New Astronomy Reviews 53 (2009) 202–208

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New Astronomy Reviews

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Gravitational lensing: From micro to nano

Alexander F. ZakharovInstitute of Theoretical and Experimental Physics, B. Cheremushkinskaya, 25, 117259 Moscow, RussiaBogoliubov Laboratory for Theoretical Physics, JINR, 141980 Dubna, Russia

a r t i c l e i n f o

Article history:Available online 11 August 2009

PACS:95.30.Sf95.75.De97.82.�j97.82.Fs98.62.Sb

Keywords:Gravitational lensingMicrolensing techniques (astronomy)Extrasolar planetsSubstellar companionsPlanetsGravitational lenses and luminous arcs

1387-6473/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.newar.2009.08.002

E-mail address: zakharov@itep.ru

a b s t r a c t

Different regimes of gravitational lensing depend on lens masses and roughly correspond to angular dis-tance between images. If a gravitational lens has a typical stellar mass, this regime is named a microlen-sing because a typical angular distance between images is about microarcseconds in the case whensources and lenses are located at cosmological distances. An angular distance depends on a lens massas a square root and therefore, if a lens has a typical Earth-like planet mass of 10�6 M� , such a regimeis called nanolensing. Thus, generally speaking, one can call a regime with a planet mass lens a nanolen-sing (independently on lens and source locations). So, one can name searches for planets with gravita-tional lens method a gravitational nanolensing. There are different methods for finding exoplanetssuch as radial spectral shifts, astrometrical measurements, transits, pulsar timing etc. Gravitational micr-olensing (including pixel-lensing) is among the most promising techniques if we are interested to findEarth-like planets at distances about a few astronomical units from the host star.

� 2009 Elsevier B.V. All rights reserved.

1 Chwolson described circular images (Chwolson, 1924) and Einstein obtained

1. Gravitational lensing: introduction

Gravitational lensing is based on a simple physical phenomenonthat light trajectories are bent in a gravitational field (in somesense a gravitating body attracts photons). For the first time thisfact was discussed by Newton (1704), but the first derivation ofthe light bending angle in the framework of Newtonian gravitywas published by Soldner (1804). Using a weak gravitational fieldapproximation in general relativity (GR) the correct bending angleis described by the following expression obtained by Einstein(1916) just after his formulation of GR

du ¼ 4GMc2p

; ð1Þ

where M is a gravitating body mass, p is an impact parameter, c is aspeed of light, G is the Newton constant. If M ¼ M� and p ¼ R� aresolar mass and radius, respectively, the angle is equal to 1.0075. In1919 this law was firstly confirmed for observations of light raybending by the Solar gravitational field near its surface (Dysonet al., 1920). Therefore, the Einstein prediction about light bendingwas confirmed by observations very soon after its appearance.

ll rights reserved.

Using Eq. (1) one can introduce the gravitational lens equation

~g ¼ Ds~n=Dd � Dds

~Hð~nÞ; ð2Þ

where Ds is a distance between a source and observer, Dd is a dis-tance between a gravitational lens and observer, Dds is a distancebetween a source and a lens, ~g;~n define coordinates in source andlens planes, respectively, and

~Hð~nÞ ¼ 4GM~n=c2n2: ð3Þ

Taking that the right hand side of Eq. (2) is zero ð~g ¼ 0Þ and substi-tuting ~H from Eq. (3), we obtain the so-called Einstein–Chwolsonradius1 (Schneider et al., 1992): n0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4GMDdDds=ðc2DsÞ

pand the

Einstein–Chwolson angle: h0 ¼ n0=Dd. If Ds � Dd, we have

h0 � 200 � 10�3 GMM�

� �1=2 kpcDd

� ��1=2

: ð4Þ

basic expressions for gravitational lensing (Einstein, 1936). Moreover, it was foundthat Einstein analyzed gravitational lensing phenomenon in his unpublished notes in1912 (Renn et al., 1997).

