# Hybrid Open Points: an Efficient Tool for Increasing Network Capacity in
Distribution Systems

Matthew Deakin, Ilias Sarantakos, David M. Greenwood, Janusz Bialek, Phil C.
Taylor, Wenlong Ming, and Charalampos Patsios M. Deakin, I. Sarantakos, D. M.
Greenwood, J. Bialek, C. Patsios are with Newcastle University, Newcastle-
upon-Tyne, UK. W. Ming is with Cardiff University. P. C. Taylor is with
University of Bristol. Email<EMAIL_ADDRESS>
###### Abstract

This letter introduces the Hybrid Open Point (HOP), a device consisting of an
electromechanical switch connected in parallel with a power converter, for the
purpose of providing additional network capacity in interconnected
distribution systems. The HOP switch is used for bulk power transfer at low-
cost, whilst the HOP converter provides targeted power transfer when the HOP
switch is open. The device can replace either a Normally Open Point (Type 1
HOP) or a Normally Closed Point (Type 2 HOP). Simple interconnection and teed
interconnection configurations are studied considering fault level and
radiality constraints, with realistic use-cases identified for both HOP types.
The HOP is shown to provide secure network capacity more cost-effectively than
the classical Soft Open Point.

###### Index Terms:

Hybrid open point, network reconfiguration, total supply capability, hosting
capacity, soft open point.

##  I Introduction

Hybrid smart grid technologies are designed to augment the low-cost and
efficient operation of AC systems with the flexibility of power electronic-
based systems to improve network performance. A wide range of these
‘semiconductor-assisted’ technologies have been proposed, including Hybrid On-
Load Tap Changers [1], Hybrid Circuit Breakers [2], Hybrid AC/DC Microgrids
[3], Doubly-Fed Induction Generators [4], and more generally Hybrid AC-DC
systems with a mix of AC and DC lines [5].

The electrification of heat and transport in Net Zero systems will require
Distribution System Operators (DSOs) to find new ways to meet secure network
capacity requirements. If a substation is fed by two circuits without
interconnection, the secure capacity is limited to just 50% of the total
rating of those two circuits. This utilisation can be increased by increasing
the transfer capacity, provided with post-fault reconfiguration to provide
interconnection between substations. Unfortunately, this does not always make
the most use of all headroom available at an adjacent substation, as the
demand that can be transferred may be limited by the available switching
devices, layout of a given substation, or fault level considerations.

To increase this power transfer between substations, a Normally Open Point
(NOP) can be replaced with a Soft Open Point (SOP), allowing for the radial
operation of the network to be maintained whilst avoiding significantly
increased fault level [6, 7, 8, 9]. All of the additional headroom from
adjacent substations can then be utilized, allowing the DSO to defer or even
avoid network upgrades. It is worth noting, however, that the SOP has to carry
not only the additional headroom, but also all of the load that the NOP would
otherwise feed under post-fault conditions [8]. This can mean that a large SOP
may need to be installed for a relatively small increase in transfer capacity.

To address this issue, this letter proposes the Hybrid Open Point (HOP). The
HOP design is presented in Section II, with Section III demonstrating how and
when a HOP can provide additional secure network capacity more cost-
effectively than a SOP (with 100% utilization of the power converters). The
lifetime benefits of HOPs are discussed in Section IV, with salient
conclusions closing the work in Section V.

##  II The Hybrid Open Point

Figure 1: Instead of replacing a Normally Open Point (NOP) with a Soft Open
Point (SOP), the Hybrid Open Point (HOP) connects a power converter in
parallel with a switch which may be normally open (Type 1) or normally closed
(Type 2). When the HOP switch is closed, the switch carries the load; when the
HOP switch is open, the HOP power converter can transfer power whilst
maintaining radial operation of the network.

The HOP consists of a switch in parallel with a power electronic converter, as
illustrated in Fig. 1. To distinguish between the configuration of the switch
during normal operation, we further differentiate between a Type 1 HOP (the
HOP switch is normally open) and a Type 2 HOP (the HOP switch is normally
closed). The HOP switch is used to provide the bulk power transfer capacity of
the integrated device, whilst the HOP power converter is used to transfer
power across the device when the switch is open.

The HOP design is based on the observation that power converters are expensive
per unit of continuous power rating compared to traditional AC technologies
[10], with changes to the network configuration being particularly cost
effective (where this is feasible) [11]. It is for this reason that it is so
crucial for the HOP power converter to be placed in parallel with the HOP
switch—the power transferred by the HOP converter is assumed to be just a
fraction of the power transferred when the switch is closed. To ensure that
costly network protection equipment does not need upgrading, the HOP converter
would be controlled in a similar way to a SOP [12], with the HOP switch in the
same normal state as the NOP or NCP it is replacing.

##  III Using HOPs to Increase Secure Capacity:
Simple and Teed Interconnection

In this section we study simple and teed interconnection between substations
to demonstrate use cases of the HOP, and compare the required rating of the
HOP power converter against a SOP providing equivalent functionality. Without
loss of generality, we consider a DSO being mandated to operate securely under
$N-1$ conditions (i.e., there should be no loss of load when one incoming HV
circuit to a substation is out of service). We ignore trivial cases when there
is already enough capacity to meet the demand, or when there is not enough
capacity to meet all demand irrespective of topology. Voltage constraints are
not considered, as it is assumed that traditional technologies are available
to address these issues more cost effectively (e.g., voltage regulators).
Following prior works, we use a nodal power balance approach, modelling the
flow in feeders as the sum of downstream demands [11].

###  III-A Simple Interconnection between Two Substations

Fig. 2 illustrates, at a high level, how two interconnected substations can
increase their firm capacity by using automatic reconfiguration. When there is
a fault on one of the circuits feeding Substation A, the NOP on circuit
$\mathcal{C}_{AB}$ is closed and Normally Closed Point (NCP) A is opened, so
the demand $D_{A,\,0}$ is transferred from Substation A to Substation B.

If there is load growth at Substation A, so that the demand
$D_{A,\,1}+D_{A,\,2}$ is greater than the circuit rating $S_{A}$, then
additional capacity must be provided. We consider the case when additional
headroom $h$ could be used to reduce net demand at Substation A. This
additional headroom would be based on (for example) thermal constraints on
circuit $\mathcal{C}_{AB}$; in this case we assume that Substation B’s total
capacity is the limiting factor.

Figure 2: Interconnection allows for increased secure capacity at Substation
A, although fault level and radiality constraints may lead to latent headroom
$h$ at Substation B being inaccessible with reconfiguration alone. This
headroom can be utilised by replacing the NOP with a SOP of size $h+D_{A,\,0}$
(Option 1), or by replacing switch at Substation A, NCP A, with a Type 2 HOP
of size $h$ (Option 2).

Depending on the configuration of Substation A and the contribution of network
components to fault level, there will be a limit to the amount of demand that
can be transferred by reconfiguration alone. When this is the case, either a
HOP or a SOP could be used to transfer the additional capacity, as follows.

  * •

Option 1: replace the NOP on circuit $\mathcal{C}_{AB}$ with a SOP, sized to
meet the feeder demand $D_{A,\,0}$ plus the additional headroom $h$. When a
fault occurs on a circuit at Substation A, the SOP injects $D_{A,\,0}+h$ to
meet the feeder demand $D_{A,\,0}$ and provide additional headroom $h$.

  * •

Option 2: install a Type 2 HOP in place of NCP A. The HOP power converter is
sized to meet the additional headroom $h$ only. When a fault occurs on the
incoming circuit at Substation A, the NOP closes, the HOP NCP opens, and the
HOP power converter injects additional headroom $h$ MVA into Substation A.

In this case, both devices are providing the additional headroom $h$, with the
HOP power converter of size $h$ whilst the SOP rating would be $h+D_{A,\,0}$,
such that

$\mathrm{HOP/SOP\,power\,rating\,ratio}=\frac{h}{h+D_{A,\,0}}\,.$ (1)

For example, for headroom $h$ that is 25% of the demand $D_{A,\,0}$ the HOP
would only be 20% the size of the equivalent SOP (although we note that a SOP
could provide other operational benefits [7]).

####  III-A1 Example

The Haxby Road network [13] has transformer size $S_{A}=15$ MVA, and is
interconnected to Huntington New Lane, $S_{B}=24$ MVA. From 2030 to 2033, the
Haxby Road demand could grow from 13.3 MVA to 16.7 MVA (the Net Zero Early
scenario in [14]), leading to a firm capacity shortfall of 1.7 MVA. Feeder B3
has load of $D_{A,\,0}=0.8$ MVA and is connected through a 5.5 MVA rated cable
[13] with load $D_{B,\,0}=0.6$ MVA. Huntington New Lane still has sufficient
headroom $h>1.7$ MVA for $D_{A,\,0}$ to be transferred via reconfiguration,
plus the additional 0.9 MVA required to meet the projected load growth.
Therefore, as in Fig. 3, this load growth could be met either with a 1.7 MVA
SOP installed at the end of Feeder B3 (Option 1), or an 0.9 MVA Type 2 HOP
installed at the head of Feeder B3 (Option 2). The nodal power balance method
(i.e., ignoring losses) was found to have less than 1% relative error as
compared to a full non-linear power flow solution, justifying its use in these
calculations.

