Probability of simultaneous parallax detection for free-floating planet microlensing events near Galactic Centre

Euclid is expected to launch in 2022ⁱⁱiAs of 25th February 2020 retrieved from Euclid mission website http://sci.esa.int/euclid/ and orbit around Lagrange Point 2 (L2) with a period of 6 months (European Space Agency, 2011). In the terrestrial sky, the angular distance of Euclid from L2 is no more than 33 deg, and the solar aspect angle (SAA) must be kept within 90^∘<SAA<120^∘. It limits the possible observation period to $\sim$30 days around the equinoxes.

The Wide Field Infrared Survey Telescope (WFIRST) will be launched sometime around 2024 (Spergel et al., 2015). The geosynchronous orbit around the Earth with a distance of 40,000 km and the Halo orbit around L2 were considered in the planning stage, and the Halo orbit was decided. In our simulation, both orbits are taken and will be compared. In case of the geosynchronous orbit, the orbital path inclines 28.5 deg from the celestial equator and a node located at RA=175 deg. The distance and inclination avoid occultations by the Earth when targeting at the hot spot. In case of the Halo orbit at L2, some trajectories have been discussed and is expected to be similar to the Euclid trajectory (Folta et al., 2016; Bosanac et al., 2018). We assume it shares the Euclid orbital period with the orbital radius of $\sim 0.75\times 10^{6}$ km (Webster & Folta, 2017) in our simulation. WFIRST allows wider SAA as 54^∘<SAA<126^∘ which completely covers the Euclid observation period. Thus, Euclid will determine the observation period of simultaneous parallax detection for the Euclid-WFIRST combination.

The Large Synoptic Survey Telescope (LSST) is an 8m-class telescope being built at Cerro Pachón in Chilé which will start operations in 2022 (Ivezić et al., 2008). It has not scheduled the high cadence, continuous operation for the microlensing event in the campaign as of today. Nonetheless, we selected this telescope to exemplify ground-based surveys. The sensitivity has enough potential to operate microlensing observations for FFPs down to the Earth-mass size whilst the current microlensing missions such as MOA and OGLE are relatively difficult to detect such low-mass lens events. Unlike the space-based telescopes described above, the day-night time and airmass will limit the observation period. Targeting at the Galactic centre, we assume LSST can on average perform for 7.5 hours per nightⁱⁱⁱⁱiiThis value is derived from http://www.eso.org/sci/observing/tools/calendar/airmass/html which we assumed a site of La Silla Observatory is a close proxy location. The airmass 1$\leq$sec(z)$\leq$8 was taken. during the Euclid observation period. Moreover, we assume the fine weather for photometric observation is $\sim$65.89% by taking an average of the climate data from 1991 to 1999 at La Silla observatory.ⁱⁱⁱⁱⁱⁱiiihttp://www.eso.org/sci/facilities/lasilla/astclim/weather/tablemwr.html We predict that, even with the sensitivity of LSST, we cannot cover the event detections as reliably as the space-based surveys. Hence, we simulate the Euclid-LSST combination as an indication of the ground-based sensitivity limitation and for comparison with the Euclid-WFIRST combination.

Figure 1 shows a 3D image of the telescope motion in the Galactic Coordinate System (GCS). We assume that the WFIRST phase is $90^{\circ}$ ahead of the Euclidphase in the x-y sky frame centred on L2. The barycentric motion of the Earth and Moon is ignored since the barycentre exists within the Earth and our Monte-Carlo based simulation (the detail of sample data pick-up is explained later) moderates the uncertainty due to the barycentric motion. An appropriate distance between two telescopes is an important factor of simultaneous parallax observation. We expect the combinations of Euclid-WFIRST and of Euclid-LSST would show a geometrical factor on their parallax detectability.

We apply the Near Infrared Spectrometer and Photometer (NISP) $H$-filter of Euclid and $W149$-filter of WFIRST for a space-based survey in our simulation (European Space Agency, 2011; Spergel et al., 2015). These filters have very similar transmission curves, so closely approximate each other. For comparison to the LSST $z$-band filter (Ivezić et al., 2008), we use the approximation of the Johnson $I$-band filter. The source magnitude is treated with the Johnson-Cousins photometric system in our simulation; hence we approximate the bands to the filters. Table 1 summarises the telescope parameters for FFP surveys. The formula for microlensing amplitude is defined as

where $A(t)$ the amplitude of the detected flux and $u(t)$ is the impact parameter in units of Einstein radii. We assume Euclid, WFIRST and LSST are sensitive enough to start a microlensing observation when the lens is approaching the source with the projected distance of 3 times larger than the Einstein radius. Thus, the maximum impact parameter is set as $u_{\rm max}$=3 for the point source case. $u_{\rm max}$=3 corresponds to the minimum amplitude of $A_{min}$$\sim$1.02. For the finite source case, however, Eq.(1) is no longer sufficient.

Figure 2 shows the distribution of maximum magnification in the finite source case, along with different impact parameters and source surface angular sizes in units of Einstein radius ($\rho$) retrieved from our previous paper (Ban et al., 2016). The contour labelled $1.017$ is the amplitude limit that we are assuming for our parallax simulation (i.e. $A_{min}$$\sim$1.02) and the -0.5$\leq$$log_{10}\rho$$\leq$1.0 regime shows the strong “boost” of threshold impact parameter to yield $A_{min}$$\sim$1.02. It implies that some specific events (i.e. finite source with -0.5$\leq$$log_{10}\rho$$\leq$1.0) allow the telescopes to observe them even though the minimum approach of the lens is larger than $u_{\rm max}$. In our simulation, this finite source effect is taken into account. The sky brightness ($m_{\rm sky}$) and the full width at half-maximum size of a point spread function ($\theta_{\rm psf}$) become dimmer and smaller for the space-based surveys because of the atmosphere scattering for the ground-based survey. The zero-point magnitude ($m_{\rm zp}$) and exposure time ($t_{\rm exp}$) are used to count photons as a signal. $m_{\rm sky}$ and $m_{\rm zp}$ are adjusted to the Johnson-Cousins photometric system.

