The Extragalactic Distance Scale without Cepheids IV
Lachlan Hislop, Jeremy Mould, Brian Schmidt, Michael S. Bessell, Gary Da Costa, Paul Francis, Stefan Keller, Patrick Tisserand, Sharon Rapoport, Andy Casey
aa r X i v : . [ a s t r o - ph . C O ] A p r The Extragalactic Distance Scale without Cepheids IV.
Lachlan Hislop & Jeremy Mould
School of Physics, University of Melbourne, Vic 3010, Australia
Brian Schmidt, Michael S. Bessell, Gary Da Costa, Paul Francis, Stefan Keller, PatrickTisserand, Sharon Rapoport, & Andy Casey
Research School of Astronomy & Astrophysics, Australian National University
ABSTRACT
The Cepheid period-luminosity relation is the primary distance indicator usedin most determinations of the Hubble constant. The tip of the red giant branch(TRGB) is an alternative basis. Using the new ANU SkyMapper Telescope, wecalibrate the Tully Fisher relation in the I band. We find that the TRGB andCepheid distance scales are consistent.
Subject headings: galaxies: distances and redshifts – cosmology: distance scale
1. Introduction
The extragalactic distance scale of the Hubble Space Telescope (HST) Key Project(Kennicutt, Freedman & Mould 1995) is based on the Cepheid period-luminosity (PL) re-lation and secondary distance indicators, such as the Tully-Fisher relation (Sakai et al.2000), the supernova standard candle (Gibson et al. 2000), surface brightness fluctuations(Ferrarese et al. 2000), and the fundamental plane (Kelson et al. 2000). It has been crit-icized recently (Tammann et al. 2008; Sandage & Tammann 2006, 2008) on the groundsthat the PL relation may not be unique. Indeed, the finite width of the Cepheid instabilitystrip in the HR diagram implies that nuisance parameters such as metallicity, helium abun-dance, and star formation history may play a role in determining the PL relation. Metallicitywas considered as a second parameter by Freedman et al. (2001), Sakai et al. (2004), andMacri et al. (2006), and linear corrections were made based on measured values of [O/H]from the galaxies’ HII regions.The classical extragalactic distance scale continues to be important because measure-ments of the Hubble Constant with WMAP (Larson et al. 2010) are model dependent: theyintegrate the scale factor all the way to high redshift. 2 –It is of interest, therefore, to see how well the distance scale can be measured withoutreference to Cepheids at all. In Papers I, II, and III (Mould & Sakai 2008, 2009a,b) wecalibrated the H-band Tully-Fisher relation, surface brightness fluctuations, the FundamentalPlane, and Type Ia supernovae using the tip of the red giant branch (TRGB) and found adistance scale compatible with that of the H Key Project (Mould et al. 2000). In thispaper we use the TRGB distance indicator to examine the I-band Tully-Fisher relation. Asdiscussed in Paper I, the TRGB is a good standard candle because it results from the heliumflash on the red giant branch, which theory suggests is relatively immune to metallicity effectsin old stellar populations. Rizzi et al. (2007) find hundredth of a magnitude systematicsin TRGB for old metal poor stellar populations. Salaris & Girardi (2005) show that thisdispersion grows substantially for young and metal rich populations.
2. The Calibration Sample
The calibration sample from which a Tully-Fisher relation will be constructed consistsof 13 galaxies, which are presented in Table 1. This sample is limited to galaxies that haveconsistent methods by which their distances, I-band apparent magnitudes, and rotationalvelocities are measured. Each of these criteria is outlined below.
TRGB distance moduli for the calibration galaxies were taken from the ExtragalacticDistance Database (EDD) and are listed in Table 1. The procedure for obtaining distancemoduli is described in Jacobs et al. (2009). CMDs for the galaxies were produced from HSTobservations, which then allows the apparent magnitude of the TRGB to be measured fol-lowing detection using the maximum likelihood algorithm of Makarov et al. (2006). TRGBdistance moduli can then be obtained using the absolute magnitude of the TRGB as cali-brated by Rizzi et al. (2007), and are expressed as: µ = m TRGB − A I + 4 . − . V − I ) − ( A V − A I ) − .
6] (1)for the Johnson-Cousins I and V filters, which have been corrected for galactic extinctionsA I and A V . No corrections have been made for extinction within the host galaxy, but theobserved RGB populations have been chosen such that they lie in the outer halo, where dust http://edd.ifa.hawaii.edu/ σ total errors are presented.There have been multiple measurements of the TRGB distance modulus for many ofthe individual calibration galaxies, but we adopt the EDD values due to their consistentcalibration. Calibration of the Tully-Fisher relation using TRGB distance moduli cannot be per-formed without first assessing the consistency of distance measurements to nearby galaxiesfrom different methods. Firstly, we perform a two-fold consistency check between the EDDlisted values for the TRGB distance moduli and Cepheid-based distances, as well as addi-tionally determined TRGB measurements. We do this by plotting in Figure 1 histograms ofthe independently determined Cepheid and TRGB distance moduli for each galaxy, alongwith the EDD measurement. It can be seen that consistency to within 0.2 mag is achievedfor most calibrator galaxies.It can also be seen for NGC 3368, NGC 3621 and NGC 3627, that these µ TRGB mea-surements represent distinct lower estimates for their respective distances. Furthermore, theadopted EDD distance modulus for NGC 3351 differs from the smallest published Cepheidmeasurement by a further 0.5 mag, and differs even more significantly from the µ TRGB pub-lished in Rizzi et al. (2007), which marks it as a clear outlier. We found that the origin ofthis large difference between the distance moduli of Jacobs et al. (2009) and Rizzi et al.(2007) is the difference between I-band apparent magnitude measurement of the TRGB it-self, published values of which differ by ∼ µ TRGB = 29 .
