Free-floating planets in the Milky Way

Gravitational microlensing is a powerful method to search for and characterize exoplanets, and it was first proposed by Paczyński in 1986. We provide a brief historical excursus of microlensing, especially focused on the discoveries of free-floating planets (FFPs) in the Milky Way. We also emphasize that, thanks to the technological developments, it will allow to estimate the physical parameters (in particular the mass and distance) of FFPs towards the center of our Galaxy, through the measure of the source finite radius, Earth or satellite parallax, and/or astrometric effects.

shock could cause them to become gravitationally unstable and collapse into objects with the mass of giant planets [27], thus implying the possible presence of a large number of FFPs in galaxies. Recently, Mróz et al. [28] reported on the analysis of a large sample of microlensing events discovered during the years 2010 ÷ 2015 by OGLE collaboration. They found no excess of events with timescales smaller than 2 days, but detected a few possible ultrashort-timescale events (with timescales of less than 0.5 day), which may indicate the existence of Earth-mass and super-Earth-mass FFPs. Therefore, knowing the FFP population in our Galaxy is still an open question and the gravitational microlensing is the only method capable of exploring the entire population of FFPs down to Mars-mass objects.
Based on the results obtained by Sumi et al. [23] and the capabilities of different telescopes, which are performing microlensing observations and/or are planned for the near future, we have considered the issue of the FFP detection by gravitational microlensing in the Milky Way. We estimate the detection efficiency (that is ratio between the number of events for which each second-order effect is detectable and the number of simulated events) of the finite source effect, orbital or satellite parallax and astrometric effect in microlensing events caused by FFPs towards the Galactic bulge. These effects are very important for the determination of the FFP physical parameters (mass and distance) due to the microlensing degeneracy. Here, we also note that microlensing events toward the Galactic bulge can be caused by different object such as stars, BD, and FFPs, which are in the field of view of the telescope. For these objects, we have considered the following density distributions: (a) exponential thin and thick disk and (b) triaxial bulge [29]. In addition, for the their velocity distribution, we have assumed a Maxwellian distribution [30,31]. Based on Sumi et al. [23], the FFP and BD mass functions are considered as power law with indexes: α PL = 1.3 +0. 3 −0.4 and α BD = 0.49 +0.24 −0.27 , respectively. The abrupt change from α PL 1.3 to α BD 0.49 favors the idea of two separate populations, as if the FFP formation process is different from that of stars and BDs. The optical depth and the microlensing rate of events caused by FFPs were calculated in Ref. [20] where it was found that these events are much fewer with respect to those due to main sequence stars, but more numerous than those due to BDs. Considering the lower limit of the FFPs per star, we found that by space-based telescopes (in particular by Euclid or WFIRST) will be detected about 100 microlensing events caused by FFPs during a month (see Ref. [20] for details). Moreover, since the FFPs are not expected to be surrounded by a plasma, we have not considered its effect in the gravitational microlensing events [32]. However, whether a microlensing event is caused by a primordial black hole, modeling the plasma-induced effects on the light curve is compelling (see [33]). In the next sections, we review the bases of microlensing and in particular on the photometric aspects. Then, in Sect. 3, we discuss astrometric microlensing and, in Sect. 4, we show how to break the degeneracy in microlensing events caused by FFPs. Our conclusions are presented in Sect. 5.

