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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed12January2010 (MNLATEXstylefilev2.2) Galaxy Zoo: A correlation between coherence of galaxy ⋆ spin chirality and star formation efficiency 0 Raul Jimenez1†, Anˇze Slosar2,3,4, Licia Verde1, Steven Bamford5, Chris Lintott3, 1 0 Kevin Schawinski7,8, Robert Nichol6, Dan Andreescu9, Kate Land3, Phil Murray10, 2 n M. Jordan Raddick9, Alex Szalay11, Daniel Thomas5, Jan Vandenberg11 a J 1ICREA & Institute of Space Sciences(CSIC-IEEC), Campus UAB, Bellaterra 08193, Spain 2Berkeley Centerfor Cosmological Physics, Lawrence Berkeley Nat. Lab & Phys. Dept, Universityof California, Berkeley,CA 94720, USA 2 3Astrophysics Department, University of Oxford, Oxford, OX13RH 1 4Faculty of Mathematics & Physics, Universityof Ljubljana, Slovenia 5Centre for Astronomy and Particle Theory, Universityof Nottingham, UniversityPark, Nottingham, NG7 2RD, UK O] 6Institute of Cosmology and Gravitation, Universityof Portsmouth, Mercantile House, Hampshire Terrace, Portsmouth, PO1 2EG, UK 7Department of Physics, Yale University,New Haven, CT 06511, USA C 8Yale Centerfor Astronomy and Astrophysics, Yale University,P.O. Box 208181, New Haven, CT 06520, USA . 9LinkLab, 4506 Graystone Ave.,Bronx, NY 10471, USA h 10Fingerprint DigitalMedia, 9 Victoria Close, Newtownards, Co. Down, Northern Ireland, BT23 7GY, UK p 11Department of Physics and Astronomy, The Johns Hopkins University,Homewood Campus, Baltimore, MD 21218, USA - o r t s 12January2010 a [ ABSTRACT 2 v Wereportonthefindingofacorrelationbetweengalaxies’paststarformationactivity 4 and the degree to which neighbouring galaxies rotation axes are aligned. This is ob- 9 tainedbycross-correlatingstarformationhistories,derivedwithMOPED,andspatial 9 coherenceofspindirection(chirality),asdeterminedby the GalaxyZooproject,fora 0 sample of SDSS galaxies. Our findings suggest that spiral galaxies which formed the . 6 majority of their stars early (z > 2) tend to display coherent rotation over scales of 0 ∼10Mpc/h. The correlation is weaker for galaxies with significant recent star forma- 9 tion.Wefindevidenceforthis alignmentatmorethanthe5σ level,butnocorrelation 0 with other galaxy stellar properties. This finding can be explained within the context : of hierarchical tidal-torque theory if the SDSS galaxies harboring the majority of the v i oldstellarpopulationwhereformedinthepast,inthesamefilamentandataboutthe X sametime.Galaxieswithsignificantrecentstarformationinsteadareinthefield,thus r influenced by the general tidal field that will align them in random directions or had a a recent merger which would promote star formation, but deviate the spin direction. Key words: cosmology:theory - galaxies 1 INTRODUCTION ertiesofgalaxieswiththesamestellarmassexistandneedto beexplained.Oneofthemostobviousphysicalmechanisms In our progress to understand how galaxies have formed toexplainthissecondparameteristhespinofthedarkmat- andevolved,ithasbecomerecentlyclear(e.g.Jimenez et al. terhalo.Itisnow60yearssinceHoyle’sseminalpaperonthe (2008)) that thepresent stellar mass of galaxies determines subject (Hoyle 1949) that showed how angular momentum most of the galaxy properties. It is thus possible to predict in galaxies can be generated via the tidal field of the other on average what the star formation, metallicity, environ- galaxies.Lateron,Doroshkevich (1970)developedthetidal ment,etc.agalaxyhasbysimplymeasuringitscurrentstel- torque theory within the framework of hierarchical galaxy larmass.However,secondorderdifferencesamongtheprop- formation that determines the amplitude and direction of thespinofadarkmatterhalobasedonthesurroundingdark ⋆ This publication has been made possible by the partici- matter field (see Schaefer (2008) for a recent review). Nu- pation of more than 100,000 volunteers in the Galaxy Zoo mericalN-bodysimulationsproduceresultsthatareingood project. Their individual contributions are acknowledged at agreement with the theoretical predictions, although linear http://www.galaxyzoo.org/Volunteers.aspx theoryisnotalwayssufficienttodeterminethefinalangular † email:[email protected] 2 Jimenez et al. momentumof acollapsed object (e.g., Barnes & Efstathiou §3presentstheresults.In§4wepresentsomediscussionand (1987);Porciani et al.