Fig. 1. Image formation for a circular source S with a radius r ¼ 0:1 and for twodifferent distances d between a source center and gravitational lens GL on thecelestial sphere: d ¼ 0:11 (top panel) and d ¼ 0:09 (bottom panel), where I1 and I2

are images and E is the Einstein–Chwolson ring.

A.F. Zakharov / New Astronomy Reviews 53 (2009) 202–208 203

1.1. Regimes of gravitational lensing

There is a number of reviews and books on gravitational lensing(Schneider et al., 1992; Wambsganss, 1993; Refsdal and Surdej,1994; Zakharov, 1997b; Roulet and Mollerach, 2002; Claeskensand Surdej, 2002). Gravitational lensing in the strong gravitationalfield approximation was also analyzed (Frittelli et al., 2000; Bozzaet al., 2001; Virbhadra and Ellis, 2002; Virbhadra and Keeton, 2008;Virbhadra, 2009).

As it is shown below, in the framework of the simplest point-like lens model (the Schwarzschild lens) angular distances be-tween images are about 2h0 and the angle is proportional to thesquare root of the lens mass for fixed other parameters (includingdistances). So, if a gravitational lens has a typical galactic mass ofabout 1012 M�, a typical distance between images is about a fewangular seconds (it corresponds to the standard gravitationalmacro-lensing regime); if a gravitational lens has a typical stellarmass of about M�, a typical distance between images is about10�6 arcsecond (it corresponds to the gravitational microlensingregime); if a gravitational lens has a typical Earth-like planet massof about 10�6 M�, a typical distance between images is about10�9 arcsecond (it corresponds to the gravitational nanolensing re-gime). Really, 10�9 arcsecond is very small angle and to imagine itone can try to take a look at one inch coin from the distance ofabout 4:5� 109 km (or about 30 AU), which is roughly equal tothe distance between Sun and Neptune.

Naturally, at the moment there is no way to resolve micro- andnano- images but there is a way to discover photometrical featuresof the phenomena by monitoring light curves of backgroundsources (Byalko, 1970). Moreover, there are projects planning toreach angular resolutions at a microarcsecond level (in differentspectral bands) such as NASA Space Interferometry Mission(SIM), ESA Global Astrometric Interferometer for Astrophysics(Gaia) (Lindegren and Perryman, 2000), NASA MicroArcsecondX-Ray Imaging Mission (MAXIM) (Cash et al., 2000; White, 2000),Russian RadioAstron. It is planned to reach even a nanoarcsecondlevel in mm band with space–ground interferometry techniquewith Millimetron mission.2

If a gravitational lens is one of the closest galaxies at a distanceDd ¼ 100 kpc with mass M ¼ 1012 M�, we have h0 � 20000. If a grav-itational lens is a star in our Galaxy at a distance 1 kpc, we haveM ¼ M�, and h0 � 200 � 10�3 (similar, if a lens is a planet at thesame distance with a mass about M ¼ 10�6M� thenh0 � 200 � 10�6). According to a standard terminology proposedmany years ago, if a lens mass is about M� ðM ¼ 10�6 M�Þ we callthis lensing regime a microlensing (nanolensing) independently onlocations of sources and lenses. More generally speaking, searchesfor planets through their impacts on gravitational lensing may benamed a gravitational nanolensing.

We could introduce dimensionless variables

~x ¼~n=n0; ~y ¼ Ds~g=ðn0DdÞ; ~a ¼ ~HDdsDd=ðDsn0Þ; ð5Þ

then we have gravitational lens equation in the dimensionless form:

~y ¼~x�~að~xÞ or ~y ¼~x�~x=x2: ð6Þ

The gravitational lens effect may lead to a formation of severalimages instead of one (see, for instance, Schneider et al., 1992;Zakharov, 1997b). We have two images (or one ring) for theSchwarzschild point lens model, as one can see in Fig. 1. The totalarea of the two images is larger than a source area. The ratio of asum of these two image areas and a source area is called gravita-tional lens amplification A and it is a result of gravitational

2 See, http://www.asc.rssi.ru.

focusing. For example, if a circular source with a radius r and itsarea pr2 is located near a position of gravitational lens on a celes-tial sphere then an area of ring image is equal to 2prREC (the widthof the ring is r and its circumference is 2pREC , since we express alldistances in Einstein–Chwolson radii – REC) and, therefore, magni-fication is 2REC=r (thus one could calculate an asymptote for a mag-nification in a limit r ! 0 by the geometrical way). That is a reasonto call gravitational lensing as gravitational focusing. As one cansee the angular distance between two images is about angular sizeof so-called the Einstein–Chwolson cone with the angle 2h0 (it cor-responds to the Einstein–Chwolson diameter).

