# Discerning Parallax Amplitude in Astrometric Microlensing

Sedighe Sajadiana Arya Mahmoudzadeh  Setareh Moein  Department of Physics,
Isfahan University of Technology, Isfahan 84156-83111, Iran

###### Abstract

Gravitational microlensing is a powerful method for discovering Isolated
Stellar-Mass Black Holes(ISMBHs). These objects make long-duration
microlensing events. To characterize these lensing objects by fully resolving
the microlensing degeneracy, measurements of parallax and astrometric
deflections are necessary. Microlensing events due to ISMBHs have considerable
astrometric deflections, but small parallax amplitudes as $\pi_{\rm E}\propto
1/\sqrt{M_{\rm l}}$, where $M_{\rm l}$ is the lens mass. We numerically
investigate the possibility of inferring parallax amplitude from astrometric
deflection in microlensing events due to ISMBHs. The parallax amplitude in
astrometric deflections is proportional to the relative parallax
$\pi_{\rm{rel}}$, which means (i) does not strongly depend on $M_{\rm l}$, and
(ii) increases in microlensing observations toward the Magellanic Clouds(MCs).
We assume these events are potentially detected in upcoming microlensing
surveys-(1): the Roman observations of the Galactic bulge (GB), and (2): the
LSST observations of the Large MC(LMC)-, and the Extremely Large Telescope
(ELT) follows up them with one data point every ten days. We evaluate the
probability of inferring parallax amplitude from these observations by
calculating the Fisher/Covariance matrices. For GB, the efficiencies for
discerning parallax amplitudes with a relative error $<4\%$ through
astrometric, and photometric observations are $3.8\%$, and $29.1\%$,
respectively. For observations toward the LMC, these efficiencies are
$41.1\%$, and $23.0\%$, respectively. Measuring parallax amplitude through
astrometric deflections is plausible in the GB events with the lens distance
$\lesssim 2.7$kpc, and in the LMC halo-lensing. The ELT telescope by
monitoring long-duration microlensing events can detect astrometric
deflections, and their parallax-induced deviations.

gravitational lensing: micro– methods: numerical – astrometry
–parallaxes–stars: black holes

<EMAIL_ADDRESS>
##  1 Introduction

A short time after the famous paper by Paczynski (1986) concerning the
detection of Massive Compact Halo Objects (MACHOs) in the Galactic halo
through continuous observations toward the Magellanic Clouds (MCs), three
microlensing groups, i.e., the Expérience de Recherche d’ Objets Sombres
(EROS, Aubourg et al. (1993)), MACHO (Wu, 1994), and Optical Gravitational
Lensing Experiment (OGLE, Szymanski et al. (1994)), searched for ongoing
microlensing events. The first generation of microlensing surveys observed MCs
for $10$ years to find the source stars that were being lensed by massive
objects inside the Galactic halo. The most important result of these
observations was the determination of an upper limit on the contribution of
MACHO in the Galactic halo (Lasserre et al., 2000; Alcock et al., 1998,
2000a). Since MACHOs have only gravitational interactions, so studying and
characterizing these objects are possible through gravitational microlensing
observations.

The second generation of survey microlensing groups changed observational
directions from MCs to the Galactic Bulge (GB) and spiral arms to discover
extra solar planets inside the Galactic disk, and additionally probe the
Galactic structure (Kiraga & Paczynski, 1994; Albrow et al., 1996; Udalski et
al., 2000; EROS Collaboration et al., 1999; Alcock et al., 2000b; Moniez et
al., 2017). In planetary microlensing events, an extra solar planet
(exoplanet) is usually orbiting the lens object and makes a deviation in the
magnification curve (see, e.g., Mao & Paczynski, 1991; Gould & Loeb, 1992). So
far, more than $204$ extra solar planets have been confirmed in microlensing
observations222https://exoplanetarchive.ipac.caltech.edu/. This population of
exoplanets discovered by gravitational microlensing observations has the
special properties listed here. (a) These exoplanets lie mostly beyond the
snowline of their parent stars. (b) In microlensing events toward the GB,
planetary lensing systems are typically found at the radial distances greater
than $1000$ parsec from the observer. (c) These exoplanets are orbiting the
host stars which are usually very dim or dark (see, e.g., Gaudi, 2012).
Gravitational microlensing is therefore a complementary method to other
methods for discovering extrasolar planets. In other methods (e.g., transits,
radial velocimetry, astrometry), planetary systems must be close to the
observer and their host stars must be bright.

Applications of survey microlensing observations are not limited to the
mentioned cases. For example, a new population of free-floating planets (FFPs)
in the Galactic disk was discovered through densely observing the GB (Sumi et
al., 2023; Mróz et al., 2017). Discovering Isolated Stellar-Mass Black Holes
(ISMBHs) inside the Galactic disk is another outstanding application of
gravitational microlensing (Sahu et al., 2022; Lam et al., 2022; Mao et al.,
2002; Bennett et al., 2002; Agol et al., 2002; Lu et al., 2016). Such isolated
objects most likely do not emit $X$-ray emissions. Hence, they are only
discernible through probing long-duration and achromatic amplifications in the
light of background source stars.

Despite such vast and unique applications of gravitational microlensing, there
is a problem while interpreting microlensing observations, which is degeneracy
(see, e.g., Sajadian, 2023). However, microlensing degeneracies can be
resolved through measuring both (a) the parallax amplitude in magnification
factor (Gould, 1994), and (b) finite-source effect (Witt & Mao, 1994). Instead
of resolving finite-source effect, measuring either lensing-induced
astrometric deflection in source trajectory (see, e.g., Hog et al., 1995;
Miyamoto & Yoshii, 1995; Walker, 1995; Miralda-Escude, 1996; Dominik & Sahu,
2000), or the images’ distance at the time of the closest approach (Dong et
al., 2019; Zang et al., 2020), or resolving the source and lens, and measuring
proper motions with adaptive optics observations (Bhattacharya et al., 2018;
Terry et al., 2022) will help to resolve the microlensing degeneracy. The
parallax effect refers to the observer’s motion around the Sun. This effect
makes some periodic perturbations in microlensing light curves (Gould, 1994).

In long-duration microlensing events due to dark and massive lens objects
(e.g., ISMBHs), this method for resolving microlensing degeneracy is somewhat
challenging, because the normalized parallax amplitude decreases like
$\pi_{\rm E}\propto 1/\sqrt{M_{\rm l}}$, where $M_{\rm l}$ is the lens mass
(see, e.g., Karolinski & Zhu, 2020). Albeit, the scale of the astrometric
deflection in the source trajectory (i.e., $\theta_{\rm E}$ which is the
angular Einstein radius) in these events is considerable, because $\theta_{\rm
E}\propto\sqrt{M_{\rm l}}$. In this work, we discuss on the possibility of
discerning parallax amplitudes in astrometric deflections of source
trajectories. We show that parallax-induced perturbations in the astrometric
deflection are directly proportional to the relative parallax
$\pi_{\rm{rel}}$, and do not depend on the lens mass strongly. Hence,
measuring parallax amplitude is possible in the astrometric deflections of
source trajectories in microlensing events toward MCs. However, microlensing
events due to more massive lens objects are more suitable, because they have
considerable astrometric deflections in source trajectories.

