# Microlensing due to free-floating moon-planet systems

Sedighe Sajadian$\leavevmode\nobreak\ {}^{1}$, Parisa
Sangtarash$\leavevmode\nobreak\ {}^{1}$
1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111,
Iran E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Gravitational microlensing is a powerful method for detecting and
characterizing free-floating planetary-mass objects (FFPs). FFPs could have
exomoons rotating them. In this work, we study the probability of realizing
these systems (i.e., free-floating moon-planet ones) through microlensing
observations. These systems make mostly close caustic configurations with a
considerable finite-source effect. We investigate finite-source microlensing
light curves owing to free-floating moon-planet systems. We conclude that
crossing planetary caustics causes an extensive extra peak at light curves’
wing that only changes its width if the source star does not cross the central
caustic. If the source trajectory is normal to the moon-planet axis, the moon-
induced perturbation has a symmetric shape with respect to the magnification
peak, and its light curve is similar to a single-lens one with a higher
finite-source effect. We evaluate the Roman efficiency for realizing moon-
induced perturbations, which is $\left[0.002-0.094\right]\%$ by assuming a
log-uniform distribution for moon-planet mass ratio in the range
$\in\left[-9,\leavevmode\nobreak\ -2\right]$. The highest detection efficiency
(i.e., $\simeq 0.094\%$) happens for Saturn-mass planets when moon-planet
distance is $\sim 43R_{\rm p}$, where $R_{\rm p}$ is the Saturn radius.
Enhancing planetary mass extends the event’s time scale and decreases the
finite-source effect, but it reduces the projected moon-planet distance
normalized to the Einstein radius $s(R_{\rm E})$ which in turn decreases the
size of planetary caustics and takes them away from the host planet’s position
in close caustic configurations.

###### keywords:

gravitational lensing: micro, planets and satellites: detection, methods:
numerical

††pubyear: 2023††pagerange: Microlensing due to free-floating moon-planet
systems–LABEL:lastpage

##  1 Introduction

One of the known populations in our galaxy is free-floating or wide-separation
planetary-mass objects (FFPs). The first known method to discover FFPs is
infrared (or near-infrared) imaging observations. This method is sensitive to
young and massive planets rather in young, and nearby stellar clusters with
star-forming regions (see, e.g., Zapatero Osorio et al., 2000; Scholz et al.,
2012; Peña Ramírez et al., 2012; Liu et al., 2013; Miret-Roig et al., 2022).

For instance, the first low-mass and unbound object, OTS 44, was discovered in
1998 through infrared measurements from a star-forming region in the
constellation Chamaeleon (Oasa et al., 1999). Its mass was estimated as
$11.5\leavevmode\nobreak\ M_{\rm J}$ which is located on a boundary between
brown dwarfs and planets. Here, $M_{\rm J}$ is the Jupiter mass. For this low-
mass object, infrared photometry and spectroscopy observations with the
Spitzer Space Telescope, the Herschel Space Observatory, and the Very Large
Telescope have revealed the existence of an accretion and rotating disc which
proposes it can develop into a planetary system (Tamura et al., 1998; Luhman
et al., 2005; Joergens et al., 2013).

Another method to discern far and low-mass FFPs located in the Galactic disc
or even its halo is gravitational microlensing (see, e.g., Sumi et al., 2011;
Mróz et al., 2017). During a microlensing event, the light of a background
star is temporarily magnified because of its alignment with a massive and
compact (even dark) object, i.e., the so-called lens object (Einstein, 1936;
Liebes, 1964; Chang & Refsdal, 1979). The magnification factor is similar in
different filters, which causes easily discerning lensing from other
variability sources (e.g., Gaudi, 2012; Tsapras, 2018). A FFP can be the lens
object in a microlensing event. That event has a timescale as short as a few
days, and may exhibit (sometimes extreme) finite-source effects (Witt & Mao,
1994) because of large normalized source radii (Mróz et al., 2018).