(2002)andreferencestherein).Inad- draw ourconclusions. ditionmergersareexpectedtosignificantlyalterahalofinal spin (Hetznecker& Burkert 2006). The amplitude of the dark matter halo spin will in- 2 SAMPLE SELECTION AND METHOD fluence the radius where baryons will settle into a disk (Fall & Efstathiou 1980; White & Rees 1978), thus influ- IntheGalaxyZooproject,asampleof893,212galaxieswere encing its density and therefore the star formation his- visually classified by about 90,000 users. The sample was tory of the galaxy. The influence of spin on star forma- selected to be sources that were targeted for SDSS spec- tion history has been studied in detail Toomre (1964); troscopy,thatisextendedsourceswithPetrosianmagnitude Dalcanton et al. (1997); Jimenez et al. (1997); Mo et al r < 17.77. Additionally, we included objects that were not (1998); Avila-Rese et al. (1998). originallytargetedassuch,butwereobservedtobegalaxies One interesting feature of tidal-torque the- once their spectrum was taken. Where spectroscopic red- ory in hierarchical models (Heavens& Peacock shifts were available, we found that galaxies havethe mean 1988; Catelan & Porciani 2001; Catelan et al. 2001; redshiftofz =0.14andtheobjectswiththehighestredshift Crittenden et al. 2001; Porciani et al. 2002; Hahn et al. reachz∼0.5. Thegalaxies thusprobeourlocal universeat 2007) is the prediction of correlated spin directions and cosmologicalscales.Eachobjecthasbeenclassifiedabout40 that the spin direction for dark halos is strongly influenced times from a simplified scheme of 6 possible classifications: by the halo environment. Pen et al. (2000) reported a anelliptical,aclockwisespiralgalaxy,ananti-clockwisespi- detection of galaxy spin correlations at 97% confidenceand ralgalaxy,anedge-onspiralgalaxy,astar/unknownobject, Slosar et al. (2009) have measured the correlation function a merger. Various cuts (hacking attempts, browser miscon- of the spin chirality and report, for the first time, of a figurations, etc.) removed about 5% of our data. The data signal at scales < 0.5 Mpc/h at the 2 − 3σ level. This were reduced into two final catalogues based on whether subject has received renewed interest not just for being data was weighted or unweighted. In the unweighted data, a test of tidal torque theory but because the mechanism each user’s classification carried an equal weight, while in that produces angular momentum alignment is believed the weighted case, users weights were iteratively adjusted to create correlations between observed galaxy shapes, according to how well each user agreed with the classifica- introducing a potential contamination to the cosmological tionsofotherusers.Inbothcases,theaccruedclassifications weak lensing signal. werefurtherdistilledinto“super-clean”,“clean”and“clean- Observationally,thefullinformation onthespinvector ish”catalogsofobjects,forwhichwerequired95%,80%and ofadarkmatterhaloisverydifficulttoobtain.Concentrat- 60% of users to agree on a given classification. In all cases, ing on disk galaxies, the plane of the disk determines the this is a statistically significant classification with respect axis of rotation of thedisk of baryonicmatter. For galaxies torandomvoting;however,thehuman“systematical” error seeninprojection,theobservedgalaxyellipticityconstrains associatedwithitisdifficulttojudge.Inanycase,wearein thegalaxyspinaxis.Forinfinitelythinandperfectlycircular thelimitwheretakingmoredatawillnotchangeoursample disks,thespinvectoraxiswouldbeknownbutnotthespin beyond noise fluctuations as the votes are uncorrelated. A ”chirality” (the spin direction along the axis). For realistic detailed account of the reduction procedures as well as the disks,projectioneffectsmeanthatwhatcanbemeasuredre- proceduretomeasurespinorientations, andremovalofsys- liably is the projected spin axis. Information abut the spin tematicsisexplainedinLintott et al.