A simple prescription for simulating and characterizing gravitational arcs

0.64 MB·English

by  Cristina Furlanetto

Astronomy&Astrophysicsmanuscriptno.furlanetto c ESO2013
(cid:13)
January24,2013
A simple prescription for simulating and characterizing
gravitational arcs
CristinaFurlanetto1,2,Bas´ılioX.Santiago1,2,Mart´ınMakler3,2,Cle´ciodeBom3,2,CarlosH.Brandt4,3,AngeloFausti
Neto2,PedroC.Ferreira5,LuizNicolacidaCosta6,2,andMarcioA.G.Maia6,2
1 DepartamentodeAstronomia,UniversidadeFederaldoRioGrandedoSul,Av.BentoGonc¸alves,9500,PortoAlegre,RS91501-
970,Brazil
2 Laborato´rioInterinstitucionaldee-Astronomia,RuaGen.Jose´Cristino,77,RiodeJaneiro,RJ20921-400,Brazil
3 3 CentroBrasileirodePesquisasF´ısicas,RuaDr.XavierSigaud150,RiodeJaneiro,RJ22290-180,Brazil
1 4 Laborato´rioNacionaldeComputac¸a˜oCient´ıfica,Av.Getu´lioVargas,333,Petro´polis,RJ25651-075,Brazil
0 5 DepartamentodeF´ısica,UniversidadeFederaldoRioGrandedoNorte,CampusUniversita´rio,Natal,RN59072-970,Brazil
2 6 Observato´rioNacional,RuaGen.Jose´Cristino,77,RiodeJaneiro,RJ20921-400,Brazil

n Preprintonlineversion:January24,2013
a
J
ABSTRACT

3
2 Simplemodelsofgravitationalarcsarecrucialforsimulatinglargesamplesoftheseobjectswithfullcontroloftheinputparameters.
These models also provide approximate and automated estimates of the shape and structure of the arcs, which are necessary for

] detecting and characterizing these objects on massive wide-area imaging surveys. We here present and explore the ArcEllipse, a
O
simpleprescriptionforcreatingobjectswithashape similartogravitational arcs.Wealsopresent PaintArcs,which isacodethat
C couplesthisgeometricalformwithabrightnessdistributionandaddstheresultingobjecttoimages.Finally,weintroduceArcFitting,
whichisatoolthatfitsArcEllipsestoimagesofrealgravitationalarcs.Wevalidatethisfittingtechniqueusingsimulatedarcsandapply
.
h ittoCFHTLSandHSTimagesoftangentialarcsaroundclustersofgalaxies.OursimpleArcEllipsemodelforthearc,associatedto
p aSe´rsicprofileforthesource,recoversthetotalsignalinrealimagestypicallywithin10%-30%.TheArcEllipse+Se´rsicmodelsalso
- automaticallyrecovervisualestimatesoflength-to-widthratiosofrealarcs.Residualmapsbetweendataandmodelimagesreveal
o
theincidence of arcsubstructure. Theymay thusbeused asadiagnosticforarcsformed bythemergingof multipleimages. The
r
t incidenceofthesesubstructuresisthemainfactorthatpreventsArcEllipsemodelsfromaccuratelydescribingreallensedsystems.
s
a Keywords.gravitationallensing:strong–methods:analytical
[

2
v 1. Introduction Survey (DES)1, will provide large samples of gravitational arc
1 systems that will enable numerous statistical studies. Because
7 of the large area, it will be virtually impossible to visually in-
7 Gravitational lensing is an important tool for a variety of as- spectthewholesurveytolookforgravitationallylensedsources.
2 trophysical applications. In particular, gravitationally lensed Therefore,automateddetectionmethodsareultimatelyneeded.
1. arcs produced by massive galaxy clusters can provide infor- There are a few proposals for automated detection al-
1 mation on their total inner mass distribution (Hattorietal. gorithms in the literature, although none of them is con-
2 1999; Guzik&Seljak 2002; Mandelbaum&Seljak 2006; Treu sidered as the default algorithm (e.g., Lenzenetal. 2004;
1 2010; Kneib&Natarajan 2011). Statistical analyses of grav- Horeshetal. 2005; Seidel&Bartelmann 2007; Cabanacetal.
v: itational arcs provide constraints on the evolution of struc- 2007; Moreetal. 2012; Bometal. 2012). Moreover, most of
i ture with cosmic time and on the cosmological parameters these algorithms are not fully characterized in the sense that
X (Hilbertetal. 2007; Vuissozetal. 2007; Kochaneketal. 2006; they do not have a determined efficiency and contamination.
r Meneghettietal. 2004; Golseetal. 2002; Bartelmannetal. These quantities could be quantified by using samples of im-
a 1998; Estradaetal. 2007; Hennawietal. 2008; Kauschetal. ageswithsimulatedarcs,whichwouldactastruthtablesforthis
2010;Moreetal.2012).Becauseofthefluxamplificationasso- type of study. Furthermore, simulated arc images are useful to
ciatedtogravitationallensing,thesearcsarealsoaprobeforthe testincontrolledstudiesarcmeasurementtoolsandimagepre-
high-redshiftuniverse, which is unaccessible throughunlensed processingalgorithmsusedtoenhancetheirdetectability.
sources. Currently,theliteraturelackssufficientlydetailedworksre-
latedtothefastsimulationandcharacterizationofgravitational
On the other hand, gravitational arcs are very rare objects, arcs. Arc simulations have mostly been based on ray-tracing
since each square degree of the sky contains approximately techniquesgivenadistributionofsourceandlensproperties.The
one galaxy cluster massive enough to produce arcs by grav- commonpracticeistomeasurethearc’slengthL,widthW,and
itational lensing (Gladders&Yee 2005). So far an order of its distance to the center of the lens. Few authorshave tried to
hundreds of these objects were detected (Luppinoetal. 1999; constrain their shapes and orientation. Miralda-Escude (1993)
Zaritsky&Gonzalez 2003; Gladdersetal. 2003; Sandetal. introducedtheideaofmeasuringLbyfittingacirclearcthrough
2005;Cabanacetal.2007;Boltonetal.2008).Futurewide-field
surveys with sub-arcsecond seeing, such as the Dark Energy 1 http://www.darkenergysurvey.org/
1
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
thearccenteranditsextremes.Bartelmann&Weiss(1994)have thetwoextremepointswhere∆r =0.ThisleadstoL=2a.The
pioneeredtheideaofapproximatinggravitationalarcsasanel- width at ∆θ = 0 is givenby W = 2b. Therefore,the length-to-
lipse whose major axis is equal to L (the so-called ellipse fit- widthratiooftheArcEllipseissimplygivenasL/W =a/b.