The paper is organized as follows. In the first subsection of Section 2, we
first explain the formalism of astrometric microlensing events by including
the parallax effect. Then, in subsections 2.2 and 2.3, by simulating
astrometric microlensing events due to ISMBHs by considering the parallax
effect toward the GB and LMC (respectively) we statistically evaluate
parallax-induced deviations. In Section 3, we do realistic Monte Carlo
simulations based on upcoming survey microlensing observations by the
Roman(Spergel et al., 2015) and LSST (Ivezic et al., 2009) telescopes. We also
consider potential follow-up observations with the Extremely Large Telescope
(ELT) telescope by taking one data point every ten days (Vernin et al., 2011).
In these simulations, we numerically calculate the Fisher and Covariance
matrices to evaluate the probability of discerning parallax amplitudes through
astrometric and photometric observations. In Section 4, we explain the results
and conclude.

##  2 Parallax effect in astrometric mirolensing

In this section, we aim to answer this question: ”In what kind of microlensing
events are the parallax amplitudes in astrometric deflections more realizable
than those in the magnification curves?” For this aim, in Subsection 2.1 we
first review our formalism for generating astrometric microlensing events by
considering the parallax effect. Then, in Subsections 2.2 and 2.3 we simulate
these events, and statistically evaluate parallax-induced perturbations.

###  2.1 Formalism

In a gravitational microlensing event, the light of a background star is
temporarily magnified due to passing through the gravitational potential of a
collinear and foreground massive object (Einstein, 1936). In this phenomenon
from a background star two images are formed whose angular positions and
magnification factors are (respectively):

$\displaystyle\bm{\theta}_{\pm}=\frac{u\pm\sqrt{u^{2}+4}}{2}\theta_{\rm
E}\hat{u},~{}~{}A_{\pm}=\frac{1}{2}\Big{(}\frac{u^{2}+2}{u\sqrt{u^{2}+4}}\pm
1\Big{)},$ (1)

where, $u$ is the angular distance of the source star from the lens object
normalized to the angular Einstein radius (i.e., $\theta_{\rm E}$ the angular
radius of the images’ ring when the lens, source star and observer are
completely aligned). $\hat{u}$ is a unit vector represents the direction of
$u$ projected on the sky plane.

Hence, the locations of two images and the source star are over a straight
line in the sky plane. The angular distance of these images ($\simeq
2\theta_{\rm E}$) is too small to be resolved, because $\theta_{\rm E}$ is on
milliarcsecond scales. For instance, for a common lens object (e.g., an
M-dwarf star) in the Galactic disk ($D_{\rm l}\simeq 4$ kpc, where $D_{\rm l}$
is the lens distance from the observer) while the source star is inside the GB
(i.e., the source distance from the observer is $D_{\rm s}\simeq 8$ kpc), it
is given by

$\displaystyle\theta_{\rm E}=\sqrt{\kappa~{}\pi_{\rm{rel}}~{}M_{\rm
l}}=0.78\rm{(mas)}~{}\sqrt{\frac{M_{\rm
l}}{0.3M_{\odot}}}\sqrt{\frac{\pi_{\rm{rel}}}{0.25~{}\rm{mas}}},$ (2)

here, $\pi_{\rm{rel}}=\rm{au}\big{(}1/D_{\rm l}-1/D_{\rm s}\big{)}$ is the so-
called relative parallax, au is the astronomical unit, and
$\kappa=8.14~{}\rm{mas}~{}M_{\odot}^{-1}$ has a constant value. Therefore, we
receive the total light of images which is magnified, the so-called
gravitational microlensing event. The magnification factor due to both images
is given by:

$\displaystyle A=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (3)

In a microlensing event, in addition to the light magnification of a
background star, the brightness center of these images does not coincide to
the source’s center. The astrometric deflection in the source position is
given by (Hog et al., 1995; Miyamoto & Yoshii, 1995; Walker, 1995; Miralda-
Escude, 1996):

$\displaystyle\delta\bm{\theta}=\frac{\bm{\theta}_{+}A_{+}+\bm{\theta}_{-}A_{-}}{A_{+}+A_{-}}-\bm{u}~{}\theta_{\rm
E}=\frac{\bm{u}~{}\theta_{\rm E}}{u^{2}+2},$ (4)

which is a function of time. This shift is a dimensional parameter, and could
be used for measuring the angular Einstein radius.

By ignoring the motion of the observer around the Sun (i.e., we first assume
the observer is the Sun), the vector of the relative lens-source position in
the heliocentric frame, $\bm{u}_{\odot}$, is expressed as a function of time
$t$:

$\displaystyle\bm{u}_{\odot}=\left(\begin{array}[]{cc}(t-t_{0})/t_{\rm E}\\\
u_{0}\end{array}\right),$ (7)

where, $u_{0}$ is the lens impact parameter (the closest lens-source
distance), $t_{0}$ is the time of the closest approach, and $t_{\rm
E}=\theta_{\rm E}/\mu_{\rm{rel},\odot}$ is the so-called Einstein crossing
time, the time of crossing the angular Einstein radius. Here,
$\mu_{\rm{rel},\odot}$ is the angular lens-source relative velocity as
measured in the heliocentric frame.

We now add the Earth motion around the Sun (by assuming the observer is on the
Earth), the so-called parallax effect. In gravitational microlensing events,
the parallax effect causes the angular lens-source relative velocity alters as

$\displaystyle\bm{\mu}_{\rm{rel}}=\bm{\mu}_{\rm{rel},~{}\odot}+\frac{\pi_{\rm{rel}}}{\rm{au}}\bm{v}_{\rm
o,\perp}(t),$ (8)

where $\bm{v}_{\rm o,\perp}$ is the vector of the Earth velocity with respect
to the Sun projected on the sky plane (normal to the line of sight).
Therefore, the parallax effect changes the vector of the relative lens-source
distance as measured from the Earth as

$\displaystyle\bm{u}=\bm{u}_{\odot}+\frac{\pi_{\rm
E}}{\rm{au}}\int_{t_{0}}^{t}dt~{}\bm{v}_{\rm
o,\perp}(t)=\bm{u}_{\odot}+\pi_{\rm E}\bm{\Delta}_{\rm o,~{}n}(t),$ (9)

here, $\pi_{\rm E}=\pi_{\rm{rel}}/\theta_{\rm E}$ is the relative lens-source
parallax normalized to the angular Einstein radius. $\bm{\Delta}_{\rm o,~{}n}$
is an extra displacement in the source position with respect to the lens due
to the Earth motion, normalized to the astronomical unit.