2. Gravitational microlensing

There is a number of reviews on gravitational lensing (Wu,1994; Paczynski, 1996; Roulet and Mollerach, 1997, 2002; Zakharovand Sazhin, 1998; Mao, 1999; Jetzer, 1999; Zakharov, 2003, 2005,2008b; Mao, 2008). If a source S lies on the boundary of theEinstein–Chwolson cone, then we have A ¼ 1:34. The microlensingtime is defined typically as a half of the total time of crossing thecone T0:

T0 ¼ 3:5 months �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM

M�

Dd

10 kpc

s� 300 km=s

V;

where V is the perpendicular component of a velocity of a darkbody. If we suppose that the perpendicular component of a velocityof a dark body is equal to �300 km=s (that is a typical stellar veloc-ity in Galaxy), then a typical time of crossing Einstein cone is about3.5 months. Thus, a luminosity of a source S is changed with thattime. We will give numerical estimates for parameters of the micr-olensing effect. If the distance between a dark body and the Sun is

204 A.F. Zakharov / New Astronomy Reviews 53 (2009) 202–208

equal to �10 kpc, then the angular size of Einstein cone of the darkbody with a solar mass is equal to �0:00001 and its linear size isequal to about 10 AU. It is clear that since angular distances be-tween two images are very small, it is very difficult to resolve theimages by ground based telescopes at least in an optical band. Thiswas a reason why Einstein noted that if gravitational lenses andsources are stars and the separation angle between images is verysmall, then gravitational lens phenomenon hardly could be everhardly could ever be detectable (Einstein, 1936).3 However, re-cently, a direct method to measure Einstein angle /E was proposedto resolve double images generated by microlensing with an opticalinterferometer (say, Very Large Telescope Interferometer (VLTI))(Delplancke et al., 2001). Moreover, it was planned to launch astro-metrical space probes, such as US SIM4 and European Gaia,5 theseinstruments which will have accuracies about 10 microarcsecondand could resolve image splitting in the case of microlensing events.Applications of future space missions for astrometrical microlensingsearches are discussed by Zakharov (2006, 2008a).

Microlensing for distant quasars was considered by Gott (1981)(soon after the first gravitational lens discovery by Walsh et al.(1979) and discovered by Irwin et al. (1989) in gravitationallylensed systems since an optical depth for such systems are highest.Later on, features of microlensing in different bands are found ingravitationally lensed systems (Sluse et al., 2007; Sluse, 2008), inparticular, microlensing event signatures were found with 1.5 mRTT -150 telescope for gravitationally lensed system SBS1520+530 (Khamitov et al., 2006). An optical depth of microlensingfor distant quasars was discussed for different locations of micro-lenses (Canizares, 1982; Zakharov et al., 2004, 2005a,b). An influ-ence of microlensing on spectral lines and spectra in differentbands was analyzed (Popovic et al., 2006; Jovanovic et al., 2008).These investigations were inspired by discoveries of microlensingfeatures in X-ray band for gravitational lensed systems (Chartaset al., 2002, 2004; Dai et al., 2003, 2004). These results were ob-tained due to an excellent angular resolution in X-ray band ofthe Chandra satellite enabling us to resolve different images ofgravitationally lensed systems and study their luminositiesseparately.