(2008)andLand et al. chirality is absent in the study of galaxies ellipticity. Nev- (2008). ertheless, the chiral information is known for the sample of WeconsiderthesampleofspiralgalaxiesoftheGalaxy face-on galaxies of theGalaxy Zoo project 1. Correlation of Zooprojectforwhichthespinchirality(i.e.thedirectionof chirality therefore implies a correlation of thespin vectors. thegalaxyarmswinding)hasbeendetermined.Thispieceof An expectation of tidal torque theory is that haloes information for each galaxy is what werefer toas halospin whichformedtogether,andthusexperiencedasimilartidal chirality. In general, the gas of a galaxy should be rotating field duringtheirinitial collapse, will havesimilarly aligned in the same direction as the halo: about 4% of the galaxies spin vectors. Galaxy chirality should thus be coherent on donotobeythis(Pasha & Smirnov1982),butthereshould scales related to those of large scale structure. be a strong –although not perfect–correlation between the Hereweexplorewhetherspatially coherentspin chiral- angularmomentumvectorofgasandthatofthedarkmatter ity (and therefore spin vectors) correlate with other galaxy halo hosting a galaxy (van den Bosch et al. 2002). Here we propertiesthatdependonthegalaxystellarpopulationand assume that the chirality of the galaxy is a proxy for the starformationhistory.Ourmainfindingisthattheabsolute rotation direction of thehost dark matter halo. valueoftheaveragespindirectionofthegalaxieslocatedin Our catalog of galaxy star formation properties is ob- a spatial patch (pixel), is correlated with past star forma- tained from Panter et al. (2007). This catalog has been tion activity in these galaxies, while we find no correlation constructed from the spectroscopic main galaxy sample withothergalaxypropertieslikemetallicity,orwiththeav- of the SDSSS-DR4 data release (Adelman-McCarthy et al. erage spin itself, i.e. the universe does not have a preferred 2006). Star formation histories, metallicities, stellar masses direction. The rest of the paper is organized as follows: §2 and dust content, have been extracted using the MOPED describes thegalaxy sample selection andthemethodology, (Heavenset al. 2000) algorithm, which allows for rapid ex- traction and parameter exploration of galaxy spectra us- ing stellar population models. The method has been ex- 1 www.galaxyzoo.org plored in detail in Panteret al. (2007) and has been tested Star formation and spin chirality 3 Table 1. Centre and boundaries ((l)ower, (c)enter, (u)pper) of Spin vector pointing ``up’’ (s=1) the star formation bins inredshift (z) and look-back time tlb in Gyr x Spin vector pointing ``down’’ (s=-1) bin lz cz uz ltlb ctlb utlb x x x x x x 11 0.0007 0.001 0.00145 0.00966 0.014 0.0200 10 0.00145 0.0021 0.003 0.0200 0.029 0.0414 x x x 9 0.003 0.006 0.0063 0.0414 0.06 0.0857 x x x x x 8 0.0063 0.012 0.013 0.0857 0.12 0.1776 x x x x 7 0.013 0.0179 0.027 0.1776 0.26 0.3677 6 0.027 0.0419 0.057 0.3677 0.53 0.7614 High |S| High |S| Low |S| 5 0.057 0.0839 0.125 0.7614 1.10 1.5767 4 0.125 0.186 0.287 1.5767 2.27 3.2650 3 0.287 0.456 0.786 3.2650 4.70 6.7609 2 0.786 1.200 2.000 6.7609 8.46 10.32 1 2.000 4.000 8.000 10.32 12.09 13.00 Figure 1. Graphical depiction for a given pixel size of what it correlation when we use the full value of Sj (including its meanstohavehighvalueofsj. sign). Each galaxy in the sample (index i) has associated a stellar mass, a metallicity, a metallicity history and a star withtheVESPA(Tojeiro et al.2007)algorithmforaccuracy formation history. The star formation history is described andextentof degeneracies in therecovered parameters (see by the star formation in independent bins (index β), ψ : β,i Panter et al. (2007); Tojeiro et al. (2007) for details). The 9 equally spaced in look back time and two bins for high star formation histories catalog from VESPA(Tojeiro et al. redshift star formation (see table 1). For each galaxy, the 2009) is available online2. While for the SDSS spectra, star formation is normalized to unity. For each pixel we individual star formation histories are poorly constraint also define a deviation from the mean star formation his- (Tojeiro et al. 2007) because of thelow S/N, average prop- tory given by the average star formation in each bin over erties sample are robust and well constrained; it is these the pixel galaxies minus the global average star formation: average quantities that we use in our study. Panter et al. SFj,β = PNi=j1ψβ,i/Nj −hψi. Metalicity history is treated (2007) give a thorough and detailed explanation of the in the same way. Note that by comparing average quan- method, its shortcomings and