Microlensing, a powerful tool in the study of astrophysics, involves the gravitational bending of light by massive objects, typically stars, within the lensing galaxy. This phenomenon, predicted by Einstein’s theory of general relativity, manifests as a temporary increase in the brightness of a background source, such as a distant star, as the foreground lensing object passes in front of it. The gravitational field of the foreground object acts as a lens, magnifying and distorting the light from the background source. In lensing, the magnification, denoted by A, represents the factor by which the flux of the source is amplified during the microlensing event. Therefore it is given by the ratio of the image size to the source size: A = Image   size Source   size . (1)

For point source magnification in microlensing, where the source size is negligible compared to the Einstein radius (R _E ), the magnification can be simplified as: A = u 2 + 2 u u 2 + 4 (2) where u is the impact parameter, defined as the closest approach of the source to the lens normalized by the Einstein radius (Gaudi, 2012). The lensing system may comprise either a single object (single-lens) or two objects (binary-lens), with the possibility of additional components (Daněk and Heyrovskỳ, 2019). The presence of two lenses introduces additional complexity to the gravitational lensing phenomenon, leading to more intricate and often asymmetric light curves. Source systems can also consist of one or more objects. The conventional technique utilized in gravitational microlensing for exoplanet detection relies on the premise that the lens star hosts an exoplanet, constituting a binary lens system. Alternatively, another microlensing approach involves a scenario where the planet orbits the source star, resulting in a binary source configuration (Bagheri et al., 2019) as illustrated in Figure 1. In this study we simulate binary-source microlensing events by assuming both single-lens and binary-lens configurations. We consider the finite source effect for the source star while neglecting limb darkening.

To characterize the star mass density within the Milky Way galaxy, we rely on the Bésançon model, extensively discussed in previous works (Robin et al., 2003; Gardner et al., 2014; Spergel et al., 2015). This model presents the distribution of matter in our galaxy as a superposition of eight thin disk structures with varying ages, a thick disk component, and a central (old) bar structure consisting of two components. Our choice is the updated model proposed by (Gardner et al., 2014). We select the model parameters fitted to a two ellipsoid bar. This refined model, utilizing star distribution data from the Hipparcos catalog, has proven effective in interpreting microlensing data from the EROS collaboration in the direction of the spiral arms (Moniez et al., 2017). When generating binary lenses, the mass ratio between the two lenses is determined from the distribution function proposed by (Duquennoy and Mayor, 1991). We choose the semi-major axis s of the binary orbit of microlenses from the Öpik’s law where the distribution function for the primary-secondary distance is proportional to ρ(s) = dN/dss ⁻¹ in the range of [0.6, 30] au.

In our simulations, we categorize the characteristic parameters of exoplanets into two primary groups: rocky planets and Jovian planets. According to the conventional planet formation theory, rocky planets are situated within the snow line of their parent star (Kennedy et al., 2006; Kennedy and Kenyon, 2008). The snowline of a star, also known as the frost line or ice line, refers to the distance from the star where volatile compounds, such as water, can condense into solid form due to the decrease in temperature. Beyond the snowline, the temperature drops low enough for these volatile compounds to freeze, forming icy grains or particles (Kennedy and Kenyon, 2008). This snowline can be expressed with the snowline of the Sun as R SL , ⋆ = R SL ⊙ × M ⋆ M ⊙ , (3) where M _⋆ is the mass of the parent star (Gould et al., 2010). For Jovian planets, as they can migrate towards their parent star from distant orbits (as discussed by (Murray et al., 1998; Papaloizou and Terquem, 2005), we set the maximum distance from the parent star at 10 au. This range is consistent with observations from MOA microlensing studies, indicating at least one bound planet per star within this range, represented as a = 0.01–10 au (Sumi et al., 2011). We assume that the distribution of exoplanet semi-major axes follows the Öpik’s.

