Dark and luminous matter in THINGS dwarf galaxies
Se-Heon Oh, W. J. G. de Blok, Elias Brinks, Fabian Walter, Robert C. Kennicutt Jr
aa r X i v : . [ a s t r o - ph . C O ] A p r Draft version October 22, 2018
Preprint typeset using L A TEX style emulateapj v. 08/13/06
DARK AND LUMINOUS MATTER IN THINGS DWARF GALAXIES
Se-Heon Oh , , W. J. G. de Blok , Elias Brinks , Fabian Walter and Robert C. Kennicutt, Jr. Draft version October 22, 2018
ABSTRACTWe present mass models for the dark matter component of 7 dwarf galaxies taken from “The H i Nearby Galaxy Survey” (THINGS) and compare these with those from numerical Λ Cold Dark Matter(ΛCDM) simulations. The THINGS high-resolution data significantly reduce observational uncertain-ties and thus allow us to derive accurate dark matter distributions in these systems. We here use thebulk velocity fields when deriving the rotation curves of the galaxies. Compared to other types ofvelocity fields, the bulk velocity field minimizes the effect of small-scale random motions more effec-tively and traces the underlying kinematics of a galaxy more properly. The “
Spitzer
Infrared NearbyGalaxies Survey” (SINGS) 3.6 µ m and ancillary optical data are used for separating the baryons fromtheir total matter content in the galaxies. The sample dwarf galaxies are found to be dark matterdominated over most radii. The relation between total baryonic (stars + gas) mass and maximum ro-tation velocity of the galaxies is roughly consistent with the Baryonic Tully − Fisher relation calibratedfrom a larger sample of gas dominated low mass galaxies. We find discrepancies between the deriveddark matter distributions of the galaxies and those of ΛCDM simulations, even after corrections fornon-circular motions have been applied. The observed solid body-like rotation curves of the galaxiesrise too slowly to reflect the cusp-like dark matter distribution in CDM halos. Instead, they arebetter described by core-like models such as pseudo-isothermal halo models dominated by a centralconstant-density core. The mean value of the logarithmic inner slopes of the mass density profiles is α = − . ± .
07. They are significantly different from the steep slope of ∼− . Subject headings:
Galaxies: dark matter – galaxies: kinematics and dynamics – galaxies: halos –galaxies (individual): IC 2574, NGC 2366, Ho I, Ho II, DDO 53, DDO 154, M81dwB INTRODUCTION
The dark matter distribution at the centers of galax-ies has been intensively debated ever since the ad-vent of high-resolution ΛCDM simulations. The exis-tence of central cusps in dark matter halos was foundin numerical simulations (Dubinski & Carlberg 1991;Navarro, Frenk & White 1996, 1997; Moore et al. 1999;Ghigna et al. 2000; Klypin et al. 2001; Power et al. 2002;Stoehr et al. 2003; Navarro et al. 2004; Reed et al. 2005;Diemand et al. 2008) but challenged by the observa-tions. The latter support a core-like density distribu-tion at the centers of galaxies (Flores & Primack 1994;Moore 1994; de Blok et al. 2001; de Blok & Bosma 2002;Bolatto et al. 2002; Weldrake et al. 2003; Simon et al.2003; Swaters et al. 2003; Gentile et al. 2004; Oh et al.2008; Trachternach et al. 2008; de Blok et al. 2008, andreferences therein). A detailed observational reviewon where the “cusp/core” problem stands is given by
Electronic address: seheon [email protected] address: [email protected] address: [email protected] address: [email protected] address: [email protected] Department of Astronomy, University of Cape Town, PrivateBag X3, Rondebosch 7701, South Africa Centre for Astrophysics Research, University of Hertfordshire,College Lane, Hatfield, AL10 9AB, United Kingdom Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, 69117Heidelberg, Germany Institute of Astronomy, University of Cambridge, MadingleyRoad, Cambridge CB3 0HA, United Kingdom Square Kilometre Array South African Fellow de Blok (2010).Of particular interest has been the assumption thatthe observations suffer from various systematic un-certainties and that the central cusps can be “hid-den” this way (Swaters 1999; van den Bosch et al.2000; van den Bosch & Swaters 2001; Swaters et al.2003; Simon et al. 2003; Rhee et al. 2004). These un-certainties consist of certain observational systematic ef-fects as well as the uncertainty in the stellar mass-to-light ratios (Υ ⋆ ) of the stellar component. The obser-vational systematic effects, such as beam smearing (forlow-resolution radio observations), dynamical center off-sets (for slit observations) and non-circular motions af-fect the derived dark matter distribution in galaxies insuch a way that the apparent inner density slopes of darkmatter halos are flattened. In addition, the fairly uncon-strained Υ ⋆ also affects the derived distribution of darkmatter in galaxies (e.g., van Albada & Sancisi 1986).The best way to minimize these uncertainties is touse high-quality data of dark matter-dominated objects.High-quality data ( ∼ ′′ angular; ≤ − velocityresolution) of dwarf galaxies taken from “The H i NearbyGalaxy Survey” (THINGS; Walter et al. 2008) signifi-cantly reduce the systematic effects inherent in lower-quality data and thus provide a good opportunity foraddressing the dark matter distribution near the cen-ters of galaxies. Dwarf galaxies which are dark matter-dominated, like low surface brightness (LSB) galaxies(de Blok & McGaugh 1997), are ideal objects for thestudy of dark matter (e.g., Prada & Burkert 2002) be-cause of the small contribution of baryons to the to- Oh et al.
TABLE 1Properties of the THINGS dwarf galaxies
Name R.A. Dec.
D V sys h P . A . i h i TR i h i BTF i z Metal. M B M dyn (h m s) ( ◦ ′ ′′ ) (Mpc) (km s − ) ( ◦ ) ( ◦ ) ( ◦ ) (kpc) (Z ⊙ ) (mag) (10 M ⊙ )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)IC 2574 10 28 27.7 +68 24 59 4.0 53 53 55 46 0.57 0.20 -18.11 14.62NGC 2366 07 28 53.4 +69 12 51 3.4 104 39 63 50 0.34 0.10 -17.17 4.29Holmberg I 09 40 32.3 +71 11 08 3.8 140 45 13 10 0.55 0.12 -14.80 0.46Holmberg II 08 19 03.7 +70 43 24 3.4 156 175 49 25 0.28 0.17 -16.87 2.07M81 dwB 10 05 30.9 +70 21 51 5.3 346 311 44 59 0.09 0.21 -14.23 0.30DDO 53 08 34 06.5 +66 10 48 3.6 18 131 27 23 0.14 0.11 -13.45 0.45DDO 154 12 54 05.7 +27 09 10 4.3 375 229 66 55 0.20 0.05 -14.23 5.40 Note . — (1)(2):
Center positions derived from a tilted-ring analysis in Section 3.4. The center position of DDO 154is from Trachternach et al. (2008); (3):
Distance as given in Walter et al. (2008); (4):
Systemic velocity derived from atilted-ring analysis in Section 3.4; (5):
Average value of the position angle from a tilted-ring analysis in Section 3.4; (6):
Average value of the inclination from a tilted-ring analysis in Section 3.4; (7):
The inclination value derived from the BaryonicTully − Fisher relation (see Section 5.2); (8):
The vertical scale height of disk derived in this paper (see Section 4.1); (9):
Metallicities; (10):
Absolute B magnitude as given in Walter et al. (2008); (11):
Dynamical mass within the last measuredpoint of the bulk rotation curve derived in this paper. tal matter content. In particular the high linear res-olution of ∼ Spitzer
