Table Of Content

Draftversion January22,2013 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 CALIBRATION OF THE MID-INFRARED TULLY-FISHER RELATION Jenny G. Sorce1, H´el`ene M. Courtois1,2, R. Brent Tully2, Mark Seibert3, Victoria Scowcroft3, Wendy L. Freedman3, Barry F. Madore3, S. Eric Persson3, Andy Monson3, and Jane Rigby4 1Universit´eClaudeBernardLyonI,InstitutdePhysiqueNucleaire,Lyon,France 2InstituteforAstronomy,UniversityofHawaii,2680WoodlawnDrive,HI96822, USA 3CarnegieObservatories,813SantaBarbaraStreet, Pasadena, CA91101,USAand 4ObservationalCosmologyLab,NASAGoddardSpaceFlightCenter,Greenbelt,MD20771, USA Draft version January 22, 2013 3 1 ABSTRACT 0 2 Distance measures on a coherent scale around the sky are required to address the outstanding cosmological problems of the Hubble Constant and of departures from the mean cosmic flow. The n correlation between galaxy luminosities and rotation rates can be used to determine distances to a manythousandsofgalaxiesinawiderangeofenvironmentspotentiallyoutto200Mpc. Mid-infrared J (3.6 µm) photometry with the Spitzer Space Telescope is particularly valuable as the source of the 1 luminosities because it provides products of uniform quality across the sky. From a perch above the 2 atmosphere,essentiallythetotalmagnitudeoftargetscanberegisteredinexposuresofafewminutes. Extinction is minimal and the flux is dominated by the light from old stars which is expected to ] O correlate with the mass of the targets. In spite of the superior photometry, the correlationbetween mid-infrared luminosities and rotation C rates extracted from neutral hydrogen profiles is slightly degraded from the correlation found with . I band luminosities. A color correction recovers a correlation that provides comparable accuracy h p to that available at I band (∼ 20% 1σ in an individual distance) while retaining the advantages - identified above. Without the color correctionthe relation between linewidth and [3.6] magnitudes is o Mb,i,k,a = −20.34−9.74(logWi −2.5). This description is found with a sample of 213 galaxies in r [3.6] mx t 13clustersthat define the slopeand26 galaxieswithCepheid ortip ofthe redgiantbranchdistances s a that define the zero point. A color corrected parameter MC[3.6] is constructed that has reduced [ scatter: M = −20.34−9.13(logWi −2.5). Consideration of the 7 calibration clusters beyond 1 50 Mpc, ouCt[s3i.6d]e the domain of obvioumsxpeculiar velocities, provides a preliminary Hubble Constant v estimate of H0 =74±5 km s−1 Mpc−1. 3 3 Subject headings: Cosmologicalparameters;distancescale;Galaxies: clusters;distancesandredshifts; 8 photometry; infrared: galaxies; radio lines: galaxies 4 . 1 1. INTRODUCTION galaxies entirely. 0 3 Soon after the discovery of the power law correla- The situation dramatically changed with the launch 1 tion between the rotation rates of galaxies and their of Spitzer Space Telescope (Werner et al. 2004). With : luminosities (Tully & Fisher 1977) it was suggested v observations using IRAC, the InfraRed Array Camera (Aaronson et al. 1979) that the methodology might be i (Fazio et al.2004),the‘sky’isfarreducedfromobserva- X improvedby movingto near-infraredbands, particularly tions on the ground, now dominated by diffuse zodiacal when it is used to measure distances. Obscuration r light and the stochastic distribution of background a corrections within the hosts and due to our Galaxy are high redshift galaxies. Imaging with of order 4 minute minimized and light from old stars, which peaks in the integrations in the [3.6] band with this facility permits infrared, should optimally represent the baryonic mass area photometry at levels that reach slightly fainter that presumably couples to the rotation rate. Progress than ground-based optical imaging with comparable with infrared observations of galaxies has been difficult, exposures;ie, to levels thatinclude allbut a few percent though, because of the high and variable sky foreground of the total light of a galaxy (Sorce et al. 2012a). at near-infrared wavelengths and overwhelming thermal In addition, and a very important point, the photom- emissionat mid-infraredwavelengthswith ground-based etryhasconsistentpropertiesinalldirectionsonthesky. observations. The most serious modern attempts to use aninfraredformofthecorrelationhavedrawnontheK s Real progress on this program had to await the magnitudes of 2MASS, the Two Micron All-Sky Survey exhaustion of cryogenics on Spitzer Space Telescope. (Karachentsev et al. 2002). However, these magnitudes, During the subsequent ‘warm’ mission, observations like with the earlier work in the infrared, only register haveonlybeenpossiblewiththetwoshortestwavelength the high surface brightness components of light from passbands with the facility, at 3.6 µm and 4.5 µm, and galaxies and can actually miss low surface brightness there has been an emphasis toward large programs that can usefully work in these bands. This article results [email protected] from a commonality of interests between two of these 2 Sorce et al. programs. One of these, initiated in Spitzer proposal care about minimizing relative distance effects in the cycle 6, is named Carnegie Hubble Program (CHP). The TFR so it is more important to minimize interlopers intent of this program is to reduce systematics arising than maximize true members. Cluster members that in the determination of the Hubble Constant. A part of are ‘window outsiders’ would not be expected to lie in CHP gives attention to a mid-infrared calibration of the any preferred part of the TF diagram. The selection Cepheid Period-Luminosity relation and a second part criteria are: 1) morphological types earlier than Sa are addresses the properties of the rotation rate–luminosity excluded, 2) HI profiles with adequate signal-to-noise correlationofgalaxies,the Tully-FisherRelation(TFR). are required (see next subsection), 3) no evidence of The two parts are related since the TFR zero point confusion or tidal disruption, 4) inclinations inferior to is established by the Cepheid distance measurements. 45◦ are rejected. Tests with samples that satisfy this Freedman et al. (2011) describe the goals of CHP and limit have not revealed any distance bias with inferred Freedman et al. (2012) report on the results of the inclinations (TC12). Criteria for inclusion of zero point Cepheid calibration that gives a distance modulus calibrators are similar, with the additional requirement for the Large Magellanic Cloud of 18.48 ± 0.03. The that they have very well known distances from either second program, initiated in cycle 8, has the name Cepheid or TRGB measurements. In our earlier papers, Cosmic Flows with Spitzer (CFS). The goalsin this case the Cepheid scale had been set by a distance modulus are to acquire distances to several thousand galaxies fortheLargeMagellanicCloudof18.50(Freedman et al. using the mid-infrared TFR in order to map deviations 2001). Here we adopt the slightly modified modulus from Hubble flow. The two programs use overlapping 18.48 ± 0.03 based on mid-infrared photometry of data from Spitzer and require a similar calibration of Cepheids in the LMC and in our Galaxy, the latter the rotation rate - luminosity correlation. This paper anchored with trigonometric parallaxes (Monson et al. presents the calibration that will be used in subsequent 2012). Our TRGB distances are based on a Popula- work with both CHP and CFS. tion II calibration but have been demonstrated to be onaconsistentscale(Rizzi et al.2007;Tully et al.2008). The ensuing discussion borrows heavily on the re- cent re-calibration of the I band correlation by With completion of the CFS program toward the end Tully & Courtois (2012) (hereafter TC12). That paper of2012theentiresampleofcalibratorsusedintheI band outlines a strategy of forming a template relation us- calibrationhas been observed. Because of the overlapin ing samples from 13 galaxy clusters and the establish- interests with CHP a large fraction of the I band cal- mentofazeropointusingnearbygalaxieswithindepen- ibrators used by TC12 have already been observed in dent Cepheid period-luminosity or Tip of the Red Giant the earlier Spitzer cycle for the same purpose of a TFR Branch (TRGB) distances. It turns out that [3.6] mag- calibrationandmostothershavebeenobservedserendip- nitudes now exist for a substantial majority of the same itously in other Spitzer programs. At this time, 230 of galaxies. In this paper we use the same HI profile and 314galaxies(73%)usedintheI bandcalibration(plus9 inclination information as in the I band calibration pa- othergalaxiesintroducedwiththeextensionoftheAbell per. The only significantdifference is the replacementof 2634 sample to include Abell 2666) have Spitzer [3.6] mid-infrared for optical luminosities. It turns out that photometry, including 26 of 36 (72%) that set the zero although the new photometry has high fidelity and the point. photometry correction terms are small there is an in- The completionis greaterthan60%with eachof12 of trinsic color term in the [3.6] band TFR. Scatter in the the13templateclusters(thePiscesfilamentistheexcep- relationisreduceduponapplicationofacolorcorrection. tion). It is deemed appropriate to present a preliminary We conclude with an estimate of the Hubble Constant. calibration with the available material. In a later sec- tion there will be a review of the impact of the current 2. DATA completeness level on the small Malmquist bias that we make. 2.1. Calibrators The slope and zero point calibrator samples are 2.2. HI linewidths described in detail in TC12. The correlation slope is The Cosmic Flows project has now analyzed HI established from a template built from galaxies in 13 profiles for over 14,000 galaxies in a consistent way, clusters. The only departure in terms of an extension deriving a linewidth parameter W with suitable from the I band calibration occurs in the case of Abell m50 precision (error estimate ≤ 20 km s−1) for over 11,000 2634. The CHP program included observations of a galaxies (Courtois et al. 2009, 2011b). This parameter larger region including Abell 2666. The two clusters are is a measure of the HI profile width at 50% of the mean close in projection and, evidently, in distance. We find flux within the velocity range encompassing 90% of no discernable difference in distance between galaxies the total HI flux. The newly measured HI profiles of closest on the sky to A2634 versus those closest to thousands of galaxies are available for public use at A2666. We propose to average over the entire complex. the Extragalactic Distance Database (EDD) website1. This observed parameter W is transformed into the Each cluster sample is comprised of galaxies likely to m50 more physically motivated parameter Wi through beatsimilardistances. Therewasanattempttoinclude mx steps that are justified in Courtois et al. (2009, 2011b) all galaxies with suitable properties down to a defined and reviewed by TC12. Wi statistically approximates faint luminosity level to have an unbiased sampling of mx the cluster volume to a magnitude limit. Candidates are chosen out of a projection-velocity window. We 1 http://edd.ifa.hawaii.edu;catalog‘AllDigitalHI’ [3.6] TF Relation 3 twice the maximum rotation velocity of a galaxy. at the level of 0.03) which is probably attributable to sky settings. We