###  III-B Simple Interconnection with a Fault Level Constraint

Fig. 3 illustrates a variation of the simple interconnected network, where the
fault current of a DG limits reconfiguration (due to, e.g., the switchgear
ratings at Substation B). In particular, it is assumed that the switch NCP DG
must be opened if the NOP is closed when there is a network outage. In this
case, an $h$ MVA Type 2 HOP installed adjacent to the DG (Option 3) can
transfer the same additional headroom $h$ as a $h+D_{\mathrm{mid}}$ MVA SOP
installed in place of the NOP (Option 4). In this case, the HOP is only
$h/(h+D_{\mathrm{mid}})$ the rating of the SOP.

Figure 3: If DG fault current limits the reconfiguration that can be done when
there is a network outage, Options 3 and 4 both allow the additional headroom
$h$ to be transferred, but Option 3 (using a Type 2 HOP) avoids oversizing the
device by placing the HOP adjacent to the DG (at NCP DG).

###  III-C Teed Interconnection between Three Substations

A teed interconnection case is shown in Fig. 4. Prior to the installation of
the HOP, under outage conditions NCP I would open, NOP C and NOP D would
close, so that demand $D_{\mathrm{I}}$ can be transferred to Substation D.
Radial constraints at the teed section prevent any headroom $h$ at Substation
E being utilized. In this case, the replacement of NOP E with a Type 1 HOP
allows for additional load $D_{h}$ to be transferred during a network outage
by the injection of $h$ MVA (Option 5), or using a suitably sized multi-
terminal SOP (Option 6). For the multi-terminal SOP sized as illustrated in
Fig. 4, the ratio of power converter ratings of the HOP and SOP is
$h/(h+D_{\mathrm{I}})$.

Figure 4: In the case of a teed section of network, replacing NOP E with a
Type 1 HOP (Option 5) allows for the controlled injection of power into the
circuit linking Substations C and D without having to install a fully rated
multi-terminal SOP (Option 6).

###  III-D Summary: Use Cases for a HOP

As outlined at the start of this section, there needs to be an active security
constraint and sufficient headroom at interconnected substations for a HOP to
potentially provide additional firm capacity. In addition, all of the thermal
ratings of any lines and cables would also need to be taken into account for
calculating available headroom. Finally, there must be sufficient physical
space at the substation (or along a feeder) for a device to be installed.

With those points in mind, the three cases envisioned for the HOP can be
summarised as follows.

  * •

When there are radiality constraints due to large lumped loads, a HOP can
still be used to meet a fraction of that lumped load when there is a circuit
outage. The lumped load could represent a large individual load, a radial
spur, or adjacent feeders (e.g., demand $D_{A,\,1}$ in Fig. 2) which cannot be
transferred to an interconnected circuit (circuit $\mathcal{C}_{AB}$) due to
the substation configuration.

  * •

When fault level constrains reconfiguration, the HOP can be installed at the
network location where the protection becomes insufficient, irrespective of
whether that point is normally open or normally closed. This constraint could
be due to DG (Fig. 3) or insufficient fault current as distance from
protection increases.

  * •

When there is a multi-terminal switching point (e.g., a teed section, as in
Fig. 4), a HOP can inject power to support the other feeders connected at that
point. In these cases, there needs to be a configuration of the network so
that the additional headroom can be utilised (e.g., NCP II can be opened in
Fig. 4 to allow $D_{h}$ to be transferred).

A Type 1 HOP with the same rating as a SOP is identical, save for the
additional HOP switch. Where a HOP that is installed is smaller than a SOP,
there will be reduced opportunities for ancillary benefits [7], although this
will only be a significant issue if the SOP is heavily dependent on these
additional value streams for profitability.

##  IV Benefits Analysis of a HOP

HOPs providing secure capacity can defer network reinforcement (or even avoid
it entirely). Reinforcement deferral provides a net benefit to society due to
the time value of money–the present value of the equivalent saving for a delay
of $N$ years with discount rate $d$ can be calculated as

$\mathrm{Cost\,Reduction}\,(\%)=100-\dfrac{100}{(1+d)^{N}}\,.$ (2)

For example, a 5-year reinforcement deferral with a 3.25% discount rate is
equivalent to a 14.8% cost reduction. HOPs could therefore provide a cost-
effective way of managing incrementally increasing demand.

The HOP can also provide ongoing operational benefits. A marginal benefit of
$B$ $/yr over a device lifetime of $N$ years has an lifetime operational
benefit (present value) of

$\mathrm{Lifetime\,Operational\,Benefit}\,(\$)=B\sum_{i=1}^{N}\dfrac{1}{(1+d)^{i}}\,.$
(3)

For example, if a 3 MVA HOP reduces the losses in a network on average by 44
kW (as for the SOP in the central case of [7]), the annual loss reduction is
385 MWh/yr; if the wholesale energy price over this period is $50/MWh, then
over ten years the present value of the benefit would be $162,320.

To determine the socially optimal investment approach, DSOs should carry out a
cost-benefit analysis, comparing the HOP against all other credible solutions
over the project lifetime. Specific further benefits that a HOP could provide,
either through reinforcement deferral (2) or via operational benefits (3)
include the following:

  * •

The HOP power converter can provide reactive power support/power factor
correction.

  * •

With the choice of a suitable power converter technology, the HOP could
provide power conditioning, reducing (for example) network unbalance or
harmonics.

  * •

A Type 1 HOP embedded within the network provides all of the benefits of an
equivalently sized SOP at that location [7].

##  V Conclusions

This letter compares the proposed Hybrid Open Point against the conventional
Soft Open Point, particularly in the context of the provision of network
capacity in interconnected systems. As a HOP can replace both normally closed
and normally open points, it provides additional flexibility for DSO planning
engineers compared to a SOP, allowing for smaller, more cost-effective devices
with higher utilisation. Future work on HOPs should focus on developing
suitable mathematical methods to determine optimal placement and operation of
these devices in the context of flexible loads and DGs alongside more
sophisticated network security standards.

AC systems are low-cost, with mature voltage transformation and protection
technologies. Conversely, DC systems based on power electronics can flexibly
control power flows to fully utilise existing assets. We conclude that only by
embracing Hybrid systems can the full potential of both of these paradigms be
realised to cost-effectively reach Net Zero.

## Acknowledgment

The authors would like to thank Dr Jialiang Yi of UK Power Networks and Anuj
Chhettri of Northern Powergrid for their helpful discussions. This work was
funded by ESPRC Grant No. EP/S00078X/1 (Supergen Energy Networks Hub 2018) and
EPSRC Grant no. EP/T021969/1 (Multi-energy Control of Cyber-Physical Urban
Energy Systems).

## References

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# Sensitivity to habitable planets in the Roman microlensing survey

Sedighe Sajadian$\leavevmode\nobreak\ {}^{1,2}$
1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111,
Iran
2Department of Physics, Chungbuk National University, Cheongju 28644, Republic
of Korea E-mail<EMAIL_ADDRESS>
###### Abstract

We study the Roman sensitivity to exoplanets in the Habitable Zone (HZ). The
Roman efficiency for detecting habitable planets is maximized for three
classes of planetary microlensing events with close caustic topologies. (a)
The events with the lens distances of $D_{\rm l}\gtrsim 7$ kpc, the host lens
masses of $M_{\rm h}\gtrsim 0.6M_{\odot}$. By assuming Jupiter-mass planets in
the HZs, these events have $q\lesssim 0.001$ and $d\gtrsim 0.17$ ($q$ is their
mass ratio and $d$ is the projected planet-host distance on the sky plane
normalized to the Einstein radius). The events with primary lenses, $M_{\rm
h}\lesssim 0.1M_{\odot}$, while their lens systems are either (b) close to the
observer with $D_{\rm l}\lesssim 1$ kpc or (c) close to the Galactic bulge,
$D_{\rm l}\gtrsim 7$ kpc. For Jupiter-mass planets in the HZs of the primary
lenses, the events in these two classes have $q\gtrsim 0.01$, $d\lesssim
0.04$. The events in the class (a) make larger caustics. By simulating
planetary microlensing events detectable by Roman, we conclude that the Roman
efficiencies for detecting Earth- and Jupiter-mass planets in the Optimistic
HZs (OHZs, which is the region between $[0.5,\leavevmode\nobreak\ 2]$ AU
around a Sun-like star) are $0.01\%$ and $5\%$, respectively. If we assume
that one exoplanet orbits each microlens in microlensing events detectable by
Roman ( i.e., $\sim 27000$ ), this telescope has the potential to detects $35$
exoplanets with the projected planet-host distances in the OHZs with only one
having a mass $\lesssim 10M_{\oplus}$. According to the simulation, $27$ of
these exoplanets are actually in the OHZs.

###### keywords:

Gravitational lensing: micro; planets and satellites: detection; planets and
satellites: terrestrial planets; methods: numerical.

††pagerange: Sensitivity to habitable planets in the Roman microlensing
survey–LABEL:lastpage

##  1 Introduction

One of the primary motivations for searching for extrasolar planets is to look
for extraterrestrial life outside of our solar system and to resolve the Fermi
paradox (see, e.g., Landis, 1998; Cirkovic, 2018; Webb, 2015; Prantzos, 2020).
According to this paradox, the likelihood of extraterrestrial civilizations
elsewhere in our galaxy is high, despite the fact that no evidence of them has
been found. Indeed, there are many planetary systems in our galaxy similar to
our solar system. Some of them must be suitable for developing intelligent
beings and we should have received signals from them up to now (e.g., see,
Huang, 1959; Kopparapu, 2013).