ting). However, we are not aware of any preceding study that In polar coordinates, if the origin of the coordinate system
hasintroducedasurfacebrightnessdistributiontosimulateand correspondstothecenterofcurvatureoftheArcEllipse,wehave
characterizegravitationalarcs. ∆r =r r and∆θ=θ θ ,whereθ isthepositionangleofthe
c 0 0
− −
Motivated by these questions, we developed a code called ArcEllipsecenter,suchthat
PaintArcs which draws geometric figures that mimic arcs and
adds them to images. It currently uses the prescription of an r (θ θ) 2
ArcEllipse,whichisananalyticalexpressionforthedeformation r =rc b 1 c 0− . (3)
± ± s − a
ofanellipsesuchthatoneofitsmainaxisbecomesacircleseg- " #
ment.Onlyahandfuloffullycontrollableparametersgointothis
TheArcEllipse is thereforedefinedby fourparameters:the
simple prescription.We use the ArcEllipse in conjunctionwith
length2a,thewidth2b,thecurvatureradiusr ,andtheposition
aSe´rsicprofiletotakeintoaccountthesurfacebrightnessdistri- c
angleθ .
bution of the simulated arc. We also present ArcFitting, which 0
Fromthesolutionabove,itissimpletoshowthattheareaof
isacodethatusestheArcEllipse+Se´rsicmodeltoestimatethe
theArcEllipseisgivenby
morphologicalandstructuralparametersofrealarcs.
The prescriptions that we adopt here for the shape and θ0+a/rc r+(θ)
structure of arcs are certainly a simplification of real gravita- A = rdrdθ=
ArcEll
tional arcs, which are often formed by the merging of individ- Zθ0−a/rc Zr−(θ)
ual multiple images of a source. One such situation are the gi- θ0+a/rc r+(θ)2 r (θ)2
ant arcs formed when a source is close to the cusp of a caus- = − − dθ=
tic. Particularly when real arcs are imaged in high resolution, Zθ0−a/rc (cid:16) (cid:17)2(cid:16) (cid:17)
theArcEllipseshapewithaSe´rsicprofilewillnotaccommodate πLW
= πab= , (4)
their substructures. Our goal here is to exactly to quantify the 4
observeddiscrepanciesfromasimplemodeland,asaresult,as-
i.e.,identicaltothatofanellipsewithsemi-axesaandb.
sesswhichimprovementsaremostpromisingandtowhatextent
Inordertorepresentanarc,wemustimposesomelimitson
these simple models can be used in controlled experimentsfor
theArcEllipseparameterssuchasb<r ,toensurethearcshape,
automaticarcdetectionalgorithms. c
anda<2πr ,toavoidclosingtheArcEllipseintoaring.
This paper is organized as follows. In 2 we present the c
§
ArcEllipse, including the addition of a surface brightness pro-
file and asymmetry. In 3 we describe the PaintArcs code, ac- 2.1.SurfacebrightnessprofileforArcEllipses
§
countingforimagepixelation,andincludingseeingeffectsand
To create a surface brightness map with ArcEllipse isophotes,
Poisson noise in the simulated image. The ArcFitting methods
fromagivenradialprofile,weneedtosimplyreplacetheargu-
are presented in 4 and their application to fit real images are
§ mentinaradialprofilebythefunction
presentedin 5.Finally,in 6,wediscusstheresultsandfuture
§ §
work.
r (θ θ ) 2 r r 2
R= c − 0 + − c , (5)
s a b
2. ArcEllipse ! (cid:18) (cid:19)
which is constantoverArcEllipses. Notice thatin this case the
TheArcEllipseconsistsofasimpleideatoanalyticallyexpress
ArcEllipsemustbe centeredat the origin(centerof curvature).
arcshapesasthedistortionofanellipsesuchthatoneofitsmain
Wecanrewritethisexpressionas
axes is bent into an arc of a circle. The resulting geometrical
figurecanbeusedtocreateasurfacebrightnessmapthatmimics 1 1
agravitationalarc. R= [rc(θ θ0)(1 e)]2+(r rc)2 = Rprof, (6)
b − − − b
Anordinaryellipseisformedbythesetofpointswhosedis-
p
tancestothexandyaxes,weightedbythesemi-axes,aandb, wheree=1 b/aistheellipticity.
−
satisfy
x 2 y 2 2.2.AsymmetricArcEllipse
+ =1. (1)
a b Sincethegravitationalarcsingeneralarenotsymmetric,wecan
(cid:18) (cid:19) (cid:18) (cid:19)
Followingthesameidea,wecandefinetheArcEllipseasthe alsoaddsomeasymmetrytotheArcEllipsemodel.Thesimplest
setofpointswhosedistancesfromapointonthecircumference waytoimplementthatistoconsidertwodifferentvaluesforthe
alongthetangentialdirection(rc∆θ)andalongtheradialdirec- semi-major axis, a1 and a2, preserving the semi-minor axis as
tion(∆r)satisfy b; in other words, two ArcEllipses with different a matched at
θ . Points whose angle θ are larger that θ will therefore have
0 0
r ∆θ 2 ∆r 2 ellipticity
c + =1, (2)
a b
b
! ! e =1 , (7)
1
where rc is the radius of curvature of the circle (and thus the − a1
curvatureradiusofthearc).
whereaspointswhoseangleθaresmallerthatθ willhave
Thissimple expressionindeedgivesa shapethatresembles 0
an arc, as we can see in the left panel of Fig. 1. We define the b
e =1 . (8)
lengthofanArcEllipseasthelengthofthearcsegmentbetween 2
− a
2
2
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
Fig.1.GeometricalfiguresproducedbytheArcEllipse.Leftpanel:symmetricArcEllipsecreatedwithe = 1 b/a=
0.9,b = 1pixel,r = 15pixels,andθ = 45o.Rightpanel:asymmetricArcEllipsecreatedwith e = 0.9,e −= 0.75,
c 0 1 2
withtheremainingparametersasintheleftpanel.