We note that in real observations the Earth motion around the Sun has been
completely known, but (i) $\pi_{\rm E}$, and (ii) the angle between the Earth
velocity projected on the sky plane and the source trajectory are unknown. For
that reason parallax effect includes two new variables in the lensing
formalism, i.e., $\pi_{\rm E}$, and $\xi$. Since, the relative source-lens
trajectory projected on the sky plane has a fixed direction (i.e., the Sun,
and other stars move on straight lines during lensing time scales), so we
define $\xi$ as the angle between $\bm{v}_{\rm o,\perp}$ and
$\bm{\mu}_{\rm{rel},~{}\odot}$ at the time of closest approach $t_{0}$.

Figure 1: Examples of astrometric deflections in source trajectories in long-
duration microlensing events toward the GB (three top panels), and LMC (three
bottom panels) due to ISMBHs, by ignoring (solid black curves) and including
(dashed blue curves) parallax effect. The dotted magenta curves are the first
sentence in the astrometric deflection given by Equation 10, i.e.,
$\delta\bm{\theta}_{0}$. Their parameters were mentioned at the top of curves.

The second term in Equation 9 changes periodically, and causes a periodic
perturbation in the magnification curve and astrometric deflection in the
source trajectory. The amplitude of this perturbation in the magnification
factor is $\pi_{\rm E}$ because the magnification factor $A$ depends only on
$u$.

The astrometric deflection in the source trajectory by considering the
parallax effect is

$\displaystyle\delta\bm{\theta}=\delta\bm{\theta}_{0}+\delta\bm{\theta}_{1}=\frac{\bm{u}_{\odot}~{}\theta_{\rm
E}}{u^{2}+2}+\frac{\bm{\Delta}_{\rm o,n}~{}\pi_{\rm{rel}}}{u^{2}+2},$ (10)

where, $u^{2}=u_{\odot}^{2}+\pi_{\rm E}^{2}\Delta_{\rm o,n}^{2}+2~{}\pi_{\rm
E}~{}\bm{u}_{\odot}.\bm{\Delta}_{\rm o,n}$. The first sentence makes an
ellipse on the lens plane (with small perturbations), and the second term
makes a periodic perturbation over that ellipse. The amplitude of these
periodic perturbations is proportional to $\pi_{\rm{rel}}$. However, the
larger $\theta_{\rm E}$, the higher detectability of these perturbations.
Because (i) the amplitude of the astrometric deflection is proportional to
$\theta_{\rm E}$, and (ii) the higher $\theta_{\rm E}$ makes lower $\pi_{\rm
E}$, and as a result, the second term in Equation 10 increases. According to
Equation 2, long-duration microlensing events due to ISMBHs close to the
observer (in comparison with the source distance) are the most suitable ones
to realize the parallax effect in their astrometric deflections instead of
magnification curves.

Briefly, the parallax effect changes both the magnification factor and the
astrometric deflection in the source position (as given by Eq. 3, and 4) in
different ways, as:

  * •

These two observing features have different time scales, as they tend to zero
by $u^{-4}$, and $u^{-1}$, respectively (see, e.g., Dominik & Sahu, 2000;
Sajadian, 2014). By getting away the source star from the gravitational
potential of the lens object, the astrometric deflection in the source
position tends to zero very slowly. The slow evolution of the astrometric
shift is beneficial to realize annual parallax amplitude.

  * •

The amplitudes of parallax-induced perturbations in the magnification factor
and the astrometric deflection are proportional to $\pi_{\rm E}$, and
$\pi_{\rm{rel}}$, respectively. Considering the fact that for realizing the
astrometric deflection itself, $\theta_{\rm E}$ should be large, in long-
duration microlensing events due to ISMBHs very close to the observer the
parallax amplitude could be discerned in astrometric deflections instead of
magnification factors.

  * •

When $u=u_{0}$, the magnification factor is maximum, whereas for $u=\sqrt{2}$
the astrometric deviation is maximum. Hence, discerning the parallax-induced
perturbation in the magnification factor, and astrometric deflection can be
done with a higher probabilities when $u\ll 1$ and $u\simeq 1$, respectively.
This point can be found in Figure 5.

So searching the parallax effect in the astrometric deflection instead of the
magnification factor would be possible in long-duration microlensing events
due to massive and close lens objects (e.g., stellar-mass or intermediate-mass
black holes). These points will be verified numerically in the next
subsections by performing simulations of microlensing events toward the GB and
LMC.

Figure 2: Scatter plots of $\delta\theta_{\rm{RMS}}$ (given by Eq. 11) for a
sample of astrometric microlensing events due to ISMBHs toward the GB (top
panels), and LMC (bottom panels) versus $D_{\rm{l}}(\rm{kpc})$,
$\log_{10}[\pi_{\rm{rel}}(\rm{mas})]$, and $\log_{10}[\theta_{\rm
E}(\rm{mas})]$ (from left to right, respectively). In the bottom panels, self-
lensing and halo-lensing events are specified with blue triangles and green
circles, respectively.

###  2.2 Astrometric microlensing toward the GB

Using the introduced formalism, we simulate possible astrometric microlensing
events toward the GB, by considering the annual parallax effect (based on
their distribution functions). We justify our simulation according to the
Roman observing strategy. We assume the Roman orbit is a circle of radius
$1.01$ au centered on the Sun (at the second Sun-Earth Lagrange point L2).
However, in the simulation we increase the observing time to $19$ years,
because the astrometric deflection tends to zero very slowly.

To simulate a microlensing event, we first choose the source distance from the
observer, $D_{\rm s}$, using the projected mass density in each line of sight
$dM(D_{\rm s},~{}l,~{}b)/dD_{\rm s}=d\Omega~{}D_{\rm s}^{2}~{}\big{(}\rho_{\rm
b}+\rho_{\rm d}+\rho_{\rm h}\big{)}$. Here, $(l,~{}b)$ are the Galactic
longitude and latitude, respectively. $\rho_{\rm b}$, $\rho_{\rm d}$, and
$\rho_{\rm h}$ are stellar densities of the Galactic bulge, disks, and halo,
respectively. The physical properties of source stars (absolute magnitudes,
mass, radius, type, …) are determined according to the Galactic Besançon model
333https://model.obs-besancon.fr/(Robin et al., 2003, 2012).

The distance of the lens object from the observer depends on $D_{\rm s}$, and
the given line of sight $(l,~{}b)$. We determine the lens distance from the
observer using the microlensing event rate, i.e., $\Gamma\propto R_{\rm
E}~{}\big{(}\rho_{\rm b}+\rho_{\rm d}+\rho_{\rm h}\big{)}~{}v_{\rm{rel}}$,
where $R_{\rm E}=D_{\rm l}~{}\theta_{\rm E}$, $v_{\rm{rel}}=D_{\rm
l}~{}\mu_{\rm{rel}}$, and $\mu_{\rm{rel}}$ is the size of the angular relative
lens-source velocity (given by Eq. 8). In order to simulate long-duration
microlensing events due to ISMBHs, we select the lens mass uniformly from the
range $M_{\rm l}\in[2,~{}50]M_{\odot}$ (Sicilia et al., 2022).

