# Identifying low-amplitude pulsating stars through microlensing observations

Sedighe Sajadian$\leavevmode\nobreak\ {}^{1,2}$, Richard
Ignace$\leavevmode\nobreak\ {}^{3}$, Hilding Neilson$\leavevmode\nobreak\
{}^{4}$
1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111,
Iran
2Department of Physics, Chungbuk National University, Cheongju 28644, Republic
of Korea
3Department of Physics & Astronomy, East Tennessee State University, Johnson
City, TN 37614, USA
4David A. Dunlap Department of Astronomy & Astrophysics, 50 St. George Street,
University of Toronto, Toronto, ON, M5S 3H4, Canada E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

One possibility for detecting low-amplitude pulsational variations is through
gravitational microlensing. During a microlensing event, the temporary
brightness increase leads to improvement in the signal-to-noise ratio, and
thereby better detectability of pulsational signatures in light curves. We
explore this possibility under two primary considerations. The first is when
the standard point-source and point-lens approximation applies. In this
scenario, dividing the observed light curve by the best-fitted microlensing
model leads to residuals that result in pulsational features with improved
uncertainties. The second is for transit events (single lens) or caustic
crossing (binary lens). The point-source approximation breaks down, and
residuals relative to a simple best-fitted microlensing model display more
complex behavior. We employ a Monte-Carlo simulation of microlensing of
pulsating variables toward the Galactic bulge for the surveys of OGLE and of
KMTNet. We demonstrate that the efficiency for detecting pulsational
signatures with intrinsic amplitudes of $<0.25$ mag during single and binary
microlensing events, at differences in $\chi^{2}$ of $\Delta\chi^{2}>350$, is
$\sim 50-60\%$. The maximum efficiency occurs for pulsational periods $P\simeq
0.1-0.3$ days. We also study the possibility that high-magnification
microlensing events of non-radially pulsating stars (NRPs) could be
misinterpreted as planetary or binary microlensing events. We conclude that
small asymmetric features around lightcurve peaks due to stellar pulsations
could be misdiagnosed with crossing (or passing close to) small caustic
curves.

###### keywords:

gravitational lensing: micro, stars: oscillations (including pulsations),
methods: numerical, software: simulations

††pubyear: 2021††pagerange: Identifying low-amplitude pulsating stars through
microlensing observations–LABEL:lastpage

##  1 Introduction

Variable stars are classified into two main categories: extrinsic and
intrinsic ones. The extrinsic variables are either eclipsing binaries or
rotating variables. The variability features for stars in the second class are
due to their interior evolution which changes their luminosity with time. Two
main subclasses of intrinsic periodic variables include RR Lyrae and Cepheids.
The RR Lyrae stars have periods in the range of a few hours to a few days.
These variable stars are mostly sub-giants, located on the horizontal branch,
and frequently found in globular clusters. They are similar to the classical
Cepheid variables, but with shorter periods and lower luminosities (see, e.g.,
Cox, 1974; Kahn, 1969; Percy, 2007). Cepheid variables have periods in the
range of $P\in[2,\leavevmode\nobreak\ 60]$ days and luminosities
$L\sim[300,\leavevmode\nobreak\ 40000]L_{\odot}$, whereas RR Lyraes have a
lower average luminosity of around $80\leavevmode\nobreak\ L_{\odot}$. Indeed,
they are population II stars with low-metallicity values. While frequently
observed in globular clusters, they are also common in other old stellar
populations, such as the Galactic bulge. RR Lyrae stars are often used as
distance indicators as well as tracers of old stellar populations (see, e.g.,
Jameson, 1986; Smith, 1984). Because of the faintness of RR Lyraes , they are
challenging to detect as extragalactic sources, or within the Galaxy when of
small-amplitude pulsations.

In any given stellar survey some variables go undetected, either because of
the intrinsic faintness of the stars (i.e., low luminosity or great distance),
or owing to the low amplitude of the variations relative to the detection
threshold for the survey. Among currently operating ground-based surveys that
have produced large numbers of variable star detections is the Optical
Gravitational Lensing Experiment (OGLE, Udalski et al. (2015a)). The OGLE
group has discovered more than $56,000$ RR Lyrae stars toward the Galactic
disk and bulge, and many others towards the Magellanic clouds, using the
$1.3$m Warsaw telescope. Most have amplitudes larger than $0.1$ mag
111http://ogledb.astrouw.edu.pl/~ogle/OCVS/catalog_query.php (e.g., Soszyński
et al., 2014, 2019; Udalski et al., 2018b). In 2019, the Gaia second data
release published a catalog of variable stars throughout the whole sky with
amplitudes larger than $0.1$ mag (Clementini et al., 2019). Lower-amplitude RR
Lyrae variables are barely detectable through these photometric observations,
but could be more easily and confidently identified if the stars’ brightnesses
were magnified through gravitational microlensing. Microlensing produces a
temporary enhancement in the stellar brightness when closely aligned with a
foreground and object of sufficient mass (Einstein, 1936; Liebes, 1964; Chang
& Refsdal, 1979; Paczynski, 1986; Sajadian & Rahvar, 2015).

In 1997, the first microlensing event of a variable source star was discovered
towards the Small Magellanic Cloud (SMC), namely
$\rm{MACHO}$-$97$-$\rm{SMC}$-$1$ (Alcock et al., 1997). In this event, the
blending-parallax degeneracy was resolved using the variability feature of the
source star (Assef et al., 2006). Another microlensing candidate from a
variable and bright source star was reported recently. For this event, the
asteroseismic analysis was done to find the intrinsic variability curve of the
source star from the baseline data (Li et al., 2019). The OGLE collaboration
announced 137 candidates of microlensing events of variable source stars
(Wyrzykowski et al., 2006). According to their results, for $10\%$ of total
detected microlensing events, the source stars show variability signatures at
their baselines. Here, “baseline” refers to the lightcurve as unaffected by
the lensing, i.e., before and after the event. A non-variable star would have
a flat baseline light curve. The OGLE result indicates that a significant
fraction of total microlensing events involve variable stars and hints at
there being more low-amplitude variables that went undetected.