2194697
Nowthelengthisgivenby +O(n 5),
−
− 30690717750n4
1 1
L=a1+a2 =b 1 e + 1 e (9) whichis betterthanonepartin 10−4 forn > 0.36.Note that
− 1 − 2! if r is defined in a differentway≈, the expressionfor b will be
e n
andEq.(4)stillholds. different.
IntherightpanelofFig.1weshowanexampleofanasym- For an arc created with ArcEllipse we choose b = r . In
e
metricArcEllipse.Ithasthesamevaluesofr andθ asthesym- otherwords,wedistortthecircularprofileonlyinthetangential
c 0
metricarcshownontheleftpanel.Butnowwesete =0.9and direction,preservingitseffectivesize(r )intheradialdirection.
1 e
e =0.75. Inthiscase,theArcEllipsewithSe´rsicprofileisgivenby
2
I(r,θ) = I R (r,θ) = (11)
S prof
3. PaintArcs:animplementationofArcEllipse
(cid:16) (cid:17) [r (θ θ )(1 e)]2+(r r )2 1/n
PaintArcsis a code written in Pythonthatimplementsa model = I exp b c − 0 − − c ,
0 n
bttorroignsihomtmnueilcasastelpiromobfiajeglecetsws.hwItoispteherrafaordcrimamlsaotrrhgpeuhmodlieogngittyailsaizgnaidtvieoandndbosyfttthhheeemEsuqtor.f(aa6cs)e-. whereI0 =Ieebn.− p re  
Italsoperformspointspreadfunction(PSF)convolution,toadd We need to normalizethe intensity of the ArcEllipse I(r,θ)
theeffectoftheatmosphereandinstrument,andaddstheeffect suchthatthetotalsignalofthearc,givenby
of Poisson fluctuations on the image counts. The code makes
use offunctionsfromSLtools2, a libraryforimage processing, = 2π ∞I(r ,θ)r dr dθ, (12)
′ ′ ′ ′ ′
catalogmanipulation,andstrong-lensingapplications. L
Z0 Z0
PaintArcs is currently set to use the ArcEllipse model to
isequaltothetotalsignalofthearcintheimage,whichisgiven
mimicgravitationalarcshapes.Butitisbuiltinamodularway
by
that can accommodate other prescriptions. The surface bright-
ness distribution currently uses the Se´rsic (1968) model, for Stotal =10−0.4(m−mzpt), (13)
whichtheintensityprofileisgivenby
where m and m are the totalmagnitudeof the arc to be sim-
zpt
r 1/n ulated and the zero point of the magnitude scale of the image.
I (r)=I exp b 1 , (10)
S e − n re! −  Therefore,theconstantI0canbefoundas
where Ie is the intensity at theeffective radius re that encloses I0 = 10−0.4(m−mzpt) .(14)
halfofthetotallightfromthemodel.Theconstantb isdefined 1/n
pinrotfierlem.sThoefvthaeluepaorfabmectaenr bne, cwohmicphutdeedsncruimbeesritchaellyshnoarpoebtoafinthede 02π 0∞exp−bn"√[rc(θ′−θ0)(1re−e)]2+(r′−rc)2# r′dr′dθ′
using analytical exprenssions that approximate its value. In this R R  
workweusetheapproximationofCiotti&Bertin(1999),given For asymmetric ArcEllipses, the denominator of Eq. 14
shouldbereplacedbya sum of two integrals,associated to the
by
twovaluesofellipticity(e ande ,see 2.2),eachoneoneither
1 2
§
1 4 46 131 sideofθ0.
bn(n) ≈ 2n− 3 + 405n + 25515n2 + 1148175n3 − To illustrate the method presented above, we show an arc
created followingthe processdescribed abovein the left panel
2 TheSLtoolslibraryisavailableathttp://che.cbpf.br/sltools/ ofFig.2.
3
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
Fig.2. Example of an ArcEllipse created by PaintArcs using a Se´rsic profile. The left panel shows the pure arc, created using
r = 17 ,r = 0.6 ,θ = 40o,n = 1,e = 0.85,e = 0.75,m = 21,andm = 31.83.Thepixelscaleoftheimageis0.154 .The
c ′′ e ′′ 0 1 2 zpt ′′
middlepanelshowstheapplicationofa Gaussianconvolution(FWHM=1.1 ) onthepurearcimage.Therightpanelshowsthe
′′
effectofaddingPoissonnoisetotheconvolvedarc.
3.1.Pixelation 3.2.Addingtheseeingeffect
To create a digitizedimage froma continuousintensity model, To add the seeing effect to a simulated image created with
weneedtointegratetheobjectsignaloveradiscretenumberof PaintArcs,weconvolveitwitha2DnormalizedGaussianPSF,
pixels. We thus need to controlthe changein signal caused by givenby
thepixelationandtrytokeepitlowerthansometolerancevalue,
ǫ.Inatypicalpositioninourimage,closetotheSe´rsiceffective exp x2 + y2
radiusr ,theprofileintensityisexpectedtovaryby PSF =g(x,y)= − 2σ2x 2σ2y , (18)
e (cid:26) (cid:20) (cid:21)(cid:27)
exp x2 + y2 dxdy
− 2σ2x 2σ2y
∂I R R (cid:26) (cid:20) (cid:21)(cid:27)
dI = ∂rr=rc+re,θ=θ0δ, (15) wThheeroepσtixonanσdσ,yaσreathlleowstsanadnaerdllidpetivciaaltiGonauinssxiaann.dydirections.
x y
Assuminga circularGaussian(σ = σ = σ ),thestandard
x y
where δ is the linearsize of the pixel.Therefore,the fractional deviation can be obtainedfrom the FWHM of an actualimage
errorinthepixelintensityis through
FWHM =2√2ln2σ. (19)
dI δ ∂I
= . (16)
I I ∂rr=rc+re,θ=θ0 WeimplementedaPythonmoduletoconvolveimageswith
(r=rc+re,θ=θ0) aGaussiankernelwhosesizeisamultipleofthestandarddevi-
ation(n ).
