Microlensing towards the LMC revisited by adopting a non-Gaussian velocity distribution for the sources

0.09 MB·English

by  Luigi Mancini

Astronomy&Astrophysicsmanuscriptno.Mancini˙RN˙Long c ESO2009
(cid:13)
January29,2009
Microlensing towards the LMC revisited by adopting a
non–Gaussian velocity distribution for the sources
(ResearchNote)
9 L.Mancini1,2
0
0
2 1 DipartimentodiFisica“E.R.Caianiello”,Universita`diSalerno,viaS.Allende,Baronissi(SA),Italy
e-mail:[email protected]
n 2 IstitutoNazionalediFisicaNucleare,SezionediNapoli,Italy
a
J

8
2 ABSTRACT

] Aims.WediscusswhethertheGaussianisareasonableapproximationofthevelocitydistributionofstellarsystemsthatarenotspher-
A icallydistributed.
G Methods.Byusinganon-GaussianvelocitydistributiontodescribethesourcesintheLargeMagellanicCloud(LMC),wereinvesti-
gatetheexpectedmicrolensingparametersofalenspopulationisotropicallydistributedeitherintheMilkyWayhaloorintheLMC
.
h (selflensing).WecompareourestimateswiththeexperimentalresultsoftheMACHOcollaboration.
p Results.Aninterestingresultthatemergesfromouranalysisisthat,movingfromtheGaussiantothenon-Gaussiancase,wedonot
- observeanychangeintheformofthedistributioncurvesdescribingtherateofmicrolensingeventsforlensesintheGalactichalo.
o Thecorrespondingexpectedtimescalesandnumberofexpectedeventsalsodonotvary.Conversely,withrespecttotheself-lensing
r case,weobserveamoderateincreaseintherateandnumber ofexpected events.Weconcludethattheerrorintheestimateofthe
t
s most likely value for the MACHO mass and the Galactichalo fraction in form of MACHOs, calculated with aGaussian velocity
a distributionfortheLMCsources,isnothigherthan2%.
[
Keywords.Gravitationallensing–Darkmatter–Galaxy:halo–Galaxies:magellanicclouds

1
v
0
1. Introduction roughly20%ofthehalomassmustbeintheformofMACHOs
9
4 (Alcock et al. 2000; CalchiNovatietal.2005). However, the
4 Galaxies are complex, collisionless, gravitationally-boundsys- interpretation of their data is controversial because of the in-
. tems formed by secular gravitational self-interaction and col- sufficient number of events detected, and the existing degen-
1 lapse of its constituents. Significant progress has been made eration among the parameters. Discordant results have been
0
in both observational and theoretical studies developed to im- reported by other experimental teams (Tisserandetal.2007;
9
0 prove our understanding of the evolutionary history of galax- deJongetal.2004).
ies and the physical processes driving their evolution, leading
:
v to the Hubble sequence of galaxy type that we observe today. Accurate theoretical estimates of the microlensingparame-
i However, many aspects of their features, such as morphology, ters,supportedbystatisticalanalysis,arefundamentaltothein-
X
terpretationoftheexperimentalresults.However,therearemany
compositions,andkinematics,stillremainunclear.Inparticular,
r uncertainassumptionsintheadoptedlensmodels.Theseuncer-
a it is notobvioushow the velocitiesof their constituent(in par-
tainties,thatcouldleadtowardsanincorrectinterpretationofthe
ticularstellar)componentscanbedescribed,becausewecannot
data, are mostly related to the shape of the individual galactic
considerthemtobeisotropicallydistributedatanypoint.Little
componentsandthekinematicsofthelenspopulation.
isknownaboutthevelocitydistribution(VD)ofthestellarpop-
ulationsofgalacticcomponents.Whilethedistributionofstellar One of the first problems to be raised by the scientific
velocitiesinanellipticalgalaxyisgenerallyreasonablycloseto community concerned the shape of the Galactic dark halo.