choose to average over CHP and CFS These transformations remove a slight relativistic photometric values. broadening and a broadening due to finite spectral resolution, adjust to twice the projected maximum Uncertainties on apparent magnitudes have been rotation velocity and de-project to edge-on orientation. shown to be very small, cumulatively ±0.05 (SCT12). Linewidth error estimates are based on the level of the The photometry reaches isophotal levels that require signal, S, at 50% of mean flux divided by the noise, onlyafewpercentextrapolationtogivetotalmagnitudes N, measured beyond the extremities of the signal. andthescaleisstabletobetterthan0.01magacrossthe Profiles with error estimates smaller than 20 km s−1 sky (IRAC Instrument Handbook V2.0, 2011). Setting are retained. These profiles meet a minimum flux per theskyremainsadominantuncertaintyatalevelof0.04 channel requirement of S/N ≥ 2 and acceptance after mag. IRAC ch.1 [3.6] luminosities receive the following visual inspection. corrections: 1) A[3.6]: galactic extinction (Cardelli et al. 1989; b Uncertainties in the rotation rate parameter are illus- Schlegel et al. 1998), trated in the error bars of the figures presented in the 2) A[3.6]: internal extinction (Giovanelli et al. 1995, next section. It will be seen that errors in the linewidth i 1997b; Tully et al. 1998), parameter dominate observational uncertainties. Errors 3) A[3.6]: shift in the flux due to Doppler effect in the logarithmiclinewidth parametertend to be larger k (Oke & Sandage 1968; Huang et al. 2007), forslowrotatorssinceatypicalmeasurementuncertainty of 10−20 km s−1 causes a larger fractional uncertainty 4) A[a3.6]: extended emission from the Point Spread withanarrowprofile. Thelargestuncertaintiesareasso- Functionouterwingsandfromscattereddiffuseemission ciated with more face-on galaxies, those toward the 45◦ across the IRAC focal plane (Reach et al. 2005). cutoff. At this limit, a 5◦ error in inclination results in These corrections are all discussed in SCT12. The re- an 8% error in linewidth. sulting apparent magnitude in the AB system is 2.3. [3.6] Photometry [3.6]b,i,k,a =[3.6]−A[3.6]−A[3.6]−A[3.6]+A[3.6]. (1) b i k a The photometric data has all been obtained with IRACch.1,passbandcenter3.55µm. CHP,theCarnegie The data that are used in the following discussion are Hubble Program (Freedman et al. 2011), provides 60% collected into Table 1. This table, the complete version of the data. In addition, S4G, the Spitzer Survey of givenwiththeonlinepublication,includesCFSandCHP total magnitudes, each including the 4 adjustments just Stellar Structure in Galaxies (Sheth et al. 2010), gives described, and averages of the two methods. The table 17%, and SINGS, the Spitzer Infrared Nearby Galaxies also gives inclination and linewidth information drawn Survey (Dale et al. 2005, 2007), giving 9%. This third from TC12 and color terms for color corrections de- program was carried out during the cryogenic phase scribed in sub-section 3.3. The galaxies in Table 1 are while the first two were conducted during the warm either partofthe zero pointcalibration(sample ZeroPt) Spitzer mission. The new Cosmic Flows with Spitzer or a member of a cluster contributing to the slope tem- program has only contributed 3% of the current data. plate. Smaller programs during the cryogenic mission supply us with information on the remaining galaxies. The information comes from a multitude of programs but 3. [3.6]BANDCALIBRATION the integration times are the same within a factor two The TFR calibration requires the definition of a slope (mostly 240 sec, occasionally 120 sec). The integrations and the establishment of an absolute scale. The slope is are sufficiently deep to reach a surface brightness 26.5 the trickiest item because there is a correlation between mag/sq. arcsec (AB) even with the shorter exposures. its value and a form of Malmquist bias. Given two Details on programs and exposure times are included galaxies at the same distance with the same linewidth, at the Extragalactic Distance Database website by the brighter galaxy might be chosen but not the fainter selecting the catalog Spitzer [3.6] Band Photometry. one. The potential bias depends on the slope of the correlation because with a relatively flat slope most The photometric reductions were carried out by two intrinsically luminous galaxies lie above the correlation independent procedures. The method utilized by the while with a very steep slope these same galaxies tend CHP uses software developed for the GALEX Large to lie below the correlation. Consider a target for a Galaxy Atlas (Seibert et al. 2012, in prep.). The distance measurement in the field that intrinsically method developed in anticipation of the arrival of CFS lies above the assumed mean relation, the trend for data is based on the Archangel photometry package distant galaxies if the relation is flat. With the distance (Schombert 2007) described by Sorce et al. (2012a) measurementthe targetis assignedthe mean luminosity (hereafter SCT12) and earlier by Courtois et al. (2011a) of the correlation at the target’s linewidth so given a in the context of optical photometry. In a comparison distancethatistoosmall. Thisbiashasrepeatedlybeen of 171 galaxies (SCT12), the two procedures result in discussed at length, most recently by TC12. The salient agreement at the level of 0.01 mag with rms scatter point is that the so-called ‘inverse’ relation (ITFR), of 0.052. Partitioned equally, the internal uncertainty the least squares regression where errors are taken to (reductions of the same data by different methods and be in linewidth only, gives results that are close to individuals) is ±0.037 mag. There are marginal differ- bias free. Willick (1994) pointed out that, while in his ences for galaxies brighter than [3.6]=11 (CHP brighter experiments the ITFR bias was reduced by a factor 6 4 Sorce et al. from that incurred using the direct relation, yet a small 8 15 Fornax 32 UMa bias remained because the sample selection was not 10 made in the band he considered. We have the same 12 problem. Our strategy is to use the ITFR and then evaluate the bias with simulations anticipatingthat, like 14 with the I band calibration, the effects will be small. 11 Antlia 11 Centaurus The bias tests are discussed in a later section. 10 The calibration process has been described in detail 12 by Tully & Pierce (2000) and TC12. With the I band 14 relation there is no clear evidence for scatter due to a 16 third parameter but the situation at [3.6] is different. A 10 12 Pegasus 14 Hydra color term is found and that matter will be discussed. 12 3.1. Relative distances and ITFR Slope 14 The measurement of distances requires the hypothesis 16 ofauniversalcorrelation. To begin,we makeinversefits 23 Pisces 11 Cancer to each one of the clusters separately. Dotted lines in 10 Figures 1 and 2 illustrate the inverse fits of the TFR for 12 each cluster. Slopes are quite similar between clusters. Slopes and their uncertainties are given for each cluster 14 in Table 2. The individual fits are consistent with 16 the soon to be derived best fit and hence with the 10 16 Coma 7 Abell 400 universal correlation hypothesis. As cluster distances 12 increase, the faint luminosity limits increase. However, no dependence of the slope with distance is seen, as 14 would be a marker of Malmquist bias (we still make a 16 tiny correctionfor bias to cluster moduli as described in Section 3.4). 10 19 Abell 1367 18 Abell 2634/66 12 14 16 2 2.5 2 2.5 3 Fig. 2.—Tully-Fisherrelation inthe [3.6] band for the Fornax, Ursa Major, Antlia, Centaurus, Pegasus, Hydra, Pisces, Cancer, Coma, Abell 400, Abell 1367 and Abell 2634/66 clusters. Solid linesgivetheinversefitoftheuniversaltemplatecorrelation. Dot- tedlinesaretheinversefitsofthecorrelationforeachcluster. cluster in question. Apparent magnitude zero points confirm that Virgo, Fornax and Ursa Major are the closest clusters. Then come Antlia-Centaurus-Pegasus, then Hydra-Pisces- Cancer, and finally Coma and the three Abell clusters A1367, A400 and A2634/66. To establish the best uni- versalslopeandthebestrelativedistancesbetweenclus- ters, we follow an iterative procedure. We initially con- siderthenearestthreeclustersbecausetheyareobserved to comparable depths in intrinsic magnitude. The For- nax and Ursa Major magnitude scales are shifted ac- cording to the difference in zero point with respect to Virgo. A least squares fit of the ITFR is made to this Fig.1.— Tully-Fisher relation in the [3.6] band for the Virgo ensemble. The new slope is assumed in a fit to the 3 Cluster. Thesolidlinegivestheinversefitoftheuniversaltemplate correlation. Thedottedlineistheinversefitofthecorrelationfor individual clusters with only the zero point as a free pa- theVirgoClusteralone. rameter in each case. Given the new zero point offsets the cycle is repeated,leading to rapidconvergence. This The next step is to combine the 13 individual cluster procedure is repeated with the addition of each distance correlations by vertical translations. The Virgo Cluster group in turn. Again, convergence is rapid. It is to is used as a reference. Each preliminary zero point from be stressed that this procedure works because, following the individual fits provides us with a first estimate of expectations, the slope of the ITFR is not affected by the relative distance between the Virgo Cluster and the themagnitudeleveloftruncation. Thisprocedurewould [3.6] TF Relation 5 manifestlynotworkwiththedirectorbi-variaterelations wheretheslopesvarywiththeleveloftruncation. Inthe end we obtain a slope of −9.74±0.22 for the template ITFR. Zero point offsets with this ‘universal’ slope are shown in Figure 3 and give relative distance moduli of clusters referenced to the Virgo Cluster. The universal slope of the ITFR is displayed in Figure 3 as well as by the solid lines in Figures 1 − 2. 24 Virgo 15 Fornax + 0.33 32 UMa + 0.44 11 Antlia + 2.07 11 Centaurus + 2.17 12 Pegasus + 2.47 14 Hydra + 2.90 23 Pisces + 3.30 11 Cancer + 3.37 16 Coma + 4.04 19 Abell 1367 + 4.10 7 Abell 400 + 4.12 18 Abell 2634/66 + 4.43 Fig. 4.—TFR forthe 26galaxies withdistances established by observationsofCepheidstars(circles)ortheTipoftheRedGiant Branch (squares). The solid line is the least squares fit with the slopeestablishedbythe13clustertemplate. Thezeropointofthe TFRissetatthevalueofthisfitatlogWi =2.5. mx gether as seen in Figure 5. The ITFR expression in the [3.6]-band is given by: Mb,i,k,a =−(20.34±0.10)−(9.74±0.22)(logWi −2.5) [3.6] mx Fig.3.— Template Tully-Fisher relation in the [3.6] band ob- (2) tained with data from 213 galaxies in 13 clusters. Offsets given withrespect to the VirgoCluster represent distance modulus dif- ferences between each cluster and Virgo. The solid line is a least squaresfittoallthegalaxieswitherrorsentirelyinlinewidths,the ITFR. 26 Zero Point Calibrators 24 Virgo 3.2. Zero Point and Absolute Distances 15 Fornax 32 UMa Presently, [3.6] photometry is available for 26 nearby 11 Antlia galaxies with suitable morphologies, inclinations, and 11 Centaurus 12 Pegasus linewidths that also have well measured distances from 14 Hydra 23 Pisces either the Cepheid period-luminosity or tip of the red 11 Cancer giant branch methodologies. These 26 are a subset 16 Coma 19 Abell 1367 of the 36 absolute calibrator galaxies used in the I 7 Abell 400 band calibration (TC12). Their luminosity-linewidth 18 Abell 2634/66 correlationis seen in Figure 4 where now the ordinate is absolutemagnitudesfromtheestablisheddistances. The line is a leastsquaresfitwith the slope −9.74prescribed by the template. The zero point is −20.34±0.10. The most deviant point is the fastest rotator, NGC 2841, with a deviation of 2.7σ with respect to the template dispersion. This galaxy was a 2.3σ deviant in the I band calibration. There is nothing unusual about this galaxy other than its extreme rotation rate so we see no reason to disregard it as a calibrator. The distance to the Virgo Cluster is given by the zero point of the constrained slope shown in Figure 3 minus Fig. 5.