The existence of a liquid bio-solvent is one of the necessary factors for
life. Liquid water is the best candidate that exits in a wide temperature
range, i.e., $[273,\leavevmode\nobreak\ 373]$ K (Huang, 1959; Kasting et al.,
1993; Kopparapu et al., 2019; Forget, 2013). The range of orbits around each
star that can support liquid water over the surface of its orbiting planets is
the so-called ’Habitable Zone’ (HZ). However, the existence of liquid water
additionally needs sufficient atmospheric pressure over the planet’s surface.

The equilibrium temperature over the planet’s surface, $T_{\rm p}$, is
determined by the Bond albedo, $A_{\rm B}$, which is the fraction of the
incident radiation to the planet’s surface that is scattered back into space,
and the re-radiation factor, $f$, and is given by (see, e.g., Peters, 2014;
Catling & Kasting, 2017):

$\displaystyle T_{\rm p}=T_{\ast}\sqrt{\frac{R_{\ast}}{s}}\Big{[}f(1-A_{\rm
B})\Big{]}^{1/4}$ (1)

where, $T_{\ast}$ and $R_{\ast}$ are the effective temperature and the radius
of the host star, respectively, $s$ is the planet-host distance. Here, the
planet is considered as a black body being heated only by its host star. For
rocky planets, the Bond albedo and re-radiation factor are $A_{\rm B}\simeq
0.2$ and $f\simeq 0.35$ which result the Earth temperature equals to
$T_{\oplus}\simeq 289$ K.

Two kinds of HZs are defined (narrower and wider zones): Conservative (CHZ)
and Optimistic (OHZ). In our solar system, the CHZ and OHZ represent the
circumstellar zones with boundaries $[0.95,\leavevmode\nobreak\ 1.67]$ AU and
$[0.5,\leavevmode\nobreak\ 2]$ AU (including Venus and Mars orbits),
respectively (Kopparapu et al., 2013, 2014; Chandler et al., 2016). In the CHZ
and OHZ of the Sun, the surface temperature of orbiting planets can be between
$[228,\leavevmode\nobreak\ 294]$ K and $[203,\leavevmode\nobreak\ 405]$ K,
respectively. In order to determine the outer and inner boundaries of the OHZ,
a thick atmosphere with a strong greenhouse effect and a thin atmosphere with
a weak greenhouse effect are assumed, respectively. Therefore, the first step
to probing extraterrestrial life is searching for extrasolar planets whose
orbits stay in the HZs, such as the Earth in our solar system.

Up to now, more than $4017$ extrasolar planets in $3016$ planetary systems
have been detected, according to the Habitable Zone
Gallery111http://www.hzgallery.org (Kane & Gelino, 2012). Among them, there
are $141$ exoplanets with orbits entirely within the OHZs, the so-called
habitable planets. The majority of these exoplanets have been discovered via
radial velocity measurements or transit methods (e.g., Borucki et al., 2011;
Gilbert et al., 2020). One of these habitable planets has been discovered
through microlensing observations which were done by the Microlensing
Observations in Astrophysics (MOA) microlensing group (Bond et al., 2001; Sumi
et al., 2003; Batista et al., 2014).

The highest sensitivity to exoplanet detection through microlensing
observations occurs on planets beyond the snow line. These planetary systems
frequently make resonant or close-to-resonant caustic topologies (e.g., Griest
& Safizadeh, 1998; Di Stefano, 1999; Park et al., 2006; Dominik et al., 2008;
Gaudi, 2012; Sajadian et al., 2016; Yee et al., 2021). For that reason, only
one habitable exoplanet, in the microlensing event MOA-2011-BLG-293Lb, has
been discovered through microlensing observations up to now (Batista et al.,
2014; Yee et al., 2012). However, the recent analysis of this event, based on
high-angular resolution follow-up imaging observations, has shown that the
excess flux at the source position in this event likely does not come from the
host microlens object (Koshimoto et al., 2020). Their results suggest a lower
mass for the host star.

Habitable planetary systems most often form binary-lens systems with close
caustic topologies characterized by three small caustic curves. The
probability of passing through these small caustics or covering such short
caustic-crossing features is low. Nonetheless, the new generation of
telescopes and microlensing surveys, such as The Nancy Grace Roman Space
Telescope (Roman) survey with the improved cadence (i.e., $\simeq 15$ min) and
better resolutions potentially may discover more habitable planets through
microlensing observations. However, one of the main goals of the Roman
Galactic Exoplanet Survey is to detect a large sample of free-floating or
bound, cool and low-mass exoplanets and measure the mass functions of these
planets, which differs somewhat from searching exclusively for exoplanets in
the HZs. We emphasize that the microlensing method can detect planetary
systems at farther distances with respect to the observer (even in the
Galactic bulge) than those detectable through transit or radial velocity
measurements.

In this paper, we first revisit the subject of probing habitable planets
through microlensing observations. Then, we will do a Monte-Carlo simulation
of planetary microlensing events to evaluate the Roman efficiency for
detecting habitable planets and probe the special microlensing configurations
with the maximum Roman sensitivity to habitable planets. Finally, we calculate
how many habitable planets could be discovered during the mission.

The paper is organized as follows. In section 2, we explain some points for
detecting habitable planets through microlensing observations. In section 3,
we study the properties and statistics of the realizable microlensing events
owing to habitable planetary systems by Roman. In the two next sections, 4 and
5, we discuss on and summarize the results.

Figure 1: The scatter plot of confirmed exoplanets through microlensing
observations up to now. The data was taken from the NASA Exoplanet Archive.
The blue stars represent the events with non-degenerate models. For the other
events, shown with magenta stars, there are two (or more) degenerate models
which are connected with gray lines. The solid and dashed black lines
represent the boundaries of the projected OHZs and CHZs, respectively. The
taken projection factor is $f_{\mathcal{P}}=0.79$, as given by Eq. 10.

##  2 Detecting habitable planetary systems by microlensing

The subject of ”microlensing abilities for detecting habitable planets” has
been studied in several references. Di Stefano (1999) has noticed that
habitable Earth-mass planets around Sun-like stars make most often resonant
caustic configurations with the highest sensitivity. In this reference, the
author has studied the advantages and disadvantages of the blending effect of
these bright host stars. The sensitivity to habitable Earth-mass planets
through microlensing observations has also been studied by Park et al. (2006)
and they concluded that roughly $3\%$ of Earth-mass planets detectable in
microlensing observations are inside the HZs. We note that they have used an
estimation of the caustic size as the criterion for detecting planetary
signatures. This choice caused an overestimation for detectable habitable
planets. In this section, I theoretically revisit the mentioned subject.

Figure 2: In these plots, the OHZ and CHZ boundaries projected on the sky
plane, $s_{\mathcal{P},\leavevmode\nobreak\ \rm{HZ}}(\rm{AU})$, versus the
host mass are represented with solid and dashed blue lines, respectively. By
assuming Earth-mass (left panel) and Jupiter-mass (right panel) planets
orbiting each host star as microlens systems, their typical Einstein radii
(with $D_{\rm l}=6.5$ kpc, $D_{\rm s}=8$ kpc) are shown with the magenta
dashed line. The edges of projected OHZs normalized to the Einstein radius,
$d_{\rm{HZ}}$, (specified on the second $y$-axis) are plotted with solid black
lines. Accordingly, the habitable planetary systems inside the green-colored
(lime-colored) parts make resonant (close-to-resonant) caustic topologies. To
better providing, the bifurcation values between close, resonant and wide
topologies, $d_{\rm c}$ and $d_{\rm w}$ (given by Eq. 4), are displayed with
dotted and dot-dashed dark-green lines. Their corresponding values due to
close-to-resonant topologies (Yee et al., 2021) are shown with lime color.

Up to now, more than $100$ exoplanets have been confirmed by surveys and
follow-up microlensing observations, as they have been archived in the
Extrasolar Planets Encyclopedia 222http://exoplanet.eu/. We take the
parameters of these planets from the NASA Exoplanet Archive
333https://exoplanetarchive.ipac.caltech.edu/. In Figure 1 the distribution of
these exoplanets is shown. For some of these microlensing events, two or more
best-fitted and degenerate models have been reported, which are shown by
magenta stars in this plot. For each microlensing event, its degenerate models
are connected with a gray line. Other discovered planetary microlensing events
with non-degenerate confirmed models are displayed with blue stars. In this
plot, the axes are the planet-host distance projected onto the sky plane
$s_{\mathcal{P}}(\rm{AU})$ (the index $\mathcal{P}$ refers to the projection
on the sky plane) and the mass of the host star $M_{\rm h}$.

As explained in the previous section, the inner and outer radii of the CHZ and
OHZ around a Sun-like star are $s_{\rm{HZ,\leavevmode\nobreak\
i,\leavevmode\nobreak\ \odot}}=0.95,\leavevmode\nobreak\ 0.5$ AU and
$s_{\rm{HZ,\leavevmode\nobreak\ o,\leavevmode\nobreak\
\odot}}=1.67,\leavevmode\nobreak\ 2.0$ AU, respectively (e.g., Kopparapu et
al., 2013). Thereupon, for a host star with the luminosity $L_{\rm
h}(L_{\odot})$ the inner and outer edges of its habitable zone are given by:

$\displaystyle s_{\rm{HZ},\leavevmode\nobreak\ i}(\rm{AU})$ $\displaystyle=$
$\displaystyle s_{\rm{HZ},\leavevmode\nobreak\ i,\leavevmode\nobreak\
\odot}\sqrt{L_{\rm h}(L_{\odot})},$ $\displaystyle
s_{\rm{HZ},\leavevmode\nobreak\ o}(\rm{AU})$ $\displaystyle=$ $\displaystyle
s_{\rm{HZ},\leavevmode\nobreak\ o,\leavevmode\nobreak\ \odot}\sqrt{L_{\rm
h}(L_{\odot})}.$ (2)

In Figure 1, we show the projected CHZ and OHZ edges with dashed and solid
black lines. For the projection on the sky plane, we multiply the edges by the
projection factor, i.e., $s_{\mathcal{P},\leavevmode\nobreak\
\rm{HZ}}=f_{\mathcal{P}}\leavevmode\nobreak\ s_{\rm{HZ}}$, and
$f_{\mathcal{P}}=0.79$, as given by Eq. 10.