Wekeepthiserrorwithinatolerancelimitbyestimatingthe σ
In the middle panel of Fig. 2 we show the result of the
intensity at a numberof subpixels. We can find the numberof
Gaussianconvolutiononthearcimagepreviouslypresented(arc
subpixelsn byimposingdI/I =ǫ asfollows
p
intheleftpanelofthisfigure),whereweuseaseeingFWHMof
1.1 ,whichcorrespondstoσ=0.47 ,andn =4.
′′ ′′ σ
1 1 ∂I Attesting the code’s flexibility and modularity, new imple-
n = = . (17)
p δ ǫI(r=rc+re,θ=θ0)∂rr=rc+re,θ=θ0 mentations of PaintArcs are being developedto use alternative
seeingkernels,suchasa2DMoffatfunction.
Thenumbern representstheminimumintegernumberofsub-
p
divisionsofapixelrequiredtomaketheerroronthepixelation
3.3.AddingPoissonnoise
lessthanǫ.
IntheexampleofFig.2,weusedǫ = 0.1astheupperlimit Ina quantumdetectorsuchasaCCD thephotoncountsfollow
fortheerroronthedigitalizationofthearcsignal.Inthis case, aPoissondistribution,
np = 4, which means that each pixel was subdivided into 16 e µ
parts.Theexactvalueofthedigitalizationerror,computedafter- P(x,µ)=µx − , (20)
x!
wardsbycomparingthe totalsignalto be simulated(S , see
total
Eq.13)withthesignalthatwaseffectivelydistributed,isonthe whereµisthemeanofthedistribution.
orderof10 3. Thisvalueismuchlowerthanthe adoptedǫ, at- We add Poisson noise to an ArcEllipse produced by
−
testingthatthecriteriumgivenbyEq.(17)isbasedonaprofile PaintArcs by randomly picking values from the distribution
region where the variation in the arc intensity is stronger than givenbyequation(20)withthemeanequaltothesignalineach
average,whiletheexactvalueoftheerroriscomputedusingall pixelofthesimulatedimageafterconvolutionwiththePSF.This
pixels.Thereforetheactualpixelationerrorismuchsmallerthan isdonewith a separatemodulethatgeneratesrandomnumbers
ǫ. fromaPoissondistribution.
4
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
In the right panel of Fig. 2 we show the result of adding InSection2 we showedthatthe ArcEllipsedependsonthe
Poissonnoisetothearcofthemiddlepanelofthesamefigure. followingparameters(seeEq.6)
– x :xpositionofthepointwherethearciscentered(pixel)
0
3.4.ApplyingPaintArcs – y :ypositionofthepointwherethearciscentered(pixel)
0
– r :distanceofthearccenterrelativeto(x ,y )(pixel)
After the pixelation of the surface brightness profile of the arc c 0 0
– θ :orientationofthearccenterrelativetothexaxis(radians)
created with PaintArcs with a Se´rsic profile and adding seeing 0
– e ande :ellipticitiesofthearc
and Poisson noise to its image, PaintArcs can paste this simu- 1 2
– b:minorsemi-axesofthearc(pixel).
latedarcontoarealimage(forexampleonecontainingagalaxy
cluster),whichcanbespecifiedinitsinputconfigurationfile.In IfthesurfacebrightnessdistributionofthearcfollowsaSe´rsic
thiscase,PaintArcsaddsthearcmodelparameterstotheimage profile,weaddthreemoreparameterstothefitting:
headerforbookkeepingpurposes.
Figure 3 shows an example of an arc simulated with – re: radius that encloses half of the total luminosity of the
PaintArcs. It was added to the image of the galaxy Se´rsicprofile(pixel)
cluster SOGRAS0328+0044, which was imaged by the – n:indexthatdescribestheshapeoftheSe´rsicprofile
SOAR Gravitational Arc Survey (SOGRAS, Furlanettoetal. – I0:intensityoftheSe´rsicprofileatthecenterofthearc(r =
2012).Two real arc candidates are also found around the clus- 0).
ter’sbrightestgalaxy(indicatedbyarrows),andmayserve asa
If we choose b = r , as we do in PaintArcs, we ob-
visualguidelineforthe accuracyof the PaintArcsoutputs. The e
tain nine fitting parameters. Therefore, the set of parameters
simulated ArcEllipse is circled in the figure andwas generated
that represents the ArcEllipse with a Se´rsic profile is p =
withtheparametersm=21.5,e =0.9,e =0.8,r =7.7 ,θ =
1 2 c ′′ 0 (x ,y ,r ,θ ,e ,e ,r ,n,I ).
40o,n=3,andr =0.8 .ItwasconvolvedwithaGaussianPSF 0 0 c 0 1 2 e 0
e ′′
whoseFWHMwasobtainedfromtheimage(FWHM = 0.7 ).
′′
The origin of the coordinate system of the arc coincides with 4.1.Creatinganarcimagetobefitted
thecenterofthecluster’sbrightestgalaxy.Thepixelsizeofthis
TheArcFittingcodeworksonapostagestampimageofanob-
imageis0.154 .
′′
servedarc(i.e.,animagewherethepixelsnotbelongingtothe
arc are set to zero),which we call the arc image. To create the
arcimagewefirstrunSExtractor2.8.6(Bertin&Arnouts1996)
ontheoriginalimagethatcontainsthearc.SExtractorthen cre-
ates a segmentation image, which assigns a single value (cor-
respondingtothesequentialnumberoftheobjectintheoutput
image catalog)to all pixelsthat it associates to the object.The
arcimageisthencreatedbycopyingallarcpixelsdefinedbythe
segmentationimage,butusingtheirvaluesasintheoriginalim-
age.Allotherpixelsinthearcimageareassignedazerovalue.
