Prediction of Astrometric and Timing Microlensing Events with Pulsars by ATNF Catalog and Gaia DR3

Determining the mass of neutron stars is crucial for understanding their formation, evolution, and interior structure. Currently, only a few dozen neutron stars have had their masses measured, and most of them belong to binary systems. However, there are a huge number of isolated neutron stars with unknown masses. Microlensing events with neutron stars provide unique opportunities for knowing these compact objects. Astrometric microlensing with a background source lensed by a neutron star might be used to determine the neutron star's mass by measuring the deviation of the motion of the centroid of the images from its unlensed one. We search and predict these recent and future events based on the Australia Telescope National Facility Pulsar Catalog and Gaia DR3. We find 60 candidate astrometric microlensing events caused by neutron stars and the probability distributions of their observables by the Monte Carlo sampling. We also find four candidate “timing microlensing” events with a pulsar lensed by a foreground object that might be detected by timing measurements. While some of these events may be verified by future astrometric missions or pulsar-timing observations, we note that our prediction of these events is significantly restricted by the uncertainties of the available astrometric and timing measurements after assessing and comparing our results with previous works.


Introduction
Mass measurements of neutron stars are vitally important for our understanding of their formation, evolution, and interior structure (Özel & Freire 2016).While progress has been made in measuring masses of pulsars in binary systems (Kramer & Stairs 2008), the mass remains unknown for isolated neutron stars, which account for about 90% of radio pulsars (Özel & Freire 2016) and may form in various channels (Beniamini & Piran 2016).
Astrometric microlensing can be used to determine masses of stars or stellar remnants (Paczynski 1995;Miralda-Escude 1996;Paczynski 1998;Gould 2000).In such events, a background source will be lensed by the intervening mass, and the motion of the lensed images will deviate from their unlensed one.By observing such shifts over time, the gravitational mass of the lens might be measured in a modelindependent manner (Miyamoto & Yoshii 1995;Boden et al. 1998).Recently, the astrometric microlensing technique has been successfully applied in determining the masses of a white dwarf (Sahu et al. 2017) and Proxima Centauri (Zurlo et al. 2018), detecting a possible free-floating stellar-mass black hole (Lam et al. 2022;Sahu et al. 2022), and investigating masses and distances of some dark lenses (Dehghani & Rahvar 2022).For now, positive detection of astrometric microlensing events is still rare because it demands dedicated and well-organized observing campaigns.To ensure success in detecting their astrometric signals, it is necessary to prepare and schedule observations based on robust and accurate prediction of possible lensing events (Refsdal 1964;Proft et al. 2011).
Various studies have searched for candidates of gravitational microlensing events regarding different astrometric missions (Gould 2000;Salim & Gould 2000).Bramich (2018) and Bramich & Nielsen (2018) searched for microlensing events during the Gaia mission and beyond based on Gaia DR2.Klüter et al. (2018Klüter et al. ( , 2022) ) predicted the astrometric microlensing events where background sources are lensed by highproper-motion stars from Gaia DR2 and eDR3.Following their work, Su et al. (2024) applied a modified strategy to extend the predictions for more types of lens stars.For astrometric microlensing events by local population of compact objects, Harding et al. (2018) estimated that the event rates for white dwarfs may reach tens per decade and at least one isolated neutron star has possibly produced detectable events in the past several decades, respectively.Paczynski (2001) predicted an astrometric microlensing event caused by the isolated neutron star RX J185635-3754.An event by nearby white dwarfs was predicted with Tycho-Gaia Astrometric Solution catalog (McGill et al. 2018), and two other events were found with VISTA Variables in the Via Lactea and Gaia DR2 (McGill et al. 2019).Using version 1.58 of the Australia Telescope National Facility (ATNF) Pulsar Catalog (Manchester et al. 2005) and Gaia DR2 (Gaia Collaboration et al. 2016, 2018), Ofek (2018) searched for encounters between pulsars and Gaia sources and found two likely lensing events that might have occurred in recent years.Ofek (2018) also tried to search for possible "timing microlensing events" in which a pulsar is lensed by a foreground object and might be detected by timing measurements, but no good candidate events were found.
In this paper, we predict astrometric and timing microlensing events with pulsars from J2010.0 to J2070.0 by making use of enlarged and better data sets and improved methods.We adopt version 1.67 of the ATNF Pulsar Catalog 3 (Manchester et al. 2005), which has 3320 pulsars up to date through 2022 June 29, nearly 700 more entries than version 1.58.We use Gaia DR3 (Gaia Collaboration et al. 2016, 2023) to search for pulsar-source pairs because the data release is based on a longer data collection and has more sources and reduced astrometric uncertainties.To extend previous predictions by representing probability distributions of the lensing observables, we employ the approach of Klüter et al. (2022) but with some modifications and improvements.We derive equations to calculate the time and angular separation at the closed approach for a lenssource pair in which the lens and source have their own, different epochs.When sampling the Einstein ring's radius and angular separation at the closest approach, we use their truncated normal distributions instead of the commonly used normal ones to prevent the sampling process from generating unphysically negative values of these two quantities due to large uncertainties of the astrometric parameters of the lens and source.We find 60 candidate astrometric microlensing events caused by pulsars with astrometric shifts of background sources more than 1 μas, eight of which have expected shifts above 10 μas.We also find four candidate timing microlensing events caused by foreground sources, which might change the second time derivative of the Shapiro delay in pulsars' times of arrival.We expect that some of these events could be verified by archival data or future observations.The rest of this paper is structured as follows.In Section 2, we briefly review the basics of astrometric and timing microlensing.We introduce the data sets used in our predictions and obtain preliminary lens-source pair candidates in Section 3.1.We describe our method for the microlensing predictions in Section 3.2.We discuss some individual cases of our findings about the astrometric and timing events, respectively, in Sections 4 and 5. Finally, we draw the conclusions and discuss the results in Section 6.