We used the stellar data from the Besançon model version 1112, which was created by Robin et al. (2004, 2012a); Robin et al. (2012b). The model comprises the stellar distributions of four populations: thin disc, thick disc, bulge, and spheroid. Each population is modelled with a star formation history, and initial mass function and kinematics are set according to an age-velocity dispersion relation in the thin disc population. For the bulge population, the kinematics are taken from the dynamical model of Fux (1999). In the bulge, a triaxial Gaussian bar structure is taken to describe its density law. The reddening effect of the interstellar medium (ISM) is considered using the 3D distribution derived by Marshall et al. (2006). The model is the same as we used in our previous simulation, and the details are described in Ban et al. (2016). Here we only mention the parameters of our target area.

Penny et al. (2013) found the discrepancy of the microlensing optical depth value toward the Galactic bulge between the Besançon model (ver.1106) and observed data in $I$-band and applied a correction factor of 1.8 to their results. In our parallax simulation, we simulate the same observational target $(l,b)=(1^{\circ},-1.^{\circ}75)$ as Penny et al. and also apply the correction factor 1.8. This survey target is very close to the “hot-spot” of microlensing observation by a ground-based survey (Sumi et al. 2013). Table 2 shows the parameters of the Besançon model for our research. The catalogues offer a population of about 15.6 million stars within 0.25 $\times$0.25 deg² region centring at our survey target. To gain a statistically reasonable number of stellar data, we divided the stellar catalogues into 4 depending on the magnitude. Since the luminosity function of stars increases significantly towards fainter magnitudes, by invoking a larger solid angle for the simulated brighter stars, we can ensure that they are sampled in the simulated data set, without creating a computationally unfeasible number of fainter stars. Figure 3 visualises the stellar luminosity functions throughout four catalogues along with the source magnitude bin of 0.1. The average threshold amplitudes based on the Euclid and WFIRST sensitivity for $H$-band and the LSST sensitivity for $I$-band are plotted together. The Besançon data is successfully providing a smooth population throughout four catalogues. Our criteria of event detectability seem to require that catalogue D stars are strongly amplified to be detected by the telescopes. The details of the detectability test are described in the next section.

Objects from the four catalogues were used for the source and lens properties, and all combinations of a source and lens were tested. We assumed three FFP mass cases: Jupiter-mass, Neptune-mass, and Earth-mass. Hence, the lens mass was fixed and replaced by these three whilst the distance and proper motion values were taken from the catalogues.

For every source-lens pair, the signal-to-noise ratio was calculated. We defined the detected signal as the number of photons received by a telescope during the exposure time. The noise came from the flux of background unresolved stars, nearby star blending, sky brightness and the photon shot noise from the event itself. Thus, the equation of the signal-to-noise ratio becomes,

where $m_{*}$ is the apparent magnitude of the source star, $A(t)$ is the microlensing amplitude factor at time $t$, and $m_{\rm zp}$, $t_{\rm exp}$ and $m_{\rm sky}$ are sensitivity-dependent parameters for every observer shown in Table 1. $\Omega_{\rm psf}$=$\pi$$\theta_{\rm psf}^{2}/4$ is the solid angle of the survey point spread function (PSF) where $\theta_{\rm psf}$ is also in Table 1. $m_{\rm stars}$ is the combined magnitude contribution of all unresolved sources within the survey target angle (0.25$\times$0.25 deg²), and $m_{\rm blend}$ is the combined magnitude contribution of nearby bright stars around a given target. To determine $m_{\rm stars}$, we have to find the boundary between the resolved and unresolved regimes of our catalogue. Suppose if the baseline magnitude of a given star ($j$) attributes to the flux ($F_{j}$) and fainter stars than the given star are unresolved, the background noise ($\sqrt{B_{res}}$) can be calculated as the combined flux of those unresolved stars. Subsequently, another signal-to-noise equation is

where $\Omega_{\rm cat}$ is the solid angle of the Besançon data catalogue, and the depth of summation ($\sum_{m_{i}>m_{j}}$) is dependent on the given star ($j$). We defined the resolved stars satisfy $F_{j}\sqrt{B_{res}}>3$ for the PSF noise contribution from the unresolved stars. Sorting the stars by magnitude throughout the catalogues, we find the boundary star ($j_{lim}$) which is the faintest resolved star satisfying $F_{j}\sqrt{B_{res}}>3$. The brighter sources easily satisfy the condition even though the number of fainter stars counted into the noise increases. Once the boundary star ($j_{lim}$) is found, the background noise of the unresolved stars is converted to $m_{\rm stars}$;

Thus, the $m_{\rm stars}$ value is attributed to the Besançon data distribution and is found for every telescope sensitivity. $m_{\rm blend}$ is also found from the combined flux of stars, but this time, the stars which are brighter than the given target and within PSF range are summed up.

where the summation limit of “nearby” is defined as the brighter stars within the PSF of the target star. Thus, all resolved stars ($m_{i}<=m_{j_{lim}}$) within the PSF area of the given star are counted. All source stars in the catalogues have the individual value of $m_{\rm blend}$ for every telescope sensitivity. Once we initialise $m_{\rm stars}$ and list $m_{\rm blend}$, the round-robin pairing of the source and lens properties from the Besançon data is carried out to simulate the detectable events.