— The template of the new [3.6] band - HI linewidth the zero point of the absolute calibration shown in Fig- correlation is built with 213 galaxies in 13 clusters extending in ure 4. Application of this shift allows both cluster tem- range from 1000 to 10,000 km s−1 with the absolute magnitude plate and zero pointcalibratorgalaxiesto be plotted to- scalesetby26zeropointcalibrators. 6 Sorce et al. The TFR scatter in magnitudes (relevant for distance will rise more than bluer galaxies. It follows that red measurements) is given by: and blue galaxies cannot be well mixed in the TFR at all wavelengths. The trends that could be anticipated χ2 areshowninFigure7(onlyaportionofthesamplehave σ = (3) TF rN −1 photometric measurements at B band). The compari- sonof fluxes at four bands fromB to [3.6]for individual where χ2 is the minimum of sources given in Figure 8 confirms the well known link- age between galaxy type and color. Early type galaxies (M −(a+b(logWi −2.5)))2 i mx have relatively more infrared flux relative to late type X galaxies. This point was also illustrated with the repre- with a and b the zero point and slope of the ITFR sentative spectral energy distribution plots in Figure 1 respectively and N −1 are the degrees of freedom. The of SCT12. Galaxies that are more luminous and earlier scatter for the entire cluster template sample is ±0.49 intype aredominatedby older,moremetalenrichedred magfromtheuniversalITFR,correspondingtoascatter giant stars emitting more in the infrared. in distance of 25%. The scatter for the 26 zero point calibrators is a similar 0.44 mag. Dispersion increases towardfaintermagnitudesaswelldocumentedatI band by Giovanelli et al. (1997a). The sample presented here is still limited but the dispersion is consistent with a Gaussian distribution. With large samples (Tully et al. 2008)onefindsabout3%ofcandidatesaremoredeviant than anticipated by Gaussian statistics. The causes are not always evident. Scatter may arise from: 1) measurement uncertain- ties affecting magnitudes, inclinations, and linewidths, 2) correctionuncertainties applied to measured parame- ters, and 3)‘cosmic’ scatter, e.g. cluster depth effects or interlopers, deviations from disk planarity, other gravi- tational and photometric asymmetries, variations in the stellar population make-up, variations in disk-to-bulge ratios,etc. Whateverthe sources,wehaveastandardto meetsetbytheI bandanalysis. Thesamplesusedinthe currentanalysisinvolve80%ofthesamplesusedintheI- Fig.6.— TFR in B,R,I and [3.6] bands. B and R bands data bandcalibration(TC12). Inclinationsandlinewidthsare arefrom Tully&Pierce(2000) , I band data are from TC12 and the same, the factors mentioned associated with cosmic [3.6]banddataarefromSCT12. Linewidthsarethesameasused byTC12. Theslopes steepen frombluetored, withvalues −7.27 scatter are the same, corrections to photometric param- atB,−7.65atR,−8.81atI,and−9.74at[3.6]. eters are reduced in the mid-infrared, and the integrity ofthemagnitudemeasurementsmustbeatleastasgood There have been long standing suggestions that the orbetterwiththeSpitzerobservationssinceobservations dispersion in the TFR might be reduced by inclusion of are made all-sky with the same instrumental configura- additionalparameters. Inanearlyinstance(Rubin et al. tion. Error bars on magnitudes are reduced in Figures 1985), when only photographic or photoelectric magni- 1−5 compared with those on equivalent plots in the I tudes were available, the case was framed in terms of bandcalibrationpaper (TC12)to the degree that obser- galaxy types which are strongly correlated with color. vational errors in magnitudes are a minimal component Masters et al. (2006) have maintained the use of a type of uncertainties. Yet the scatter found at I band is less: separation with I band work. Tully & Pierce (2000) ac- ±0.41mag for the cluster template sample, lowerwith a knowledgedthe hint of a type dependence in the I band significance of 2σ, and 0.36 mag for the zero point cal- relation but concluded that the evidence remained too ibrators. As much as half of the increase in magnitude weaktowarrantaddingcomplexitytotheTFRanalysis. scatter will occur because the slope of the correlation is The situation changes with the mid-infrared infor- steeper in the mid-infrared. However there could be an mation. In spite of superior photometry the scatter additionalexplanationfor the increasedscatter foundat in the TFR is increased and there is a significant [3.6]. color signature. The variations in spectral energy distributionimplicitintherangeofrepresentativecolors 3.3. A Color Term shown in Figure 8 provide a natural explanation given Ithas long beenknownthatthe TFRsteepens toward theextendedleverarmfromtheopticaltothe[3.6]band. longerwavelengths(Tully et al.1982). Theeffectisseen in Figure 6. (Note: in the discussions in this section There is also the possibility that some flux in the [3.6] allopticalphotometryvalueshavebeentransferredfrom band may come from other than old stars. Meidt et al. VegatoAB zeropoints.) Thereisastrongcolorcorrela- (2012) determined that 12 ± 5% of [3.6] flux arises tion with linewidth, more rapidly rotating galaxies tend from hot dust, PAH emission, or young to intermediate to be redder, so at longer wavelengths the high rotation age stars in 6 representative spiral galaxies observed end of the TFR rises with respect to the low rotation with Spitzer Space Telescope. However the variance of end. Within a small linewidth interval, redder galaxies 0.05 mag is small compared with the ITFR scatter. [3.6] TF Relation 7 Fig.8.—RepresentationoffluxesatB,R,I,[3.6]bandsnormal- ized to unity at I band. Type Sa: red; types Sb-Sc: green; types Sd-Sm: blue. Theextrema aredefined bymembers ofour sample andcolor swathsindicate thedomainsdominated bythedifferent types. Moreover, it can be anticipated that the galaxies most affected by manifestations of star formation are later, bluertypes,whenceaugmentedfluxwilltendtodiminish a color term arising from old stars. Whatever the cause, it can be anticipated that the scattercanbe decreasedwiththe introductionofacolor correction. Toaddressthisissueweconsiderthestraight line fits included in the top three panels of Figure 7. The fits are least squares minimizations on the ordinate parameter; the difference in magnitude of a target from the meanTFR. The bottompanelshowsthe concordant variation of color with linewidth. Faster rotators tend to be redder. In the mid-infrared case, the offset for an individual galaxy from the mean fit in the figure is: ∆Mcolor =Mb,i,k,a+20.34+9.74(logWi −2.5). (4) [3.6] [3.6] mx An equivalent correction can be constructed with ap- parent magnitudes rather than absolute magnitudes, ∆[3.6]color =∆Mcolor, with an appropriate replacement [3.6] of the zero point constant in Eq. 4. The correctionterm commensuratewith the fit in the third panel ofFigure 7 is ∆[3.6]color =∆Mcolor =−(0.47±0.11)[(I−[3.6])+0.77]. [3.6] (5) Weintroduceanewcoloradjustedmagnitudeparameter C =[3.6]b,i,k,a−∆[3.6]color wherethedistinctnomen- [3.6] clature emphasizes the composite nature of this pseudo- magnitude. Next, the analysis discussed in Section 3.2 leadingtothe constructionofFigure4isrepeated. Like- wise,the adjustmentsareappliedtothe calibratorswith independently established distances and the procedures Fig.7.—Top3panels: DeviationsfromthemeanITFRrelation asafunctionofI−[3.6]color. Solidanddottedlinesarebestfits are repeated that lead to Figure 5. The adjusted re- and95%probabilitylimits. Top: AtBbandredgalaxiestendtolie lations are shown in Figure 9. The new correlation is belowthemeanrelationship. Topmiddle: AtIbandthereisahint described by the formula: that redgalaxies lielow although the correlation fit isdominated byafewextremecases. Bottommiddle: At[3.6]bandthesenseof M =−(20.34±0.08)−(9.13±0.22)(logWi −2.5) the correlation has flipped and red galaxies tend to lie above the C[3.6] mx mean relation. Bottom: The correlation between linewidth and (6) color. Theflatteningoftheadjustedrelationcomesaboutsince 8 Sorce et al. 3.4. Bias 24 Virgo 15 Fornax + 0.37 Willick (1994) showed that a small Malmquist bias 32 UMa + 0.45 11 Antlia + 1.95 existsinthe useofthe ITFR,althoughreducedfromthe 11 Centaurus + 2.08 direct TFR by a factor of 6 in the situation he explored 12 Pegasus + 2.40 14 Hydra + 2.86 (Willick et al. 1995), reducing the bias reflected in the 23 Pisces + 3.19 Hubble Constant from 17% to 3%. The bias arises from 11 Cancer + 3.29 16 Coma + 4.01 two effects. First, sample selection departs from an 19 Abell 1367 + 4.01 7 Abell 400 + 4.01 idealized case of a flat magnitude limit because samples 18 Abell 2634/66 + 4.35 have been selected in blue bands and color terms trans- late to a slope in the limiting magnitude in the infrared: slower rotators which tend to be bluer are favored for inclusion over faster rotators which tend to be redder (see Figure 7, bottom). Second, the shape of the galaxy luminosityfunctioncontributestothebiasbecausethere aremoreintrinsicallyfaintergalaxiesthatscatterbright- ward through errors than intrinsically brighter galaxies that scatter faint-ward (Eddington 1913). The bias increases with distance as the effect of the exponential cutoffoftheluminosityfunctionplaysanincreasingrole. The amplitude ofthe bias fromthe twoeffects wasex- plored with the calibration at I band (TC12). The sit- uation now with the [3.6] band sample is slightly worse than at I because the wavelength interval from selec- 26 Zero Point Calibrators tion at B is larger. The bias analysis carried out in the 24 Virgo 15 Fornax case of the I band calibration is repeated here, tailored 32 UMa to the current situation. We first combine the Virgo, 11 Antlia 11 Centaurus Fornax, and Ursa Major samples to improve statistics 12 Pegasus and include contributions from a range of environments. 14 Hydra 23 Pisces This ensemble is described by a Schechter (1976) func- 11 Cancer tion with faint end slope α = −0.9 and a bright end 16 Coma 19 Abell 1367 cutoff at M⋆ = −22. Then we randomly populate 7 Abell 400 [3.6] 18 Abell 2634/66 an artificial TFR to match the observed [3.6] band re- lation, drawing from the Schechter luminosity function. Thefaintlimitisdeterminedempiricallytoroughlyobey the relation Mlim =C −2.70(logWi −1.8) where C [3.6] ℓ mx ℓ couples with distance. The artificial TFR and the cut- off for the nearest clusters is shown in the top panel of Figure 10. The dashed blue line indicates the cut- off experienced at a distance modulus of 31. The cut- off, characterized by C , slides to brighter (more nega- ℓ tive) magnitudes linearly with increasing distance mod- ulus. The bias < ∆M > is determined at in- measured tervalsof C correspondingto increasingdistance. Here, ℓ <∆M > is the averagedeviationfromthe fidu- measured cial relation where < ∆M > = 0 by construction. Fig.9.