Three examples of astrometric deflections in the source trajectories are
depicted in three top panels of Figure 1. In these plots, solid black and
dashed blue curves are astrometric deflections without and with considering
the parallax effect. Dotted magenta curves represent the first term in
Equation 10. These magenta curves show that parallax-induced perturbations are
mostly generated by the second term of Equation 10. At the top of plots the
relevant parameters are reported. These plots confirm the results in
Subsection 2.1.

Concerning the detectability of parallax-induced perturbations in astrometric
deflections, we evaluate an statistical parameter. For each microlensing
event, in the time interval $[-1.5~{}t_{\rm E},~{}1.5~{}t_{\rm E}]$, we
numerically calculate the root mean square (RMS) of deviations in astrometric
deflections, i.e.,

$\displaystyle\delta\theta_{\rm{RMS}}=\sqrt{\left<\big{(}\delta\theta(t)-\delta\theta_{\odot}(t)\big{)}^{2}\right>_{t}},$
(11)

where $\delta\theta_{\odot}(t)$ is the astrometric deflection without parallax
effect. This parameter shows the scale of parallax-induced deviations in
astrometric deflections. We simulate a large number of astrometric
microlensing events due to massive lens objects toward the GB and extract the
detectable events in the Roman observations. Then, we determine
$\delta\theta_{\rm{RMS}}$ for each detectable event.

In three top panels of Figure 2, we represent the scatter plots of
$\delta\theta_{\rm{RMS}}$ versus $D_{\rm l}$,
$\log_{10}\big{[}\pi_{\rm{rel}}(\rm{mas})\big{]}$, and
$\log_{10}\big{[}\theta_{\rm E}(\rm{mas})\big{]}$, from left to right
respectively. Accordingly, the parallax-induced perturbations in astrometric
deflections are considerable when the lens is very close to the observer.
Although the size of these perturbations does not depend on the lens mass,
they are detected in the events due to more massive lenses with a higher
probability. Because astrometric deflections are scaled with $\theta_{\rm
E}\propto\sqrt{M_{\rm l}}$. Two last panels of Figure 2 manifest that
$\delta\theta_{\rm{RMS}}\propto\pi_{\rm{rel}}\propto\theta_{\rm E}^{2}$.

For the microlensing observations toward the Magellanic Clouds (MCs), the
parallax perturbations in astrometric deflections should be even higher.
Because the lens object can be in the Galactic halo ($D_{\rm l}\sim 2$ kpc)
and the source stars are inside the Large Magellanic Cloud ($D_{\rm s}\sim 50$
kpc), which results a large relative parallax, i. e., $\pi_{\rm{rel}}\sim
0.48$ mas. We study this point in the next subsection.

###  2.3 Astrometric microlensing toward LMC

Microlensing observations toward LMC were first proposed by Paczynski (1986)
to determine the contribution of MACHOs in the Galactic halo. The upcoming
LSST telescope will also monitor LMC with a $3$-day cadence in 6 filters,
$ugrizy$, during its mission. This telescope will discern a considerable
number of long-duration microlensing events, because of its long observing
time (i.e., 10 years). Although, its long cadence is such that it is simply
not going to be able to resolve the hours/day anomalies typical of planetary
microlensing events. In this subsection, we simulate astrometric microlensing
events toward LMC by considering the parallax effect.

The LMC celestial coordinate is
$(\rm{RA},~{}\rm{DEC})=(80.9^{\circ},~{}-68.2)^{\circ}$, and its distance from
the observer is $49.97$ kpc. For simulating potential microlensing events
toward LMC, we include stellar spacial distributions due to the LMC’s disk,
bulge, and halo, as given by Gyuk et al. (2000), and rewritten in Appendix (A)
of Sajadian (2021a). We assume the photometric properties of the LMC stars are
the same as those in our galaxy. The extinction toward LMC is ignorable except
its central part (see, e.g., Dobashi et al., 2008). For the central part of
LMC, we consider the $V$-band extinction in the range of
$A_{V}\in[1.6,~{}2.1]$ mag uniformly. We simulate microlensing events toward a
square of angular side $4$ degree centered on the LMC’s center.

Three examples of simulated astrometric deflections in microlensing events
toward LMC are represented in bottom panels of Figure 1. The last one is a
self-lensing event (its lens object is inside LMC), and the two others are
halo-lensing ones (their lens objects are in the Galactic halo). Accordingly,
the parallax effect is realizable in astrometric deflections of halo-lensing
events made by massive lens objects.

In bottom panels of Figure 2, we show the scatter plots of
$\delta\theta_{\rm{RMS}}$ versus the lens distance,
$\log_{10}\left[\pi_{\rm{rel}}(\rm{mas})\right]$, and
$\log_{10}\left[\theta_{\rm E}(\rm{mas})\right]$. In these plots, halo-lensing
events are specified with green circles and self-lensing events are denoted
with blue triangles. On average, relative parallax amplitude
($\pi_{\rm{rel}}$), angular Einstein radius, and as a result
$\delta\theta_{\rm{RMS}}$ in halo-lensing events are larger than those due to
self-lensing events by more than two orders of magnitude.

Also, by changing the observational direction from the GB to LMC,
$\delta\theta_{\rm{RMS}}$ enhances more than one order of magnitude. The
astrometric accuracy of ELT444https://elt.eso.org/(Vernin et al., 2011; Tamai
& Spyromilio, 2014), which is under construction, for a bright star with a
$K$-band apparent magnitude $18-19$ mag reaches $50~{}\mu$as. Hence, this
telescope by following up long-duration events toward LMC can detect not only
astrometric deflections in source trajectories, but also their parallax-
induced deviations. Measuring $\theta_{\rm E}$, and $\pi_{\rm{rel}}$ will
specify the lens mass, and its distance uniquely. We study this point in the
next section quantitatively.

##  3 Simulations of astrometric microlensing by Roman and LSST

In previous section, we found that the parallax effect could make considerable
deviations in lensing-induced astrometric deflections due to massive and close
lens objects. Here, we aim to evaluate detectability of these parallax-induced
deviations and their characterizations.

Extracting parallax amplitudes depend on the observing photometric and
astrometric accuracies, observing cadence, and the observing time interval.
Therefore, evaluating efficiency for discerning and characterizing parallax
amplitudes in astrometric deflections or microlensing light curves needs
comprehensive simulations of these events by generating synthetic data points
based on real observations.