For each event, to determine the event detectability, we assume the event must show $S/N$>50 at a peak, and this limit offers the threshold amplitude ($A_{t}\geq A_{min}$) of the event; hence, threshold impact parameter ($u_{\rm t}$). We can ignore the blending influence due to the lens itself since we assume FFP lenses.

Once the event is confirmed to be detectable by satisfying above conditions ($S/N>50$), the angular Einstein radius $\theta_{\rm E}$ and event timescale for the given source ($j$) and lens ($i$) pair is given by

where $M_{i}$ is the lens mass and $G$ and $c$ are the gravitational constant and the speed of light. $D_{\rm i}$ and $D_{\rm j}$ represent the lens and source distances, respectively, and $\mu_{ij}$ is the relative lens-source proper motion. To find the mean timescale of all detectable event, we define an event occurrence weight ($W_{ij}$), which is a factor of physical lens size and lens speed.

Since we tested all possible source-lens pairs overall four catalogues, the population difference defined by the solid angle ($\Omega_{cat}$) should also be considered:

$P_{FFP,l}$ is a population ratio of FFPs per star for each catalogue. We assume $P_{FFP}$=1 for all FFP mass cases (i.e. one Jupiter-mass, one Neptune-mass, and one Earth-mass planet per stars), and for all catalogues. This is discussed further in §4.1. These equations perform overall lenses ($i$) drawn from catalogue ($l$) and all sources ($j$) drawn from catalogue ($s$). Thus, the mean timescale ($\langle t\rangle$) is

Note that $\langle t\rangle$ becomes the mean “Einstein” timescale $\langle t_{E}\rangle$ when $u_{\rm max}=1$.

The optical depth for a given source ($\tau_{j}$) is defined as compiling possible lenses between the source and an observer. The population difference of four catalogues is also considered here. Therefore, the optical depth for a given source ($j$) is

where the equation performs overall lenses ($i$) drawn from catalogue ($l$). $D$ and $\Omega_{\rm cat}$ are the distance and solid angle from the catalogue, respectively. The final optical depth ($\tau$) is the mean value over all possible sources weighted by the “effectivity” of the microlensing. Here we define the effectivity as the impact parameter factor ($U_{j}^{(N)}$) of a given source ($j$) where $N$ varies by the proportionality of the target parameter to the Einstein radius:

Since $\tau\propto\theta_{E}^{2}$, $N=2$ is applied and the final optical depth for all possible sources is expressed as

where the equation is calculated over all sources ($j$) drawn from catalogue ($s$), and we have already assumed $u_{\rm max}$=3 for our telescopes.

Finally, the source-averaged event rate is given by the optical depth and mean timescale calculated above. The standard formula of the event rate ($\Gamma$) is

So far, we have factorised the timescale and optical depth by the maximum impact parameter to reflect the telescope sensitivity into the event rate. However, there is another way to do this; Eq. 14 can be rewritten as,

where $\tau_{1}$ is the optical depth for the $u_{\rm max}$=1 case and $\langle t_{E}\rangle$ is the mean Einstein timescale. Thus, Eq.(15) is that the event rate for the $u_{\rm max}=1$ case factored by any maximum impact parameter $u_{\rm max}$. Finally, the actual number of events per year ($\tilde{\Gamma}$) is given by $\Gamma$$\times$$N_{*}$ where $N_{*}$ is the number of source stars for the observation period counted as

$U_{j}{(1)}$ is given by Eq.(12) with $N=1$.

In Ban et al. (2016), we simulated a 200 $deg^{2}$ survey field and mapped the results. The maximum predicted microlensing event rate was recovered in a “hot-spot” (Table3), which defines our simulated field. The ground-based (LSST) sensitivity expects a higher noise level originating from the larger PSF and the unresolved background stars, leading to a lower event rate. We take in to account the rate-weight of the events and the uncertainty of the parameters when computing every formula above. The farther the source is, the more lenses pass the front so that the uncertainty of the optical depth per source gets large. The calculated uncertainty of the event rate is less than 0.01% for every band and FFP mass so that it is omitted from Table 3. However, this uncertainty does not include the complexities of modelling real-world events, which may increase the sensitivity above what is modelled in unpredictable ways.

In real observation, however, the interference of any other objects and events cannot be treated so simply, hence our method is just ignoring such unexpected interferences and only assuming the ubiquitous causes of uncertainties. Moreover, we assumed S/N>50 at the peak, and the event may be too short to have sufficient indicidual exposures above the threshold signal-to-noise at which an event can be identified. Therefore, as the next step of the microlensing event simulation, we create the time-dependent observation model and simulate parallax observations. The time-dependent model is expected to yield the probability of simultaneous parallax observation that provides the parallax event rate per year multiplied by the annual number of detectable FFP microlensing events.

In the parallax simulation, the source and lens properties are randomly selected unlike a round-robin pairing done in Ban et al. (2016). The given pair is thrown into the signal-to-noise detectability test ($S/N>$50, Eq.2). If it passes the test, the time-dependent observation model is applied to the event as the simultaneous parallax observation is tried by the expected telescope combinations. The process is repeated until 30,000 events are detected in parallax for every fixed FFP mass (i.e. Jupiter-mass, Neptune-mass, and Earth-mass) to offer a statistically plausible probability of simultaneous parallax. In the next section, we describe the detail of the parallax simulation process and some calculations for further analyses of successful parallax observations by Euclid and either LSST or WFIRST.