92 as published 4 –in Rizzi et al. (2007), since this exhibits a more realistic discrepancy between Cepheid andTRGB-based distance measurements.A similar discrepancy between TRGB distance moduli measurements exists for the An-tennae galaxies (NGC 4038/39), where the values published by Schweizer et al. (2008) andSaviane et al. (2008) differ by 0.9 mag. Schweizer et al. suggest that the difference is likelydue to the contaminating presence of AGB stars, which led Saviane et al. to misidentifythe TRGB in the CMD. It is possible that this issue of contamination may also apply tomeasurements of the TRGB for NGC 3351.When considering consistency between different measurements of the TRGB distancemodulus, µ TRGB , it is important to note that methods by which the tip itself is detected differbetween groups, with recently employed methods being the maximum likelihood approachesof Mendez et al. (2002) and Makarov et al. (2006), and the Gaussian-smoothed Sobel edge-detection filter used by Bellazini, Ferraro, & Pancini (2001) and Sakai et al. (2004). Whileit is beyond the scope of this paper to determine the reliability of one method over another,it is imperative that we adopt distance measurements that have been consistently calibrated.We have favored more recent calibrations of M
T RGBI in light of criticisms from Rizzi et al.(2007) that a significantly more populated CMD is required to accurately detect the TRGBthan those used by Lee, Freedman, & Madore (1993). And while Madore & Freedman(1995) suggest a minimum of ∼
100 stars for accurate detection, this number has been re-vised to the far more conservative 400 - 500 stars suggested by Madore, Mager, & Freedman(2009). Madore, Mager, & Freedman have attempted to resolve these statistical limitationsby expressing the I-band apparent magnitude of the TRGB as a function of color (and hencemetallicity), such that individual stars can be corrected for color and essentially transferredto the same TRGB magnitude. The calibration of M
T RGBI by Madore, Mager, & Freedmanagrees well with that of Rizzi et al. .Interestingly, Dalcanton et al. (2009) have published multiple TRGB distance modulifor a number of different fields for a range of nearby galaxies as part of the ACS NearbyGalaxy Survey Treasury (ANGST). Detections of the TRGB are determined using a com-bination of techniques from Mendez et al. (2002) and Sakai, Madore, & Freedman (1996).Where the measurement of M
T RGBI differs by less than 0.02 mag for each field within a galaxy,we are able to briefly assess the extent to which the choice of field within a galaxy to measurethe TRGB magnitude affects the yielded distance modulus. We expect neither variations inmetallicity nor internal extinction amongst halo RGB populations, where metallicities here 5 –are typically low and internal extinction is assumed to be negligible. The extent of crowd-ing and hence contamination of the CMB by AGB stars may differ slightly between fields,but again we do not expect the populations of the RGB fields to differ significantly enoughto render measurements of the TRGB inconsistent between fields. Upon inspection of theresults for a number of galaxies, distance moduli for each galaxy vary at most by 0.1 mag,which are consistent to within errors for an uncertainty of 0.05 mag, which suggests that anyvariations can be largely attributed to random photometric errors. Uncertainties introducedby adopting the same foreground reddening value for each field may also contribute to thedispersion of distance moduli measurements (Rizzi et al. 2007), although this suggestionhas not been fully supported.
The total sample of photometric data has been sourced from SDSS data and our ownobservations performed using the ANU SkyMapper Telescope, along with data taken fromthe literature.
Photometry was obtained for four galaxies from the calibration sample (NGC 55, NGC247, NGC 253, and NGC 7793) from observations performed using the ANU SkyMapper1.35m telescope (Keller et al. 2007), located at Siding Spring Observatory, on September23 and 24, 2010. Observations were performed in the g and i bands and the images werebias-corrected and flat-fielded using bias frames and twilight flats taken on the night. Thesewere applied using the IRAF imarith package. Integration times were set at 60 s for allobjects.For each galaxy, calibration from an i -band magnitude to a standard I-band was per-formed using observations of standard stars from Graham (1982). 6 – Photometry for galaxies NGC 3351 and NGC4826 was obtained from the SDSS cata-logue , and was separately calibrated using the included observations of Landolt standards(Landolt 1992) .The calibration of SkyMapper and SDSS data is discussed in Appendix 1. Photometryfor the remaining calibration galaxies (NGC 300, NGC 891, NGC 2403, NGC 3368, NGC3621, NGC 3627, and NGC 4258) was obtained from Tully & Pierce (2000). Apparent magnitudes were measured using both the IRAF ellipse routine, and theARCHANGEL photometry package developed by James Schombert . ARCHANGEL per-forms the core routines of sky determination, ellipse fitting, and surface brightness profilegeneration which are common to all photometry software packages. Following Schombert(2007), the procedure for measuring apparent magnitudes is outlined below.Images are first cleaned of foreground contaminating objects, and a preliminary max-imal radius out to which ellipses are fit is then determined. This allows for accurate skybackground determination that avoids both foreground objects and the outer regions of thegalaxy. In order to measure the background, boxes of a set size are placed randomly in theimage, and a mean count is obtained for each box. The final magnitude of the backgroundis then taken as the mean of each of these mean sky box values.Isophotal contours are then fit to the image as ellipses, which simply act as tracersfor the stellar material. The quality of these can be manually assessed by observing rapidchanges in parameters such as position angle and eccentricity towards the outermost ellipses.Once ellipses are fitted, a surface brightness profile is produced and a disk fit can be appliedto yield apparent