For more realistic simulations, we apply observing strategies due to upcoming
microlensing surveys. We consider three strategies which are summarized in
Table 1, and explained in the following

  * •

A: Survey observations with the Roman telescope toward the GB during six
62-day seasons (its total observing time is $5$ years) with a $15$-min
cadence. We also consider some extra observations (one hour observation every
ten days) during the Roman large seasonal gap and when the bulge is visible
for this telescope. The same observing strategy was introduced to study
detecting ISMBHs by the Roman telescope in Sajadian & Sahu (2023).

  * •

B: Survey observations with the Roman telescope and follow-up observations
with the ELT telescope in $K$-band toward the Galactic bulge. The ELT
telescope will start observations for each event when $A>1.34$. We assume this
telescope will take one data point every $10$ days during its observing
seasons.

  * •

C: Survey observations with the Vera C. Rubin Observatory’s Legacy Survey of
Space and Time (LSST) telescope toward the LMC, and follow-up observations
with the ELT telescope. The LSST cadence is planned to be $3$ days, and its
mission will take $10$ years.

We tune our simulations based on these observing strategies. In the following,
we explain the details of these Monte Carlo simulations, and the results.

Figure 3: Two examples of simulated microlensing events which are detectable with the Roman telescope. For each event its light curve and astrometric deflection are represented alongside (left and right panels). The synthetic green and dark red data points are taken by the Roman telescope during its observing seasons, and its large seasonal gap, respectively. Their parameters can be found at the top of plots. Table 1: The descriptions of Monte Carlo simulations (A), (B), and (C) which are explained in Section 3.  Simulation | Observations | Telescope | $T_{\rm{obs}}(\rm{yrs})$ | Cadence | Filter
---|---|---|---|---|---
(A) | Survey | Roman | 5 |  $15$min | W149
| Follow-up | Roman | 5 |  $10$ days | W149
(B) | Survey | Roman | 5 |  $15$min | W149
| Follow-up | ELT | 5 |  $10$ days | $K$
(C) | Survey | LSST | 10 |  $3$ days | $ugrizy$
| Follow-up | ELT | 10 |  $10$ days | $K$
Table 2: The results of Monte Carlo simulations (A), (B), and (C) which are
explained in Tab 1.

$\rm{Simulation}$ | $\epsilon_{\theta_{\rm E}}$ | $\epsilon_{\pi_{\rm E}}$ | $\epsilon_{\pi_{\rm E},~{}\rm{Phot}}$ | $\epsilon_{\pi_{\rm E},~{}\rm{Ast}}$ | $\epsilon_{M_{\rm l}}$ | $\epsilon_{D_{\rm l}}$ | $\epsilon_{t_{\rm E}}$ | $\epsilon$
---|---|---|---|---|---|---|---|---
| $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$
Detection Threshold = $1\%$
(A) | $70.65$ | $7.80$ | $7.72$ | $0.36$ | $5.48$ | $21.76$ | $29.73$ | $5.71$
(B) | $76.71$ | $11.35$ | $11.30$ | $0.78$ | $9.18$ | $29.87$ | $49.78$ | $9.75$
(C) | $62.14$ | $25.92$ | $8.38$ | $25.08$ | $8.03$ | $18.69$ | $20.88$ | $10.20$
Detection Threshold = $4\%$
(A) | $97.47$ | $21.94$ | $21.68$ | $2.16$ | $21.05$ | $46.67$ | $58.40$ | $20.94$
(B) | $98.25$ | $29.18$ | $29.09$ | $3.76$ | $28.41$ | $59.57$ | $85.78$ | $28.83$
(C) | $77.12$ | $42.00$ | $22.98$ | $41.09$ | $22.80$ | $47.67$ | $54.64$ | $27.44$
Detection Threshold = $7\%$
(A) | $99.36$ | $29.81$ | $29.44$ | $4.05$ | $29.19$ | $57.57$ | $69.29$ | $28.93$
(B) | $99.61$ | $38.61$ | $38.46$ | $6.55$ | $38.18$ | $71.18$ | $93.34$ | $38.46$
(C) | $82.78$ | $47.27$ | $29.93$ | $46.10$ | $29.78$ | $59.76$ | $67.97$ | $35.04$

Note. — $\epsilon_{a}$ refers to the probability of measuring the parameter
$a$ so that its relative error be less than the given threshold. The indices
’Phot’ and ’Ast’ in the probability of measuring parallax amplitudes refer to
infer parallax from photometric and astrometric data with a given detection
threshold, respectively.

A: Survey Microlensing with Roman: We first tune our simulations for potential
survey observations toward the Galactic bulge with the Roman telescope. We
have done similar simulations in Sajadian & Sahu (2023). In that paper, we
showed that a small number of additional observations, i.e., one hour of
observations every 10 days, when the Galactic bulge is observable during the
large seasonal gap will improve the Roman efficiency for detecting and
characterizing ISMBHs. In the simulation (A), we assume these extra and sparse
data points are taken with Roman.

In the simulation, we take the time of the closest approach $t_{0}$ uniformly
in the range $[0,~{}5]$ years. The lens impact parameter is also chosen
smoothly from the range $[0,~{}1]$. The finite source effect for microlensing
events due to massive lens objects is ignorable because the normalized source
radius projected on the lens plane is $\rho_{\star}\propto 1/\sqrt{M_{\rm
l}}$, and very small. We consider the parallax effect while calculating light
curves, and astrometric deflection, to compare the errors while extracting the
parallax amplitude from each of them. Since the Roman telescope will orbit the
Sun from the second Sun-Earth Lagrange point (L2), the radius of its orbit is
$1.01$ au.

In Monte Carlo simulations from microlensing events detectable by Roman we
remove events with time scale $t_{\rm E}>2000$ days. Because in most of these
events magnification factors do not reach to the baseline during the Roman
$5$-year mission.

For calculating the blending effect in all Monte Carlo simulations, we first
calculate the average number of background stars inside a stellar PSF as

$\overline{N_{\star}}=\Omega_{\rm{PSF}}~{}f_{\rm b}~{}\int_{0}^{D_{\rm
s}}\Big{(}\frac{\rho_{\rm b}(x)}{\overline{M_{\rm b}}}+\frac{\rho_{\rm
d}(x)}{\overline{M_{\rm d}}}+\frac{\rho_{\rm h}(x)}{\overline{M_{\rm
h}}}\Big{)}x^{2}dx,$

where, $f_{\rm b}=2/3$ is the binary fraction, $\overline{M_{\rm b}}$,
$\overline{M_{\rm d}}$, and $\overline{M_{\rm h}}$ are the average masses of
stars in the Galactic bulge, disk, and halo, respectively.
$\Omega_{\rm{PSF}}=\pi\Big{(}\rm{FWHM}/2\Big{)}^{2}$ is the area of a typical
stellar PSF in the Roman observations. We determine the number of stars using
$N_{\star}=\overline{N_{\star}}+\sigma_{\rm N}$, where $\sigma_{\rm N}$ is
chosen using a normal distribution
$\mathcal{N}(0,~{}\sqrt{\overline{N_{\star}}})$. A common value for FWHM is
$3\times$pixel size, i.e., $0.33$ arcsec for the Roman observation.

