# Predicting Astrometric Microlensing Events from Gaia DR3

Jie Su,1,2,3,4 Jiancheng Wang,1,2,3 Yigong Zhang5 Xiangming Cheng1,2,3,4 and
Lei Yang1,2,3
1Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China
2Key Laboratory for the Structure and Evolution of Celestial Objects,Chinese
Academy of Sciences, Kunming 650216, China
3Center for Astronomical Mega-Science, Chinese Academy of Sciences, Beijing
100012, China
4 University of Chinese Academy of Sciences, Beijing 100049, China
5 College of Information Engineering, Kunming University, Kunming 650214,
China E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Currently astrometric microlensing is the only tool that can directly measure
the mass of a single star, it can also help us to detect compact objects like
isolated neutron stars and black holes. The number of microlensing events that
are being predicted and reported is increasing. In the paper, the potential
lens stars are selected from three types of stars, high-proper-motion stars,
nearby stars and high-mass stars. For each potential lens star, we select a
larger search scope to find possible matching sources to avoid missing events
as much as possible. Using Gaia DR3 data, we predict 4500 astrometric
microlensing events with $\delta\theta_{+}>0.1mas$ that occur between J2010.0
and J2070.0, where 1664 events are different from those found previously.
There are 293 lens stars that can cause two or more events, where 5 lens stars
can cause more than 50 events. We find that 116 events have the distance of
background stars from the proper motion path of lens stars more than
$8^{\prime\prime}$ in the reference epoch, where the maximum distance is
$16.6^{\prime\prime}$, so the cone search method of expanding the search range
of sources for each potential lens star can reduce the possibility of missing
events.

###### keywords:

astrometry – gravitational lensing: micro – methods: data analysis

††pubyear: 2015††pagerange: Predicting Astrometric Microlensing Events from
Gaia DR3–Predicting Astrometric Microlensing Events from Gaia DR3

##  1 Introduction

Gravitational lensing describes the deflection and magnification of background
sources when a massive object (lens) passes in front of them. When the lens is
the stellar-mass object, the deflection is less than a few microarcseconds
(mas) and the effect is referred to as microlensing. Microlensing describes
the positional deflection (astrometric microlensing) and magnification
(photometric microlensing) of a background source over time (Paczynski, 1986;
Hog et al., 1995; Miyamoto & Yoshii, 1995; Walker, 1995). The photometric
microlensing has been investigated by surveys such as the Optical
Gravitational Lensing Experiment (OGLE) (Udalski, 2003) or the Microlensing
Observations in Astrophysics (MOA) (Bond et al., 2001), whereas the
astrometric microlensing was detected for the first time only recently (Sahu
et al., 2017; Zurlo et al., 2018; McGill et al., 2023).

Astrometric microlensing provides the possibility to directly measure the mass
of a single star (Paczynski, 1995; Kains et al., 2017), it also can detect
faint and compact lens such as isolated neutron stars and black holes because
the luminosity of the lens is not necessarily measured. Recently, an isolated
stellar-mass black hole was detected by Sahu et al. (2022) through astrometric
microlensing based on the Hubble Space Telescope (HST) astrometry and the
ground-based photometry, and for the microlensing events Lam et al. (2022)
proposed that the lens star is a compact object.

The microlensing events are intrinsically rare occurrences, e.g. the all-sky
averaged value of the astrometric optical depth is $2.5\times 10^{-5}$
(Belokurov & Evans, 2002), that depend on the alignments of source star and
lens star, so predicting when and where they will occur is highly advantageous
for the collection of data throughout an event.

Gaia Data has been proven to be ideal for predicting astrometric microlensing
events (Gaia Collaboration et al., 2016), and about 25 000 sources will have a
significant variation of the centroid shift during the Gaia mission period
(Belokurov & Evans, 2002). Using Gaia DR1 (Gaia Data Release 1), McGill et al.
(2018) predicted an astrometric microlensing event with the white dwarf LAWD
37 as a lens star,and recently McGill et al. (2023) measured the astrometric
deflection of the background source and obtained the mass of lens star LAWD37.
After Gaia DR2 (Gaia Data Release 2) was released, many research works have
been done. Bramich & Nielsen (2018) and Bramich (2018) predicted the
microlensing events between 25th July 2014 and the end of the century. In
addition, McGill et al. (2019b) used the data of Gaia DR2 and Vista Variables
in the Via Lactea Infrared Astrometric Catalog (Smith et al., 2018) to predict
two astrometric microlensing events,and McGill et al. (2019a) searched for two
upcoming photometric microlensing events. Mustill et al. (2018) predicted the
photometric microlensing events between 2015.5 and 2035.5. Ofek (2018)
searched for the astrometric microlensing events between pulsars and stars in
Gaia DR2. Klüter et al. (2018a, b) predicted 3914 astrometric microlensing
events caused by 2875 different lenses between 2010 and 2065. In 2020, Gaia’s
early Data release 3 (Gaia eDR3) data was released, Klüter et al. (2022)
updated their prediction results and added 1758 new microlensing events
between 2010 and 2066. In addition, Luberto et al. (2022) searched for
astrometric microlensing events by nearby brown dwarfs, but this work did not
reveal any upcoming microlensing events. Wyrzykowski et al. (2022) identified
363 photometric microlensing events in Gaia Data Release 3 (Gaia DR3) covering
the years of 2014 - 2017 in all over the sky, and Jablonska et al. (2022)
found that one of the events discovered by Wyrzykowski et al. (2022) may lead
to a measurable astrometric signal.

In June 2022, Gaia Data Release3 (Gaia DR3) was released, and its astrometry
and broad band photometry content are the same as those of Gaia EDR3. Gaia DR3
contains 585 million sources with five-parameter astrometry (two positions,
the parallax, and two proper motion components), and about 882 million sources
with six-parameter astrometry, including an additional pseudocolour parameter
given by Collaboration et al. (2023b). Gaia DR3 also released many new data
products, such as the astrophysical parameters (mass and age) of 128 million
stars (Creevey et al., 2023; Fouesneau et al., 2023), and the mass estimation
of some lens stars used in this paper.

In this paper, we search for microlensing events using Gaia DR3, and potential
lens stars are selected from the three types of stars, high-proper-motion
stars (HPMS), nearby stars (NS) and high-mass stars (HMS). It should be noted
that NS do not contain HPMS.

In Section 2, we briefly outline the theoretical background for astrometric
microlensing. In Section 3, we present the lens and background star catalogs
used in the paper. In Section 4, we detail the methods of searching for
background star matched with lens star, calculate the source-lens closest
approach, and estimate lens masses. We present our results in Section 5. In
Section 6, we give summary and conclusions.

##  2 Theoretical background

###  2.1 Astrometric Microlensing Signals

The theory of astrometric microlensing is described in detail in the
literature (Belokurov & Evans, 2002; Dominik & Sahu, 2000; Paczynski, 1996)
and we briefly introduce important concepts and equations directly relevant to
this paper.

When the background source (S), the lens (L), and the observer are perfectly
collinear, the lensed image of the source will form a so-called Einstein ring.
The characteristic size of this ring is given by the Einstein radius as

$\theta_{\mathrm{E}}=\sqrt{\frac{4GM_{L}}{c^{2}}\frac{D_{S}-D_{L}}{D_{L}D_{S}}},$
(1)

where $G$ is the gravitational constant, $c$ is the speed of light, $M_{L}$ is
the mass of the lens, $D_{S}$ and $D_{L}$ are the distances between the
observer and the background source or the lens.

The angular position vector $\boldsymbol{\varphi}$ on the celestial sphere for
star is given:

$\boldsymbol{\varphi}=\left(\begin{array}[]{l}\alpha_{0}\\\
\delta_{0}\end{array}\right)+\left(\begin{array}[]{c}\mu_{\alpha^{*}}/\cos\delta_{0}\\\
\mu_{\delta}\end{array}\right)\cdot\left(t-t_{ref}\right)+\varpi\cdot\boldsymbol{P}(t),$
(2)

where $\boldsymbol{P}(t)$ is expressed by

$\boldsymbol{P}(\mathrm{t})=\left(\begin{array}[]{c}{\left[X(t)\sin\alpha_{0}-Y(t)\cos\alpha_{0}\right]/\cos\delta_{0}}\\\
X(t)\cos\alpha_{0}\sin\delta_{0}+Y(t)\sin\alpha_{0}\sin\delta_{0}-Z(t)\cos\delta_{0}\end{array}\right),$
(3)

and $\alpha_{0}$, $\delta_{0}$,$\mu_{\alpha}$, $\mu_{\delta}$ and $\varpi$
represent the right ascension, declination, proper motion in right ascension
direction, proper motion in declination direction and annual parallax
respectively. $t_{ref}$ is the reference epoch, $X(t)$, $Y(t)$ and $Z(t)$ are
the Cartesian barycentric Solar-system coordinates in au of the earth on the
ICRF at time $t$. In this paper, the above coordinate are calculated by the
astropy PYTHON package from NASA JPL’s Horizons Ephemeris (Astropy
Collaboration et al., 2013, 2018; Collaboration et al., 2022).