The theoretical framework of microlensing involving NRPs was first introduced
in Bonanno & Sereno (2004). Recently, the characterizations of (single and
binary) microlensing lightcurves from radial and non-radial pulsating stars
have been studied extensively in Sajadian & Ignace (2020a, b). As an extension
of these two papers, this study explores the following two issues pertaining
to microlensing of variable stars. (1) How gravitational microlensing can
enhance the detectability of these low-amplitude pulsating stars. (2) How
microlensing lightcurves of these low-amplitude variables could in some cases
masquerade and be misinterpreted as other second-order effects associated with
microlensing of non-variable stars. We conduct Monte-Carlo simulations of
single and binary microlensing events of radially and non-radially pulsating
variable stars and generate synthetic light curves relevant to the OGLE
(Udalski et al., 2015b) and the Korea Microlensing Telescope Network (KMTNet,
Kim et al. (2016)) surveys. A description of our simulations are given in
section 2. Then, in section 3, we study the efficiency for detecting stellar
variability during microlensing events. In section 4, we explore the
conditions under which microlensing lightcurves of low-amplitude and non-
radially pulsating variables may be degenerate with planetary/binary
lightcurves. Finally, we summarize our conclusions in section 5.

##  2 Simulating Microlensing of low-amplitude variables

We explain the details of our Monte-Carlo simulation of single and binary
microlensing events toward the Galactic bulge in subsection 2.1. We assume
that source stars in the simulation have some low-amplitude pulsation
features. In order to study their detectability, we generate synthetic data
points for simulated events based on the OGLE and KMTNet surveys as explained
in subsection 2.3.

###  2.1 Simulating microlensing events

We have constructed a source star population using the Galactic Besançon model
222https://model.obs-besancon.fr/ for luminosity classes I, II, III, IV and V
(supergiants through main-sequence stars) (Robin et al., 2003, 2012) with the
same method as explained in (Sajadian & Poleski, 2019; Moniez et al., 2017;
Sajadian, 2015). The distances of the source stars from the observer ($D_{\rm
s}$) are determined using the cumulative mass versus distance for each given
direction $dM/dD_{\rm s}(l,\leavevmode\nobreak\ b)$, where
$(l,\leavevmode\nobreak\ b)$ represent the Galactic longitude and latitude,
respectively (see, e.g., Sajadian, 2021).

We have generated another stellar population for the lens stars. The distance
of each lens with respect to the observer ($D_{\rm l}$) is taken from the
microlensing event rate versus distance $d\Gamma/dD_{\rm l}\propto\rho_{\rm
t}(D_{\rm l},\leavevmode\nobreak\ l,\leavevmode\nobreak\
b)\leavevmode\nobreak\ R_{\rm E}$. Here, $\rho_{t}$ is the cumulative mass
density in the given direction and distance from the observer, and $R_{\rm E}$
is the Einstein radius (the ring of the image for perfect alignment).

Our simulation is pertinent for the Galactic bulge over a region with Galactic
longitude $l\in[3,\leavevmode\nobreak\ 6^{\circ}]$ and Galactic latitude
$b\in[-6,\leavevmode\nobreak\ -4^{\circ}]$. We exclude events within which the
baseline magnitude of the source stars (by considering the blending effect) is
fainter than $22$ mag in the $I$-band. Events with Einstein crossing times
longer than $300$ days are also ignored.

For binary microlensing events, the mass ratio of the two stars and the
angular separation of the components are uniformly sampled from the ranges
$q\in[0.4,\leavevmode\nobreak\ 1.0]$ and $d\in[0.6,\leavevmode\nobreak\ 1.4]$,
where $d$ is the projected angular separation of the lenses on the sky as
normalized to the angular Einstein radius. We evaluate the magnification
factor in binary microlensing events with the well-developed $\rm{RT}$-model
333http://www.fisica.unisa.it/gravitationastrophysics/RTModel.htm by V. Bozza
(Bozza et al., 2018; Bozza, 2010; Skowron & Gould, 2012). The lens angular
impact parameter in the sky, normalized to the angular Einstein radius, is
also sampled uniformly, in the range $u_{0}\in[0,\leavevmode\nobreak\ 1]$.

Figure 1: The photometric uncertainties (in magnitudes) of data points for
single and binary microlensing events as seen by OGLE (left panel) and KMTNet
(right panel) versus $I$-band apparent magnitude of the source stars. The
dashed magenta lines are the best-fitted polynomial curves to the
observations.

###  2.2 Pulsational features

To differing degrees, all stars are variable. Variations in the brightnesses
of intrinsic pulsating stars are modeled with either simple sinusoidal
functions to represent radially pulsating stars, or spherical harmonic
functions to represent non-radially pulsating stars. Details for simulating
microlensing of such stars appears in Sajadian & Ignace (2020a, b), an
approach that we also adopt here. The simulation of radially pulsating stars
requires two parameters: the pulsation period $P$ and brightness amplitude
$\delta_{\rm m}(\rm{mag})$. For these stars, the radius and surface
temperatures vary periodically. In the case of non-radially pulsating stars,
further parameters are needed: the inclination angle $i$ between the plane of
the sky and the polar axis of the star, and two numbers for determining the
harmonic modes of the stellar pulsation $l,\leavevmode\nobreak\ m$.

Then for our simulations, stellar pulsation periods are randomly selected
uniformly in the range $P\in[1\leavevmode\nobreak\
\rm{hr},\leavevmode\nobreak\ 15\leavevmode\nobreak\ \rm{days}]$. In order to
simulate low-amplitude pulsating stars, we randomly select the magnitude
amplitude in the $I$-band as uniform in the range $\delta_{\rm
m}\in[0.005,\leavevmode\nobreak\ 0.25]$ mag. We note that most detected
variable stars have amplitudes larger than $0.1$ mag. Hence, our simulations
include variables of lower amplitude than is detectable at the baseline of the
OGLE and KMTNet surveys.