Infrared Nearby GalaxiesSurvey” (SINGS; Kennicutt et al. 2003) data are avail-able for our sample galaxies. The SINGS near-IR imagesprovide virtually dust-free pictures of the old stellar pop-ulations in galaxies. This allows us to make reliable massmodels for the stellar component of a galaxy.We select 7 dwarf galaxies from THINGS that showa clear rotation pattern in their velocity fields to derivetheir rotation curves. Although some of them have beenanalyzed before, a more careful kinematic analysis is use-ful to derive a more accurate dark matter distributionin these slowly rotating galaxies. In the previous anal-ysis (e.g., Martimbeau et al. 1994; Hunter et al. 2001;Bureau & Carignan 2002 etc.), the intensity-weightedmean velocity field which is most likely affected by non-circular motions in galaxies was used and the asymmetricdrift correction was usually not addressed. Both non-circular motions and pressure support tend to induce alower observed rotation velocity than the true one.In general four different types of non-circular motionsin galaxies can be distinguished on the basis of velocityfields (Bosma 1978): (1) Motions associated with spiralarms. The streaming motions caused by the arms dis-tort the velocity field in a regular fashion (e.g., M81);(2) Large-scale symmetric deviations. The radial changeof the kinematical major axis’ position angle distortsthe velocity field, while still having a central symmetry.These velocity distortions are known as “oval” distortionswhen encountered in the inner region, and as a “warp”when they occur in the outer region, respectively; (3)Large-scale asymmetries. The tidal interaction with aneighbouring galaxy causes asymmetries mainly in theouter regions of galaxies (e.g., M81; Yun et al. 1994); (4)Small-scale asymmetries. Various sources, such as super-nova (SN) explosions and stellar winds from young stars(e.g., OB associations), locally stir up the bulk motionof gas and give rise to random motions. These are usu-ally visible as “kinks” in iso-velocity contours of velocityfields. Of these, small-scale random motions can be classifiedas additional components of the velocity profiles in theH i data cube and result in asymmetric profiles. There-fore, a single Gaussian function cannot properly modelthese (non-Gaussian) velocity profiles. To minimize theeffect of these random non-circular motions in our sam-ple galaxies, we use the “bulk” velocity fields described inOh et al. (2008). We compare the bulk rotation curveswith those derived from other types of velocity fields,such as the intensity-weighted mean, peak, single Gaus-sian fit and hermite h . In addition, we correct for theasymmetric drift for the galaxies where the pressure sup-port is significant with respect to the circular rotation.We then obtain dark matter mass models of the galaxiesusing Υ ⋆ as derived in Oh et al. (2008). From this, weaddress the “cusp/core” problem by comparing the de-rived dark matter distribution of our galaxies with thatof ΛCDM simulations.This paper is set out as follows. In Section 2, we givea general description of the data used. In Section 3, wepresent the rotation curves of the THINGS dwarf galax-ies. The mass models for the baryons are presented inSection 4. The measured dark matter fractions of thegalaxies are given in Section 5, and their relation to thegalaxy properties is discussed. In Section 6, the deriveddark matter distribution of the galaxies is discussed withrespect to the fit quality of the halo models used, therotation curve shape and the inner density slope. Lastly,the main results of this paper are summarized in Sec-tion 7. Data and kinematic analysis of individual galax-ies are presented in the Appendix. DATA
We use high-resolution H i data of 7 nearby ( ∼ Very Large Array (VLA) to derive the dark mat-ter distribution in these systems. Basic properties of oursample galaxies are listed in Table 1. See Walter et al.(2008) for a detailed description of observations and the The National Radio Astronomy Observatory is a facility ofthe National Science Foundation operated under cooperative agree-ment by Associated Universities, Inc. ark and luminous matter in THINGS dwarf galaxies 3data reduction. SINGS IRAC 3.6 µ m data (with a res-olution of ∼ ′′ ) are used to separate the contributionof stars from the total kinematics. In addition, ancillaryoptical broadband ( B , V and R ) images of the samplegalaxies taken with the 2.1m telescope at Kitt Peak Na-tional Observatory (KPNO) as part of the SINGS surveyare used. The data used in this paper are presented inthe Appendix. IC 2574 and NGC 2366 have already beenpublished in Oh et al. (2008). However, we here makefurther use of the plots by extending the analysis. For aconsistency with other galaxies presented in this paper,we show the old plots again together with some new re-sults. Some of the galaxies (e.g., Ho I and DDO 53) havelow inclinations ( < ◦ ). The effect of inclination on therotation curves will be discussed in Section 3.4. ROTATION CURVES
Velocity field types
As a first step towards deriving the rotation curve ofa galaxy, we need to extract the velocity field from thedata cube. The velocity field contains the entire 2D dis-tribution of velocities and is therefore less prone to sys-tematic uncertainties in deriving rotation curves, e.g.,due to pointing offset and non-circular motions, thanone-dimensional long-slit spectra (Zackrisson et al. 2006;de Blok et al. 2008).A velocity field can be derived in many different ways.The most popular ones are the intensity-weighted mean(IWM), peak, single Gaussian fit and hermite velocityfields (see de Blok et al. 2008). For a highly resolvedgalaxy that is not affected by non-circular motions, thesevelocity fields are nearly identical to each other and therotation curves derived are also similar. However, for agalaxy with dynamics severely affected by non-circularmotions, the resulting rotation curves from the differ-ent types of velocity fields show significant differences.Therefore, we have to examine the various types of ve-locity fields for a galaxy and determine which is the leastaffected by non-circular motions and the most appropri-ate for deriving an accurate rotation curve. In the fol-lowing sections, we briefly introduce the velocity fieldsmentioned above, as well as the bulk velocity field firstproposed by Oh et al. (2008).
Intensity-weighted mean velocity field (1 st moment map) The IWM velocity field has been the most widely usedvelocity field tracing intensity-weighted velocities alongthe line-of-sight through a galaxy (Warner et al. 1973).The intensity-weighted mean velocity of a profile in adata cube at a given line-of-sight for a galaxy is given as, V IWM ( x, y ) = R ∞−∞ dvI ( x, y, v ) v R ∞−∞ dvI ( x, y, v ) , (1)where I ( x, y, v ) is the flux of the profile in the data cubeat a given sky position ( x, y ) and is a function of velocity v . Mapping the velocities weighted by I ( x, y, v ) over theentire area of a galaxy gives the IWM velocity field. Asthis method does not depend on profile fitting, it pro-vides a robust estimate of velocity even for asymmetric NGC 2366 is not targeted in SINGS observations but retrievedfrom the
Spitzer archive
Fig. 1.—
Schematic H i profiles with different asymmetries. Thegray dashed lines represent the bulk motion and the light-graydotted lines indicate an additional non-circular component. Theblack solid lines are the resulting profiles combining both the bulkand non-circular motions. As the non-circular component increasesfrom the top to bottom panels the asymmetry of the resulting pro-file also increases. The long light-gray arrows in all panels indicatethe central velocity of the bulk motion profile. The short black ar-rows in all panels indicate the derived velocity from the IWM, peak,single Gaussian fit, and hermite h polynomial fit. The larger theasymmetry of a profile, the larger the velocity deviation from thebulk motion. In case the non-circular motion does not dominatethe bulk motion (upper two rows), the derived velocity is close tothat of the bulk motion and peak and hermite h velocities give anequally good result. However, if the non-circular motion dominatesthe bulk motion (lower two rows), the derived velocity deviatessignificantly from the bulk velocity even if we use the hermite h polynomial fit. profiles with a low S/N. If a profile in the data cube issymmetric with respect to its central velocity, then theIWM field properly traces the central velocity at whichthe peak flux is found. However, it begins to deviatefrom the central velocity of a profile, as the asymmetryof the profile increases. A schematic example of this isshown in Fig. 1. Peak velocity fields
Tracing the velocities at which the peak fluxes of theprofiles in a data cube are found can be an alternativeway of determining the line-of-sight velocities of a galaxy.This type of velocity field is called a peak-intensity ve-locity field. Since no fitting procedure is required, thismethod is simple and fast. Unlike the IWM method,this method is able to trace the velocities at which thehighest fluxes are found, even for profiles showing signif-icant asymmetries. In this respect, the peak velocity isthe preferred velocity compared to the ones derived us-ing other methods. However, this method is sensitive tothe noise in profiles with low S/N in which case it fails toextract proper line-of-sight velocities. See de Blok et al.(2008) for more discussions.