Let $\varphi_{S}$ and $\varphi_{L}$ represent the angular positions of the
source and the lens, respectively. One can define the dimensionless distance
vector as

$\mathbf{u}=\frac{\boldsymbol{\varphi_{S}}-\boldsymbol{\varphi_{L}}}{\theta_{E}}.$
(4)

Usually two images of the source are observed when the source is not perfectly
aligned with the lens. The bright image (+) is close to the source and the
faint image (-) is close to the lens. Their distance relative to the lens are
given by

$\theta_{\pm}=\frac{u\pm\sqrt{(u^{2}+4)}}{2}\cdot\theta_{E},$ (5)

where $u=|\mathbf{u}|$. When the separation of the images is too small to be
resolved, only the centroid of light formed by the images can be measured.
This can be described by

$\theta_{C}=\frac{u^{2}+3}{u^{2}+2}u\cdot\theta_{E},$ (6)

and the corresponding shift is the astrometric signal given by

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

The maximum shift of the center of light occurs at $u=\sqrt{2}$.

When the lens is luminous and unresolved from the source , the shift between
lensed and unlensed position (position of the combined center of light) can be
determined by

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

where $f_{LS}$ is the flux ratio between the lens and the source. When the
separation between the lens and the source is large enough for lens and the
brighter image (+) to be resolved, the brighter image (+) will be measured.
The corresponding shift compared to the unlensed position can be expressed by

$\delta\theta_{+}=\frac{\sqrt{u^{2}+4}-u}{2}\cdot\theta_{E}.$ (9)

Therefore $\delta\theta_{C,lum}$, $\delta\theta_{C}$ and $\delta\theta_{+}$
are the astrometric microlensing signals.

###  2.2 The change of astrometric microlensing signal with time

Refer to the derivation method of Dominik & Sahu (2000), we deduce the
relationships between the diurnal variation of the astrometric signals
($\delta\theta_{C,lum}$, $\delta\theta_{C}$ and $\delta\theta_{+}$) and the
parameters of $u$, $f_{LS}$ , $\mu_{LS}$ and $\varpi_{LS}$, where $\mu_{LS}$
and $\varpi_{LS}$ are the relative proper motion and parallax between the
source and the lens, respectively. First we obtain the following equation from
the equations (4) and (9):

$\begin{array}[]{c}\frac{d\delta\theta_{+}}{d\varphi}=\frac{d\delta\theta_{+}}{du}\cdot\frac{du}{d\varphi}=\frac{u}{2\cdot\sqrt{u^{2}+4}}-\frac{1}{2},\end{array}$
(10)

and then we have the equations from the equations (2) and (3):

$\begin{array}[]{l}\frac{d\varphi}{dt}\approx\left|\frac{d\boldsymbol{\varphi}}{dt}\right|=\sqrt{(\Delta\alpha_{\mu,\varpi})^{2}+(\Delta\delta_{\mu,\varpi})^{2}},\\\
\Delta\alpha_{\mu,\varpi}=\mu_{\alpha_{LS}^{*}}+\varpi_{LS}\cdot\Delta
P_{\alpha^{*}},\\\
\Delta\delta_{\mu,\varpi}=\mu_{\delta_{LS}}+\varpi_{LS}\cdot\Delta
P_{\delta}\par\end{array}$ (11)

where $\Delta P_{\alpha^{*}}=\Delta x\cdot sin(\alpha_{0,L})-\Delta y\cdot
cos(\alpha_{0,L})$ , $\Delta P_{\delta}=\Delta x\cdot cos(\alpha_{0,L})\cdot
sin(\delta_{0,L})+\Delta y\cdot sin(\alpha_{0,L})\cdot
sin(\delta_{0,L})-\Delta z\cdot cos(\delta_{0,L})$, and $\Delta x=-0.0171AU$,
$\Delta y=-0.0036AU$, $\Delta z=-0.0016AU$ are the daily variation of x, y, z
respectively. We take the daily variation from 2455199.5JD to 2455200.5JD near
the perihelion, and set
$\mu_{\alpha_{LS}^{*}}=\mu_{\alpha^{*}_{L}}-\mu_{\alpha^{*}_{S}}$,
$\mu_{\delta_{LS}}=\mu_{\delta_{L}}-\mu_{\delta_{S}}$, and
$\varpi_{LS}=\varpi_{L}-\varpi_{S}$. It should be noted that since the
projection on the parallactic motion is position depended, the equation
underestimates the effect for stars with $\alpha\sim 0^{\circ}$ or $\alpha\sim
180^{\circ}$. However, the effect only causes a small change in
$\frac{d\varphi}{dt}$, so it does not affect our conclusion. Then we obtain

$\begin{array}[]{c}\frac{d\delta\theta_{+}}{dt}=\frac{d\delta\theta_{+}}{d\varphi}\cdot\frac{d\varphi}{dt}=\left(\frac{u}{2\cdot\sqrt{u^{2}+4}}-\frac{1}{2}\right)\cdot\frac{d\varphi}{dt}.\end{array}$
(12)

According to the equations (7), we then have

$\frac{d\delta\theta_{C}}{dt}=\frac{d\delta\theta_{C}}{d\varphi}\cdot\frac{d\varphi}{dt}=\frac{2-u^{2}}{\left(u^{2}+2\right)^{2}}\cdot\frac{d\varphi}{dt},$
(13)

According to the equations (12),(13)), we obtain that the diurnal variation of
the astrometric signal depending on $\mu_{LS}$ and $\varpi_{LS}$ varies with
$u$. For $\delta\theta_{C}$ and $\delta\theta_{+}$ , their maximum diurnal
variations increase with $\mu_{LS}$ and $\varpi_{LS}$. The maximum diurnal
variation of $\delta\theta_{+}$ can exceed 0.1mas only when $u<5$. Therefore,
for most astrometric microlensing events, the change of their astrometric
signals within 24 hours (or even longer) cannot be detected by space
telescope, e.g. Hubble Space Telescope (Bellini et al., 2011) and James Webb
Space Telescope (Gardner et al., 2023), we do not need high accuracy in
finding the closest approach ($u_{0}$ and $t_{0}$) because the shift doesn’t
vary by an appreciable amount on a small timescale (e.g. a daily timescale).
We just need to encrypt the calculation points appropriately during certain
periods.

In this paper, according to the characteristic of that the astrometric signals
of microlensing events change slowly with time, we only take a small number of
data points when calculating the stellar trajectory, and can find the closest
approach ($u_{0}$ and $t_{0}$). More detailed description is given in Section
4.3.

##  3 Data Sources

###  3.1 Lens star selection

High proper motion stars are typically selected as potential lenses (Klüter et
al., 2018b, 2022; McGill et al., 2019b) because they traverse large areas of
the sky in a given time period which increases the rate of close alignments to
background sources. In addition, according to equation (1), for the events
with $D_{S}\gg D_{L}$, the closer the lens star is to the Earth or the larger
its mass is, the larger the radius of the Einstein ring is. For a given lens-
source angular separation, the larger $\theta_{E}$ corresponds to a larger
astrometric signal, the larger the observed signal is, the higher the
probability observed by the telescope is. Therefore, in this paper, the
potential lens stars are selected from three types of stars, HPMS, NS and HMS
(as shown in Table 1).

The potential lens star comes from Gaia DR3 and must have position, parallax,
and proper motion. To ensure a good astrometric solution, we require the
parallaxes with relatively small errors parallax_over_error>5\. However, the
source with parallax_over_error>5 may also contain about 1.6% spurious
astrometric solutions (Fabricius et al., 2021). In order to classify valid and
spurious astrometric solutions, Rybizki et al. (2022) train a neural network
to get a parameter of "astrometric fidelity" taken between 0 and 1 for 1.47
billion sources (e.g. five-parameter solutions or six- parameter solutions) in
Gaia eDR3. A value of 1.0 means a perfectly trustable solution, and the lowest
value of 0.0 indicates a lot of issues in the astrometric solution. Using this
parameter to eliminate spurious astrometric solutions is more effective than
simple cutting. Rybizki et al. (2022) suggested "In most regimes, the use of
the astrometric fidelitiy should yield a purer and more complete sample of
sources with reliable astrometric solutions." Because the astrometric data of
Gaia DR3 is the same as that of Gaia eDR3, the "astrometric fidelity"
parameter is still applicable in Gaia DR3. In this paper, the lens with the
"astrometric fidelity" parameter greater than 0.8 (fidelity_v2>0.8) are
selected. This limit of the parameter is the condition that all potential lens
stars need to meet.

For HPMS, $\mu\geq 100mas/yr$, where $\mu$ is the total proper motion, their
number is about 470000.

For NS, $\varpi>10mas$, and $\mu<100mas/yr$ is also needed to avoid the
overlap with HPMS. Their number is about 160000.

For HMS, their mass provided by Gaia DR3 is limited as $M_{L}>5M_{\odot}$, and
$\mu<100mas/yr$ and $\varpi<10mas$ are needed to avoid the overlap with HPMS
and NS. Their number is about 190000.

