et al., 2018). The motion can also be seen by comparing VVV
images from different epochs. Hence, G-2025 and Gaia eDR3-4053455379465036800
are forming a binary (or multiple stars) system.

In $2025.17\pm 0.12$ it will pass by a $G=20.25\,\mathrm{mag}$ (Gaia
eDR3-4053455379420641152; VVV 176850869) background star with an impact
parameter of $d_{min}=(111\pm 69)\,\mathrm{mas}$. The large uncertainties are
due to the missing five parameter solution for the background star in Gaia
eDR3. Hence we used the displacement between Gaia eDR3 and DR2 for the proper
motion. for the proper motion. We note that the so derived proper motion does
not agree with the proper motion provided by VVV. Using the the VVV proper
motion will lead to a ${\sim}10\%$ smaller minimal angular separation and thus
to a ${\sim}10\%$ larger shift, larger shift, but with an epoch of the closest
approach about 6 months earlier (We can find large discrepancies between Gaia
and VVV proper motion also for other stars close-by stars). The fourth Gaia
data release might provide proper motion and parallax for this source, which
will help in the analysis after a successful observation of the event. In
Figure 17, the background star is indicated by the red circle. For the
background star VVV list J, H and Ks magnitudes of $J=15.7\,\mathrm{mag}$,
$H=14.8\,\mathrm{mag}$ and $Ks=14.4\,\mathrm{mag}$, respectively.

We determined an Einstein radius of $\theta_{E}=(11.6\pm 1.5)\,\mathrm{mas}$.
This results in an expected maximum shift of the major image of
$\delta\theta_{+}=1.2^{+2.0}_{-0.5}\,\mathrm{mas}$ If it is not possible to
resolve lens and source, only the combined center of light can be observed.
For G-Band observation, we then expect a shift of
$\delta\theta_{lum}=0.11^{0.18}_{-0.05}\,\mathrm{mas}$. Since in the near
infrared, the source is brighter than the lens, a larger shift of the center
of light can be observed. Using the Ks filter a maximum shift of
$\delta\theta_{lum,Ks}\simeq 0.87\,\mathrm{mas}$ is expected (red dotted line
in Fig. 18). JWST101010James Webb Space Telescope, https://www.jwst.nasa.gov
will be able to measure this small shift. Both, the shift of the major image
and the shift of the centre of light, as well as the angular separation
between lens and source star, is shown in Figure 18as a function of time. The
expected shift for the major image will be above $0.5\,\mathrm{mas}$ over a
period of about 11 months (about 5.5 years for
$\delta\theta_{+}>0.1\,\mathrm{mas}$).

We found that G-2025 will pass by to further BGS. One in early 2028, with an
expected impact parameter of$d_{min}=(1300\pm 60)\,\mathrm{mas}$. For this
event, we expect a shift of the major image of $\delta\theta_{+}=(0.10\pm
0.03)\,\mathrm{mas}$, which is only slightly above our selection criterion.
And one in 2036, with an expected impact parameter of $d_{min}=(260\pm
3)\,\mathrm{mas}$. This will lead to a shift of the major image of
$\delta\theta_{+}=0.52\pm 0.13\,\mathrm{mas}$. The BGS for the later event is
marked by a cyan hexagon in Fig. 17.