Astrometric Microlensing
Astrometric microlensing changes the motion of the centroid of the images from its unlensed one.For a pointlike lens, there will be two images at each side of the lens.The image at the same side of the lens as the source is called the primary image, with position θ + > 0; the one at the opposite side is the secondary one, with position θ − < 0. Their deviations from the position of the source β can be found as (Einstein 1936;Refsdal 1964;Paczynski 1986;Miyamoto & Yoshii 1995) where the angular radius of the Einstein ring θ E is with π L and π S being the parallaxes of the lens and the source, respectively, and u = β/θ E being the normalized angular separation between the lens and the source.Because of the lack of mass measurements of most pulsars, we assume their mass to be 1.4 M e with uncertainty of 10% throughout this work, as did Ofek (2018).
When these two images cannot be resolved, we can then observe only their centroid, the weighted average of their angular position by their magnification.For a dark lens, the deviation of the centroid from the source position is (Miyamoto & Yoshii 1995;Walker 1995;Dominik & Sahu 2000) and the total magnification of these two images is For a luminous lens, its flux contribution to the centroid cannot be neglected.With the flux ratio of the luminous lens to the background source f LS , the deviation of the centroid then becomes (Boden et al. 1998;Dominik & Sahu 2000) In this work, we assume that the pulsars are dark lenses in the optical astrometric microlensing events due to their faintness.
As the lens and the source are widely separated, the two images might be resolved, but the secondary one would be too faint to see.In practice, we can observe only the primary image, making it almost indistinguishable from the centroid for the dark lens.Based on Equations (1) and (3) and condition u ? 1, we can have We will search for astrometric microlensing events with maximum deviation d + max exceeding the astrometric threshold q m = 1 as min , which might be achieved by spaceborne astrometric missions in the future (Hobbs et al. 2021;Malbet et al. 2021).
The observational signatures of a microlensing event might be characterized by several timescales.As the most commonly used one, the Einstein timescale t E indicates the duration of a microlensing event that is (Paczynski 1986) where μ rel is the relative proper motion of the source to the lens.As  q q E min , an astrometric microlensing event might have a longer duration t aml when the image deviation δ + is larger than the astrometric precision threshold q min .It is defined as (Honma 2001) [ ( )]| and tells how long δ C takes to change by q min with respect to its maximum at the time of closest approach.We find a general expression for q t min as Such a timescale might be suitable for assessing the observability of the events and planning the measurements around the time of closest approach .