Thesizeofthearcimageinthexandydirectionsaremultiples
ofthearc’smajorandminoraxisvaluesasfoundbySExtractor.
The procedure described above to create the arc image makes
useoffunctionsfromSLtools.
4.2.Parameterdeterminationfromthearcimage
Fig.3. Example of the application of PaintArcs on a real
image: the circled arc was created by the code using an In the following we describe the measurements performed by
ArcEllipse+Se´rsicmodelwithm=21.5,e =0.9,e =0.8,r = ArcFittingtofindtheArcEllipseparameters.ButtheArcFitting
1 2 c
7.7 ,θ = 40o,n= 3,r = 0.8 ,FWHM = 0.7 andwasadded is not only restricted to the model parameters, which currently
′′ 0 e ′′ ′′
to the image of the galaxyclusterSOGRAS0328+0044,where are the nine ArcEllipse+Se´rsic parameters listed above. It also
tworealarccandidates(indicatedbyarrows)werefound. attemptstomeasureothercommonlyusedarcparameters,such
asitslengthLandwidthW.
4.2.1. DeterminingtheArcEllipseparameters
First, ArcFitting finds the two farthest points of the arc image,
4. ArcFitting
p and p . For that, it uses any reference pointto find the first
1 2
Motivated by the need for tools for measuring arc parameters, extremeasthefarthestpointfromthereferenceone.Itthenfinds
such as its length, width, curvature,and surface brightness, we thefarthestpointfromthisfirstextremethroughthefarthest-of-
alsodevelopedatoolthatfitsthesepropertiestoreal(orrealisti- farthestmethod.Thecodethennamestheextremepointwhose
callysimulated)gravitationalarcs.WenameditArcFitting. angleislarger(smaller)thanθ (seeEq.22below)as p (p ).
0 1 2
Currently,ArcFittingisbasedontheArcEllipsemodelwith After that, ArcFitting obtains the points along the bisectrix
aSe´rsic profile.Therefore,itperformsmeasurementsonactual betweenp andp .Thislineisperpendiculartothelinesegment
1 2
arc images that lead to estimates of the ArcEllipse parameters. thatconnectsthesetwoextremepoints.Itthenfindsp ,whichis
3
Similarly to PaintArcs, ArcFitting is also a Python-based code themeanpointofthearcalongthebdirection.
that was designed in a modular way to allow different shape- Thisprocessisrepeatedtwicetoobtainthepoints p andp .
4 5
andprofilemodelstobeadoptedinthefuture. p is the meanarc positionalongthe bisectrix line between p
4 1
5
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
and p , whereas p isthemeanarcpositionalongthebisectrix lackinginpreviousattemptstomeasurearcpropertiesisthesur-
3 5
line between p and p . This method is an application of the facebrightnessdistribution.Thisisthetopicofthenextsubsec-
2 3
Mediatrixmethod(Bometal.2012)forn = 2,wheren isthe tion.
b b
numberofbisections.Thelength(L)ofthearcisthencomputed
asthesumofthelinesegmentsconnectingthefivepoints
4.2.2. DeterminingtheSe´rsicparameters
L= p1p4+p4p3+ p3p5+p5p2. (21) Weuseχ2 minimizationtodeterminetheSe´rsicprofileparame-
ters,I ,nandr .Thisχ2minimizationworksasfollows.
Inthe leftpanelofFig.4, weillustrate thisprocessoftrac- 0 e
We considera data set thatrepresentsthe informationof N
ing the arc image and determining its length. The five points
pixels in an arc image. We denote I (j) as the count of the
p,i=1,5areshownalongthearcimage.Theimageusedasan img
i jth pixel of the arc image and σ2 as its associated uncertainty,
exampleinthefigureistakenfromanHSTarcsamplethatwill j
bepresentedin 7. where j = (1,N). We also denote Iarc(j,p) as the theoretical
TheArcEllip§separameters x ,y andr arefoundbydeter- countofthe jth pixelfromArcEllipsewithaSe´rsicprofile.This
0 0 c
miningtheuniquecirclethatpassesthroughtheextremepoints, theoretical intensity depends on the pixel position and on the
p1and p2,andthroughthepointwithmaximumintensity, pmax. set of parameters p. Except for I0, n and re, whose values are
Thelatterpointisfoundasthearcpointwithmaximumintensity unknown, the other parameters are fixed by the procedure de-
amongthosethatliewithin0.5σbfromthelinesegmentsofEq. scribedinSect.4.2.1.AssumingthatthepixelcountsIimg(j)are
21, where σ is the dispersionof the pointsalongthe bisectrix independentfromeachother,wemustminimizethequantity
b
line containing p . This procedure is intended to eliminate arc
3 2
imagepixelslocatedfarawayfromthetracedarc,whichmaybe I (j,p) I (j)
arc img
associated to a cosmic ray,orto a residualfromthe processof χ2 = − (23)
(cid:16) σ2 (cid:17)
objectsegmentationanddeblending,asthesepixelswouldbias j j
X
the p estimate. The determinationof p is depictedin the
max max toestimatetheremainingparametersthatprovidethebestfitto
rightpanelofFig.4.Thepixelofmaximumintensityintheen-
thedata.
tire image, m , failed to satisfy the 0.5σ distance criterion, in
1 b Ingeneral,weusethereducedχ2
this case measured from the p p segment. The second maxi-
1 4
mum,m2,however,iswell-placedalongtheobservedarc. χ2
The width of the arc is taken as W = 4σ . The ArcEllipse χ2 = (24)
b red N 1
semi-minoraxisbisthentakenashalfofthewidth(b=W/2 =
−
2σb). asameasureoftheconcordancebetweenthemodelandthedata.
Theangleθ0isobtainedusingtheexpression Ifthisvalueismuchhigherthanunity,themodelisnotconsid-
eredagooddescriptionofthedata.