a Gaussian, analyses of the line-of-sight(l.o.s.) velocity distri- Unfortunately,informationthatcan be extractedfrom observa-
butions of disk galaxies have shown that these distribution are tionsofhigh-velocitystarsandsatellitegalaxiesdoesnotplace
highlynon-Gaussian(Binney&Merrifield1998). strong constraints on its shape. In the absence of precise data,
Today, one of the most important problems regarding the weareaidedbycomputationalmodelsoftheformationofgalax-
compositionoftheMilkyWay(MW)concernstheexistenceof ies,whichsuggestthatthedarkhalosaremoreorlessspherical
dark,compactagglomeratesofbaryonsintheGalactichalo,the (Navarroetal.1996). However, Griest (1991) showed that, re-
so-calledMACHOs(MAssiveCompactHaloObjects).Fromthe ferringtoMACHOs,theopticaldepthisrelativelyindependent
experimentalpointofview,severalobservationalgroupshaveat- ofassumptionsaboutthecoreandcutoffradiioftheMWhalo.
temptedtodetecttheseobjectsbyperformingmicrolensingsur- Sackett & Gould (1993) first investigated the role of the MW
veys in the directions of the Large Magellanic Cloud (LMC), halo shape in Magellanic Cloud lensing, finding that the ratio
Small Magellanic Cloud, and M31. Two groups(MACHO and of the optical depths towards the Small and Large Magellanic
POINT-AGAPE) reported similar conclusions, despite the fact Clouds was an indicator of the flattening of the Galactic dark
that they observed different targets (LMC and M31), that is halo.Alcocketal.(2000)consideredawidefamilyofhalos,be-
2 L.Mancini:MicrolensingtowardstheLMCrevisitedbyadoptinganon–Gaussianvelocitydistributionforthesources(RN)
sidesthesphericalone,rangingfromamassivehalowitharising 2.1.SuperpositionofGaussiandistributions
rotationcurvetomodelswithmoremassivedisksandlighterha-
N-body simulations of different processes of galaxy formation
los. On the other hand, the problemrelated to the shape of the
wereperformedbyIguchietal.(2005).Asaresultoftheirsim-
LMChalowasdefinedbyMancinietal.(2004).Theseauthors
ulations,theseauthorsfoundstationarystatescharacterizedbya
also explored the consequences of different LMC disk/bar ge-
velocitydistributionthatiswelldescribedbyanequallyweighed
ometriesapartfromthecoplanarconfiguration.Allthesestudies
superpositionofGaussiandistributionsofvarioustemperatures,
demonstrated that the estimate of the microlensing parameters
a so-called democratic temperature distribution (DT distribu-
were noticeably affected by the shape of the Galactic halo and
tion),thatis
theotherGalacticcomponents.
For the kinematics of the lenses, the expression of their
1 2 v
r(a2n0d0o6m),-wmhootiocnonvseildoecrietydwthaesLreManCalybzuelkdbmyoCtioalncshiinNcoluvdaitnigettahle. fDT(v)= σ2 rπσe−v2/(2σ2)−|v|"1−Erf √|2|σ!#, (2)
driftvelocityofthediskstars.Thisstudyindicatesthatthemean whereErf(x)istheerrorfunction.TheconclusionisthattheDT
rotationalvelocityoftheLMCstarsisirrelevanttoestimatesof
velocity distribution is a universal property of self-gravitating
theMACHOmicrolensingparametersduetothepreponderance
structuresthatundergoviolent,gravitationalmixing.Theorigin
ofthebulkmotionoftheLMC.Fortheself-lensing,itisslightly
ofsuchuniversalityremains,however,unclear.
significantforlenseslocatedin thebarandsourcesinthedisk.
WeemphasizethattheVDoftheLMCsourceshasalwaysbeen
modeledbyaGaussian.Thisassumptionisjustafirstapproxi- 2.2.Universalvelocitydistribution
mationandwasadoptedforpracticalreasons.Inthispaper,we
investigatewhetherthishypothesisisacceptableornotfordiffer- Hansenetal.(2006)performedasetofsimulationsofcontrolled
ent source/lensconfigurationsor at least providea quantitative collisionexperimentsofindividualpurelycollisionlesssystems
measure of the effectiveness and accuracy of the Gaussian hy- formedbyself-gravitatingparticles.Theyconsideredstructures
initially isotropic as well as highly anisotropic. After a strong
pothesis.Toachievethispurpose,were-examinetheframework
perturbationfollowed by a relaxation, the final structures were
ofmicrolensingtowardstheLMC,andinparticularwerecalcu-
notat all sphericalor isotropic. The VD extractedfrom the re-
latethenumberofexpectedeventsbyassumingthatthesource
sults of the simulations was divided into radial and tangential
velocitiesarenolongerGaussiandistributed.BoththeMACHO
parts.Inthisway,theyfoundthattheradialandtangentialVDs
andtheself-lensingcasesareconsidered.