Timing Microlensing
It is also possible for pulsars to be lensed by foreground objects (Larchenkova & Doroshenko 1995;Wex et al. 1996).If so, the variation in the time of arrivals of a pulsar might be measured through timing, providing a unique approach to detect dark objects.
For a lensed pulsar, Ofek (2018) discussed its lensed timing signatures.A constant time delay introduces a shift to the timing signal.The first time derivative of the time delay multiplied by the period adds a constant shift in the pulsar period.Neither of them can be measured by timing alone because their unlensed values are unknown.The second time derivative Dt ̈multiplied by the pulsar period introduces a shift to the first derivative of the pulsar period, which is detectable by timing.For a stellar lens, we call it "timing microlensing." Considering a background pulsar is traveling rectilinearly with a relative proper motion μ rel to the lens and the minimal angular separation β 0 at the closest approach t 0 , we can have their angular separation β at any time t as The contribution of microlensing to the second derivative of Shapiro time delay (Shapiro 1964) Dt ̈is found to be At the closest approach, Dt ̈reaches its maximum as in which the lens-source separation β satisfies the condition θ E = β 0 < β = 1.Being far away from the closest approach, i.e., |(t − t 0 )| ?t E , Dt ̈behaves as which decreases quadratically with respect to the time span from closest approach.For a solar-mass lens with an angular separation β 0 ∼ mas and a relative proper motion μ rel ∼ mas yr −1 , we can find Dt ̈would manifest itself in the pulsar timing similar to the spin-down rate  P P because they share the same physical dimension.Dt ̈has a nonlinear evolution with the timescale t E of a few years for a lens with typical mass and proper motion in Equation (14) (Ofek 2018), while  P P is controlled by the inverse of its characteristic age of about 10 million years because of magnetic braking (Lyne & Graham-Smith 2012). P P is down to the level of 10 −19 s −1 for the pulsars in the ATNF catalog, and their uncertainties range from 10 −23 to 10 −16 s −1 .Because they have dramatically different timescales, we think it might be possible to separate Dt ̈from  P P with better timing capabilities, good enough predictions of the events, and well-organized observations in the future, such as by the Square Kilometre Array.Considering the currently measured  P P close to 10 −19 s −1 and its future improvements, we will try to predict the timing microlensing events with D ~t 10 20 | | s −1 in the work by following Ofek (2018).

Data
We take the pulsars from the ATNF Pulsar Catalog (Manchester et al. 2005) and predict the astrometric and timing microlensing events with them.We adopt its recent available update-v1.67,with 3320 pulsars up to date through 2022 June 29; it has nearly 700 new entries and receives updates and corrections of pulsar data, in comparison to v1.58 used in Ofek (2018).Among these pulsars, 421 have proper motion and distance measurements, whose uncertainties have been improved overall.The distances of a pulsar might be determined through annual parallax and/or dispersion measures.When the parallax is available for a pulsar in the catalog, we use that to estimate the distance and its uncertainty.Otherwise, we use the distance DIST_DM in the ATNF database, which is inferred from the dispersion measure, and set its uncertainty to be 20%.
For the other component in a microlensing event besides the pulsar, we choose from Gaia DR3 (Gaia Collaboration et al. 2023), whose astrometric uncertainties are significantly improved compared with those of Gaia DR2.For each of the 421 pulsars with distance measurements in the ATNF catalog, we use the ADQL interface in the Gaia DR3 archive to search for sources within 50″ of the pulsars, and find a total of 68,730 pairs of the ANTF pulsars and Gaia sources.
We will search for astrometric and timing microlensing events with these pulsars and Gaia sources from J2010.0 to J2070.0.