magnitudes. http://abyss.uoregon.edu/ ∼ js/archangel/ − . ∼ The output from photometry programs such as ARCHANGEL represents an initial,raw measurement for galactic apparent magnitudes. This is corrected by k times the air-mass where k is the atmospheric extinction coefficient, which must be determined for eachtelescope site (Sung & Bessell 2000). Measurement errors on the observed magnitudes aretypically given by the propagated uncertainty in the sky background measurement, whichbecomes the dominant source of error where the faint edges of the galaxy are difficult todetermine (Schombert 2007). Sky backgrounds are determined using the skybox routine inARCHANGEL, which places boxes throughout a frame, albeit away from the galaxy andforeground stars, and determines a mean sky value for each box. The final sky value forthe image is then taken as the mean of each box mean, and the uncertainty taken as the1/ √ n of the rms dispersion. We then apply a calibration offset to each of these measuredmagnitudes, as outlined in Appendix 1, and propagate the uncertainty in the calibrationoffset along with the aforementioned uncertainty to determine a final measurement error.Measured apparent magnitudes are then corrected for foreground Galactic extinctionusing the E(B − V) redenning values of Schlegel et al. (1998), which serves as an estimatefor the amount of light from distant objects that is scattered by dust within our own galaxyalong its line of sight. Magnitudes were k-corrected using k I = 0 . z , as per Han (1992).Magnitudes are also corrected for internal extinction using ∆m i = − γ log( b/a ) as perGiovanelli et al. (1994), where b/a is the axial ratio of the galaxy - the ratio of the minor andmajor axes. This internal extinction correction provides an estimate for an inclination de-pendent extinction, whereby light emitted within galaxies that are inclined edge-on ( i ∼ ◦ )undergoes a larger amount of internal scattering relative to galaxies that are inclined face-on( i ∼ ◦ ). Methods of internal extinction corrections are numerous, and do not agree as tohow the γ parameter should be parametrised (see Sakai et al. 2000 for outline). Further- 9 –more, some methods will incorporate a morphology-dependent offset, whereas others willnot. The correction used by Giovanelli et al. (1994) is perhaps ideal then, since it ignoresthe additional morphology dependence, in light of the criticism from Sakai et al. regardingthe uncertainties involved in morphology classification. We have chosen to evaluate the γ parameter as per Tully et al. (1998) in favor of Giovanelli et al. .The final expression for the corrected apparent magnitudes is that given by Giovanelli et al.(1997a): m c = m obs − A I + k I − ∆m i (2)Following Giovanelli et al. , the total measurement error on the corrected apparent mag-nitudes is the quadrature sum of the measurement error and the error associated with theinternal extinction correction. The error on the internal extinction correction is given by thepropagation of the errors in γ and axial ratio b/a . For our calibration sample galaxies, we adopt the rotational velocity measurements pre-sented in Courtois et al. (2009), a database of homogeneously determined line widths usingthe profiles taken from HI catalogues published by Koribalski et al. (2004), Springob et al.(2005), Huchtmeier et al. (2005), Theureau et al. (2006), Giovanelli et al. (2007), Saintonge et al.(2008), and Kent et al. (2008). Courtois et al. define a window over the profile which ex-cludes a small portion of the integrated flux in the wings, such that a total flux can bedetermined within that window, and a mean flux taken over the number of spectral chan-nels. The line widths are then estimated at 50% of the mean flux density per channel.For galaxies with multiple profiles, a line width is measured for each available profile.For consistency, we adopt line widths derived from profiles taken from the same source asthe cluster sample (Springob et al. ), but this is not possible for our calibration sample. Formost of the calibration sample galaxies, Courtois et al. line widths have been measured us-ing the either the HIPASS BGC profiles (Koribalski et al. 2004), or from profiles presentedin Springob et al. (2005). For NGC247, a W m line width was determined using the profilepresented in Carignan & Puche (1990). For NGC 891 and NGC 2403, profiles were takenfrom Rots (1980).Measurement uncertainties for each line width are limited to the spectral resolution 10 –after smoothing. Courtois et al. provide conservative error estimates, and apply a thresh-old error of 20 km s − to indicate galaxies suitable for distance applications, in line withTully & Pierce (2000).Other possible line width measurements are listed in Springob et al. (2005) and Giovanelli et al.(1997a), such as widths measured as 50% of the peak flux, W . Where consistency betweenthe calibration and cluster samples is preserved, the choice of line width is largely irrelevantand should be motivated by breadth and accuracy of available data. We choose not to usethe line widths included in the SFI++ sample, as we are unable to apply the same correctionfor instrumental effects to our calibration sample line widths. In accordance with Sakai et al. (2000), line widths are corrected for inclination andredshift. W c = W obs (sin i )(1 + z ) (3)We forgo the corrections for instrumental effects and turbulence since this cannot beapplied consistently across both data sets. Errors in the corrected line widths are given bythe propagation of the measurement error on the observed line width and the error in theinclination. One of the key parameters in Tully-Fisher applications is the axial ratio, b/a , from whichan inclination is derived, which allows apparent magnitudes to be corrected for internalextinction, and to project line-of-sight rotational velocities. Following our ellipse fittingroutines, the axial ratio is determined at the I ∼