The Roman photometry accuracy $\sigma_{\rm m}$ is a function of stellar
apparent magnitude in the W149 filter and was given in Fig. (4) of Penny et
al. (2019). Stellar absolute magnitudes in W149 are estimated by
$M_{\rm{W149}}\simeq(M_{H}+M_{J}+M_{K})/3$, where $M_{H}$, $M_{J}$, and
$M_{K}$ represent the stellar absolute magnitudes in the standard filters $H$,
$J$, and $K$, respectively (see, e.g., Sajadian, 2021b). We note that the
Galactic Becançon model gives the stellar absolute magnitudes in the standard
filters $BVRIK$. We specify the absolute magnitudes in $H$-, and $J$-bands
using Dartmouth Isochrones555http://stellar.dartmouth.edu/(Dotter et al.,
2008; Feiden et al., 2011).

We use the 3D extinction map offered by Marshall et al. (2006), and use the
following transformation relations to specify extinctions in other bands:
$A_{V}=8.47A_{K}=8.21A_{K_{\rm s}}=1.67A_{I}=5.43A_{H}=4.44A_{\rm{W149}}$
(Cardelli et al., 1989). We determine the astrometric accuracy of Roman
according to stellar apparent magnitude, and using Jitter simulations done by
S. C. Novati. We assume the probability of regular observations with Roman is
$90\%$ during its observing seasons.

In Figure 3, two simulated astrometric microlensing events detectable by Roman
are represented. The sparse data points during its large seasonal gap are
shown with dark red colour, and the Roman data during its observing seasons
are depicted with green colour. In these figures the sparse data points rather
help to discern astrometric deflections, and as a result the Einstein angular
radius. The parallax amplitude can be extracted from the Roman photometric
data.

Figure 4: The normalized distributions of relative errors in the parallax
amplitudes, as measured through photometric (green) and astrometric
observations (purple ones) based on simulations (A), (B), and (C), from top to
bottom, respectively. The dashed black lines determine the threshold amount
$\sigma_{\pi_{\rm E}}/\pi_{\rm E}=4\%$. The fractions of events with the
relative parallax errors less than $4\%$ are mentioned inside the panels.

After making a big ensemble of these events, we extract the detectable events
in the Roman observations. We have two criteria for detectability as follows.
(i)
$\Delta\chi^{2}=\left|\chi^{2}_{\rm{real}}-\chi^{2}_{\rm{base}}\right|>800$,
where $\chi^{2}_{\rm{real}}$, and $\chi^{2}_{\rm{base}}$ are $\chi^{2}$s from
fitting the real model and the baseline, respectively. (ii) Three data point
should be above the baseline by at least $4\sigma$, where $\sigma$ is the
photometric accuracy.

We numerically calculate the photometric and astrometric Fisher matrices
separately, i.e., $\mathcal{A}$, and $\mathcal{B}$ for each detectable event.
In this regard, observable parameters that affect on magnification curves and
astrometric deflections are $t_{0},~{}u_{0},~{}t_{\rm E},~{}\xi,~{}f_{\rm
b},~{}m_{\rm{base}},~{}\pi_{\rm E}$, and $\theta_{\rm E},~{}\pi_{\rm
E},~{}\bm{\mu}_{\rm s}$, respectively. We embed $\pi_{\rm E}$ into both lists
to compare the photometric and astrometric errors while extracting the
parallax amplitude.

In fact, real astronomical data points determine the projected source
trajectory on the sky plane, which is

$\displaystyle\bm{\theta}_{\rm s}(t)=\bm{u_{0}}~{}\theta_{\rm E}+\bm{\mu}_{\rm
s}(t-t_{0})-\pi_{\rm s}\bm{\Delta}_{\rm o,~{}n}(t)+\bm{\delta\theta}(t),$ (12)

where, $\bm{\mu}_{\rm s}$ is the angular velocity of source star, $\pi_{\rm
s}=1.01~{}\rm{au}/D_{\rm s}$ is the so-called source parallax amplitude when
the observer is Roman. The source parallax amplitude ($\pi_{\rm s}$) is very
small and ignorable. Hence, the parallax-induced perturbations mostly alter
the last term (i.e., the astrometric deflection). Throughout the paper and in
Figures 1, 3, 6, and 7, we only show the last term (the astrometric
deflection) so that the parallax-induced deviations get highlighted. However,
while calculating the astrometric Fisher matrix ($\mathcal{B}$) we numerically
calculate derivatives of the source trajectory $\bm{\theta}_{\rm s}(t)$ with
respect to its parameters.

The results from this simulation are reported in Table 2. In this table,
$\epsilon_{a}$ is the probability (or efficiency) of measuring the parameter
$a$ with the relative error less than the given thresholds (i.e., $1\%$,
$4\%$, and $7\%$). The indices ’Phot’ and ’Ast’ for $\epsilon_{\pi_{\rm E}}$
refer to measuring parallax amplitudes from photometric and astrometric data,
respectively. The last column is the probability of simultaneously measuring
three parameters $\theta_{\rm E}$, $\pi_{\rm E}$, and $t_{\rm E}$ with the
relative errors less than the given threshold. These parameters uniquely offer
the lens mass, the lens distance, and the lens-source relative velocity. The
relations to calculate errors can be found in Sajadian & Sahu (2023), and we
do not repeat them here.

The normalized distributions of relative errors in the parallax amplitude,
$\sigma_{\pi_{\rm E}}/\pi_{\rm E}$, extracted from Covariance matrices
$\mathcal{A}^{-1}$ (based on photometric data), and $\mathcal{B}^{-1}$ (based
on astrometric data) are depicted in the first panel of Figure 4 with green
and purple colours, respectively.

Accordingly, in long-duration microlensing events toward the Galactic bulge
detectable by the Roman telescope, the parallax can be took out from light
curves much more than from astrometric deflections ($10$ times). This point
can be discovered in Figures 3.

Figure 5: The normalized distributions (NDs) of $\log_{10}[\theta_{\rm
E}(\rm{mas})]$, $u_{0}$, and $\log_{10}[\pi_{\rm{rel}}(\rm{mas})]$ from
simulations (A), (B), and (C) (from top to bottom, respectively) are
represented by green color. In these panels, red step lines represent the NDs
of these parameters due to events in which $\pi_{\rm E}$s can be measured
(with relative error less than $4\%$) through either photometric or
astrometric data. Blue and black step curves are NDs due to events with
measurable parallax amplitudes through photometric data, and astrometric data,
respectively. Inside each panel, the average values of parameters
($\theta_{\rm E}$, $u_{0}$, $\pi_{\rm{rel}}$) from their distributions are
indicated.