###  2.3 Synthetic data points

To evaluate the detectability of pulsation properties from microlensing light
curves, we have generated synthetic lightcurves relevant to groups that
currently conduct large surveys of stellar populations. We have produced
ensembles of cadences and photometric uncertainties to replicate
characteristics of single and binary microlensing events of giant stars that
have been detected with OGLE 444http://ogle.astrouw.edu.pl/ogle4/ews/ews.html
and KMTNet555http://kmtnet.kasi.re.kr/ulens/ programs 666We do not include the
The Microlensing Observations in Astrophysics (MOA, Sako et al. (2008))
observations, because of its filter $R$-band used to observe microlensing
events. The microlensing light curves from variable stars are chromatic. These
lightcurves in different passbands do not have similar shapes, so we can not
convert these data with just shifting. The photometric uncertainties depend on
the source brightness, with better uncertainties for brighter sources. In the
simulations photometric uncertainties are determined from the source apparent
magnitude. Figure 1 shows a scatter plot of photometric uncertainties in
magnitude units versus the source magnitude for data from OGLE (left panel)
and KMTNet (right panel) surveys. We fit two second-order polynomial functions
to the data sets, as shown by the magenta dashed lines. For the OGLE survey,
the photometric uncertainty is $\sim 0.003$ mag for stars brighter than $15$
mag in $I$-band; for fainter stars, the average uncertainty is given by:

$\sigma_{m}=0.01\left[0.86+0.68(m_{I}-16.7)+0.23(m_{I}-16.7)^{2}\right].$ (1)

For KMTNet data, the uncertainty for stars brigher than $14$ mag in the
$I$-band is $\sim 0.0038$ mag; for fainter sources, the average uncertainty is
given by:

$\displaystyle\sigma_{m}=0.01\left[7.4-1.6(m_{I}-5.6)+0.1(m_{I}-5.6)^{2}\right].$
(2)

The above trends for photometric errors are incorporated into our simulations
for $I$-band light curves with microlensing. Although the microlensing
monitoring surveys are conducted in the $I$-band filter, we additionally
calculate magnification light curves in the $BVR$-bands.

The time interval between data points also mimics actual observations. We
downloaded observational data for several lightcurves taken by the OGLE and
KMTNet surveys. Accordingly, we made two ensembles of the observing cadences.
While simulating sythetic data points, their cadence is determined from those
ensembles.

Figure 2: Examples of single and binary microlensing light curves of pulsating stars. Lightcurves (magnification factor) are for standard $BVRI$-bands as blue, green, orange and dark-red colors, respectively. The black dashed curves represent the calculated microlensing light curves without pulsation. Synthetic data points are plotted with uncertainties and cadences appropriate for OGLE (lime) and KMTNet (grey) surveys. The residuals display differential magnitudes in $BVRI$-bands due to the stellar pulsation, given by Equation 2.3. The parameters of the light curves can be found in Tables 1 and 2. For binary events (Fig. 2 and 2), the caustic curves (black curves) and the source trajectories over the lens plane (red lines) are shown in insets. In the last three events, the source stars are non-radially pulsating variables, and their surface temperature variations are represented in insets with colored schemes. Table 1: Parameters of microlensing lightcurves shown in Figure 2 from radially pulsating stars. Here, $m_{\rm{base}}$ is the apparent magnitude due to the average stellar flux over the baseline and $\left<\sigma_{\rm{m}}\right>/\left<\sigma_{\rm{base}}\right>$ is the ratio of the averaged photometric error bar during lensing to that from the baseline amount. The last column indicates if the stellar pulsations are detectable during microlensing (labeled by $\rm{D}$) or not (labeled by $\rm{ND}$). $\rm{Fig.\leavevmode\nobreak\ No.}$ | $\delta_{\rm m}$ | $P$ | $u_{0}$ | $\rho_{\star}$ | $t_{\rm{E}}$ | $m_{\rm{base}}$ | $\Delta\chi^{2}_{\rm{lens}}$ | $\left<\sigma_{\rm{m}}\right>/\left<\sigma_{\rm{base}}\right>$ | $q$ | $d$ | $\rm{ND}/\rm{D}$
---|---|---|---|---|---|---|---|---|---|---|---
| $\rm{(mmag)}$ | $\rm{(days)}$ |  |  | $\rm{(days)}$ | $\rm{(mag)}$ |  |  |  | $\rm{(R_{E})}$ |
2 | $83.58$ | $14.34$ | $0.17$ | $0.001$ | $15.4$ | $14.8$ | $38170$ | $0.9$ | $--$ | $--$ | $\rm{D}$
2 | $124.13$ | $2.52$ | $0.07$ | $0.001$ | $15.2$ | $14.9$ | $11182$ | $0.9$ | $--$ | $--$ | $\rm{D}$
2 | $75.47$ | $7.79$ | $0.65$ | $0.002$ | $8.9$ | $19.3$ | $42$ | $1.0$ | $0.9$ | $1.1$ | $\rm{ND}$
Table 2: The parameters of microlensing lightcurves shown in Figure 2 from non-radially pulsating stars. $\rm{Fig.\leavevmode\nobreak\ No.}$ | $\delta_{\rm m}$ | $P$ | $i$ | $l$ | $m$ | $u_{0}$ | $\rho_{\star}$ | $t_{\rm{E}}$ | $m_{\rm{base}}$ | $\Delta\chi^{2}_{\rm{lens}}$ | $\left<\sigma_{\rm m}\right>/\left<\sigma_{\rm{base}}\right>$ | $q$ | $d$ | $\rm{ND}/\rm{D}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $\rm{(mmag)}$ | $\rm{(days)}$ | $\rm{(deg)}$ |  |  |  |  | $\rm{(days)}$ | $\rm{(mag)}$ |  |  |  | $\rm{(R_{E})}$ |
2 | $25.62$ | $9.6$ | $47.8$ | $1$ | $1$ | $0.22$ | $0.002$ | $5.8$ | $18.4$ | $95.5$ | $0.7$ | $--$ | $--$ | $\rm{ND}$
2 | $265.97$ | $4.05$ | $82.5$ | $2$ | $0$ | $0.06$ | $0.001$ | $8.3$ | $20.5$ | $1151$ | $0.8$ | $--$ | $--$ | $\rm{D}$
2 | $44.04$ | $8.45$ | $22.6$ | $1$ | $0$ | $0.49$ | $0.001$ | $8.4$ | $21.6$ | $5.9$ | $0.9$ | $0.4$ | $1.3$ | $\rm{ND}$

