Advances in Astronomy

Volume 2015, Article ID 402303, 8 pages

http://dx.doi.org/10.1155/2015/402303

## Estimating Finite Source Effects in Microlensing Events due to Free-Floating Planets with the Euclid Survey

^{1}Department of Physics, University of Tirana, Tirana, Albania^{2}Department of Mathematics and Physics “Ennio De Giorgi’” and INFN, University of Salento, CP 193, 73100 Lecce, Italy

Received 24 April 2015; Accepted 5 August 2015

Academic Editor: Zdzislaw E. Musielak

Copyright © 2015 Lindita Hamolli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years free-floating planets (FFPs) have drawn a great interest among astrophysicists. Gravitational microlensing is a unique and exclusive method for their investigation which may allow obtaining precious information about their mass and spatial distribution. The planned Euclid space-based observatory will be able to detect a substantial number of microlensing events caused by FFPs towards the Galactic bulge. Making use of a synthetic population algorithm, we investigate the possibility of detecting finite source effects in simulated microlensing events due to FFPs. We find a significant efficiency for finite source effect detection that turns out to be between 20% and 40% for a FFP power law mass function index in the range [0.9, 1.6]. For many of such events it will also be possible to measure the angular Einstein radius and therefore constrain the lens physical parameters. These kinds of observations will also offer a unique possibility to investigate the photosphere and atmosphere of Galactic bulge stars.

#### 1. Introduction

Recent years have witnessed a rapid rise in the number of planetary objects discovered in the Milky Way with mass that are not bound to a host star [1]. These objects are called free-floating planets (FFPs) or also rogue planets, nomads, or orphan planets (see [2] and references therein). Examples of objects of this kind are WISE 0855-0714, about 2 pc away from the Earth and Cha 110913-773444 (see http://exoplanet.eu). Due to their intrinsic faintness, it is very hard to detect them by direct imaging at distances larger than a few tens of parsecs. The only way to detect FFPs further away relies on the gravitational microlensing technique. This method has recently allowed the detection of ten FFPs towards the Galactic bulge, by selecting the microlensing events characterized by duration shorter than 2 days [3]. Indeed the event duration is proportional to the square root of the lens mass and therefore, based on statistical grounds, shorter events are caused by lower-mass lenses.

A consistent boost in the number of detectable FFPs is expected to occur by using space-based microlensing observations, in particular with the planned Euclid telescope. Euclid is a Medium Class mission of the European Space Agency (ESA), which is scheduled to be launched in 2018-2019. Nowadays, the possibility to perform, by using the Euclid satellite, microlensing observations towards the Galactic bulge for about ~10 months [4] is under study. It is only marginally relevant, indeed, if the 10 months of observation is consecutive or not.

A gravitational microlensing event occurs when a massive object passes close enough to the line of sight to a distant source star and is described by three parameters: the time of maximum amplification , Einstein time (where is the Einstein radius and is the transverse velocity between the lens and the source), and the impact parameter (the minimum value of the separation between the lens and the line of sight in units of ). However, of these parameters only the Einstein radius crossing time, , contains information about the lens and this gives rise to the so-called parameter degeneracy problem. One of the ways to break, at least partially, the microlensing parameter degeneracy is by considering the finite source effects induced in the microlensing light curves due to the finite extension of the source stars (that is when the source cannot be considered point-like, as in the Paczynski model) [5]. Finite source effects are nonnegligible when the value of becomes comparable to the source radius projected on the lens’s plane in Einstein radius and the point-source approximation is not valid anymore [6]. These microlensing events are of particular importance due to various reasons. First, an event with a lens passing over a source star provides a rare chance to measure the brightness profile of a remote star. For such an event, different parts of the source star are magnified by different amounts. The resulting lensing light curve deviates from the standard form of a point-source event [7] and the analysis of the deviation may enable measuring the limb-darkening profile of the lensed star [8]. Second, in these kinds of events it is possible to measure the Einstein radius of the lens. The light curve at the moment of the entrance (exit) of the lens into (from) the source disk exhibits inflection of its curvature. The duration of the lens transit over the source , as measured by the interval between the entrance and exit of the lens over the surface of the source star, is [9]where is the normalized source radius (in units of ). Once is measured and if and are known, the normalized source radius can be estimated through the relation in (1). With the additional information about the angular source size, , the angular Einstein radius is measured as . Since the Einstein radius does not depend on , the physical parameters of the lens can be better constrained. Third, these events provide a chance to spectroscopically study remote Galactic bulge stars. Most stars in the Galactic bulge are too faint for spectroscopic observations even with large telescopes. However, the enhanced brightness of lensed stars of high-magnification events may allow performing spectroscopic observations, enabling population study of Galactic bulge stars [10]. These events might also allow performing polarimetric observations of selected ongoing events, for example, by using the VLT telescope, with the aim of further characterizing both the source and the lens system [11–13]. In the present paper we mainly concentrate on finite source effects on microlensing events caused by FFPs with the aim of obtaining a realistic treatment of the events, which are expected to be observable by the Euclid telescope. The great difference between space-based telescopes like the Euclid and usual ground-based microlensing observations is that the amplification threshold detectable by Euclid telescope is that, in turn, implies that the maximum value of turns out to be (much larger than the corresponding value for ground-based observation ).

In this respect, it is also important to emphasize that finite source effects are expected to occur and possibly be observable in a large number of microlensing events due to FFPs. Indeed, the smaller the lens mass is (therefore the smaller ), the more likely it is that these events involve source star disk crossing. The plan of the paper is as follows: in Section 2 we have shown the finite source microlensing equation considering different source limb-darkening profiles. Our main results are presented and discussed in Section 3, and we have summarized the main conclusions of this work in Section 4.

#### 2. Finite Source Effects in FFP Microlensing Events

In order to make realistic treatment of the microlensing event rate due to FFPs expected to be observable by the Euclid telescope, it is important to consider finite source effects in the microlensing event light curves. Each light curve is calculated therefore by the intensity-weighted magnification averaged over the source disk [14]. For example, under the simplified assumption of a uniform source brightness and using polar coordinates centered at the source center, the magnification of an event by a finite source can be expressed asor, in extended form, bywhere is the projected source radius in units of the Einstein radius , is the physical source size, represents the normalized separation between the lens and the center of the source star, and (the radial coordinate in units of the Einstein radius) and are polar coordinates of a point on the source star surface with respect to the source star center. and are the source-observer and lens-observer distance, respectively. Figure 1(a) shows the magnification curve of a selected microlensing event when the source is assumed to be point-like (continuous line) and when it is considered extended, with a uniform brightness profile (dashed line calculated by (3)). Inside the Einstein radius, the residuals between the two curves may be significant. Therefore, by measuring the residuals with respect to the point-like case, one can estimate the angular Einstein radius of the microlensing event which may allow constraining the lens physical parameters.