Table 1: The conditions of the potential lens stars HPMS | NS | HMS
---|---|---
$\mu\geq$100mas/yr |  $\varpi$>10mas |  $\varpi$<10mas
|  $\mu$<100mas/yr |  $\mu$<100mas/yr
|  | $M_{L}>5M_{\odot}$
$\sim$470000 |  $\sim$160000 |  $\sim$190000
parallax_over_error>5
phot_g_mean_mag IS NOT NULL
astrometric_params_solved=31(OR astrometric_params_solved=95)
fidelity_v2>0.8
Notes. $\mu$ is the total proper motion, $\varpi$ is parallax,
$M_{L}$ is the mass estimation of star (mass_flame) in the appendix
Astrophysical parameters provided by Gaia DR3,
parallax_over_error is parallax divided by its standard deviation,
phot_g_mean_mag is G-band mean magnitude,
astrometric_params_solved is astrometric solutions,
astrometric_params_solved=31 is five-parameter solutions,
astrometric_params_solved=95 is six-parameter solutions,
fidelity_v2 is "astrometric fidelity" parameter from Rybizki et al. (2022)

###  3.2 Background star selection

Background star (BGS) data are also taken from Gaia DR3. The selection
conditions are listed in Table 2. The condition is fidelity_v2>0.7 for
background stars with five - and six-parameter solutions. For background stars
with two-parameter solution, Rybizki et al. (2022) did not provide
"astrometric fidelity" parameter, and Gaia DR3 did also not provide the ruwe
parameter (Lindegren et al., 2021). However, we can estimate the ruwe
parameter according to the Gaia DR3 documentation (see the section 20.1.1 of
Gaia DR3
111https://gea.esac.esa.int/archive/documentation/GDR3/Gaia_archive/chap_datamodel/sec_dm_main_source_catalogue/ssec_dm_gaia_source.html)
, where the equations include

$\begin{array}[]{c}astrometric\\_gol\\_al=(9\cdot
v/2)^{1/2}\cdot\left[ruwe^{2/3}+2/(9\cdot v)-1\right],\end{array}$ (14)

and

$\begin{array}[]{c}v=astrometric\\_n\\_good\\_obs\\_al-N.\end{array}$ (15)

Then the ruwe parameter is given by

$\begin{array}[]{c}ruwe=\left(\frac{astrometric\\_gof\\_al}{\sqrt{9\cdot
v/2}}-\frac{2}{9\cdot v}+1\right)^{3/2},\end{array}$ (16)

where $astrometric\\_gof\\_al$ is the goodness of fitting statistic of model
along-scan observations, $v$ is the number of degrees of freedom for a source
update, and $N=5$. Therefore, the condition for background stars with two-
parameter solution is $ruwe<2$ and
$\sqrt{\sigma_{\alpha^{*}}^{2}+\sigma_{\delta}^{2}}<10mas$. In the subsequent
calculations, we set the parallax and proper to be 0 for background stars with
two-parameter solution and assumed standard errors of
$\sigma_{\alpha^{*}_{S},\delta_{S}}=5mas/yr$ and $\sigma_{\varpi_{S}}=2mas$.
Roughly 90% BGSs with five or six parameter solution have the standard errors
in proper motion and parallaxe below the above values.

Table 2: The conditions of background stars two-parameter solution | five - and six-parameter solutions
---|---
| ruwe<2,
---
$\sqrt{\sigma_{\alpha^{*}}^{2}+\sigma_{\delta}^{2}}<10mas$
fidelity_v2>0.7

##  4 Predicting microlensing events

Figure 1: Illustration of the selection processes. They are the selection of
potential lens stars (blue), the selection of background stars (green), the
determination of the closest approach, and the exclusion of co-moving stars
and the estimation of the expected microlensing effect (red). Figure 2: Color-
magnitude Diagrams. Upper-left panel: Color-magnitude diagram of the third
case potential lens with $G,G_{BP},G_{RP}$ from Section 4.2, $M_{G_{BP}}$ is
the absolute magnitude at $G_{BP}$ band, the stars below the green line are
considered to be WDs, and the stars above the red line are considered to be
RGs. Upper right panel: Hertzsprung-Russell diagram of $M_{G}$ versus
$G_{BP}-G_{RP}$ for all potential lens with $G,G_{BP},G_{RP}$ from Section
4.2. The red dots indicate the first case, the yellow indicate the second case
and the blue indicate the third case. Lower left panel: Same as the upper
right panel for all potential lens with $G,G_{BP},G_{RP}$. The lens masses are
indicated by the colour of the points (see the scale at the right of the
panel). Lower right panel: Same as lower left panel, the grey indicate all
potential lens with $G,G_{BP},G_{RP}$ and the colour dots indicate the lenses
of predicted events.

The selection process is shown in Figure 1. First, the potential lens (blue)
are selected, then the background stars are matched (green), and finally the
predicted events that meet the conditions are selected (red).

###  4.1 Initial lens-source matching

For $\sim 820000$ potential lens stars, we query for all sources using the
cone search method, where the search time range is J2010.0 $\sim$ J2070.0. The
center of the circle is the position of the lens at the reference epoch
(J2016.0) of Gaia DR3. For HPMS, the search radius is
$T\cdot\mu+6^{\prime\prime}$, and $\sim 1240000$ star pairs are found. For NS
and HMS, the search radius is $T\cdot\mu+8^{\prime\prime}$, and $\sim 260000$
and $\sim 760000$ star pairs are found respectively. $T=54yr$ is from J2070.0
to J2016.0. To avoid missing events as much as possible, we use the cone
search method that has larger searching area than that of the rectangular
search method (Klüter et al., 2018b, 2022; McGill et al., 2019b), and the
subsequent steps can further constrain the star pairs.

Now the number of founded pairs are still large, we further select the pair of
stars according to the characteristics of their relative motion. The relative
motion of the lens-background pair can be approximated as linear motion
(without considering parallax), and the relative position
($\boldsymbol{\Delta\varphi}$) of the star pair with time can be given by the
equation (2) as

$\begin{array}[]{c}\boldsymbol{\Delta\varphi}\approx\left(\begin{array}[]{c}\alpha_{0L}-\alpha_{0S}\\\
\delta_{0L}-\delta_{0S}\end{array}\right)+\left[t-t_{ref}\right]\left(\begin{array}[]{c}{\left[\mu_{\alpha^{*}L}-\mu_{\alpha^{*}S}\right]/\cos\delta_{0S}}\\\
\mu_{\delta L}-\mu_{\delta S}\end{array}\right)\\\
=\left(\begin{array}[]{c}\alpha_{0,\text{ LS }}\\\ \delta_{0,\text{ LS
}}\end{array}\right)+\left[t-t_{\text{ref
}}\right]\left(\begin{array}[]{c}\mu_{\alpha^{*},\text{ LS
}}/\cos\delta_{0S}\\\ \mu_{\delta,\text{ LS }}\end{array}\right),\end{array}$
(17)

where $\alpha_{0L}$, $\delta_{0L}$, $\alpha_{0S}$ and $\delta_{0S}$ are the
right ascensions and declinations of the lens star and the background star at
the reference epoch, respectively. $\mu_{\alpha^{\ast}L}$, $\mu_{\delta L}$,
$\mu_{\delta S}$ and $\mu_{\alpha^{\ast}S}$ are the proper motions in right
ascension and declination directions of the lens star and the background star.
$\alpha_{0,LS}$ and $\delta_{0,LS}$ are the relative right ascension and
relative declination of the star pair, $\mu_{\alpha^{\ast},LS}$ and
$\mu_{\delta,LS}$ are proper motions in right ascension and declination
directions of the star pair.

Therefore, the relative angular distance ($\theta_{sep}$) of the star pair can
be approximated as

$\begin{array}[]{c}\begin{aligned}
\theta_{sep}\approx\sqrt{\left(\Delta\alpha\cdot
cos\delta_{0S}\right)^{2}+\left(\Delta\delta\right)^{2}},\par\end{aligned}\end{array}$
(18)

where
$\Delta\alpha=\alpha_{0,LS}+\left(t-t_{ref}\right)\cdot\mu_{\alpha^{*},LS}/cos\delta_{0S}$
and $\Delta\delta=\delta_{0,LS}+\left(t-t_{ref}\right)\cdot\mu_{\delta,LS}$.
Based on $\frac{\partial\theta_{sep}}{\partial t}=0$, we can get the minimum
value $\theta_{sep,min}$ of $\theta_{sep}$ and its corresponding time
$t_{min}$ as

$\begin{array}[]{c}\theta_{sep,min}=\left|\frac{\mu_{\delta,LS}\cdot\alpha_{0,LS}\cdot\cos\delta_{0S}-\mu_{\alpha^{*},LS}\cdot\delta_{0,LS}}{\sqrt{\mu_{\alpha^{*},LS}^{2}+\mu_{\delta,LS}^{2}}}\right|,\end{array}$
(19)

and

$\begin{array}[]{c}t_{min}=-\frac{\alpha_{0,LS}\cdot\cos\delta_{0S}\cdot\mu_{\alpha^{*},LS}+\delta_{0,LS}\cdot\mu_{\delta,LS}}{\mu_{\alpha^{*},LS}^{2}+\mu_{\delta,LS}^{2}}+t_{ref}.\end{array}$
(20)

For all star pairs searched above, we use the equation (20) to calculate
$t_{min}$. If $t_{min}$ is within the range of J2010.0-J2070.0, the equation
(19) is used to calculate $\theta_{sep,min}$. If $t_{min}>2070$, the relative
angular distance $\theta_{sep,2070}$ is calculated at $t=2070$ according to
the equation (18). If $t_{min}<2010$, the relative angular distance
$\theta_{sep,2010}$ is calculated at $t=2010$ based on the equation (18). If
$\theta_{sep,min}$, $\theta_{sep,2070}$ or $\theta_{sep,2010}$ is less than
$\frac{20}{mas}\cdot\theta^{2}_{E}$, e.g. $\delta\theta_{+,mas}>0.05mas$
($\delta\theta_{+}\sim\frac{\theta_{E}}{u}=\frac{\theta_{E^{2}}}{\varphi}$),
where $\delta\theta_{+,max}$ is the approximate value of the maximum value of
$\delta\theta_{+}$ in the range J2010.0-J2070.0, the star pair can be
initially matched, otherwise the star pair is excluded. $\theta_{E}$ is
calculated according to the equation (1), where the lens star masses are
estimated in Section 4.2 .