For simulating light curves, we have primarily three key timescales for a
single lens, and four when considering a binary lens. The foreground lens and
background source have a relative proper motion, $\mu_{\rm rel}$, with respect
to each other. The lens has a characteristic angular Einstein radius,
$\theta_{\rm E}$ (which is the angular radius of the images ring for the
prefect alignment), and the source has an angular stellar radius,
$\theta_{\ast}$. One key timescale is the crossing of the Einstein radius, as
given by $t_{\rm E}=\theta_{\rm E}/\mu_{\rm rel}$. This sets the duration of
the microlensing event (e.g., Gaudi, 2012). The next timescale is the crossing
time of the star, $t_{\ast}=\theta_{\ast}/\mu_{\rm rel}$. For source transits
or near transits by the lens, $t_{\ast}$ sets the timescale for which finite
source effects are relevant and for the polarimetry signals in microlensing
events (Witt & Mao, 1994; Simmons et al., 1995; Sajadian, 2016, 2019). Then
for a variable star, the pulsation period is the third relevant timescale. For
a binary lens, the fourth timescale is the orbital period (e.g., Sajadian,
2014). For reference, typical values of $t_{\rm E}$ can be months, $t_{\ast}$
can be hours to days, and $P$ can be hours to days for pulsating stars. With
these fiducials in mind, the orbital period for a binary lens is not likely
relevant unless of order a year or less, since the orientation of the mass
components will be fixed in the sky for longer periods. With these scales in
mind, we have computed synthetic lightcurves in the time interval
$[-0.9,\leavevmode\nobreak\ 0.9]\leavevmode\nobreak\ t_{\rm E}$. For
convenience, the time of closest approach is set to zero.

Figure 2 shows examples of six simulated light curves. The first three,
(a)–(c), are for radial pulsators, whereas the last three, (d)–(f), the source
stars are for non-radial pulsators. Tables 1 and 2 list model parameters
specific to these plots. In the figures, the black dashed curves represent the
calculated microlensing light curve without pulsation. Light curves in the
$BVRI$ bands with pulsations are shown in blue, green, orange and dark-red,
respectively. Measurement uncertainties are represented as data points with
black for the KMTNet survey and lime for the OGLE survey. In each of the six
examples shown (a)–(f), the upper panel is the light curve, and the lower
panel shows residuals when the black-dashed curve (i.e., for a non-pulsating
star) is subtracted.

Plots for (c) and (f) are for binary lens cases, and these display inset
geometrical figures, with the caustic curve and the source trajectory shown as
black curves and red straight lines, respectively. For the case of NRPs, the
pattern of surface temperature variations (i.e., $l,\leavevmode\nobreak\ m$
modes) are also shown as insets.

All the simulated light curves are shown in terms of the magnification factor,
$A$. The magnification of the source stars is the modeled flux,
$F_{\star}(t)$, normalized to the flux average at the baseline. This means
that for a non-varying star with no microlensing, the light curve would
ideally be a horizontal line with magnification of unity. The differing
timescales along the horizontal axis signify the differing values of $t_{\rm
E}$.

The residuals are residual fluxes which are displayed in differential
magnitude. To obtain the residuals, we find a best-fit magnification model for
a non-pulsating star, $A_{\rm{best}}(t)$, and then calculate the differential
magnitude for the residuals with

$\displaystyle\Delta
m(mag)=-2.5\log_{10}\left[\frac{F_{\star}(t)}{\left<F_{\star,\rm{base}}\right>\leavevmode\nobreak\
A_{\rm{best}}(t)}\right],$ (3)

where, $\left<F_{\star,\rm{base}}\right>$ is the average stellar flux over the
baseline entering the source PSF. The real microlensing lightcurve of a non-
pulsating source star ($A_{\rm{best}}(t)$) is shown by the black dashed curves
in the magnification panels. Again, all the parameters adopted for these
lightcurves are given in Tables 1 and 2. In these tables, $m_{\rm{base}}$ is
the apparent magnitude due to the average stellar flux over the baseline, and
$\left<\sigma_{\rm{m}}\right>/\left<\sigma_{\rm{base}}\right>$ is the ratio of
the averaged photometric error bar during lensing to that from the baseline
amount (see the next subsection). For binary events (Fig. 2 and 2), the
caustic curves (black curves) and the source trajectories over the lens plane
(red lines) are shown in insets. In the last three events, the source stars
are non-radial pulsating variables, and their surface temperature variations
are represented with insets using the colored schemes.

Unless the lens transits or nearly transits the source star, the magnification
curves from microlensing of pulsating stars derive from a multiplication of a
simple magnification curve for microlensing of a non-varying star and the
variability curve for the pulsations. For example, a classic result from the
point-lens, point-source approximation for microlensing is the absence of
chromatic signatures. This means that the residuals from the various filter
bands will all be in phase. In cases of transits or near transits, the
residuals will not generally be strictly periodic, as for example, see the
residual plots of Figure 2. Such effects occur when the point-source
approximation is no longer valid, and persist in the light curve over the
relatively short time scale of order $t_{\star}$, the crossing time of the
source radius by the lens. However, if the source trajectory is tangential to
the caustic line for a binary, the duration of such effects can be longer (Han
et al., 2000).