areuniversalsincetheydependonlyonthe radialortangential
dispersionandthelocalslopeofthedensity;thedensityslopeα
isdefinedastheradialderivativeofthedensityα dlnρ/dlnr.
2. Non-Gaussianvelocitydistributions
≡
Thepointsobtainedbythesimulations,whichdescribetheuni-
Ifweconsiderasphericallysymmetricdistributionofstarswith versaltangentialVD,aredescribedwellbythefollowingfunc-
densityρ,thenwecandescribethedynamicalstateofthesystem tionalform,
byadistributionfunctionofthefollowingform
v v 2 q/(1−q)
F(E)= 2πσρ2 3/2 eE/σ2, (1) wftahne(rvetaσn)=iks2tπhtσaen2tatnang1e−nt(ia1l−veql)o ckitσytatndanis!persion,,v is the tw(o3-)
(cid:0) (cid:1) tan tan
whereE = Ψ v2/2isthebindingenergyperunitmass,andΨ dimensionalvelocitycomponentprojectedontheplaneorthog-
istherelativeg−ravitationalpotential(Binney&Tremaine1987). onaltothel.o.s.,whileqandkarefreeparameters.Hansenetal.
It is well known that the structure of a collisionless system of (2006) reported the universal tangential VD for three different
stars,whosedensityinphasespaceisgivenbyEq.(1),isiden- valuesofthedensityslopeα.Hereweusetheintermediatecase
tical to the structure of an isothermal self-gravitating sphere where α equals -2. This VD has a characteristic break, where
of gas. Therefore,the velocity distribution at each point in the vtan = 1.6σtan is taken to be the transition velocity. The low
stellar-dynamicalisothermalsphereistheMaxwelliandistribu- energypartisdescribedby q = 5/3andk = 0.93.By compar-
tion f(v) = Ne−12v2/σ2, which equals the equilibrium Maxwell- ison,forthe highenergytails, theparametersareq = 0.82and
k=1.3.
Boltzmann distribution given by the kinetic theory. If we con-
sider a stellar system that is far from having a spherical distri-
bution (for example, a galactic flattened disk, a triaxial bulge,
3. MicrolensingtowardstheLMCrevisited
or an elongated bar), we do not expect that it is correct to use
aMaxwelliandistributiontodescribeitsvelocityprofile.Inthe ConcerningtheHubblesequencetype,theNASAExtragalactic
same way,we mustask if itis corrector notto use a Gaussian DatabaseconsiderstheLMCasIrr/SB(s)m.TheLMCisformed
shape f(v) exp( (v2/σ2)) to describe the l.o.s. or projected of a disk and a prominent bar at its center, suggesting that it
∼ −
velocityprofilesofnon-spheroidalgalacticcomponents.Weat- may be considered as a small, barred, spiral galaxy. Different
tempt to answer this question by using non-Gaussian VDs ob- observational campaigns towards the LMC (MACHO, EROS,
tainedbysimulations.Numericalsimulationsofcollapseandre- OGLE,MOA,SUPERMACHO)havebeenperformedwiththe
laxationprocessesofself-gravitatingcollisionlesssystems,sim- aim of detecting MACHOs. Among these, only the MACHO
ilar to galaxies, are useful in identifying their general trends, and EROS groupshave publishedtheir results. The EROS col-
suchasdensityandanisotropyprofiles.Twostudiesoftheveloc- laborationdetectednoevents(Tisserandetal.2007).Incontrast,
itydistributionfunctionofthesesystemsbymeansofnumerical theMACHOProjectdetected16microlensingevents,andcon-
simulationswereperformed,showingthatthevelocitydistribu- cludedthatMACHOsrepresentasubstantialpartoftheGalactic
tion of the resultant quasi-stationary states generally becomes halo mass, but is not the dominant component (Alcock et al.
non-Gaussian(Iguchietal.2005;Hansenetal.2006). 2000). The maximumlikelihoodestimate of the mass m of the
L.Mancini:MicrolensingtowardstheLMCrevisitedbyadoptinganon–Gaussianvelocitydistributionforthesources(RN) 3
lensing objects was 0.5 M , whereas the fraction f of dark
matter in the form o≈f MACH⊙Os in the Galactic halo was esti- 0.03
matedtobe 20%.