Basic criteria for microlensing event identification are that alight curve should be symmetrical and achromatic. If we considera spherically symmetric gravitational field of a lens, a point sourceand a short duration of microlensing event then the statementabout the symmetrical and achromatic light curves will be a cor-rect claim, but if we consider a more complicated distribution ofa gravitational lens field or an extended source then some devia-tions of symmetric light curves may be observed and (or) the micr-olensing effect may be chromatic (Zakharov, 1997b).

Many years ago it was found that density of visible matter isabout a few percent of total matter density in galactic halos (Oort,1932; Zwicky, 1933) and the invisible component is called darkmatter (DM). It is now known that the matter density (in criticaldensity units) is Xm ¼ 0:3 (including baryonic matter Xb �0:05—0:04, but luminous matter Xlum 6 0:005), K-term densityXK ¼ 0:7 (Komatsu et al., 2009; Astier et al., 2006; Zakharovet al., 2009). Thus, baryonic density is a small fraction of total den-sity of the Universe. Probably galactic halos are ”natural” places tostore not only baryonic DM, but non-baryonic DM as well. If DMforms objects with masses in the range 10�5—10M�, microlensing

3 However, the microlensing effect is observed analyzing luminosity variations of abackground source as it was originally proposed by Byalko (1970) using gravitationalfocusing the light.

4 http://sim.jpl.nasa.gov/whatis/.5 http://astro.estec.esa.nl/gaia, see also (Lindegren and Perryman, 2000; Perryman

et al., 2001, 2008).

could help us to detect such objects. Thus, before intensive micro-lensing searches it was a dream that microlensing investigationscould help us to solve DM problem for Galactic halo at least.

As it was mentioned before, a possibility to discover microlen-sing by monitoring background stars for the first time was pro-posed by Byalko (1970) (however, to increase a probability in theoriginal paper it was proposed to detect very faint flashes for thebackground star light curves and in this form the idea is hardlyever realizable). Systematic searches of dark matter using typicalvariations of light curves of individual stars from millions observa-ble stars started after Paczynski’s discussion of the halo dark mat-ter discovery by monitoring stars from Large Magellanic Cloud(LMC) (Paczynski, 1986). At the beginning of the nineties new com-puter and technical facilities providing the storage and processingcapabilities for the huge volume of observational data appearedand enabled the rapid realization of Paczynski’s proposal (the situ-ation was different in time of Byalko’s paper). Griest (1991) sug-gested to call the microlenses as Machos (Massive AstrophysicalCompact Halo Objects). Besides, MACHO is the name of theUS–English–Australian collaboration project which observed theLMC and Galactic bulge using 1.3 m telescope of Mount Stromloobservatory in Australia.6 Since one can monitor several millionstars for several years by the microlens searches, the ongoing sear-ches have focused on two targets: (a) stars in the Large and SmallMagellanic Clouds (LMC and SMC) which are the nearest galaxieshaving lines of sight which go out of the Galactic plane and wellacross the halo; (b) stars in the Galactic bulge which allow us to testthe distribution of lenses near the Galactic plane. The first papersabout the microlensing discovery were published by the MACHO col-laboration (Alcock et al., 1993) and the French collaboration EROS(Expérience de Recherche d’Objets Sombres) (Aubourg et al., 1993).7

First papers about the microlensing discovery toward Galacticbulge were published by the US – Polish Optical Gravitational LensExperiment (OGLE) collaboration, which used 1.3 m telescope atLas Campanas Observatory. Since June 2001, after second majorhardware upgrade OGLE entered into its third phase, OGLE IIIand as a result the collaboration observed more than 200 millionstars regularly once every 1–3 nights. During last years OGLE III de-tected several hundred microlensing event candidates each year(Udalski et al., 2003, 2005). The OGLE-III phase has ended onMay 3rd, 2009.8 During the previous observing seasons the EarlyWarning System (EWS) of OGLE-III discovered a number of micro-lensing event candidates (see Table 2).