Basics of photometric microlensing
Gravitational microlensing method is a well-known technique for detecting compact objects in the disk, bulge, and/or halo of our Galaxy via the time-dependent amplification of the light from background sources. In the simplest case, when the point-like approximation for both lens and source is assumed, and the relative motion among the observer, lens, and source is uniform and linear, individual images cannot be resolved due to their small separation, but the total brightness of the images is larger with respect to that of the unlensed source, leading to a specific time-dependent amplification of the source [4,34]. For a source at angular distance θ S from a point-like gravitational lens with mass M, the positions θ of the two images with respect to the lens are obtained by solving the lens equation (see Ref. [35]): where u = θ S /θ E andũ = θ/θ E are the dimensionless distances and θ E is the Einstein angular radius. When the observer, the lens, and the source are perfectly aligned, the source image becomes a ring, called the Einstein ring. The angular Einstein ring radius can be expressed as follows: Here, D S and D L are the distances to the source and the lens, respectively. The solutions of Eq. (1) are as follows:ũ which give the locations of the positive and negative parity images (+ and −, respectively) with respect to the lens position. Note that, in the lens plane, the + image resides always outside the Einstein ring centered on the lens position, while the − image is always within this ring. Due to the conservation of the surface brightness, the amplification of the background source is simply given by the ratio between the area of the images to the area of the source. Therefore, the time-dependent amplification of the distorted images can be calculated by the following: If we assume the relative lens-source motion to be rectilinear, u can be decomposed into the components parallel and perpendicular to the direction of the relative lens source motion. Thus, u(t) and A(t) can be calculated as follows: where t 0 and u 0 are the time and impact parameter at the closest approach. Here, t E is the Einstein timescale, which is defined as the time required for the lens to transit the Einstein radius: where μ rel is the relative lens-source velocity, R E is Einstein radius and v T is the lens-sourceobserver relative transverse velocity. During a microlensing event, the source position projected in the lens plane encounters the Einstein ring when the projected separation is u = 1, where the source amplification takes the value A th = 1.34 called the threshold amplification. For space-based observations, due to the absence of seeing effects, the amplification threshold may be much smaller than 1.34, with a corresponding much larger value for the parameter u. Assuming a photometric error ∼ 0.1%, the threshold amplification turns out to be A th = 1.001 and the maximum value of u from Eq. (5) turns out to be u max = 6.54 [20].
One has to consider at this point that, from the event lightcurve, three parameters can be defined: the time of maximum amplification t 0 , the Einstein time t E , and the impact parameter u 0 . However, of these parameters, only t E contains information about the lens and this gives rise to the so-called parameter degeneracy problem, since there are only two observable quantities. This degeneracy cannot allow to infer the lens parameters uniquely, thus making the interpretation of microlensing results somewhat ambiguous. To break this degeneracy, we consider the second-order effects, which are the finite source effects (see Refs. [36][37][38][39]) and the parallax effect. In principle, there are two ways to observe the shift caused by the parallax effect. First, the orbital motion of the Earth (annual parallax) creates on the light curve a shift relative to the simple straight motion between the source and lens (see Refs. [40,43,44]). Second, two observers at different locations looking contemporarily towards the same event can compare their observations [14]. These second-order effects induce small deviations in the light curve (with respect to the Paczyński profile), which may be extremely useful to break, at least partially, the parameter degeneracy problem in microlensing observations.

Basics of astrometric microlensing
It is well known that, in addition to the photometric lightcurve, a gravitational microlensing event gives rise also to an astrometric deflection, as the event unfolds. This is because the images produced by the lens are not symmetrically distributed, leading to a typical elliptic pattern traced by the centroid, which was studied by many authors [45][46][47][48][49]. The centroid of the image pair can be defined as the average position of the + and − images weighted by the associated magnifications [50,51]: so that, by symmetry, the image centroid is always aligned with the lens and the source. The measurable quantity is the displacement of the centroid of the image pair relative to the source, that is: which, of course is a function of time, since u depends on time. While the image magnification, A is a dimensionless quantity which depends only on the dimensionless separation u, and is a function of both u and the angular Einstein ring radius θ E , so that the observed centroid shift is directly proportional to θ E : It is straightforward to show that during a microlensing event of a single lens on a single source, the centroid shift, , traces an ellipse. The ellipse semi-axes a and b, which depend on both the lens impact parameter u 0 and θ E , are given by the following: The Einstein angular radius in microlensing events caused by FFPs is typically of the order of a few μas. For this reason, the astrometric signal is expected to be detected more efficiently through space-based observations, as those by the Gaia satellite. Its precision for astrometric observations depends on the visual magnitude of the star. Eric Hog [52] has determined the astrometric precision of the Gaia telescope and found that it can be as low as 4 μas for stars with visual magnitude in range from 6 to 13 (see table A in Ref. [52]).