∼
In the numerical estimates of the microlensing parameters,
0.025
usefulin studyingthe fraction of the Galactic halo in the form
ofMACHOs,aGaussianshapevelocitydistributionisstillcom-
monly used to describe the projected velocity distribution for
0.02
the lenses as well as the source stars, although they are not
spherically distributed (Jetzeretal.2002; Mancini et al. 2004; Lvtan
Awasssetfoeuttailli.z2e0t0h6e;nCoanl-cGhaiuNsosivaantiveetloacl.it2y0p0r6o)fi.lHeserdee,socurirbiendteinntitohne Hftan0.015
previous section for the sources, instead of the usual Gaussian
shape,andshowhowthemicrolensingprobabilitieschangeac- 0.01
cordingly.Asaconcretecase,inSect.3.1weanalyzedtwomain
parameters of the microlensing towards the LMC, the rate and
0.005
thenumberofexpectedmicrolensingeventsgeneratedbyalens
population belonging to the MW halo as well as one belong-
ingtotheLMCitself.Theresultsofourmodelwerecompared
withtheMACHOcollaborationobservationalresults(Alcocket 20 40 60 80 100
al. 2000). Finally, the method of maximum likelihood is used vtan
inSect.3.2tocalculatetheprobabilityisocontoursinthe m, f
{ }
plane. Fig.1. The velocity profiles used to describe the kinematics of
the LMC stars: VDs derived from the simulations of Iguchi et
al.(2005)(dashedcurve)andHansenetal.(2006)(dottedline).
3.1.Microlensingrateandnumberofexpectedevents ThesolidblacklinerepresentsaGaussianprofile.
We restricted our analysis by consideringa homogeneoussub-
setof12Paczyn´ski-likeeventstakenfromtheoriginallargerset
B reported by MACHO; we did not consider the Galactic disk
events,thebinaryevent,andallcandidateswhosemicrolensing Efield 0∞(dΓ/dTE)E(TE)dTE, where Efield is the field exposure,
origin had been placed in doubt. In our calculations, we used dΓ/dTR isthedifferentialratewithrespecttotheobservedevent
E
the models presented in Mancini et al. (2004) to represent the duration,T istheEinsteintime,and (T )isthedetectionef-
E E
E
various Galactic components: essentially an isothermal sphere ficiencyoftheexperiment.Thedifferentialrateisdefinedtobe
for the Galactic halo, a sech2 profile for the LMC disk, and a (Mancinietal.2004;CalchiNovatietal.2006)
triaxialboxy-shapefortheLMCbar.vanderMareletal.(2002)
measuredthevelocitydispersionoftheLMCsourcestarstobe dΓ 2π 2π π/2
20.2km/s.Thismeasurementwascompletedasusualbyaquan- = dα dϕ cosθdθ ∞ f (v )dv
s s
titativeanalysisoftheabsorptionlinesintheLMCspectrum,by dTE Z0 Z0 Z π/2 Z0 ×
−
assumingaGaussianformfortheVD.Inprinciple,toobtainan v2 v2+x2v2+2xv v cosϕ
etbisyotniam,pwaptleeyhionafgvteahtenoovrnee-lpGoecaaiuttytshsdieaissnpaamelrgesoiaornnitahflmoyrs.iasToonfoatnhfi-erGsLtauMaspsCpiralonixndiemipsatrrtoiibfiounle-, × Z0∞µma2xπRℓσE2ℓdnex(xp)d−µℓ 1xdxs 2σdℓminρℓ (sD )DddvℓD×, (4)
weignoredthissubtletyandsimplyassumedthatthedispersion × Z dµ Z Z s os os os
of Gaussian and non-Gaussian VDs were equal. Fixing in this µmin N 0 dmin
waythevaluesofthevelocitydispersions,wedrawinFig.1the
where ρ is the source density, f(v ) represents the two-
universaltangentialVD (dottedline) togetherwith the TD dis- s s
dimensional transverse velocity distribution of the sources, x
tribution(dashedcurve),anda classical Gaussianprofile(solid
is the ratio between the observer-lens distance D and the
line). ol
observer-sourcedistance D , whereasµ isthelensmassinso-