Table 3: The results from simulating single and binary microlensing events involving main sequence (MS) and giant (Giant) pulsating stars. The second and third columns, $\epsilon_{\rm{low}}$ and $\epsilon_{\rm{high}}$,are efficiencies for detecting pulsating signatures with low and high sensitivities, respectively. Here, $\left<f_{\rm b}\right>,\leavevmode\nobreak\ \left<A_{\rm{m}}\right>,\leavevmode\nobreak\ \left<\rm{SNR}\right>$ are the average blending parameter, maximum magnification factors as measured in $I$-band, and the signal-to-noise ratio (SNR) at the maximum magnification over the simulated events with detectable pulsation signatures, respectively. | $\epsilon_{\rm{low}}[\%]$ | $\epsilon_{\rm{high}}[\%]$ | $\left<\sigma_{\rm{m}}\right>/\left<\sigma_{\rm{base}}\right>$ | $\left<t_{\rm{s}}\right>$ | $\left<P\right>$ | $\left<\delta_{\rm{m}}\right>$ | $\left<f_{b}\right>$ | $\left<m_{\rm{base}}\right>$ | $\left<u_{0}\right>$ | $\left<A_{\rm{m}}\right>$ | $\left<\rm{SNR}\right>$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | $\rm{(days)}$ | $\rm{(days)}$ | $\rm{(mmag)}$ |  | $\rm{(mag)}$ |  |  |
$\rm{Single}\leavevmode\nobreak\ \rm{Microlensing}$
$\rm{Total}$ | $50.89$ | $37.73$ | $0.84$ | $38.29$ | $7.03$ | $0.16$ | $0.73$ | $19.43$ | $0.46$ | $8.99$ | $85.6$
$\rm{MS}$ | $46.23$ | $33.83$ | $0.85$ | $39.20$ | $7.06$ | $0.16$ | $0.69$ | $19.92$ | $0.46$ | $8.22$ | $60.8$
$\rm{Giant}$ | $81.77$ | $63.55$ | $0.80$ | $34.88$ | $6.93$ | $0.14$ | $0.88$ | $17.63$ | $0.49$ | $11.85$ | $178.5$
$\rm{Binary}\leavevmode\nobreak\ \rm{Microlensing}$
$\rm{Total}$ | $62.35$ | $49.30$ | $0.81$ | $39.26$ | $6.98$ | $0.14$ | $0.73$ | $19.58$ | $0.326$ | $129.65$ | $385.2$
$\rm{MS}$ | $57.96$ | $45.17$ | $0.82$ | $38.35$ | $6.94$ | $0.14$ | $0.69$ | $20.05$ | $0.322$ | $124.96$ | $255.8$
$\rm{Giant}$ | $92.21$ | $77.35$ | $0.76$ | $43.13$ | $7.17$ | $0.13$ | $0.89$ | $17.59$ | $0.344$ | $149.67$ | $937.9$

Ignoring effects of transits and near-transits, the expectation is that the
light curve, after dividing by the best-fitted simple microlensing model, will
display the intrinsic periodic variability of the pulsating star. Of course,
the pulsating modes exist in the baseline data as well; magnification by
microlensing serves to decrease the photometric errors of the data points
taken during lensing event, owing to the enhanced brightness of the star. The
improvement in photometric uncertainties by decreasing the stellar magnitude
for real observations is shown in Figure 1. These effects are apparent in the
residuals panels of Figures 2 around $t$ of zero. The improved signal-to-noise
enhances the probability of detecting stellar variability of low-amplitude
pulsating stars, which is the topic of the next section.

##  3 Observing stellar pulsations during microlensing

To evaluate the effect of microlensing on detection of low-amplitude stellar
pulsation, we calculate (i) the detection efficiency and (ii) the improvement
in visibility of stellar pulsation, which are explained in the following.

###  3.1 Detection efficiency

Modeling of microlensing events proceeds generally as follows. The first step
is to assume simple microlensing models without consideration of second-order
effects. Residuals are then analyzed, and any remaining trends are assessed in
relation to established second-order effects. An example of a second-order
effect is the parallax effect from Earth motion that can skew the simple
microlensing model from being time symmetric about peak magnification (see,
e.g., Gould, 1994; Shvartzvald et al., 2019; Zang et al., 2020; Hirao et al.,
2020; Ryu et al., 2018).

However, pulsational variability is not one of the standard second-order
effects that is considered, as variable stars have traditionally been excluded
from potential stellar microlensing samples. Our goal here is to reassess the
value of using variable stars, either to expand the pool of microlensing
events, or as a tool for stellar astrophysics.

We adopt this same strategy in our theoretical study to determine the fraction
of simulated events in which stellar pulsations will be detectable. To do so,
we apply two main criteria. The first (hereafter, “I.”) relates to the quality
of the data, expressed as a threshold on the $\chi^{2}$ values of the
residuals to indicate that second-order effects may be detected. The second
(hereafter, “II.”) is the practical consideration that the pulsation period
cannot be longer than the characteristic timescale of the microlensing event.
Both criteria are quantified in what follows.

To test for the presence of stellar pulsations in lightcurves, we adopt for
the first criterion (I.) $\Delta\chi^{2}_{\rm{lens}}>350$. Here,
$\Delta\chi^{2}_{\rm{lens}}=\chi^{2}_{\rm{pulse}}-\chi^{2}_{\rm{nonpulse}}$,
which is the difference in $\chi^{2}$ values from fitting the real model
(microlensing of a pulsating source star, based on assumed parameters) and
fitting a simple microlensing model for a non-pulsating star,
$A_{\rm{best}}(t)$. The simple model for the non-pulsator uses the average
observed flux of the pulsator from the baseline segments of the synthetic
lightcurve, $\left<F_{\star,\rm{base}}\right>$. The fit from such simple
models are plotted as the black dashed curves in Figures 2, which is
$A_{\rm{best}}(t)$.

This threshold amount (i.e., $350$) will include microlensing events with
lower magnification factor (e.g., microlensing events that do not have obvious
perturbations due to caustic-crossing features or transiting the source
surface), but the stellar pulsations can be discerned (see, e.g., Figure 2).
However, this threshold is higher than those reported for planet detection in
no caustic-crossing events (see, e.g., Han et al., 2021), because the time
scales of planetary perturbations is shorter than that due to pulsation
features of the source stars.

The above allows us to test for the presence of a second-order effect in the
synthetic microlensing lightcurves. To ensure that periodic trends in the
residuals can be detected, we additionally demand that the pulsating period be
smaller than the magnification time scale, (II.) $P<t_{\rm{s}}$ In that case
at least one pulsation cycle occurs during the lensing event. The
magnification time scale, $t_{\rm{s}}$, is the time interval with the
magnification factor larger than $3/\sqrt{5}=1.34$, which corresponds to the
time interval in which the source star crosses the Einstein ring. Note that
for single microlensing events, the timescale when the magnification factor
exceeds $1.34$ is $t_{s}=2\leavevmode\nobreak\ t_{\rm E}\sqrt{1-u_{0}^{2}}$
(for events with $|u_{0}|<1$).

Figure 3: The efficiency for detecting stellar pulsation versus the Einstein
crossing time, the apparent magnitude of the source star, the pulsating period
and its amplitude. The solid and dotted curves are efficiencies for detecting
pulsation signatures in single and binary lens events, respectively. The blue
and red curves are for main-sequence (MS) and giants source stars.