advantages and we refer the tities in each pixel rather than total quantities we are not interested reader to this paper for a detailed description of directlysensitivetothegalaxylocaldensity.Correlationbe- the star formation catalog used in the present work. The tweenchiralityanddensityhasbeenexploredinLand et al. catalog contains about half million galaxies. (2008). Tocreateourfinalsamplewematchtheabovetwocat- Wethereforehaveseveralpixelizedmaps:amapofspin alogs, MOPED andGalaxyZoo (keepingonly galaxies that chirality, and 24 maps of galaxy star formation properties are classified as clean), to obtain a total of 12897 galaxies (stellar mass, metallicity, 11 maps of star formation at dif- at redshift z < 0.2. The redshift distribution of our sam- ferentlook backtimes,andsimilarly 11mapsformetalicity ple is fairly narrow (the redshift interval 0.012 < z < 0.13 history). encloses 90% of the galaxies, the maximum of the redshift Toquantifyapossiblecorrelationbetweengalaxyprop- distribution is zm = 0.08. It is this sample that we use in ertiesandspincoherenceweusethePearsoncorrelationco- our study. Note that the galaxies in the sample are spirals, efficientandexplorehowitvariesasafunctionofpixelsize, and therefore the downsizing effect is not as extreme as in lag, and star formation properties. Errors are estimated by ellipticals. repeating theprocedurewith randomized spins. Thiscross- Weuniformlypixelize thesurveyusingarangeofpixel correlationinpixelsofafinitesizeessentiallycontainsinfor- sizes. In what follows we use a flat concordance LCDM mation on integrated cross-correlation function below some model to convert between redshifts and distances. In each pixelsize.Ifcertain propertiescorrelateintwo-point statis- pixelwedefineapixelspinchiralitySj givenbytheaverage ticsup toacertain scale, ourstatistical test will pickit up. spinchiralityforthegalaxies inthepixel:Sj =PNi=j1si/Nj where Nj denotes the number of galaxies in pixel j and si denotesagalaxyspinchirality;sicanonlytakevaluesof+1 3 RESULTS or −1 while −1 < Sj < 1. For uncorrelated spin direction, within the errors Sj = 0, only for correlated spin direction We are interested in exploring if galaxies with a particu- the average spin chirality will be significantly non-zero (see lar property have their spins aligned or randomly oriented Fig.1).Notethatwhenwecomputethecross-correlation of with respect to their neighbours. In this respect we do not galaxies quantities with the pixel spin chirality Sj we will needtoworryaboutprojectingthespinvectoronacommon use the absolute value, |Sj|. We will show that there is no reference frame for all galaxies as was done by Land et al. (2008); Slosar et al. (2009) as we are only interested in the cross-correlation between the spatially averaged chirality of 2 http://www-wfau.roe.ac.uk/vespa/ thegalaxiesandtheirstarformationproperties.Wealsoin- 4 Jimenez et al. Figure 2. Relation between pixel-averaged star formation frac- Figure 4. Same as Fig. 3 but for three different mass ranges. tionintheoldestMOPEDbin(1,seeTable1)andpixel-averaged In this cases we use the projected map in order to increase the absolutespin|Sj|(forapixelsizeof10Mpc/h)forallthosegalax- signal-to-noise.Notethatthesignalatbins1−2isdominatedby iesthathaveformedmorethan30%oftheirpresentstellarmass galaxiesinthemassrange1010−1011 M⊙. at z > 2.0. Due to the large number of pixels, instead of show- ingascatter plotwesow themean relation,error-barsshow the standard deviation around the mean relation. Note that spiral galaxies thatformedmorethan30% oftheirstarsatz>2have terpret the chirality of the galaxy spin as a proxy for the preferentially a coherent spin chirality. If we focus on those spi- halo spin chirality. ral galaxies that formedmorethan 80%, denoted by the dashed line, of their stars at z > 2 then virtually all of them have co- Spins of the galaxies in our sample are spatially corre- herentspinchirality.Thedotted anddash-dotted linesshowthe lated (Slosar et al. 2009) and star formation properties are pixel-averaged star formation but for choices of the pixel size of also spatially correlated with the large-scale cosmological 2Mpc/h and 50Mpc/h respectively. For these cases the value of structures (e.g., Shethet al. (2006)). The cross correlation