20 mag arc sec − level, where our plots of b/a versus ellipse radius converged. Of the 6 galaxies for which we measured inclinations,these values agree with values published in Pierce & Tully (1992), Springob et al. (2007)and the RC3 catalogue (de Vaucouleurs et al. 1995) to within 1-2 ◦ , except for NGC 55 andNGC 3351, where published values vary between 7 ◦ and 5 ◦ repsectively. For this study, wehave adopted our measured values. 11 –
3. Fitting Procedures and the Tully-Fisher Relation
For our calibration sample, we apply direct, inverse and bivariate least squares fits asoutlined in Giovanelli et al. (1997b). Each of these approaches assumes a typical straightline fit for the Tully-Fisher relation of the form, M = a + b log W , where a and b are thezero-point and slope respectively, but each fitting method weights the residuals of the meritfunction χ = N X i =1 (cid:20) M i − M (log W i ; a, b ) σ i (cid:21) (4)differently. For the direct fit, residuals in absolute magnitude are minimised such that σ i represents the total error in the absolute magnitude.For the inverse fit, line widths are expressed as a function of absolute magnitude, suchthat the dependent and independent variables have switched roles between the direct andinverse fits. Our model Tully-Fisher relation is now expressed as log W = M/b − a/b , andresiduals are minimised with respect to the errors in the line width.For the bivariate fit, values for the slope and zero-point are determined according to M = a + b log W as before, but instead the merit function is now minimised according to or-thogonal residuals, where σ i = σ M,i + b σ x,i , such that errors in absolute magnitude and linewidth are weighted equally. Bivariate fits are performed following the method of Press et al.(1992). The use of the direct, inverse, and bivariate fits are not uncommon to Tully-Fisher appli-cations (e.g Giovanelli et al. 1997b, Sakai et al. 2000), and it is certainly true that all threefitting procedures satisfy the basic function of providing a prediction of a variable when itsdependent variable is known. But each regression fit will result in a different measurementof the slope and zero-point in cases where errors are present in both variables and are uniquefor each data point, which becomes important where H exhibits particular sensitivity toeither of these parameters. Furthermore, Isobe et al. (1990) argue that each different fittingmethod represents a result with theoretically distinct implications, and Feigelson & Babu 12 –(1992) reiterate that values of H derived from different fitting techniques are not directlycomparable. These concerns are nevertheless somewhat muted in applications of the Tully-Fisher relation, since its origin is purely empirical (see Tully & Fisher 1977 and sourcestherein) such that insights into the physical relation between absolute magnitude and rota-tional velocity remain secondary to its practicality.While we are purely interested in using the Tully-Fisher relation to provide absolutemagnitudes for galaxies with measured rotational velocities, in which instance we would bequite satisfied in applying the direct fit as a simple predictive tool, we must nevertheless takeinto consideration the errors in line width when fitting data, as these remain the dominantcontribution to the overall error at smaller distances. At larger distances, the statisticaluncertainties in the distance dominate.The direct and inverse fits are applied purely for comparison’s sake to previous calibra-tions of the Tully-Fisher relation (e.g, Sakai et al. 2000,Tully & Pierce 2000). Isobe et al.(1990) contend that these fitting methods are applicable where the cause of the scatter aboutthe fitted relation is unknown, but also note that a direct fit applicable only where the inde-pendent variable (rotational velocity in this instance) is measured without error. We adoptthe bivariate fitting method as the most robust estimate of the Tully-Fisher parameters,where measurement errors in both absolute magnitude and line width are treated as equalcontributers to the overall scatter. Figure 4 shows the Tully-Fisher relation derived from 13 calibration galaxies, where thedirect, inverse and bivariate fits are superimposed on the single plot, represented by thedot-dashed, dashed, and solid lines respectively.The Tully-Fisher relation for our calibration sample using the bivariate fit is thus: M I = ( − . ± . W − . − (21 . ± .
07) (5)Values for the slopes, zero-points and observed scatter for each fit are listed in Table 2. 13 –As can be seen, there is a clear discrepancy between the values of the zero-point for the di-rect and inverse fit when compared with the value for the bivariate fit. But when comparingto the Tully-Fisher parameters of Giovanelli et al. (1997b) derived using the same fittingtechniques for similarly sized samples, we can see that such disagreements are not uncommon.
We measure the scatter about the Tully-Fisher relation by determining an RMS disper-sion of the magnitude residuals, which corresponds to a total observed scatter, σ obs . Thisscatter will have a contribution from the error measurements in apparent magnitude anddistance, but what is typically found is that the measurement scatter does not account forthe total observed scatter (e.g., Sakai et al. (2000); Meyer et al. (2008)). In this way, anintrinsic scatter is implied. The intrinsic scatter is of particular interest as it illustrates anunderlying variation between luminosity and rotational motion which can be used to explorehow these properties are physically related.Following Meyer et al. (2008), an upper limit to the intrinsic scatter can be determinedby including in the sample to which a Tully-Fisher relation is fit only galaxies whose mea-surement errors are effectively minimised. From this reduced sample, the intrinsic scattercan be measured using, σ obs = σ meas + σ int . However, the lack of a robust calibration samplefor this study means that in addition to our Tully-Fisher relation being poorly constrained,we have only a poor estimate of the intrinsic scatter. Since we require a measure of theintrinsic scatter in order to propagate this uncertainty into a final error on our Tully-Fisherdistance moduli, we instead adopt the intrinsic scatter of 0.25 mag via Sakai et al. (2000),which was taken from their Cepheid-calibrated I-band Tully-Fisher relation.