Solving the parameter degeneracy
As mentioned above, from the parameters obtained by the light curve of a microlensing event, only t E contains information about the lens. Therefore, to infer the lens properties uniquely, we have considered the secondorder effects, from which the microlens parallax π E and the angular Einstein radius θ E can be determined. Gould [53] pointed out that the microlens parallax is given by the following: where r E is the Einstein radius projected on the observer plane. If r E is measured, then the mass of the lens can be determined without ambiguity by the following: where κ ≡ 4G c 2 AU = 8.144 mas M . The microlens parallax can be derived from the orbital parallax, which is caused by the orbital motion of the observer (Earth) around the Sun [54] and by the simultaneous observation of the source microlensing event by two telescopes at different locations. Indeed, during a microlensing event, the deviations of the light curve from the symmetrical shape, due to the Earth orbital motion, can be observed [40][41][42]. The information of microlens parallax can be obtained by modeling and fitting the tiny asymmetry in the light curve. In addition, the microlens parallax can be detected by analyzing the photometric curves detected by two telescopes that are far away from each other [14]. The value of the angular Einstein radius can be obtained by measuring finite source effects, from high-resolution imaging and from astrometric measurements. More detailed descriptions of each method are provided in the following sections.
Before closing this section, we note that in our numerical simulations, we have assumed that a microlensing event can be detected if, in its light curve, there are at least 8 points in which the amplification is bigger than the threshold amplification. For space-based observations, the threshold amplification is assumed A th = 1.001. The parallax effect and finite source effects can be detected on a light curve when the residuals with respect to the Paczyński curve are larger than 0.001.

π E in microlensing events caused by FFPs
The parallax effect, due to the motion of the Earth around the Sun, may leave in the light curve of microlensing events observable features, which can be used to constrain the lens parameter. Alcock et al. [40] have presented the first detection of parallax effects in a gravitational microlensing event. Their description of the parallax effect in the light curves was obtained by expanding the Earth trajectory up to the first order in the eccentricity. For relatively long events (with duration about a few months), the deviations by the Earth motion may be consistent, and so, one can determine microlens parallax, π E . However, in the case of space observatories (like WFIRST or Euclid positioned at the L2 point), the satellite acceleration around the Sun can produce a parallax effect that can be detectable also for short-duration microlensing events as those caused by FFPs [20]. Following the analysis by Dominik [43] and based on the capabilities of space telescopes, we found that nearly 30% of the events caused by FFPs towards the Galactic bulge may have detectable orbital parallax effect. We also found that the best period during the year to observe the parallax effect is June, due to the orientation of the Earth orbital plane with respect to the line of sight towards the bulge [44].
Another way to estimate the microlens parallax can be provided using two telescopes: one located on the Earth and the other one in space, provided that they observe the same event contemporarily. The possibility to measure the microlens parallax through the simultaneous observations of the same microlensing event by two telescopes distant enough from each other has been set by Refsdal [6] and later developed by Gould [55]. The era of space-based microlensing parallax observations started much later using Spitzer with the analysis of an SMC event [14] and, later on, continued with the ongoing Spitzer observational campaign started in 2014 for the follow-up of the microlensing events detected towards the Galactic bulge [56][57][58][59]. This observational campaign has already led to several important results assessing clearly the importance of these kinds of measurements. By the two photometric curves, the shift time at the peak t 0 =| t 0,⊕ − t 0,sat | and u 0 =| u 0,⊕ − u 0,sat | can be measured, and consequently, π E = AU u D ⊥ can be estimated, being u = ( t 0 t E , u 0 ) and D ⊥ the projected separation between the two telescopes in the observer plane. Since D ⊥ is known, one can determine the microlens parallax. In the case of large values of D ⊥ , the light curves seen from two observers will exhibit noticeable difference in the parallax effect (see Ref. [13] for details). Considering the photometric observations towards the Galactic bulge, by the Earth (OGLE) and the space telescopes (K2C9 and Spitzer), we calculated the probability that a microlensing event is detected by two telescopes simultaneously. This probability depends on the mass function index α PL and the space distribution of the FFPs. It is larger at the beginning of the compaign, while it decreases towards the end of it. The detection probability of a microlensing event by OGLE-K2 pair of the telescopes is bigger than by OGLE-Spitzer pair. Moreover, it depends on their threshold amplification and their projected separation.