In this figure, we use the known mass-luminosity relation for main-sequence
stars (e.g., Kuiper, 1938) to determine the HZ edges for a given host mass.
This relation is (Salaris & Cassisi, 2005; Duric, 2003):

$\displaystyle L_{\rm h}(L_{\odot})=\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\begin{cases}0.23\left[M_{\rm h}(M_{\odot})\right]^{2.3}&M_{\rm
h}(M_{\odot})\leq 0.43,\\\ \left[M_{\rm h}(M_{\odot})\right]^{4}&0.43<M_{\rm
h}(M_{\odot})\leq 2,\\\ 1.4\left[M_{\rm h}(M_{\odot})\right]^{3.5}&2<M_{\rm
h}(M_{\odot})\leq 55,\\\ 32000M_{\rm h}(M_{\odot})&M_{\rm
h}(M_{\odot})>55.\end{cases}$ (3)

The only habitable planet discovered in microlensing observations, i.e.,
MOA-2011-BLG-293Lb, is located over the green area in the plot. This planetary
system contains a super-Jupiter orbiting an M dwarf with the mass $M_{\rm
h}=0.86\leavevmode\nobreak\ M_{\odot}$ at the distance $7.7$ kpc from us,
close to the Galactic bulge (Batista et al., 2014). The reason for such a
small number of habitable planets detected by gravitational microlensing is
that its sensitivity is maximized for planets beyond the snow lines (out of
the HZs). These planetary systems make intermediate (resonant) or close-to-
intermediate (close-to-resonant) caustic curves (Yee et al., 2021).

In Figure 2, we show the OHZ and CHZ boundaries projected on the sky plane,
$s_{\mathcal{P},\leavevmode\nobreak\ \rm{HZ}}(\rm{AU})$, versus the mass of
the host star with solid and dashed blue lines, respectively. By assuming
Earth-mass (left panel) and Jupiter-mass (right panel) planets orbiting host
stars as microlens systems, we calculate typical values of their Einstein
radius (with the lens and source distances from the observer $D_{\rm l}=6.5$
kpc, $D_{\rm s}=8$ kpc, respectively), as shown with the magenta dashed lines.
The edges of projected OHZs normalized to the Einstein radius,
$d_{\rm{HZ}}=s_{\mathcal{P},\leavevmode\nobreak\ \rm{HZ}}/R_{\rm E}$,
(specified on the second $y$-axis) are plotted with solid black lines.

These habitable planetary systems make close, intermediate or wide caustic
topologies. We determine these topologies using two bifurcation values of the
projected planet-host distance normalized to the Einstein radius, (i) $d_{\rm
c}$, between close and intermediate caustics, and (ii) $d_{\rm w}$ which
determines the transition between intermediate and wide caustic topologies.
These two bifurcation values depend only on the planet-host mass ratio $q$ as
follows (Erdl & Schneider, 1993; Dominik, 1999; Cassan, 2008; Tsapras, 2018):

$\displaystyle d_{\rm c}^{8}$ $\displaystyle=$
$\displaystyle\frac{\big{(}1+q)^{2}}{27q}\Big{(}1-d_{\rm c}^{4}\Big{)}^{3},$
$\displaystyle d_{\rm w}^{2}$ $\displaystyle=$
$\displaystyle\frac{\Big{(}1+q^{1/3}\Big{)}^{3}}{1+q}.$ (4)

In Figure 2, we plot the bifurcation values, $d_{\rm c}$ and $d_{\rm w}$, with
dotted and dot-dashed dark-green lines. Recently, Yee et al. (2021) have shown
that majority of discovered planetary microlensing events are found inside a
region between $d_{\rm c}^{3}$ and $d_{\rm w}^{1.8}$, the so-called close-to-
resonant (or near-resonant) zone. These two new edges (due to near-resonant
region) are shown with lime color. Accordingly, habitable planetary systems
located in the green-colored (lime-colored) parts of these plots make resonant
(close-to-resonant) caustic topologies. For these planetary systems in
resonant (green-colored) part, some values of $d_{\rm HZ}$ (between two solid
black lines) are inside the range $[d_{\rm c},\leavevmode\nobreak\ d_{\rm
w}]$.

In this figure we use a typical value for the lens distance $D_{\rm l}=6.5$
kpc, but the lens object can be very close to the observe or very close to the
source star. Hence, we change the lens distance in the range $D_{\rm
l}\in[0.08,\leavevmode\nobreak\ 7.92]$ kpc and study the properties of
detectable habitable planetary systems in different distances from the
observer through microlensing observations. Accordingly, we list some key
points in the following.

  * •

Habitable planets around host stars with $M_{\rm h}\lesssim 0.5M_{\odot}$,
generate close caustic topologies, for all values of the lens distance in the
range $D_{\rm l}\in[0.08,\leavevmode\nobreak\ 7.92]$ kpc. However, the size of
these planetary caustics depends on the mass ratio $q$. For Earth and Jupiter-
mass planets around main-sequence stars, the resulting mass ratios are
$\log_{10}[q]\lesssim-5.0,\leavevmode\nobreak\ -2.5$, respectively (as given
on the second $x$ axes of Figure 2). We note that Jupiter-mass planets can
have habitable moons orbiting them.

  * •

When $s_{\mathcal{P},\leavevmode\nobreak\ \rm{HZ}}\simeq R_{\rm E}$, the
planetary system makes resonant caustic curves, as specified with green-
colored zone for the lens distance $D_{\rm l}=6.5$ kpc in Figure 2. Generally,
these configurations happen for the host stars with the mass of $M_{\rm
h}\in[0.64,\leavevmode\nobreak\ 5.62]M_{\odot}$ and $M_{\rm
h}\in[0.61,\leavevmode\nobreak\ 6.03]M_{\odot}$ for Earth- and Jupiter-mass
planets, respectively (by assuming $D_{\rm l}\in[0.08,\leavevmode\nobreak\
7.92]$ kpc). We note that these intermediate caustic topologies happen for
some special values of $s_{\mathcal{P},\leavevmode\nobreak\ \rm{HZ}}$ and
$D_{\rm l}$, but not for all values. For instance, a Jupiter-mass planet in
the HZ of the host star with $M_{\rm h}=0.6\leavevmode\nobreak\ M_{\odot}$
should be at the distances $D_{\rm l}\simeq 7.92$ kpc to produce intermediate
caustic curves.

  * •

When the lens system is very close to or very far from the observer, i.e.,
$x<0.1$ or $x>0.9$, where $x=D_{\rm l}/D_{\rm s}$, and for host stars with
$M_{\rm h}\gtrsim 0.6M_{\odot}$ (orbiting by Earth or Jupiter-mass planets in
the HZs) the Einstein radius decreases and the intermediate caustic topologies
form. However, for very close lens systems with $x<0.1$, the blending effect
of the bright host star increases. Hence, very far lens systems with $x>0.9$
are more detectable and they occur more frequently. We note that the habitable
planetary system discovered by the MOA microlensing observations, i.e.,
MOA-2011-BLG-293Lb, is at a distance $D_{\rm l}=7.7$ kpc from us with $x\simeq
0.96$.

  * •

Although the more massive host stars have larger Einstein radii, the HZs
around them are at larger distances as well. According to the mass-luminosity
relation, given by Equation 3, the projected planet-host distance in the HZs
normalized to the Einstein radius, $d_{\rm{HZ}}$, is proportional to:

$\displaystyle d_{\rm{HZ}}=\frac{s_{\mathcal{P},\leavevmode\nobreak\
\rm{HZ}}}{R_{\rm E}}\propto\begin{cases}\left[M_{\rm
h}(M_{\odot})\right]^{0.65},&M_{\rm h}(M_{\odot})\leq 0.43,\\\ \left[M_{\rm
h}(M_{\odot})\right]^{1.5},&0.43<M_{\rm h}(M_{\odot})\leq 2.\\\ \end{cases}$
(5)

Hence, more massive host stars have larger normalized lens distances and, as a
result, bigger caustic curves.

  * •

The size of the Einstein radius for the events toward the Galactic bulge is
$R_{\rm{E}}\simeq 2.2$ AU, (as specified over the $y$-axis of the plots) and
primarily larger than the HZ location. One way to decrease the Einstein radius
is microlensing observations toward the Galactic disc. In the Galactic disc,
the source distance is on average less than $8$ kpc (the Galactic center
distance), resulting in a smaller Einstein radius (e.g., Moniez et al., 2017).
We expect higher efficiency for detecting planets within the HZs towards the
Galactic disc than that for the Galactic bulge. In addition, towards the
Galactic disc, the lens-source relative velocity is small and the Einstein
crossing times are longer (e.g., see Fig. (4) of Sajadian & Poleski, 2019).