Figure 17: H-J-Y colored cutout from VVV. centered on the position of the background source (red circle) The high-proper-motion lens star is indicated as by the green square (November 2010). Its motion between 2010 and 2030 is indicated by the yellow line, and its J2016.0, and J2021.5 position are given by the green dot and star, respectively. The small wobbling is due to its parallax. The binary companion is marked by the black cross. The background star for the close encounter in 2036 is marked by the cyan hexagon. Figure 18: Top: Expected shift of the major image (blue line), for the combined center of light using the G flux ratio (red solid line), and for the combined center of light using the Ks flux ratio (red dotted line). The $0.1\,\mathrm{mas}$ and $0.5\,\mathrm{mas}$ level is indicated by the dashed lines. Bottom: angular separation between lens and source. Table 1: Number of predicted astrometric-microlensing events: all events, events which lead to an shift including blending by a luminous lens, events with a White Dwarf as lens, and potential photometric microlensing signature. For each subset, three different ranges for the expected effect are given, and the first line is the sum of the three lines below. The number are also given for different magnitude differences between lens and source, where the last column gives the total number of events | number of events with
---|---
| $\Delta G\leq 3$ | $3<\Delta G\leq 6$ | $\Delta G>6$ | total
all events |  |  |  |
$\delta\theta_{+}>=0.1\,\mathrm{mas}$ | 1246 | 1436 | 2160 | 4842
$0.5\,\mathrm{mas}>\delta\theta_{+}>=0.1\,\mathrm{mas}$ | 1000 | 1121 | 1716 | 3837
$1\,\mathrm{mas}>\delta\theta_{+}>=0.5\,\mathrm{mas}$ | 135 | 166 | 231 | 532
$\delta\theta_{+}>=1\,\mathrm{mas}$ | 111 | 149 | 213 | 473
luminous lens |  |  |  |
$\delta\theta_{lum}>=0.1\,\mathrm{mas}$ | 380 | 13 | 0 | 393
$0.5\,\mathrm{mas}>\delta\theta_{lum}>=0.1\,\mathrm{mas}$ | 330 | 13 | 0 | 343
$1\,\mathrm{mas}>\delta\theta_{lum}>=0.5\,\mathrm{mas}$ 35 | 0 | 0 | 35 |
$\delta\theta_{lum}>=1\,\mathrm{mas}$ | 15 | 0 | 0 | 15
events by White Dwarfs |  |  |  |
$\delta\theta_{+}>=0.1\,\mathrm{mas}$ | 409 | 154 | 62 | 625
$0.5\,\mathrm{mas}\ >\delta\theta_{+}>=0.1\,\mathrm{mas}$ | 333 | 129 | 43 | 505
$1\,\mathrm{mas}>\delta\theta_{+}>=0.5\,\mathrm{mas}$ | 41 | 11 | 8 | 60
$\delta\theta_{+}>=1\,\mathrm{mas}$ | 35 | 14 | 11 | 60
photometric events |  |  |  |
$\delta m>=1\,\mathrm{mmag}$ | 99 | 37 | 3 | 139
$10\,\mathrm{mmag}>\delta m>=1\,\mathrm{mmag}$ | 59 | 28 | 3 | 90
$100\,\mathrm{mmag}>\delta m>=10\,\mathrm{mmag}$ | 31 | 9 | 0 | 40
$\delta m>=100\,\mathrm{mmag}$ | 9 | 0 | 0 | 9

##  5 Summary and Conclusion

From Gaia eDR3 we found $136,000$ close passages of background stars by high
proper motion (foreground) stars with proper motions larger than 100 mas per
year. We defined a set of quality criteria to avoid spurious entries. Further,
we sorted out potentially co-moving star pairs using the cross-match between
Gaia DR2 and eDR3. By forecasting the motion of HPMS and BGS we derived the
angular separations and epochs of the closest approach for each case. Using
approximate masses for the lens stars, we estimated the expected microlensing
effects, both astrometric and photometric.

We give these predictions for 4842 events by 3791 distinct HPMS, where we
expect a shift larger than $0.1\,\mathrm{mas}$. 3084 of these events had
already been predicted from Gaia DR2. For those we give updated parameters
based on Gaia eDR3. For most of the events, we find a good agreement for the
angular separation and epoch of the closest approach. Significant differences
are mainly due to a significant changes of the proper motion values between
Gaia DR2 and eDR3.

Compared to K18b, we improved the exclusion of spurious sources and the
detection of co-moving star pairs, especially in cases when Gaia eDR3 does not
provide a proper motion for the BGS. Due to the higher precision in the proper
motions, the uncertainties for the expected epochs and minimum separations
decrease roughly by a factor of 2. However, the uncertainties of the expected
shifts stay roughly constant since they are dominated by the errors of the
assumed masses. This should be improved for future studies. However, measuring
the masses is the ultimate aim of observing the astrometric-microlensing
events. For that purpose the unlensed angular separation is of special
interest, since it cannot directly be observed but is essential to determine
the deflection. And this parameter does not depend on the assumed mass.

The observation of the predicted events and the subsequent mass determination
will lead to a better understanding of stellar masses for isolated stars.
Based on the prediction from Gaia DR2, several observing programs started.
With the present work, it is possible to further improve the selection of
promising events. The 625 events caused by 473 different WD might be of
particular interest. On the one hand, these are easier to observe, due to the
lower magnitude difference (caused by the high mass-to-luminosity ratio). On
the other hand, such observations will lead to a better understanding of WDs,
and thus of the final phase of the evolution of low- to medium-mass stars. It
might also be valuable to observe multiple events caused by the same lens
star. This can strongly reduce the uncertainties of the mass determination for
such cases.