Single Gaussian velocity fields
Oh et al.It is possible to fit a single Gaussian function to thevelocity profiles. A Gaussian function depends on threeparameters and is given as follows: V Gauss ( v ) = A exp (cid:18) − ( v − v ) σ (cid:19) , (2)where v and σ are the central velocity and velocity dis-persion of a profile. Due to the assumption on the shapeof the profiles (i.e., Gaussian function), this is less sensi-tive to the noise or (modest) asymmetries of profiles. Inaddition, the least squares fit procedure provides robustestimates of velocities, even for profiles with low S/Nvalues. The single Gaussian velocity field is best used inprofiles where the FWHM is comparable to the velocityresolution (de Blok et al. 2008). However, this methodstill suffers from significant profile asymmetries inducedby non-circular motions or projection effects of a galaxy.As shown in Fig. 1, the derived velocities from the sin-gle Gaussian fit can deviate from the peak velocities ofasymmetric profiles, although this method provides bet-ter results than the IWM method. Hermite h polynomials It is also possible to use the Gauss-Hermite polynomial(van der Marel & Franx 1993) to model the skewness ofa non-Gaussian profile. In addition, the Gauss-Hermitepolynomial also has a parameter called h , which mea-sures the kurtosis of a profile. However, to minimizethe number of free parameters, this term is not usuallyused when fitting the function. As the skewness is builtinto the profile, it is efficiently applicable to profiles withsignificant asymmetries. Compared to the peak velocityfield, hermite h polynomials give more stable results,even for profiles with low S/N values. Hermite h poly-nomials have been used to extract the velocity fields ofthe galaxies from THINGS (de Blok et al. 2008). Bulk velocity fields
A velocity profile in a data cube can consist of multiplecomponents if there are additional components movingat different velocities with respect to the underlying ro-tation of a galaxy. Until now, we have assumed thatthe underlying rotation of a galaxy is the dominant mo-tion in a galaxy. This assumption leads us to choose thepeak-intensity velocity in a fitted or raw velocity profileas the most representative velocity. This, however, onlyholds for a case where the majority of the gas movesat this velocity. Any additional components present ina velocity profile are then considered to be non-circularmotions that deviate from the bulk rotation. But this isnot true for a profile where non-circular motions domi-nate the kinematics of a galaxy. In this case, even if aprofile is decomposed successfully with multiple compo-nents, no clues exist as to which component is the bulkmotion and which ones are the non-circular motions. Wetherefore need additional constraints to distinguish thebulk motion and non-circular motions among such de-composed components.To this end, Oh et al. (2008) proposed a new methodto extract circularly rotating velocity components fromthe H i data cube and derive a so-called bulk velocity field.This type of velocity field efficiently separates small-scalerandom motions from the underlying rotation of a galaxy and extracts the bulk velocity. See de Blok et al. (2008)for the comparison of the various types of velocity fields.This method has been successfully used for two galaxiesthat are significantly affected by non-circular motions:IC 2574 and NGC 2366 (Oh et al. 2008).We extract the various types of velocity fields (i.e.,IWM, single Gaussian, hermite h polynomial, and thepeak and bulk velocity fields) from the H i data cubesof our sample galaxies and use them to derive rota-tion curves, as described in the following section. Thenatural-weighted cubes are used for this and no resid-ual scaling, primary beam correction or blanking is ap-plied to preserve the noise characteristics. The extractedvelocity fields of the 7 THINGS dwarf galaxies are pre-sented in the Appendix. Tilted-ring analysis
Using rotcur in GIPSY (Begeman 1989), we fit atilted-ring model to the bulk velocity field of the galax-ies to derive the ring parameters that best describe theobserved velocity fields. We then apply these tilted-ringmodels obtained from the bulk velocity field to the othervelocity fields to examine the effect of the type of velocityfield on the derived rotation curve. We show the rota-tion curves derived from different types of velocity fieldsof the sample galaxies in the Appendix. We will compareand discuss these rotation curves in Section 3.4.
Asymmetric drift correction
Pressure support plays an important role in galaxieswhose velocity dispersions are large enough compared totheir maximum rotation velocities (Bureau & Carignan2002). This is the case for the galaxies in our samplewhose typical maximum rotation velocities ( V max ) areless than ∼
35 km s − , except for IC 2574 ( ∼
80 km s − )and NGC 2366 ( ∼
60 km s − ). In order to obtain morereliable rotation velocities for these galaxies, we need tocorrect for the asymmetric drift. Following the methoddescribed in Bureau & Carignan (2002), we correct forthe asymmetric drift as follows, V = V + σ , (3)where V rot is the rotation velocity derived from the simplefit of a tilted-ring model to the velocity field and V cor isthe asymmetric drift corrected velocity. The asymmetricdrift correction σ D is given as, σ = − Rσ ∂ ln( ρσ ) ∂R = − Rσ ∂ ln(Σ σ ) ∂R , (4)where σ and ρ are the velocity dispersion and volumedensity of H i , and R is the radius of a galaxy. In partic-ular, ρ can be converted to the H i surface density Σ byassuming an exponential distribution in the vertical di-rection and a constant scale height (for a 1 st approxima-tion). For the surface density Σ and velocity dispersion σ , we use the integrated H i (0 th moment) and velocitydispersion (2 nd moment) maps, respectively. Using thetilted-ring model derived earlier from the bulk velocity Except in the determination of the bulk velocity field. SeeOh et al. (2008) for a detailed description. ark and luminous matter in THINGS dwarf galaxies 5field, we obtain the corrected radial profiles of Σ and σ .To avoid large fluctuations in the derivative in Eq. 4, wefit Σ σ with an analytical function,Σ σ ( R ) = I ( R + 1) R + e αR , (5)where I and R are the fitted values in units ofM ⊙ pc − km s − and arcsec, respectively. α is given inunit of arcsec − . The resulting profiles of σ D and Σ σ ( R )for the galaxies where the asymmetric drift correctionsare needed are shown in the Appendix. The rotation curves of THINGS dwarf galaxies
The resulting rotation curves of the individual galaxiesderived using different types of velocity fields and theircomparison are given in the Appendix. Below we discussthe rotation curves derived using the bulk velocityfield and corrected for asymmetric drift, where needed,in order to examine the effect of small-scale randomnon-circular motions. • IC 2574:
IC 2574 is affected by non-circular mo-tions (Walter & Brinks 1999) and this is clearly seen as“kinks” in the iso-velocity contours of the velocity fieldsas shown in Fig. A.1. The spatial locations of these small-scale random motions are also found in the non-circularmotion velocity field (hereafter NONC velocity field) asshown in panel (k) of Fig. A.1. As described in Oh et al.(2008), the NONC velocity field only contains the veloci-ties of the primary (i.e., strongest intensity) componentsamong the decomposed ones at the positions where theseprimary components were found to track the non-circularmotions.For a quantitative analysis of non-circular motions,we expand the velocity fields into harmonic termsup to the 3 rd order, c n and s n ( n = 1 , c , s and s ; corrected for inclination) decom-posed using the hermite h velocity field are ∼
10 km s − in the inner regions. However, the results from the bulkvelocity fields are less than 5 km s − over all radii.In general, small-scale random motions tend toresult in a lower rotation velocity than the true oneas they make the velocity gradients along the recedingand approaching sides of a galaxy less steep. This isparticularly prominent for the rotation velocity derivedusing the IWM velocity field which is most affected bynon-circular motions. As shown in Fig. 2, the rotationcurves derived from the other types of velocity fieldare largely consistent with each other. At ∼ h curves are about ∼
11 km s − and ∼ − ,respectively. For the kinematic analysis of IC 2574, wetherefore use the bulk rotation velocity which is lessaffected by these random motions and thus providesa better description of the underlying kinematics. Werefer to Oh et al. (2008) for a complete discussion onthe rotation curve analysis. • NGC 2366:
In Fig. A.5, the distorted iso-velocitycontours of the velocity fields indicate that most distur-bances caused by non-circular motions are present in theouter regions ( > ∼
10 km s − ) of harmonic terms in Fig. A.6.However, these disturbances are largely removed in thebulk velocity field as shown in Fig. A.5.In Fig. A.7, we compare the derived rotationcurves with those from the literature (Swaters 1999;Hunter et al. 2001; van Eymeren et al. 2009). TheHunter et al. (2001) and the THINGS IWM curves aresystematically lower than the bulk rotation curve. Thisis not due to different inclination assumption since aninclination of ∼ ◦ which is similar to our value ( ∼ ◦ )was used for the Hunter et al. (2001) curve. Instead,the velocity difference can be due to non-circularmotions in the galaxy. This idea is supported by asignificant velocity difference beyond ∼ h curve with that derived using the same THINGS h velocity field by van Eymeren et al. (2009). They useda slightly different center position ( ∼ ′′ in declination)and a lower systemic velocity (98 km s − ) but similarinclination ( ∼ ◦ ) and position angle ( ∼ ◦ ). Thevan Eymeren et al. (2009) curve agrees well with ourhermite h curve but is systematically lower than thebulk rotation curve. As in the case of IC 2574, weadopt the bulk rotation curve for the mass modeling ofNGC 2366. We refer to Oh et al. (2008) for a completediscussion on the rotation curve analysis. • Ho I:
The inclination of Ho I is the lowest amongour galaxies. Therefore, the projected velocities, V sin i (where V and i are the circular rotation velocity andthe inclination) of the galaxy are small, and more sen-sitive to the effect of non-circular motions. As can beseen in Fig. A.9, the iso-velocity contours of the velocityfields are severely distorted, particularly in the centraland north-western regions of the galaxy. The NONCvelocity field in Fig. A.9 also indicates the presence ofstrong non-circular motions in these regions as confirmedby inspection of position-velocity cuts along the kinemat-ical major and minor axes (Ott et al. 2001). In addition,the harmonic analysis of the bulk velocity field which isalready corrected for non-circular motions shows largeamplitudes ( ∼
10 km s − ) of the decomposed harmonicterms in Fig. A.10.To minimize the effect of the non-circular motions, wederive the rotation curve using the bulk velocity field inFig. A.9. The derived ring parameters, such as the kine-matic center, the systemic velocity and the position an-gle are consistent with those found by Ott et al. (2001).The rotation curve keeps increasing out to ∼ ◦ .However, the small inclination value of 14 ◦ impliesconsiderable uncertainty in the rotation curve. To checkthis we compare the rotation curves derived using in- An IWM velocity field was used.