4. Application of the Tully-Fisher Relation to Distant Cluster Samples
With our fitted Tully-Fisher relation, we now have the I-band absolute magnitude ex-pressed as a function of the logarithm of rotational velocity, such that distances to objectscan be obtained where their rotational velocities and apparent brightnesses remain observ-able.In order to measure H , we require a large sample of galaxies across a range of distances 14 –such that a Hubble diagram, in which an object’s distance is plotted against its recessionalvelocity, can be constructed. However, we are instead interested in measuring distancesto local and more distant clusters, rather than individual galaxies. For clusters at a largeenough distance, we can safely neglect the physical extent of the cluster and assume thatall galaxies assigned to a particular cluster lie at the same distance, where the foregroundand background residuals are averaged out. Furthermore, for a cluster containing n suitablegalaxies, we have n independent measures of the clusters distance, which allows us to reducethe statistical uncertainty in the cluster’s distance by a factor of 1/ √ n .By similarly determining an average recessional velocity for each cluster, we can obtaina value of H for each cluster. The use of multiple clusters allows us to compensate forcluster motions relative to the Hubble flow, i.e peculiar motion, such that for a distributionof clusters across the sky, these motions can be averaged out. The use of multiple clustersadditionally allows the consistency of the value of H to be assessed over a large range indistance. The SFI++ catalogue (Springob et al. 2007), contains I-band photometry along withoptical and 21cm line widths of ∼ measurement applications.Included in this catalogue are measurements of the recessional velocities, V CMB , for eachgalaxy, where these velocities have been transformed to the CMB reference frame to correctfor motions relative to the CMB to approximate Hubble flow.In order to maintain consistent treatment between the calibration and cluster data sam-ples, we adopted from this catalogue the galaxies whose line widths has been measured byCourtois et al. (2009). In emphasising this consistency, we reject galaxies for which W m line widths were not available in Courtois et al. . Line widths were assigned measurementerrors as a function of their signal to noise, where an error of 20 km s − or less signals aprofile that is appropriate for Tully-Fisher applications. As a preliminary cut to the data,all line width profiles with an assigned error larger than 20 km s − were rejected. 15 –We adopt all other measurements from the SFI++ catalogue where used, apart fromthe line widths for the reasons outlined above. Photometry and line width measurements forthese galaxies are corrected in a similar manner to that of the calibration sample galaxies,given by equations (2) and (3) respectively. Following Sakai et al. (2000), we apply the following cuts to the cluster data:1. Galaxies with inclinations i ≤ ◦ are rejected as uncertainties in the inclination mea-surement begin to dominate line width errors. The deprojection of the line width toan edge-on measurement becomes increasingly large and uncertain as the inclinationangle decreases.2. Both the calibration sample and the cluster should encompass the same rotationalvelocity distribution. Since the lowest rotational velocities of the calibration sampleapproach 180 km s − , we reject galaxies from the cluster sample with smaller velocities,as indicated in Figure 5.3. An upper limit to the internal extinction correction of 0.75 mag is applied, correspond-ing to highly inclined galaxies which have more than half of their light is subject toself-scattering.Regarding the internal extinction cut-off, Sakai et al. emphasise that the distributionof internal extinction values of the cluster data sample should correspond to that of thecalibration sample. In reproducing their Figure 8 for our own data samples in Figure 6, wecan see that an extinction cut-off to 0.75 mag remains appropriate.Lastly, only clusters with 5 or more galaxies and with V CMB ≥ − were usedin the determination of H . This final sample, referred to as the Hubble Sample, consists of261 galaxies across 15 clusters. For each cluster, individual distance measurements to the constituent galaxies wereobtained by applying the Tully-Fisher parameters to their corrected line widths to yield 16 –absolute magnitudes, from which distance moduli could be determined in conjunction withthe SFI++ apparent magnitude measurements. The cluster distance modulus is then simplytaken as the mean of these distances, with the RMS dispersion as the corresponding uncer-tainty.From the recessional velocities of each individual galaxy, we can determine an overallrecessional velocity for the cluster by taking the mean of these values. Whilst an uncer-tainty can be similarly adopted as the RMS dispersion about the mean of these values,we instead adopt a larger uncertainty of 300 km s − for each cluster’s recessional velocity.This uncertainty corresponds to the dispersion of peculiar velocities for clusters as found byGiovanelli et al. (1998), and allows us to account for any large scale peculiar motions andanisotropy in the distribution of clusters across the sky.Both of these quantities are of course dependent on the assumption that the cluster is aself-contained object with little substructure. This can be verified for each cluster by plot-ting histograms of distance moduli and recessional velocity measurements, where we expectthese quantities to be normally distributed about some mean value. The presence of outliersare not likely to be indicators of measurement errors, but instead may point to errors in themethod by which galaxies are assigned to a particular cluster. For the galaxies in the SFI++template sample, cluster membership assignment follows that of Giovanelli et al. (1997a),in which galaxies are assigned membership based on spatial proximity to a clearly definedcentral region, and consistency with recessional velocity measurements. Galaxies that sat-isfy both of these criteria for a particular cluster constitute the in sample. Galaxies