(y y )
θ =arctan c− 0 , (22) We assume that the pixel counts follow a Poisson distribu-
0 (xc−x0)! tion,suchthat
where(xc,yc)arethecoordinatesofthe pmaxpoint. σ2 =I (j), (25)
The asymmetric ArcEllipse ellipticities e and e are ob- j img
1 2
tained using Eqs. 7 and 8, respectively,where a1 is the size of andEq.(24)becomes
the line segment p p and a is the size of the line segment
1 max 2
p p . 2
maxTh2eprocedureoutlinedaboveisdifferentcomparedtoprevi- χ2 = 1 Iarc(j,p)−Iimg(j) . (26)
ous works. Some studies, particularly of observed arcs, quote red N 1 (cid:16) Iimg(j) (cid:17)
− j
L, L/W, or other arc parameters, without clearly mentioning X
howtheyaredetermined.Othersourcesusenomorethanthree The χ2 function minimization is carried out with
red
points to infer L, either using a sum of arc segments, as in Pyminuit3, which is an extension for Python modules of the
eq. 21, or fitting a circular arc through them (Oguri 2002; Minuit4functions.
Dalaletal. 2004; Lietal. 2005; Sandetal. 2005; Horeshetal. TheinitialguessesforI andr aredeterminedthroughmea-
0 e
2005; Meneghettietal. 2008; Horeshetal. 2010; Moreetal. surements on the arc image. We consider the intensity of the
2012). Lenzenetal. (2004) and Kauschetal. (2010) measured point p astheinitialestimate for I .Thesemi-minoraxisb,
max 0
momentaoverarcimagestoderivetheirsizes. whose estimate was describedabove,is the initial guessfor r .
e
Width measurements are usually based on the area of Given the lack of a direct method to estimate the parameter n
the arc divided by its length assuming some arc geometry, fromtheimage,theinitialguessofthisparametersisfound by
usually a rectangle. Meneghettietal. (2008), however, mea- usingthealgorithScanfromPyminuit,whichcalculatesχ2 for
sured widths at several different positions along the arc length arangeofnvalueskeepingallotherparametersinthe psetrfiedxed,
and assumed a median value. Some authors also account i.e.,themorphologicalparametersasdeterminedintheprevious
for seeing effects when measuring W (Meneghettietal. 2008; subsection,andI andr fixedattheirinitialguesses.
0 e
Zaritsky&Gonzalez2003).
Afterdeterminingtheinitialguessesfor I ,nandr ,weuse
0 e
Arc orientation and curvature are not as common to find.
thePyminuitalgorithmMigrad,whichisanoptimizedgradient-
OnesuchreferenceisLuppinoetal.(1999),whousedthechord basedminimumsearchalgorithm,toperformtheχ2 minimiza-
length from the arc extremes and its sagittal depth to estimate red
tionandtofindthebest-fittingvaluesfortheseparameters.
r . Estimates of θ can also be found in Oguri (2002) and
c 0
Miralda-Escude(1993). 3 http://code.google.com/p/pyminuit/
Weareunawareofpreviousstudiesthataccommodateapos- 4 http://seal.web.cern.ch/seal/snapshot/work-
sible asymmetry in the arc’s shape. Finally, another element packages/mathlibs/minuit/
6
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
Fig.4.Illustrationoftheprocessoftracingarealarcinitsimage.Theleftpanelshowsthefive ppointsusedtocomputethe
arclength.Therightpanelshowsthehigh-intensitympoints(dashedcircles)usedtofind p forthesamearc.Thecenter
max
andradiusofcurvaturearedeterminedasthecirclecontainingthepoints p ,p ,and p = m .Thewidthisbasedonthe
1 2 max 2
linethroughthecentralpoint p ,whereastheasymmetricmajor-axes a anda arethelengthsofthesegmentsconnecting
3 1 2
m = p tothearcextremes p and p .
2 max 1 2
The approach outlined here to estimate the shape parame- Table1.InputandfittedArcEllipseandSe´rsicparametersforan
ters directly from the image and to apply a χ2 minimization arccreatedwithPaintArcsusingtheArcEllipseprescriptionand
to the Se´rsic profile proved to be the most effective. We tried aSe´rsicsurfacebrightnessprofile.
otheralternatives,suchasχ2minimization(usingMigrad)ofall
shape- and Se´rsic parameters,or an iterative method that com- Parameter Truevalue Recoveredvalue
bines Migrad for some parameters and brute force (Scan) for x (pix) 100.00 98.50
0
others.Inmanyofthecaseswetried,thesemethodsdidnotcon- y (pix) 100.00 96.88
0
verge, or resulted in a final χ2 much larger than unity. These rc(pix) 50.00 52.85
red
failuresareprobablycausedbyacombinationofstrongdegen- re(pix) 5.20 5.96
θ (rad) 0.70 0.72
eraciesamongtheArcEllipseandSe´rsic modelparametersand 0
I (counts) 100.00 102.81
bythelimiteddegreesoffreedomwhenconsideringthenumber 0
n 1.00 1.06
ofavailablepixelsinthearcimage.
e 0.90 0.88
1
e 0.80 0.77
2
4.3.TestingtheArcFittingalgorithm
TotestandvalidatetheArcFittingcode,weappliedittoimages and the reduced χ2, given by Eq. (26), as metrics for the con-
ofpurearcs(noiselessandwithnoseeingconvolution)produced cordancebetween the ArcEllipse+Se´rsic model,resultingfrom
withPaintArcs,usingtheArcEllipse+Se´rsicmodel.