Method
With these pairs of pulsars and Gaia sources, we use the method of Klüter et al. (2022) to predict the microlensing events and find the distributions of their observables, but we also make some modifications and improvements for this method to deal with our specific circumstances with the pulsars.
First, we loosen the parallax and proper motion selection criteria of Klüter et al. (2022) for the pulsars and Gaia sources in the predicting of the timing microlensing events to expand our search, while we adopt the same quality filters for the Gaia sources in our prediction of astrometric microlensing events.The second derivative of time delay Dt ̈Equation (11) is For convenience, we introduce new quantities , , L, S .16 Therefore, we can rewrite the differences of coordinates and proper motion as After having these two approximated values, we predict the apparent positions of the pulsars and Gaia sources by taking their parallaxes into account and determine the minimal angular distance between them and its corresponding epoch within 2 yr around the approximated epoch by a nested-intervals algorithm.See Klüter et al. (2022) for more details.
Third, we use the truncated normal distributions of Einstein ring radius θ E and minimal angular separation β 0 , instead of their normal distributions in Klüter et al. (2022), to find the probability distributions of lensing observables, such as maximum deviation of the primary image and the total magnification, by Monte Carlo sampling.When there are large uncertainties in the astrometric parameters of the pulsar or/and Gaia source in a pair, their θ E and β 0 might have significant uncertainties as well, which happens very often, especially for the pulsars without well-measured parallaxes.Some negativevalued samples of θ E and β 0 would be sampled from their spread normal distributions, generating results without physical meaning.To mitigate this situation, we adopt their truncated normal distributions for sampling.For a normal distribution with mean of μ and standard deviation of σ, its truncated normal distribution would be (Johnson et al. 1994, chapter 13) where f is normal distribution function and Φ is its cumulative distribution function.
Finally, we select the astrometric microlensing events with d q m > =

Summary
We find 60 candidate astrometric microlensing events caused by the pulsars that take place between 2010 and 2070; see Table 1 for details. Figure 1 shows their maximum deviation d + max with respect to their time of closest approach t 0 .Among them, 24 events have the pulsars with parallax measurements, while the remaining 36 events have the dispersion measures only.Inferring the distance of a pulsar from its dispersion measure depends highly on the Galactic electron density model, which might result in some inconsistency from the parallax-based distance estimation (Deller et al. 2019).Therefore, cautions should be taken for those pulsars with their distances inferred from dispersion measure.
While we can see that the event rate with d m > + 1 as max is about 1 yr −1 , we also note that there are 14 events between 2022 and 2027, yielding a higher event rate.There are eight events having d m > + 10 as max , five of which might happen between 2021 and 2031.These events will require further investigations and observations for potential detection.However, there are six pulsars of these eight events without parallaxes but have the dispersion measure only.The most significant event we found has PSR J1622-0315 as the lens and Gaia DR3 4358428942492430336 as the background source, which might happen at J2016.0 with a deviation of the primary image of about 800 μas.Although this event has high uncertainties and we think it is very likely a false alarm (see more details and discussion in Section 4.3), it needs archival data or future observations to verify.
Figure 2 shows the violin plots of logarithmic probability distributions of d + max for all 60 predicted events.Table 1 lists the details of these events.The distributions of d + max are mostly asymmetric due to their nonlinear dependency on θ E and β 0 .The distributions of some events might head further toward the Note.An asterisk ( * ) indicates that the pulsar is in a binary system, according to the BINARY field in the ATNF database.
(This table is available in its entirety in machine-readable form.) high end of d + max , such as the event by PSR J0846-3533, while some might have long tails, such as the event by PSR J1622-0315.
Figure 3 shows the scatterplot matrix of the deviation maximum of the primary image d + max , the astrometric threshold timescale from the time of closest approach t μas , and the Einstein ring radius θ E .Different from timescales t E and t aml (see Section 2.1), the threshold timescale q t min 9 characterizes the shortest time span to detect the astrometric signals of an event.In this work, we take the astrometric threshold as q m = 1 as min and denote º m q m = t t as 1 as min for short.As the distribution of d + max shown in Figure 3(a), most of the events have their d + max between 1 and 10 μas, and the event rate drops significantly as d + max increases.Figure 3(e) shows that the threshold timescale t μas lies between 1 and 10,000 yr and is shorter for larger d + max , and indicates that detection of the shifts of the events with d m + 1 as max impractically requires more than 100 yr.The radius of the Einstein ring θ E for our predicted events varies from about 1 to 10 mas, see Figure 3(i), and its distribution shows multimodal, especially for the pulsars with parallaxes, which is partially because some of the pulsars cause more than one event.For example, PSR J1856-3754 might cause nine events with θ E ≈ 9 mas, leading to the rightmost peak in Figure 3(i).We can see high correlation between t μas and d + max from their scatterplot in Figure 3(b), because Equation (6) yields min .Other scatterplots of Figures 3(c) and (f) are absent of any distinct trend.