To determine in what kind of simulated events, parallax amplitudes could be
measured through astrometric observations, in three top panels of Figure 5 we
show the normalized distributions (NDs) of $\log_{10}[\theta_{\rm
E}(\rm{mas})]$, $u_{0}$, and $\log_{10}[\pi_{\rm{rel}}(\rm{mas})]$ due to
events which are detectable by Roman with green filled distributions. In these
figures, red step lines represent the NDs of these parameters due to the
events in which $\pi_{\rm E}$s can be measured (with relative errors less than
$4\%$) through either photometric or astrometric data. Blue and black step
curves are NDs due to events with measurable parallax amplitudes through
photometric data, and astrometric data, respectively. Inside each plot the
average values of parameters from their distributions are mentioned.

Accordingly, in the events with $\pi_{\rm E}\gtrsim 0.25$mas (or the lens
distance $D_{\rm l}\lesssim 2.7$ kpc from the observer) measuring parallax
amplitudes through astrometric observations is possible. The events with
larger $u_{0}$ are more favorable to infer parallax amplitudes from
astrometric data, whereas in high-magnification events parallax amplitudes can
be measured rather from light curves. In fact, the magnification factor tends
to zero fast as $u^{-4}$, whereas the astrometric deflection maximizes when
$u=\sqrt{2}$.

B: GB Survey observations by Roman with the ELT follow-up: The astrometric
precision of the ELT telescope is planned to be better than 50 $\mu$as. In
this simulation, we consider potential follow-up astrometric observations with
the ELT telescope from long-duration microlensing events that will be alarmed
by Roman.

We assume this telescope will take one data every ten days, which is suitable
for long-duration microlensing events. For each microlensing event the ELT
observation is started when its magnification factor reaches $1.34$.

For the ELT astrometric observations, we assume it observes microlensing
events in $K$-band. In the regard of the ELT astrometric precision, the
statistical astrometric accuracy of a telescope with the aperture $D$, and in
the observing wavelength $\lambda$ is given by (see, e.g., Trippe et al.,
2010):

$\displaystyle\sigma_{\rm
a,~{}1}=\frac{\lambda}{\pi~{}\rm{D}}\frac{1}{\rm{SNR}}=34\mu\rm{as}\frac{\lambda}{2.2\mu
m}\frac{42m}{\rm{D}}\frac{100}{\rm{SNR}},$ (13)

where, SNR is the signal to noise ratio. Hence, for the ELT observations with
SNR$=100$ in $K$-band by the Multi-adaptive optics Imaging CAmera for Deep
Observations (MICADO666https://elt.eso.org/instrument/MICADO/, Davies et al.
(2021)) camera, the statistical astrometric precision will be $\simeq
34\mu$as. We add the systematic error ($\sim 10\mu$as) to this statistical
accuracy. Therefore, for a source star with the apparent magnitude
$m_{K}\simeq 18$-$19$ mag, the MICADO astrometric accuracy reaches $\sim
50~{}\mu$as. Fiorentino et al. (2020) have modeled the images would be taken
by MICADO/ELT from globular clusters and found a more practical relation
between the apparent magnitude of stars in the $K$-band filter and the MICADO
astrometric precision adjusted for a $20$-min exposure time, which was given
in their second table. Throughout the paper, we use their results to estimate
the ELT astrometric accuracy in the $K$-band filter.

We consider a $5$-month seasonal gap annually for ELT. Also, this telescope
can take data from stars with the $K$-band apparent magnitude in the range
$m_{K}\in[12,~{}23]$ mag (its saturation and detection thresholds). We assume
both probabilities for suitable weather, and doing regular observations with
the ELT telescope are $90\%$.

In this simulation, extracting lensing parameters and simulating the Roman
data are similar to ones explained in Simulation (A).

Figure 6: Same as Figure 3, but dark red points are hypothetically taken by
the ELT telescope in the $K$-band with a $10$-day observing cadence.

In Figure 6 we show two examples of simulated astrometric microlensing events
detected with Roman and ELT. Here, dark red points are taken with the ELT
telescope. Comparing Figures 3 and 6, one can find the large difference
between astrometric accuracies of Roman and ELT. Although the ELT data are
sparse, they manifest astrometric deflections well. In the first event,
discerning the parallax effect in the astrometric deflection is doable because
of the ELT astrometric data.

We mention the results from performing Monte Carlo Simulation (B) in Table 2.
Accordingly, the ELT observations will improve the efficiency in measuring
both $\theta_{\rm E}$, and $\pi_{\rm E}$. Although the efficiency for
discerning parallax amplitudes from astrometric deflections is low for
observations toward the Galactic bulge, some sparse data points with the ELT
telescope double this efficiency. This improvement can be noticed by comparing
the two first panels of Figure 4.

In the middle panels of Figure 5, the normalized distributions of
$\log_{10}[\theta_{\rm E}\rm{(mas)}]$, $u_{0}$, and
$\log_{10}[\pi_{\rm{rel}}(\rm{mas})]$ resulted from Simulation (B) are
depicted. They are similar to top panels (owing to Simulation (A)) with small
changes.

Figure 7: Same as Figure 3, but for observations toward the LMC with LSST. The
simulated data points taken by LSST in $ugrizy$ filters are shown with purple,
blue, dark green, yellowish green, orange, and red, respectively. The data
taken with ELT are depicted with dark red color. The observing cadences for
the LSST and ELT data are 3 and 10 days, respectively.

C: Survey and follow-up observations with LSST and ELT:  The upcoming LSST
telescope which is under construction in the Vera C. Rubin observatory, Chile
is planned to survey the whole sky with a $3$-day cadence (Ivezic et al.,
2009). This telescope will observe LMC during its $10$-year mission and
potentially detect several long-duration microlensing events in this
direction.

For the LSST observations toward the LMC, we assume follow-up astrometric
observations with the ELT telescope with a $10$-day cadence. The details of
this follow-up astrometric observation are the same as those explained in
Simulation (B).

The LSST data will be taken in 6 filters $ugrizy$ which are similar to the
filters used by the Sloan Digital Sky Survey (SDSS) telescope (Fukugita et
al., 1996; Ivezic et al., 2009). We determine the photometric errors in these
filters using the relations explained in Section (3) of Ivezic et al. (2009).
The astrometric accuracy of LSST is also a function of stellar apparent
magnitude (see, e.g., Eyer et al., 2012). We assume that both probabilities of
suitable weather for observations and doing regular observations with LSST are
$90\%$. The LSST seasonal gap lasts $5$ months.

Since, the LSST observing time is $10$ years, so we ignore microlensing events
with $t_{\rm E}>4000$ days. In these events magnification factors do not reach
the baseline during the LSST observing time. Also, the time of the closest
approach is uniformly chosen from the range $t_{0}\in[0,~{}10]$ years. Other
lensing parameters are determined in the same way as explained in Simulation
(A). We note that for the LSST observations the blending parameter is
considerable, because of the LSST’s observing depth (see, e.g., Sajadian &
Poleski, 2019).