$\rm{Targets}$ | $\log_{10}\big{[}q\big{]}$ | $d$ | $P$ | $\left<t_{\rm E}\right>$ | $\log_{10}[\left<R_{\rm E}\right>]$ | $\log_{10}[\left<s\right>]\pm\log_{10}[\sigma_{s}]$ | $\left<\rho_{\star}\right>$ | $\log_{10}[\left<\Delta_{\rm c}\right>]$ | $\log_{10}[\left<\Delta_{\rm p}\right>]$
---|---|---|---|---|---|---|---|---|---
|  | $(R_{\rm p})$ | $\rm{(days)}$ | $\rm{(days)}$ |  | $\rm{(R_{\rm E})}$ |  |  |
$\rm{Earth}$-$\rm{Moon}$ | $-1.910$ | $60.34$ | $27.29$ | $0.104$ | $-2.260$ | $-0.377\pm-0.590$ | $0.555$ | $--$ | $-1.621$
$\rm{Mars}$-$\rm{Phobos}$ | $-7.778$ | $2.77$ | $0.32$ | $0.035$ | $-2.749$ | $-1.501\pm-1.713$ | $1.711$ | $-10.634$ | $-7.927$
$\rm{Mars}$-$\rm{Deimos}$ | $-8.636$ | $6.92$ | $1.27$ | $0.034$ | $-2.749$ | $-1.104\pm-1.316$ | $1.678$ | $-10.651$ | $-7.164$
$\rm{Jupiter}$-$\rm{Europa}$ | $-4.597$ | $9.60$ | $3.55$ | $1.878$ | $-1.013$ | $-1.383\pm-1.592$ | $0.032$ | $-7.210$ | $-5.972$
$\rm{Jupiter}$-$\rm{Ganymede}$ | $-4.108$ | $15.31$ | $7.16$ | $1.869$ | $-1.013$ | $-1.179\pm-1.390$ | $0.031$ | $-6.289$ | $-5.125$
$\rm{Jupiter}$-$\rm{Callisto}$ | $-4.247$ | $26.93$ | $16.69$ | $1.872$ | $-1.012$ | $-0.935\pm-1.150$ | $0.031$ | $-4.063$ | $-4.470$
$\rm{Jupiter}-\rm{Io}$ | $-4.327$ | $6.03$ | $1.77$ | $1.860$ | $-1.011$ | $-1.586\pm-1.798$ | $0.031$ | $-7.356$ | $-6.455$
$\rm{Jupiter}-\rm{Amalthea}$ | $-8.960$ | $2.59$ | $0.50$ | $1.873$ | $-1.013$ | $-1.951\pm-2.162$ | $0.032$ | $-12.722$ | $-9.864$
$\rm{Saturn}$-$\rm{Titan}$ | $-3.626$ | $20.98$ | $15.95$ | $1.021$ | $-1.274$ | $-0.861\pm-1.071$ | $0.056$ | $-3.806$ | $-3.922$
$\rm{Saturn}$-$\rm{Rhea}$ | $-5.392$ | $9.05$ | $4.52$ | $1.016$ | $-1.274$ | $-1.228\pm-1.437$ | $0.058$ | $-7.677$ | $-5.904$
$\rm{Saturn}$-$\rm{Iapetus}$ | $-5.498$ | $61.15$ | $79.34$ | $1.026$ | $-1.274$ | $-0.396\pm-0.610$ | $0.056$ | $-1.972$ | $-3.475$
$\rm{Saturn}$-$\rm{Dione}$ | $-5.715$ | $6.48$ | $2.74$ | $1.021$ | $-1.274$ | $-1.370\pm-1.580$ | $0.058$ | $-8.302$ | $-6.495$
$\rm{Saturn}$-$\rm{Tethys}$ | $-5.964$ | $5.06$ | $1.89$ | $1.017$ | $-1.274$ | $-1.481\pm-1.693$ | $0.056$ | $-8.779$ | $-6.956$
$\rm{Saturn}$-$\rm{Enceladus}$ | $-6.721$ | $4.09$ | $1.37$ | $1.011$ | $-1.274$ | $-1.570\pm-1.775$ | $0.058$ | $-9.714$ | $-7.592$
$\rm{Saturn}$-$\rm{Mimas}$ | $-7.181$ | $3.19$ | $0.94$ | $1.019$ | $-1.274$ | $-1.681\pm-1.891$ | $0.058$ | $-10.399$ | $-8.161$
$\rm{Uranus}$-$\rm{Titania}$ | $-4.407$ | $17.19$ | $8.70$ | $0.400$ | $-1.683$ | $-0.899\pm-1.112$ | $0.148$ | $-5.443$ | $-4.439$
$\rm{Uranus}$-$\rm{Oberon}$ | $-4.451$ | $23.01$ | $13.47$ | $0.395$ | $-1.683$ | $-0.773\pm-0.987$ | $0.147$ | $-0.805$ | $-4.081$
$\rm{Uranus}$-$\rm{Umbriel}$ | $-4.833$ | $10.49$ | $4.14$ | $0.400$ | $-1.683$ | $-1.113\pm-1.321$ | $0.149$ | $-6.860$ | $-5.282$
$\rm{Uranus}$-$\rm{Ariel}$ | $-4.841$ | $7.53$ | $2.52$ | $0.404$ | $-1.682$ | $-1.259\pm-1.471$ | $0.146$ | $-7.197$ | $-5.731$
$\rm{Uranus}$-$\rm{Miranda}$ | $-6.132$ | $5.10$ | $1.41$ | $0.398$ | $-1.682$ | $-1.428\pm-1.637$ | $0.147$ | $-8.838$ | $-6.879$
$\rm{Neptune}$-$\rm{Triton}$ | $-3.680$ | $14.41$ | $5.88$ | $0.436$ | $-1.647$ | $-1.025\pm-1.233$ | $0.135$ | $-5.471$ | $-4.438$
$\rm{Neptune}$-$\rm{Proteus}$ | $-6.367$ | $4.78$ | $1.12$ | $0.432$ | $-1.646$ | $-1.506\pm-1.718$ | $0.135$ | $-9.233$ | $-7.235$
$\rm{Neptune}$-$\rm{Nereid}$ | $-6.519$ | $223.54$ | $359.19$ | $0.439$ | $-1.646$ | $0.164\pm-0.040$ | $0.137$ | $--$ | $--$
$\rm{Pluto}$-$\rm{Charon}$ | $-0.917$ | $16.49$ | $6.37$ | $0.005$ | $-3.556$ | $-0.391\pm-0.692$ | $10.177$ | $--$ | $--$
$\rm{Pluto}$-$\rm{Hydra}$ | $-5.436$ | $54.48$ | $40.53$ | $0.005$ | $-3.580$ | $0.150\pm-0.153$ | $10.837$ | $--$ | $--$
Table 1: The lensing parameters for the known moon-planet configurations in
our solar system. Here, $q$ is the moon-planet mass ratio, $d$ is the moon-
planet distance normalized to the planet radius ($R_{\rm p}$), $P$ is the moon
orbital period, and $s$ is the projected moon-planet distance on the sky plane
which is normalized to the Einstein radius ($R_{\rm E}$). $\sigma_{s}$ is the
Standard Deviation of the $s$ distribution. Two last columns are rough
estimations of the central and planetary caustic sizes in the logarithmic
scale, as derived by $\Delta_{c}=q\left(s-1/s\right)^{-2}$, and $\Delta_{\rm
p}=\sqrt{q}\leavevmode\nobreak\ s^{3}$.