MOA (Microlensing Observations in Astrophysics) is collabora-tion involving astronomers from Japan and New Zealand (Bondet al., 2001).9

To investigate Macho distribution in another direction onecould use searches toward M31 (Andromeda) Galaxy lying at725 kpc, which is the closest galaxy for an observer in the Northernhemisphere (Crotts, 1992; Baillon et al., 1993; Ansari et al., 1996,1997). On the other hand, there are several suitable telescopes con-centrated in this Earth hemisphere. In nineties two collaborationsAGAPE (Andromeda Gravitational Amplification Pixel Experiment,Pic du Midi, France)10 and Vatican Advanced Technology Telescope(VATT) started to monitor pixels instead of individual stars (Moniez,2001; Le Du, 2001). These teams reported discoveries of severalmicrolensing event candidates (Calchi Novati et al., 2005, 2009).

6 MACHO stopped since the end of 1999.7 EROS experiment stopped in 2002 (Moniez, 2001).8 http://www.astrouw.edu.pl/ogle/ogle3/ews/ews/html. OGLE collaboration plans

to start the phase OGLE IV.9 http://www/roe.ac.uk/%7Eiab/alert/alert/alert/html.

10 The POINT-AGAPE collaboration started in 1999 with the 2.5 m Isaac NewtonTelescope (INT) (Kerins et al., 2001; Belokurov et al., 2005), the new robotic projectAngstrom was proposed as well (Kerins et al., 2006).

A.F. Zakharov / New Astronomy Reviews 53 (2009) 202–208 205

Results of Monte Carlo simulations for these observations and differ-ences between pixel and standard microlensing are discussed by DePaolis et al. (2005), Ingrosso et al. (2006, 2007), Riffeser et al. (2008).

Concerning microlens detections one can say that even manyyears ago there was no doubt about this issue (Paczynski, 1996).However, it is impossible to say exactly which part of the microlen-sing event candidates is actually connected with the effect, sincethere are probably some variable stars among the event candi-dates, or it could be stellar variability of an unknown kind.11 Belowwe will list the most important results. Observed light curves areachromatic and their shapes are interpreted very well by simple the-oretical expressions, however, there is not a complete consent about‘‘very well interpretation”, since even for the event candidate MA-CHO # 1 the authors of the discovery proposed two fits. Dominikand Hirshfeld (1994, 1995)) suggested that this event could be fittedperfectly in the framework of the binary lens model, but one can alsoassume that this microlensing event candidate could be caused by anon-compact microlens (Gurevich et al., 1996a,b; Zakharov andSazhin, 1996a,b; Zakharov, 1998, 1999, 2001a,b).

Using photometric observations of the caustic-crossing binarylens microlensing event EROS BLG-2000-5, Probing Lensing Anom-alies NETwork (PLANET) collaboration reported the first microlensmass determination, namely the masses of these components are0.35 M� and 0.262 M� and the lens lies within 2.6 kpc from theSun (An et al., 2002).

Gravitational microlensing events due to stellar mass blackholes have been discovered by Bennett et al. (2002). The lensesfor events MACHO-96-BLG-5 and MACHO-96-BLG-6 are the mostmassive, with mass estimates M=M� ¼ 6þ10

�3 and M=M� ¼ 6þ7�3,

respectively. However, it was established later that event MA-CHO-99-BLG-22 is a strong BH candidate (78%), MACHO-96-BLG-5is marginal BH candidate (37%), and MACHO-96-BLG-6 is a weakBH candidate (2%) (Poindexter et al., 2005).

The optical depth towards the Galactic bulge is equal to�3� 10�6, so it, which is larger than the initially estimated value(Alcock et al., 2000a), so that there is an additional feature for abar like structure for the Galactic bulge.