After the application the above criteria, we find 19401, 3742 and 3914 pairs
for HPMS, NS and HMS respectively. It is noted that there is no restriction on
the background star at this time.

###  4.2 Mass estimation of lens stars

The masses of these potential lens stars are estimated in three cases :

First case, the masses are searched in the "Astrophysical parameters" table of
128 million stars released by Gaia DR3 (Creevey et al., 2023; Fouesneau et
al., 2023). The mass estimation of all HMS comes from this table. In addition,
the masses of about 40000 HPMS and NS are given in this table (see upper-right
panel of Figure 2, the mass estimation of the red dots in the panel comes from
this table).

Second case, the masses are estimated by matching the white dwarf catalog
(Gentile Fusillo et al., 2021). The catalog provides the parameter ’PWD’ as a
measure of the probability of the source being a white dwarf. In this paper,
the potential lens stars that do not belong to the first case are matched with
the white dwarf star catalog. About 13000 potential lens stars are matched
with the catalog (see upper-right panel of Figure 2, the mass estimation of
the yellow dots in the panel comes from the white dwarf star catalog), where
99% potential lens stars with ’PWD’>0.7 . Gentile Fusillo et al. (2021) listed
three types of estimated mass: M_H, M_He and M_mix corresponding to the mass
estimated by pure-H, pure-He, mixed hydrogen/helium (H/He) compositions model,
respectively. The order of use for mass estimation is M_H, M_He and M_mix. If
there is no mass estimation in the first order, the second order is used, and
so on. The lens ratios using M_H, M_He and M_mix are $\sim 96\%$, $\sim 1\%$,
and $\sim 2\%$, respectively. According to the statistics of lens, $\sim 93\%$
of lens stars have a difference between M_H and M_He less than $3\sigma$, and
$\sim 70\%$ of lens stars have a difference between M_H and M_mix less than
$3\sigma$. It is noted that about 200 potential lens stars have not mass
estimations, we then use the classical mass of white dwarf stars as their
masses, $M_{WD}=(0.65\pm 0.15)M_{\odot}$.

Third case, the masses are estimated by the mass-luminosity relations. For
potential lens stars (about 270000) that do not belong to the above two cases,
we use the method of Klüter et al. (2022) to estimate the mass. These lens
stars are divided into white dwarf, red giant, brown dwarf and main sequence
stars (see upper-left panel of Figure 2). The masses of white dwarfs, red
giants and brown dwarfs are estimated as $M_{WD}=(0.65\pm 0.15)M_{\odot}$,
$M_{RG}=(1.0\pm 0.5)M_{\odot}$ and $M_{BD}=(0.07\pm 0.03)M_{\odot}$
respectively. The masses of the main sequence stars are estimated using the
mass-luminosity relations, where a 10 per cent error is assumed.

It should be noted that mass estimation is important as it can affect follow
up planning decisions (McGill et al., 2019a). If we can find a more accurate
mass estimate from the literature, we prefer to choose it for improving the
accuracy of the prediction.

###  4.3 the source-lens closest approach

Figure 3: Flow chart of calculating the source-lens closest approach.

In this paper, the source-lens closest approach are searched by calculating
the relative motion of the star pair. According to Section 2.2, the signal of
astrometric microlensing events changes slowly with time. Therefore, we only
take a small number of data points when calculating the trajectory of the
star, and only encrypted sampling is carried out in a short period of time
when the signals change rapidly. The specific steps are the following as (see
Figure 3):

Step 1: we use the equations (2) and (3) to calculate the source-lens
separation ($u$) every 30 days, where there are 732 data points in the range
of J2010.0-J2070.0. From these data points we find the minimum source-lens
separation ( $u=u_{01}$) and the time of closest approach ( $t=t_{01}$)
between the source and the lens. If the difference between the source-lens
separation ($u_{i}$) and the minimum source-lens separation $u_{01}$ is less
than $\frac{5mas}{\theta_{E}}$, we record its corresponding time as $t_{i}$,
where i represents the ith data point.

Step2: we calculate the source-lens separation every day from 30 days before
$t_{01}$ to 30 days after $t_{01}$, and from 30 days before $t_{i}$ to 30 days
after $t_{i}$ , where there are 60 $\sim$ 120 data points. From these data
points we find the minimum source-lens separation ($u=u_{02}$) and the time of
closest approach ($t=t_{02}$) between the source and the lens. According to
the equations (13) and (12), we calculate $\frac{d\delta\theta_{C}}{dt}$, and
$\frac{d\delta\theta_{+}}{dt}$ with $u=u_{02}$. If the absolute value of any
one of them is greater than $80\mu as/day$, we go to step 3. Otherwise, the
calculation ends, and we obtain the results of $u_{0}=u_{02}$ and
$t_{0}=t_{02}$.

Step3: we calculate the source-lens separation every two hours from 24 hours
before $t=t_{02}$ to 24 hours after $t=t_{02}$, where there are 24 data
points. From these data points we find the minimum source-lens separation
($u=u_{0}$) and the time of closest approach ($t=t_{0}$) between the source
and the lens.

###  4.4 Checking for microlensing events

Figure 4: The proper motion difference between the lens star and the
background star. The red points are star pairs that do not meet the conditions
(21) and (22). The blue dots are the final selected star pairs.

Using $u_{0}$ and $t_{0}$ obtained in Section 4.3, we calculate the
observation signal $\delta\theta_{+}$ through the equations (1), (4) and (9),
and find out the events with $\delta\theta_{+}>0.1mas$ and the background star
meeting the conditions in Table 2. For these events, if the background star
has a five - or six-parameter solution, it is necessary to exclude binary
stars or co-moving stars. We refer to the conditions of excluding these star
pairs used by many authors (Bramich & Nielsen, 2018; Bramich, 2018; Klüter et
al., 2022) and the method of selecting binary stars by El-Badry et al. (2021),
and retain the star pairs meeting the following conditions (as shown in Figure
4):

$\begin{array}[]{c}\frac{\Delta\varpi}{\sigma_{\Delta\varpi}}=\frac{\varpi_{L}-\varpi_{S}}{\left(\sigma_{\varpi_{L}}^{2}+\sigma_{\varpi_{S}}^{2}\right)^{1/2}}>3,\end{array}$
(21)

$\begin{array}[]{c}|\boldsymbol{\mu_{S}-\mu_{L}}|>0.7\cdot\mu_{L},\\\
\mu_{S}<0.8\cdot\mu_{L}.\end{array}$ (22)

In addition, McGill et al. (2020) indicates that events with bright background
sources($G<18mag$) which are predicted to happen during the Gaia mission are
likely not genuine. For the background star with a two-parameter solution, we
check these events using a method similar to that of Klüter et al. (2022).
First, we search for matched sources by using the gaiadr3.dr2_neighbourhood
catalog (Torra et al., 2021). For one-to-many match case, we only consider the
DR3-DR2 pair with the smallest angular distance. Then we only select the match
that meets the following criteria as

$\begin{array}[]{c}\Delta\varphi_{match}<400mas,\\\ \Delta
G_{match}<1mag,\end{array}$ (23)

where $\Delta\varphi_{match}$ and $\Delta G_{match}$ are the position and
magnitude of source in DR3-DR2, respectively. We notice that 40 source have
different source_id in DR3-DR2 release data. To ensure a correct match, we
check these sources again. Specifically, we match their dr2_ source_id to
dr3_source_id and find that one of them is an error match. We have obtained
1281 sources with good match and 1117 sources with bad or no match. Second, we
estimate the proper motion ($\mu_{S}$) of background stars with good match as

$\begin{array}[]{l}\mu_{S}=\left[\left(\begin{array}[]{c}\alpha_{S,DR3}\\\
\delta_{S,DR3}\end{array}\right)-\left(\begin{array}[]{c}\alpha_{S,DR2}\\\
\delta_{S,DR2}\end{array}\right)\right]\cdot\begin{bmatrix}\cos\delta_{S,DR3}\\\
1\end{bmatrix}/\Delta t,\end{array}$ (24)

where $\alpha_{S,DR3}$, $\delta_{S,DR3}$, $\alpha_{S,DR2}$ and
$\delta_{S,DR2}$ are the positions of the BGS in Gaia DR3 and Gaia DR2 at
reference epoch, respectively, $\Delta t$ is the difference between the
catalog epochs of Gaia eDR3 and Gaia DR2 taken as 0.5 yr. We determine whether
these events satisfy the equations (22).