Three microlensing surveys, including the Optical Gravitational Lensing
Experiment (OGLE, Udalski et al. (2015)), The Microlensing Observations in
Astrophysics (MOA, Sako et al. (2008)), Korea Microlensing Telescope Network
(KMTNet, Kim et al. (2016)), have frequently reported (and report) detecting
FFPs in our galaxy (see, e.g., Sumi et al., 2011; Mróz et al., 2017, 2018,
2019, 2020a, 2020b; Ryu et al., 2021; Kim et al., 2021). To derive a precise
estimation of their abundance, The Nancy Grace Roman Space Telescope (Roman)
telescope is planned to monitor the Galactic bulge during six $62$-day
observing seasons in its $5$-year mission. Johnson et al. (2020) reported that
the Roman telescope would detect $\sim 250$ FFPs with masses down to that of
Mars (including $\sim 60$ with masses $\lesssim m_{\oplus}$).

Generally, FFPs could be either ejected from their host planetary systems
(even with their moons) owing to planet-planet scattering and dynamic
instability (Lissauer, 1987; Laughlin & Adams, 2000; Boss, 2006; Veras &
Raymond, 2012; Hong et al., 2018; Rabago & Steffen, 2019), or formed through
core accretion inside protoplanetary discs in a similar way to stars or brown
dwarfs (Ida & Lin, 2004; Mordasini et al., 2009; Luhman, 2012). In the first
mechanism, Earth-mass planets are ejected more frequently than Jupiter-like
ones (Pfyffer et al., 2015; Ma et al., 2016).

We note that FFPs could have exomoons orbiting them which can be produced by
either planet-planet collision or co-accretion, and capture (Debes &
Sigurdsson, 2007; Barr, 2016). Exomoons orbiting FFPs even potentially
maintain liquid water on their surface which increases the probability of life
formation over them (Reynolds et al., 1987; Scharf, 2006; Dobos et al., 2017;
Ávila et al., 2021). This property of free-floating moon-planet systems
promotes the importance of detecting them. In the regard of discovering such
systems Limbach et al. (2021) investigated the probability of detecting
exomoons transiting FFPs, and concluded that detection of Io and Titan-like
exomoons is possible by James Webb Space Telescope (JWST) telescope.

In this work, we study the possibility of discerning FFPs orbiting exo-moons
through microlensing observations, especially with the Roman telescope. In
this regard, Han & Han (2002) noticed the disruptive finite-source effect on
realization of moon-induced perturbations in planetary lensing events. Also,
Liebig & Wambsganss (2010) simulated lensing events from triple systems
(including a moon, planet, and host star) and concluded discerning such
systems would be possible through frequent and highly precise monitoring of
the Galactic bulge. Detecting such triple systems will be improved with the
Roman’s highly precise monitoring of the Galactic bulge (see, e.g., Bachelet
et al., 2022).

In Section 2 we first review the known moon-planet systems orbiting the Sun.
We assume they are free and make microlensing events, and study the properties
of their microlensing events. In Section 3 we investigate properties of
caustic-crossing features in short-duration microlensing events due to free-
floating moon-planet systems. In Section 4, we evaluate the Roman efficiency
for detecting these systems during its Galactic Bulge Time Domain Survey.
Finally, in Section 5 we summarize the results and conclusions.

##  2 Known moon-planet systems

There are several moon-planet systems in our solar system. These known lunar
systems are a good sample of possible free-floating moon-planets in our
galaxy. Hence, we calculate the properties of their microlensing events that
they would make if they were free. We assume these lunar systems are in our
galaxy and in lines of sight toward the Galactic bulge and cause short-
duration lensing effects for collinear and background stars located in the
Galactic bulge. In order to have a correct sense of their lensing parameters,
we calculate their average values over big ensembles of their microlensing
events. To simulate microlensing events due to each known moon-planet system,
masses of planet and moon and their distance are fixed to the real values.
Other parameters are determined according to the related distributions. For
instance, the source photometry properties are chosen from the Besançon model
111https://model.obs-besancon.fr/ (Robin et al., 2004, 2012). The lens impact
parameter, and the angle of source trajectory with respect to moon-planet axis
are chosen uniformly. Other details can be found in previous papers (see,
e.g., Sajadian & Poleski, 2019; Moniez et al., 2017; Sajadian, 2015).

In Table 1, the average values of lensing parameters due to the known moon-
planet systems (mentioned in the first column) are reported. In this table $q$
is the moon-planet mass ratio, $d$ is the moon-planet distance normalized to
the planet radius ($R_{\rm p}$), $P$ is the moon orbital period, $\left<t_{\rm
E}\right>$ is the average Einstein crossing time on different possible
microlensing events, $\left<R_{\rm E}\right>$ is the average Einstein radius
(the radius of images ring at the complete alignment), and $\left<s\right>$
and $\sigma_{s}$ are the average projected moon-planet distance on the sky
plane and normalized to the Einstein radius and the Standard Deviation from
the average value, respectively 222We use the notation proposed by Skowron et
al. (2011).. $\left<\rho_{\star}\right>$ is the average projected source
radius on the lens planet and normalized to the Einstein radius.

Two last columns are the length scales of central and planetary caustics in
the logarithmic scale which are roughly estimated by $\Delta_{\rm
c}=q\left(s-1/s\right)^{-2}$, and $\Delta_{\rm p}=\sqrt{q}\leavevmode\nobreak\
s^{3}$ (respectively), for close caustic configurations and when $q\ll 1$
(Bozza, 2000; An, 2005; Chung et al., 2005). To determine their caustic
configurations, in Figure 1 we show positions of these moon-planet systems
over 2D $q$-$s$ plane. The errors for normalized distance ($s$) are the
Standard Deviation from the average values (seventh column of Table 1).
Accordingly we list some key points in the following.

  * •

Most of these lunar systems make close caustic configurations with relatively
small values of $q$.