that donot satisfy spatial proximity, but have a redshift that matches that of the assigned membersform a combined sample with the in galaxies, referred to as the in+ sample (Giovanelli et al.1997a). For the sake of preserving as large a sample as possible, we have utilised the in+ sample in this study.Nevertheless, since we are relying on in+ sample, our cluster sample is more prone tooutliers as we limit the sample to galaxies for which corrected magnitudes can be reliablydetermined. This becomes particularly problematic for galaxies with less than 10 galaxies,from which a mean distance for the cluster must be measured. However, for clusters withfewer galaxies, the random error measurement on the distance will be larger, and hencesuitably reflect the loose constraint.With respect to measuring a mean cluster recessional velocity, we are only limited to 17 –galaxies that have a reliable cluster assignment, since the redshift for each galaxy can beaccurately determined, independent of the properties that limit our sample by suitability ofmagnitude corrections. Again however, this method relies on the assumption that the clusteris a gravitationally bound system supported by the random motion of its galaxies. Underthis assumption, the line of sight motion of each galaxy varies about some mean recessionalmotion of the cluster corresponding to random motions undergone within the cluster. Anyevidence of substructure, in which distinct groups that have yet to coalesce would have theirown mean cz values, would be revealed by a cz histogram with multiple distinct peaks.For this study, recessional velocity measurements for each cluster have been determinedusing the entire SFI++ template sample, which were then compared with values publishedin Masters et al. (2006) and Giovanelli et al. (1997b). This is to ensure that the samplefrom which we are working is suitably large enough for accurate measurements to be made.The immediate limitation of course in using only these galaxies is that we have only sam-pled the spiral galaxies in each cluster, and ignored the large number of elliptical galaxies.We may be able to take some solace in the fact that since we are sampling spiral galaxiesonly, which are found in greater numbers in the outer regions of most clusters, we avoid thelarge infall motions of galaxies closer to cluster cores which can induce large dispersions inan overall cz measurement (Mould et al. 2000). All our measurements are consistent withpublished values, apart from cluster A2634, where we instead adopt the published value infavour of our own. For A2634, our measurement is based on 22 galaxies, whereas the valuefrom Masters et al. is based on 200 galaxies, thereby providing a better sample for analysis.Final distances, recessional velocities, and values for H for each cluster are presentedin Table 3. For each parameter, 1 σ random errors are included.
5. Hubble Diagram and the Value of the Hubble Constant
With distance and recessional velocity measurements, H values can be determined foreach individual cluster, and a weighted mean can be adopted, where the random errors areused as the weights. For the data sample presented in Figure 7, the value of H arrived atvia a TRGB-calibrated Tully-Fisher relation is 79 ± − Mpc − .We can further construct a Hubble Diagram (Figure 8). 18 – Random and systematic errors are propagated independently into two final uncertaintiesfor H . Random errors can be reduced statistically with relative ease, but systematic errorscannot be reduced in this manner, and hence require refinement of existing methods, or newmethods altogether. These errors must be tracked carefully through each stage of the cal-culation from the calibration sample magnitudes and line widths until a value of H is derived.When determining a total error for a particular quantity in which we are propagatingmultiple individual errors, these are typically added in quadrature since these measurementuncertainties are independent of one another. The linear addition of errors instead indicatesa correlation between errors, such that an overestimate of one quantity implies an overesti-mate in the other. But where error measurements are independent, quadrature addition ofuncertainties provides a smaller, and more appropriate overall uncertainty (Taylor 1997).The following error propagation recipe follows closely that of Sakai et al. (2000). Random errors in the calibration scale are propagated as an uncertainty in H throughthe intrinsic dispersion of the Tully-Fisher relation. Systematic errors arise in the calibrationscale through the zero-point error in M T RGBI , and the zero-point uncertainty in the Tully-Fisher relation. As determined by Rizzi et al. (2007), the systematic uncertainty in M
T RGBI is 0.02 mag.
Random errors in the corrected apparent magnitude are given by the quadrature addi-tion of the uncertainty in the measurement and the internal extinction correction, the latterof which propagates the errors in the ellipticity and γ parameters. As per Giovanelli et al.(1997a), we assume an uncertainty of 25% for γ . Uncertainties in the redshift correction andgalactic extinction are sufficiently small so as to be ignored. The final expression for theerror in m c is given by: ǫ m = ǫ obs + (0 . − e )) + (cid:18) . γ (1 − e ) ǫ e (cid:19) (6) 19 –Errors in the corrected line widths are purely random, and are given by the quadratureaddition of the uncertainty in the measurement and the inclination: ǫ W = (cid:18) z )sin i ǫ W,meas (cid:19) + (cid:18) − W obs cos i (1 + z )(sin i ) ǫ i (cid:19) (7)where the uncertainty in the inclination, ǫ i is the propagated uncertainty in the ellip-ticity: ǫ i = 1 − e (1 − q ) h − (1 − e ) − q − q i h (1 − e ) − q − q i ǫ e (8)The uncertainty in the ellipticity is given by the empirically determined relation, ǫ e =0 . − . e + 0 . e , as presented in Giovanelli et al. (1997b). It should be noted thatthis evaluation of ǫ e has the potential to underestimate the error for low ellipticities, butfor the most part these are not present in the sample, since low ellipticities translate toapproximately face-on galaxies which are excluded from the sample. Thus, a final error onlogW is given by: ǫ logW = ǫ W . W c (9) For each individual galaxy, the random error in the distance modulus is the quadratureaddition of the errors in the apparent magnitude measurement and the random error in theabsolute magnitude. The random error in the absolute magnitude consists of the uncertain-ties propagated by the Tully-Fisher relation: the intrinsic dispersion, σ int , and the line widtherror, where the error in logW is multiplied by the slope of the Tully-Fisher relation, b , tobe expressed in magnitudes. ǫ µ,rand = ǫ m + bǫ logW + σ (10)For each cluster, a mean distance modulus is adopted as the cluster distance, for which theuncertainty is simply taken as the RMS dispersion. There is a 1/ √ n dependence on thiserror, and hence this random error component is reducible for clusters with a larger numberof galaxies. 