In order to evaluate the Roman ability to discern habitable planets in its
microlensing observations toward the Galactic bulge, we perform a Monte-Carlo
simulation by taking into account all relevant factors, e.g., the blending
effect of the lens, observing cadence, seasonal gap, photometric accuracy,
etc. Based on this simulation, we access the likelihood of discerning
habitable planets using the next generation of microlensing observation by the
Roman telescope and study the properties of these detectable events. This
simulation and its results are explained in the following section.

Figure 3: Two simulated microlensing lightcurves (solid green curves) from
habitable planetary systems detectable by the Roman telescope. The solid black
curves are the best-fitted single lightcurves. Magenta points display the
simulated synthetic data points taken by the Roman telescope. The source
trajectories projected on the lens plane (cyan lines) and caustic curves
(black curves) are shown in the right-hand insets. The parameters of these
lightcurves are given at the top of the plots.

##  3 Detecting the HZ planets in the Roman mission

The Roman telescope will potentially detect many microlensing event toward the
Galactic bulge during its mission (Penny et al., 2019; Johnson et al., 2020).
In order to accurately evaluate the Roman ability for discovering planets in
the HZs, a comprehensive Monte-Carlo simulation of such events is necessary,
which will be done in this section. Many of the planetary microlensing
lightcurves are created first. Then, we simulate synthetic observing data
points for the generated lightcurves and finally extract the events with the
detectable planetary signatures. These steps in the simulation are explained
in the following.

The simulation is done toward the subfields which will be observed by Roman.
The centers of these subfields, as shown in Fig. (7) of Penny et al. (2019),
are $(l^{\circ},\leavevmode\nobreak\ b^{\circ})=(1.3,\leavevmode\nobreak\
-0.89)$, $(0.9,\leavevmode\nobreak\ -0.89)$, $(1.3,\leavevmode\nobreak\
-1.63)$, $(0.9,\leavevmode\nobreak\ -1.63)$, $(0.5,\leavevmode\nobreak\
-1.63)$, $(0.1,\leavevmode\nobreak\ -1.63)$, and $(-0.3,\leavevmode\nobreak\
-1.63)$. The size of each sub-field is $\simeq 0.75\times
0.4\leavevmode\nobreak\ \rm{deg}^{2}$. Towards each given direction, the
source distance, $D_{\rm s}$, is determined from the cumulative mass density
versus distance from the observer in our galaxy:

$\displaystyle\frac{d^{2}M}{d\Omega\leavevmode\nobreak\ dD_{\rm s}}=D_{\rm
s}^{2}\leavevmode\nobreak\ \sum_{i=1}^{4}\rho_{i}(D_{\rm
s},\leavevmode\nobreak\ l,\leavevmode\nobreak\ b),$ (6)

here, the summation is done over all structures, i.e., the Galactic bulge,
thin and thick discs, and the stellar halo. The mass densities, $\rho_{i}$,
are given by the Besançon model 444https://model.obs-besancon.fr/ (Robin et
al., 2003, 2012). For each source star, we determine its absolute magnitude,
mass, radius from different stellar populations of our galaxy as given by this
model (see, e.g., Sajadian & Rahvar, 2010; Sajadian, 2015a). The source radius
determines $\rho_{\ast}$ which is the projected source radius on the lens
plane and normalized to the Einstein radius. $\rho_{\ast}$ significantly
changes the magnification factor in high-magnification or caustic-crossing
microlensing events (Witt & Mao, 1994; Nemiroff & Wickramasinghe, 1994).

In the next step, the lens objects are simulated. The lens location is chosen
from the range $[10\rm{pc},\leavevmode\nobreak\ D_{\rm s}]$, according to the
microlensing event rate:

$\displaystyle\Gamma(D_{\rm l})\propto\sqrt{\frac{D_{\rm l}(D_{\rm s}-D_{\rm
l})}{D_{\rm s}}}\leavevmode\nobreak\ \sum_{i=1}^{4}\rho_{i}(D_{\rm
l},\leavevmode\nobreak\ l,\leavevmode\nobreak\ b),$ (7)

where, the first factor is related to the Einstein radius. In this
relationship and when characterizing the lens distance, we ignore variations
in the lens-source relative velocity $v_{\rm{rel}}$ with respect to the lens
distance. $v_{\rm{rel}}$ is the difference of the lens and source velocities
as measured by the observer (in the observer coordinate system), while both
velocities are projected onto the sky plane (see, e.g., Eq. (2) of Sajadian,
2021). This relative velocity determines the Einstein crossing time, $t_{\rm
E}=R_{\rm E}/v_{\rm{rel}}$, which specifies the timescale of microlensing
events. Other parameters of the lens object, i.e., their photometric
properties, mass, radius, effective temperature are taken from different
stellar populations in the Besançon model. We explain how the planets are
characterized in subsections 3.1 and 3.2.

The lens impact parameter is chosen uniformly from the range
$u_{0}\in[0,\leavevmode\nobreak\ 1]$. The angle between the source trajectory
and the binary axis is taken from the range $\xi\in[0,\leavevmode\nobreak\
360]$ deg. Total observing time of the Roman mission is $T_{\rm{obs}}=5$ yrs
which includes $6$ $72$-day seasons. Three observing seasons will occur at the
start of the mission with $110$-day seasonal gaps and three other seasons will
occur at the end of the mission. We randomly take the time of the closest
approach of simulated events, $t_{0}$, in the range of
$[0,\leavevmode\nobreak\ T_{\rm{obs}}]$. To calculate the magnification factor
in binary and planetary microlensing events we use the $\rm{RT}$-model
555http://www.fisica.unisa.it/gravitationastrophysics/RTModel.htm which was
well developed by V. Bozza (Bozza et al., 2018; Bozza, 2010; Skowron & Gould,
2012).

Table 1: The Roman efficiencies for detecting planets within the HZs with different mass values and properties of these detectable events. $M_{\rm p}$ | $\epsilon_{\rm{OHZ,\leavevmode\nobreak\ LS}}$ | $\epsilon_{\rm{OHZ,\leavevmode\nobreak\ HS}}$ | $\epsilon_{\rm{CHZ,\leavevmode\nobreak\ LS}}$ | $\epsilon_{\rm{CHZ,\leavevmode\nobreak\ HS}}$ | $\left<D_{\rm l}\right>$ | $\left<M_{\rm h}\right>$ | $\left<f_{\rm l}\right>$ | $\left<m_{W149}\right>$ | $\left<t_{\rm E}\right>$ | $\left<d\right>$ | $f_{\rm c}:f_{\rm i}:f_{\rm w}$ | $f_{\rm{si}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ | $\rm{(kpc)}$ | $\rm{(M_{\odot})}$ |  | $\rm{(mag)}$ | $\rm{(days)}$ |  | $[\%]$ | $[\%]$
$M_{\oplus}$ | $0.02$ | $0.01$ | $0.01$ | $0.01$ | $7.5$ | $0.83$ | $0.35$ | $18.0$ | $21.2$ | $0.78$ |  $85$ : $1$ : $14$ | $5$
$10\leavevmode\nobreak\ M_{\oplus}$ | $0.14$ | $0.09$ | $0.13$ | $0.09$ | $7.2$ | $0.58$ | $0.26$ | $18.8$ | $18.3$ | $0.63$ |  $84$ : $4$ : $12$ | $11$
$100\leavevmode\nobreak\ M_{\oplus}$ | $1.75$ | $1.5$ | $1.86$ | $1.60$ | $6.8$ | $0.31$ | $0.14$ | $19.2$ | $16.7$ | $0.28$ |  $96$ : $2$ : $2$ | $4$
$M_{\rm J}$ | $5.20$ | $5.01$ | $5.47$ | $5.29$ | $6.7$ | $0.28$ | $0.12$ | $19.1$ | $17.7$ | $0.22$ |  $97$ : $1$ : $2$ | $3$
$5\leavevmode\nobreak\ M_{\rm J}$ | $17.03$ | $16.98$ | $17.94$ | $17.89$ | $6.5$ | $0.28$ | $0.11$ | $19.0$ | $19.0$ | $0.17$ |  $98$ : $1$ : $1$ | $2$
$10\leavevmode\nobreak\ M_{\rm J}$ | $25.43$ | $25.41$ | $26.79$ | $26.77$ | $6.5$ | $0.29$ | $0.10$ | $19.0$ | $19.7$ | $0.15$ |  $98$ : $1$ : $1$ | $2$

During a microlensing event, the angular Einstein radius, $\theta_{\rm E}$, is
much smaller than the angular size of the FWHM (Full width at Half Maximum) of
the target PSF (Point Spread Function), $\theta_{\rm PSF}$. In the Roman
microlensing survey, $\theta_{\rm PSF}\simeq 0.33$ arcs, i.e., around three
times of the Roman pixel size which is $0.11$ arcs (Penny et al., 2019).
Hence, the lens flux is always blended with the source flux and we consider
this effect in the simulation. In order to evaluate the blending effect owing
to the lens flux, we define the lens blending factor as:

$\displaystyle f_{\rm l}=\frac{F_{\rm l}}{F_{\ast}+\sum_{i}F_{i}},$ (8)

which is the ratio of the lens flux $F_{\rm l}$ to the total flux due to the
source star itself $F_{\ast}$ and other blending stars (which their angular
distances from the source star are smaller than the radius of the source PSF)
at the baseline. The greater the $f_{\rm l}$, the greater the contamination
caused by the lens brightness.