The planetary mass range in our simulation is 0.002–10M _J (Schlaufman, 2018). For planets with masses > 100 M ⊕ orbiting host stars with masses greater than 0.6 solar masses, we utilized the correlation between giant planet occurrence and host star mass and metallicity as outlined by (Johnson et al., 2010; Fulton et al., 2021). Following the occurrence dependencies discussed in (Santerne et al., 2016; Wittenmyer et al., 2020; Fulton et al., 2021; Winn and Petigura, 2024), we adopted their result of planet occurrence. We summarized the frequencies of various exoplanet types from our simulation in Table 1.

Generally, exoplanets exhibit a wide range of mass-radius relationships, reflecting their diverse compositions and evolutionary histories. However, a commonly used empirical relationship for estimating the radius of exoplanets based on their mass and incident flux is the following power-law scaling relation R p R ⊕ = 1.78 M p M ⊕ 0.53 F e r g s 1 c m 2 0.03 for M p < 150 M ⊕ R p R ⊕ = 2.45 M p M ⊕ − 0.039 F e r g s 1 c m 2 0.094 for M p > 150 M ⊕ (4) where F is the bolometric incident flux from the parent star and can be calculated by F = σ T eff 4 R ⋆ 2 a 2 1 1 − e 2 , (5) where R _⋆ is the stellar radius, T _eff is the effective stellar temperature, a is the semi-major axis, and e is the orbital eccentricity (Weiss et al., 2013).

To simulate the detection process involving the Roman telescope for surveying microlensing events and subsequent follow-up observations with the SKA telescope, a model is necessary to estimate the emission of exoplanets based on their characteristics and the attributes of their parent stars. In general, the flux received from a planet contains thermal radiation due to its intrinsic temperature, as well as the reflection radiation from the parent star and radio emission if it has a magnetic field due to the electron cyclotron maser instability (ECMI) (Melrose and Dulk, 1982; Dulk, 1985). The thermal and reflection radiation can be readily computed from the planet’s characteristics and its parent star (Bagheri et al., 2019). The ECMI radiation is characterized by strong circular polarization and high anisotropy. ECMI stands out as the most efficient mechanism for radio emission, superseding other mechanisms like beam-plasma instabilities. Consequently, it plays a dominant role in the generation of radio emissions. This dominance is exemplified in the brightness of auroral radio emissions from planets in the Solar System, particularly Jupiter, which rival the intensity of solar radio bursts.

Based on observations of the magnetized Solar system planets, we know that all auroral radio emissions power is related to incident energy flux of the stellar wind (Zarka et al., 2001; Zarka, 2004; Zarka, 2007; Zarka et al., 2018). In all cases in our Solar System, about 2 × 10⁻³ of the electrons’ energy goes into radio waves (Zarka et al., 2018). Therefore, the average ECMI radiation can be linked to both the total power of the solar wind and the magnetic field strength of the planet, utilizing the RBL model (Zarka et al., 2001; Zarka, 2004; Zarka, 2007). The gyrofrequency can be expressed mathematically as: ν = e B 2 π m (6) where ν is the emitted pick frequency, e is the charge of the emitting particles (in this case, electrons), B is the magnetic field strength, and m is the mass of the emitting particles (electron mass in this case). This equation shows that the emitted frequency is directly proportional to the magnetic field strength of the planet. Therefore, observations of this emission type can provide an indirect means of measuring and classifying the planetary magnetic field. This quantification of the magnetic field gives us constraints on internal structure models and planetary rotation, informing us about the evolutionary history of the planet and host system (Hess et al., 2011).