  * •

Comparing the moon orbital periods and the average lensing timescales
(mentioned in the fourth and fifth columns of Table 1), for most systems with
large $q$ the lens orbital motion effect (see e.g., Sajadian, 2014) is
ignorable in their binary lensing events, albeit they make close caustic
configurations.

  * •

Similar to microlensing from FFPs, lensing due to lunar FFPs suffer from
considerable finite-source effect. In all of these events the size of
normalized source radius projected on the lens plane $(\rho_{\star})$ is at
least 1-2 orders of magnitude larger than the size of caustic curves
(comparing three last columns of Table 1). So, microlensing events due to
free-floating moon-planet systems will not have obvious caustic-crossing
features and the moon-induced perturbations due to passing over small caustic
curves should be very small.

In the next section, we characterize and evaluate the moon-induced
perturbations in microlensing light curves.

Figure 1: The green points represent locations of the known moon-planet
systems orbiting the Sun (as mentioned in Table 1) over 2D $s$-$q$ plane. The
error bars in $s$ are the Standard Deviation from the average values in their
simulated ensembles. $s_{\rm c}$ and $s_{\rm w}$ are the bifurcation values of
$s$ between close-intermediate and intermediate-wide caustic topologies.

Figure 2: Four examples of microlensing light curves due to free-floating
moon-planet systems with $q=0.03$, $s(R_{\rm E})=0.7$, $t_{\rm E}=0.1$ days,
and $m_{\rm{base},\leavevmode\nobreak\ \rm{W149}}=21$ mag (solid blue curves).
The corresponding single microlensing light curves are depicted by dashed
black curves. The synthetic data points are simulated according to the Roman
observing strategy. The residual parts show moon-induced perturbations. The
insets represent caustic maps, the source trajectory (dashed lines), and the
normalized source sizes (solid black lines) projected on the lens plane.

##  3 Caustic-crossing features

The Roman telescope is planned to observe the Galactic bulge during six
$62$-days seasons with the observing cadence $15.16\leavevmode\nobreak\ $min
333https://roman.gsfc.nasa.gov/galactic_bulge_time_domain_survey.html. During
its mission, this telescope is predicted to detect $\sim 250$ FFPs with masses
down to that of Mars (including $\sim 60$ with masses $\leq m_{\oplus}$) in
short-duration and finite-source microlensing events (Johnson et al., 2020).
In these events, the most significant second-order effect is finite-source, as
a result of small Einstein radii.

We note that short-duration microlensing events with extreme finite-source
effect suffer from a continuous degeneracy which was first discerned by Mróz
et al. (2020a), and then semi-analytically evaluated in Johnson et al. (2022).
Because of this degeneracy, there is a strong correlation between the lensing
parameters $\rho_{\star}$, $t_{\rm E}$, $u_{0}$, and $f_{\rm b}$, so that they
can not be realized uniquely. Here, $f_{\rm b}$ is the blending factor which
is the ratio of the source flux to the total flux at the baseline, and $u_{0}$
is the lens impact parameter.

Additionally, if these FFPs host moon-like objects, another second-order
perturbation disturbs their light curves. According to the previous section,
free-floating moon-planet systems mostly make close caustic configurations
that their caustic sizes are small compared to the normalized source radii.
Hence, their light curves should be very similar to single and finite-source
microlensing ones, but with small perturbations. Close caustic topologies
include two planetary and one central caustic curves. Planetary caustics are
larger and form at the opposite side of the secondary object’s location and at
the distance $\sim 1/s-s$ from the location of primary lens along binary axis
(see, e.g., Schneider et al., 1992; Gaudi, 2012).

In order to study the properties of such finite-source microlensing events due
to free-floating moon-planet systems, we consider two moon-planet systems as a
lens object and make several animations of their light curves due to planetary
and central caustic crossings by varying (i) the lens impact parameter, (ii)
the angle of source trajectory with respect to the binary axis, and (iii) the
projected source size normalized to the Einstein radius. In these simulations,
we make binary lensing light curves due to extended source stars using RT-
model 444http://www.fisica.unisa.it/gravitationastrophysics/RTModel.htm
developed by V. Bozza (Bozza, 2010; Skowron & Gould, 2012; Bozza et al.,
2018). All animations made in this work can be found at this address:
https://iutbox.iut.ac.ir/index.php/s/NYnnowMoLbFnZeD.

In these animations, for each microlensing event two light curves are plotted
in left panels: (a) real light curve of finite-source microlensing due to
free-floating moon-planet systems (solid blue curve), and (b) single and
finite-source microlensing light curve due to a lens with the total mass of
planet and moon (dashed black curve). Their residual versus time normalized to
the Einstein crossing time is represented in the bottom part. The caustic map,
the projected source trajectory over it (black dashed line), and the source
size projected on the lens plane and normalized to the Einstein radius
$\rho_{\star}$ (solid black circle) are shown in the right panel. In these
animations, in order to have a correct sense with respect to moon-induced
perturbations and the difference between light curves, we assume that these
events are detected by Roman in the W149 filter with the cadence $15.16$ min.
The simulated data points are shown with magenta color. We generate the
animations for two set of lensing parameters (I) $q=0.03$, $s(R_{\rm E})=0.7$,
$t_{\rm E}=0.1$ days, $\rho_{\star}=0.4$, $m_{\rm{base},\leavevmode\nobreak\
\rm{W149}}=21$ mag (the source magnitude at the baseline ), and (II) $q=0.01$,
$s(R_{\rm E})=0.42$, $t_{\rm E}=0.1$ days, $\rho_{\star}=0.56$,
$m_{\rm{base},\leavevmode\nobreak\ \rm{W149}}=21$ (same as Moon-Earth system).