Oh et al.
Fig. 2.— (a):
Comparison of the bulk rotation curve of IC 2574 with rotation curves derived using other types of velocity fields asdenoted in the panel (i.e., IWM, hermite h , single Gaussian and peak velocity fields). The Martimbeau et al. (1994) curve was derivedusing an IWM velocity field with a lower resolution. They adopted a large value for the inclination ( ∼ ◦ ). The IWM rotation curve whichis most likely affected by random non-circular motions is the lowest among the others. See Section 3.4 for more details. (b)(c): The radialmass surface density profiles of the stellar and gas components of IC 2574, respectively. (d)(e):
The resulting rotation velocities of thestellar and gas components of IC 2574 derived from the surface density profiles given in the panels (b) and (c), respectively. More detailscan be found in Section 4. clinations deviating ± ◦ from our adopted value (seethe INCL panel of Fig. A.10). In the VROT panel ofFig. A.10, we find significant differences between them.In particular, the rotation curve derived using the lowinclination value ( ∼ ◦ ) significantly deviates from ourpreferred curve. However, the value of ∼ ◦ falls at theextreme lower end of the inclinations derived from thetilted-ring fits. For reference, we show a fit result withonly the INCL left free (indicated by gray dots in theINCL panel of Fig. A.10). The values are systematicallylarger than those (open circles) derived keeping all pa-rameters free. From this we conclude that it is unlikelythat Ho I has an inclination as small as ∼ ◦ and we adopta value of 14 ◦ for the remainder of this paper. Notwith-standing the low inclination, we will derive the rotationcurves, fully keeping in mind the uncertainties due toinclination. To avoid our conclusions being skewed bythis galaxy, we will present our analysis both with andwithout Ho I.The H i velocity dispersions in Fig. A.9 show highvalues of ∼
12 km s − in the north western part (seealso Ott et al. 2001). Compared to the derived rota-tion velocity of ∼
20 km s − in the outer region, themagnitude of this velocity dispersion is significant. Wetherefore correct the rotation curve for asymmetric driftas described in Section 3.3. The corrected curve ispresented in Fig. A.10. We adopt this corrected bulkrotation curve for the kinematic analysis of Ho I. Weagain stress that in all further analysis we consider ourresults with and without Ho I. • Ho II:
We derive the rotation curve using the bulkvelocity field shown in Fig. A.13. As shown in Fig. A.14, all ring parameters are well determined and are consis-tent with the results by Bureau & Carignan (2002). The2 nd moment map shows rather large velocity dispersionscompared to the circular rotation velocity. Therefore,we make a correction for pressure support. The asym-metric drift corrected bulk rotation curve is presented inFig. A.14; it is rather flat and increases slightly beyond4 kpc, compared to the uncorrected one. In Fig. A.14,the amplitudes of the harmonic terms derived from thehermite h velocity field are less than 5 km s − over mostradii, although slightly larger than those from the bulkvelocity field.In Fig. A.15, we compare our rotation curve withthat from the literature. The ∼ ′′ resolution IWMrotation curve by Bureau & Carignan (2002) not onlyfalls below the asymmetric drift corrected bulk rotationcurve but also below the THINGS IWM one, despitethe correction for asymmetric drift by the authors. Thedifference between the respective tilted-ring models isnot enough to explain this velocity difference. It is likelythat the Bureau & Carignan’s lower beam resolutiondata ( ∼ ′′ ) and the derived rotation curve with a largerring width ( ∼ ′′ ) which smooth small-scale “wiggles”caused by non-circular motions in the galaxy (e.g., at ∼ ′′ is slightly lower than the one with 12 ′′ .We use the asymmetric drift corrected bulk rotationcurve for the mass modeling of Ho II. • DDO 53:
As shown in Fig. A.17, DDO 53 shows aclear rotation pattern in its velocity field. However, thedistorted iso-velocity contours imply the presence of non-ark and luminous matter in THINGS dwarf galaxies 7circular motions. In particular they are prominent in theouter regions as confirmed by the extracted NONC veloc-ity field and the harmonic analysis as shown in Figs. A.17and A.18, respectively. To minimize the effect of thesenon-circular motions, we extract the bulk velocity fieldas shown in Fig. A.17. Compared to other types of veloc-ity fields, the bulk velocity field is noisier but the overallrotation pattern is better visible. In addition, as shownin Fig. A.18, the amplitudes of the harmonic terms de-composed from the bulk velocity field are close to zero incomparison to those from the hermite h velocity field.The derived rotation curves using the bulk velocity fieldare shown in Fig. A.18, and most ring parameters exceptinclination are well determined. The inclination showsa large scatter as a function of radius, especially in theinner regions.We examine the effect of inclination on the rotationvelocity by changing it by ± ◦ and performing tilted-ring fits while keeping other ring parameters the same.Although the rotation curve derived using the lower in-clination value ( ∼ ◦ ) is higher by up to ∼
10 km s − ,this low inclination value seems not plausible for DDO53. Like the case of Ho I, we show a fit result with onlythe INCL free as indicated by gray dots in the INCLpanel of Fig. A.18. They are larger than those (open cir-cles) from the very first run with all ring parameters free.The lower inclination value ( ∼ ◦ ) can be regarded as alower limit.The maximum bulk rotation velocity is ∼
18 km s − which is comparable to the values found for the velocitydispersion in the outer regions of the galaxy (seeFig. A.17). This demands an asymmetric drift correc-tion, and the corrected curve is shown in Fig. A.18. Thecorrected curve keeps increasing to ∼
34 km s − at 2 kpc.We use this rotation velocity for the mass modeling ofDDO 53. • M81dwB:
The extracted velocity fields themselvesshow little difference with respect to each other as shownin Fig A.21. The NONC velocity field shows no signif-icant non-circular motions in the galaxy, except in thevery outer regions. Moreover, as shown in Fig. A.22, theamplitudes of the harmonic terms (corrected for inclina-tion) decomposed from both the IWM and hermite h velocity fields are less than 5 km s − over most radii. Inaddition, the difference between the harmonic terms de-rived from the IWM and hermite h velocity fields is alsonegligible. This implies that the effect of non-circularmotions on the velocity fields is not significant. There-fore, we can use the hermite h velocity field to derivethe rotation curve. As already discussed in Section 3.1,the hermite h velocity field gives a robust estimate forthe underlying circular rotation of a galaxy in which non-circular motions are insignificant. In Fig. A.22, the de-rived hermite h rotation curve keeps increasing out to0.5 kpc and then stays flat to 1 kpc. Beyond that, thecurve rapidly declines but this is mainly due to the smallnumber of pixels that contain signal and the large uncer-tainties in the velocities of the outer rings.The large velocity dispersions ( ∼
15 km s − ) in theouter regions are significant compared to the maximumrotation velocity of ∼
28 km s − . We therefore performthe asymmetric drift correction for the circular rotationafter which the corrected curve keeps increasing out to 1 kpc as shown in Fig. A.22. • DDO 154:
The complete description of the dataand the mass modeling including the tilted-ring analysisis given in detail in de Blok et al. (2008). It shows aregular rotation pattern in the hermite h velocity fieldin Fig. A.25 (see also Fig. 81 in de Blok et al. 2008). Inaddition, no significant non-circular motions were foundin the galaxy from a harmonic analysis of the velocityfield (Trachternach et al. 2008). We therefore use thehermite h rotation curve as in the case of M81dwB. Asdescribed in de Blok et al. (2008), the resulting rotationcurve resembles that of a galaxy with solid-body rotationbut increases more steeply in the inner regions thanprevious determinations (e.g., Carignan & Freeman1988; Carignan & Purton 1998) for which the IWMvelocity fields with lower beam resolutions were used. Inthis paper, we use the hermite h rotation curve derivedin de Blok et al. (2008) for the mass modeling of DDO154, and refer to their paper for a complete discussion.In summary, the rotation velocities derived from thebulk velocity fields of the THINGS dwarf galaxies (exceptM81dwB and DDO 154 where a hermite h velocity fieldwas used) generally show the most rapid increase com-pared to those from the other types of velocity fields, suchas the IWM, peak, single Gaussian fit and hermite h .The IWM velocities show the slowest increase, especiallyin the inner region of the galaxy. The rotation velocitiesderived from the peak, single Gaussian fit and hermite h velocity fields show an increasingly steeper gradientthan the IWM velocity, but somewhat less steep thanthe bulk velocity. This is due to their different abilitiesto take asymmetries of profiles affected by non-circularmotions into account. The IWM velocity field is the onemost affected by non-circular motions. The random non-circular motions induce a smaller velocity gradient acrossthe IWM velocity field, which results in a rotation veloc-ity that increases more slowly. In contrast, the bulk ve-locity field minimizes the effect of random motions, andproperly extracts the underlying circular rotation.We also examine the sensitivity of the rotation curvesto the exact value of inclination. Of our galaxies, therotation curves of Ho I and DDO 53 (whose inclinationvalues are ∼ ◦ and ∼ ◦ , respectively) are most sen-sitive to changes in inclination. However, the adoptedinclination values from the tilted-ring fits appear to beplausible. In addition, they also agree well with thosederived independently from the Baryonic Tully − Fisherrelation as will be discussed in Section 5.2 later. MASS MODELS OF BARYONS
The rotation curve reflects the dynamics of the totalmatter content in a galaxy, including the baryons andthe dark matter. We therefore subtract the dynamicalcontribution of baryons from the total dynamics to de-termine the dark matter component only. To this end,we first derive radial distributions of the baryons in ourgalaxies and derive mass models for them.