We applied some random values for the parameters in our simulation: source-lens pair selection, zero-time ($t_{event}=0$) reference and minimum impact parameter. The randomness was controlled by the proper probability if the distribution is not uniform.

Figure 4 shows the observational setup for simultaneous parallax observation. The left panel is a schematic 2D image of the event object arrangement, and the right panel shows sample light curves with their difference (residual light curve). The parallactic angle $\gamma$ helps to identify the lens mass and distance from the light curves. The value $\gamma$ is determined as

where $\vec{u_{1}}(t)$ and $\vec{u_{2}}(t)$ are impact parameters of every observer in a function of time. In real observations, the observed amplitude factors of each telescope represent these impact parameters. Gould & Horne (2013) stated the mass and distance equations with microlensing parallax ($\pi_{E}$) and their equations are rewritten by our parallactic angle as

and

where these symbols correspond to those in Figure 4. $D_{S}$ is a standardised distance of the source from the Sun. Note that the impact parameters and parallactic angle time-dependent quantities that will be computed from the light curves. The observer distance ($D_{T}$) causes a time gap ($\Delta t$) of the arriving signal. As we described in §4.1, we set the reference observer on the Earth, and the time gap for space-based observers are carefully considered in our time-dependent model.

To analyse the parallax observation, we define a parallax signal ($S$) from the residual light curve as

where $\Delta A_{max}$ is the maximum absolute value of the differential amplitude, and $T$ is the duration over which the amplitude seen by both telescopes exceeds $A_{t}$, in hours. Besides, $T$ should be at least 1 hour for given telescopes’ cadences of 10-20 minutes. Note that a “cadence” for microlensing surveys is usually used in the meaning of the interval between observations/shuttering. Thus, we can instead state that both telescopes require at least three exposures with amplitude above $A_{t}$ to identify a simultaneous detection of an event. We also consider the noise level of differential amplitude calculated as

where $A_{i}(t)$ is the observed magnification for each telescope. $m_{*,i}$ and $m_{zp,i}$ are the source magnitude and zero-point magnitude in the corresponding photometric band of each telescope taken from Table 1. We assume $\sigma_{A_{i}(t)}\geq 3\times 10^{-4}$ for every moment $t$ and require D > 5 for a detectable parallax event.

Finally, we will find the parallax event rate by

where $\tilde{\Gamma}_{FFP}$ is the event rate only from the FFP simulation described in $\S$3.2, and $P_{parallax}$ is the rate weighted probability of detectable FFP parallax observation to all tested FFP events.

where $W_{ij}$ is the rate weight value of detectable parallax (see Eq.8). Note that the SAA limit will constrain detectable events to within two 30-day periods around the equinoxes of each year. $W_{all}$ is the rate weight of all detectable FFP microlensing event through a year which satisfy the S/N>50 limit. Hence, the summation of $W_{all}$ contains events out of SAA, events observed by either telescope within SAA (i.e. no parallax observation), events observed by both telescopes within SAA but failed our parallax detectability limit, and detectable parallax events. The input catalogues enough contain data to operate Monte-Carlo method by repeated random selection of a source-lens pair followed by the random selection of positioning angles and minimum impact parameter.

The population of catalogue stars is almost a continuous distribution in magnitude, but it is true that the faint star population (main-sequence (MS) stars) is much larger than bright stars (i.e. Red Giant Branch (RGB) stars). The faint sources, therefore, dominate the 30,000 simulated parallax events. Figure 5 shows the distribution of the fraction of detectable events in Einstein timescale and source magnitude in $H$-band. Because both combinations provided similar distribution maps, here we only show the maps of the Euclid-WFIRST combination for the top panels of Figure 5 and the residual fraction between two combinations (i.e. the fraction recovered in the Euclid-WFIRST (EW) combination subtracted from the Euclid-LSST (EL) combination) for the bottom panels to see the distribution difference more effectively.

Events associated with each FFP mass concentrate near a particular Einstein timescale. The timescale corresponds to the mean Einstein timescale for every FFP mass with which event is observed solely (Ban et al., 2016). For example, the canonically assumed Jupiter-mass FFP event will have $\theta_{E}\sim 0.03$ mas and $t_{E}\sim$2 days, and these canonical values are proportional to the square root of the lens mass (Sumi et al., 2011). The mean value from our simulation is shorter than the canonical timescale. The reason is that our simulation likely provided a smaller lens than the canonical size. As Eq.(6) shows, the source and lens distance combination ($(D_{j}-D_{i})/D_{j}D_{i}$) determines the size of the Einstein radius for a fixed lens mass. A canonical Jupiter-mass event with $t_{E}$$\sim$2 days is usually assumed to have $D_{j}=8$ kpc and $D_{i}=4$ kpc; which gives $(D_{j}-D_{i})/D_{j}D_{i}$=0.125 kpc^-1. We confirmed that about 97% of our 30,000 detectable parallax events gave less than 0.125 kpc^-1. The same tendency appears for the Neptune-mass and Earth-mass cases. Further discussion about the source and lens distance relation will be made in the later paragraph with the lens distance distribution maps.