We separately consider microlensing events involving main sequence (MS) and
giant source stars. For each of them, by applying these conditions, we
evaluate the efficiencies to discover stellar (radial and non-radial)
pulsations in single and binary microlensing events. Here, detection
efficiency refers to the fraction of the events with detectable pulsation
signatures (i.e., referring to criteria I and II) to the total simulated
events. These values are noted in the second column of Table 3. We do not
separate the results for radial and non-radial pulsators, because they are
similar for the same ranges of pulsation amplitudes and periods. We identify
efficiencies for detecting stellar pulsations with low sensitivity, labelled
$\epsilon_{\rm{low}}[\%]$ in the table. By considering two tighter criteria
(higher sensitivity) as (I) $\Delta\chi^{2}_{\rm{lens}}>500$ and (II)
$2\leavevmode\nobreak\ P<t_{\rm{s}}$ indicating a minimum of two pulsation
cycles during the lensing, the detection efficiencies, $\epsilon_{\rm{high}}$,
decrease, with values given in the third column of Table 3. The threshold of
$500$ is usually considered as a robust criterion for planet detection in
microlensing events (see, e.g., Gould et al., 2010; Udalski et al., 2018a).

Note that the table also provides the average parameter values for the
pulsating source stars whose pulsation signatures are detectable with low
sensitivity. Here, $\left<f_{\rm b}\right>$ is the average blending parameter
over the simulated events with detectable pulsation signatures. The blending
parameter $f_{\rm b}$ refers to the fraction of the source flux to the total
fluxes entering into the source PSF. This parameter is unity if there is no
blending stars close to the source star. $\left<A_{\rm{m}}\right>$ is the
average of maximum magnification factors as measured in $I$-band over the
simulated events with detectable pulsation signatures. The last column of this
table is the average of the signal-to-noise ratio (SNR) at the maximum
magnification over the events with detectable pulsating signatures. The SNR is
evaluated as the square root of the number of photons entered to the source
PSF (see, e.g., Sajadian & Rahvar, 2012):

$\displaystyle\rm{SNR}=\sqrt{t_{\rm{exp}}10^{0.4m_{\rm{zp}}}\left[10^{-0.4m_{\rm{base}}}A_{\rm{m}}+\Omega
10^{-0.4\mu_{\rm{sky}}}\right]},$ (4)

where, $t_{\rm{exp}}$ is the exposure time, $\Omega$ is the angular area of
the source PSF, $\mu_{\rm{sky}}$ is the sky brightness and $m_{\rm{zp}}$ is
the zero point magnitude. For the OGLE observation,
$\Omega=0.79\leavevmode\nobreak\ \rm{arcs}^{2}$, $t_{\rm{exp}}=120$ sec,
$\mu_{\rm{sky}}=19.7$ mag and $m_{\rm{zp}}=21.7$ mag (Wyrzykowski et al.,
2015; Sajadian, 2021).

Generally, pulsations can be detected for events as long as $\left<t_{\rm
s}\right>\sim 40$ days, which corresponds to $\left<t_{\rm E}\right>\sim 25.5$
days in single microlensing events (because in these events, we have
$\left<t_{\rm s}\right>=\pi/2\leavevmode\nobreak\ \left<t_{\rm E}\right>$). In
our simulations the average value of $t_{\rm E}$ over all simulated single
events is $\sim 19.6$ days, which is shorter than that reported for OGLE
microlensing events (see, e.g., Mróz et al., 2019; Wyrzykowski et al., 2015).
In fact, combining the OGLE and KMTNet data increases the probability of
detecting short-duration events, and as a result decreases the averaged
timescale of detectable events. The corresponding timescale for binary
microlensing is more complicated, depending on the caustic configuration and
the source trajectory.

Since giant source stars are on average brighter and the blending effect is
reduced (see the last column of Table 3), the efficiency of detecting
intrinsic pulsations is about twice that of main sequence stars.

For binary microlensing events, the simulated events mostly have caustic-
crossing features. Additionally, in these events the time scale associated
with the magnification threshold is somewhat longer than single-lens events.
For these reasons, their efficiencies are higher than those due to single
events with similar amplitudes. However, it is the amplitude of the pulsations
that dominates the detection efficiency. The pulsation features are mostly
detectable in single and binary microlensing events with magnification peaks
around $10$ and $130$, which result in SNR values of $\sim 80$ and $400$,
respectively.

We plot in Figure 3 the detection efficiency curves with the Einstein crossing
time, the apparent magnitude of the source stars in the $I$-band, the period
of the stellar pulsation and the pulsation amplitude. In each panel, four
curves are given. Blue curves are efficiencies for detecting the pulsation
signatures of main-sequence (MS) source stars, for single (solid) and binary
(dotted) microlensing events. Red curves are for giant sources.

As expected, the probability for detecting pulsations is higher in longer
microlensing events from intrinsically brighter source stars, e.g., caustic-
crossing binary microlensing events from giant source stars. Higher pulsation
amplitudes lead to larger detection efficiencies. We note that the efficiency
for the amplitudes $0.05<\delta_{\rm m}<0.1$ mag is not zero. However,
identifying these stellar pulsations with current surveys is almost
impossible, but during the microlensing events the efficiencies for discerning
these weak signatures are not zero and even reaches to $80\%$ in binary events
from giant source stars.

The detection efficiency is maximized in the range of stellar periods
$0.1\lesssim P\lesssim 0.3$ days and it drops slowly with increasing period.
Variable stars with pulsation periods around this peak of detection efficiency
include RR Lyrae stars, which are giants on the horizontal branch. These
variable stars are radially pulsating and more than $10\times$ brighter than
the Sun. Their pulsation amplitudes are generally larger than $0.03$ mag.
$\delta$-Sct and $\gamma$-Dor pulsating stars are fainter than RR Lyrae
variable stars and have the pulsation periods less than 1 day (Poleski et al.,
2010; Guzik et al., 2016). These intrinsic variable stars can be discerned
through microlensing with the highest efficiencies.

Figure 4: Microlensing lightcurves from non-radially pulsating stars (red
solid lines) and non-pulsating stars with average luminosity at the baseline
(black dashed curves). We show the best-fitted binary/planetary microlensing
models for each light curve with solid blue curves. The insets are source
trajectories (solid blue), caustic curves (gray curves) and the source disks
(black circle) due to the best-fitted models. The (black) caustic curves and
(red solid) source trajectories for the two bottom lightcurves are for the
real models. Also, the gray caustic and blue source trajectory is for the
best-fitting models. The chromatic schemes inside the lightcurves represent
the variations of stellar temperature over their surfaces. In two top
lightcurves, some parts of lightcurves with significant deviations with
respect to black dashed curves are zoomed. The parameters for these
lightcurves are given in Table 4.