Stellar component
We derive the mass models for the stellar componentsof our sample galaxies following the method described inOh et al. (2008; see also de Blok et al. 2008). Firstly, we Oh et al.derive the luminosity profiles of the galaxies by apply-ing the tilted-ring models derived in Section 3.2 to theIRAC 3.6 µ m images from SINGS to derive radially av-eraged surface brightness profiles. These are shown inthe Appendix. We then convert the luminosity profilesto mass density profiles in units of M ⊙ pc − using anempirical Υ ⋆ relation derived from population synthesismodels, as described in Oh et al. (2008). The empiricalrelation derives Υ ⋆ in the IRAC 3.6 µ m band (Υ . ⋆ ) fromthe Υ ⋆ in K band (Υ K⋆ ) which in turn is determined us-ing optical colors and metallicity of a galaxy, as given inBell & de Jong (2001). The optical ( B , V and R ) surfacebrightness profiles and colors ( B − V and B − R ) used fordetermining Υ K⋆ are shown in the Appendix. Here, weuse constant average colors for our galaxies, except forIC 2574 where the radial distribution of colors can be de-rived. We also show the metallicity of the sample galaxiesin Table 1. From this we compute Υ . ⋆ for the samplegalaxies, as shown in the Appendix. Leroy et al. (2008)use an empirical K -to-3.6 µ m calibration to derive K -band fluxes from the IRAC 3.6 µ m images for a numberof THINGS galaxies. They then derive the stellar diskmasses adopting a fixed Υ K⋆ =0 .
5. de Blok et al. (2008)make a comparison between the stellar disk masses de-rived using our method and the approach by Leroy et al.(2008), and find that they agree well with each other.Using the Υ . ⋆ values, we derive the mass density pro-files of stellar components of the sample galaxies (pre-sented in the Appendix). We then calculate the rota-tion velocities for the stellar components from the massdensity profiles, assuming a vertical sech (z) scale heightdistribution. We use h/z =5, the ratio between the ver-tical scale height z and the radial scale length h of disk,as determined in van der Kruit & Searle (1981; see alsoKregel et al. 2002). The derived z values are given inTable 1. The average value is z ≃ .
32 kpc. We con-struct the final mass models for the stellar componentsof the sample galaxies using the rotmod task in GIPSY,the results of which are shown in the Appendix.
Gas component
The H i surface density profile in M ⊙ pc − units can bedirectly derived from the observed H i column density. Inorder to calculate the radial H i distribution of the samplegalaxies, we apply the tilted-ring models derived in Sec-tion 3.2 to the integrated H i maps to derive azimuthallyaveraged radial H i profiles. We scale the derived H i sur-face density profile by a factor of 1.4 (de Blok et al. 2008)to account for helium and metals and calculate the rota-tion velocities for the gas component. For this we assumean infinitely thin disk and use the task rotmod imple-mented in GIPSY. The gas surface density profiles andthe gas rotation velocities of our galaxies are presentedin the Appendix. DARK MATTER HALO AND LUMINOUS MATTER
In general, dwarf galaxies are dark matter-dominatedthroughout due to the small contribution of baryonsto the total dynamics, as is the case in LSB galax-ies (e.g., de Blok & McGaugh 1997; Prada & Burkert2002). Therefore, dwarf galaxies have been consideredto be ideal objects for studying dark matter propertiesin galaxies. Of particular interest is testing the dark
Fig. 3.—
The dark matter fraction γ dm (as described in Eq. 6)of the 7 THINGS dwarf galaxies. Most galaxies are dark matterdominated across all radii except the inner region of DDO 53 andthe outer region of Ho II, respectively. This is discussed in detailin Section 5.1 matter distribution as predicted from cosmological sim-ulations. In this section, we calculate the dark matterfraction of our galaxies and verify if dark matter indeeddominates the total dynamics of these systems. Fur-thermore, we also examine the relationship between thedark matter fraction and other galaxy properties, such asthe dynamical mass, the absolute B magnitude and theBaryonic Tully − Fisher (BTF) relation (Bell & de Jong2001; Verheijen 2001; McGaugh 2004; De Rijcke & et al.2007).
Dark matter fraction and galaxy properties
We derive the radial dark matter fraction of our galax-ies using, γ dm = M DM M tot = V − V − V V , (6)where V tot is the observed total rotation velocity, and V star and V gas are the rotation velocities of stars and gas,respectively. For this measurement, we use V tot , V star and V gas of our galaxies as derived in Sections 3.4 and 4(see the Appendix).In Fig. 3, we plot γ dm as a function of radius. Theradii are normalized to the maximum radius ( R max ) atwhich the last data point is measured. Most of our galax-ies show large values of γ dm of about 0.7 over the radialrange. This implies that they are indeed dark matter-dominated over most of their radial range. Ho II andDDO 53 show radial gradients. The value of γ dm for HoII is ∼ R max , but decreases to ∼ γ dm value of DDO 53 is ∼ < R max ), but increases to ∼ B magnitude and theark and luminous matter in THINGS dwarf galaxies 9 Fig. 4.—
Left:
The relationship between the dark matter fraction and the absolute B magnitude of 19 THINGS dwarf and spiralgalaxies. <γ dm > is determined by radially averaging γ dm values of each galaxy. For the spiral galaxies, <γ dm > values are calculated overthree regions, splitting a galaxy into three annuli (inner, middle and outer) as indicated by different symbols. Right:
The relationshipbetween the mean dark matter fraction <γ dm > and the dynamical mass of the same galaxies. See Section 5.1 for more discussions. dynamical mass of 12 spiral galaxies from THINGS andthe 7 dwarf galaxies from the current sample. In the leftpanel of Fig. 4, we plot the radial average of γ dm againstthe absolute B magnitude of individual galaxies. Forthe spiral galaxies where the inner and outer regions aretotally dominated by baryons and dark matter, respec-tively, we calculate average γ dm values over three regions,splitting a galaxy into three annuli (inner, middle andouter). For this, we choose an inner radius at which therotation curve reaches its flat part, and split the regionbeyond it into two equal-size radial bins for the middleand outer annuli. The calculated γ dm values within theannuli for each spiral galaxy are indicated by differentsymbols in Fig. 4. As expected, the outer region of thespiral galaxies is more dark matter dominated than theinner region. In addition it is likely that the dark mat-ter fraction in the outer region of the spiral galaxies issimilar to that of the dwarf galaxies.In the right panel of Fig. 4, we show the relation-ship between the dark matter fraction and the dynami-cal mass of the galaxies. Likewise, for the spiral galaxieswe calculate average γ dm values over three regions, split-ting a galaxy into three annuli (inner, middle and outer).Considering that more luminous galaxies are in generalmore massive (e.g., Guo et al. 2010; Dutton et al. 2010),this is largely consistent with the relationship between γ dm and absolute B magnitude, as shown above. The baryonic Tully − Fisher relation
We also examine whether the THINGS dwarf samplegalaxies follow the Baryonic Tully − Fisher (BTF) rela-tion. There have been several efforts to calibrate theBTF relation using a sample of gas dominated low masssystems (McGaugh et al. 2000; McGaugh et al. 2005;Stark et al. 2009). These found that the broken conti- nuity of the classical Tully − Fisher relation of low-massgalaxies can be restored by using their total baryonicmass (i.e., including not only stars but also the gascomponent). As shown in Fig. 5, we plot the baryonic(stars + gas) mass of our galaxies derived in Section 4against the maximum rotation velocity at the last mea-sured point. They are roughly consistent with the BTFrelation (indicated as the dashed line) from Stark et al.(2009) within the uncertainty but systematically slightlyhigher than the line except M81dwB. This could be ow-ing to the underestimated maximum rotation velocities ofthe galaxies. Some of the rotation curves derived in Sec-tion 3.4 still keep increasing at the last measured point,which implies a larger maximum rotation velocity.As already discussed in Section 3.4, the rotation curvesof some of our sample galaxies are sensitive to the exactvalue of inclination (e.g., Ho I and DDO 53). As a sanitycheck, we therefore derive inclinations based on the BTFrelation. The observed line-of-sight velocity of a galaxyat the last measured point R max can be expressed bythe following equation (if we only consider the azimuthalvelocity component), V obs ( R max ) = V sys + V TRmax × sin i TR × cos θ, (7)where V sys is the systemic velocity, θ is the position angle, i TR is the inclination and V TRmax is the maximum rotationvelocity derived from tilted-ring analysis. The BTF re-lation yields an estimate of maximum rotation velocity V BTFmax at a given baryonic mass. Therefore, in Eq. 7 V TRmax can be substituted with V BTFmax and the corresponding in-clination value i BTF which gives the same V obs ( R max ) canbe calculated using the following formula, i BTF = arcsin (cid:18) V TRmax V BTFmax × sin i TR (cid:19) . (8)0 Oh et al. Fig. 5.—
The baryonic Tully − Fisher relation of the THINGSdwarf galaxies. The baryonic mass includes the stellar and gascomponents derived in Section 4. The long dashed-line indicatesthe BTF relation calibrated using a sample of gas dominated galax-ies in Stark et al. (2009), and the short dashed-lines indicate theuncertainty in the relation. See Section 5.2 for more details.