ArcFitting, and the data. Both quantities are computed using
As an example, we considered an arc with true parameters
onlythepixelsthatbelongtothearcasdescribedin 4.1.
asshowninthemiddlecolumnofTable1.Thevaluesrecovered §
byArcFittingaredisplayedintherightcolumn,showinganex-
cellentagreementwiththeinputvaluesusedbyPaintArcs.This 5.1.ApplyingArcFittingonarcssimulatedwithAddArcs
situationistypicalofthevalidationtests.Indeed,ArcFittingwas
For this first application we used a set of arcs simulated with
verysuccessfulinrecoveringtheArcEllipseandSe´rsicparame-
theAddArcscode(Brandtetal.,inprep.),whichwasoriginally
tersinthisidealizedcase.ThisisexpectedsincebothPaintArcs
designedforDark EnergySurveyimagesimulations.The code
and ArcFitting use the same analytical expression for the arc
uses as basic inputs a catalog of lenses, a catalog of sources,
shapeandsurfacebrightnessprofile,butdemonstratesthevalid-
the cosmological parameters, and the properties of the images
ity of the whole process of determining the morphologicalpa-
onwhichthearcswillbesimulated(e.g.,pixelscale,magnitude
rameters and using the initial guesses for scanning and fitting
zeropoints,etc.).Forthesimulationsusedinthiswork,thein-
theSe´rsicparameters.
putlenscatalogsweregivenbytheCarmenLasDamasN-body
cosmological simulation5 (McBrideetal. 2011) and the lenses
5. ArcFittingapplicationtorealandsimulatedarcs aremodeledasaprojectedNavarro–Frenk–White(Navarroet al.
1996, 1997, hereafter NFW) profile with elliptical mass distri-
We applied the ArcFitting code to three different sets of arcs.
bution. The redshift and mass of each lens are taken from the
The first one consists of realistically simulated arcs. The other
catalog,buttheNFWconcentrationparameteristakenfromthe
twosetsareofrealarcs,onefromground-basedimagesandthe
fitsofNetoetal.(2007),sincethesimulationsdonothavesuffi-
otherfromHSTimages.
cientprecisiontodetermineitaccurately.Thesourcesare mod-
Weusedtheresidualsignalcontrast∆S/S,definedas
eled as elliptical Se´rsic brightnessdistributionswhose parame-
tersareobtainedfromfitstogalaxiesintheHubbleUltra-Deep
I (j) I (j,p)
∆S j img − arc Field (UDF) givenbyCoeetal. (2006), whichprovidesthe in-
= , (27)
S P(cid:16) I (j) (cid:17)
img 5 http://lss.phy.vanderbilt.edu/lasdamas/
j
P
7
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
putsourcecatalog.Thissame catalogcontainsthe photometric
redshiftinformationfrommulti-wavebanddataontheUDF.
Thesourcesaredividedintoredshiftbinswhosespatialdis-
tribution is uniform in each bin. For each lens (i.e., dark mat-
terhalo)inthecatalog,anumberofsourcesislensedusingthe
Gravlens(Keeton2001)software.Thenumberofsourcestobe
lensedisobtainedfromthesurfacedensitydistributionofthein-
putsourcecatalog.Afirstselectionofsourcesismadeusingthe
localmagnificationcomputedwith Gravlensandthenthesame
codeisusedtoproduceafullbrightnessdistributionbylensing
finite Se´rsicsourcesfromtheUDFcatalog.Finally,images are
identified and are dimensionally estimated by SExtractor, and
only elongated images are selected by the code. We selected a
subset of nine arcs generated from these simulations to have a
similarsamplesizeasthetwoothersamples.
WeranArcFittingonthearcsproducedwithAddArcstoin-
fer their ArcEllipse+Se´rsic parameters and recreate them with
PaintArcs. In Fig. 5 we show some examples of this applica-
tionoftheArcFittingcode.Intheleftpanelsweshowtheorig-
inal simulated arcs (noiseless and without seeing convolution)
fromAddArcs.InthemiddlepanelweshowtheArcEllipsede-
rived from ArcFitting and reproduced with PaintArcs. In the
right panel we show the residual image resulting from the dif-
ferencebetweenthetwopreviousones.Allpanelsusethesame
grayscale.Somesubstructuresarevisibleintheselatterimages,
indicatingthelimitationsoftheArcEllipseprescription.Butthe
residualsaremuchlessintensethanthefittedarcs.
We computed the residual signal contrast ∆S/S, as given
in Eq. 27, between each AddArcs simulation and its fitted Fig.7. Selected ArcFitting results for HST arcs. Left panels:
ArcEllipse+Se´rsic. In the left panel of Fig. 6 we show the re- originalarc;middlepanels:arccreatedwithPaintArcsusingthe
sulting distribution of ∆S/S values. The median ∆S/S value parametersderivedfromArcFittingappliedtotheoriginal arcs;
is 0.09, indicating that the ArcFitting process based on the right panels: difference between original and reproduced arcs.
Arc−Ellipse+Se´rsicmodeltypicallyreproduces91%ofthesignal Fromtoptobottom:arcC8foundinAbell68;arcH0ofAbell
oftheinputarcimage.IntherightpanelofFig.6wepresentthe 963;arcM3ofAbell2218,andarcB0/B1/B4ofAbell383.
distributionof χ2 resultingfromthe applicationofArcFitting
red
onarcssimulatedwithAddArcs,whosemedianisχ2 =5.27.
red
5.2.ApplyingArcFittingonarcsobservedwiththeHST
arcswithmanysubstructures,suchasthefirstandthelastofthat
WeheretestthereconstructionofHSTarcsusingtheArcFitting
figure,ArcFittingcannoteffectivelyreproducethearcbrightness
code basedon the ArcEllipse+Se´rsicmodel. A sample of HST
profiles.
arcswasselectedfromtheworkbySmithetal.(2005).Wechose
bright and isolated arcs with spectroscopic redshift measure- We again used the residual signal contrast ∆S/S, given by
mentsthatconfirmtheirnatureaslensedsources.A totalof12 Eq. 27, and the χ2 , givenby Eq. 26, to assess the agreement
red
arcs were selected. We use the same notation here as the orig- betweenmodelanddata.Thedistributionof∆S/S valuesforall
inal authors. The arcs are C4, C8, and C12 from the Abell 68 12arcsinourHST sampleisshownin theleftpanelof Fig.8,
galaxycluster;H0,H1,H2,andH3fromAbell963;B0/B1/B4 while the distributionof χ2 is shown in the rightpanelof the
red
from Abell 383; M1 and M3 from Abell 2218; and P0 and P2 samefigure.Themedianofthe∆S/S distributionis 0.13,but
from Abell 2219. Arc propertiessuch as magnitude, L/W, and values as high as 0.32 are found. The median χ2 −is 18.39,
− red
distance to the BCG of these arcs are presented in Sandetal. butthedistributionhasalongtailstowardmuchhighervalues.