Using the upcoming Gaia data releases will allow to further improve the
analysis. Also, the future Gaia releases will provide orbital parameters of
binaries. These can strongly improve the selection of co-moving stars and
might also be included in the forecast of the paths. Further improvements from
future releases will be achieved due to the more precise astrometric
parameters and due to the expected increase in the effective angular
resolution. Finally, with the upcoming data releases, more detailed knowledge
of the spurious sources is expected, which leads to better selections of BGS.
New event predictions will be mainly added with epochs of closest approach
during the Gaia mission or in the near future. These will be of special
interest once the individual Gaia measurements are published.

Finally, we showed that the WD Gaia eDR3-4053455379420641152 will cause a
promising astrometric-microlensing event in 2025. This event is ideal for a
JWST observation, since the background star is brighter in the near infrared
wavelength regime, and a measurable shift of the center of light is expected
even if it is not possible to resolve the lens and source stars.

## Acknowledgements

We would like to thank the referee who provided useful and detailed comments,
which helps to improve this paper. We gratefully acknowledge the technical
support we received from staff of the e-inf-astro project (BMBF
Förderkennzeichen 05A20VH5).
This work has made use of results from the ESA space mission Gaia, the data
from which were processed by the Gaia Data Processing and Analysis Consortium
(DPAC). Funding for the DPAC has been provided by national institutions, in
particular the institutions participating in the Gaia Multilateral Agreement.
The Gaia mission website is: http://www.cosmos.esa.int/Gaia.
Based on data products from observations made with ESO Telescopes at the La
Silla or Paranal Observatories under ESO programme ID 179.B-2002.
This publication makes use of data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation.
The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation Grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.
This research has made use of the SIMBAD database and the “Aladin sky atlas”
(Bonnarel et al., 2000) both developed and operated at CDS, Strasbourg
Observatory, France