Four examples of these simulated light curves (which are shown in the
animations) are represented in Figure 2. In the top of these light curves, the
change in Full Width at Half Maximum (FWHM) of blue light curves with respect
to black dashed ones normalized to the Einstein crossing time
$\Delta\rm{FWHM}(t_{\rm E})$, and $\Delta\chi^{2}$ which is the difference
between $\chi^{2}$ from fitting two (solid blue and dashed black) light curves
to synthetic data points are reported. According to the animations, we list
some key points in the following.

  * •

Generally moon-induced perturbations are very wide and their light curves are
similar to single-lens ones without obvious perturbations.

  * •

If the source star passes over planetary caustic in addition to crossing the
central one, a very wide peak will appear in the wing of light curve (see Fig
2). Although the closest distance of the source trajectory to the central
caustic determines the magnification peak, crossing the planetary caustic
could change the light curve width.

  * •

If the source star passes only over (or close to) planetary caustics, the peak
location changes from the time of the closest approach to the central caustic.
In that case, the moon-induced perturbations in light curves are small and
barely realizable. One example light curve is shown in Figure 2. However,
these perturbations change the light curves width which leads to wrong
estimations of $t_{\star}=t_{\rm E}\rho_{\star}$, i.e., the time of crossing
the source radius.

  * •

The least perturbations due to crossing planetary caustics happen when the
source trajectory passes normal to the binary axis which leads to symmetric
light curves with respect to the magnification peak. The width of these light
curves is proportional to the normal distance between planetary caustics which
is $\propto 2q^{1/2}\sqrt{s^{-2}-1}$ (Han, 2006). This distance is
significantly larger than the width of their corresponding single-lens light
curve, i.e., $\propto t_{\star}\sqrt{1-u_{0}^{2}/\rho{\star}^{2}}$. Hence,
these light curves are degenerate with single microlensing light curves by
higher finite-source effects. For instance, in Figure 2, one example of such
microlensing events is shown. The degenerate model is plotted with red solid
curve. The lensing parameters of this degenerate model is $\rho_{\star}=1.2$,
$u_{0}=1.07$, and $t_{\rm E}=0.095$ days, whereas the lensing parameters of
the real light curve (blue dashed curve) are $\rho_{\star}=0.4$, $u_{0}=0.7$,
$t_{\rm E}=0.1$ days, $q=0.03$, and $s=0.7$. The difference between $\chi^{2}$
values for the real and the degenerate light curves is only
$\Delta\chi^{2}=66$. Therefore, when the source trajectory passes normal to
the binary axis, realizing the real model is almost impossible.

  * •

If free floating moon-planet system makes intermediate caustic, the highest
moon-induced perturbation happens when the source trajectory is parallel with
the binary axis. In that case, the light curve is not symmetric with respect
to the magnification peak, although this asymmetry does not change the light
curve width. One example is shown in Figure 2.

Accordingly, the moon-induced perturbations are mostly small and barely
detectable. Nevertheless, the Roman telescope with improved cadence $15.16$
min and high photometry accuracy may realize such small perturbations. We note
that discerning a low-mass moon orbiting a free-floating planet far from us is
only doable through gravitational microlensing observations. In next section,
we study the Roman efficiency for discerning moon-induced perturbations in
finite-source microlensing light curves due to free-floating moon-planet
systems.

$M_{\rm p}$ | $\left<t_{\rm E}\right>$ | $\left<u_{0}\right>$ | $\log_{10}\left[\left<q\right>\right]$ | $\log_{10}\left[\left<s\right>\right]$ | $\log_{10}\left[\left<\rho_{\star}\right>\right]$ | $\left<m_{\rm{base}}\right>$ | $\left<d\right>$ |  $f_{\rm c}$:$f_{\rm i}$:$f_{\rm w}$ | $\epsilon(\%)$
---|---|---|---|---|---|---|---|---|---
| $(\rm{days})$ |  |  | $(\rm{R_{\rm E}})$ |  | $(\rm{mag})$ | $(R_{\rm p})$ |  |
$M_{\oplus}$ | $0.42\pm 0.45$ | $0.21\pm 0.25$ | $-2.19\pm-2.63$ | $-0.18\pm-0.87$ | $-1.19\pm-1.43$ | $19.6\pm 1.4$ | $37.8\pm 8.6$ |  $90.9$:$0.0$:$9.1$ | $0.00224$
$5\leavevmode\nobreak\ M_{\oplus}$ | $1.54\pm 1.68$ | $0.33\pm 0.28$ | $-2.39\pm-2.53$ | $-0.19\pm-0.75$ | $-1.14\pm-0.87$ | $20.7\pm 1.3$ | $36.8\pm 6.0$ |  $100.0$:$0.0$:$0.0$ | $0.00243$
$10\leavevmode\nobreak\ M_{\oplus}$ | $2.23\pm 3.52$ | $0.26\pm 0.30$ | $-2.18\pm-2.60$ | $-0.25\pm-0.62$ | $-1.10\pm-1.02$ | $19.8\pm 1.7$ | $42.7\pm 8.7$ |  $91.7$:$0.0$:$8.3$ | $0.00405$
$50\leavevmode\nobreak\ M_{\oplus}$ | $2.05\pm 2.69$ | $0.30\pm 0.28$ | $-2.17\pm-2.66$ | $-0.25\pm-0.78$ | $-0.96\pm-0.98$ | $19.9\pm 1.9$ | $41.7\pm 8.3$ |  $92.9$:$0.0$:$7.1$ | $0.04276$
$100\leavevmode\nobreak\ M_{\oplus}$ | $2.14\pm 2.91$ | $0.41\pm 0.31$ | $-2.22\pm-2.66$ | $-0.21\pm-0.63$ | $-0.99\pm-0.92$ | $19.6\pm 1.8$ | $42.9\pm 6.1$ |  $95.7$:$4.3$:$0.0$ | $0.08964$
$150\leavevmode\nobreak\ M_{\oplus}$ | $2.37\pm 4.48$ | $0.34\pm 0.29$ | $-2.18\pm-2.64$ | $-0.21\pm-0.62$ | $-1.05\pm-1.11$ | $19.8\pm 1.6$ | $43.0\pm 7.8$ |  $90.3$:$2.8$:$6.9$ | $0.09411$
$200\leavevmode\nobreak\ M_{\oplus}$ | $1.50\pm 2.08$ | $0.26\pm 0.25$ | $-2.16\pm-2.62$ | $-0.13\pm-0.49$ | $-1.04\pm-1.10$ | $20.5\pm 1.3$ | $42.1\pm 8.8$ |  $77.8$:$14.8$:$7.4$ | $0.07919$
$M_{\rm J}$ | $3.76\pm 3.50$ | $0.45\pm 0.36$ | $-2.17\pm-2.69$ | $-0.22\pm-0.82$ | $-1.32\pm-1.37$ | $19.5\pm 2.1$ | $48.0\pm 3.2$ |  $100.0$:$0.0$:$0.0$ | $0.01841$
$5\leavevmode\nobreak\ M_{\rm J}$ | $43.60\pm 43.07$ | $0.70\pm 0.08$ | $-2.07\pm-3.19$ | $-0.33\pm-0.89$ | $-0.87\pm-1.98$ | $19.4\pm 0.8$ | $53.7\pm 0.3$ |  $100.0$:$0.0$:$0.0$ | $0.00254$
Table 2: The average lensing parameters of microlensing events due to free-
floating moon-planet systems with discernible moon signatures by Roman. In the
simulation, we consider discrete values for mass of planets as specified in
each row. The last column is the Roman efficiency for discerning moon-induced
perturbations in these systems.