thestar-formationfractionasafunctionofabsolutespinisconsis- asdefinedin§2willindicatewhichgalaxypropertyismostly tentwithzeroaserrorbarsaresimilartothe10Mpc/hpixel-size correlatedwiththelarge-scaletidalfieldresponsibleforspin caseandarenotplottedforclarity(seetextformoredetails). alignment. Wefindnocorrelationoftheabsolutevalueofthepixel spinchiralitywithstellarmass,metallicityormetallicityhis- tory but we find significant correlation with star formation history. Further, when we correlate the pixel spin chirality (not its absolute value but keeping the sign) with the star formationhistorywefindnocorrelation ortobeprecisethe Pearson coefficient we find is of order 4% at all look-back times, which is below the error in the coefficient itself (see below). Fig.2showstherelationbetweenthepixelaveragedstar formation fraction (SFj,1)intheoldest MOPEDbin(1,see Table 1) and the absolute value of the pixel-averaged spin |Sj| for all those galaxies that have formed more than 30% oftheirpresentstellarmassatz>2.0.Becauseofthelarge number of pixels a scatter plot is unclear; here the points indicatethemeanrelationandtheerrorbarsshowthestan- darddeviationaroundtherelation.Notethatspiralgalaxies thatformedmorethan30%oftheirstarsatz >2havepref- Figure3.Pearsoncorrelationcoefficientbetweenabsolutevalue erentiallyacoherentspinchirality.Ifwefocusonthosespiral of pixel spin chirality and past star formation as a function of galaxiesthatformedmorethan80%,denotedbythedashed thestartformationinlook-backtimebinnumber(seeTable1for conversion between bin number and look-back time). The pixel line, of their stars at z > 2 then virtually all of them have sizeis10Mpc/h.Thepointsarethemeanofthecorrelationcoeffi- coherentspinchirality.Thisimpliesthatthosegalaxiesthat cientforseveralredshiftslices.Errorbarsshowjacknifeestimates wereinplaceatz>2rotateonthesamedirectiononscales of the correlation coefficient rms by using several redshiftslices, of ∼10Mpc. thedashedlineshowsthe1−σcorrelationlevelobtainedwitha To quantify our results we find that the Pearson cor- randomizedspinmap.Galaxieswithstarformationintheoldest relation coefficient between absolute value of the pixel spin bin (1-3) show strong correlation with spin chirality (and there- chiralityandpaststarformation mapsismaximumatzero- forewith spinalignment). The correlation decreases for galaxies lagforapixelphysicalsizeof∼10Mpc/h.Forsmallerpixel withmorerecentstarformation. sizes(<2Mpc/h)thenumberofgalaxiesperpixeldecreases veryrapidly.Thecorrelationdisappearsforpixelsizeslarger Star formation and spin chirality 5 presentandtotalstellarmassinthegalaxy.InFig.4weshow Table 2. Per cent deviation in each look-back time star bin for the quantity |Sj| from zero, which means the universe has no theabovecorrelationbutforthreemassranges.Thedashed preferred direction. Note that there is a negative bias at the % lineshowsthecorrelationlevelforamapwheretheabsolute levelbutthiserrorisbelowtheerrorfromthecorrelationanalysis values of pixel spins chirality have been redistributed ran- (seetext). domlyandthereforeisameasureofthenoiselevel.Wenote that the correlation at early times (bins 1-2) is dominated SFbin 11 10 9 8 7−1 by galaxies in the mass range 1010−1011 M⊙. (Recall that % -0.7 -2.5 -3.4 -2.9 -3.8 since the sample is approximately volume limited thenum- ber of galaxies in each for the mass bins decreases rapidly withincreasingmass,sothetrendisnotdrivenbythenum- than ∼50 Mpc/h. See also Fig. 1 where we show the value berofgalaxiesineachmassbin).Thisisincontrastwiththe of the correlation for different pixelsizes. strongcorrelationfoundbetweenmassandstarformationin We find that the correlation is greatest for galaxies the past by Heavenset al. (2004); Panter et al. (2007). We which formed most of their stars in the past and decays conclude then that the observed correlation between spin for galaxies with most of their stars beingformed recently. chiralityandearly starformation isnotdrivenbymassbut Fig. 3 shows the Pearson correlation coefficient for the bystar formation activity. absolute value of the pixel spin chirality as a function of SofarwehaveusedthegalaxiesfromGalaxyZooclassi- the star formation bin (see Table 1). The points have been fiedascleanwhichlimitsoursampletoonly12897galaxies. obtained by