— The ITFR after adjustments for the color term. Top: true The growth of the bias as a function of cutoff magni- Coloradjustedapparentmagnitudestranslatedtotherelativedis- tanceoftheVirgoCluster. Bottom: Coloradjustedabsolutemag- tude is seen in the bottom panel of Figure 10. The solid nitudeswiththeabsolutedistancescaleestablishedbythegalaxies curve,normalizedto unityatadistancemodulus µ=31 withindependentdistances representedbylargeopencircles. whereeventhefaintestofusefulcandidatesareincluded, reddersystemsmovedownwardandreddergalaxiestend isdescribedbytheformulabetweenbias,b,anddistance to havelargerlinewidths. The overallmagnitude scatter modulus, µ: inthenewrelationis±0.44mag(correspondingtoascat- ter in distance of 22%), down from 0.49 mag before ad- b=−0.0065(µ−31)2. (7) justment,andcomparablewith0.41foundatI bandwith an otherwise comparable analysis (TC12). The compa- By comparison, the coefficient in the case of the I band rable numbers for the zero point calibrators alone are a analysisis−0.005. Thelettersatthebottomofthefigure scatter of 0.37 with the adjusted parameter C , 0.44 are codes for the 13 calibrating clusters (see Table 4 to [3.6] beforetheadjustment, and0.36atI band. The compar- deciphercodes)andtheir horizontalplacements indicate isonsbetween [3.6]and I havesome imprecisionbecause therespectivesamplelimitsandprojectionupwardgives thesamplesizesforthelatterare25%greater. TheTFR the corresponding biases. These biases are recorded in parameters derived from alternative samples and band- Table 2 and are reflected in the adjusted cluster moduli passes are summarized in Table 3. and distances. For a galaxy in the field, the corrected [3.6] TF Relation 9 distance modulus µc can be expressed as 4. THEHUBBLECONSTANT µc =(C −M )+0.0065[(C −M )−31]2 (8) ThelastcolumninTable2recordsthe‘Hubbleparam- [3.6] C[3.6] [3.6] C[3.6] eter’ for each cluster: the velocity of the cluster in the CMB frame divided by the measured distance. These quantities are plotted against distance in Figure 11. A similar figure was presented as a summary of results from the I band calibration with the same 13 clusters (TC12: distances compared in Table 4). Here, as there, we see a large scatter in the Hubble parameter for the nearer clusters and small scatter for the more distant clusters. It can be anticipated that the measures for the nearerclustersarestronglyaffectedbypeculiarmotions. The5clusterswithin40Mpcareallpartofourextended supercluster complex: either within the historic Local Supercluster or the so-called Great Attractor region. The low scatter among the 7 clusters more distant than 50 Mpc (V > 4000 km s−1) suggests that CMB the relative contributions of peculiar velocities have a modest effect on redshifts at such large distances. InthecaseoftheI bandcalibration,themeanvalueof theHubbleparameterforthe7mostdistantclusterswas 75.1±2.7 km s−1 Mpc−1 where the erroris just the rms scatterofthe7contributions. Thatvaluewouldincrease to75.8withtherevisedLMCdistancefromMonsonetal (2012). With the present calibration, including the new LMC distance, the fit shown in Figure 11 gives a value of H = 73.8 with an rms scatter of 1.1 and a standard 0 deviation of 0.4 km s−1 Mpc−1 for the same 7 clusters considered previously. If the fit is extended to include the Pegasus Cluster at 44.5 Mpc then H = 74.4 and 0 the scatter is 2.0 km s−1 Mpc−1. The effect of a deviant radial motion of 200 km s−1 is illustrated in the figure as a function of distance. UF V CPee PCia AnAH2 Co A1 A4 Fig.10.— Top: Simulated TFR drawing randomly from a Schechter luminosity function with slope α = −0.9 and cutoff M⋆ =−22. TheITFR has slope−9.13andscatter 0.4mag. The dashed blue slanting line illustrates the color dependence at the Fig.11.— Hubble parameter as a function of distance. The fBwaihianiscth<liimn∆cirtMeraes>seusmlwteianistguhrfderdiosmtaasnsacaemf.upBnlcelaticsokenletrcoitfaionanbglseionsl:uttflheaetmbfaalguinneti.tluiBmdeoittlt;iomrmeidt: s(MVolpCidcM−lBi1n.e>iCs4ua0r0fiv0etdtkomdcoltsut−set1de)r.lipnToeihsnetislfiluattsgtdrivaisettesanHdce0evsi=agtri7eo3an.t8ser±inth1va.1enlok5cm0itMys−po1cf circles: faint limit increasing with increasing linewidth in accor- 200kms−1 fromthefit. dance withblue lineintop panel. Solidcurve: the empirical bias fit b = −0.0065(µ−31)2. Letters at the bottom: codes for the The uncertainty from the fit in Figure 11 is given by 13calibratingcluster(see Table4fortranslationofcodes). Their horizontal positions indicate sample limits and vertical intercepts thestatisticsofthedeviationsofthe7contributionsand withthesolidcurvegivethecorrespondingbiases. is unrealistically low. This error is what is expected if there is perfect Hubble expansion. If peculiar mo- 10 Sorce et al. tions of 200 km s−1 are the norm, and given the ex- same sample at I band (TC12). pected statistical errors in the distance of each cluster, the anticipated scatter around the mean Hubble value At the expense of the requirement of extra knowledge is ±2.6 km s−1 Mpc−1. We consider this to be our 1σ in the form of a color, the TFR in the [3.6] band can random error. We have several sources of systematic er- be reformulated in a form with scatter that matches the ror. The dominant component, creating almost 4% un- best optical formulations. The correction is small and certainty in H , comes from the uncertainty in the TFR not acutely dependent on the color measurement. The 0 zero point with just 26 calibrators. Combined with a appropriate ITFR equation for the measurement of dis- small uncertainty from the finite population of the tem- tances is plate,theuncertaintyinH associatedwiththeTFRcal- 0 M =−(20.34±0.08)−(9.13±0.22)(logWi −2.5) ibration(assumingthezeropointcalibratordistancesare C[3.6] mx perfect) is ±2.9 km s−1 Mpc−1. The zero point calibra- where M is derived from the corrected apparent tordistancesarenotperfect. Freedman et al.(2012)and C[3.6] magnitude [3.6]b,i,k,a of a source minus the color term Riess et al.(2011) reportthat with new Milky Way par- allaxes for Cepheid stars and mid-infrared Spitzer pho- ∆[3.6]color =−0.36−0.47(I−[3.6]). tometry the uncertainty in the Cepheid scale is in the range ±[1.9 − 2.5] km s−1 Mpc−1. The TRGB zero The slope of this formulation has been derived from point calibration which concerns 4 of the 26 calibra- a sample of 213 galaxies distributed in 13 clusters, tors, has similar or smaller systematics. The cumulative while the zero point is established from 26 calibrators systematic error in H is ±[3.5− 3.8] km s−1 Mpc−1. with Cepheid or tip of the red giant branch distances. 0 Combining random and systematic components we find The rms scatter in distances found with these galaxies H =73.8±2.6(ran)±[3.5−3.8](sys) km s−1 Mpc−1. (cluster template and zero point calibrators combined) 0 of 0.42 mag, 21% in distance, is insignificantly different 5. CONCLUSIONS from the accuracy found with the strongly overlapping Agreatconcernwithstudiesofmotionsonlargescales I band study. with the TFR has been the possibility that system- atic errors in photometry could create spurious flows. Distance measures derived with this calibration Small offsets between different observers, instruments, are subject to a small Malmquist bias, requiring the conditions, hemispheres, or seasons could be sky-sector distance modulus correction µc = µ+0.0065(µ−31)2. dependent. Probably the single most important ad- After application of bias and color corrections, a vantage of the use of space-based photometry such as preliminary estimate of the Hubble Constant can be offeredbythe Spitzermissioncomesfromtheconfidence made from the velocities and distances to 7 clusters at that measurements are on the same scale at better than VCMB >4000 km s−1. Accounting for all error sources, 1% in all parts of the sky. There are other advantages. the determination is H0 = 74 ± 5 km s−1 Mpc−1. Obscuration is minimal both within targets and from The difference between the value determined with this our Galaxy. This latter point is especially significant mid-IR analysis compared with the I band value found because studies of galaxy flow patterns can now reach with the same procedures and a strongly overlapping high levels of completion across the sky. Then it is sample (TC12) is ∆H0 = −2 km s−1 Mpc−1, not a a considerable advantage that the great majority of formally significant difference. We reiterate that the flux at [3.6] band arises from old stars, mainly those greatstrength of the presentcalibration is the high con- on the red giant branch. It can be surmised from the fidence in uniformity over the entire sky. Nevertheless modest scatter in the TFR that there is a close coupling the presentsample ofonly 7 clusters beyondthe domain betweenthe massin starsandthe dynamicalmass. And of known extreme peculiar velocities is unsatisfactorily there is an advantage, at least vis `a vis ground infrared small. In a subsequent paper (Sorce et al. 2012b) the observations, with the sensitivity achieved because of [3.6] band calibration is extended to a calibration of the very low sky noise. All but a few percent of the total Type Ia supernova scale, analogous to what has been flux ismeasuredwithinisophotsresolvedfromthe noise. done at I band (Courtois & Tully 2012), permitting a determination of H at z ∼0.1. 0 A small disadvantage with the mid-IR TFR calibra- tion has been revealed with the documentation of a colorterm. This color termis understood as the natural consequence of the correlation between galaxy rotation The data use in this paper are available at the Extra- rate or luminosity and color (Tully et al. 1982). At a galactic Distance Database.2 The photometric data are given linewidth, red galaxies progressively get brighter found by selecting the catalog Spitzer [3.6] Band Pho- relative to blue galaxies as one considers the TFR at tometryandthenagalaxyofchoicewhile theHI profiles longer wavelengths. Evidence is accumulating that are found in the catalog All Digital HI. We thank Tom intrinsic scatter in the simple two parameter TFR is Jarrett, part of the Cosmic Flows with Spitzer collab- minimalwith photometryatabout1µm. Aconsequence oration, for advice with Spitzer photometry and James of the color dependence is a steepening of the TFR Shombert for his development of Archangel. Much of toward the infrared. If one is interested in the physical the data used here comes from the Spitzer Space Tele- implicationsoftheTFRratherthanitsuseasadistance scope archive. We thank Kartik Sheth for the contribu- tool then the bivariate fit is of interest. Our template tion of his program Spitzer Survey of Stellar Structure sample has the bivariate dependence M ∝ W3.8±0.1 [3.6] which is 0.4 steeper than was found with almost the 2 http://edd.ifa.hawaii.edu

ebook img

Calibration of the Mid-Infrared Tully-Fisher Relation PDF

0.47 MB·English
by  Jenny G. Sorce
#additional_collections #journals #arxiv
Save to my drive
Quick download
Download
Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Calibration of the Mid-Infrared Tully-Fisher Relation