The RBL model is frequently used to estimate the radio emission of extrasolar planets (Farrell et al., 1999; Zarka et al., 2001; Joseph et al., 2004; Zarka, 2007; Christensen, 2010). The simplicity of this model has some advantages, but it does not provide a complete picture of all processes involved, in particular for close-in exoplanets as discussed in (Bagheri et al., 2024b) (for Examples of studies that extend beyond the RBL model for calculating radio emission (Vidotto, 2017; Vidotto and Donati, 2017; Dong et al., 2018; Li et al., 2023)). However, the RBL model facilitates comparative analyses between the radio emissions of exoplanets and those observed within the Solar System, thereby enhancing our contextual understanding of planetary magnetospheric dynamics. This capacity for comparative analysis underscores the utility of the RBL model in advancing our comprehension of exoplanetary environments and their potential habitability while also serving as a foundation for future investigations into the broader astrophysical implications of planetary radio emissions. Thus, in this paper, we follow the approach outlined by (Joseph et al., 2004) to estimate the radio flux density of exoplanets. As described by (Farrell et al., 1999), the radio power from an exoplanet is related to the incident power of the stellar winds. Following (Farrell et al., 1999), the Jovian decametric component is considered at least partly related to the solar wind kinetic energy input and is used as the base power level to give P rad = 4 × 1 0 18 e r g s − 1 M ̇ ion 1 0 − 14 M ⊙ y r − 1 0.8 v ∞ 400 k m s − 1 2 a 5 a u − 1.6 ω ω J 0.8 M p M J 1.33 , (7) where a is the semi-major axis of the planet’s orbit in au, M ̇ ion is the stellar ionized mass-loss rate, v _∞ is the terminal velocity of the stellar wind, and ω is the corotation speed. For the terminal velocity of stellar wind, we use v ∞ = 0.75 × 617.5 × R ⋆ / M ⋆ , (8) where R _⋆ and M _⋆ are the host star radius and mass in solar radius and mass units (O’Gorman et al., 2018). Eq. 8 serves as a mere approximation for low-mass stars, however, recent study (Mesquita and Vidotto, 2020) indicates that the terminal velocity of M dwarf stellar winds either aligns well with predictions from Parker wind models, in which stellar winds are driven by radiation pressure mechanism, or exceed these model estimates. As a result, employing this equation may yield conservative estimations of stellar winds terminal velocity and the overall count of planets with detectable signals. Then, the radio flux density (S _ν ) emitted by an exoplanet can be estimated using the following equation S ν = P rad Δ ν Ω d 2 , (9) where P _rad is the total radio emission power of the exoplanet (7), d is the distance between the exoplanet and the observer, and ν is the frequency of the radio emission, ν c = 23.5 MHz ω ω J M p M J 5 / 3 R p R J 3 , (10) where R _p is the planetary radius. Substituting (9) into (7) gives. S ν = 7.6 mJy ω ω J − 0.2 M p M J − 0.33 R p R J − 3 Ω 1.6 s r − 1 d 10 p c − 2 × a 1 a u − 1.6 (11) M ̇ ion 1 0 − 11 M ⊙ y r − 1 0.8 v ∞ 100 k m s − 1 2 , (12) where ω _J , M _J , and R _J , are Jupiter’s corotation speed, mass, and radius and Ω is the beaming solid angle of the emission. In deriving this expression, we have followed (Farrell et al., 1999) and assumed that the planet will emit ECMI emission between the frequencies 0.3ν _c and ν _c , where ν _c is the maximum radiation frequency.

The planetary corotational speed (ω) is a crucial parameter in estimating radio flux from exoplanets. Exoplanets orbiting closely around their host stars experience significant tidal dissipation, resulting in considerably slowed corotation or complete tidal locking. This phenomenon has a profound impact on the production of auroral radiation due to the permanent orientation of one side of the planet towards the star, causing asymmetrical ionospheric conductance (Zarka et al., 2001; Seager and Hui, 2002). To consider the effect of tidal locking, we assume all the exoplanets with orbits < 0.1 au are tidally locked, so the orbital period is equal to their corotation period. For all other exoplanets, we use the Darwin-Radau relation to relate corotation and oblateness (Murray and Dermott, 2000) ω = f G M p R e q 2 5 2 1 − 1.5 C 2 + 2 5 (13) where C is 0.4 for rocky planets and for gas giant planets C = 0.25 (Hubbard, 1984). In this equation, f is the planet’s oblateness; the oblateness of planets refers to the degree to which a planet deviates from a perfect sphere, taking on a slightly flattened shape due to the centrifugal force generated by its corotation. This deviation from sphericity is more pronounced for rapidly rotating planets. The oblateness of a planet is quantified by f = R e q − R pole R e q (14) where R _eq and R _pole are the equatorial and polar radii of the planet. In our simulation, for the Jovian planet with orbital distance a > 0.1 au, we use the oblateness of Jupiter (f = 0.064), and for rocky planets, we use the Earth’s value which is f = 0.00335 (Barnes and Fortney, 2003). In Our simulation we also assume no planets rotate faster than Jupiter.