We then find 1070 events satisfying the above conditions. Third, we still need
to match the external catalogues for the events (42 events) with two-parameter
solution BGS that are brighter than 18 mag and satisfy the equation (22). It
is also required that the angular distance between the source of Gaia DR3 and
the external catalogues is less than 1 arcsecond, and the lens and background
star can match the same catalogues. Gaia DR3 catalogue includes pre-computed
cross-matches with optical/near infrared photometric and spectroscopic surveys
(Marrese et al., 2017, 2019). These external catalogues matched with Gaia DR3
are Pan-STARRS1 DR1 (Flewelling et al., 2020), SkyMapper DR2(Onken et al.,
2019a), SDSS DR 13(Albareti et al., 2017), URAT1(Zacharias et al., 2015),
Tycho2(Hog et al., 2000), Hipparcos-2(van Leeuwen, 2007), 2MASS(Skrutskie et
al., 2006), AllWISE(Mainzer et al., 2011), APASS DR9(Henden et al., 2016), GSC
2.3(Lasker et al., 2008), RAVE DR5(Kunder et al., 2017) and RAVE DR6(Steinmetz
et al., 2020). We exclude 33 events that cannot match any of the above
external catalogues. Fourth, we exclude 993 events that cannot match any of
the above external catalogues from 1117 events of that their background stars
do not match with Gaia DR3.

In addition, we recalculate 360 events occurred before 2010 or after 2070 to
estimate the exact epoch, and remove 75 events occurred before 2005 or after
2075.

For the final star pair, we determine the uncertainties of these predictions
using a Monte Carlo method, where we draw 1000 samples from appropriate
gaussian distribution for the lens with position, proper motion and parallax.
We do not include any co-variances between different input parameters. It
should be noted that for 95 events with $u_{0}<10$ and
$\sigma_{u_{0}}/u_{0}>5$, we provide the lower confidence level (16%) of
$\delta\theta_{+}$, $\delta\theta_{C}$, and $\delta\theta_{C,lum}$
respectively, rather than the standard deviation.

##  5 Results

###  5.1 Predicted Astrometric-microlensing Events

Figure 5: The statistical diagram of multiple events caused by a single lens
star. The abscissa is the number of lens stars, and the ordinate is the number
of events caused by a single lens star. It is noted that there are five lens
stars to cause more than 50 events. Figure 6: The histogram of
$\delta\theta_{+}$ for all events. 457 events have $\delta\theta_{+}>1mas$,
and $48\%$ events have $0.1mas<\delta\theta_{+}<0.2mas$ . Figure 7: Cutouts
for the events with $\delta\theta_{+}>10mas$ obtained using online observation
tool: the ESAsky (Baines et al., 2017; Giordano et al., 2018) ($\sim$ J2012
for PanSTARRS DR1 color(i,r,g) and $\sim$ J1990 for DSS2 color) and the Legacy
Surveys(Collaboration et al., 2023a) ($\sim$ J2017 for DECaPS 2 images(riY)).
The lens is shown in red rectangular, and the source is indicated in green
circles. There are 7 events with $\delta\theta_{+}>10mas$. We do not show
HPMS862 and HPMS3274, their times of closest approach are after J2070.0, and
their cutouts can be found through the ESAsky and Legacy Surveys. Figure 8:
Top panel: the histogram of the closest time for star pairs. The green, yellow
and cyan bars show the events with two-, five- and six-parameter solutions
BGS, respectively. Bottom panel: the histogram of the closest time for HMPS
and NS. The blue and red bars show the events with HMPS and NS lens,
respectively.  Figure 9: The $\Delta t_{0}$-statistic diagram of overlap
events with the five- or six-parameter background source stars. About 92% of
events have $\Delta t_{0}<3d$. Figure 10: Comparison of
$\left|\Delta\delta\theta_{+}\right|$ for overlap events with the five- or
six-parameter background source stars. Upper-left panel: the relationship
diagram of $\left|\Delta\delta\theta_{+}\right|$ and $\Delta\theta_{sep}$.
Upper right panel: the relationship diagram of
$\left|\Delta\delta\theta_{+}\right|$ and $M_{R}$. There are four events with
$M_{R}>2$ (HPMS1160, HPMS3441, HPMS1742 and HPMS1703), and their mass
estimates are from the white dwarf catalogue (Gentile Fusillo et al. (2021)).
For the four events, $\Delta\delta\theta_{+}$ are 5.174 mas, 1.412 mas , 0.827
mas and 0.562 mas, respectively. Lower left panel: the relationship diagram of
$\left|\Delta\delta\theta_{+}\right|$ and $u_{0}$. For events with large
difference in $\delta\theta_{+}$, $u_{0}$ is very small. Lower right panel:
the relationship diagram of $\left|\Delta\delta\theta_{+}\right|$ and $\Delta
t_{0}$. There is not obvious difference in $\delta\theta_{+}$ for events with
large difference of $t_{0}$.  Figure 11: The
$\left|\Delta\delta\theta_{+}\right|$-statistic diagram of the overlap events,
where about 71% of events have $\left|\Delta\delta\theta_{+}\right|\leq
0.010mas$. Figure 12: Comparison of $\delta\theta_{+}$ for overlap events with
the two-parameter background source stars. Upper-left panel: The
$\left|\Delta\delta\theta_{+}\right|$-statistic diagram. The contents of other
three panels are similar with those of Figure 10. Figure 13: Top panel: the
relationship diagram of $t_{0}$ and $\delta\theta_{+}$ for microlensing events
by NS.The green, red and blue dots show the events with two-, five- and six-
parameter solutions BGS, respectively. Bottom panel:the histogram of the
timescales for star pairs by NS.Four events with a timescale of more than 35
years are not shown in the figure. Figure 14: The changes of $u$ (black) and
$\delta\theta_{+}$ (blue) with time for the events HMS1. Figure 15: The
distances of background stars from the proper motion paths of lens stars for
all astrometric microlensing events. There are 116 events with the distances
more than 8 arcseconds.

In this paper, the searching time range is J2010.0-J2070.0, and the result is
provided as a spreadsheet available online, where each event has 43 columns
and a single row. Table 3 is a sample table consisting of 5 rows of result
data. Column meanings in Table 3 are defined in Table 8. Finally, 4500 events
caused by 3558 lens stars are found, where 4279 events are caused by HPMS
(including 907, 1453 and 1919 events with two, five and six astrometric
parameter for the background stars, respectively). There are 220 events caused
by NS (including 71, 50 and 99 events with two, five and six astrometric
parameter for background stars, respectively. Only one event is caused by HMS
and has two parameter background star. The details are shown in Table 9. In
Section 5.1.1, Section 5.1.2 and Section 5.1.3, our results are compared with
those of Klüter et al. (2022) in detail.

In the paper, the masses of the lens stars are mainly distributed in
$M_{L}<1.2M_{\odot}$ for many events. No event is found for the mass of the
lens star distributed in $3.2M_{\odot}<M_{L}<5M_{\odot}$. For the events found
by the authors (Bramich & Nielsen, 2018; Bramich, 2018; Klüter et al., 2018b,
2022), the masses of their lens stars are less than $2.5M_{\odot}$. The mass
threshold of HMS is set to be $M_{L}>5M_{\odot}$, and only one event is
finally determined for HMS, more detailed description is given in Section
5.1.3. Therefore our future work can reduce the mass threshold of lens stars
to $3M_{\odot}$ which could help us to find more events.

There are 293 lens stars that can cause two or more events, as shown in Figure
5. Surprisingly, there are five lens stars to cause more than 50 events. In
section 5.2, we will discuss two of these lens stars and their corresponding
events.

About $48\%$ of events have $0.1mas<\delta\theta_{+}<0.2mas$, 457 events have
$\delta\theta_{+}>1mas$, and 7 events have $\delta\theta_{+}>10mas$, as shown
in Figure 6 and Figure 7. The events around the Gaia DR3 epoch of J2016.0 are
the least because of the angular resolution limitation of Gaia DR3. From 2018
to 2035, the events will gradually increase and reach at the rate of about
$96events/yr$, as shown in Figure 8.

####  5.1.1 The overlap events

There are 2836 events overlapped, we calculate $\Delta
t_{0}=|t_{0,ext}-t_{0}|$,
$\Delta\delta\theta_{+}=\delta\theta_{+,ext}-\delta\theta_{+}$,
$M_{R}=M_{L}/M_{L,ext}$ and
$\Delta\theta_{sep}=\theta_{sep,ext}-\theta_{sep}$, where the values with
subscript ’ext’ are the results of Klüter et al. (2022) and $\theta_{sep}$ is
our estimated distance at closest approach.