5.7 years analysis of photometry of 11.9 million stars in LMC byMACHO collaboration revealed 13–17 microlensing events (Alcocket al., 2000b). The optical depth towards the LMC is equal tosð2 < t < 400 daysÞ ¼ 1:2þ0:4

�0:3 � 10�7, so, it is smaller than the ini-tially estimated value based on an assumption that the halo darkmatter is concentrated in Machos. The maximum likelihood analy-sis gives a Macho halo fraction f ¼ 0:2. Estimates of the followingprobabilities Pð0:08 < f < 0:5Þ ¼ 0:95 and Pðf ¼ 1Þ < 0:05 are gi-ven. The most likely Macho mass M ¼ 0:15—0:9M�, dependingon the halo model and total mass in Machos out 50 kpc is foundto be 9þ4

�3 � 1010M�. EROS collaboration gives a consistent conclu-sion. Namely, this group estimates the following probabilityPðM 2 ½10�7;1M�&f > 0:4Þ < 0:05 (Lasserre et al., 2000; Lasserre,2001). Recently, this collaboration concluded that the optical depthtoward LMC is s < 0:36� 10�7 (with 95% confidence level) whichmeans that Macho contribution to halo mass is less than 7%(Tisserand et al., 2007).

On the other hand, OGLE collaboration claims that the fractionof mass of compact dark matter objects in the Galactic halo couldbe 8 6% (Wyrzykowski, 2009). Their results indicate a non-detec-tion of Machos lensing towards the LMC with an upper limit fortheir abundance in the Galactic halo of 19% for M ¼ 0:4M� and10% for masses between 0.01 and 0.2 M� (Wyrzykowski, 2009).

11 The microlensing event candidates, which were proposed earlier by the EROScollaboration (#1 and #2) and by the MACHO collaboration (#2 and #3) are nowconsidered as the evidence of a stellar variability (Paczynski, 1996).

However, these conclusions are based on assumptions aboutmass and spacial distributions of microlenses but such distribu-tions are not known very well. In principle, microlensing searchesare realistic ways to improve our knowledge, but in this case weneed thousands of events.

When new observational data would be collected and the pro-cessing methods would be perfected, probably some microlensingevent candidates would loose their status, but perhaps new micr-olensing event candidates would be extracted among analyzedobservational data. Thus, the following general conclusion maybe made: the very important astronomical phenomenon was dis-covered, but some quantitative parameters of microlensing willbe specified in future. However, the problem about a content of80% (or even 93% according to EROS point of view) of DM in thehalo of our Galaxy is still open (before microlensing search thepeople hoped that it could give an answer for this problem). Thus,describing the present status Kerins wrote adequately that now wehave ‘‘Machos and clouds of uncertainty” (Kerins, 2001). It meansthat there is a wide field for studies, in particular, pixel microlen-sing, microlensing of gravitationally lensed systems and extrasolarplanet searches seem to be the most promising issues.

3. Methods for exoplanet searches

Mao and Paczynski (1991) evaluated a probability to find a pla-net among microlensing events and they noted that if massivemicrolensing searches toward the Galactic bulge will be carriedon then the first planetary system may be found among microlen-sing events. In spite of the fact that the first planetary system wasfound around the millisecond pulsar PSR1257+12 (Wolszczan andFrail, 1992), the prediction by Mao and Paczynski was correct be-cause: (a) extensive microlensing searches were not realized dueto a time shortage between the first planet discovery and a publi-cation of the article by Mao and Paczynski, but in principle micro-lensing searches may start a little bit earlier since they do not usesuper-advanced technology; (b) nowadays we know that microlen-sing is rather efficient method for exoplanet searches.

At the moment one of the most fruitful technique to find extra-solar planets is based on measurements of radial velocities withthe High Accuracy Radial velocity Planet Searcher (HARPS) spectro-graph. These facilities are installed at the ESO 3.6 m telescope at LaSilla Observatory. A typical uncertainty is about 1 m/s with a fullrange in the 0.7–2 m/s interval depending on weather conditions(Mayor et al., 2009). A summary for radial velocities searches is gi-ven in Table 1 by Perryman et al. (2005). More than 300 planetswere discovered by this method.

Around 60 planets were discovered by transit methods (seeTable 2 in Perryman et al. (2005)), where ground and space facilitiesare listed. A recent launch of Kepler mission significantly increasesexpectations to find new interesting objects with the transit tech-nique. We remind that a diameter of Kepler mirror is more than 3times larger than the diameter of the COnvection ROtation andplanetary Transits (CoRoT) telescope mirror and a field of view ofKepler is more than 100 times larger, but even CoRoT discoveredvery interesting planetary systems such as CoRoT-7b which radiusis about 2 Earth radii (ESA press release on February 3, 2009).