The derived i BTF values of the THINGS dwarf galax-ies sample are given in Table 1, and they are smallerthan those derived from tilted-ring analysis except forM81dwB. This is because the inferred V BTFmax values ofour galaxies are larger than the V TRmax values as shownin Fig. 5. Given the uncertainties in the estimates, theinclination values derived from both the BTF relationand the tilted-ring analysis are not significantly differ-ent from each other except for Ho II. However, as canbe seen from not only the tilted-ring analysis (includingthe position-velocity diagram) but also the comparisonof rotation velocities in the Appendix it is unlikely thatHo II has an inclination ( ∼ ◦ ) as low as that inferredfrom the BTF relation. DARK MATTER DISTRIBUTION
In this section, we compare the derived dark matterdistribution of the THINGS dwarf galaxies with thatinferred from structure formation N-body simulationsbased on the ΛCDM paradigm. For this we use an NFWhalo model (Navarro et al. 1996, 1997) which is given as, ρ NFW ( R ) = ρ i ( R/R s )(1 + R/R s ) , (9)where ρ i is the initial density of the Universe at the timeof the collapse of the halo and R s is the characteristic ra-dius of the dark matter halo. This gives a “cusp” featurehaving a power law mass density distribution ρ ∼ R − towards the centers of galaxies. The corresponding rota-tion velocity induced by this potential has the followingform, V NFW ( R ) = V s ln(1 + cx ) − cx/ (1 + cx ) x [ln(1 + c ) − c/ (1 + c )] , (10)where c is the concentration parameter defined as R /R s . V is the rotation velocity at radius R where the mass density contrast exceeds 200 and x isdefined as R/R .In addition, we also use an observationally motivatedpseudo-isothermal halo model as an extreme represen-tation of “core-like” halo models (e.g., Begeman et al.1991). It has the following form, ρ ISO ( R ) = ρ R/R C ) , (11)where ρ and R C are the core-density and core-radius ofthe halo, respectively. This gives rise to a mass distribu-tion with a sizable constant density-core ( ρ ∼ R ) at thecenters of galaxies. The rotation velocity induced by themass distribution is given as, V ISO ( R ) = vuut πGρ R C " − R C R atan RR C ! . (12)Using these two halo models, we examine which modelis preferable to describe the observed dark matter distri-bution of our galaxies. Dark matter mass modeling
We subtract the dynamical contribution of the baryonsfrom the total kinematics and construct mass models ofthe dark matter halos of our galaxies. We fit the two halomodels, i.e., the NFW and pseudo-isothermal models (seee.g. Oh et al. 2008), to the bulk rotation curves derivedin Section 3, taking into account the mass models of thebaryons. When performing the fits, we use various as-sumptions for Υ ⋆ , such as “maximum disk”, “minimumdisk” and “minimum+gas disk” as well as the model Υ . ⋆ value as described in Section 4.1. The maximum diskassumes that the observed rotation curve in the innerregions of a galaxy is almost entirely due to the stel-lar component (van Albada & Sancisi 1986). Therefore,the dark matter properties derived using this assumptionwill provide a lower limit to its mass distribution. Incontrast, the minimum disk hypothesis ignores the con-tribution of baryons and attributes the rotation curve tothe dark matter component only (van Albada & Sancisi1986). This yields a robust upper limit on the proper-ties of dark matter. The minimum+gas disk ignores thestellar component but includes the gas component.The fit results of individual galaxies are presented inthe figures (“Mass modeling results”) and Tables in theAppendix. We find that in terms of fit-quality (i.e., χ red )pseudo-isothermal halo models are mostly preferred overNFW halo models in describing the dark matter distri-bution of our galaxies. In addition the mean value ofthe logarithmic central halo surface density log( ρ R C ) inunits of M ⊙ pc − of our sample galaxies is ∼ . ± . ρ R = 2 . ± . ⋆ (i.e., maximum, minimum, minimum+gas and modelΥ . ⋆ disk), giving negative (or close to zero) c val-ues. Even if the fits are feasible (e.g., Ho II), pseudo- a r k a nd l u m i n o u s m a tt e r i n T H I N G Sd w a r f ga l a x i e s Fig. 6.—
Left:
The shape of the total rotation curves (not corrected for baryons) of the 7 THINGS dwarf galaxies. The rotation curves are scaled with respect to the rotation velocity V . at R . where the logarithmic slope of the curve is d log V/d log R = 0 . V ranging from 10 to110 km s − . The dotted lines indicate the NFW models with V less than 110 km s − . The scaled rotation curves of the best fit pseudo-isothermal halo models (denoted as ISO) arealso overplotted. See Section 6.2 for more details. Right:
The shape of the dark matter rotation curves of the 7 THINGS dwarf galaxies. These are corrected for baryons and are scaledwith respect to the rotation velocity V . at R . . Same legends as in the left panel. Compared to the total rotation curves in the left panel, the dark matter rotation curves increase lesssteeply but they are similar due to the low baryonic fraction of the galaxies as discussed in Section 5.1. See Section 6.2 for more discussions. Fig. 7.—
The dark matter density profiles of the 7 THINGS dwarf galaxies. The profiles are derived using the scaled rotation curves(assuming minimum disk) as described in Section 6.2 (see also Fig. 6). The dotted lines represent the mass density profiles of NFW models( α ∼− .
0) with V ranging from 10 to 110 km s − . The dashed lines indicate the mass density profiles of the best fit pseudo-isothermalhalo models ( α ∼ . ark and luminous matter in THINGS dwarf galaxies 13isothermal halo models are still slightly better in de-scribing the rotation curves irrespective of the all as-sumptions on Υ ⋆ . We also fit the NFW model to therotation curves with only V as a free parameter afterfixing c to 9 which is similar to typical values (e.g., 8–10;McGaugh et al. 2003) predicted from ΛCDM cosmology.However, as shown in the Tables in the Appendix, thebest-fit χ red values are even larger than those from thefits with both c and V as free parameters. Moreover, atthe inner regions of the rotation curves, the fitted NFWhalo models are too steep. This will be further discussedin the following section.It is also interesting how well the “minimum disk” as-sumption on Υ ⋆ provides a good description of the bary-onic mass distributions of the galaxies. As shown inthe Appendix, the best-fit χ red values are close to theones obtained assuming the model Υ . ⋆ . This confirmsthat the THINGS dwarf galaxies are indeed dark-matter-dominated. Rotation curve shape
The divergent density profiles (e.g., ∼ R α where α ∼− . V from 10 to 110 km s − . The concentra-tion parameter c corresponding to a particular value of V is determined by the following empirical c − V re-lation from the WMAP observations in McGaugh et al.(2007; see also de Blok et al. 2003),log V = 2 C − log[ g ( c )] − log( h , (13)where g ( c ) = c ln(1 + c ) − c/ (1 + c ) , (14) h = H / − Mpc − and C = 1 .
61 for the 3 yearWMAP parameters (Spergel et al. 2007). We adopt h =0 .