In this paper, the arrangement considered for the lens, source star, and planet entails that the observed light curve of microlensing events results from a combination of two distinct light curves: one originating from the source star and the other from the planet. Consequently, the final light curve results from merging these two light curves. To calculate host stars’ radio flux at the desired frequency, we use the relationship between the luminosity of the host star in the X band and the radio band. Stars emit X-rays primarily through high-energy processes such as magnetic reconnection events, coronal heating, and interactions between fast-moving charged particles and the stellar atmosphere. The X-ray luminosity of a star is thus influenced by its magnetic activity level, mass, age, and evolutionary stage. On the other hand, radio emission from stars is more associated with non-thermal processes, such as ECMI. The strength of radio emission depends on factors including the density and strength of the stellar magnetic field, the presence of energetic particles, and the efficiency of acceleration mechanisms. While there is no universal scaling relationship between X-ray and radio luminosity for all types of stars, correlations have been observed within specific stellar populations, such as active stars with strong magnetic fields or those undergoing rapid evolution. One can use empirical relationships based on observations for the radio luminosity of main sequence stars. One such relationship is the Gudel-Benz relation, which relates the X-ray and radio luminosities: L X = L R × 1 0 15 ± 1 (15) where L _X is the X-ray luminosity and L _R is the radio luminosity (Guedel and Benz, 1993). Notably, ECMI radiation exhibits strong circular polarization, whereas solar or stellar plasma radiation lacks polarization and occurs sporadically. Additionally, ECMI radiation is beamed anisotropically, resulting in significant modulation by the planetary rotation (Zarka et al., 2014). Therefore, polarization and temporal variations should enable the differentiation between stellar and exoplanetary radio emissions, as gravitational lensing does not change the polarization of the light rays.

The NASA Roman Space Telescope is a next-generation space observatory designed to investigate various astrophysical phenomena using infrared observations. Named in honor of the astronomer Nancy Grace Roman, often regarded as the “Mother of Hubble” for her pivotal role in the development of the Hubble Space Telescope (Roman, 2019), this advanced instrument is slated to launch in the mid-2020s. With its wide field of view, the telescope is designed to conduct large-scale surveys to explore dark energy, dark matter, and the formation and evolution of galaxies with unprecedented detail (Rose et al., 2021). The Roman Space Telescope will also be crucial in advancing our knowledge of exoplanets using microlensing events. Roman W 149 filter (0.9–2 μm) will have a significantly low detection threshold, with a zero-point magnitude of 27.61. The microlensing survey of Roman will monitor 1.97deg² of the Galactic bulge (in the direction of b = −1.5 and l = 0.5 in the Galactic coordinate) with 15-min cadence, over six 72-day per season (Spiegel et al., 2005), potentially detecting thousands of exoplanets via the perturbations that they produce on the microlensing light curves (Bagheri et al., 2019). Since microlensing events usually have a duration of a few days up to a few weeks, we propose to use the Roman telescope as a survey observer to detect a microlensing event and then alert that event to a ground-based radio telescope for a follow-up observation of that microlensing event in the radio band. The most suitable ground-based telescope to detect planetary radio emission is the Square Kilometre Array (SKA) telescope. As the world’s largest radio telescope, the SKA comprises two arrays: SKA1-Low, optimized for low-frequency observations between 50 MHz and 350 MHz, and SKA1-Mid, designed for mid-frequency observations between 350 MHz and 14 GHz. The vast collecting area of the SKA, totaling one square kilometer when fully operational, enables it to detect faint radio signals from cosmic phenomena spanning a wide range of scales, from nearby planets to distant galaxies and beyond. Furthermore, the SKA’s innovative design incorporates advanced signal processing techniques and data analysis algorithms, maximizing its scientific output while minimizing data volumes and processing requirements.