Draftversion January22,2013 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 CALIBRATION OF THE MID-INFRARED TULLY-FISHER RELATION Jenny G. Sorce1, H´el`ene M. Courtois1,2, R. Brent Tully2, Mark Seibert3, Victoria Scowcroft3, Wendy L. Freedman3, Barry F. Madore3, S. Eric Persson3, Andy Monson3, and Jane Rigby4 1Universit´eClaudeBernardLyonI,InstitutdePhysiqueNucleaire,Lyon,France 2InstituteforAstronomy,UniversityofHawaii,2680WoodlawnDrive,HI96822, USA 3CarnegieObservatories,813SantaBarbaraStreet, Pasadena, CA91101,USAand 4ObservationalCosmologyLab,NASAGoddardSpaceFlightCenter,Greenbelt,MD20771, USA Draft version January 22, 2013 3 1 ABSTRACT 0 2 Distance measures on a coherent scale around the sky are required to address the outstanding cosmological problems of the Hubble Constant and of departures from the mean cosmic flow. The n correlation between galaxy luminosities and rotation rates can be used to determine distances to a manythousandsofgalaxiesinawiderangeofenvironmentspotentiallyoutto200Mpc. Mid-infrared J (3.6 µm) photometry with the Spitzer Space Telescope is particularly valuable as the source of the 1 luminosities because it provides products of uniform quality across the sky. From a perch above the 2 atmosphere,essentiallythetotalmagnitudeoftargetscanberegisteredinexposuresofafewminutes. Extinction is minimal and the flux is dominated by the light from old stars which is expected to ] O correlate with the mass of the targets. In spite of the superior photometry, the correlationbetween mid-infrared luminosities and rotation C rates extracted from neutral hydrogen profiles is slightly degraded from the correlation found with . I band luminosities. A color correction recovers a correlation that provides comparable accuracy h p to that available at I band (∼ 20% 1σ in an individual distance) while retaining the advantages - identified above. Without the color correctionthe relation between linewidth and [3.6] magnitudes is o Mb,i,k,a = −20.34−9.74(logWi −2.5). This description is found with a sample of 213 galaxies in r [3.6] mx t 13clustersthat define the slopeand26 galaxieswithCepheid ortip ofthe redgiantbranchdistances s a that define the zero point. A color corrected parameter MC[3.6] is constructed that has reduced [ scatter: M = −20.34−9.13(logWi −2.5). Consideration of the 7 calibration clusters beyond 1 50 Mpc, ouCt[s3i.6d]e the domain of obvioumsxpeculiar velocities, provides a preliminary Hubble Constant v estimate of H0 =74±5 km s−1 Mpc−1. 3 3 Subject headings: Cosmologicalparameters;distancescale;Galaxies: clusters;distancesandredshifts; 8 photometry; infrared: galaxies; radio lines: galaxies 4 . 1 1. INTRODUCTION galaxies entirely. 0 3 Soon after the discovery of the power law correla- The situation dramatically changed with the launch 1 tion between the rotation rates of galaxies and their of Spitzer Space Telescope (Werner et al. 2004). With : luminosities (Tully & Fisher 1977) it was suggested v observations using IRAC, the InfraRed Array Camera (Aaronson et al. 1979) that the methodology might be i (Fazio et al.2004),the‘sky’isfarreducedfromobserva- X improvedby movingto near-infraredbands, particularly tions on the ground, now dominated by diffuse zodiacal when it is used to measure distances. Obscuration r light and the stochastic distribution of background a corrections within the hosts and due to our Galaxy are high redshift galaxies. Imaging with of order 4 minute minimized and light from old stars, which peaks in the integrations in the [3.6] band with this facility permits infrared, should optimally represent the baryonic mass area photometry at levels that reach slightly fainter that presumably couples to the rotation rate. Progress than ground-based optical imaging with comparable with infrared observations of galaxies has been difficult, exposures;ie, to levels thatinclude allbut a few percent though, because of the high and variable sky foreground of the total light of a galaxy (Sorce et al. 2012a). at near-infrared wavelengths and overwhelming thermal In addition, and a very important point, the photom- emissionat mid-infraredwavelengthswith ground-based etryhasconsistentpropertiesinalldirectionsonthesky. observations. The most serious modern attempts to use aninfraredformofthecorrelationhavedrawnontheK s Real progress on this program had to await the magnitudes of 2MASS, the Two Micron All-Sky Survey exhaustion of cryogenics on Spitzer Space Telescope. (Karachentsev et al. 2002). However, these magnitudes, During the subsequent ‘warm’ mission, observations like with the earlier work in the infrared, only register haveonlybeenpossiblewiththetwoshortestwavelength the high surface brightness components of light from passbands with the facility, at 3.6 µm and 4.5 µm, and galaxies and can actually miss low surface brightness there has been an emphasis toward large programs that can usefully work in these bands. This article results [email protected] from a commonality of interests between two of these 2 Sorce et al. programs. One of these, initiated in Spitzer proposal care about minimizing relative distance effects in the cycle 6, is named Carnegie Hubble Program (CHP). The TFR so it is more important to minimize interlopers intent of this program is to reduce systematics arising than maximize true members. Cluster members that in the determination of the Hubble Constant. A part of are ‘window outsiders’ would not be expected to lie in CHP gives attention to a mid-infrared calibration of the any preferred part of the TF diagram. The selection Cepheid Period-Luminosity relation and a second part criteria are: 1) morphological types earlier than Sa are addresses the properties of the rotation rate–luminosity excluded, 2) HI profiles with adequate signal-to-noise correlationofgalaxies,the Tully-FisherRelation(TFR). are required (see next subsection), 3) no evidence of The two parts are related since the TFR zero point confusion or tidal disruption, 4) inclinations inferior to is established by the Cepheid distance measurements. 45◦ are rejected. Tests with samples that satisfy this Freedman et al. (2011) describe the goals of CHP and limit have not revealed any distance bias with inferred Freedman et al. (2012) report on the results of the inclinations (TC12). Criteria for inclusion of zero point Cepheid calibration that gives a distance modulus calibrators are similar, with the additional requirement for the Large Magellanic Cloud of 18.48 ± 0.03. The that they have very well known distances from either second program, initiated in cycle 8, has the name Cepheid or TRGB measurements. In our earlier papers, Cosmic Flows with Spitzer (CFS). The goalsin this case the Cepheid scale had been set by a distance modulus are to acquire distances to several thousand galaxies fortheLargeMagellanicCloudof18.50(Freedman et al. using the mid-infrared TFR in order to map deviations 2001). Here we adopt the slightly modified modulus from Hubble flow. The two programs use overlapping 18.48 ± 0.03 based on mid-infrared photometry of data from Spitzer and require a similar calibration of Cepheids in the LMC and in our Galaxy, the latter the rotation rate - luminosity correlation. This paper anchored with trigonometric parallaxes (Monson et al. presents the calibration that will be used in subsequent 2012). Our TRGB distances are based on a Popula- work with both CHP and CFS. tion II calibration but have been demonstrated to be onaconsistentscale(Rizzi et al.2007;Tully et al.2008). The ensuing discussion borrows heavily on the re- cent re-calibration of the I band correlation by With completion of the CFS program toward the end Tully & Courtois (2012) (hereafter TC12). That paper of2012theentiresampleofcalibratorsusedintheI band outlines a strategy of forming a template relation us- calibrationhas been observed. Because of the overlapin ing samples from 13 galaxy clusters and the establish- interests with CHP a large fraction of the I band cal- mentofazeropointusingnearbygalaxieswithindepen- ibrators used by TC12 have already been observed in dent Cepheid period-luminosity or Tip of the Red Giant the earlier Spitzer cycle for the same purpose of a TFR Branch (TRGB) distances. It turns out that [3.6] mag- calibrationandmostothershavebeenobservedserendip- nitudes now exist for a substantial majority of the same itously in other Spitzer programs. At this time, 230 of galaxies. In this paper we use the same HI profile and 314galaxies(73%)usedintheI bandcalibration(plus9 inclination information as in the I band calibration pa- othergalaxiesintroducedwiththeextensionoftheAbell per. The only significantdifference is the replacementof 2634 sample to include Abell 2666) have Spitzer [3.6] mid-infrared for optical luminosities. It turns out that photometry, including 26 of 36 (72%) that set the zero although the new photometry has high fidelity and the point. photometry correction terms are small there is an in- The completionis greaterthan60%with eachof12 of trinsic color term in the [3.6] band TFR. Scatter in the the13templateclusters(thePiscesfilamentistheexcep- relationisreduceduponapplicationofacolorcorrection. tion). It is deemed appropriate to present a preliminary We conclude with an estimate of the Hubble Constant. calibration with the available material. In a later sec- tion there will be a review of the impact of the current 2. DATA completeness level on the small Malmquist bias that we make. 2.1. Calibrators The slope and zero point calibrator samples are 2.2. HI linewidths described in detail in TC12. The correlation slope is The Cosmic Flows project has now analyzed HI established from a template built from galaxies in 13 profiles for over 14,000 galaxies in a consistent way, clusters. The only departure in terms of an extension deriving a linewidth parameter W with suitable from the I band calibration occurs in the case of Abell m50 precision (error estimate ≤ 20 km s−1) for over 11,000 2634. The CHP program included observations of a galaxies (Courtois et al. 2009, 2011b). This parameter larger region including Abell 2666. The two clusters are is a measure of the HI profile width at 50% of the mean close in projection and, evidently, in distance. We find flux within the velocity range encompassing 90% of no discernable difference in distance between galaxies the total HI flux. The newly measured HI profiles of closest on the sky to A2634 versus those closest to thousands of galaxies are available for public use at A2666. We propose to average over the entire complex. the Extragalactic Distance Database (EDD) website1. This observed parameter W is transformed into the Each cluster sample is comprised of galaxies likely to m50 more physically motivated parameter Wi through beatsimilardistances. Therewasanattempttoinclude mx steps that are justified in Courtois et al. (2009, 2011b) all galaxies with suitable properties down to a defined and reviewed by TC12. Wi statistically approximates faint luminosity level to have an unbiased sampling of mx the cluster volume to a magnitude limit. Candidates are chosen out of a projection-velocity window. We 1 http://edd.ifa.hawaii.edu;catalog‘AllDigitalHI’ [3.6] TF Relation 3 twice the maximum rotation velocity of a galaxy. at the level of 0.03) which is probably attributable to sky settings. We choose to average over CHP and CFS These transformations