os
In general,the velocityofthe lenses v consists ofa global
ℓ lar mass units. The normalizationfactor is the integralover
rotationplusadispersivecomponent.Sinceweassumedthatthe N
the l.o.s. of the sources. The distribution dn(x)/dµ represents
MW halo has a spherical form, we considered that the lenses the number of lenses with mass between µ and µ + dµ at a
aresphericallydistributed.Inthiscase,therotationalcomponent
given point in the Galactic component considered. Assuming
could be neglected,and at the same time we could safely con-
thefactorizationhypothesis,wecanwritedn(x)/dµastheprod-
siderthedistributionofthedispersivecomponenttobeisotropic
uctofa distributiondn /dµdependingonlyonµandthe perti-
0
and Maxwellian (deRu´julaetal.1995). This assumption was
nentdensityprofile(deRu´julaetal.1995;Mancinietal. 2004)
also supportedby an analysisof the kinematicsof nearly2500 dn(x) = ρℓ(x)dn0, where ρ is the lens density. Concerning the
Blue Horizontal-Branch Halo stars at z 4 kpc, and with dµ M dµ ℓ
distances from the Galactic center up |to| ≥ 60 kpc extracted functionalf⊙ormofdn0/dµ,wesupposedthatforthelensesinthe
from the Sloan Digital Sky Survey, where∼the observed distri- halothemassfunctionispeakedataparticularmassµ0,sothat
butionofl.o.s.velocitiesiswell-fittedbyaGaussiandistribution itcouldbedescribedbyadeltafunctiondn0/dµ=δ(µ µ0)/µ0.
−
(Xueetal.2008). For lenses in the LMC disk/bar, we utilized the exponential
It is well known that the number of events N is the form dn0/dµ = Aµ−αe−(µ0/µ)β (Chabrier2001), where α = 3.3,
sum, N = Nfield, of the number of events expected for µ0 = 716.4,β = 0.25,whereas A is obtainedfromthe normal-
each monitorePd field of the experiment defined to be Nfield = izationcondition 01.008Aµ1−αe−(µ0/µ)βdµ=1.
R
4 L.Mancini:MicrolensingtowardstheLMCrevisitedbyadoptinganon–Gaussianvelocitydistributionforthesources(RN)
3.1.1. LensesintheGalactichalo 1.2
We calculated the differential rate of the microlensing events 1
with respect to the Einstein time, along the lines pointing to-
wards the events found by the MACHO collaboration in the 0.8
LMC and for different values of µ . We used a Gaussian VD LŸ
0 M 0.6 ´
for f(v )aswellasthenon-GaussianVDs,givenbyEqs.(2)and H
s m
(3).Asµ andthel.o.s.change,wedidnotobserveanysubstan-
0 0.4
tial reduction in the height of the distribution curve of the mi-
crolensingeventrate,andthecorrespondingexpectedtimescale 0.2
did not vary among the cases considered. With respect to the
numberof events,the situation did notchange.Takinginto ac- 0.2 0.4 0.6 0.8 1
count the MACHO detection efficiency and the total exposure, f
we calculatedthe expectednumberof events,summedoverall
Fig.2.LikelihoodcontoursforMACHOmassmandHalofrac-
fields examined by the MACHO collaboration in the case of a
tion f foratypicalsphericalHalo.Thecontoursencloseregion
haloconsisting(100%)ofMACHOs.BothintheGaussianand
the non-Gaussiancase, we achievedthe well-knownresult that of 34%, 68%, 90%, and 95% probability. The cross shows the
the expected number of events is roughly 5 times higher than maximum-likelihoodestimate. Two different curves are shown
observed. foreachcontouraccordingtothevelocityprofileadoptedforthe
sources:aGaussianshape(solidline)andauniversalVD(gray
dottedline).