Main-sequence stars can show periodic variability as well. The primary sources
of periodic behavior can come from magnetic activity, stellar spots, eclipses,
or planetary transits. The pulsation periods of these stars depend on stellar
rotational speed or the separation between the binary components. Even though
the shapes of these stellar variations are different from radial and non-
radial pulsations (as simulated in this work), the detection efficiency
through microlensing based on the definition in this work and $\Delta\chi^{2}$
values, should depend on only their amplitudes and periods.

###  3.2 Improvement in detectability of stellar pulsations

Here, we study the effects of microlensing on the detection of stellar
pulsations from another point-of-view. Decreasing the photometric errors
during the lensing improves detectability of the stellar pulsations. To
evaluate this improvement, we focus on the residuals of the light curve fits
and the associated errors. We note that the photometric uncertainties in the
synthetic data, $\sigma_{\rm m}$, decrease because of the light magnification
during lensing. On the other hand, subtracting the best-fitted microlensing
model from the data increases the uncertainty in each data point by the factor
$\sqrt{2}$.

The improvement in detectability of stellar pulsations by lensing
magnification depends on how much the photometric error bars improve during
lensing. In this regard, we calculate the ratio of the averaged photometric
error bar during lensing to that from the baseline amount, $\left<\sigma_{\rm
m}\right>/\left<\sigma_{\rm{base}}\right>$. The smaller this ratio, the higher
the reliability for discerning the stellar pulsations. The average value of
this ratio over all simulated events is $\sim 0.75-0.85$. Values for stars
with (radial and non-radial) pulsations with single or binary microlensing
events are given in the fourth column of Table 3. Decreasing the photometric
error bars is the main reason for higher efficiency in detecting stellar
pulsations during lensing events, because SNR is improved during a lensing
event. The relation between the photometric error bars and the SNR is

$\displaystyle\sigma_{\rm m}=-2.5\log_{10}\left[1+\frac{\delta
F(t)}{F(t)}\right]=-2.5\log_{10}\left[1+\frac{1}{\rm{SNR}}\right],$ (5)

where $F(t)$ and $\delta F(t)$ are total flux and its fluctuation at any given
time. Assuming $\left<\sigma_{\rm
m}\right>=0.8\left<\sigma_{\rm{base}}\right>$ and using Equation 5, the SNR
improves $\simeq 25\%$ (i.e., $\rm{SNR}\simeq 1.25\leavevmode\nobreak\
\rm{SNR}_{\rm{base}}$).

With real observational data, the true interpretation of perturbations in the
residuals depends on many factors. For instance, if the period of pulsation is
comparable to the magnification time scale, the periodic features of the
perturbations may be missed. Also, in high-magnification microlensing events
involving a transit by a single lens, or during caustic-crossings, the
magnification curve does not follow the standard simple shape with a non-
variable source, which likely causes wrong best-fitted models. In the next
section, we discuss the prospect of misinterpreting the microlensing of
pulsating source stars in terms of other second-order effects.

##  4 Misinterpreting pulsation-induced perturbations

Some non-radially pulsating source stars have very small pulsation amplitudes,
although the amplitudes of variation in their temperature and radius are
large. But overall their stellar luminosities may not change significantly.
For instance, when the stellar polar axis is toward the observer and $m>0$,
variations in the stellar surface temperatures are averaged across the face of
the stellar surface. The result is very small variation of stellar luminosity
as seen by the observer (see, Sajadian & Ignace, 2020b). But if in this
orientation, a single lens were to transit across the face of surface or if
the source star passes a caustic curve, there could be detectable
perturbations to the residuals from a standard microlensing fit, but they
would not be periodic and might be attributed to some other effects. If
detected, the perturbations might be misattributed to some other second-order
effect.

Figure 4 displays some examples of the events just described. In each panel,
the real lightcurves in the $I$-band magnitude are shown with red solid
curves. The real lightcurves from non-pulsating source stars are shown with
black dashed curves. For each lightcurve, a best-fit binary or planetary
microlensing model is displayed (blue solid curve) that assumes non-pulsating
source stars and a search within a grid of predetermined models, as developed
by C. Han (see, e.g., Han et al., 2020b, a)
777http://astroph.chungbuk.ac.kr/~cheongho/. These fitted models are shown
with blue solid curves. In the residuals panels, we show the deviation of red
and blue solid lightcurves with respect to the black dashed ones (real model
without pulsation features).

The input parameters for the real microlensing lightcurves are given in Table
4. For each lightcurve, the parameters of the best-fitted models are given in
the second row. In this table, $\xi$ is the angle between the binary axis and
source trajectory in binary lensing events, and the last column “$\rm{No.}$”
is the number of their synthetic (OGLE and KMTNet) data points. The number of
data points can be used to evaluate the normalized $\chi^{2}$ values.

The first two lightcurves are single lens and high-magnification microlensing
events from non-radially pulsating source stars with amplitudes of less than
$1$ mmag. In lightcurve 4, the periodic features around the peak of the light
curve model arise from the close passage by a small central caustic due to a
wide binary system. However, the best-fitting model compensates only for two
pulsation features around the peak, its $\chi^{2}$ value is larger than the
real model by only $\sim 800$. We note that the number of data points for this
lightcurve is $105$.

The second lightcurve, 4, is a single lens transit microlensing event also
involving a non-radially pulsating source star with small pulsation amplitude.
While the lens is transiting the source surface (i.e., $u<\rho_{\star}$ where
$u$ is the lens-source distance normalized to the Einstein radius), the
lightcurve is not symmetric in time owing to the pulsational modulation of the
temperature distribution across the surface. The asymmetric feature in the
lightcurve can be seen by zooming around the time of peak magnification. In
the modeling process, this asymmetric feature results from crossing over a
caustic curve that is more compact than the source size (shown as the left
insets), which can arise from a low-mass companion around the microlens.
However, the difference between $\chi^{2}$ values is high, because the data
points around the peak are not well reproduced with the best-fitted model. The
number of data points in this lightcurve is $369$.