Events Caused by PSR J0846-3533 and J1856-3754
Two of our predicted events have already been found before in Ofek (2018).One is the event by PSR J0846-3533 on Gaia DR3 5626369190251744384 and the other is caused by PSR J1856-3754 on Gaia DR3 6730688466980426624.Tables 2 and 3 list their parameters and our predictions.After taking the uncertainties of these parameters into account and by using Monte Carlo sampling the truncated normal distributions of β 0 and θ E (see Section 3.2), we find the distributions of the parameters and observables of these events and report them in the tables as well.For better understanding the difference between our predictions and those in the previous work, we also list the data sets and results of Ofek (2018) in the same tables.
Our predictions have some differences from those of Ofek (2018); see Tables 2 and 3 for detailed comparison.One of them is about t 0 for the event by PSR J0846-3533, in which the difference is about 24 yr.We think this is largely due to the update of the astrometric parameters of the pulsar, especially its epoch.PSR J0846-3533's coordinates and proper motion barely changed in ATNF v1.58 and v1.67 (see Figure 4), whereas its epoch is significantly updated from MJD 48719 (Hobbs et al. 2004) in v1.58 to MJD 57600 (Jankowski et al. 2019) in v1.67 by an increment of about 24 yr.Furthermore, its distribution of d + max tells us that our prediction of this event has large ) and its error bar are shown with respect to their time of closest approach t 0 .The red points indicate the events where the distances of the pulsars are inferred from dispersion measure, while the blue ones show those with their distances deduced from the parallaxes.| |.This situation also happens with our prediction about the event caused by PSR J1856-3754 (see Figure 5), where its coordinates remain unchanged but its epoch is updated from J1997.0 to J2003.5, so that it brings about 7 yr variation in our predicted t 0 .Therefore, we think these discrepancies in our predictions originate purely from data updates instead of physical reasons.
In the following parts of this section, we will discuss three most significant astrometric microlensing events in our new predictions.max , the maximum deviation of the primary image, for the 60 predicted astrometric microlensing events.For each sub-figure, the thick black bar indicates the confidence interval between the 15.87th and 84.13th percentile, while the white dot within the bar indicates the median of its probability distribution.As in Figure 1, the colors indicate the type of inferred distance of the pulsars, i.e., the red color for DM-inferred distances and the blue color for parallax-inferred distances.