computing the mean for several redshift slices BecausetherearemanymoregalaxiesintheMOPEDcata- of width ∼100 Mpc and the error bars shown are from the logwehaveimplementedthefollowingalgorithmtoincrease mean dispersion in the different, ∼ 10, redshift slices. The theGalaxy Zoosample. Weweight thechirality ofagalaxy dashedlineshowstheerrorinthecorrelation coefficientob- bythenumberofvotesfromthepublic.Ifmostpeoplevoted tained from randomly redistributing the values of the |Sj| foronedirection thisalgorithm convergestotheclean sam- map.Wecanseethatforthosegalaxieswithsignificantstar pleweusedbefore.Theresultsofthecorrelationsoobtained formation intheoldest bins(1−4)thecorrelation with ab- are similar to Fig. 1 within theerrors bars. solutevalueofpixelspinchiralityis6σabovethenoiselevel, while it decreases to slightly weaker levels for galaxies with significant star formation in their recent bins (8-11). This 4 DISCUSSION AND CONCLUSIONS findingindicatesthatspiralgalaxies withmost oftheirstar formation in the past have most of their galaxies rotating Halo spin directions correlate with the large-scale struc- in the same direction with coherence length of 10 Mpc/h. turesee e.g. Bailin & Steinmetz(2005); Hahn et al. (2007); To explore the correlation for separation smaller that 10 Trujillo et al. (2006): halos in sheets tend to have their an- Mpc/h we have repeated the analysis with the projected gular momentum parallel to the sheet, a weaker indication catalog. The catalog depth implies that in the line of sight isobserved for halos in filamentstohavetheirspin perpen- direction thecell size, R is always ∼343 Mpc/h thusdilut- dicular to thefilament direction. ing the signal if the signal coherence length is smaller than Spins directions are also correlated, as seen in obser- R. The higher density of objects however enables us to ex- vations (Slosar et al. 2009; Pen et al. 2000) and as pre- ploresmallerseparations.Wefindthatthesignalismaximal dicted by tidal torque theory (Catelan & Porciani 2001; for cell sizes of 10 Mpc/h (given bythe transversal cell size Catelan et al. 2001; Crittenden et al. 2001; Porciani et al. at zm) and decreases rapidly for cell sizes greater than 50 2002). Mpc/h. As before, also in this case when considering the Wehavefoundsignificantcorrelationbetweenpaststar fullvalueofthepixelspinchirality,i.e.keepingthesign,we formation activity and the degree of coherence of spin chi- obtain no statistically significant correlation. rality for galaxies in a spatial patch (pixel spin chirality). These results indicate that the universe does not have While we have only used the chiral information of nearly a preferred direction and/or more importantly that human face-on galaxies, thedetection ofanon-zerosignal canonly biases in classifying therotation direction donot enterinto be produced if spin orientations are correlated. The corre- our analysis. lation is highest for regions in which galaxies have formed Table2showsforthe11look-backtimes,thepercentage most of their stars in the past. We do not find significant deviation from zero in the value of the pixel spin chirality correlations with spin coherence for any of the other mea- when we consider the sign and demonstrates that a small suredgalaxypropertiesthatwehaveconsidered(metalicity, bias at the 3−4% level remains, toward negative values of stellar mass, metalicity history). S but below the intrinsic error. To calculate the values of We note that using chiral information rather than axis the bias in each look-back time bin we calculate PS for inferred from the projected ellipsoid is safer from the per- β all the galaxies in that bin that contribute to more than spective of potential systematic effects. The reason for this 90% of the bin star formation. We have also verified that is that inferred properties of stellar population are likely to the strength of the correlation signal does not depend on be affected by the galaxy being viewed face-on or edge-on. how manygalaxies are included in thesample byrandomly These properties, on the other hand, cannot be affected by sub-sampling the catalog, the error-bars of course increase thegalactic arms winding on way or another. when less galaxies are considered. 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