75. We refer the reader to McGaugh et al. (2007) formore details.To be able to compare any discrepancies in shape, wescale both the rotation curves of our galaxies and thoseof the adopted CDM halos to the velocity V . at the The Wilkinson Microwave Anisotropy Probe (Spergel et al.2003; Spergel et al. 2007) radius R . , where R . is the radius where the loga-rithmic slope of the curve is d log V /d log R =0.3. As dis-cussed in Hayashi & Navarro (2006), the NFW curves arewell resolved at the scaling radius R . (correspondingto ∼ . R s where R s is given in Eq. 9) as their asymp-totic slopes are about d log V /d log R =0.5. In addition,this scaling radius is also well determined in observedrotation curves since it lies between the inner linear( d log V /d log R =1) and the outer flat ( d log V /d log R =0)regions of the rotation curves of most disk galaxies(Hayashi & Navarro 2006). This also holds for our sam-ple galaxies, except for IC 2574 where the outermost log-arithmic slope is still larger than 0.3. In the case of IC2574, we scale the rotation curve to the maximum ra-dius R max where the last data point is measured, andcorresponding maximum rotation velocity.We plot the scaled rotation curves in the left panelof Fig. 6. These rotation curves are not corrected forbaryons, and assume the minimum disk model as de-scribed in Section 6.1. Similarly, in the right panel ofFig. 6, we plot the scaled rotation curves corrected forbaryons derived from Section 4. Although the rotationcurves corrected for baryons increase less steeply thanthe curves assuming a minimum disk, they are very sim-ilar. This directly shows that using the minimum diskassumption gives a good description of the dark mat-ter distribution in our galaxies. In Fig. 6, the CDM ro-tation curves with V less than 110 km s − are rep-resented by dotted lines. Of our sample galaxies, IC2574 has the largest maximum rotation velocity of about80 km s − . Therefore, the CDM rotation curve with V =110 km s − is a hard upper limit for our galaxiesassuming that V max ∼ V . We also overplot the bestfits of pseudo-isothermal models (dashed lines) derivedusing the minimum disk assumption and derived Υ . ⋆ inFig. 6.As can be seen in Fig. 6, the rotation curve shapesof the galaxies are similar and consistent with those ofpseudo-isothermal halo models. However, they are in-consistent with those of ΛCDM simulations. The ro-tation curves of ΛCDM simulations rise too steeply tomatch the observations. The difference in rotation curveshapes between our galaxies and ΛCDM simulations isparticularly prominent in the inner regions of galaxiesi.e., at radii less than R . . This difference is further en-hanced for the CDM rotation curves with V less than110 km s − . In conclusion, the solid body-like rotationcurves of our galaxies rise too slowly to reflect the cusp-like dark matter distribution in CDM halos. Dark matter density profile
Direct conversion of the galaxy rotation curve to themass density profile allows us to examine the radial mat-ter distribution in the galaxy. In particular, the mea-sured inner slope of the density profile is critical forresolving the “cusp/core” problem at the galaxy cen-ter. The Poisson equation ( ∇ Φ = 4 πGρ , where Φ = − GM/R ) can be used for the conversion under the as-sumption of a spherical mass distribution. The mass den-sity ρ is directly derived from the rotation curve V ( R ),as follows (see de Blok et al. 2001 for more details), ρ ( R ) = 14 πG " VR ∂V∂R + VR ! . (15)4 Oh et al. Fig. 8.—
The inner slope of the dark matter density profile plot-ted against the radius of the innermost point. The inner densityslope α is measured by a least squares fit to the inner data point asdescribed in the small figure. The inner-slopes of the mass densityprofiles of the 7 THINGS dwarf galaxies are overplotted with earlierpapers and they are consistent with previous measurements of LSBgalaxies. The pseudo-isothermal model is preferred over the NFWmodel to explain the observational data. Gray symbols: open cir-cles (de Blok et al. 2001); triangles (de Blok & Bosma 2002); openstars (Swaters et al. 2003). See Section 6.3 for more discussions. Using Eq. 15, we directly convert the total rotationcurves into mass density profiles. Here, we use the mini-mum disk hypothesis (i.e., ignores baryons). As alreadydiscussed in Section 5.1, our galaxies are mostly darkmatter-dominated and this “minimum disk” assumptionis a good approximation in describing their dynamics.Particularly useful is the fact that it gives a hard upperlimit to the dark matter density.In this way, we derive the mass density profiles of the7 THINGS dwarf galaxies and present them in the Ap-pendix. We also derive the mass density profiles usingthe scaled rotation curves derived assuming minimumdisk in Fig. 6, and plot them in Fig. 7. The best fits ofthe NFW and pseudo-isothermal models are also over-plotted. Despite the scatter, the derived mass densityprofiles are more consistent with the pseudo-isothermalmodels as shown in Fig. 7.To quantify the degree of concentration of the darkmatter distribution towards the galaxy center, we mea-sure the logarithmic inner slope of the density profile.For this measurement, we first need to determine abreak-radius where the slope changes most rapidly. Theinner density slope is then measured by performing aleast squares fit to the data points within the break-radius. For the uncertainty, we re-measure the slopetwice, including the first data point outside the break-radius and excluding the data point at the break radius.The mean difference between these two slopes is adoptedas the slope uncertainty ∆ α . The measured slope α and slope uncertainty ∆ α of the galaxies are shown inthe Appendix. In addition, we overplot the mass den-sity profiles of NFW and pseudo-isothermal halo mod- els which are best fitted to the rotation curves of thegalaxies. From this, we find that the mean value of theinner density slopes for the galaxies is α = − . ± . − . ± .
07 without Ho I which has a low incli-nation. See Section 3.4 for details). These rather flatslopes are in very good agreement with the value of α = − . ± . ∼− . h rotation curve instead of thebulk one. For this, we use IC 2574 which shows strongnon-circular motions close to the center. As shown in the“Mass density profile” panel of Fig. A.3, the mass den-sity profile derived using the hermite h rotation curveis found to be slightly lower than that from the bulk ro-tation curve at the central regions. This is mainly dueto the lower hermite h rotation velocity, resulting insmaller velocity gradients ∂V /∂R in Eq. 15 and thussmaller densities. The measured inner density slope is α =0 . ± .
19 which is similar, within the error, to that( α =0 . ± .
07) based on the bulk rotation curve. Thissupports earlier studies that suggest that the effect ofsystematic non-circular motions in dwarf galaxies is notenough to hide the central cusps (e.g., Gentile et al. 2004;Trachternach et al. 2008; van Eymeren et al. 2009).In Fig. 8, we plot the logarithmic inner density slope α against resolution of a rotation curve. At high resolu-tions ( R in < R in ∼ ∼ α − R in rela-tions of NFW and pseudo-isothermal halo models as solidand dotted lines, respectively. The highly-resolved rota-tion curves of our galaxies (i.e., R in ∼ R in ∼ α − R in trend of our galaxies is more consistent with those ofpseudo-isothermal halo models. CONCLUSIONS
In this paper we have presented high-resolution massmodels of the 7 dwarf galaxies, IC 2574, NGC 2366,Ho I, Ho II, DDO 53, DDO 154 and M81dwB from theTHINGS survey, and examined their dark matter distri-bution by comparison with classical ΛCDM simulations.The THINGS high-resolution data significantly reduceobservational systematic effects, such as beam smear-ing, center offset and non-circular motions. When deriv-ing the rotation curves, we used various types of veloc-ity fields, such as intensity-weighted mean, peak, singleGaussian, hermite h and bulk velocity fields, and com-pared the results. In particular the bulk velocity fieldark and luminous matter in THINGS dwarf galaxies 15was able to efficiently remove small-scale random mo-tions and allowed us to better determine the total kine-matics of the galaxies.We also found that the relation between the total bary-onic mass (stars + gas) and the maximum rotation veloc-ity of the galaxies is roughly consistent with the BaryonicTully − Fisher (BTF) relation calibrated from a largersample of low mass galaxies. Especially, the inclinationvalues derived if one takes the BTF relation at face valueare not significantly different from those derived from atilted-ring analysis. This implies that the BTF relationcan be used as an alternative way for deriving inclinationsof galaxies for which it is difficult to apply a tilted-ringanalysis.We derived the mass models of baryons, and subtractedthem from the total kinematics. For the stellar compo-nent, we used SINGS 3.6 µ m and optical data determinedby the stellar mass–to–light ratio Υ ⋆ for the 3.6 µ m band.For the purpose of our study we use the 3.6 µ m Spitzer images to estimate the mass of the old stellar popula-tion in our target galaxies. Even though this band maycontain some dust emission features, we consider it tobe the best consistent tracer of the stellar masses (seediscussion in Leroy et al. 2008, de Blok et al. 2008 andOh et al. 2008). These therefore allow us to estimatethe old stellar population that dominates the stellar con-tinuum emission in the infrared regime. Although oursample dwarf galaxies are dark matter dominated as in-dicated by their low baryonic fraction, the populationsynthesis Υ . ⋆ values gave slightly better or similar fitsthan not only the maximum disk but also the minimum(+gas) disk assumptions in describing the stellar compo-nent.With the help of the well determined total kinemat-ics and the mass models of baryons, we were able toaccurately constrain the dark matter distribution in thegalaxies. From this, we found a significant discrepancyin the dark matter distribution between the THINGSdwarf galaxies and classical dark-matter-only cosmologi-cal simulations both in the rotation curve shape and theinner slope α of the mass density profiles. The rotationcurves of the galaxies rise less steeply to be consistentwith the cusp feature at the centers. In addition themean value of the inner slopes of the mass density pro-files is α = − . ± .