(2005). These results clearly attest that there is a stronger discrepancy
In Figure 7 we present some results of the application of between model and HST data than for the simulated AddArcs
ArcFittingtoourselectedHSTarcs.ThepanelsofFigure7are images.Forallarcs,∆S isnegative,whichmeansthatArcFitting
similartothoseofFig.5.TheleftpanelsshowtheoriginalHST isoverestimatingthearcsignal.Highvaluesinbothdistributions
arcimages,themiddlepanelsshowthefittedArcEllipse+Se´rsic correspondtothearcsthatpresentmanysubstructures(example:
modelarcs,whichwasreproducedwithPaintArcs,andtheright lastarcofFig.7,whichhas∆S/S = 0.32andχ2 =76.26).
− red
panelsshowresidualimages. InFig.9weshowacomparisonoftheL/Wvaluesmeasured
From the middle panels we can see that ArcFitting is effi- by Sandetal. (2005) to those obtained with ArcFitting. Since
cientinrecoveringarccurvatureandorientation.Insomecases, the arcs H1, H2, and H3 from Abell 963 are considered as a
asforthesecondandthirdarcinthefigure,apartfromtheeffi- single object in Sandetal. (2005), these arcs are not included
ciencyinrecoveringthearcshape,ArcFittingisalsoefficientin inthiscomparison.Nostrongsystematicsisseen,whichshows
reproducingthearcbrightnessdistributions,sothattheresidual thatArcFittingprovidesarobustandautomaticmeasureofL/W
imagescontain only little substructureand noise. However,for ratioinrealgravitationalarcs.
8
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
Fig.5. ArcFittingresults forAddArcsarcs. Leftpanels:original arc;middle panels:arc reproducedwith PaintArcsusing the pa-
rametersprovidedbytheapplicationofArcFittingonoriginalarcs;rightpanels:differencebetweenoriginalandreproducedarcs.
Fig.6.Distributionofresidualsignalcontrast∆S/S (leftpanel)andofχ2 (rightpanel)resultingfromapplyingArcFittingonarcs
red
simulatedwithAddArcs.
5.3.ApplyingArcFittingonarcsobservedfromtheground from the differencebetween the previoustwo. We can see that
ArcFittingisefficientinrecoveringarcshapeandbrightnessdis-
We also ran the ArcFitting code to reconstruct arcs observed
tributions.
fromtheground.Weselectedbrightandisolatedarcsincluster
and galaxy group lenses from the CFHTLS Strong Lensing
LegacySurvey(SL2S),presentedintheworkbyCabanacetal. In the left panel of Fig. 11 we present the residual signal
(2007). Following the notation of the original authors, the contrastdistributionfortheselectedCFHTLSarcsample.Asin
seven selected arcs that compose our CFHTLS sample are the HST arcs, we can see that ArcFitting overestimatesthe arc
SL2SJ021416-050315, SL2SJ021807-051536, SL2SJ085446- signal,since∆S < 0forallarcsofthissample.Themedianof
012137, SL2SJ141912+532612, SL2SJ142258+512440, the∆S/S distributionis 0.11and∆S/S = 0.20intheworst
− −
SL2SJ143001+554647,andSL2SJ143140+553323. case.Intherigthpanelofthesamefigure,weshowthedistribu-
The results of applying ArcFitting on three arcs of our tionofχ2 ,whosemedianis10.76.IncontrasttotheHSTarcs,
red
CFHTLSsampleareshowninFig.10.Asinthepreviouscom- no high χ2 tail is seen; the only outlier in the χ2 distribu-
red red
parisons,theleftpanelsshowtheoriginalarcimages,themiddle tioncorrespondstothearcSL2SJ085446-012137,thesamearc
panelsshowthefittedArcEllipse+Se´rsicmodelreproducedwith thatpresentsthehighestvalueinthe∆S/S distribution.Theim-
PaintArcs, and the right panels show residual images resulting provementin ArcFitting of CFHTLS arcs with respect to HST
9
CristinaFurlanettoetal.:Asimpleprescriptionforsimulatingandcharacterizinggravitationalarcs
Fig.8.Residualsignalcontrast(∆S/S)distribution(leftpanel)andχ2 distribution(rightpanel)resultingfromapplyingArcFitting
red
onourselectedsampleofarcsfromtheHST.
Fig.11.Residualsignalcontrast(∆S/S)distribution(leftpanel)andχ2 distribution(rightpanel)resultingfromapplyingArcFitting
red
onourselectedsampleofarcsfromCFHTLS.
arcs is likely caused by the smearing of the arc substructures
causedbyseeingeffectsontheground-basedset.
6. Conclusionsandfutureperspectives
WehavepresentedandexploredindetailtheArcEllipse,which
is a simple analyticalmodelthat approximatelyreproduces the
shapeofrealgravitationalarcs.ArcEllipsesareordinaryellipses
whose major axes are bent along an arc of a circle and which
thereforemimictheshapeoftangentialarcs.Theyarefullychar-
acterizedbya limitednumberof parametersthatquantifytheir
curvature, length and width, degree of deformation, and ori-
entation. We also devised a simple prescription to incorporate
asymmetriestotheirshapes.Thisdescriptionofarcmorphology
is more comprehensivethan previously found in the literature.
We also incorporated a Se´rsic light profile distribution to the
Fig.9.ComparisonbetweentheL/Wmeasurementspresentedin ArcEllipses in an attempt to explore a rough structural model
Sandetal.(2005)andthoseobtainedwithArcFitting.Thesolid forlensedgalaxies.
linerepresentstheidentityfunction. We also implemented PaintArcs, which is a tool that cur-
rently generates simulated ArcEllipse images, and ArcFitting,
which is a code that derives model parametersfrom images of
real arcs. Both are Python codes built in a modular way, thus
10