According to J. Schneider database12 (updated on 25 November2008) four planetary systems (with 7 planets and 2 multiple planetsystems) are found with pulsar timing.

At the moment, astrometrical features of exoplanets were foundonly for one system (see, Jet Propulsion Laboratory press release onMay, 28, 200913), but there is a hope that future missions such as

12 See web-site http://www.exoplanet.eu (developing by J. Schneider).13 http://www.jpl.nasa.gov/news/news.cfm?release=2009-090.

Table 1Different regimes of gravitational lensing Wambsganss, 1993.

Prefix/name Deflection angle (arcsecond) Mass m=M� Lens Time delay

Kilo-lensing 103 1018 Supercluster

Macro-lensing 100 1012 Galaxy Months

Milli-lensing 10�3 106 MBH min

Micro-lensing 10�6 100 Star 10�4 secNano-lensing 10�9 10�6 Planet 10�10 secPico-lensing 10�12 10�12 ??? 10�16 secFemto-lensing 10�15 10�18 Comet 10�20 sec

Table 2Microlensing event candidates discovered in the observational campaign of OGLE-III.

Year of observations Number of event candidates

2002 About 3502003 About 4502004 About 6002005 About 5502006 About 6002007 About 6002008 About 650

206 A.F. Zakharov / New Astronomy Reviews 53 (2009) 202–208

James Webb Space Telescope (JWST), SIM, Gaia will provide excel-lent facilities to discover a number of planetary systems with astro-metrical measurements.

An important aspect of exoplanet searches is a potential possi-bility to use different methods to verify conclusions about plane-tary system existence made with only one technique. Forexample, radial velocity measurements and transits or (and) astro-metrical measurements may be complementary (see for instance,observations of extrasolar planet Gliese 876b with the HubbleSpace Telescope and high precision radial velocity measurements)(Benedict et al., 2002).

Much more information about different methods to find extra-solar planets are given in Perryman (2000), Perryman et al. (2005).

4. Exoplanet searches with gravitational microlensing

Since an existence of planets leads to a violation of circular sym-metry of lens (star) and as a result to a formation of fold and cusptype caustics (Schneider et al., 1992; Zakharov, 1995, 1997a), onecan detect extra peaks due to caustic crossing by a background staras a result of its proper motion.

As it was noted above, (Mao and Paczynski, 1991) pointed outthat a probability to discover planetary systems by microlensingis rather high (see also (Gould and Loeb, 1992; Bolatto and Falco,1994)). These conclusions were practically confirmed by furtherobservations.

A list of planets discovered with microlensing searches towardthe Galactic bulge is given in Table 3 (Bennett, 2009; Bennett

Table 3Exoplanets discovered with microlensing (Bennett, 2009, 2008a,).