3.1.2. Self-lensing
Werepeatedthesameanalysisfortheself-lensingconfiguration, mostsignificantvariationisintheestimateofthelensmass,but
thatis whereboth the lenses and the sourcesare locatedin the thatthis is nothigherthan 2%forthe 95%probabilitycontour
disk/baroftheLMC.Byvaryingthel.o.s.,wefoundingeneral line.
thatthemicrolensingdifferentialrateforthenon-Gaussiancase
washigherthanthatfortheGaussiancase.Wenotedthattheex-
pectedtimescalealsovaried.BetweentheGaussianandthenon- 4. Discussionandconclusion
Gaussian case, we also observed that the median value of the
We have investigated the limits of the validity of the Gaussian
asymmetricdistributionsdecreasesofroughly20%.Concerning
approximationusedtodescribethekinematicsofasourcepop-
the expected number of microlensing events, we estimated the
ulation in a microlensing context. This hypothesis, due to its
same variation for sources with a non-Gaussian VD, that is an
practicality, has always been adopted without any check of its
increaseofroughly20%fromthevalueof 1.2eventsobtained
∼ plausibility.Wehaveremediedthisdeficiencyinconfirmationby
withaGaussianVD(Mancinietal.2004).
anexhaustiveanalysis.Todescribethe motionofa stellarpop-
ulation with a non-spheroidal distribution as correctly as pos-
3.2.MACHOHalofractionandmass sible, we utilize two VDs (Sect. 2.1, Sect. 2.2) extracted from
numericalsimulationsofcollisionslesssystemsformedbyself-
FollowingthemethodologyofAlcocketal.(2000),namelythe gravitatingparticles.TheseVDsaresubstantiallydifferentthan
methodofmaximumlikelihood,weestimatedthehalofraction
foraGaussianone.Weconsideredstarsinthediskandbarcom-
f in theformofMACHOs andthemostlikely MACHO mass.
ponentsoftheLMCandinvestigatedtheirpotentialtobesources
Thelikelihoodfunctionis
oflensingbytransientlenses.Inthisframework,werecalculated
Nobs dΓ themainmicrolensingparameters,includingtheMACHOhalo
L(m, f)=exp(−Nexp) "EE(TEi)dT TEi,m # , (5) fractionandthemostlikelyvalueforthelensmass.
Yi=1 E (cid:0) (cid:1) For a configurationin which the lenses and sourcesbelong
where N is thetotalnumberofexpectedevents,anddΓ/dT to the target galaxy, we detected an increase in the differential
exp E
is the sum of the differential rates of the lens populations rate of microlensing events towards the LMC when we used a
(MACHOs, LMC halo, LMC disk+bar). The MACHO contri- non-GaussianVD to describe the motionof its stars instead of
bution is multiplied by f. The product applies to the N ob- aGaussianone.Thisincreaseisreflectedintheestimateofthe
obs
served events. The resulting likelihood contours are shown in number of expected events, which is roughly 20% higher than
Fig.2,wheretheestimateofthedifferentialratewasperformed the1.2eventsfoundfortheGaussiancase.