##  4 Roman Efficiency

Up to now, no FFPs with rotating exomoons has been firmly confirmed through
microlensing observations. However, the microlensing event MOA-2011-BLG-262Lb
has a short time scale ($t_{\rm E}=3.8$ days) and a caustic-crossing feature.
Although one possible solution for its lens is a free-floating moon-planet
system, this event has another accidentally degenerate solution (a planetary
system) which is favorable according to the Bayesian analysis (Bennett et al.,
2014). We expect that detecting microlensing events due to free-floating moon-
planet systems considerably increases in the Roman era. In this section, we
focus on moon-planet systems and determine how the Roman telescope is
efficient to realize moon-induced perturbations.

In this regard, we make big ensembles of microlensing events due to free-
floating moon-planet systems. Then, synthetic data points taken by the Roman
telescope are generated for each of them. Finally, we evaluate moon-induced
perturbations in light curves and their detectability by Roman.

We have little information about distribution functions of moon-planet
systems, so we get the known moon-planet systems orbiting the Sun to help. In
Table 1, parameters of 25 moon-planet configurations in our solar system were
reported. In Appendix A we plot the fractional distributions of their
$d/R_{\rm p}$, and $q$ in the logarithmic scale. In these plots, the average
values and Standard Deviation from the averages are reported in plots with
green color, and additionally depicted with dashed green (vertical) lines.
These average values are close to the averages of log-uniform distributions
(which are shown with black vertical lines in these figures). However, these
two parameters have a correlation. This correlation is shown in the last panel
of Figure 4. Accordingly, there is a linear relation between
$\log_{10}[d/R_{\rm p}]$ and $\log_{10}[q]$ as given by Equation 2. Hence, in
simulations we first choose the moon-planet mass ratio uniformly in the
logarithmic scale from the range $[-9,-2]$. Then, we determine the boundaries
of all possible values for $\log_{10}[d/R_{\rm p}]$ (as depicted with magenta
filled area in the last panel of Figure 4, and given by Equation 2). The final
value for $\log_{10}[d/R_{\rm p}]$ is a random value between boundaries.

To project the moon-planet distance on the sky plane, we consider a projection
angle $\theta$. We choose $\cos\theta$ uniformly from the range
$[0,\leavevmode\nobreak\ 1]$. The projected moon-planet distance is
$d\leavevmode\nobreak\ \sin\theta$.

We take other parameters including the photometry properties of source stars,
extinction map, source distances, the lens distance, and their velocity
components from the known distributions as explained in the previous papers
(see, e.g., Sajadian & Poleski, 2019; Moniez et al., 2017; Sajadian, 2015). To
generate synthetic data points we determine the photometry error bars of data
points according to the apparent magnitude of source stars in the W149 filter
(Johnson et al., 2020; Sajadian, 2021). The stellar absolute magnitude in W149
is roughly one-third of the summation of its absolute magnitudes in K, H, and
J-bands (Bagheri et al., 2019; Sajadian & Salehi, 2020). We assume that these
events fully occur in one observing season.

In previous section, we conclude the moon-induced perturbations should be
small and very wide. Most of these perturbations make asymmetric features in
light curves unless the source trajectory passes normal to the binary axis.
Hence, to extract detectable moon-induced perturbations we apply two criteria
which are (i) the difference between $\chi^{2}$ from fitting real models
($\chi^{2}_{\rm{real}}$), and single-lens and finite-source microlensing
models (due to an object with the total mass of planet and moon
($\chi^{2}_{\rm{ESPL}}$) should be large, $\left|\Delta\chi^{2}\right|>800$,
and (ii) two sides of light curves with respect to the magnification peak
should be different (i.e., asymmetric features). We evaluate these asymmetric
features using:

$\displaystyle\Delta\mathcal{A}=\sum_{i=1}^{N}\Big{(}\frac{A(t_{i})-A(t^{\prime}_{i})}{\sigma_{i}}\Big{)}^{2}-N$
(1)

where, $A$ is the magnification factor, $t_{i}$, and $t^{\prime}_{i}$ are two
mutual times with the same time interval with respect to the time of
magnification peak, and $N$ is the number of data points taken in one side of
light curves. The error in the magnification factor is specified as
$\sigma_{i}=A(t_{i})\big{[}10^{-0.4\sigma_{\rm{m}}}-1\big{]}$, where
$\sigma_{\rm{m}}$ is the Roman photometric precision (see, Fig. (4) of Penny
et al., 2019).