The lower limit of the source magnitude increases as the FFP mass decreases because a low-mass source requires a small threshold impact parameter to yield a large amplitude and therefore the less-massive lens cannot pass our parallax detection limit of $T>$1 hour. The diagonal cut of the bottom-left edge of the distribution occurs for the same reason, and the effect of short timescale is more strictly appearing. The differential distributions shows that the Euclid-LSST combination rises the lower-limit of the source magnitude because the ground-based sensitivity of LSST requires a smaller threshold impact parameter (i.e. higher amplitude) to be observed. Due to the more strict cut-off by the LSST sensitivity and the stellar population, the distribution of the fraction of detectable events concentrates more on $H\sim 20$ sources in the Euclid-LSST combination than that of the Euclid-WFIRST combination. The differential distributions of Earth-mass lenses specifically show how the limiting timescale differs between two telescope combinations; the Euclid-WFIRST combination can detect shorter-timescale events than the Euclid-LSST combination. The influence of the ground-based sensitivity and our parallax detectability limit of $T>$1 hour caused such a clear boundary between two regimes where the distribution is concentrated by the Euclid-WFIRST combination and the Euclid-LSST combination. Consequently, the Jupiter-mass lens and Neptune-mass lens cases do not clearly show the gap of the timescale limit between two combinations.

Figure 6 shows the distribution of the fraction of detectable events in Einstein timescale and parallax signal ($S$ in logarithmic scale, Eq.(20)) combination. The fraction distribution shows that the range of $S$ does not vary depending on the lens mass despite the lens mass determining the Einstein radii, hence Einstein timescale. The lower limit of $S$ can be attributed to our duration limit for the parallax detectability ($T>$1 hour). On the other hand, the upper limit can vary with the Einstein timescale, but we require enough “distinguishability” of parallax light curves compared to the noise level (D > 5, see Eq.(21)). The larger the Einstein radius and the longer the duration, the more difficult it is to identify differences in the two light curves. Consequently, none of the FFP lens-mass cases exceeds $log_{10}S\sim 2.5$. The differential distributions indicates that the Euclid-LSST combination is sensitive to larger parallactic angles than the Euclid-WFIRST combination. The higher noise of the ground-based survey requires a larger differential amplitude between the Euclid light curve and LSST light curve to satisfy the noise level limit (D > 5) in our simulation. The boundary between the dominant regions of the Euclid-WFIRST combination and the Euclid-LSST combination illustrates the relation of $S\propto t_{E}$.

Figure 7 shows the relation between $H$-band magnitude and $S$ (top), and the position of source stars on the $I-H$ colour–magnitude diagram (bottom). The fraction distribution shows that $S$ reaches a maximum at a source magnitude of $H\sim$20.5 and decreases to both brighter and fainter source regimes. The peak in source magnitude can be explained using Figure 3. Sources with $H>$20.5 mag numerically dominate the source population, but an event requires strong magnification to become detectable. Sources with $H<$20.5 mag are comparatively rare, and finite source effects decreases the detectability of $S$. The near the $H\sim$20.5 mag boundary are common and can easily satisfy the S/N$>$50 criterion without strong amplitude (i.e. without a small impact parameter).

Figure 8 shows the distribution of the fraction of detectable events in Einstein timescale and source mass combination. Figure 9 is a supportive plot, showing the stellar mass and the luminosity function from the whole Besançon catalogue. Detectable events involving Jupiter-mass FFPs typically involve less-massive source stars than events involving Earth-mass FFPs. Since the MS stars have the power-law relationship between luminosity and mass, low mass sources are faint sources which require significant amplitude to satisfy the signal-to-noise detectability limit (S/N>50). The Earth-mass lens events for these faint sources are likely cut-off by our parallax duration limit ($T>$1 hour) due to the small lens size. Hence, the fraction of detectable events involving massive source stars increases for Earth-mass FFPs whilst the fraction for Jupiter-mass FFPs reflects the source population more directly. According to Figure 9, the majority of faint stars from catalogue C and D were $\leq 1M_{sun}$ whose population was much larger than the brighter stars. The differential distributions shows almost the same tendencies as Figure 5; the Euclid-WFIRST combination can reliably detected parallax-induced differences in the light curves from lower-mass lenses than the Euclid-LSST combination, and the Earth-mass lens case shows the timescale limit of the Euclid-LSST combination due to the ground-based sensitivity and the parallax detectability limit of $T>$1 hour.

Figure 10 shows the distribution of the fraction of detectable events in Einstein timescale and lens distance combination. Figure 11 is a supportive plot, showing the stellar distance and the luminosity function from the whole Besançon catalogue. Note that both source and lens distances were taken from the same Besançon datasets. For lenses up to 6 kpc from the Sun, the lens properties tend to be taken from the bright-stellar data ($H<9$) whose population is very low. Moreover, the disc-disc microlensing dominated the event weights because the small relative proper motion of the disc-disc microlensing offers longer events than the disc-bulge microlensing. So the fraction distribution spreads more on the long timescale regime. From 6 to 9 kpc, the timescale range spreads and the most likely distance for a detectable FFP lens is at $\sim$7 kpc for all FFP lens masses. Stars between 6-9 kpc dominate the objects simulated in our Besançon catalogue, and occupy a wide range of $H$-band magnitudes, hence we expect a lens will typically form part of this dense population, but will normally be closer to us than the Galactic Centre (8kpc).

Sources with $H<$23 tended to be located closer to the lens (both would be in bulge). They tend to exhibit finite source effects. The motion of bulge stars varied so that events occur with different timescales. Stars beyond 9 kpc are also included in our simulation. It is obvious from Figure 11 that these events only had fainter sources ($H>$24). Since we considered the background noise due to unresolved stars and the extinction decreases the apparent population, these fainter sources were quite difficult to observe, and therefore, the likelihood diminishes rapidly beyond 9 kpc. According to Figure 9, stars with $H>$26 are more massive than stars $H\sim 24$, and their mean mass covers the canonical white dwarf mass. Most of the population is white dwarfs, and a few events are detectable only with a Jupiter-mass lens (Figure 7).