There are 2352 events with five or six astrometric parameter for background
stars. For $\Delta t_{0}$, about 92% of events have $\Delta t_{0}<3d$ (as
shown in Figure 9), but there are two events with $\Delta t_{0}\approx 140d$,
where their $\Delta\delta\theta_{+}$ are very small ($-0.002mas$ and
$-0.031mas$ respectively, as shown in lower-right panel of Figure 10). For
$\Delta\delta\theta_{+}$, about 71% of events have
$\left|\Delta\delta\theta_{+}\right|\leq 0.010mas$ (as shown in Figure 11).
However, about 5.9 % of events have $\left|\Delta\delta\theta_{+}\right|\geq
0.1mas$, and we note that $u_{0}$ of these events is relatively small,
especially when $\left|\Delta\delta\theta_{+}\right|\geq 0.5mas$, $u_{0}$ is
less than 10 (see lower-left panel of Figure 10). It can be seen from formula
(9) that $\delta\theta_{+}$ is related to $u$ and $\theta_{E}$, and
$\delta\theta_{+}$ changes rapidly with $u$ when $u$ is small. There are four
events with $M_{R}>2$ (HPMS1160, HPMS3441, HPMS1742 and HPMS1703), and their
mass estimates are from the white dwarf catalogue (Gentile Fusillo et al.
(2021)), $M_{L}$ are $(0.87\pm 0.09)M_{\odot}$, $(0.44\pm 0.16)M_{\odot}$,
$(1.32\pm 0.01)M_{\odot}$ and $(1.38\pm 0.06)M_{\odot}$, respectively. For the
four events, $\Delta\delta\theta_{+}$ are 5.174 mas, 1.412 mas , 0.827 mas and
0.562 mas, respectively (see upper right panel of Figure 10 ). For these
events with $\left|\Delta\delta\theta_{+}\right|\geq 0.1mas$, our estimated
lens masses are not the same as that of (Klüter et al., 2022) (out of which
67% the mass difference is more than 10% ) to cause the differences in
$\theta_{E}$. But, 54 % of events with
$\left|\Delta\delta\theta_{+}\right|\geq 0.1mas$ still have $\delta\theta_{+}$
differences within $1\sigma$.

For 484 events with two parameter background source stars, the differences of
both predicted results are obvious, where the parallax and proper motion of
background stars used are different in addition to lens estimated mass.
(Klüter et al., 2022) used the estimated values of the parallax and proper
motion, but we set them to 0. The comparisons of both results are shown in
Figure 12. It is noted that there are 106 events with $\Delta t_{0}<5d$ and
215 events with $\left|\Delta\delta\theta_{+}\right|\leq 0.05mas$.

####  5.1.2 The events only searched by Klüter et al. (2022)

2006 events in Klüter et al. (2022) are not included in our results, where
1578 are excluded because they do not meet the selecting conditions in Table
1, Table 2, or the equation (21). There are 166 events that do not satisfy the
selection of two-parameter background stars in Section 4.4. In addition, we
find that $\delta\theta_{+}$ of 262 events do not reach the threshold of 0.1
mas. These differences could be caused by lens mass estimation, the parallax
and proper motion of background stars. See Table 10 for details. However,
$\sim 80\%$ of these 262 events have a difference of $\delta\theta_{+}$ within
$1\sigma$.

####  5.1.3 The events only searched by us (the new predicted event)

We have searched 1664 new predicted event, where about 86 % of the events are
not included in the search range of Klüter et al. (2022), including the lens,
background star and $t_{0}$ of event. It is noted that the sources of some
events are not within the rectangular box set by Klüter et al. (2022) during
the initial search. Other events could be related to the differences of
selecting parameters, such as lens mass estimation, the parallax and proper
motion of background stars. See Table 11 for details.

We will discuss these new events in detail based on our classification of
lens.

(1) Microlensing events by NS

There are 220 events caused by NS (see Table 9), where 45 events with
$\delta\theta_{+}>0.5mas$, out of which 20 events with $\delta\theta_{+}>1mas$
(including 4, 5 and 11 events with two, five and six parameter BGS,
respectively). 27 events will occur between J2010.0 and J2035.0, of which 2
events with $\delta\theta_{+}>1mas$. After J2035.0, the number of events will
increase, about 5 events per year (see Figure 8 and Figure 13). For timescales
of the events with $\delta\theta_{+}>0.1mas$, 62 events are within 5 years, 67
events are between 5 years and 10 years, 91 events are longer than 10 yr (see
Figure 13).

NS180. This event pick will occur in $2032.21\pm 0.03$, the timescale is $\sim
6.2yr$ and the maximum shift of the major image is $(1.45\pm 0.24)mas$. The
lens mass is estimated to be $0.32M_{\odot}$ by the mass-luminosity relations
in the third case of Section 4.2. Its total proper motion and parallax are
$95.777mas/yr$ and $11.334mas$, respectively. The lens has $G\approx 14.77mag$
which is $\sim 2.78mag$ brighter than the source star with six parameter
solutions. This event is listed in the first row of Table 3.

(2) Microlensing events by HMS

Initially, for HMS we obtained 558 pairs with $\delta\theta_{+}>0.1mas$
(including 4 and 554 events with two, five or six parameter BGS,
respectively). For 554 events with five or six parameter BGS, 487 pairs can
not meet the conditions of table 2 (fidelity_v2>0.7), and the remaining 67
pairs did not pass the criteria in equation 21. For 4 events with two
parameter BGS, 3 events have bad match with Gaia DR2.In the end, we only get
an event (HMS1) that satisfies all the conditions.

HMS1. The lens star (source_id=4090028395985895552) is a single star, its
spectral type is B and its mass is estimated as
$5.160^{+0.098}_{-0.089}M_{\odot}$,taken from Gaia DR3 astrophysical
parameters. Its parallax, proper motion in right ascension direction and
declination direction are $(0.489\pm 0.017)mas$, $(-22.466\pm 0.018)mas/yr$
and $(-38.676\pm 0.015)mas/yr$, respectively. The lens has $G\approx 12.77mag$
which is $\sim 3.1mag$ brighter than the source star. The event has a time of
closest lens-source approach of $2046.0\pm 3.3yr$ and the Einstein radius of
$(4.54\pm 3.19)mas$. The maximum shift of the major image is $0.53mas$ and its
lower confidence level (16%) is $0.05mas$. The changes of $u$ and
$\delta\theta_{+}$ with time is shown in Figure 14, the timescale is $\sim
9yr$. More details are shown in the second row of Table 3.

(3) Microlensing events by HPMS

We find 1443 new events by HPMS,where 240 events with
$\delta\theta_{+}>0.5mas$, out of which 123 events with
$\delta\theta_{+}>1mas$ (including 83 events with five or six parameter BGS).
There are 845 events occurred between J2010.0 and J2067.0, of which 100 will
occur within the next 10 years. In 116 events, the distance of background
stars from the proper motion track of the lens star exceeds
$8^{\prime\prime}$, as shown in Figure 15, where the maximum is
$16^{\prime\prime}$, and $\sim 97\%$ of these events have a Einstein radius of
more than 30 mas, with a maximum of $\sim 40.66mas$, so these lens can produce
microgravitational effects with $\delta\theta_{+}>0.1mas$ on the background
stars far away, and $\sim 83\%$ of these lens can cause ten or more events
because of high proper motion. However, since $\delta\theta_{+}$ decreases
with the increase of $\theta_{sep}$, the expected shift is small
($\delta\theta_{+}<0.2mas$).

There are three examples listed in Table 3:

HPMS1816. The maximum shift of the major image is more than 10 mas
($\delta\theta_{+}=10.94mas$) and its lower confidence level (16%) is
$0.79mas$. The event has a time of closest lens-source approach of $2068.48\pm
1.62yr$. The lens (CD-29 5220) has $G\approx 9.95mag$ which is $\sim 11mag$
brighter than the source star and the source star only have two parameter
solutions. The lens mass is estimated to be $0.65M_{\odot}$ by the mass-
luminosity relations in the third case of Section 4.2. Its total proper motion
and parallax are $162.919mas/yr$ and $23.662mas$, respectively.

HPMS3274. The maximum shift of the major image is the largest of all events
($\delta\theta_{+}=21.38mas$) and its lower confidence level (16%) is
$4.82mas$. The Einstein radius is $(30.54\pm 0.89)mas$. However, the event
peak will occur in $2073.23\pm 0.23$ and the lens (V∗ V2215 Oph) is very
bright ($G\approx 5.89mag$) which is $\sim 14.87mag$ brighter than the source
star with two parameter solutions. The lens star is a single star, its
spectral type is K and its mass is estimated as
$0.682^{+0.040}_{-0.040}M_{\odot}$, taken from Gaia DR3 astrophysical
parameters.

HPMS4600. The maximum shift of the major image is $(6.72\pm 2.17)mas$, the
event peak will occur in $2069.47\pm 0.12$ and the timescale is $\sim 6.4yr$.
The lens mass is estimated as $(0.42\pm 0.02)M_{\odot}$ from the white dwarf
catalogue (Gentile Fusillo et al. (2021)) labeled as ’WDJ221800.59+560214.92’.
The lens with $G\approx 18.03mag$ is $\sim 2.44mag$ brighter than the source
star with six parameter solutions.