## References

  * Andersen (1991) Andersen, J. 1991, A&A Rev., 3, 91, doi: 10.1007/BF00873538
  * Bonnarel et al. (2000) Bonnarel, F., Fernique, P., Bienaymé, O., et al. 2000, A&AS, 143, 33, doi: 10.1051/aas:2000331
  * Bramich (2018) Bramich, D. M. 2018, A&A, 618, A44, doi: 10.1051/0004-6361/201833505
  * Bramich & Nielsen (2018) Bramich, D. M., & Nielsen, M. B. 2018, Acta Astron., 68, 183, doi: 10.32023/0001-5237/68.3.1
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560.  https://arxiv.org/abs/1612.05560
  * Chwolson (1924) Chwolson, O. 1924, Astronomische Nachrichten, 221, 329
  * Dominik & Sahu (2000) Dominik, M., & Sahu, K. C. 2000, ApJ, 534, 213, doi: 10.1086/308716
  * Einstein (1936) Einstein, A. 1936, Science, 84, 506, doi: 10.1126/science.84.2188.506
  * Fabricius et al. (2020) Fabricius, C., Luri, X., Arenou, F., et al. 2020, A&A, doi: 10.1051/0004-6361/202039834
  * Gaia Collaboration et al. (2020a) Gaia Collaboration, Brown, A. G., Vallenari, A., Prusti, T., & de Bruijne, J. H. 2020a, A&A, doi: 10.1051/0004-6361/202039657
  * Gaia Collaboration et al. (2020b) Gaia Collaboration, Smart, R., Sarro, L. M., et al. 2020b, A&A, doi: 10.1051/0004-6361/202039498
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98, doi: 10.3847/1538-3881/aafc33
  * Gould (1992) Gould, A. 1992, ApJ, 392, 442, doi: 10.1086/171443
  * Harding et al. (2018) Harding, A. J., Stefano, R. D., Lépine, S., et al. 2018, MNRAS, 475, 79, doi: 10.1093/mnras/stx2985
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hog et al. (1995) Hog, E., Novikov, I. D., & Polnarev, A. G. 1995, A&A, 294, 287
  * Honma (2001) Honma, M. 2001, PASJ, 53, 233, doi: 10.1093/pasj/53.2.233
  * Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90
  * Klüter et al. (2018a) Klüter, J., Bastian, U., Demleitner, M., & Wambsganss, J. 2018a, A&A, 615, L11, doi: 10.1051/0004-6361/201833461
  * Klüter et al. (2018b) —. 2018b, A&A, 620, A175, doi: 10.1051/0004-6361/201833978
  * Lépine & DiStefano (2012) Lépine, S., & DiStefano, R. 2012, ApJ, 749, L6, doi: 10.1088/2041-8205/749/1/L6
  * Lindegren et al. (2020) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2020, A&A, doi: 10.1051/0004-6361/202039709
  * McGill et al. (2020) McGill, P., Everall, A., Boubert, D., & Smith, L. C. 2020, MNRAS, 498, L6, doi: 10.1093/mnrasl/slaa118
  * McGill et al. (2019a) McGill, P., Smith, L. C., Evans, N. W., Belokurov, V., & Lucas, P. W. 2019a, MNRAS, 487, L7, doi: 10.1093/mnrasl/slz073
  * McGill et al. (2018) McGill, P., Smith, L. C., Evans, N. W., Belokurov, V., & Smart, R. L. 2018, MNRAS, 478, L29, doi: 10.1093/mnrasl/sly066
  * McGill et al. (2019b) McGill, P., Smith, L. C., Evans, N. W., Belokurov, V., & Zhang, Z. H. 2019b, MNRAS, 483, 4210, doi: 10.1093/mnras/sty3344
  * Minniti et al. (2010) Minniti, D., Lucas, P. W., Emerson, J. P., et al. 2010, New A, 15, 433, doi: 10.1016/j.newast.2009.12.002
  * Miralda-Escude (1996) Miralda-Escude, J. 1996, ApJ, 470, L113, doi: 10.1086/310308
  * Miyamoto & Yoshii (1995) Miyamoto, M., & Yoshii, Y. 1995, AJ, 110, 1427, doi: 10.1086/117616
  * Mustill et al. (2018) Mustill, A. J., Davies, M. B., & Lindegren, L. 2018, A&A, 617, A135, doi: 10.1051/0004-6361/201833527
  * Nielsen & Bramich (2018) Nielsen, M. B., & Bramich, D. M. 2018, Acta Astron., 68, 351, doi: 10.32023/0001-5237/68.4.3
  * Paczyński (1986a) Paczyński, B. 1986a, ApJ, 301, 503, doi: 10.1086/163919
  * Paczyński (1986b) —. 1986b, ApJ, 304, 1, doi: 10.1086/164140
  * Paczyński (1991) —. 1991, ApJ, 371, L63, doi: 10.1086/186003
  * Paczyński (1995) —. 1995, Acta Astron., 45, 345
  * Paczyński (1996) —. 1996, Acta Astron., 46, 291
  * Proft et al. (2011) Proft, S., Demleitner, M., & Wambsganss, J. 2011, A&A, 536, A50, doi: 10.1051/0004-6361/201117663
  * Refsdal (1964) Refsdal, S. 1964, MNRAS, 128, 307, doi: 10.1093/mnras/128.4.307
  * Rybizki et al. (2021) Rybizki, J., Green, G., Rix, H.-W., et al. 2021, arXiv e-prints, arXiv:2101.11641.  https://arxiv.org/abs/2101.11641
  * Sahu et al. (2017) Sahu, K. C., Anderson, J., Casertano, S., et al. 2017, Science, 356, 1046, doi: 10.1126/science.aal2879
  * Salim & Gould (2000) Salim, S., & Gould, A. 2000, ApJ, 539, 241, doi: 10.1086/309196
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163, doi: 10.1086/498708
  * Smith et al. (2018) Smith, L. C., Lucas, P. W., Kurtev, R., et al. 2018, MNRAS, 474, 1826, doi: 10.1093/mnras/stx2789
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * The Astropy Collaboration et al. (2013) The Astropy Collaboration, Robitaille, Thomas P., Tollerud, Erik J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Torra et al. (2020) Torra, F., Castañeda, J., Fabricius, C., et al. 2020, A&A, doi: 10.1051/0004-6361/202039637
  * Torres et al. (2010) Torres, G., Andersen, J., & Giménez, A. 2010, A&A Rev., 18, 67, doi: 10.1007/s00159-009-0025-1
  * Walker (1995) Walker, M. A. 1995, ApJ, 453, 37, doi: 10.1086/176367
  * Zurlo et al. (2018) Zurlo, A., Gratton, R., Mesa, D., et al. 2018, MNRAS, 480, 236, doi: 10.1093/mnras/sty1805