In simulation, we noticed that $\Delta\mathcal{A}>1200$ is sufficient to
exclude symmetric light curves. This second criterion is necessary to exclude
light curves with symmetric moon-induced perturbations (e.g., Figure 2). Such
events are mostly interpreted as single-lens and finite-source microlensing
events.

We repeat simulations for different values of planets mass as $M_{\rm
p}=M_{\oplus},\leavevmode\nobreak\ 5M_{\oplus},\leavevmode\nobreak\
10M_{\oplus},\leavevmode\nobreak\ 50M_{\oplus},\leavevmode\nobreak\
100M_{\oplus},150M_{\oplus},\leavevmode\nobreak\
200M_{\oplus},\leavevmode\nobreak\ M_{\rm J},\leavevmode\nobreak\ 5M_{\rm J}$.
In Table 2, results from these simulations are reported.

Figure 3: The fractional histograms of four parameters $d/R_{\rm p}$,
$\log_{10}[q]$, $\log_{10}[s(R_{\rm E})]$, and $\log_{10}[\rho_{\star}]$ for
all simulated events (green filled histograms, specified on the left vertical
axes) and detected events (black step ones, specified on the right vertical
axes). These histograms are plotted for simulated events due to a planet with
$M_{\rm p}=150M_{\oplus}$.

According to Table, the Roman telescope will detect moons around Earth-mass,
super-Earth, Jupiter-mass, and super-Jupiter planets with efficiencies $\simeq
0.002\%,\leavevmode\nobreak\ 0.004-0.094\%,\leavevmode\nobreak\
0.02\%,\leavevmode\nobreak\ 0.002\%$, respectively. So the highest efficiency
happens for moons orbiting Saturn-mass planets, while the moon-planet mass
ratio is $\log_{10}\left[q\right]\simeq-2.18$, and their distance is $\simeq
43R_{\rm p}$.

For detecting moon-induced perturbations in microlensing events due to free-
floating moon-planet systems, four parameters have more impacts, which are (i)
the moon-planet distance, (ii) the moon-planet mass ratio, (iii) the size and
distance of planetary caustics from central caustics (which are functions of
$s(R_{\rm E})$), and (iv) finite-source effect ($\rho_{\star}$). We show these
points in Figure 3. For events with $M_{\rm p}=150M_{\oplus}$, histograms of
$d/R_{\rm p}$, $\log_{10}[q]$, $\log_{10}[s(R_{\rm E})]$, and
$\log_{10}[\rho_{\star}]$ are plotted in each panel for all simulated events
(green filled histograms) and detected events (black step ones). In these
figures, green and black dashed lines determine the average values for green
filled and black step histograms, respectively.

Increasing planets’ mass reduces the finite-source effect and extends the
event timescale, but it decreases the normalized moon-planet distance
projected on the sky plane and as a result reduces the planetary caustic
sizes. For that reason, the highest detection efficiency happens for planets
with neither lowest nor highest masses.

We note that the size of planetary caustics is proportional to
$q^{1/2}\leavevmode\nobreak\ s^{3}$, their normal distance to the binary axis
and their horizontal distance from the host location are $\propto
2q^{1/2}\sqrt{s^{-2}-1}$, and $\propto s^{-1}-s$, respectively. We note that
dividing $s$ and $q$ by two decreases planetary caustic sizes by $0.13$, and
$0.7$, respectively. This shows a higher impact of $s$ than $q$ on planetary
caustic topologies.

##  5 Results and Conclusions

Free-floating planets (FFPs) could temporarily magnify the light of background
and collinear source stars. The caused microlensing light curves have short
timescales and usually are affected by finite-source effect. However, these
FFPs can have moons orbiting them and even these exomoons potentially maintain
liquid water. This ability increases the worth of detecting exomoons around
FFPs.

The Roman telescope will do a time domain Galactic bulge survey during six
62-day seasons with a $15.16$ min cadence. The number of FFPs will be detected
through the Roman observations is $\sim 250$, as estimated by Johnson et al.
(2022). Some of these FFPs could have exomoons orbiting them developed by
either planet-planet collision or co-accretion, and capture (Debes &
Sigurdsson, 2007; Barr, 2016). In this work, we have studied how the Roman
telescope is efficient for realizing moon-induced perturbations in these
microlensing light curves.

We first studied the lensing parameters due to possible microlensing events
made from 25 known moon-planet systems orbiting teh Sun. We concluded that
most of them make close caustic configurations, with ignorable orbital motion
effect of lenses, and considerable finite-source effect. So, these light
curves should be similar to finite-source and single-lens events with very
small moon-induced perturbations.

By making several animations of light curves due to different configurations
of free-floating moon-planet systems, we concluded that these perturbations
are very wide, especially when source stars pass over (or close to) planetary
caustic (see Figure 2). These wide moon-induced perturbations are barely
detectable if the source star does not cross central caustic (low
magnification). In that case, crossing only planetary caustics changes light
curves’s width, e.g., Figure 2.

If the source star passes normal to moon-planet axis, the caused light curves
are symmetric and very similar to (degenerate with) single-lens ones with
higher finite-source effect. One example is shown in Figure 2. In this plot,
the red solid curve is a degenerate and single-lens microlensing light curve
that is well fitted to the synthetic data points. We note that the width of
these light curves is proportional to the normal distance between planetary
caustics. If free-floating moon-planet systems make intermediate caustic
configurations, the largest asymmetric perturbations happen when the source
star passes parallel with the binary axis.