Star mass Planet mass Semi-major axis

0:63þ0:07�0:09M� 830þ250

�190M� 4:3þ2:5�0:8 AU

ð0:46 0:04ÞM� ð1100 100ÞM� ð4:4 1:8Þ AU

0:22þ0:21�0:11M� 5:5þ5:5

�2:7M� 2:6þ1:5�0:6 AU

0:49þ0:14�0:18M� 13þ4:0

�5:0M� 3:2þ1:5�1:0 AU

ð0:50 0:04ÞM� ð226 25ÞM� ð2:3 0:2Þ AUð0:50 0:04ÞM� ð86 10ÞM� ð4:6 0:5Þ AU

0:060þ0:028�0:021M� 3:3þ4:9

�1:6M� 0:62þ0:22�0:16 AU

0:30þ0:19�0:12M� 260:54þ165:22

�104:85M� 0:72þ0:38�0:16 AU

or 6:5þ3:2�1:2 AU

et al., 2008a; Dong et al., 2008). For the last planetary systemtwo probable regions for distances between a planet and star aregiven due to degeneracies in a determination of planet–star lensparameters (Dominik, 1999; Bennett, 2009). Reports about thesediscoveries were described by (Bond et al., 2004; Udalski et al.,2005; Beaulieu et al., 2006; Gould et al., 2006; Gaudi et al., 2008;Bennett, 2009; Bennett et al., 2008a; Dong et al., 2008). It isremarkable, that the first planet was discovered by MOA-I collabo-ration with a small telescope having an aperture about 0.6 m (Bondet al., 2004; Bennett, 2009). This microlensing event was discov-ered by the OGLE collaboration, but MOA collaboration had a largerfield of view and they produced about five exposures per night foreach of their fields. It was a crucial advantage in spite of muchworse weather conditions in comparison with OGLE. Five giantstars and three super-Earth planets (with masses about 10M�)have been discovered. Therefore, from these data one can claimthat microlensing is very efficient if we wish to find planets withmasses about the Earth mass at distances around a few AU.

Among the most important discoveries one should point outplanet detections made with the microlensing technique (Abeet al., 2004) such as discovery of a planet with 5.5 Earth masses be-cause at that moment it was the lightest discovered extrasolar pla-net, taking into account all the techniques used for extrasolarplanet searches.14 It means that existence of cool rocky planets iscommon phenomenon in the Universe (Beaulieu et al., 2006; Domi-nik, 2006; Dominik et al., 2006).

Pixel-lensing in M31 may provide an efficient tool to searchexoplanets in this galaxy (Ingrosso et al., 2009) (a detailed discus-sion of the issue is far beyond a format of the article). Source starsfor pixel-lensing towards M31 are basically red giants, and there-fore, their typical diameters are comparable to the caustic sizes(Ingrosso et al., 2009), and one has to take into account the sourcefiniteness, similarly to microlensing in quasars (Agol and Krolik,1999; Popovic et al., 2006; Jovanovic et al., 2008). As it is well stud-ied (Witt and Mao, 1994; Bogdanov and Cherepashchuk, 1995a,b;Gaudi and Gould, 1999; Bogdanov and Cherepashchuk, 2000,2002; Dominik, 2005; Heyrovsky, 2007; Pejcha and Heyrovsky,2009) the amplification for finite source cases taking into accountlimb darkening is different from the standard Paczynski amplifica-tion. Giant stars have large angular sizes and relatively high prob-ability to touch planetary caustics (see Ingrosso et al., 2009, fordetails).

5. Conclusions

Searches for exoplanets are connected with searches for life inthe Universe. Clearly, from this point of view the most interestingand exciting planetary systems have planets with masses around

14 Very recently it was reported about a discovery of very light planet with a massabout mp sin i ¼ 1:94M� (Mayor et al., 2009) at the distance about 0.03 AU from thehost star in the GJ 581 multiple planetary system.

A.F. Zakharov / New Astronomy Reviews 53 (2009) 202–208 207

the Earth mass and distances between planets and main sequencestar have to be about AU. Gravitational microlensing is a very effi-cient method for discovering such planetary systems. In this con-text Microlensing Planet Finder (MPF) mission may be veryfruitful and comparable with other space missions for exoplanetsearches (see Fig. 2 in Bennett et al., 2008b and Fig. 1.9 in Bennett,2009).

For distant planetary systems discovered with microlensing, anusage of complementary methods may be rather difficult (at leastat the moment) because they could not be sensitive for such plan-etary systems. However, a potential direct observations of star (forinstance with a space telescope) in a planetary system (Bennettet al., 2006) may be very useful to reduce uncertainties in determi-nation of planetary system parameters.

Acknowledgements

It is a pleasure to ackhowledge S. Calchi Novati, F. De Paolis, G.Ingrosso, P. Jovanovic, Ph. Jetzer, A.A. Nucita, L.C Popovic and M.V.Sazhin for useful discussion and clarifications. The author is grate-ful to profs. M. Dimitrijevic and L.C Popovic for their kind attentionto this contribution. The author thanks comments of an anony-mous referee that helped to improve the quality of the paper.

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