Using the specified parameters, e.g., $\rho_{\ast},\leavevmode\nobreak\ t_{\rm
E},\leavevmode\nobreak\ u_{0},t_{0},...$, we make the microlensing
lightcurves. For each simulated lightcurve, we generate synthetic data points
taken during the Roman observing program, as described in Bagheri et al.
(2019); Sajadian & Salehi (2020). The cadence of data is fixed at $15.16$ min.
The photometric error bars of the synthetic data points, $\sigma_{\rm m}$, are
specified based on the magnified apparent magnitudes of the targets, as shown
in Figure (4) of Penny et al. (2019).

We first extract the detectable microlensing events from simulated events
using three criteria which have been explained in Sajadian & Poleski (2019).
Then, we evaluate the detectability of the planetary signatures in extracted
events. In order to realize the planetary signatures, we exert two extra
criteria. Because the planet-host mass ratio, $q=M_{\rm p}/M_{\rm h}$, is
small, the resulting planetary signatures from caustic-crossing features (or
passing close to caustic curves) are small and short. For these events, the
best-fitted single models are the real event by considering only primary
lenses (ignoring their planets), which are located at $(-d\leavevmode\nobreak\
q/(1+q),\leavevmode\nobreak\ 0)$. Here, $d=s_{\mathcal{P}}/R_{\rm E}$ is the
planet-host distance projected on the sky plane and normalized to the Einstein
radius. Our criteria are as follows. (i) There are at least three consecutive
data points that are $4\sigma_{\rm m}$ above (or below) the best-fitted single
models. (ii) $\Delta\chi^{2}_{\rm p}>300,\leavevmode\nobreak\ 800$,
corresponding to low and high sensitivities to planetary signatures (LS and
HS), where $\Delta\chi^{2}_{\rm
p}=\chi^{2}_{\rm{real}}-\chi^{2}_{\rm{single}}$, the difference between
$\chi^{2}$ values obtained by fitting the real models and the best-fitted
single-lens models. Here, we assume that the best-fitted planetary models are
real models. We note that the threshold of $500$ for $\Delta\chi^{2}_{\rm p}$
is often adjudged as a robust criterion for detecting planetary signatures in
microlensing events (see, e.g., Gould et al., 2010; Udalski et al., 2018; Ryu
et al., 2018; Hirao et al., 2020). The first criterion ejects symmetric events
that may be misinterpreted as single microlensing events, unless the deviated
data points exactly occur at the time of the closest approach, which is rare.
Our aim is to evaluate the detectability of planetary signatures and we do not
pay attention to any other possible degenerate solutions.

We aim to respond two main questions from this simulation. (1) If there is an
exoplanet with a given mass in the HZ, what is the Roman efficiency for
detecting this planet? Furthermore, for what lensing configurations is this
efficiency maximized? (2) During the Roman mission, how many habitable planets
are detectable? On average what properties do these detectable planets within
the HZs have? These two issues are referred to in the following subsections.

###  3.1 Detection efficiency

Here, we study the Roman efficiency for detecting HZ planets during its
mission by doing a Monte-Carlo simulation. For each simulated microlensing
event, we assume that one exoplanet is located in the OHZ of the lens object
and choose the semi-major axis of its orbit uniformly from the range of
$s\in[s_{\rm{OHZ},\leavevmode\nobreak\ i},\leavevmode\nobreak\
s_{\rm{OHZ},\leavevmode\nobreak\ o}]$. We assume the planet’s orbit around its
host star is circular and disregard the orbital motion effect of microlens
systems (see, e.g., Sajadian, 2014). Hence, for projecting the planetary orbit
onto the sky plane, we need one inclination angle, $i$, as (see, e.g., Gould &
Loeb, 1992; Batista et al., 2015):

$\displaystyle s_{\mathcal{P}}=s\leavevmode\nobreak\ \sin(i).$ (9)

In fact, if we assume the host star at the center of a sphere, its orbiting
planet can be at any random position over the surface of that sphere. Its
location is specified with two angles in the spherical coordinate system which
are the azimuthal and polar angles. The second one is the inclination angle of
the planet orbit with respect to the line of sight toward the observer (polar
axis). To uniformly choose initial position of the planet over the sphere
surface, $\cos(i)$ is chosen uniformly from a range of
$[0,\leavevmode\nobreak\ 1]$ in the simulation. Therefore, on average the
ratio of the projected planet-host distance to their real distance,
$f_{\mathcal{P}}=\big{<}s_{\mathcal{P}}/s\big{>}$, is

$\displaystyle
f_{\mathcal{P}}=\left<\sin(i)\right>=\int_{0}^{1}\sqrt{1-x^{2}}\leavevmode\nobreak\
dx=\frac{\pi}{4},$ (10)

where $x=\cos(i)$.

Two example simulated lightcurves are represented in Figure 3. The planets are
located in the HZs of their host stars. In these plots, the best-fitted and
single-lens (solid black curves) and the real (solid green) lightcurves are
displayed. The planetary signatures, as zoomed in the left-hand insets, are
covered by the synthetic data points (magenta points) and are detectable. In
the left and right panels, the planetary systems make close and wide caustic
topologies, respectively. The source trajectories projected on the lens plane
in these events (cyan lines) pass close to and over the planetary caustic
curves. In the second lightcurve, although the blending effect due to the host
lens is high, the planetary signature is detectable during its caustic-
crossing features.

We perform the Monte-Carlo simulation toward $(l,\leavevmode\nobreak\
b)=(1.0^{\circ},\leavevmode\nobreak\ -1.5^{\circ})$ for planet masses of
$M_{\rm p}=M_{\oplus},\leavevmode\nobreak\ 10\leavevmode\nobreak\
M_{\oplus},\leavevmode\nobreak\ 100\leavevmode\nobreak\
M_{\oplus},\leavevmode\nobreak\ M_{\rm{J}},\leavevmode\nobreak\
5\leavevmode\nobreak\ M_{\rm J},\leavevmode\nobreak\ 10\leavevmode\nobreak\
M_{\rm{J}}$, i.e., Earth-mass, Super-Earth, Saturn-mass, Jupiter-mass and
Super-Jupiter planets (the index $\rm{J}$ refers to Jupiter). Accordingly, the
inputs of the simulation are HZ planetary systems acting as microlenses and
outputs are the ones that make the planetary microlensing events detectable by
Roman. In Table 1, we report the detection efficiency values, $\epsilon$, and
the average parameters of output events for the different masses of planets as
specified in different rows. The indexes LS and HS for the efficiencies refer
to different criteria for detecting planetary signatures (i.e., the thresholds
$300$ and $800$ for $\Delta\chi^{2}_{\rm p}$). In this table,
$m_{W149}=-2.5\log_{10}\left[F_{\ast}\leavevmode\nobreak\
+\leavevmode\nobreak\ F_{\rm l}\leavevmode\nobreak\ +\leavevmode\nobreak\
\sum_{i}F_{i}\right]$ is the apparent magnitude of the targets at the
baseline. $f_{\rm c}$, $f_{\rm i}$, $f_{\rm w}$ and $f_{\rm{si}}$ are
percentages of detectable planetary systems that make close, resonant, wide
and close-to-resonant (with $d_{c}^{3}<d<d_{\rm w}^{1.8}$) caustic topologies,
respectively.

Figure 4: The Roman efficiencies for detecting planets in the OHZs with LS
criteria ($\epsilon_{\rm{OHZ},\leavevmode\nobreak\ \rm{LS}}$) in terms of the
lens distances from the observer, the mass of the host lenses, the lens
blending effect, and the target apparent magnitude at the baseline are shown
in different panels. Each panel contains six efficiency curves which are
corresponding to planets with masses $10\leavevmode\nobreak\ M_{\rm J}$
(magenta), $5\leavevmode\nobreak\ M_{\rm J}$ (yellow) $M_{\rm J}$ (green),
$100\leavevmode\nobreak\ M_{\oplus}$ (cyan) and $10\leavevmode\nobreak\
M_{\oplus}$ (red) and $M_{\oplus}$ (blue).

Additionally, in Figure 4 we show the dependence of the LS efficiencies for
detecting planets in the OHZs, $\epsilon_{\rm{OHZ},\leavevmode\nobreak\
\rm{LS}}$, on the distance of the planetary systems from the observer $D_{\rm
l}$, the mass of the primary lenses $M_{\rm h}$, the lens blending effect
$f_{\rm l}$, and the target apparent magnitude at the baseline $m_{W149}$,
represented in different panels. In each panel, six given curves are related
to different planet masses, namely, $10\leavevmode\nobreak\ M_{\rm J}$
(magenta), $5\leavevmode\nobreak\ M_{\rm J}$ (yellow), $M_{\rm J}$ (green),
$100\leavevmode\nobreak\ M_{\oplus}$ (cyan), $10\leavevmode\nobreak\
M_{\oplus}$ (red), and $M_{\oplus}$ (blue). We list some key points here.

Figure 5: Map of the LS Roman efficiency for detecting Jupiter-mass planets
around host stars with masses $M_{\rm h}$ at $D_{\rm l}$ from the observer.

  * •

Generally, the Roman detection efficiency increases by enhancing the mass of
planets. Their relation is almost linear for planets with the mass $M_{\rm
p}\lesssim M_{\rm J}$. By increasing the planet mass by two orders of
magnitude, from $M_{\oplus}$ to $100\leavevmode\nobreak\ M_{\oplus}$, the
efficiency increases by two orders of magnitude ($0.01\%$ to $1\%$) as well.
But, it tends toward a constant value for the heavier planets.