Event by PSR J1622-0315
The most significant event among our all predictions involves PSR J1622-0315 passing by Gaia DR3 4358428942492430336 at t 0 = J2016.0with a minimal angular separation of β 0 = 3.3 mas and an Einstein ring radius of θ E = 1.7 mas, which might lead to an astrometric shift about d m = + 789 as max .Table 4 lists the astrometric parameters of the pulsar and Gaia source, as well as the observables of this event.
PSR J1622-0315 is a redback millisecond pulsar with a spin period of P = 3.86 ms (Sanpa-Arsa 2016), which belongs to a binary system with an orbital period of 3.9 hr and a companion star of 0.1 M e .The projected semimajor axis is 0.22 lightsecond or 4.4 × 10 −4 au (Sanpa-Arsa 2016), which corresponds to an angular size of 0.3 μas at its distance.Gaia DR3 4358428942492430336 has its G = 19.2,G RP = 18.7, and G BP = 19.7.According to the mass determination method in Klüter et al. (2018), its B abs = 8.65 and G − G RP = 0.5 make it located between the main sequence and white dwarf in a colormagnitude diagram.According to Gaia DR3, its has RUWE = 0.91 and goodness of fit (GoF) of the five astrometric parameters GoF = − 1.98, indicating a relatively poor fit to the observations.
It is the only event in our predictions that has its minimal angular separation comparable to its Einstein ring radius, β 0 ≈ 2θ E , which gives the magnification about 0.07-0.7 mag of this event at t 0 = J2016.0and might trigger a photometric microlensing event.However, the 363 photometric microlensing candidates published in Gaia DR3 (Gaia Collaboration et al. 2023;Wyrzykowski et al. 2023) do not include this event, implying that its brightness variation might not pass the sample cuts and selections process.An adequate explanation would demand the intermediate data, which is not publicly available for now.(2018).Our result is predicted based on the data set from ATNF v1.67 and Gaia DR3 (blue), while the other is from ATNF v1.58 and Gaia DR2 (red).The squares and circles stand for the positions of Gaia source and pulsars, respectively.The filled and unfilled data points mark their positions at the initial epoch T and at the time of closest approach t 0 , respectively.Dashed lines and arrows show the trajectory and proper motion under the assumption of linear proper motion.The error bars are marked for the uncertainties of the pulsar's position at T, which affects the event prediction significantly.
Our prediction of this event also has some uncertainties.Evidence suggests that the pulsar and the Gaia source could be the components of a single binary system.The coordinates of PSR J1622-0315 and the Gaia source at the time of closest approach are separated by only a few mas, much less than the rest of our predicted events with the minimal separation at the order of arcsecond.The high uncertainty of the Gaia source's parallax π S = 0.624 ± 0.303 mas makes its distance with a very spread distribution, even comparable with the pulsar's distance d L = 1.141 kpc in some cases.The optical observation of PSR J1622-0315 in R band reveals a periodic modulation around 19.3 mag (Sanpa-Arsa 2016), which is consistent with the magnitude of the Gaia source.The Gaia source with the indicator ipd_frac_multi_peak = 0 does not show any multipeak structure because the pulsar is very faint in G band.The subpar GoF of the Gaia source could also be explained by its orbital motion.In agreement with the conclusion of a previous work (Liu et al. 2023), we also think the Gaia source is very likely to be the optical counterpart of the companion of PSR J1622-0315 based on their proximity in the sky and in physical distance, similar brightness in the optical observations, and the properties revealed by the Gaia indicators.

Event by PSR J1741-2054
We predicted that PSR J1741-2054 would encounter the Gaia DR3 4118158340230056704 by a angular distance of 1 80 at J2025.7, causing an astrometric shift of about 20 μas.
Table 5 lists the astrometric parameters and the observables of this event.
PSR J1741-2054 has a spin period of 413 ms (Abdo et al. 2009), DM = 4.7 pc cm −3 (Camilo et al. 2009), and a distance estimated as 0.273 kpc (Yao et al. 2017).It was the least luminous radio pulsar at the time of its discovery (Camilo et al. 2009).This pulsar has a large total proper motion of μ tot = 109 mas yr −1 (Auchettl et al. 2015).The Gaia DR3 source has modest uncertainty in its position, about a few tens of μas, and in its proper motion about several tens of μas yr −1 .Its distance is about 8.4 kpc estimated from its parallax π = 0.12 mas.The quality of this Gaia source's astrometric solution is good, indicated by its GoF = 0.22 and RUWE = 1.0068.Based on its magnitudes in different bands of G = 16.1 mag, G BP = 17.1 mag, G RP = 15.1 mag, and by the method in Klüter et al. (2022), we infer that this Gaia source might be a main-sequence star.
Given the modest uncertainties of the pulsar and Gaia source, we think this prediction might be relatively reliable and verified by future observations.