Table 2: Description of the columns in the online table. This part list all the important parameters directly taken from Gaia eDR3. It is continued in Table3), where all the from us derived columns are described. | Name | Unit | Description
---|---|---|---
1 | event_id |  |  | Unique identifier of this event. This is the decimal representation of the
---
Gaia eDR3 source_id of the lens, and a disambiguator for the lensed object
2 | lens_id |  | Gaia eDR3 source_id of the lens
3 | lens_ra | deg | Gaia eDR3 RA of the lens
4 | lens_dec | deg | Gaia eDR3 Declination of the lens
5 | lens_err_ra | mas | Error of lens__ra from Gaia eDR3
6 | lens_err_dec | mas | Error of lens__dec from Gaia eDR3
7 | lens_pmra | mas/yr | Proper motion in RA of the lens from Gaia eDR3
8 | lens_pmdec | mas/yr | Proper motion in Declination of the lens from Gaia eDR3
9 | lens_err_pmra | mas/yr | Error of lens_pmra from Gaia eDR3
10 | lens_err_pmdec | mas/yr | Error of lens_pmdec from Gaia eDR3
11 | lens_parallax | mas | Parallax of the lens from Gaia eDR3
12 | lens_err_parallax | mas | Standard error of the parallax the lens from Gaia eDR3
13 | lens_phot_g_mean_mag | mag | Mean magnitude of the lens in the integrated G band from Gaia eDR3
14 | lens_phot_rp_mean_mag | mag | Mean magnitude of the lens in the integrated RP band from Gaia eDR3
15 | lens_phot_bp_mean_mag | mag | Mean magnitude of the lens in the integrated BP band from Gaia eDR3
16 | ob_id |  | Gaia eDR3 source_id of the lensed object
17 | ob_ra | deg | Gaia eDR3 RA of the lensed object
18 | ob_dec | deg | Gaia eDR3 Declination of the lensed object
19 | ob_err_ra | mas | Error of ob__ra from Gaia eDR3
20 | ob_err_dec | mas | Error of ob__dec from Gaia eDR3
21 | ob_pmra | mas/yr | Proper motion in RA of the lensed object from Gaia eDR3
22 | ob_pmdec | mas/yr | Proper motion in Declination of the lensed object from Gaia eDR3
23 | ob_err_pmra | mas/yr | Error of ob_pmra from Gaia eDR3
24 | ob_err_pmdec | mas/yr | Error of ob_pmdec from Gaia eDR3
25 | ob_parallax | mas | Parallax of the lensed object from Gaia eDR3
26 | ob_err_parallax | mas | Standard error of the parallax the lensed object from Gaia eDR3
27 | ob_phot_g_mean_mag | mag | Mean magnitude of the lensed object in the integrated G band from Gaia eDR3
28 | ob_phot_rp_mean_mag | mag | Mean magnitude of the lensed object in the integrated RP band from Gaia eDR3
29 | ob_phot_bp_mean_mag | mag | Mean magnitude of the lensed object in the integrated BP band from Gaia eDR3
30 | ob_displacement_ra_doubled | mas |  Doubled displacement in RA between DR2 and DR3, cos($\delta$) applied
31 | ob_displacement_dec_doubled | mas | Doubled displacement in Dec between DR2 and DR3
Table 3: Continuation of Table2. Description of the columns estimated in this analysis | Name | Unit | Description
---|---|---|---
32 | star_type |  |  | Type of the lensing star:
---
WD = White Dwarf, MS = Main Sequence, RG = Red Giant, BD = Brown Dwarf
33 | mass | solMass | Estimated mass of the lens from Gaia eDR3
34 | err_mass | solMass | Error in the mass of the lens from Gaia eDR3
35 | theta_e | mas | Einstein radius of the event
36 | err_theta_e | mas | Error in the Einstein radius of the event
37 | t_aml | yr |  Approximate duration of the event (i.e., shift $\gtrsim 0.1\,\mathrm{mas}$)
37 | t_0_1mas | yr |  | Approximate duration on which shift_plus changes by 0.1 mas from its maximum
---
( abs(shift_plus_vector(tca)-shift_plus_vector(tca-t_0_1_mas)) = 0.1 mas )
37 | t_50pc | yr |  |  Approximate duration on which shift_plus changes by $50\%$ from its maximum
---
( abs(shift_plus_vector(tca)-shift_plus_vector(tca-t_50pc)) = 50% * shift_plus
)
40 | tca | yr | Estimated time of the closest approach
41 | err_tca | yr | Error in tca
42 | dist | mas | Estimated distance at closest approach
43 | err_dist | mas | Error d_min
44 | u |  | Estimated distance at closest approach in Einstein radii
45 | u_error_m |  | Left 67% confidence interval of u
46 | u_error_p |  | Right 67% confidence interval of u
47 | shift | mas | Maximal astrometric shift of the center of light
48 | shift_error_m | mas | Left 67% confidence interval of shift