remove a slight relativistic photometric values. broadening and a broadening due to finite spectral resolution, adjust to twice the projected maximum Uncertainties on apparent magnitudes have been rotation velocity and de-project to edge-on orientation. shown to be very small, cumulatively ±0.05 (SCT12). Linewidth error estimates are based on the level of the The photometry reaches isophotal levels that require signal, S, at 50% of mean flux divided by the noise, onlyafewpercentextrapolationtogivetotalmagnitudes N, measured beyond the extremities of the signal. andthescaleisstabletobetterthan0.01magacrossthe Profiles with error estimates smaller than 20 km s−1 sky (IRAC Instrument Handbook V2.0, 2011). Setting are retained. These profiles meet a minimum flux per theskyremainsadominantuncertaintyatalevelof0.04 channel requirement of S/N ≥ 2 and acceptance after mag. IRAC ch.1 [3.6] luminosities receive the following visual inspection. corrections: 1) A[3.6]: galactic extinction (Cardelli et al. 1989; b Uncertainties in the rotation rate parameter are illus- Schlegel et al. 1998), trated in the error bars of the figures presented in the 2) A[3.6]: internal extinction (Giovanelli et al. 1995, next section. It will be seen that errors in the linewidth i 1997b; Tully et al. 1998), parameter dominate observational uncertainties. Errors 3) A[3.6]: shift in the flux due to Doppler effect in the logarithmiclinewidth parametertend to be larger k (Oke & Sandage 1968; Huang et al. 2007), forslowrotatorssinceatypicalmeasurementuncertainty of 10−20 km s−1 causes a larger fractional uncertainty 4) A[a3.6]: extended emission from the Point Spread withanarrowprofile. Thelargestuncertaintiesareasso- Functionouterwingsandfromscattereddiffuseemission ciated with more face-on galaxies, those toward the 45◦ across the IRAC focal plane (Reach et al. 2005). cutoff. At this limit, a 5◦ error in inclination results in These corrections are all discussed in SCT12. The re- an 8% error in linewidth. sulting apparent magnitude in the AB system is 2.3. [3.6] Photometry [3.6]b,i,k,a =[3.6]−A[3.6]−A[3.6]−A[3.6]+A[3.6]. (1) b i k a The photometric data has all been obtained with IRACch.1,passbandcenter3.55µm. CHP,theCarnegie The data that are used in the following discussion are Hubble Program (Freedman et al. 2011), provides 60% collected into Table 1. This table, the complete version of the data. In addition, S4G, the Spitzer Survey of givenwiththeonlinepublication,includesCFSandCHP total magnitudes, each including the 4 adjustments just Stellar Structure in Galaxies (Sheth et al. 2010), gives described, and averages of the two methods. The table 17%, and SINGS, the Spitzer Infrared Nearby Galaxies also gives inclination and linewidth information drawn Survey (Dale et al. 2005, 2007), giving 9%. This third from TC12 and color terms for color corrections de- program was carried out during the cryogenic phase scribed in sub-section 3.3. The galaxies in Table 1 are while the first two were conducted during the warm either partofthe zero pointcalibration(sample ZeroPt) Spitzer mission. The new Cosmic Flows with Spitzer or a member of a cluster contributing to the slope tem- program has only contributed 3% of the current data. plate. Smaller programs during the cryogenic mission supply us with information on the remaining galaxies. The information comes from a multitude of programs but 3. [3.6]BANDCALIBRATION the integration times are the same within a factor two The TFR calibration requires the definition of a slope (mostly 240 sec, occasionally 120 sec). The integrations and the establishment of an absolute scale. The slope is are sufficiently deep to reach a surface brightness 26.5 the trickiest item because there is a correlation between mag/sq. arcsec (AB) even with the shorter exposures. its value and a form of Malmquist bias. Given two Details on programs and exposure times are included galaxies at the same distance with the same linewidth, at the Extragalactic Distance Database website by the brighter galaxy might be chosen but not the fainter selecting the catalog Spitzer [3.6] Band Photometry. one. The potential bias depends on the slope of the correlation because with a relatively flat slope most The photometric reductions were carried out by two intrinsically luminous galaxies lie above the correlation independent procedures. The method utilized by the while with a very steep slope these same galaxies tend CHP uses software developed for the GALEX Large to lie below the correlation. Consider a target for a Galaxy Atlas (Seibert et al. 2012, in prep.). The distance measurement in the field that intrinsically method developed in anticipation of the arrival of CFS lies above the assumed mean relation, the trend for data is based on the Archangel photometry package distant galaxies if the relation is flat. With the distance (Schombert 2007) described by Sorce et al. (2012a) measurementthe targetis assignedthe mean luminosity (hereafter SCT12) and earlier by Courtois et al. (2011a) of the correlation at the target’s linewidth so given a in the context of optical photometry. In a comparison distancethatistoosmall. Thisbiashasrepeatedlybeen of 171 galaxies (SCT12), the two procedures result in discussed at length, most recently by TC12. The salient agreement at the level of 0.01 mag with rms scatter point is that the so-called ‘inverse’ relation (ITFR), of 0.052. Partitioned equally, the internal uncertainty the least squares regression where errors are taken to (reductions of the same data by different methods and be in linewidth only, gives results that are close to individuals) is ±0.037 mag. There are marginal differ- bias free. Willick (1994) pointed out that, while in his ences for galaxies brighter than [3.6]=11 (CHP brighter experiments the ITFR bias was reduced by a factor 6 4 Sorce et al. from that incurred using the direct relation, yet a small 8 15 Fornax 32 UMa bias remained because the sample selection was not 10 made in the band he considered. We have the same 12 problem. Our strategy is to use the ITFR and then evaluate the bias with simulations anticipatingthat, like 14 with the I band calibration, the effects will be small. 11 Antlia 11 Centaurus The bias tests are discussed in a later section. 10 The calibration process has been described in detail 12 by Tully & Pierce (2000) and TC12. With the I band 14 relation there is no clear evidence for scatter due to a 16 third parameter but the situation at [3.6] is different. A 10 12 Pegasus 14 Hydra color term is found and that matter will be discussed. 12 3.1. Relative distances and ITFR Slope 14 The measurement of distances requires the hypothesis 16 ofauniversalcorrelation. To begin,we makeinversefits 23 Pisces 11 Cancer to each one of the clusters separately. Dotted lines in 10 Figures 1 and 2 illustrate the inverse fits of the TFR for 12 each cluster. Slopes are quite similar between clusters. Slopes and their uncertainties are given for each cluster 14 in Table 2. The individual fits are consistent with 16 the soon to be derived best fit and hence with the 10 16 Coma 7 Abell 400 universal correlation hypothesis. As cluster distances 12 increase, the faint luminosity limits increase. However, no dependence of the slope with distance is seen, as 14 would be a marker of Malmquist bias (we still make a 16 tiny correctionfor bias to cluster moduli as described in Section 3.4). 10 19 Abell 1367 18 Abell 2634/66 12 14 16 2 2.5 2 2.5 3 Fig. 2.—Tully-Fisherrelation inthe [3.6] band for the Fornax, Ursa Major, Antlia, Centaurus, Pegasus, Hydra, Pisces, Cancer, Coma, Abell 400, Abell 1367 and Abell 2634/66 clusters. Solid linesgivetheinversefitoftheuniversaltemplatecorrelation. Dot- tedlinesaretheinversefitsofthecorrelationforeachcluster. cluster in question. Apparent magnitude zero points confirm that Virgo, Fornax and Ursa Major are the closest clusters. Then come Antlia-Centaurus-Pegasus, then Hydra-Pisces- Cancer, and finally Coma and the three Abell clusters A1367, A400 and A2634/66. To establish the best uni- versalslopeandthebestrelativedistancesbetweenclus- ters, we follow an iterative procedure. We initially con- siderthenearestthreeclustersbecausetheyareobserved to comparable depths in intrinsic magnitude. The For- nax and Ursa Major magnitude scales are shifted ac- cording to the difference in zero point with respect to Virgo. A least squares fit of the ITFR is made to this Fig.1.— Tully-Fisher relation in the [3.6] band for the Virgo ensemble. The new slope is assumed in a fit to the 3 Cluster. Thesolidlinegivestheinversefitoftheuniversaltemplate correlation. Thedottedlineistheinversefitofthecorrelationfor individual clusters with only the zero point as a free pa- theVirgoClusteralone. rameter in each case. Given the new zero point offsets the cycle is repeated,leading to rapidconvergence. This The next step is to combine the 13 individual cluster procedure is repeated with the addition of each distance correlations by vertical translations. The Virgo Cluster group in turn. Again, convergence is rapid. It is to is used as a reference. Each preliminary zero point from be stressed that this procedure works because, following the individual fits provides us with a first estimate of expectations, the slope of the ITFR is not affected by the relative distance between the Virgo Cluster and the themagnitudeleveloftruncation. Thisprocedurewould [3.6] TF Relation 5 manifestlynotworkwiththedirectorbi-variaterelations wheretheslopesvarywiththeleveloftruncation. Inthe end we obtain a slope of −9.74±0.22 for the template ITFR. Zero point offsets with this ‘universal’ slope are shown in Figure 3 and give relative distance moduli of clusters referenced to the Virgo Cluster. The universal slope of the ITFR is displayed in Figure 3 as well as by the solid lines in Figures 1 − 2. 24 Virgo 15 Fornax + 0.33 32 UMa + 0.44 11 Antlia + 2.07 11 Centaurus + 2.17 12 Pegasus + 2.47 14 Hydra + 2.90 23 Pisces + 3.30 11 Cancer + 3.37 16 Coma + 4.04 19 Abell 1367 + 4.10 7 Abell 400 + 4.12 18 Abell 2634/66 + 4.43 Fig. 4.—TFR forthe 26galaxies withdistances established by observationsofCepheidstars(circles)ortheTipoftheRedGiant Branch (squares). The solid line is the least squares fit with the slopeestablishedbythe13clustertemplate. Thezeropointofthe TFRissetatthevalueofthisfitatlogWi =2.5. mx gether as seen in Figure 5. The ITFR expression in the [3.6]-band is given by: Mb,i,k,a =−(20.34±0.10)−(9.74±0.22)(logWi −2.5) [3.6] mx Fig.3.— Template Tully-Fisher relation in the [3.6] band ob- (2) tained with data from 213 galaxies in 13 clusters. Offsets given withrespect to the VirgoCluster represent distance modulus dif- ferences between each cluster and Virgo. The solid line is a least squaresfittoallthegalaxieswitherrorsentirelyinlinewidths,the ITFR. 26 Zero Point Calibrators 24 Virgo 3.2. Zero Point and Absolute Distances 15 Fornax 32 UMa Presently, [3.6] photometry is available for 26 nearby 11 Antlia galaxies with suitable morphologies, inclinations, and 11 Centaurus 12 Pegasus linewidths that also have well measured distances from 14 Hydra 23 Pisces either the Cepheid period-luminosity or tip of the red 11 Cancer giant branch methodologies. These 26 are a subset 16 Coma 19 Abell 1367 of the 36 absolute calibrator galaxies used in the I 7 Abell 400 band calibration (TC12). Their luminosity-linewidth 18 Abell 2634/66 correlationis seen in Figure 4 where now the ordinate is absolutemagnitudesfromtheestablisheddistances. The line is a leastsquaresfitwith the slope −9.74prescribed by the template. The zero point is −20.34±0.10. The most deviant point is the fastest rotator, NGC 2841, with a deviation of 2.7σ with respect to the template dispersion. This galaxy was a 2.3σ deviant in the I band calibration. There is nothing unusual about this galaxy other than its extreme rotation rate so we see no reason to disregard it as a calibrator. The distance to the Virgo Cluster is given by the zero point of the constrained slope shown in Figure 3 minus Fig. 5.