Event by PSR J1125-6014
We predicted that PSR J1125-6014 would pass by Gaia DR3 5336020987242787328 at J2047.0 with a separation of 0 23, causing a deviation d + max of about 20 μas.See Table 6 for details.uncertainty (Reardon et al. 2021).We assume the pulsar mass as 1.4 M e with an uncertainty of 10% in our event prediction, which is close to the value determined by Shapiro delay measurement, so that such an assumption will not affect our prediction significantly.A variation of mass changes the predicted θ E and d + max , which are listed at the end of Table 6 for a comparison, but the time t 0 and the angular separation β 0 of the closest approach stayed the same, because they depend on only the astrometric parameters of the lens and the background source.
Gaia DR3 5336020987242787328 is a G = 19.8star with a highly uncertain parallax of π = 0.236 ± 0.391 mas.The quality of the astrometric solution for this source demands further improvements with GoF = 3.47 and RUWE = 1.13.Its GoF deviates from the standard normal distribution by more than 3σ, indicating a bad fit of the data.
Because of the significant uncertainties in the parallaxes of both the pulsar and the background source, we think our prediction of this lensing event is not reliable.

Predicted Timing Microlensing Events
As described in Section 2.2, the gravitational lensing by a foreground object on a pulsar can change the second derivative of Shapiro time delay in the pulsar's timing signal.In this section, we make predictions for such timing microlensing events by selecting the Gaia objects as lenses and the ANTF pulsars as background sources, and estimate their observables at the closest approach D = t t t 0 | Equation (14).We find four events with D 7 for details).Among them, three lensed pulsars, PSR J0437-4715 (Reardon et al. 2016), PSR J1959+2048 (Arzoumanian et al. 1994) and PSR J2039-5617 (Clark et al. 2021), are millisecond pulsars located in binary systems.In these events, the angular separations between the pulsars and the Gaia sources are at the level of ∼mas, and their proper motion and distances inferred with either parallax or dispersion measure are consistent within 2σ.Therefore, as pointed out in the work of Liu et al. (2023), we argue that the Gaia sources in these three predicted events are very likely to be optical counterparts of these pulsars' companion stars.
In our fourth prediction, we found that PSR J1829-1751 would be encountered by a Gaia source at J2052.4 with a separation at the closest approach of about 70 mas, leading to D ~--t 10 s 20 1 ̈.Although PSR J1829-1751 is not indicated as a component of a binary in the ATNF catalog, it suffers highly uncertain astrometric parameters.The uncertainties of its R.A. and decl.reach 0.03 s and 0 3 (Hobbs et al. 2004), respectively, and the uncertainties of its proper motion components are close to their absolute values (Zou et al. 2005).They significantly degrade the reliability of our prediction, making the time of closest approach and its minimal angular separation highly uncertain.

Conclusions and Discussion
With the ATNF pulsar catalog and Gaia DR3, we predict the astrometric microlensing events caused by pulsars and timing microlensing events on pulsars and find their observables, including the minimal separation at the closest approach, the deviation of the lensed image, and the second derivative of the Shapiro time delay.Our method is based on the one proposed by Klüter et al. (2018Klüter et al. ( , 2022)), but we also make some modifications and improvements, such as extending the linear approximated formulae for the minimal angular separation at the closest approach by taking the different epochs of the lens and the source and using the truncated normal distributions of the Einstein ring's radius and the angular separation at the closest approach instead of the commonly used normal ones to prevent the sampling process from generating unphysically negative values.
We find 60 candidate astrometric microlensing events with d m > + 1 as max between J2010.0 and J2070.0, in which there are 14 events between 2022 and 2027.There are eight events with d m > + 10 as max , five of which might happen between 2021 and 2031 and could be potentially verified.We also find two events that were previously found by Ofek (2018), but we predict some different observables due to the updates of the astrometric parameters for the pulsar catalog and Gaia data release.Although we think some of the most significant events in our predictions are very likely false alarms, especially for those pulsars in binary systems, and are highly uncertain due to their poorly determined astrometric parameters, it might still be probable to test some of these prediction by astrometric missions in the future (Hobbs et al. 2021;Malbet et al. 2021).
We also predict four candidate timing microlensing events with D > -- t 10 s 21 1 | | .However, three of them might be false alarms because the Gaia sources could be the optical counterparts of these pulsars' companion stars.The fourth predicted event has high uncertainties, making it difficult to verify.Note.Notations are similar to Table 2.
We suggest that it is necessary to improve the astrometric parameters of the lenses and sources for obtaining more solid predictions on these lensing events.We also think that the method we use in this work could be further improved in the future.For example, although it might be computationally expensive, it would be necessary to apply the Monte Carlo sampling directly on the astrometric parameters of the lenses and sources for find more consistent predicted observables.
have the same form as those ofKlüter et al. (2022) but with new defined quantities at different epochs.When the two epochs coincide with each other T L = T S , they naturally return to those inKlüter et al. (2022).