Table 3:  5 rows of the predicted astrometric-microlensing events in this paper, and the rest are available online. Event_name | ref_epoch | lens_source_id | lens_ra | lens_dec | lens_parallax | lens_pm | lens_pmra | lens_pmdec
---|---|---|---|---|---|---|---|---
- | yr | - | deg | deg | mas | mas/yr | mas/yr | mas/yr
NS180 | 2016 | 5255174721885285248 | 157.677070264 | -59.942343487 | 11.334 | 95.777 | -80.225 | -52.318
HMS1 | 2016 | 4090028395985895552 | 277.984338089 | -22.029599100 | 0.489 | 44.727 | -22.466 | -38.676
HPMS1816 | 2016 | 5597168291724032512 | 119.328289919 | -30.147903097 | 23.662 | 162.919 | -15.912 | -162.140
HPMS3274 | 2016 | 4109034455276324608 | 259.053294100 | -26.551146095 | 167.962 | 1222.339 | -479.573 | -1124.332
HPMS4600 | 2016 | 2006217676803960960 | 334.503341666 | 56.038507792 | 25.406 | 256.966 | 111.256 | 231.632
Notes. NS180 pick will occur in 2032, the timescale is $\sim 6.2yr$, the
maximum shift of the major image is $(1.45\pm 0.24)mas$,
For HMS1, the lens estimation mass is more than $5M_{\odot}$.
For HPMS1816 and HPMS3274, the maximum shift of the major image are more than
10 mas.
For HPMS4600, the lens is White Dwarf, the maximum shift of the major image is
$(6.72\pm 2.17)mas$.
Table 4: continued

5 rows of the predicted astrometric-microlensing events in this paper, and the
rest are available online.  lens_g_mean lens_M WDJ_name Pwd type
lens_fidelity_v2 source_source_id source_ra mag solMass - - - - - deg 14.774
0.32 - - MS 1.000 5255174721854095872 157.676416918 12.775 5.16 - - - 1.000
4090028400306349568 277.984125875 9.952 0.65 - - MS 1.000 5597168296018750080
119.328018818 5.890 0.68 - - - 1.000 4109028510993111040 259.044797669 18.034
0.42 WDJ221800.59+560214.92 0.985 WD 1.000 2006218535799265024 334.506387815

Table 5: continued

5 rows of the predicted astrometric-microlensing events in this paper, and the
rest are available online.  source_dec source_parallax source_pm source_pmra
source_pmdec source_g_mean source_ruwe(new) source_fidelity_v2 deg mas mas/yr
mas/yr mas/yr mag - - -59.942597575 0.274 9.248 -7.977 4.680 17.554 - 0.832
-22.029916621 0.000 0.000 0.000 0.000 15.878 1.4 - -30.150261708 0.000 0.000
0.000 0.000 20.965 0.8 - -26.569040325 0.000 0.000 0.000 0.000 20.761 1.1 -
56.041993407 1.078 4.041 -3.024 -2.681 20.476 - 0.987

Table 6: continued

5 rows of the predicted astrometric-microlensing events in this paper, and the
rest are available online.  sep e_sep E e_E u0 e_u0 d+ e_d+ d+_lower dC e_dC
dC_lower dC_lum e_dC_lum mas mas mas mas - - mas mas mas mas mas mas mas mas
18.53 2.78 5.38 0.27 3.44 0.55 1.45 0.24 - 1.34 0.18 - 0.11 0.02 38.32 91.18
4.54 3.19 8.45 52.11 0.53 - 0.05 0.52 - 0.04 0.03 - 0.50 153.89 11.18 0.60
0.04 14.36 10.94 - 0.79 3.95 - 0.78 0.00 - 22.25 166.31 30.54 0.89 0.73 5.52
21.38 - 4.82 10.80 - 4.70 0.00 - 5.73 19.37 9.14 0.26 0.63 2.13 6.72 2.17 -
3.23 0.78 - 0.77 0.22

Table 7: continued

5 rows of the predicted astrometric-microlensing events in this paper, and the
rest are available online.  dC_lum_lower t0 e_t0 new events mas yr yr -
2032.2067 0.0277 * 0.00 2046.0000 3.3156 * 0.00 2068.4830 1.6180 * 0.00
2073.2950 0.2298 * - 2069.4857 0.1241 *

Table 8: Column meanings in Table 3 . Column Number | Column Name | Description
---|---|---
1 | Event_name | The name of the predicted astrometric microlensing event.
2 | ref_epoch | Reference epoch
3 | lens_source_id | Gaia DR3 source_id of the lens
4 | lens_ra | Right ascension of the lens
5 | lens_dec | Declination of the lens
6 | lens_parallax | Parallax of the lens
7 | lens_pm | Total proper motion of the lens
8 | lens_pmra | Proper motion in right ascension direction of the lens
9 | lens_pmdec | Proper motion in declination direction of the lens
10 | lens_g_mean | G-band mean magnitude of the lens
11 | lens_M | The estimated mass of the lens
12 | WDJ_name | The name of a white dwarf from ’the white dwarf catalog’ (Gentile Fusillo et al., 2021)
13 | Pwd | the probability of the source becoming a white dwarf from Gentile Fusillo et al. (2021)
14 | type |  | Type of the lensing star: WD = White Dwarf, MS = Main Sequence, RG = Red Giant,
---
BD = Brown Dwarf
15 | lens_fidelity_v2 | Astrometric fidelity of the lens from Rybizki et al. (2022)
16 | source_source_id | Gaia DR3 source_id of the background source stars
17 | source_ra | Right ascension of the background source stars
18 | source_dec | Declination of the background source stars
19 | source_parallax | Parallax of the background source stars
20 | source_pm | Total proper motion of the background source stars
21 | source_pmra | Proper motion in right ascension direction of the background source stars
22 | source_pmdec | Proper motion in declination direction of the background source stars
23 | source_g_mean | G-band mean magnitude of the background source stars
24 | source_ruwe(new) |  Renormalised unit weight error for the background source star with two-parameter solution, and can be solved by formula(16)
25 | source_fidelity_v2 | Astrometric fidelity of the background source stars from Rybizki et al. (2022)
26 | sep | Estimated distance at closest approach
27 | e_sep | Error in sep
28 | E | Einstein radius of the event
29 | e_E | Error in Einstein radius of the event
30 | u0 | Estimated distance at closest approach in Einstein radii
31 | e_u0 | Error in u0
32 | d+ | Maximal astrometric shift of brighter image
33 | e_d+ | Error in d+
34 | d+_lower | Lower confidence level (16%) of d+
35 | dC | Maximal astrometric shift of center of light
36 | e_dC | Error in dC
37 | dC_lower | Lower confidence level (16%) of dC
38 | dC_lum | Maximal astrometric shift including lens-luminosity effects
39 | e_dC_lum | Error in dC_lum
40 | dC_lum_lower | Lower confidence level (16%) of dC_lum
41 | t0 | Estimated time of the closest approach
42 | e_t0 | Error in t0
43 | new events | Marked as * represents a new event different from Klüter et al. (2022)
Table 9: Statistical results of events caused by three types of lens stars and their corresponding background stars.  | solutions | HPMS | NS | HMS
---|---|---|---|---
BGS | two-parameter solutions | 907 | 71 | 1
five-parameter solutions | 1453 | 50 | -
six-parameter solutions | 1919 | 99 | -
Table 10: Reasons why the results in Klüter et al. (2022) are not included in this paper.  | BGS
---|---
Exclusion reasons |  | two-parameter
---
solutions
| five or
---
six-parameter
solutions
Lens fidelity _ v2 <0.8 | - | 2
BGS fidelity _ v2 <0.7 | - | 1560
No satisfying equations (21) | - | 16
| No matching with DR2
---
or external catalogues.
166 | -
$\delta_{+}<0.1mas$ | 173 | 89
Table 11: The reason why the results of this paper are not included in Klüter et al. (2022).  |  | BGS
---|---|---
Reasons |  |  | two-parameter
---
solutions
| five or
---
six-parameter
solutions
Not in the search range | lens | 159 | 473
BGS | 0 | 12
time | 109 | 484
rectangular box | 41 | 119
Data Differences |  | 186 | 81

###  5.2 Multiple astrometric microlensing events caused by a single lens star

Figure 16: The information of 71 astrometric microlensing events caused by 61
CYG A. Upper-left panel: the distance of background stars in the reference
epoch from the proper motion path of lens stars. The black line is the proper
motions of the lens star in J2010.0-2070.0, the red asterisks are the
background stars (reference epoch time position) in the result of Klüter et
al. (2022), and the blue dots are the background star (reference epoch time
position) that are not present in the result of Klüter et al. (2022). Upper
right panel: the histogram of the distance between the background star and the
proper motion track of the lens star. Lower left panel: the histogram of
$\delta\theta_{+}$ for 71 events, with 3 events $\delta\theta_{+}>1mas$, and
the maximum value of $\delta\theta_{+}$ is 1.51mas. Lower right panel: the
histogram of $t_{0}$. Figure 17: Cutouts for 61 CYG A and its source stars
(70) are obtained using the ESAsky online observation tool.The source of the
HPMS4722 is not marked in these images because it is too close to HPMS4717 and
dim. The lens are indicated as red rectangular, and the sources are indicated
as green circles."HPMS47" + number (next to the source star) is the event
name. Top panel: 2MASS color JHK image (epoch $\sim$ J2000) with 8
events.Middle panel:PanSTARRS DR1 color(i,r,g) image (epoch $\sim$ J2012) with
57 events.Bottom panel:DSS2 color image (epoch $\sim$ J1990) with 5 events.
Figure 18: The information of 52 astrometric microlensing events caused by
LAWD 37. The format of the figure is the same as in Fig. 16 Figure 19: Cutouts
for LAWD 37 and its source stars (52).The lens are indicated as red
rectangular, the sources are indicated as green circles, HPMS2079 and HPMS2080
($\delta\theta_{+}>8mas$) is indicated as yellow circles. "HPMS" + number
(next to the source star) is the event name. Top panel: 2MASS color JHK image
(epoch $\sim$ J2000) with 10 events are obtained using the ESAsky online
observation tool.Middle panel:SkyMapper image (epoch $\sim$ J2016) with 11
events are obtained through the website SkyMapper(Wolf et al., 2016, 2018;
Onken et al., 2019b).Bottom panel:DECaPS 2 images(riY)(epoch $\sim$ J2017)
with 31 events are obtained using the Legacy Surveys online observation tool.
Figure 20: The data of the events HPMS2079 and HPMS2080. Top panel: the tracks
of the lens star and the background star for two events in J2010-J2070. The
black dots are the tracks of the lens star, the blue dots and the red dots are
the tracks of the background star of the events HPMS2079 and HPMS2080
respectively. Middle panel: the change of the $\varphi$ of two events with
time. The blue curve and the red curve represent the events HPMS2079 and
HPMS2080 respectively. Bottom panel: the change of $\delta\theta_{+}$ with
time, the blue curve and the red curve represent the events HPMS2079 and
HPMS2080, respectively.