20 –For individual cluster measurements, the random error in H is comprised of the errorsin the distance scale and the error in the adopted recessional velocity, ǫ cz , of the cluster. Theuncertainty on the measured cz is simply the dispersion of the individual member galaxy cz values, divided by the square root of the number of galaxies in the cluster. However, thisuncertainty reflects only the extent to which the averaged motion of the galaxies representsmotion of the cluster as a whole. As per Masters et al. (2006), we adopt an uncertainty of300 km s − for the recessional velocity of each cluster to account for the peculiar motionof each cluster, which is in accordance with Giovanelli et al. (1998). The error in H for aparticular cluster is then given by: ǫ H,rand = (cid:16) ǫ cz d (cid:17) + (0 . ǫ µ,rand ) (11)where d is the distance to the cluster in Mpc, determined from the average distance modulus.We adopt as a final random error for H as the RMS dispersion about the weightedmean, reduced by a factor of 1/ √ n , where we have used n = 15 individual measurements ofH taken from 15 clusters. The systematic error in the calibration scale, consisting of the zero-point error on M
T RGBI and the zero point error in the Tully-Fisher relation propagate in to a final systematicuncertainty in H as a systematic distance error. This is defined as: ǫ H,syst = 0 . ǫ µ,syst (12) Following the above procedures, we determine our final value for H :H = 79 ± ± − Mpc − to Changes in the Tully-Fisher Parameters Our least squares regression fits to our calibration sample are of course susceptible to thesmall number of calibrating points, where for such small samples outliers have a significant 21 –impact on the fit. As such, were this calibration sample to be extended, we would expectthe slope and zero-point to change non-negligibly. Therefore, it is important to assess howchanges in these Tully-Fisher parameters affect our final value for H . Using the values ofthe slope and zero-point for the bivariate fit, we evaluate the change in H for a 3 σ change ineach , in order to anticipate possible values of H that could be achieved with a more robustfit. For a ± σ change in the slope, our value of H changes by 2%, and for a ± σ in thezero-point, H changes by 9%. As can be seen by this quick analysis, the final value of H exhibits minimal sensitivity to the slope, but a much stronger sensitivity to the zero-point.
6. Conclusion
In this paper, we have utilised the TRGB standard candle to construct a distancescale by which the expansion rate of the universe, or the Hubble constant, is measured.By determining apparent magnitudes for 11 galaxies with known TRGB distance moduliusing observations from the SDSS catalogue, the ANU SkyMapper telescope along with dataadopted from the literature, we have constructed a Cepheid-independent I-band Tully-Fisherrelation: M I = ( − . ± . W − . − (21 . ± . . (13)This Tully-Fisher relation was then applied to the SFI++ Tully-Fisher template datafrom Springob et al. (2007), which initially consisted of 807 galaxies across 31 clusters. Afterapplying cuts for line width and line width measurement error, internal extinction, inclina-tion, and a minimum cluster number, we maintained a sample of 261 galaxies across 15clusters. Applying the Tully-Fisher relation to line widths of these galaxies yielded absolutemagnitudes for each galaxy which, when combined with their listed apparent magnitudes,gave a distance to each galaxy. For each cluster, an average distance could then be obtained,in addition to an average recessional velocity from measurements of individual galaxy red-shifts. These two measurements provide the basis for measuring the Hubble constant viaHubble’s Law, cz = H r .Random and systematic errors have been propagated through each step of the distancescale into a final uncertainty for H . Systematic errors continue to dominate distance-scalebased measurements of H , which are principally introduced in the calibration scale, al-though we have reduced the systematic error effectively using the well constrained absolute The 3 σ do not represent confidence intervals, but simply possible variations.
22 –magnitude of the TRGB in favour of the Cepheid distance scale. The final value of H istaken to be 79 ± ± − Mpc − , which represents an uncertaintyof 4%. While our measurement of H represents a upper estimate relative to other recentdeterminations (74.2 ± − Mpc − (Riess et al. 2009); 73 ± ± − Mpc − (Freedman & Madore 2010); 73 ± − Mpc − (Mould & Sakai2008)), but it is still consistent given the uncertainties.Agreement with the results of Freedman et al. (2001) implies that the TRGB andCepheid distance scales are consistent.Thanks go to James Schombert for providing the ARCHANGEL Photometry Package,along with generous and timely support. A big thank you is also extended to Sean Crosby,Christina Magoulas, Ta¨ıssa Danilovich, and Loren Bruns Jr for their ongoing assistance.Special thanks to the whole SkyMapper team for making it all possible in the first place.This research has made use of IRAF, which is distributed by NOAO. NOAO is operated byAURA under a cooperative agreement with NSF. 23 – Appendix: Calibration of Photometric Data
As outlined in Bessell (2005), the idea behind calibration is to simply place photometricmeasurements from multiple instruments onto a single scale.