using a Gaussian VD (solid line) and a universal VD (dashed ThepredictionforahalothatconsistsentirelyofMACHOs
line).TheprobabilitieswerecomputedusingaBayesianmethod isafactorof 5abovetheobservedrates.Thesituationdoesnot
∼
withaprioruniformin f andm.Asphericalisothermaldistribu- changeinanoticeablewayifweconsidera non-GaussianVD,
tionwasusedtodescribethelensdensityintheMWandLMC sincewehavefoundthatthenumberofeventsexpectedispracti-
haloes.We foundthatthemostprobablemassfortheGaussian callyequaltothepreviouscase.Theresultsremainvalidforboth
case is m = 0.60+0.40 M , where the errors are 68% confi- theDTandtheuniversalVD. Themaindifferencebetweenthe
0.33
denceintervals,and−f =23⊙%witha95%confidenceintervalof velocitydispersionoftheLMCstarsandtheMACHOs,practi-
10% 47%.We notethatthesevaluesareslightlyhigherthan, callyneutralizesanypossiblevariationduetothedifferentshape
−
although fully compatible with, the original result reported by oftheVDofthesources.Themaximum-likelihoodanalysispro-
Alcocketal.(2000).Themismatchisduetosomedifferencesin videsvaluesformand f thatarequitesimilarfortheGaussian
themodellingandthefactthatthesetoftheeventsconsideredis andthenon-Gaussiancase.Weconcludethattheerrorinthees-
smaller.IfweconsiderthatthevelocitiesofthestarsintheLMC timateofthemostprobablevaluefortheMACHOmassaswell
arenon-Gaussian-distributed,thelikelihoodcontourshavemini- as for the Galactic halo fraction in the form of MACHOs, cal-
maldifferencesfromthoseofthepreviouscase.Wenotethatthe culatedwithaGaussianVDfortheLMCsources,isroughlyof
L.Mancini:MicrolensingtowardstheLMCrevisitedbyadoptinganon–Gaussianvelocitydistributionforthesources(RN) 5
the order of 1 2%. This fact implies that, in the study of the
−
MW halocompositionby microlensing,a Gaussian profile isa
reasonableapproximationforthe velocitydistributionofa sys-
temofsourcestars, eveniftheyarenotsphericallydistributed.
On the other hand,for self-lensing,the Gaussian doesnotpro-
vide a good description of the kinematics of a non-spherically
distributedstellarpopulation,inasimilarwaytothediskorthe
baroftheLMC.Toensureaccuratemicrolensingpredictions,it
is thus necessary to replace the Gaussian VD by a more phys-
ically motivated one, which takes into account the real spatial
distributionofthesourcestars.
Acknowledgements. TheauthorwishtothankValerioBozza,GaetanoScarpetta,
and the anonymous referee for their contribute toimprove the quality ofthis
work, and Steen Hansen and Sebastiano Calchi Novati for their useful sug-
gestions andcommunications. Theauthor acknowledge support for this work
byfundsoftheRegioneCampania,L.R.n.5/2002,year2005(runbyGaetano
Scarpetta),andbytheItalianSpaceAgency(ASI).
——————————————————————
References
Alcock,C.,Allsman,R.A.,Alves,D.R.,etal.2000,ApJ,542,281
Assef,R.J.,Gould,A.,Afonso,C.,etal.2006,ApJ,649,954
Bennett,D.P.2005,ApJ,633,906
Binney,J.,&Tremaine,S.1987,GalacticDynamics(PrincetonUniv.Press)
Binney,J.,&Merrifield,M.1998,GalacticAstronomy(PrincetonUniv.Press)
CalchiNovati,S.,Paulin-Henriksson,S.,An,J.,etal.2005,A&A,443,911
CalchiNovati,S.,DeLuca,F.,Jetzer,Ph.,&ScarpettaG.2006,A&A,459,407
Chabrier,G.2001,A&A,554,1274
deJong,J.T.A.,Kuijken,K.,Crots,A.P.S.,etal.2004,A&A,417,461
deRu´jula,A.,Giudice,G.F.,Mollerach,S.,etal.1995,MNRAS,275,545
Griest,K.1991,ApJ,366,413
Hansen,S.H.,Moore,B.,Zemp,M.,&StadelJ.2006,JCAP,01,014
Iguchi,O.,Sota,Y.,Tatekawa,T.,etal.2005,Phys.Rev.E71,016102
Jetzer,Ph.,Mancini,L.,&Scarpetta,G.2002,A&A,393,129
Mancini,L.,CalchiNovati,S.,Jetzer,Ph.,&Scarpetta,G.2004,A&A,427,61
Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1996,ApJ,462,563
Sackett,P.D.,&Gould,A.1993,ApJ,419,648
Tisserand,P.,LeGuillou,L.,Afonso,C.,etal.2007,A&A,469,387
vanderMarel,R.P.,Alves,D.R.,Hardy,E.,etal.2002,AJ,124,2639
Xue,X.-X.,Rix,H.-W.,Zhao,G.,etal.2008,ApJ,684,1143