We also examined the Roman efficiency for detecting moon-induced perturbations
in light curves due to free-floating moon-planet systems by generating
synthetic events and data points taken by Roman in W149 filter. We consider
discrete values for planet masses, i.e., $M_{\oplus},\leavevmode\nobreak\
5M_{\oplus},\leavevmode\nobreak\ 10M_{\oplus},\leavevmode\nobreak\
50M_{\oplus},\leavevmode\nobreak\
100M_{\oplus},150M_{\oplus},\leavevmode\nobreak\
200M_{\oplus},\leavevmode\nobreak\ M_{\rm J},\leavevmode\nobreak\ 5M_{\rm J}$
and chose the moon-planet mass ratio from the range
$\log_{10}\left[q\right]\in\left[-9,\leavevmode\nobreak\ -2\right]$ uniformly
in the logarithmic scale. The moon-planet distances are chosen in the
logarithmic scale according to $\log_{10}[q]$ (using Eq. 2). We extracted
detectable events by applying two criteria, (i) $\Delta\chi^{2}>800$ from
fitting real and extended-source and single-lens models, and (ii) some
asymmetry features should be in light curves, as defined in Equation 1.

We found that the detectability of moon-induced perturbations depends on
several factors (i) the finite-source effect, (ii) the events’ duration, (iii)
the moon-planet mass ratio, and (iv) the size and distance of planetary
caustics from central caustics. The first and second ones decrease by
increasing the planet’s mass. On the other hand, it reduces the projected
moon-planet distance normalized to the Einstein radius. As a result,
increasing the planet mass reduces the size of planetary caustics and enhances
the distances of planetary caustics from central caustics. Moons orbiting
Saturn-mass planets with $\log_{10}\left[q\right]\simeq-2.18$, and at the
distance $\simeq 43R_{\rm p}$ have the highest efficiency to be realized in
the Roman observations. These free-floating moon-planet systems make
microlensing events with time scales of around $2.4$ days.

## Acknowledgements

We specially thank the anonymous referee for her/his careful and useful
comments.

## DATA AVAILABILITY

The animation made in this work can be found in
https://iutbox.iut.ac.ir/index.php/s/NYnnowMoLbFnZeD. All simulations have
been done for this paper are available at:
https://github.com/SSajadian54/Binary_FFPs_microlensing.

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Figure 4: Two first panels are the normalized (fractional) distributions of
$\log_{10}[d/R_{\rm p}]$, and $\log_{10}[q]$ of the known moon-planet binaries
in our solar systems. The average values (depicted with dashed and green
lines) and Standard Deviation from the averages are reported in plots with
green color. The dashed and black lines are the average of log-uniform
distributions of $\log_{10}[d/R_{\rm p}]$ in the range
$[0.4,\leavevmode\nobreak\ 1.8]$, and $\log_{10}[q]$ in the range
$[-8.9,\leavevmode\nobreak\ -0.9]$. The characteristic of these binary systems
are reported in Table 1. The last panel shows the dispersion of these systems
over the 2D $\log_{10}[q]$-$\log_{10}[d/R_{\rm p}]$ plane.

##  Appendix A Distributions of the known moon-planet systems

In our solar systems there are many known moons orbiting planets. For
instance, Earth has one moon, Mars has two moons (Phobos, Deimos), Jupiter and
Saturn have 80 and 83 moons, respectively. A full list of these moons can be
found in https://solarsystem.nasa.gov/moons/overview/. We list 25 of largest
ones in Table 1. In Section 4, we use their distributions to simulate
microlensing events due to free-floating moon-planet systems. The normalized
(fractional) distributions of the moon-planet distance normalized to the
planet radius, and the moon-planet mass ratio in the logarithmic scale are
shown in two first panels of Figure 4, respectively. For each distribution,
the average and Standard Deviation values are reported in plots with green
color, and depicted with dashed and green lines. If we assume
$\log_{10}[d(R_{\rm p})]$ and $\log_{10}[q]$ are uniformly distributed, their
average and Standard Deviation values are $-4.94\pm 2.23$, and $1.10\pm 0.40$
(shown with dashed black lines). Comparing these average and Standard
Deviation values, we can assume $d(R_{\rm p})$, and $q$ in the logarithmic
scale are uniformly distributed. The Kolmogorov-Smirnov(K-S) test (Karson,
1968) for uniformity of these two distributions (of $\log_{10}[q]$ and
$\log_{10}[d/R_{\rm p}]$) returns: K-S scores $0.25,\leavevmode\nobreak\
0.16$, and p-values $0.07,\leavevmode\nobreak\ 0.48$, respectively. It means
that K-S test confirms the uniformity of $\log_{10}[d/R_{\rm p}]$. But for
distribution of $\log_{10}[q]$ its p-value is small (albeit it is higher than
$0.01$) and K-S test can not firmly confirm its uniformity. However, the
number of entrances is low (only 25). Additionally, we note that there is a
correlation between these two parameters.

According to the last panel of Figure 4, the dispersion of these moon-planet
systems in 2D $\log_{10}[q]$-$\log_{10}[d(R_{\rm p})]$ plane is not uniform.
On average, moons with larger mass ratios rotate in larger orbits and moons
with mass ratios have smaller orbital radii. However, the scattering of data
is relatively high and the number of entrances is low. Hence, this correlation
is weak. We fit a linear relation between $\log_{10}[d(R_{\rm p})]$ and
$\log_{10}[q]$ as:

$\displaystyle\log_{10}[d/R_{\rm p}]=0.14(\pm
0.03)\log_{10}\left[q\right]+1.76(\pm 0.19).$ (2)

Considering the errors of coefficients, for each given value of
$\log_{10}\left[q\right]$ there is a range of possible values for
$\log_{10}\left[d/R_{\rm p}\right]$, which is specified with a filled magenta
region in the last panel of Figure 4. In simulations of Section 4, we use
these distributions to simulate microlensing events due to free-floating moon-
planet systems in our galaxy.