49 | shift_error_p | mas | Right 67% confidence interval of shift
50 | shift_lum | mas | Maximal astrometric shift including lens-luminosity effects
51 | shift_lum_error_m | mas | Left 67% confidence interval of shift_lum
52 | shift_lum_error_p | mas | Right 67% confidence interval of shift_lum
53 | shift_plus | mas | Maximal astrometric shift of brighter image
54 | shift_plus_error_m | mas | Left 67% confidence interval of shift_plus
55 | shift_plus_error_p | mas | Right 67% confidence interval of shift_plus
56 | magnification | mag | Maximal magnification
57 | magnification_error_m | mag | Left 67% confidence interval of magnification
58 | magnification_error_p | mag | Right 67% confidence interval of magnification
59 | l2_tca | yr | Estimated time of the closest approachas seen from Earth-Sun L2 point
60 | l2_err_tca | yr | Error in l2_tca
61 | l2_dist | mas | Estimated distance at closest approachas seen from Earth-Sun L2 point
62 | l2_err_dist | mas | Error l2_d_min
63 | l2_u |  | Estimated distance at closest approach in Einstein radiias seen from Earth-Sun L2 point
64 | l2_u_error_m |  | Left 67% confidence interval of l2_u
65 | l2_u_error_p |  | Right 67% confidence interval of l2_u
66 | l2_shift | mas | Maximal astrometric shift of the center of lightas seen from Earth-Sun L2 point
67 | l2_shift_error_m | mas | Left 67% confidence interval of l2_shift
68 | l2_shift_error_p | mas | Right 67% confidence interval of l2_shift
69 | l2_shift_lum | mas | Maximal astrometric shift including lens-luminosity effectsas seen from Earth-Sun L2 point
70 | l2_shift_lum_error_m | mas | Left 67% confidence interval of l2_shift_lum
71 | l2_shift_lum_error_p | mas | Right 67% confidence interval of l2_shift_lum
72 | l2_shift_plus | mas | Maximal astrometric shift of brighter imageas seen from Earth-Sun L2 point
73 | l2_shift_plus_error_m | mas | Left 67% confidence interval of l2_shift_plus
74 | l2_shift_plus_error_p | mas | Right 67% confidence interval of l2_shift_plus
75 | l2_magnification | mag | Maximal magnificationas seen from Earth-Sun L2 point
76 | l2_magnification_error_m | mag | Left 67% confidence interval of l2_magnification
77 | l2_magnification_error_p | mag | Right 67% confidence interval of l2_magnification
Figure 19: Cutouts from DSS, PanSTARRS, 2MASS and VVV used in the visual
inspections. The sources are indicated as red circles and the lens as green
squares epoch. All closeby Gaia eDR3 sources are marked by a yellow hexagon.
If Gaia list a proper-motion, we show the positions at the epoch of the image
(${\sim}2012.5$ for PanStarrs, ${\sim}1990.0$ for DSS, ${\sim}2015.75$ for VVV
, and ${\sim}1998.5$ for 2MASS). The first image (a) shows a ”candidate” which
was identified as binary star using the displacement between Gaia DR2 (black
cross) and eDR3 (yellow cross). The Gaia DR2 positions are shown as a black
cross. We plotted the ”BGS” also as square, and use the displacement to
determine the position at the image epoch ${\sim}2013$. The other two images
(e & i) in the first column are labelled as bad images since the source should
be visible in the PanSTARRS or DSS images. The second column contains images
where no we were not able to confirm or rule out the existence of the
background source. These are either highly blended by or within the saturated
core of the lens stars (b & f), to faint to be seen in DSS image (j). Further,
in some cases, the field is highly crowded to identify the source. For all
three cases, we removed the event from our results. In the third column, we
show examples where the source could be identified. Those are contained in our
result. Finally, in the fourth column, we show a comparison between DSS (d),
PanSTARRS(h) and VVV (l) for the same microlensing event. The BGS is visible
in the VVV’s H band images but blended by the HPMS in the DSS and PanSTARRS
images. The Gaia sources do perfectly align with sources or features seen in
the VVV image even though they are in the saturated areas of the PanStarrs and
DSS image. There is a second candidate marked by the magenta diamond, which is
aligned with a spike of the lens and hence was removed.