A. SkyMapper Images
In calibrating the SkyMapper images, standards from the E1 region from Graham(1982) were observed only once on September 24, 2010. Three exposures each were takenof the field in SkyMapper’s g and i bands, from which an average instrumental magnitudefor each star was determined. Magnitudes were determined using the IRAF phot pack-age, and corrected for atmospheric extinction using the k-coefficients for the Siding SpringsObservatory from Sung & Bessell (2000). These were science verification observations forSkyMapper hence the limited calibration available.Despite the small sample of standards, we have obtained a calibration applicable to asuitable range in color. Fitted to the data are least squares regressions, one minimising resid-uals in the offset, and the other minimising residuals in both the offset and the color. Errorsin the color are the propagated errors from the individual magnitude measurements, wherethe errors in the instrumental magnitude are simply the RMS dispersion about the adoptedmean since multiple measurements were available. Errors in the offset measurement are thequadrature addition of the errors in the I-band magnitudes from Graham and the errors on i .Also superimposed on the plot is the weighted mean which, over the plotted range incolor, is consistent with the least squares regressions to within 0.05 mag. Added to thefact that there is no clear trend with color, we adopt a constant offset allowing conversionbetween SkyMapper’s instrumental i magnitude and the standard I magnitude. Thus, wedetermine an overall calibration offset constant of - 3.31 ± B. SDSS Images
Standard fields for the SDSS catalogue were obtained through the SDSS Data ArchiveServer and I-band magnitudes from Landolt (1992) were used. Atmospheric extinctioncoefficients for the Apache Point Observatory (where the SDSS observations were performed)are taken from Hogg et al. (2001).In an identical manner to the SkyMapper calibration, Sloan instrumental colors g − i for each star were plotted against an offset, I − i , in Figure 10. Superimposed on this plotare the weighted mean along with a least squares regression.As this figure indicates, the mean is consistent with the least squares fits to 0.005 magover the covered color range, such that we can suitably apply a mean offset of 2.39 ± http://das.sdss.org
25 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
Table 1.
Calibration sample of galaxies
NGC α δ z
E(B-V) T e W m µ TRGB
I(J2000) (J2000) (km s − ) (mag)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)55 00 14 53.60 -39 11 47.9 0.00043 0.013 9 0.87 181 ±
13 26.62 ± ± ±
11 27.84 ± ± ±
18 27.83 ± ± a ±
19 26.59 ± ± a
891 02 22 33.4 +42 20 57 0.001761 0.065 3 0.77 a ±
22 29.98 ± ± a a ±
22 27.5 ± ± a ±
16 29.92 ± ± a ± ± ± a a ±
13 29.08 ± ± a a ±
15 29.59 ± ± a a ±
20 29.41 ± ± a ±
16 28.2 ± ± ±
13 27.79 ± ± a are taken from Tully & Pierce (2000).Note. — In column (1), the galaxy’s NGC number is listed. In columns (2) and (3) are theright ascension and declination of the object in J2000 coordinates. The redshift, foreground reddening value (taken from Schlegel et al. ), and the morphological T code are listed incolumns (4), (5), and (6). Ellipticity values are listed in column (7), with the correspondingsource code. The measured rotational velocities adopted from Courtois et al. (apart fromNGC 247, NGC 891 and NGC 2403) are listed in column (8). Column (9) contains theTRGB moduli as taken from the EDD (apart from NGC 3351). Column (10) contains themeasured I-band magnitudes, with the corresponding source code. 31 – Table 2. Tully-Fisher parameters.
Fit b a σ obs (mag)Direct -7.37 ± ± ± ± ± ± Table 3. Cluster data
Cluster N µ D V
CMB V CMB /D(mag) (Mpc) (km s − ) (km s − Mpc − )N383 21 33.95 ± ± ± ± ± ± ± ± ± ± ± ± ± ±
20 9746 91 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Note. — Presentation of the final Hubble sample cluster data. For each cluster, its average distancemodulus and corresponding distance in Mpc, along with CMB recessional velocity and the resultant valueof H are listed.
33 –Fig. 1.— Distribution of distance moduli measurements for each of our calibrator galaxies.Cepheid-based measurements are plotted in blue, TRGB-based measurements are plottedin red, and our adopted EDD TRGB values are plotted in black. Width corresponds tomeasurement error, and vertical scaling is arbitrary. 34 –Fig. 2.— Calibrated integrated magnitudes are plotted as a function of isophotal (semi-major) radius for each galaxy. The dashed line shows the final adopted magnitude. Fits foreach galaxy were performed in IRAF. 35 –Fig. 3.— Integrated magnitudes determined using IRAF are plotted on the left, withARCHANGEL magnitudes on the right. The dashed line indicates the adopted raw mag-nitude. For NGC 7793, we have additionally plotted the integrated magnitudes determinedusing the ARCHANGEL extremelsb fitting routine in red, which accommodates faint galax-ies. 36 –Fig. 4.— Measured I-band Tully-Fisher Relation. 37 –Fig. 5.— Distribution of corrected line widths for the cluster sample. The vertical dashedline marks 180 km s − . 38 – N Calibration SampleCluster Sample
Fig. 6.— Comparison of internal extinction distributions for the calibration and clustersamples. 39 –Fig. 7.— Values for the Hubble constant for each individual cluster (with 1 σ random errors),where the weighted mean is plotted as the dashed line. These values are published in Table3. 40 –Fig. 8.— Hubble diagram for the sample of clusters. The slope of the dashed line correspondsto the weighted mean for H . 41 –Fig. 9.— SkyMapper I-band calibration plot. 42 – −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 02.362.372.382.392.42.412.42 Colour (g−i) O ff s e t ( I − i ))