(1) 71 astrometric microlensing events caused by 61 CYG A for Gaia soruce_id
187204660934556480

In Gaia DR3, the lens star 61 CYG A will cause 71 astrometric microlensing
events, as shown in Figure 16 and Figure 17, and the detailed data is listed
in the Appendix (event names: HPMS4708 - HPMS4783). The lens star is the main
sequence star, its spectral type is K, taken from Gaia DR3 astrophysical
parameters. It is a binary stars, and has the annual parallax of 286mas and
the proper motion of 5282mas / yr. We compare our results with those of Klüter
et al. (2022). In 26 events searched by Klüter et al. (2022), one event is not
included in our results, because the background star cannot match Gaia DR2 and
external catalogues (see section 4.4). The mass of the lens star is estimated
by the table "Astrophysical parameters" to be $(0.680\pm 0.04)M_{\odot}$,
which is slightly larger than that of Klüter et al. (2022) ($0.621\pm
0.06)M_{\odot}$. There are 3 events with $\delta\theta_{+}>1mas$. It is noted
from Figure 16 that the distance between the background stars and the proper
motion track of the lens star exceeds $10^{\prime\prime}$ in about $45\%$ of
the events, and finding that these events can increase the completeness of the
predicted events. It should be emphasized that events with the distance
between the background stars and the proper motion track of the lens star
below $7^{\prime\prime}$ are predicted by Klüter et al. (2022) except 3 events
occurred before 2010 or after 2068.

(2) 52 astrometric microlensing events caused by LAWD 37 for Gaia source_id
5332606522595645952

In Gaia DR3, the lens star LAWD 37 will cause 52 astrometric microlensing
events, as shown in Figure 18 and Figure 19. The detailed data is shown in the
appendix (event names: HPMS2109 - HPMS2132). The lens star is the white dwarf
star, its spectral type is B, and it has the annual parallax of 216mas and the
proper motion of 2684 mas / yr, taken from Gaia DR3. Klüter et al. (2022) had
searched for 36 events, where 7 events are not included in our results,
because the "astrometric fidelity" parameter value of the background star is
less than 0.7, which do not meet the conditions in Table 2. We originally
estimated the mass of the lens star as $(0.77\pm 0.01)M_{\odot}$ from the
white dwarf catalogue (Gentile Fusillo et al. (2021)) labeled as
’WDJ114542.92-645029.46’.The mass of the lens star measured by McGill et al.
(2023) is $(0.56\pm 0.08)M_{\odot}$, so we update the mass estimation and
delete 23 events with $\delta\theta_{+}<0.1mas$ due to mass changes. While
Klüter et al. (2022) estimated the mass of the lens star as $(0.65\pm
0.15)M_{\odot}$. For 10 events, the distance between the background stars and
the proper motion track of the lens star exceeds $8^{\prime\prime}$. There are
4 events with $\delta\theta_{+}>1mas$, where 2 events (HPMS2079 and HPMS2080)
have $\delta\theta_{+}>8mas$, and the minimum separation of star pairs is
greater than 105mas, as shown in Figure 19 and 20. Events with the distance
between the background stars and the proper motion track of the lens star
below $7^{\prime\prime}$ are predicted by Klüter et al. (2022) except 4 events
occurred before 2010 or after 2068.

As seen from the above two examples, the events searched by us and Klüter et
al. (2022) are different, because the different constraints for star pairs and
mass estimation for lens star are used, specially we use the cone search
method to find more background stars. It is also seen from Figure 15 that
there are a small number of event with large distance of background stars from
the proper motion track of the lens star. Therefore, we suggest that our cone
search method is helpful to find more events. It is noted that two lens stars
will cause multiple events due to their high proper motion and parallax, the
detection of many events caused by one lens will help us to improve lens mass
accuracy.

###  5.3  The events with large astrometric signals ($\delta\theta_{+}$)

From the discussion in section 2.1, it can be seen that only when
$\theta_{sep}$ is large, for example, $\theta_{sep}>103mas$ for Gaia with
FWHM=103mas (Fabricius et al., 2016), the signal $\delta\theta_{+}$ can be
detected. Therefore, the events with large $\delta\theta_{+}$ discussed in
this section should meet the condition of $\theta_{sep}>103mas$. We find that
348 events have $\delta\theta_{+}>0.5mas$, including 97 events with
$\delta\theta_{+}>1mas$ and 2 events with $\delta\theta_{+}>8mas$ discussed in
Section 5.2. The masses of the lens stars in these events are concentrated in
$0.4M_{\odot}<M_{L}<1M_{\odot}$. However, we have not new predicted events
with $\delta\theta_{+}>1mas$ and $\theta_{sep}>103mas$.

##  6 Summary and Conclusions

In this paper, we select about 820000 potential lens stars from Gaia DR3 data
for three types of stars, HPMS, NS and HMS, and obtain them to be about
470000, 160000 and 190000, respectively. Based on the cone search method,
about 2260000 star pairs are initially selected, including 1240000, 260000 and
760000 star pairs with lens stars of HPMS, NS and HMS respectively. The
relative motion of the star pair is approximated to be linear motion (not
considering parallax). The mass of the lens star is estimated according to the
"Astrophysical parameters", the White Dwarf Catalog (Gentile Fusillo et al.
(2021)) or the mass-luminosity relations, and then the minimum angular
distance of the star pair in the period of J2010.0-J2070.0 is estimated. These
star pairs are further selected to obtain 27057 star pairs, where HPMS, NS and
HMS are 19401,3742 and 3914 respectively. For these star pairs, we search for
their minimum angular distance to reserve the star pairs with
$\delta\theta_{+}>0.1mas$ and remove unqualified star pairs according to Table
2 and the equations (21) and (22). We finally get 4500 events caused by 3558
lens stars, where 4279 events are caused by HPMS (their background stars with
two-, five- and six-parameter solutions are 907, 1453 and 1919 respectively),
220 events are caused by NS (their background stars with two-, five- and six-
parameter solutions are 71, 50 and 99 respectively), and only one event is
caused by HMS (its background star has a two-parameter solution). There are
293 lens stars that can cause two or more events, where 5 lens stars will
cause more than 50 events. Specially two lens stars (61 CYG A and LAWD 37) are
found to cause 71 and 52 events respectively. It is noted that two events,
HPMS2079 and HPMS2080, have $\delta\theta_{+}>8mas$ and $\theta_{sep}>103mas$.

Because we use the different conditions of lens star, background star, binary
stars or co-moving stars, mass estimation of lens stars and search scopes of
star pairs, the events we searched are not the same with ones searched by
Klüter et al. (2022). We obtain 2836 events that are the same with those of
Klüter et al. (2022), and 1664 new predicted event.

In the events we searched, 116 events have the distances of background stars
from the proper motion path of lens stars more than $8^{\prime\prime}$ in the
reference epoch, where the maximum distance is $16.6^{\prime\prime}$. Although
$\delta\theta_{+}$ does not exceed 0.2 mas for 116 events, they increase the
completeness of the predicted events. Therefore, the cone search method we
used can help us to find more events. In addition, we have not found the event
with lens star mass distributed in $3.2M_{\odot}<M_{L}<5M_{\odot}$, we
consider to reduce the threshold of mass for HMS in future work and expect to
find more events.

In the future, Gaia DR4 will provide the epoch
astrometry222https://www.cosmos.esa.int/web/gaia/newsletter/contents, which
can be used to measure the shift of a handful events. The predicted events are
expected to be followed up using space devices such as HST, JWST or Chinese
Space Station Telescope (CSST) that will be launched at the near future (Cao
et al., 2018; Gong et al., 2019) .

## Acknowledgements

We thank the referee for useful comments and suggestions to improve the paper.
This research work is financially supported by the National Natural Science
Foundation of China (Grant Nos. 11403101, 12173085), Training Object Project
of technological innovation talents in Yunnan Province(No. 202305AD160004) and
the China Manned Space Project. We use the data from the European Space Agency
(ESA) mission Gaia (https://www.cosmos.esa.int/gaia) and the Gaia Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

This research has made use of the data of ESASky, developed by the ESAC
Science Data Centre (ESDC) team and maintained alongside other ESA science
mission’s archives at ESA’s European Space Astronomy Centre (ESAC, Madrid,
Spain)(Baines et al., 2017; Giordano et al., 2018), the images of the DESI
Legacy Imaging Surveys consisting of three individual and complementary
projects: the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona
Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS) (Collaboration
et al., 2023a), and the images of SkyMapper (Wolf et al., 2016, 2018; Onken et
al., 2019b).

## Data Availability

The data underlying this article are available in the article and in its
spreadsheet available online.

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