— The template of the new [3.6] band - HI linewidth the zero point of the absolute calibration shown in Fig- correlation is built with 213 galaxies in 13 clusters extending in ure 4. Application of this shift allows both cluster tem- range from 1000 to 10,000 km s−1 with the absolute magnitude plate and zero pointcalibratorgalaxiesto be plotted to- scalesetby26zeropointcalibrators. 6 Sorce et al. The TFR scatter in magnitudes (relevant for distance will rise more than bluer galaxies. It follows that red measurements) is given by: and blue galaxies cannot be well mixed in the TFR at all wavelengths. The trends that could be anticipated χ2 areshowninFigure7(onlyaportionofthesamplehave σ = (3) TF rN −1 photometric measurements at B band). The compari- sonof fluxes at four bands fromB to [3.6]for individual where χ2 is the minimum of sources given in Figure 8 confirms the well known link- age between galaxy type and color. Early type galaxies (M −(a+b(logWi −2.5)))2 i mx have relatively more infrared flux relative to late type X galaxies. This point was also illustrated with the repre- with a and b the zero point and slope of the ITFR sentative spectral energy distribution plots in Figure 1 respectively and N −1 are the degrees of freedom. The of SCT12. Galaxies that are more luminous and earlier scatter for the entire cluster template sample is ±0.49 intype aredominatedby older,moremetalenrichedred magfromtheuniversalITFR,correspondingtoascatter giant stars emitting more in the infrared. in distance of 25%. The scatter for the 26 zero point calibrators is a similar 0.44 mag. Dispersion increases towardfaintermagnitudesaswelldocumentedatI band by Giovanelli et al. (1997a). The sample presented here is still limited but the dispersion is consistent with a Gaussian distribution. With large samples (Tully et al. 2008)onefindsabout3%ofcandidatesaremoredeviant than anticipated by Gaussian statistics. The causes are not always evident. Scatter may arise from: 1) measurement uncertain- ties affecting magnitudes, inclinations, and linewidths, 2) correctionuncertainties applied to measured parame- ters, and 3)‘cosmic’ scatter, e.g. cluster depth effects or interlopers, deviations from disk planarity, other gravi- tational and photometric asymmetries, variations in the stellar population make-up, variations in disk-to-bulge ratios,etc. Whateverthe sources,wehaveastandardto meetsetbytheI bandanalysis. Thesamplesusedinthe currentanalysisinvolve80%ofthesamplesusedintheI- Fig.6.— TFR in B,R,I and [3.6] bands. B and R bands data bandcalibration(TC12). Inclinationsandlinewidthsare arefrom Tully&Pierce(2000) , I band data are from TC12 and the same, the factors mentioned associated with cosmic [3.6]banddataarefromSCT12. Linewidthsarethesameasused byTC12. Theslopes steepen frombluetored, withvalues −7.27 scatter are the same, corrections to photometric param- atB,−7.65atR,−8.81atI,and−9.74at[3.6]. eters are reduced in the mid-infrared, and the integrity ofthemagnitudemeasurementsmustbeatleastasgood There have been long standing suggestions that the orbetterwiththeSpitzerobservationssinceobservations dispersion in the TFR might be reduced by inclusion of are made all-sky with the same instrumental configura- additionalparameters. Inanearlyinstance(Rubin et al. tion. Error bars on magnitudes are reduced in Figures 1985), when only photographic or photoelectric magni- 1−5 compared with those on equivalent plots in the I tudes were available, the case was framed in terms of bandcalibrationpaper (TC12)to the degree that obser- galaxy types which are strongly correlated with color. vational errors in magnitudes are a minimal component Masters et al. (2006) have maintained the use of a type of uncertainties. Yet the scatter found at I band is less: separation with I band work. Tully & Pierce (2000) ac- ±0.41mag for the cluster template sample, lowerwith a knowledgedthe hint of a type dependence in the I band significance of 2σ, and 0.36 mag for the zero point cal- relation but concluded that the evidence remained too ibrators. As much as half of the increase in magnitude weaktowarrantaddingcomplexitytotheTFRanalysis. scatter will occur because the slope of the correlation is The situation changes with the mid-infrared infor- steeper in the mid-infrared. However there could be an mation. In spite of superior photometry the scatter additionalexplanationfor the increasedscatter foundat in the TFR is increased and there is a significant [3.6]. color signature. The variations in spectral energy distributionimplicitintherangeofrepresentativecolors 3.3. A Color Term shown in Figure 8 provide a natural explanation given Ithas long beenknownthatthe TFRsteepens toward theextendedleverarmfromtheopticaltothe[3.6]band. longerwavelengths(Tully et al.1982). Theeffectisseen in Figure 6. (Note: in the discussions in this section There is also the possibility that some flux in the [3.6] allopticalphotometryvalueshavebeentransferredfrom band may come from other than old stars. Meidt et al. VegatoAB zeropoints.) Thereisastrongcolorcorrela- (2012) determined that 12 ± 5% of [3.6] flux arises tion with linewidth, more rapidly rotating galaxies tend from hot dust, PAH emission, or young to intermediate to be redder, so at longer wavelengths the high rotation age stars in 6 representative spiral galaxies observed end of the TFR rises with respect to the low rotation with Spitzer Space Telescope. However the variance of end. Within a small linewidth interval, redder galaxies 0.05 mag is small compared with the ITFR scatter. [3.6] TF Relation 7 Fig.8.—RepresentationoffluxesatB,R,I,[3.6]bandsnormal- ized to unity at I band. Type Sa: red; types Sb-Sc: green; types Sd-Sm: blue. Theextrema aredefined bymembers ofour sample andcolor swathsindicate thedomainsdominated bythedifferent types. Moreover, it can be anticipated that the galaxies most affected by manifestations of star formation are later, bluertypes,whenceaugmentedfluxwilltendtodiminish a color term arising from old stars. Whatever the cause, it can be anticipated that the scattercanbe decreasedwiththe introductionofacolor correction. Toaddressthisissueweconsiderthestraight line fits included in the top three panels of Figure 7. The fits are least squares minimizations on the ordinate parameter; the difference in magnitude of a target from the meanTFR. The bottompanelshowsthe concordant variation of color with linewidth. Faster rotators tend to be redder. In the mid-infrared case, the offset for an individual galaxy from the mean fit in the figure is: ∆Mcolor =Mb,i,k,a+20.34+9.74(logWi −2.5). (4) [3.6] [3.6] mx An equivalent correction can be constructed with ap- parent magnitudes rather than absolute magnitudes, ∆[3.6]color =∆Mcolor, with an appropriate replacement [3.6] of the zero point constant in Eq. 4. The correctionterm commensuratewith the fit in the third panel ofFigure 7 is ∆[3.6]color =∆Mcolor =−(0.47±0.11)[(I−[3.6])+0.77]. [3.6] (5) Weintroduceanewcoloradjustedmagnitudeparameter C =[3.6]b,i,k,a−∆[3.6]color wherethedistinctnomen- [3.6] clature emphasizes the composite nature of this pseudo- magnitude. Next, the analysis discussed in Section 3.2 leadingtothe constructionofFigure4isrepeated. Like- wise,the adjustmentsareappliedtothe calibratorswith independently established distances and the procedures Fig.7.—Top3panels: DeviationsfromthemeanITFRrelation asafunctionofI−[3.6]color. Solidanddottedlinesarebestfits are repeated that lead to Figure 5. The adjusted re- and95%probabilitylimits. Top: AtBbandredgalaxiestendtolie lations are shown in Figure 9. The new correlation is belowthemeanrelationship. Topmiddle: AtIbandthereisahint described by the formula: that redgalaxies lielow although the correlation fit isdominated byafewextremecases. Bottommiddle: At[3.6]bandthesenseof M =−(20.34±0.08)−(9.13±0.22)(logWi −2.5) the correlation has flipped and red galaxies tend to lie above the C[3.6] mx mean relation. Bottom: The correlation between linewidth and (6) color. Theflatteningoftheadjustedrelationcomesaboutsince 8 Sorce et al. 3.4. Bias 24 Virgo 15 Fornax + 0.37 Willick (1994) showed that a small Malmquist bias 32 UMa + 0.45 11 Antlia + 1.95 existsinthe useofthe ITFR,althoughreducedfromthe 11 Centaurus + 2.08 direct TFR by a factor of 6 in the situation he explored 12 Pegasus + 2.40 14 Hydra + 2.86 (Willick et al. 1995), reducing the bias reflected in the 23 Pisces + 3.19 Hubble Constant from 17% to 3%. The bias arises from 11 Cancer + 3.29 16 Coma + 4.01 two effects. First, sample selection departs from an 19 Abell 1367 + 4.01 7 Abell 400 + 4.01 idealized case of a flat magnitude limit because samples 18 Abell 2634/66 + 4.35 have been selected in blue bands and color terms trans- late to a slope in the limiting magnitude in the infrared: slower rotators which tend to be bluer are favored for inclusion over faster rotators which tend to be redder (see Figure 7, bottom). Second, the shape of the galaxy luminosityfunctioncontributestothebiasbecausethere aremoreintrinsicallyfaintergalaxiesthatscatterbright- ward through errors than intrinsically brighter galaxies that scatter faint-ward (Eddington 1913). The bias increases with distance as the effect of the exponential cutoffoftheluminosityfunctionplaysanincreasingrole. The amplitude ofthe bias fromthe twoeffects wasex- plored with the calibration at I band (TC12). The sit- uation now with the [3.6] band sample is slightly worse than at I because the wavelength interval from selec- 26 Zero Point Calibrators tion at B is larger. The bias analysis carried out in the 24 Virgo 15 Fornax case of the I band calibration is repeated here, tailored 32 UMa to the current situation. We first combine the Virgo, 11 Antlia 11 Centaurus Fornax, and Ursa Major samples to improve statistics 12 Pegasus and include contributions from a range of environments. 14 Hydra 23 Pisces This ensemble is described by a Schechter (1976) func- 11 Cancer tion with faint end slope α = −0.9 and a bright end 16 Coma 19 Abell 1367 cutoff at M⋆ = −22. Then we randomly populate 7 Abell 400 [3.6] 18 Abell 2634/66 an artificial TFR to match the observed [3.6] band re- lation, drawing from the Schechter luminosity function. Thefaintlimitisdeterminedempiricallytoroughlyobey the relation Mlim =C −2.70(logWi −1.8) where C [3.6] ℓ mx ℓ couples with distance. The artificial TFR and the cut- off for the nearest clusters is shown in the top panel of Figure 10. The dashed blue line indicates the cut- off experienced at a distance modulus of 31. The cut- off, characterized by C , slides to brighter (more nega- ℓ tive) magnitudes linearly with increasing distance mod- ulus. The bias < ∆M > is determined at in- measured tervalsof C correspondingto increasingdistance. Here, ℓ <∆M > is the averagedeviationfromthe fidu- measured cial relation where < ∆M > = 0 by construction. Fig.9.