Figure 12 shows the binned distribution of some event parameters with differential amplitude and Einstein timescale. The fraction of detectable events distribution (top-row) becomes an indicator to read the relative proper motion map (2nd-row), the transverse line of the projected source in the lens frame map (3rd-row), and the parallactic angle map (bottom-row). Note that parallactic angle is time-dependent throughout an event so that we picked the maximum angle of every event (corresponding to the peak of the residual light curve) and took the mean of each bin to show the distribution. These maps are for the Euclid-WFIRST combination. The figures for the Euclid-LSST combination show similar distributions.

From the top panels of Figure 12, we can identify the most-likely combination of $(log_{10}\Delta A_{max},t_{E})$ to be at $\sim$(-0.8, 0.6), $\sim$(-0.6, 0.2) and $\sim$(-0.6, 0.03) for Jupiter-mass, Neptune-mass and Earth-mass, respectively. We can interpret the remaining panels of Figure 12 through this probability distribution.

The distributions of relative proper motion (2nd-row) show a gradation along the timescale axis for all FFP lens masses. The short timescale and high differential amplitude regime corresponds to the largest proper motion. This is plausible because large relative proper motion leads to quicker events. Besides, the most likely combination of $(log_{10}(\Delta A_{max}),t_{E})$ from the top panels typically yields $\mu_{rel}\sim$7.5 mas yr^-1 for all FFP lens masses. Since we used stellar data for lens properties, replacing only their mass by planetary masses, the most likely relative proper motion is similar to the mean proper motion of disc stars in our model. FFPs possibly have higher velocity as a result of ejection from their host stars and of swing-by acceleration by encounters. In that case, $\mu_{rel}$ becomes higher and yields shorter $t_{E}$ though we did not model it.

We calculated the mean transverse line, which is defined as the angular distance of the projected source path in the lens frame and plotted the distribution (3rd row). The Jupiter-mass and Neptune-mass FFP lenses show large size is correlated with low differential amplitude. If the radius of the Einstein ring is large compared to the projected telescope baseline, both telescopes will experience similar amplification and the light curves will not be differentiable from each other. This tendency can be seen for Earth-mass FFP events to some extent but not as extreme as massive FFPs since an Earth-mass FFP lens is small enough not to let telescopes to induce similar light curves. Like the distributions of relative proper motion, the most likely combination of $(log_{10}(\Delta A_{max}),t_{E})$ from the top panels provides the transverse line $\theta_{trans}\sim 8\pm 2$ micro-arcsec for Jupiter-mass and Neptune-mass, and $\sim 3\pm 1$ mas for Earth-mass FFP. The reason is the same above; a massive lens requires a smaller threshold impact parameter to identify the differential light curve so that the source population converges to an effective transverse line. This condition indicates that a Jupiter-mass lens has the potential to observe fainter sources with which an Earth-mass lens is influenced by the finite source effect and cut-off. Thus, our criteria for detectable parallax give weight to both massive and less-massive FFPs. For the canonical Einstein radii ($\sim$0.03, $\sim$0.007, and $\sim$0.002 milli-arcsec with $D_{S}=8$ and $D_{L}=4$ for Jupiter-mass, Neptune-mass, and Earth-mass lenses, respectively), the transverse line of 8 micro-arcsec is $\sim 0.3\theta_{E}$ for Jupiter-mass, $\sim\theta_{E}$ for Neptune-mass and $\sim 1.5\theta_{E}$ for Earth-mass. Since we set the maximum impact parameter of three telescopes as $u_{max}=3$ or equivalent in the finite source case, the transverse line of $\geq\theta_{E}$ with any $u_{0}>0$ is fairly possible. However, the Earth-mass FFP events are most likely cut-off due to a finite source effect. From this point, the Neptune-mass lens is the most suitable size for the telescope separations among the three FFP masses in our model.

The distributions of the parallactic angle (bottom-row) show two regimes: the small parallactic angle regime at short timescales and the large parallactic angle spots scattered across longer timescales. The small-angle regime corresponds to our parallax detectability limit, i.e., that the event is simultaneously observable by both telescopes at least for 1 hour under the proper SAA, day/night timing, and weather. Especially for low-mass lens events, a large proper motion reduces the chance of “simultaneous” observation so that a similar line-of-sight between two observers is preferred to satisfy our detectability limit. The large parallactic angle spots are likely an artefact due to plotting the mean of each bin. We assume that the maximum differential amplitude corresponds to the maximum parallactic angle and take their maximum values individually. This assumption is plausible since the telescope separation is quite smaller than the source and lens distances; however in a spatial model of our simulation, this assumption is not always the truth. The model treats the direction of their line-of-sights in vector, and the parallax angle is found in the way of simulation whilst it would be found from the differential light curve at real observations. For the Earth-mass FFP lens, the large parallactic angle spots seem to concentrate on the high differential amplitude regime. A large differential amplitude requires relatively wider telescope separation or large difference in time of maximum amplification. In either case, a very small impact parameter is necessary for Earth-mass FFP lens to offer $log_{10}(\Delta A_{max})$>0.2 satisfying the detectable duration limit of $T>$1 hour. The most likely combination of $(log_{10}\Delta A_{max},t_{E})$ from the top panels also has similar parallactic angles for all FFP lens masses: $\gamma$$\sim$0.36$\pm$0.02 micro-arcsec. As for the threshold lens radii, this is because the requirement of threshold impact parameter and the Einstein radius concentrated parallactic angles towards a specific value across all FFP lens masses.