Table 4: The real parameters of microlensing lightcurves from non-radial pulsating stars shown in Figure 4. For each lightcurve, the parameters of the best-fitted models are given in the second row. $\xi$ is the angle between the binary axis and source trajectory in binary lensing events. The last column $\rm{No.}$ is the number of their synthetic (OGLE and KMTNet) data points. $\rm{Fig.\leavevmode\nobreak\ No.}$ | $\delta$ | $P$ | $i$ | $l$ | $m$ | $u_{0}$ | $\rho_{\star}$ | $t_{\rm{E}}$ | $\chi^{2}$ | $q$ | $d$ | $\xi$ | $\rm{No.}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $\rm{(mmag)}$ | $\rm{(days)}$ | $\rm{(deg)}$ |  |  |  |  | $\rm{(days)}$ |  |  | $\rm{(R_{E})}$ | $\rm{(deg)}$ |
4 | $0.46$ | $1.21$ | $89.3$ | $3$ | $1$ | $0.013$ | $0.009$ | $65.3$ | $131.1$ | $--$ | $--$ | $--$ | $105$
$\rm{Best}$-$\rm{fitted}$ | $--$ | $--$ | $--$ | $--$ | $--$ | $0.027$ | $0.008$ | $26.9$ | $928.6$ | $0.23$ | $5.86$ | $86.0$ | $105$
4 | $0.53$ | $2.82$ | $89.5$ | $2$ | $2$ | $0.004$ | $0.006$ | $72.9$ | $2738.5$ | $--$ | $--$ | $--$ | $369$
$\rm{Best}$-$\rm{fitted}$ | $--$ | $--$ | $--$ | $--$ | $--$ | $0.004$ | $0.006$ | $70.6$ | $8185.1$ | $0.001$ | $2.3$ | $293.3$ | $369$
4 | $0.48$ | $6.43$ | $89.6$ | $4$ | $2$ | $0.023$ | $0.011$ | $77.8$ | $4086.9$ | $0.77$ | $0.99$ | $-3.3$ | $1269$
$\rm{Best}$-$\rm{fitted}$ | $--$ | $--$ | $--$ | $--$ | $--$ | $0.023$ | $0.012$ | $76.3$ | $6085.7$ | $0.76$ | $1.01$ | $3.4$ | $1269$
4 | $44.10$ | $3.37$ | $40.7$ | $2$ | $1$ | $0.021$ | $0.064$ | $22.7$ | $1294.3$ | $0.29$ | $1.17$ | $-9.72$ | $398$
$\rm{Best}$-$\rm{fitted}$ | $--$ | $--$ | $--$ | $--$ | $--$ | $0.031$ | $0.057$ | $24.4$ | $25955.0$ | $0.25$ | $1.12$ | $-9.06$ | $398$

The other two lightcurves are for binary microlensing events of non-radially
pulsating source stars. In the plots, the inset represent the real models of
caustic curve (black one) and the source trajectory (solid red line).
Additionally, the best-fitted models of caustic curve (gray one) and the
source trajectory (blue solid line) are plotted over them. The best-fitting
models are very close to the real ones. In these events, the stellar
pulsations produce large deviations at the times of caustic crossings. In the
right-hand lightcurve, 4, these deviations around the first peak tend to alter
the inferred mass ratio, $q$.

##  5 conclusions

Several microlensing events of variable source stars have been reported. In
all of these events, the source stars were obviously variable at the baseline.
In such events, measuring the stellar variability leads to indications of the
source type, its distance and radius. By knowing the source properties, one
can determine the lens parameters including its mass and distance from the
observer. A significant benefit from stellar pulsations is the prospect of
resolving the microlensing degeneracy and determining the physical parameters
of the lenses.

However, it is possible that the pulsational amplitudes of the variable source
stars are relatively small, such that the pulsational behavior is not apparent
in the baseline data (i.e., before and after a microlensing event). Yet the
pulsations could affect the microlensing lightcurves. For these events, there
are two main points. First is whether the lensing effect could help discern
the presence of pulsational variability. Second is whether the perturbations
in the microlensing lightcurves arising from the low-amplitude stellar
pulsations are distinguishable from other effects that might perturb the
lightcurves. This paper has reported on studies of both of these issues.

Microlensing decreases photometric uncertainties of the data points by
increasing the signal-to-noise ratio owing to the enhanced brightness of the
lensed source. This improvement raises efficiencies for detecting the
pulsating signatures from variable stars. We simulated OGLE and KMTNet
observations of ongoing microlensing events from low-amplitude ($\delta_{\rm
m}\in[0.005,\leavevmode\nobreak\ 0.25]$ mag) radially and non-radially
pulsating source stars. In this regard, we determined the cadences and
photometric error bars for synthetic data points based on real observations by
these surveys. This Monte-Carlo simulation of single and binary microlensing
events revealed that on average, uncertainties during the microlensing event
decrease to around $0.75-0.85$ of their amounts at the baseline. Also,
efficiencies for detecting the stellar pulsations from low-amplitude
pulsations are around $50-60\%$ with the criterion that $\Delta\chi^{2}>350$
and a pulsation period shorter than the magnification time scale. By
considering criteria with higher sensitivity ($\Delta\chi^{2}>500$ and at
least two pulsation cycles over the magnification time scale), the efficiency
is around $40-50\%$. Efficiencies for the giant stars are as twice as large as
for the main-sequence stars.

Although microlensing can increase the detectability of the stellar
pulsations, some lightcurves of these pulsating source stars do not have
obvious periodic features in their residuals (after subtracting the best-
fitted simple microlensing model). In these cases the perturbations arising
around peak magnification could be misinterpreted with alternate sources of
perturbations in microlensing events. We have shown example lightcurves in
section 4 to find that small asymmetric perturbations around microlensing peak
could be misidentified in terms of the source star crossing over or passing
close to a caustic curve, in which the caustic is smaller than the source
disk. Not taking pulsations into account could misattribute such a signature
as arising from planetary systems or wide/close binary ones.

## Acknowledgements

S. Sajadian thanks the Department of Physics, Chungbuk National University.
She specially thanks C. Han and D. Kim to teach how to find the best-fitted
binary and planetary microlensing events. R. Ignace acknowledges support for
this work from the National Science Foundation under Grant No. (AST-1747658).
We thank the referee for the helpful comments and good suggestions.

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the
corresponding author.

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