  * •

Only $\sim 0.01\%$ and $0.1\%$ of Earth and super Earth-mass planets in
habitable zones are detectable through the Roman microlensing observations.
The properties of these detectable planetary events are: (i) Habitable Earth-
mass planets are detectable when they rotate around G and K-type stars located
at a distance beyond $7$ kpc from the observer (close to the Galactic bulge).
(ii) The brightness of their primary lenses is considerable. (iii) These
planetary systems produce rather close or wide caustic topologies, owing to
small values of $q$.

  * •

$\sim 5-25\%$ of Jupiter and super Jupiter-mass planets within habitable zones
are detectable by Roman. In these detectable events, the primary lens objects
are most often M dwarfs (as usual), resulting less blending effect from the
brightness of host objects. These planetary systems make mainly close caustic
topologies. Jupiter-mass planets can have habitable moons.

  * •

As mentioned in the previous section, the detection efficiency depends on
$D_{\rm l}$ and it maximizes for the lens distances offering smaller Einstein
radii. Generally, the Roman efficiency for detecting HZ planets is maximized
for three classes of planetary microlensing events with close caustic
topologies, which are pointed out here.

(a) The events with massive primary lenses, $M_{\rm h}\gtrsim 0.6M_{\odot}$
while their mirolens systems are very far, $D_{\rm l}\gtrsim 7$ kpc. In these
events and for Jupiter-mass planets within the HZs, the mass ratio is very
small $q\lesssim 0.001$, but $d\gtrsim 0.17$. The parallax amplitude is very
small, $\pi_{\rm{rel}}\lesssim 0.02$ mas. Here,
$\pi_{\rm{rel}}=\rm{AU}(1/D_{\rm l}-1/D_{\rm s})$. These planetary systems
have orbital periods as long as $P\gtrsim 100$ days. Therefore, the orbital
motion effect due to lenses is likely small.

The events with low-mass primary lenses, $M_{\rm h}\lesssim 0.1M_{\odot}$
while their microlens systems are either (b) very close to the observer
$D_{\rm l}\lesssim 1$ kpc or (c) very close to the Galactic bulge $D_{\rm
l}\gtrsim 7$ kpc. By assuming a Jupiter-mass planet in the HZ of the primary
lenses, these events (in both classes) have $q\gtrsim 0.01$ and $d\lesssim
0.04$. For class (b) of microlensing events, the parallax effect is
significant with $\pi_{\rm{rel}}\gtrsim 0.87$ mas. These planetary systems
have the short orbital periods with $P\lesssim 7$ days, which can potentially
make considerable orbital motion effects on caustic-crossing features.

For that reason, we have two peaks in the two first panels of Figures 4. In
Figure 5 we depict the map of the Roman efficiency for detecting Jupiter-mass
planets around primary lenses (to better display this point). Three local
maxima in this parameter space $M_{\rm h}$-$D_{\rm l}$ are manifest.

We note that the events in the classes (a) and (c) occur more frequently. The
horizontal sizes of central and planetary caustics (over the binary axis) in
close caustic topologies are proportional to $\Delta_{\rm c}\propto
4q/(d-1/d)^{2}$ and $\Delta_{\rm p}\propto\sqrt{q}\leavevmode\nobreak\ d^{3}$
(Bozza, 2000; An, 2005; Chung et al., 2005). Hence, central and planetary
caustics of the class (a) events are larger (by one and two orders of
magnitude, respectively).

  * •

The maximum detection efficiency occurs when the targets are as bright as
$m_{W149}\sim 16$-$17$ mag (at the baseline). A significant part of this
apparent brightness of the targets at the baseline is owing to the brightness
of the primary lenses.

Table 2: The properties and statistics of planetary microlensing events within the OHZs and CHZs which are detected by Roman with the LS and HS criteria. | $\left<M_{\rm p}\right>$ | $\left<D_{\rm l}\right>$ | $\left<M_{\rm h}\right>$ | $\left<f_{\rm l}\right>$ | $\left<m_{W149}\right>$ | $\left<t_{\rm E}\right>$ | $\left<d\right>$ | $\left<R_{\rm E}\right>$ | $f_{\rm c}:f_{\rm i}:f_{\rm w}$ | $f_{\rm{si}}$ | $\epsilon_{\rm p}$ | $\epsilon_{\rm{HZ}}$ | $N_{\rm HZ}$ | $\epsilon_{\rm{HZ},\leavevmode\nobreak\ \mathcal{P}}$ | $N_{\rm HZ,\leavevmode\nobreak\ \mathcal{P}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $M_{\rm J}$ | $\rm{(kpc)}$ | $\rm{(M_{\odot})}$ |  | $\rm{(mag)}$ | $\rm{(days)}$ |  | $\rm{(AU)}$ | $[\%]$ | $[\%]$ | $[\%]$ | $[\%]$ |  | $[\%]$ |
$\rm{OHZ}\leavevmode\nobreak\ \rm{planetay}\leavevmode\nobreak\
\rm{Microlensing}$
$\rm{LS}$ | $2.71$ | $6.89$ | $0.51$ | $0.24$ | $19.5$ | $23.1$ | $0.29$ | $1.63$ |  $96$ : $3$ : $1$ | $5.1$ | $3.00$ | $3.35$ | $27$ | $4.29$ | $35$
$\rm{HS}$ | $2.76$ | $6.90$ | $0.53$ | $0.25$ | $19.3$ | $24.7$ | $0.30$ | $1.70$ |  $95$ : $4$ : $1$ | $6.0$ | $2.79$ | $3.13$ | $24$ | $4.07$ | $31$
$\rm{CHZ}\leavevmode\nobreak\ \rm{planetay}\leavevmode\nobreak\
\rm{Microlensing}$
$\rm{LS}$ | $2.57$ | $6.63$ | $0.43$ | $0.22$ | $19.5$ | $18.4$ | $0.31$ | $1.43$ |  $95$ : $3$ : $2$ | $7.2$ | $3.00$ | $1.79$ | $15$ | $2.76$ | $22$
$\rm{HS}$ | $2.60$ | $6.69$ | $0.44$ | $0.23$ | $19.4$ | $19.2$ | $0.33$ | $1.47$ |  $94$ : $4$ : $2$ | $8.4$ | $2.79$ | $1.65$ | $12$ | $2.62$ | $20$

###  3.2 Statistical Calculations

In order to estimate the number of habitable planets that could potentially be
detected during the Roman mission, we will do a more realistic simulation of
all possible planetary systems detectable by this telescope and extract the
events with planets in the HZs.

For doing this realistic simulation, we choose semi-major axis of the
planetary orbit, $s$, from the Öpik’s law (Öpik, 1924), i.e.,

$\displaystyle dN/ds\propto 1/s,$ (11)

and from the range $s\in[0.01,\leavevmode\nobreak\ 100]$ AU. To extract the
lens mass ratio, we use the mass ratio function which has been resulted from
microlensing observations. One of the advantages of microlensing observations
is studying the abundance of bound planets in planetary systems. For instance,
Cassan et al. (2012) have found that $17\%$ of stars have Jupiter-mass planets
and low-mass planets are more common. From the MOA microlensing observations,
Suzuki et al. (2016) have discovered a turnover in the mass ratio function at
$q\simeq 1.7\times 10^{-4}$. Their results have been confirmed by Udalski et
al. (2018) for a larger sample and they have offered a twofold mass ratio
function which is given by:

$\displaystyle\frac{dN}{d\log_{10}q}\propto\leavevmode\nobreak\
\begin{cases}q^{0.73}&q\lesssim 2\times 10^{-4},\\\ q^{-0.93}&q>2\times
10^{-4}.\\\ \end{cases}$ (12)

In the simulation, we choose the mass ratio using this function and from the
range $q\in[10^{-5},\leavevmode\nobreak\ 0.06]$. However, we disregard the
events with $M_{\rm p}(=qM_{\rm h})>15M_{\rm J}$ which are brown dwarfs. In
the simulation, we extract planetary microlensing events which (i) have
detectable planetary signatures by applying LS and HS criteria, and (ii) their
planets are within the HZs. We perform the simulation toward all mentioned
sub-fields and the average parameters of these events are mentioned in Table
2.

We conclude that the fraction of simulated microlensing events with detectable
planetary signatures (by exerting LS criteria) during the Roman mission is
$\epsilon_{\rm p}=3.0\%$. The fractions of these planetary events that their
planets locate in the OHZs and CHZs (i.e.,
$s\in[s_{\rm{HZ},\leavevmode\nobreak\ i},\leavevmode\nobreak\
s_{\rm{HZ},\leavevmode\nobreak\ o}]$) are $\epsilon_{\rm{HZ}}\simeq
3.35,\leavevmode\nobreak\ 1.79\%$, respectively.