Figure 13 shows the probability distribution of difference in time of maximum amplification. Lower FFP masses generate longer times between the two light curve peaks. This is because the projected telescope separation becomes larger with respect to the threshold lens radii. For the Neptune-mass and Earth-mass FFP lenses, the highest probability is at $\Delta t_{0}/t_{E}$$\sim$0.05 and $\Delta t_{0}/t_{E}$$\sim$0.19, respectively. For the most-likely event timescale, the highest probability corresponds to a time difference of $\sim$33 min for Neptune-mass lens and $\sim$22 min for Earth-mass lens. These values are much larger than the maximum time-gap due to the observer separation: taking $\leq$6 sec for radiation travel between Earth and L2. Besides, it is larger than or close to the cadence of our assumed telescopes (15-20 min). The time gap is even possible to identify through real observation.

The event rate is summarised in Table 4. The yearly rate covers two 30-day-seasons around the equinoxes, as determined by the maximum SAA of Euclid ($2.1). For the Euclid-WFIRST combination, we simulated both geosynchronous orbit and Halo orbit at L2 cases of WFIRST. The parallax probability is calculated from the weighted rate of the event (Eq.(8)) where the threshold impact parameter ($u_{t}$) depends on the telescope configurations. Therefore, there might be a gap of parallax probability between the telescopes even though they are detecting the same events. It is reasonable to accept the smaller probability between the combination telescopes rather than optimising the large number. For the Euclid-LSST combination, the parallax detectability of our model considers the day-night position of LSST. Our assumption of 7.5 hours of performance per night corresponds to the zenith angle of $\sim 56^{\circ}$. Hence, the population of clear nights (65.89%) is only applied to the final result.

For the Euclid-WFIRST combination, the Halo orbit at L2 case provided a lower probability than the geosynchronous orbit case. This arises mostly from the observer separation in our simulation. On average, the geosynchronous orbit case provided the observer separation $\sim$1.6$\times 10^{6}$ km whilst the Halo orbit at L2 case provided $\sim$0.9$\times 10^{6}$ km. The narrower separation reduces the light curve gap. Moreover, the variation of the relative line-of-sight is greater for the Halo orbit at L2 case than the geosynchronous orbit case because of its motion apart from the reference observer on the Earth. Whilst the geosynchronous orbit case keeps the parallactic angle based on the SAA, the Halo orbit at L2 is more likely to yield a small parallactic angle. We confirmed that the distribution of sample event properties (i.e. lens distance, lens size, relative proper motion and threshold impact parameter) did not show a clear difference between these cases. Therefore, we can simply say that the variation of phase positions resulted in a smaller parallax event rate for the Halo orbit at L2 case than the geosynchronous orbit case. In both orbital cases, the parallax event rate derived from the WFIRST configuration is smaller than that of Euclid configuration. It indicates that more events are detected only by WFIRST in our simulation and is understandable because the WFIRST sensitivity yields the fainter zero-point magnitude and allows more faint sources which population is relatively large.

The Euclid-LSST combination yields less FFP parallax detection than the Euclid-WFIRST combination with the Halo orbit. This event rate is based on our fine-weather assumption of 65.89% and an average operation of 7.5 hours/night. Compared with the geosynchronous orbit case (the telescope separation is similar to the Euclid-LSST combination), the parallax event rate becomes smaller because the event detectability is lead by less-sensitive LSST. The event rate derived from the Euclid configuration is smaller than that from the LSST configuration, unlike the Euclid-WFIRST combination. The reason is the same as the Euclid-WFIRST combination that there are more events observed only by Euclid because of the sensitivity difference in our simulation.

The combination of WFIRST and LSST was skipped in our simulation. The zero-point magnitude difference would yield the sensitivity difference and result in different detectability, but as long as LSST is the less-sensitive ground-based survey, the LSST sensitivity determines the detectability like the Euclid-LSST combination. Besides, the geosynchronous orbit case would provide an observer separation that is too short to successfully and effectively observe parallax. For the Halo orbit case, the wider SAA of WFIRST would provide a larger event rate than the Euclid-LSST combination, and we can simply multiply the event rate by the SAA coverage ratio to gain the WFIRST-LSST combination event rate since our random selection of Earth’s positioning angle (hence, L2 position) is equally distributed within the range.

Zhu et al. (2015) and Zhu & Gould (2016) considered parallax observations by ground-based and space-based surveys. They tested the combination of OGLE-Spitzer and of KMTNet-WFIRST where the Halo orbit case of WFIRST was assumed. In both combinations, they suggested the importance of telescope separation to observe FFPs as we have discussed in relation to our results. Besides, LSST is expected to have higher sensitivity than other ground-based telescopes currently operating. Although Zhu et al. (2015) suggested the reinforcement of sensitivity by a “combination” of ground-based and space-based telescopes, the individual sensitivity is essential to observe FFP events effectively.

The mass estimation of the lens will theoretically be more precise using parallax data than a single observation. For every detectable event in our simulation, the residual light curve of parallax observation was generated. $u_{1}(t)$, and $u_{2}(t)$ are derived from the light curve. Instead of taking all $u$ for every $t$, we took $u$ at the light curve peak of each observer (see Figure 4 where is expressed as $\Delta A(t_{0,1})$ and $\Delta A(t_{0,2})$); hence we had $u_{1}(t_{0,1})$, $u_{2}(t_{0,1})$, $u_{1}(t_{0,2})$, and $u_{1}(t_{0,2})$. The relative proper motion was assumed to be $\mu_{rel}\sim 7.5\pm 1.5$ mas yr^-1, which we derived from the most likely values of ($log_{10}(\Delta A_{max}),t_{E}$; Figure 12). Then $\theta_{E}$ was calculated. The telescope separation for each combination is averaged as $D_{T}\sim 1.6\times 10^{6}$ km for the Euclid-LSST combination and $\sim 0.9\times 10^{6}$ km for the Euclid-WFIRST combination, and the fluctuation by the orbital motion is contained as the uncertainty of it. Using Eq.(17) and these parameters from the light curve, $\gamma$ was calculated. Thus, we had $\gamma(t_{0,1})$ and $\gamma(t_{0,2})$ and moved to Eq.(18) to calculate the lens mass and averaged them.