— The ITFR after adjustments for the color term. Top: true The growth of the bias as a function of cutoff magni- Coloradjustedapparentmagnitudestranslatedtotherelativedis- tanceoftheVirgoCluster. Bottom: Coloradjustedabsolutemag- tude is seen in the bottom panel of Figure 10. The solid nitudeswiththeabsolutedistancescaleestablishedbythegalaxies curve,normalizedto unityatadistancemodulus µ=31 withindependentdistances representedbylargeopencircles. whereeventhefaintestofusefulcandidatesareincluded, reddersystemsmovedownwardandreddergalaxiestend isdescribedbytheformulabetweenbias,b,anddistance to havelargerlinewidths. The overallmagnitude scatter modulus, µ: inthenewrelationis±0.44mag(correspondingtoascat- ter in distance of 22%), down from 0.49 mag before ad- b=−0.0065(µ−31)2. (7) justment,andcomparablewith0.41foundatI bandwith an otherwise comparable analysis (TC12). The compa- By comparison, the coefficient in the case of the I band rable numbers for the zero point calibrators alone are a analysisis−0.005. Thelettersatthebottomofthefigure scatter of 0.37 with the adjusted parameter C , 0.44 are codes for the 13 calibrating clusters (see Table 4 to [3.6] beforetheadjustment, and0.36atI band. The compar- deciphercodes)andtheir horizontalplacements indicate isonsbetween [3.6]and I havesome imprecisionbecause therespectivesamplelimitsandprojectionupwardgives thesamplesizesforthelatterare25%greater. TheTFR the corresponding biases. These biases are recorded in parameters derived from alternative samples and band- Table 2 and are reflected in the adjusted cluster moduli passes are summarized in Table 3. and distances. For a galaxy in the field, the corrected [3.6] TF Relation 9 distance modulus µc can be expressed as 4. THEHUBBLECONSTANT µc =(C −M )+0.0065[(C −M )−31]2 (8) ThelastcolumninTable2recordsthe‘Hubbleparam- [3.6] C[3.6] [3.6] C[3.6] eter’ for each cluster: the velocity of the cluster in the CMB frame divided by the measured distance. These quantities are plotted against distance in Figure 11. A similar figure was presented as a summary of results from the I band calibration with the same 13 clusters (TC12: distances compared in Table 4). Here, as there, we see a large scatter in the Hubble parameter for the nearer clusters and small scatter for the more distant clusters. It can be anticipated that the measures for the nearerclustersarestronglyaffectedbypeculiarmotions. The5clusterswithin40Mpcareallpartofourextended supercluster complex: either within the historic Local Supercluster or the so-called Great Attractor region. The low scatter among the 7 clusters more distant than 50 Mpc (V > 4000 km s−1) suggests that CMB the relative contributions of peculiar velocities have a modest effect on redshifts at such large distances. InthecaseoftheI bandcalibration,themeanvalueof theHubbleparameterforthe7mostdistantclusterswas 75.1±2.7 km s−1 Mpc−1 where the erroris just the rms scatterofthe7contributions. Thatvaluewouldincrease to75.8withtherevisedLMCdistancefromMonsonetal (2012). With the present calibration, including the new LMC distance, the fit shown in Figure 11 gives a value of H = 73.8 with an rms scatter of 1.1 and a standard 0 deviation of 0.4 km s−1 Mpc−1 for the same 7 clusters considered previously. If the fit is extended to include the Pegasus Cluster at 44.5 Mpc then H = 74.4 and 0 the scatter is 2.0 km s−1 Mpc−1. The effect of a deviant radial motion of 200 km s−1 is illustrated in the figure as a function of distance. UF V CPee PCia AnAH2 Co A1 A4 Fig.10.— Top: Simulated TFR drawing randomly from a Schechter luminosity function with slope α = −0.9 and cutoff M⋆ =−22. TheITFR has slope−9.13andscatter 0.4mag. The dashed blue slanting line illustrates the color dependence at the Fig.11.— Hubble parameter as a function of distance. The fBwaihianiscth<liimn∆cirtMeraes>seusmlwteianistguhrfderdiosmtaasnsacaemf.upBnlcelaticsokenletrcoitfaionanbglseionsl:uttflheaetmbfaalguinneti.tluiBmdeoittlt;iomrmeidt: s(MVolpCidcM−lBi1n.e>iCs4ua0r0fiv0etdtkomdcoltsut−set1de)r.lipnToeihsnetislfiluattsgtdrivaisettesanHdce0evsi=agtri7eo3an.t8ser±inth1va.1enlok5cm0itMys−po1cf circles: faint limit increasing with increasing linewidth in accor- 200kms−1 fromthefit. dance withblue lineintop panel. Solidcurve: the empirical bias fit b = −0.0065(µ−31)2. Letters at the bottom: codes for the The uncertainty from the fit in Figure 11 is given by 13calibratingcluster(see Table4fortranslationofcodes). Their horizontal positions indicate sample limits and vertical intercepts thestatisticsofthedeviationsofthe7contributionsand withthesolidcurvegivethecorrespondingbiases. is unrealistically low. This error is what is expected if there is perfect Hubble expansion. If peculiar mo- 10 Sorce et al. tions of 200 km s−1 are the norm, and given the ex- same sample at I band (TC12). pected statistical errors in the distance of each cluster, the anticipated scatter around the mean Hubble value At the expense of the requirement of extra knowledge is ±2.6 km s−1 Mpc−1. We consider this to be our 1σ in the form of a color, the TFR in the [3.6] band can random error. We have several sources of systematic er- be reformulated in a form with scatter that matches the ror. The dominant component, creating almost 4% un- best optical formulations. The correction is small and certainty in H , comes from the uncertainty in the TFR not acutely dependent on the color measurement. The 0 zero point with just 26 calibrators. Combined with a appropriate ITFR equation for the measurement of dis- small uncertainty from the finite population of the tem- tances is plate,theuncertaintyinH associatedwiththeTFRcal- 0 M =−(20.34±0.08)−(9.13±0.22)(logWi −2.5) ibration(assumingthezeropointcalibratordistancesare C[3.6] mx perfect) is ±2.9 km s−1 Mpc−1. The zero point calibra- where M is derived from the corrected apparent tordistancesarenotperfect. Freedman et al.(2012)and C[3.6] magnitude [3.6]b,i,k,a of a source minus the color term Riess et al.(2011) reportthat with new Milky Way par- allaxes for Cepheid stars and mid-infrared Spitzer pho- ∆[3.6]color =−0.36−0.47(I−[3.6]). tometry the uncertainty in the Cepheid scale is in the range ±[1.9 − 2.5] km s−1 Mpc−1. The TRGB zero The slope of this formulation has been derived from point calibration which concerns 4 of the 26 calibra- a sample of 213 galaxies distributed in 13 clusters, tors, has similar or smaller systematics. The cumulative while the zero point is established from 26 calibrators systematic error in H is ±[3.5− 3.8] km s−1 Mpc−1. with Cepheid or tip of the red giant branch distances. 0 Combining random and systematic components we find The rms scatter in distances found with these galaxies H =73.8±2.6(ran)±[3.5−3.8](sys) km s−1 Mpc−1. (cluster template and zero point calibrators combined) 0 of 0.42 mag, 21% in distance, is insignificantly different 5. CONCLUSIONS from the accuracy found with the strongly overlapping Agreatconcernwithstudiesofmotionsonlargescales I band study. with the TFR has been the possibility that system- atic errors in photometry could create spurious flows. Distance measures derived with this calibration Small offsets between different observers, instruments, are subject to a small Malmquist bias, requiring the conditions, hemispheres, or seasons could be sky-sector distance modulus correction µc = µ+0.0065(µ−31)2. dependent. Probably the single most important ad- After application of bias and color corrections, a vantage of the use of space-based photometry such as preliminary estimate of the Hubble Constant can be offeredbythe Spitzermissioncomesfromtheconfidence made from the velocities and distances to 7 clusters at that measurements are on the same scale at better than VCMB >4000 km s−1. Accounting for all error sources, 1% in all parts of the sky. There are other advantages. the determination is H0 = 74 ± 5 km s−1 Mpc−1. Obscuration is minimal both within targets and from The difference between the value determined with this our Galaxy. This latter point is especially significant mid-IR analysis compared with the I band value found because studies of galaxy flow patterns can now reach with the same procedures and a strongly overlapping high levels of completion across the sky. Then it is sample (TC12) is ∆H0 = −2 km s−1 Mpc−1, not a a considerable advantage that the great majority of formally significant difference. We reiterate that the flux at [3.6] band arises from old stars, mainly those greatstrength of the presentcalibration is the high con- on the red giant branch. It can be surmised from the fidence in uniformity over the entire sky. Nevertheless modest scatter in the TFR that there is a close coupling the presentsample ofonly 7 clusters beyondthe domain betweenthe massin starsandthe dynamicalmass. And of known extreme peculiar velocities is unsatisfactorily there is an advantage, at least vis `a vis ground infrared small. In a subsequent paper (Sorce et al. 2012b) the observations, with the sensitivity achieved because of [3.6] band calibration is extended to a calibration of the very low sky noise. All but a few percent of the total Type Ia supernova scale, analogous to what has been flux ismeasuredwithinisophotsresolvedfromthe noise. done at I band (Courtois & Tully 2012), permitting a determination of H at z ∼0.1. 0 A small disadvantage with the mid-IR TFR calibra- tion has been revealed with the documentation of a colorterm. This color termis understood as the natural consequence of the correlation between galaxy rotation The data use in this paper are available at the Extra- rate or luminosity and color (Tully et al. 1982). At a galactic Distance Database.2 The photometric data are given linewidth, red galaxies progressively get brighter found by selecting the catalog Spitzer [3.6] Band Pho- relative to blue galaxies as one considers the TFR at tometryandthenagalaxyofchoicewhile theHI profiles longer wavelengths. Evidence is accumulating that are found in the catalog All Digital HI. We thank Tom intrinsic scatter in the simple two parameter TFR is Jarrett, part of the Cosmic Flows with Spitzer collab- minimalwith photometryatabout1µm. Aconsequence oration, for advice with Spitzer photometry and James of the color dependence is a steepening of the TFR Shombert for his development of Archangel. Much of toward the infrared. If one is interested in the physical the data used here comes from the Spitzer Space Tele- implicationsoftheTFRratherthanitsuseasadistance scope archive. We thank Kartik Sheth for the contribu- tool then the bivariate fit is of interest. Our template tion of his program Spitzer Survey of Stellar Structure sample has the bivariate dependence M ∝ W3.8±0.1 [3.6] which is 0.4 steeper than was found with almost the 2 http://edd.ifa.hawaii.edu

See more

The list of books you might like

Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.
DMCA & Copyright: Dear all, most of the website is community built, users are uploading hundred of books everyday, which makes really hard for us to identify copyrighted material, please contact us if you want any material removed.
Copyright @ PDFdrive.to, 2022 | We love our users