θ E in microlensing event caused by FFPs
As already anticipated in section 1, the angular Einstein radius can be obtained when finite source effects in the microlensing light curve are detectable. In these events, the value of u 0 becomes comparable to the source radius projected onto the lens plane in units of the Einstein radius and the resulting light curve deviates from the standard form of a point-source event [60]. These deviations depend on the light intensity distribution throughout the source stellar disk. Different brightness profiles have been proposed and discussed in the literature. Among them, that describing the light intensity distribution in the stellar disk more accurately than any other model is the non-linear limb-darkening model [61]. By fitting the microlensing light curve with the Claret model for the source's disk limb-darkening profile, one can define the source radius projected onto the lens plane in units of the Einstein radius. If the angular size of the source may be estimated through the color and the absolute magnitude of the source, then the angular Einstein radius can be measured. For example, Zub et al. [62] have presented a detailed analysis of the highly sampled OGLE 2004-BLG-482 event and have determined the source star limb-darkening coefficients (LDCs) and the angular Einstein radius, which results to be 0.4 μas. Using the LCDs given by the Claret model, we found that the probability of the finite source effect in microlensing events caused by FFPs is about 30% [39].
In the case of bright lens (when the lens is a star), the angular Einstein radius can be detected by the highresolution imaging. Long after the microlensing event, by taking a snapshot with very high precision astrometry, one can easily calculate the relative lens-source velocity μ rel . Combined with the Einstein timescale t E obtained from the light curve, one can thus derive the Einstein radius [63]. In our calculation, we have not considered this method, because the FFPs are not bright objects. Another way to determine the Einstein radius is through astrometric microlensing. The most extensive work on astrometric microlensing was provided by Dominik and Sahu [47], who provided a thorough review of astrometric microlensing of stars. The idea of astrometric microlensing is that, although the state-of-art observatories are not able to resolve the two microlensed images, it is possible to measure the astrometric shift of the centroid of the two images with respect to the source star position. If we consider a source star in the bulge of our Galaxy (D S = 8.5 kpc) and the lens (FFP with mass in the range [10 −5 , 10 −2 ]M ) in the middle of the observer-source distance, the Einstein angular radius results to be, from Eq. (2) in the range 3 ÷ 98 μas. Since, for microlensing events with u 0 ≤ √ 2, the maximum value of the centroid shift is given by max 0.35 θ E , it results in the range 1 ÷ 35 μas. These events are astrometrically detectable if the precision of the astrometric observation is good enough. In Ref. [64], the astrometric signal in microlensing events caused by FFPs by the Gaia space telescope is discussed. These measurements, in combination with photometric observations, can be used to precisely constrain the FFP mass.
Of course, the efficiency of the astrometric effect (that is, in other words, the percentage of events with detectable astrometric shift) depends on the value of the FFP mass function index and it decreases when the value of D L is increased (see [64] for details). In Table 1, we give the efficiency values of the second-order effects in microlensing events caused by the FFPs towards Galactic bulge. Here, the FFPs are considered to be distributed in the thin Galactic disk and the values of the α PL are in the range 0.9-1.6. As one can see, the finite source efficiency is increased with α PL , while the efficiency of the orbital parallax, satellite parallax (for the pairs: Earth-K2C9 and Earth-Spitzer) and astrometric shift decreases with increasing α PL (see Refs. [13,20,39,64] for details).

Conclusions
In this paper, we have discussed and stressed that the gravitational microlensing method is the best one to obtain valuable information about the population of FFPs in the Milky Way. Both photometric and astrometric microlensing observations are very important to solve the parameter degeneracy, at least in a sub-sample of the observed events, and determine the lens parameters. We have stressed that space-based observations are particular important not only to the number of detected microlensing events caused by FFPs, but also for the detection of the second-order effects such as finite source, parallax, and astrometric effects. The first two effects may be detected in surveys as those conducted by Kepler and Spitzer telescopes or by future missions as WFIRST and Euclid, while the latter effect is within the objectives of the Gaia mission.
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