In order to estimate the number of habitable planets that can be detected by
Roman, we use predictions of detectable microlensing events by this telescope,
done by Penny et al. (2019). Accordingly, Roman will detect $N_{\rm e}\sim
27000$ microlensing events. If we assume that each of these microlenses has
one planet orbiting it at a distance of $[0.01,\leavevmode\nobreak\ 100]$ AU
(specified by the Öpik’s law), the number of habitable planets which are
detectable by Roman is evaluated as $N_{\rm HZ}\simeq N_{\rm
e}\leavevmode\nobreak\ \epsilon_{\rm p}\leavevmode\nobreak\
\epsilon_{\rm{HZ}}$. Hence, by considering LS criteria Roman will detect $27$
exoplanets in the OHZs, where $15$ of them are in the CHZs. If there is more
than one exoplanet for each microlens, the total number of habitable
exoplanets will increase. In fact, the simulations in this work are based on
the Besançon model, similar to ones done by Penny et al. (2019). We note that
the Roman efficiencies (given by our simulations) are the average values due
to all Roman observing directions. If we had had $N_{\rm
e}(l,\leavevmode\nobreak\ b)$ for each line of sight, by calculating the
efficiencies in different line of sights we could have determined $N_{\rm HZ}$
more accurately. However, $N_{\rm HZ}$ given in this paper is an acceptable
estimation, because the Roman observing directions are close to each other and
the variations of related parameters are small. For instance, the average
extinction in W149 towards the mentioned sub-fields are
$\left<A_{\rm{W149}}\right>=1.58,\leavevmode\nobreak\
1.97,\leavevmode\nobreak\ 0.79,\leavevmode\nobreak\ 0.69,\leavevmode\nobreak\
0.79,\leavevmode\nobreak\ 0.86,\leavevmode\nobreak\ 0.94$, and the average
optical depth for the observations toward these sub-fields are
$\left<\tau(\times 10^{6})\right>=10.4,\leavevmode\nobreak\
10.5,\leavevmode\nobreak\ 8.5,\leavevmode\nobreak\ 9.7,\leavevmode\nobreak\
10.6,\leavevmode\nobreak\ 10.2,\leavevmode\nobreak\ 9.9$.

By applying HS criteria for detectability of planetary signatures, the number
of detectable planets in the OHZs and CHZs is $24$ and $12$, respectively.

These planets are most often rotating around $\rm{K}$-type stars which are on
average at the distance farther than $6.7$ kpc from us. Targets in these
events are on average as bright as $19.5$ mag in the W149 filter at the
baseline. Some part of this brightness is due to the primary lens. Because
primary lenses have a high brightness (around $0.24$ of the source
brightness), follow-up observations can resolve primary lens objects.
Detecting lens objects directly helps to resolve microlensing degeneracy and
inferring parameters of planets.

Owing to the brightness of primary lens objects, around $10.5\%$ of source
stars in our sample (of detectable planetary events by Roman) are giants. In
current microlensing observation toward the Galactic bulge, this fraction is
higher, $\sim 18.4\%$ (see, e. g., Wyrzykowski et al., 2015; Sajadian, 2015b).

From our simulation, we found that $20.7\%$ of detectable planetary
microlensing events from habitable planetary systems have obvious caustic-
crossing features (the events with the minimum source distance from caustic
lines less than $5\rho_{\ast}$, where $\rho_{\ast}$ is the projected source
radius normalized to the Einstein radius). In other events, source stars pass
close to the caustic instead of crossing it, e.g., the left event in Figure 3.
Some examples of such events are reported in Han et al. (2021); Kondo et al.
(2021).

##  4 Discussion

In planetary microlensing events, one can measure the planet-host mass ratio
and planet-host distance projected on the sky plane. The mass of primary
components $M_{\rm h}$ is carefully evaluated in caustic-crossing microlensing
events, if parallax and finite-size effects are measured (see, e.g., Yoo et
al., 2004; Shvartzvald et al., 2019; Han et al., 2018). But, microlensing
observations give the projected planet-host distance onto the sky plane,
$s_{\mathcal{P}}$, which is given by Equation 9. The projected planet-host
distance is always less than the actual distance, i.e., $s_{\mathcal{P}}<s$.

For a given $s$, we choose $\cos(i)$ randomly. The distribution of resulting
projected distances normalized to real distance, $s_{\mathcal{P}}/s$, is shown
in Figure 6. On average,
$\left<s_{\mathcal{P}}\right>=0.79\leavevmode\nobreak\ s$, as given by
Equation 10. Hence, if microlensing observations find a planetary system with
the projected distance in the habitable zone, the real distance between the
planet and its host star is most likely larger than the measured amount. Since
microlensing is more sensitive to normalized planet-host distances larger than
those owing to habitable planetary systems, so microlensing observations
overestimate the number of HZ planets. In the simulation of section 3, we
found that fraction of detectable planetary microlensing events (by
considering LS criteria) for which $s_{\mathcal{P}}$ is located in the OHZs
and CHZs are $\epsilon_{\rm{HZ},\leavevmode\nobreak\
\mathcal{P}}=4.3,\leavevmode\nobreak\ 2.8\%$, respectively. These efficiencies
are greater than $\epsilon_{\rm{HZ}}$. Accordingly, Roman will detect
$N_{\rm{HZ},\leavevmode\nobreak\ \mathcal{P}}=35$ exoplanets with projected
distances in the OHZs, which $22$ of them have projected distances in CHZs, by
applying LS criteria. These numbers are given in the two last columns of Table
2 as well.

If we consider only Earth-mass and super-Earth planets, fractions of total
detectable planetary microlensing events by Roman that (i) the planet mass is
$M_{\rm p}<10M_{\oplus}$ (Earth and Super Earth-mass planets) and (ii) their
projected positions are within the OHZs and CHZs are
$\epsilon_{\rm{HZ},\leavevmode\nobreak\ \mathcal{P},\leavevmode\nobreak\
\oplus}\simeq 0.11,\leavevmode\nobreak\ 0.05\%$, respectively. Hence, only
$\sim 1$ of total detectable exoplanets in the OHZs (i.e., $35$) has $M_{\rm
p}<10M_{\oplus}$.

The orbital motion effect should be very small in detectable planetary
microlensing events with planets in the HZs. There are two reasons. (i) In
these events, the caustic-crossing features are rare and in most cases,
caustic sizes are very small and source stars are passing close to caustic
curves. (ii) Most detectable events are from the second class of microlensing
events introduced in subsection 3.1, i.e., the events with $D_{\rm l}\gtrsim
7$ kpc and $M_{\rm h}\gtrsim 0.6M_{\odot}$ with the orbital period $P\gtrsim
100$ days. These long orbital periods likely have a small effect on
microlensing lightcurves.

Figure 6: The normalized distribution of $f_{\mathcal{P}}=s_{\mathcal{P}}/s$
for different values of the projection angle.

##  5 conclusions

Detecting planets in the HZs is an important stage for probing
extraterrestrial life, which is one of the active research areas in
astrophysics and astronomy. One method for detecting exoplanets is
microlensing observations. Although the sensitivity of microlensing
observations to habitable planets is low, by this method one can probe
planetary systems farther from the Earth position than those detectable
through transit or radial velocity measurements. Hence, microlensing is a
complementary method for detecting habitable planets in terms of their
distances from us.

In this work, we evaluated the Roman efficiency to detect planets within the
HZs by doing a Monte-Carlo simulation. We concluded that the LS Roman
efficiencies for detecting planets in the HZs were $0.01,\leavevmode\nobreak\
5,\leavevmode\nobreak\ 25\%$ for Earth, Jupiter, and Super Jupiter-mass
planets, respectively. This efficiency increases linearly with the mass, for
$M_{\rm p}\lesssim M_{\rm J}$. But, it tends toward a constant value for the
heavier planets. Earth-mass planets within the HZs are detectable around
massive host stars with $\left<M_{\rm h}\right>\simeq 0.8M_{\odot}$ at the
distances farther than $7$ kpc from us.

The Roman efficiency for detecting planets in the HZs is maximized for three
classes of planetary microlensing events with close caustic topologies.

(a) The events with $D_{\rm l}\gtrsim 7$ kpc, $M_{\rm h}\gtrsim 0.6M_{\odot}$
(massive primary lenses). By assuming Jupiter-mass planets located within the
HZs of these lenses, these events have $q\lesssim 0.001$ and $d\gtrsim 0.17$
with negligible effects due to parallax and orbital motion of lenses (Their
orbital periods are longer than $100$ days).

The events with low-mass primary lenses, $M_{\rm h}\lesssim 0.1M_{\odot}$,
while their lens systems are either (b) close to the observer with $D_{\rm
l}\lesssim 1$ kpc (which have considerable parallax effects with the amplitude
$\pi_{\rm{rel}}>0.87$ mas) or (c) close to the Galactic bulge, $D_{\rm
l}\gtrsim 7$ kpc. For Jupiter-mass planets located in the HZs of primary
lenses, the events in these two classes have $q\gtrsim 0.01$ and $d\lesssim
0.04$.

In all of these microlensing events (three classes), planetary systems make
close caustic topologies. The events in class (a) occur more frequently and
make planetary and central caustics with larger sizes.

We simulated an ensemble of detectable planetary microlensing events by Roman
and concluded that in $4\%$ of these events, the projected planet-host
distances are in the OHZs. We noted that in $3\%$ of them, their planets are
in reality in the OHZs. Roman during its $5$-year mission will potentially
detect $35$ planetary microlensing events with the projected planet-host
distances, $s_{\mathcal{P}}$, in the OHZs. One of them will have $M_{\rm
p}<10M_{\oplus}$. We note that any difference between the predicted and
observed number of planets in the HZ would also provide precious constraints
either for planet formation theories and their abundance, or for the
understanding of Roman photometry, or understanding the Galactic models.

## Acknowledgements

I thank the Department of Physics, Chungbuk National University, and
especially C. Han for hospitality. I am grateful to A. Gould for his helpful
discussion and the referee for his/her useful comments and suggestions.

## DATA AVAILABILITY

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

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