In Figure 7, we depict two examples of simulated astrometric microlensing
events detected with the LSST and ELT telescopes. The data points taken by
LSST in the filters $ugrizy$ are shown with purple, blue, dark green,
yellowish green, orange, and red, respectively. The data taken by ELT are
shown with dark red color.

The first event is halo-lensing and the second one is a self-lensing event. In
the first one, the parallax effect makes considerable deviations in both
magnification factor and the astrometric deflection. Since the ELT data are
taken on the domain of light curve (because of the ELT saturation limit), they
only cover the parallax-induced deviations in the astrometric deflection. In
this event, the parallax amplitude can be only extracted from its astrometric
deflection. In the second event (a self-lensing one) the ELT data manifest the
astrometric deflection itself, and as a result, $\theta_{\rm E}$. In this
event, the parallax amplitude is not measurable.

We perform this Monte Carlo simulation, denoted (C), and make a large sample
of these astrometric microlensing events which are discernible in the LSST
observations. The detectability criteria are the same as ones mentioned for
Simulation (A). For each simulated event we numerically calculate Fisher and
Covariance matrices. The results can be found in Table 2.

Comparing the results reported in Table 2, toward the LMC the efficiency for
measuring $\theta_{\rm E}$ even by considering the ELT follow-up observations
is less than that toward the Galactic bulge by $\sim 10\%$. There are three
reasons. (i) For self-lensing events, $\theta_{\rm E}$ is too small (see,
Figure 2). (ii) The LSST telescope can discern faint source stars with the
$r$-band apparent magnitude $m_{r}\in[16,~{}24.3]$ mag, so most of source
stars in detectable microlensing events (by the LSST observations) are faint.
That causes the efficiencies for measuring parameters from photometric and
astrometric data (e.g., $u_{0}$, $t_{\rm E}$, $\pi_{\rm E}$, $\theta_{\rm E}$)
are lower than those for measuring parameters from the Roman data. (iii) The
maximum number of the LSST data during its mission with a $3$-day cadence is
$708$, whereas the maximum number of the Roman data for a microlensing event
during six $62$-day seasons is $35,700$. While calculating Fisher and
Covariance matrices, higher numbers of data points offer lower errors.

Nevertheless, efficiency for measuring $\pi_{\rm E}$ in microlensing events
toward the LMC is more than that efficiency in events toward the Galactic
bulge by $\sim 13\%$. Toward the LMC, extracting parallax amplitudes from
astrometric deflections is more efficient than taking out them from light
curves. According to last panels of Figure 4, and 5, there are two kinds of
events toward the LMC with very different $\pi_{\rm{rel}}$ values, i.e., self-
lensing and halo-lensing ones. The second class has large $\pi_{\rm{rel}}$
values and is suitable to determine parallax amplitude from astrometric data.

Hence, through observations toward the LMC by LSST, efficiencies for
extracting $\theta_{\rm E}$ and $t_{\rm E}$ is lower, and for extracting
$\pi_{\rm E}$ is higher in comparison with observations toward the GB with
Roman. For that reason, efficiencies of measuring these three parameters
simultaneously through both observations are the same.

Three last panels of Figure 5 manifest that by decreasing the lens impact
parameter, the efficiency to specify parallax amplitudes from light curves
improves by two orders of magnitude.

##  4 Conclusions

In microlensing events due to massive lens objects (e.g., ISMBHs) measuring
parallax amplitudes from their magnification curves is a challenge. Because
the parallax amplitude in these light curves decreases by the lens mass as
$\pi_{\rm E}\propto 1/\sqrt{M_{\rm l}}$. Although these massive lenses make
long-duration events in comparison to the Earth orbital period, they have
small parallax amplitudes.

In this work, we studied possibility of measuring parallax amplitudes from
astrometric deflections instead of light curves. We found that parallax-
induced deviations in astrometric deflections are proportional to
$\pi_{\rm{rel}}$, but not $\pi_{\rm E}$. However, for detecting these second-
order effects astrometric deflections themselves should be measurable. Hence,
the events with large $\theta_{\rm E}$, and $\pi_{\rm{rel}}$ are suitable for
measuring parallax amplitudes from astrometric data. The best events are long-
duration microlensing events toward the LMC due to ISMBHs inside the Galactic
halo. In these events on average parallax-induced deviations reach $\sim
10-100$ mas (see Figure 2). The largest parallax-induced deviations on
astrometric deflections occur when $u\simeq\sqrt{2}$ (at light curves’
domain).

To quantitatively study detectability of parallax amplitudes in astrometric
measurements, we have done three realistic Monte Carlo simulations based on
upcoming microlensing surveys, as follows. (A) Galactic bulge survey
microlensing observations with Roman in its $5$-year mission. In this
simulation, we assumed that the Roman telescope itself would take some sparse
data points (one hour observation every $10$ days) during its large seasonal
gap and when the Galactic bulge is observable. (B) Galactic bulge survey
microlensing observations with the Roman telescope and follow-up observations
with the ELT telescope in $K$-band. We assumed that ELT would take one data
point every 10 days. (C) The LSST survey observations towards the LMC during
its $10$-year mission, and follow-up observations with ELT by taking one data
point every $10$ days.

In simulations, for each event we numerically calculated Fisher and Covariance
matrices based on synthetic photometric and astrometric data points, and then
estimated errors. We included the parallax effect in both magnification
factors and astrometric deflections in source trajectories. The results from
these simulations were reported in Table 2.

Accordingly, for observations toward the Galactic bulge and in events due to
ISMBHs at the distances $D_{\rm l}\lesssim 2.7$ kpc from the observer,
measuring parallax amplitudes through astrometric deflections is possible.
However, for microlensing events toward the Galactic bulge, the efficiency to
measure parallax amplitudes (with relative errors less than $4\%$) from
astrometric deflections is only $2\%$. This efficiency gets double by adding
the ELT sparse data points.

In the LSST observations toward the LMC, the efficiency for measuring the
parallax amplitudes increases by more than $13\%$. Toward the LMC and
specially in halo-lensing events, extracting parallax amplitudes from
astrometric deflections is more efficient (by $18\%$) than extracting them
from light curves.

Briefly, through observations toward the LMC by LSST, efficiencies for
extracting $\theta_{\rm E}$, and $t_{\rm E}$ are lower (because LSST will
mostly detect microlensing events of faint source stars, and its cadence is
long), and for extracting parallax amplitudes its efficiency is higher in
comparison to observations toward the Galactic bulge with Roman. Hence, the
efficiencies for measuring three parameters simultaneously through both
observations are in the same range.

The source codes have been developed for this work can be found in the GitHub
and Zenodo addresses: https://github.com/SSajadian54/Parallax_Astrometry, and
https://zenodo.org/record/8342045(Sajadian et al., 2023).

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