The detectablity of the radio signal of exoplanets depends on, first, the detection of the microlensing event by the Roman telescope, and, second, the frequency and the intensity of the radio emission of the star-planet system. To address the first condition, we calculate the host star and planet magnitude in Roman’s W149 filter. For the flux received from the host stars, we consider both the distance-dependent effects of the source stars and the reddening of their apparent magnitudes due to interstellar dust. Using the host stars’ position and stellar type, we estimate the extinction caused by interstellar dust along the line of sight, using the comprehensive 3D extinction map provided by (Marshall et al., 2006). We use the relationships outlined in (Nishiyama et al., 2008; Nishiyama et al., 2009) to convert the extinction values from V bands to the band of W149 filter (Bagheri et al., 2019; Penny et al., 2019). The flux received from a planet comprises both thermal radiation originating from the planet’s intrinsic temperature and reflected radiation from its parent star. The planet’s temperature can be calculated by assuming that the planet’s thermal emission adheres to blackbody radiation principles (López-Morales and Seager, 2007). This calculation incorporates the absorption of radiation from the parent star by the planet, which is then re-emitted following Boltzmann’s law, T p = T eff R ⋆ a f × 1 − A B 1 / 4 (19) where T _p indicates the temperature of the planet, A _B denotes the albedo (0.15 and 0.52 for rocky and Jovian planets), while f characterizes the proportion of re-radiated energy absorbed by the planet (Harrington et al., 2006; Knutson et al., 2007). Since hot Jupiters are the most frequently detected, we adopt a value of 2/3 for f across all planets. Assuming blackbody radiation, the thermal flux (F_th ) at the specified frequency ν can be expressed using Planck’s law as follows: I ν , T p = 2 h ν 3 c 2 1 exp h ν / k T p − 1 (20) where, I(ν, T _p ) denotes the emitted power per unit area of the emitting surface, per unit solid angle, and unit frequency. Integrating it across the energy range of the W149 filter and over a hemisphere yields the thermal flux of the planet as observed by Roman. The flux of the planet resulting from the reflection (F _ref ) of the parent star’s light is determined by F ref = A g g Φ F ⋆ R p a (21) where g(Φ) denotes the fraction of the illuminated portion of the planet visible to the observer, A _g represents the geometric albedo. We utilize a geometric albedo formula: A _g = 2/3 A _B (Bagheri et al., 2019). While this relationship may not hold true for all phase angles, its alteration does not significantly impact results. This is because the thermal radiation emitted by planets is much weaker than the flux of host stars in the Roman W149 filter. Thus, any changes in this relationship would only affect the total flux received by the Roman telescope at the order of 10^–4 at most. Consequently, the overall flux received from the planet (F _p ) can be calculated as the combined sum of the thermal flux and the reflection flux. At the final stage we add realistic noise. The total noise consists of the intrinsic Poisson fluctuation in the flux received from the microlensing event in a single exposure, the intrinsic Poisson fluctuation within the point spread function (PSF) representing the background sky flux in a single exposure, the read-out and dark noise (Bagheri et al., 2019). Finally, we determine the range within which the noisy signal exhibits magnitudes below the zero-point magnitude of Roman and above the saturation magnitude of 14.8 (Penny et al., 2019).

For the follow-up observation by the SKA, we consider the SKA1-low and SKA1-mid frequency ranges up to 890 MHz. The image noise is given by, σ = S D S E F D η S η pol ν Δ τ , (22) where S _D = 2.5 is a degradation factor relative to the natural array sensitivity for the specific target Gaussian FWHM resolution of the image, η _S = 0.9 is a system efficiency that takes account of the finite correlator efficiency, and η _pol = 2 is the number of contributing polarizations and for SKA1 with orthogonal linear polarizations (Braun et al., 2019). The imaging sensitivity for SKA1-Low and SKA1-mid up to 890 MHz is listed in Table 2 by assuming a continuum observation with fractional bandwidth of Δν/ν _c ≈ 0.3, together with an integration time Δτ = 1 h.

We use the SKA image sensitivity for different ranges of frequencies as summarized in Table 2. Therefore, any event with flux density greater than the SKA sensitivity for at least 4 h of imaging is considered a detected event.