Figure 1 .
Figure 1.For our predictions of 60 astrometric microlensing events caused by the pulsars, their expected maximum deviation of the primary image d m > + 1 as max () and its error bar are shown with respect to their time of closest approach t 0 .The red points indicate the events where the distances of the pulsars are inferred from dispersion measure, while the blue ones show those with their distances deduced from the parallaxes.
uncertainties coming from this pulsar's insufficiently measured proper motion with s m m

Figure 2 .
Figure 2. Violin plots of probability distributions of d +max , the maximum deviation of the primary image, for the 60 predicted astrometric microlensing events.For each sub-figure, the thick black bar indicates the confidence interval between the 15.87th and 84.13th percentile, while the white dot within the bar indicates the median of its probability distribution.As in Figure1, the colors indicate the type of inferred distance of the pulsars, i.e., the red color for DM-inferred distances and the blue color for parallax-inferred distances.

Figure 3 .
Figure3.Scatterplot matrix of the deviation maximum of the primary image d + max , the astrometric threshold timescale from the time of closest approach t μas , and the Einstein ring radius θ E .As in Figure1, the colors indicate the type of inferred distance of the pulsars.

Figure 4 .
Figure 4. Comparison of our prediction about the event by PSR J0846-3533 on Gaia DR3 5626369190251744384 as the background source (BGS) with the prediction ofOfek (2018).Our result is predicted based on the data set from ATNF v1.67 and Gaia DR3 (blue), while the other is from ATNF v1.58 and Gaia DR2 (red).The squares and circles stand for the positions of Gaia source and pulsars, respectively.The filled and unfilled data points mark their positions at the initial epoch T and at the time of closest approach t 0 , respectively.Dashed lines and arrows show the trajectory and proper motion under the assumption of linear proper motion.The error bars are marked for the uncertainties of the pulsar's position at T, which affects the event prediction significantly.

Table 1
Klüter et al. (2022)c Microlensing Events Caused by ATNF Pulsars independent of the parallaxes of the Gaia sources and the pulsars, but merely requires the pulsar to be more distant than the Gaia source.We can see that Dt ̈relies on the relative proper motion only so that, unlike the selection inKlüter et al. (2022), we do not apply proper motion filters for the pulsars or the Gaia sources.Second, we extend the linear approximated formulae to estimate the minimum angular separation β 0 and the epoch of the closest approach t 0 by including the different epochs of the pulsars and the Gaia sources.Suppose the lens position is (α L , δ L ) at epoch T L with its proper motion m m

Table 2
Parameters and Observables of the Event by PSR J0846-3533 on Gaia DR3 5626369190251744384 We show estimated values of t 0 and β 0 with and without parallax effect (p.e.) and by sampling the normal (Norm) or the truncated normal (TrunNorm) distributions (see Section 3.2 for details).

Table 3
Parameters and Observables of the Event by PSR J1856-3754 on Gaia DR3 6730688466980426624