To consider the accuracy of our lens mass estimation, we calculate the uncertainty ($\epsilon$) from the parameter errors of $u$, $\mu_{rel}$, and $D_{T}$ and the discrepancy of the estimated mass from the input mass ($|Delta|$=$|M_{FFP}-M_{input}|$/$M_{input}$). If $M_{FFP}\pm\epsilon$ does not cover the input mass, it is obvious that the uncertainty is underestimated. However, even though $M_{FFP}\pm\epsilon$ covers the input mass, the very large $\epsilon$ case is not acceptable from the view of the estimation accuracy. Table 5 summarises the percentage likelihood among our 30,000 events that $M_{FFP}\pm\epsilon$ covers the input mass. It indicates that our parameter errors are underestimated in the majority case. It is because we assumed a certain value of relative proper motion (7.5 mas yr^-1) with an error value that was derived from the Besançon data. Considering the distribution of detectable parallax events, the error is flexible in dependance on the differential maximum-amplitude. Figure 14 visualises the percentage likelihood of the discrepancy among 30,000 events. For example, $|\Delta|$<100% means the estimated mass was up to twice as large as the input mass. Since $|\Delta|$<50% regime is less than 50% for all FFP masses and telescope combinations, we can regard that our calculation likely overestimates the FFP mass. One cause was that the parallactic angle was underestimated in most cases because we derived it from the differential light curves, not from the spatial components in our model. The other cause was that we theoretically estimated the impact parameter to derive the parallactic angle from the light curve using Eq.(1) assuming a point source. Thus, the parallactic angle variation was wider than what we generalised through our mass estimation process, and our calculation method biased to make the least parallactic angle.

Table 6 shows the Gaussian means and standard deviations derived from the data of each discrepancy: <10%, <50%, and <100%. As we mentioned in the last paragraph, the mean estimated masses tend to become equal to or larger than the input mass because of the underestimation of the parallactic angle. However, it is not true for the Neptune-mass case in the Euclid-WFIRST combination with the geosynchronous orbit and the Euclid-LSST combination. One possible reason is because of the interaction between Einstein radius and the Einstein parallax (i.e. the telescope separation). In §5.2, we confirmed that the Neptune-mass lens is the most suitable to our telescope combinations because of the effectiveness of the transverse line for identify the differential light curve. In other words, the upper limit of the minimum impact parameter, which satisfies our parallax detectability limit, was maximised for the Neptune-mass lens. Thus, Table 6 indicates that the telescope separation of $1.6\times 10^{6}$ km allowed relatively larger $u_{0}$ and underestimated the Einstein radius, which we derived from the transverse line and the detected duration reading from the light curves.

The discrepancy of estimated FFP lens mass correlates with the accuracy of Einstein timescale ($t_{E}$) and Einstein parallax ($\pi_{E}$) estimated from light curves. Note that Einstein parallax was calculated from the reciprocal of observer separation corresponding to the full lens size. We confirmed that the accuracy of Einstein timescale estimation was quite acceptable; 74.5% of events had Einstein timescales reproduced to within 10% of the input timescale, and almost all of the rest stayed within 50% discrepancy. On the contrary, Einstein parallax estimation was as inaccurate as the FFP mass estimation. Hence, we need to improve the Einstein parallax estimation from light curves. In our simulation, the 3D positioning model was used, and the measurement of observer separation was done in vectorial 3D space. However, photometric light curves were drawn with the scalar value of impact parameters since we never know the vectorial impact parameters on the sky during real observations, since we never know the rotation angle that the transect the source star makes behind the FFP lens.

We computed the distribution of the angular difference between two vectorial impact parameters observed by two telescopes. Here we define the angular difference as $\alpha_{u}$ that two vectorial impact parameters $\vec{u_{1}}$ and $\vec{u_{2}}$ form between.

The angle $\alpha_{u}$ is not evenly distributed. The most likely case was the opposite vectorial direction (i.e. $\alpha_{u}\sim\pi$), but the case was just 11-29% of detectable parallax events depending on the observer combinations in our simulation, which allowed $\alpha_{u}$ to take from $-\pi/2$ to $\pi/2$. Compared to the event rate values in Table 4, we also found that the percentile likelihood of $\alpha_{u}\sim\pi$ decreased for the higher event rate. One reason for this tendency was that the relative positions between two telescopes varied more for the Euclid-WFIRST combination than for the Euclid-LSST combination. Another reason was that the Euclid-LSST combination required a larger amplitude difference between two light curves due to the lower sensitivity of ground-based partner (LSST).

Thus, the Einstein parallax estimation using a scalar value of impact parameters was not enough accurate in the commonest scenario, and this issue affected our mass estimation accuracy as we discussed above. We also tried to compute a distribution of parallax time gap (i.e. difference of the light curve peak time between two observers divided by the Einstein timescale; $\Delta t_{0}/t_{E}$) for every $\alpha_{u}$. The range of possible parallax time gap showed a convex curve for $-\pi/2\leq\alpha_{u}<\pi/2$, but the distribution peak depends on the telescope combinations and FFP masses. Thus, the parallax time-gap was not sufficient evidence to identify the angle between two impact parameters. Besides, the error in the time of peak magnification ($\epsilon(t_{0})$) should be smaller than the parallax time gap otherwise the fraction error in Einstein parallax exceeds unity.