07 (and − . ± .
07 without Ho Iwhich has a low inclination), significantly deviating from ∼− − body + Smoothed Particle Hydrodynamic (SPH)simulations by Governato et al. (2010), of dwarf galaxiesthat include effects of baryonic feedback processes whichresult in shallower slopes of α .SHOH acknowledges financial support from the SouthAfrican Square Kilometre Array Project. The work ofWJGdB is based upon research supported by the SouthAfrican Research Chairs Initiative of the Department ofScience and Technology and National Research Founda-tion. This research has made use of the NASA/IPACExtragalactic Database (NED) which is operated by theJet Propulsion Laboratory, California Institute of Tech-nology, under contract with the National Aeronauticsand Space Administration. This publication makes useof data products from the Two Micron All Sky Sur-vey, which is a joint project of the University of Mas-sachusetts and the Infrared Processing and Analysis Cen-ter/California Institute of Technology, funded by the Na-tional Aeronautics and Space Administration and theNational Science Foundation. REFERENCESBegeman, K. 1989, A&A, 223, 47Begeman, K., Broeils, A. H., & Sanders, R. H. 1991, MNRAS, 249,523Bell, E. F. & de Jong, R. S. 2001, ApJ, 550, 212Bolatto, A. D., Simon, J. D., Leroy, A., & Blitz, L. 2002, ApJ, 565,238Bosma, A. 1978 (PhD Thesis, University of Groningen)Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000Bureau, M. & Carignan, C. 2002, AJ, 123, 1316Burkert, A. 1995, ApJ, 447, L25Carignan, C. & Freeman, K. C. 1988, ApJ, 332, L33Carignan, C. & Purton, C. 1998, ApJ, 506, 125de Blok, W. J. G. 2010, AdAst, 2010, 1de Blok, W. J. G. & Bosma, A. 2002, A&A, 385, 816de Blok, W. J. G., Bosma, A., & McGaugh, S. S. 2003, MNRAS,340, 657de Blok, W. J. G. & McGaugh, S. S. 1997, MNRAS, 290, 533de Blok, W. J. G., McGaugh, S. S., Bosma, A., & Rubin, V. C.2001, ApJ, 552, 23 de Blok, W. J. G., Walter, F., Brinks, E., Trachternach, C., Oh,S.-H., & Kennicutt, R. C. 2008, AJ, 136, 2648De Rijcke, S. & et al.. 2007, ApJ, 659, 1172Diemand, J., Kuhlen, M., Madau, P., Zemp, M., Moore, B., &Potter, D. Stadel, J. 2008, Nature, 454, 735Donato, F., Gentile, G., Salucci, P., Frigerio Martins, C.,Wilkinson, M. I., Gilmore, G., Grebel, E. K., Koch, A., & Wyse,R. 2009, MNRAS, 397, 1169Dubinski, J. & Carlberg, R. G. 1991, ApJ, 378, 496Dutton, A. A., Conroy, C., van den Bosch, F. C., Prada, F., &More, S. 2010, arXiv:1004.4626v1Flores, R. A. & Primack, J. R. 1994, ApJ, 427, L1Gentile, G., Salucci, P., Klein, U., Vergani, D., & Kalberla, P. 2004,MNRAS, 351, 903Ghigna, S., Moore, B., Governato, F., Lake, G., Quinn, T., &Stadel, J. 2000, ApJ, 544, 616Governato, F., Brook, C., Mayer, L., Brooks, A., Rhee, G.,Wadsley, J., Jonsson, P., Willman, B., Stinson, G., Quinn, T., &Madau, P. 2010, Nature, 463, 203
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Appendix.
DATA AND KINEMATIC ANALYSIS
In the following we present the data and kinematic analysis of 7 dwarf galaxies from “The H i Nearby Galaxy Survey(THINGS)”. The kinematic analysis includes (1) the tilted ring model, (2) the harmonic analysis, (3) the mass modelsof baryons and dark matter and (4) the dark matter density profile. The following are general descriptions of the figures. • Data − We show the total intensity maps in
Spitzer
IRAC 3.6 µ m, optical B , R -bands and H i
21 cm. The lattercan be used to directly derive the H i surface density. The stellar surface density is based on the Spitzer µ m mapand information about the optical colors (see main text for details). The 2 nd moment map showing the velocitydispersions of the H i profiles is also given. We then compare the five types of velocity fields extracted from the H i datacube: The intensity-weighted mean (IWM), the Peak-intensity (PEAK), the Single Gaussian profile (SGFIT), thehermite h (HER3) and the bulk velocity fields (BULK). Additionally, we show the extracted velocity field of strongnon-circular motions (NONC). We also show a major-axis position-velocity (P-V) diagram overlayed with the derivedbulk rotation curve corrected for inclination. For the extraction of the integrated H i map, the intensity-weightedmean velocity field, and the 2 nd moment map, the natural-weighted data cube is used. See Walter et al. (2008) for adetailed description of the data cubes. • Rotation curves − The tilted ring model derived from the bulk (or hermite h for M81dwB) velocity field. Notethat the black solid lines are not the fits to the gray open circles. The open gray circles indicate the fit made with allring parameters “free”. The final rotation curves (black solid lines) are derived after several iterations. • Asymmetric drift correction − For galaxies where the velocity dispersion is comparable to the maximumrotation velocity, we correct for the asymmetric drift following the method described in Bureau & Carignan (2002).See Section 3.3 for a detailed description. • Harmonic analysis − Harmonic expansion of the hermite h and bulk velocity fields. Gray circles and blackdots represent the results from the hermite h and the bulk velocity fields, respectively. c and c are the systemicand the rotation velocities. c , c , s , s , and s components quantify non-circular motion components. In thebottom-rightmost panel, we show a global elongation of the potential, ǫ pot sin 2 φ calculated at each radius asdescribed in Schoenmakers et al. (1997) and Schoenmakers (1999). This measurement can be used as an additionaltest for CDM halos (e.g., Trachternach et al. 2008). The black solid and gray dashed lines indicate the average valuesof the potential derived using the bulk and hermite h velocity fields, respectively. • Mass models of baryons − (a): Azimuthally averaged surface brightness profiles in the 3.6 µ m, R , V , and B bands (top to bottom) derived assuming the tilted-ring parameters derived as above. These are not corrected forinclination except for the 3.6 µ m. The lines shown are fits to the data which are partly filled. (b)(f ): Derived valuesof Υ ⋆ in the K and 3.6 µ m bands from Bruzual & Charlot (2003) population synthesis models. The dotted and dashedlines are computed using optical colors ( B − R and B − V ) in (e) and the mean value (solid line) is adopted as thefinal Υ ⋆ . The relationships between Υ K⋆ and optical colors (e.g., B − R , B − V ) are adopted from the models ofBell & de Jong (2001). For the conversion of Υ K⋆ to Υ . ⋆ , Eq. 6 in Oh et al. (2008) is used. (e): The optical colors( B − R and B − V ) derived from the surface brightness profiles in (a) . Where B − R was not available, only B − V is given. (c)(d): Mass models for the stellar component. The stellar mass surface density is derived from the 3.6 µ msurface brightness (inclination corrected) in (a) using the Υ . ⋆ values shown in (f ) . The resulting expected rotationvelocity for H i if it were to move in circular orbits in the potential corresponding to the optical mass density only isthen derived from this. (g)(h): The mass model for the gas component. The radial mass surface density distributionof neutral gas is scaled by 1.4 to account for He and metals. • Comparison of rotation curves − Comparison of the rotation velocity derived from the bulk velocity field withthose from the other types of velocity fields (i.e., IWM, hermite h , single Gaussian fit and peak velocity fields) andthe literature in case other measurements are available. For the bulk rotation velocity, we derive rotation velocities forreceding and approaching side only, by keeping the ring parameters the same. These are indicated as the gray (inverse)triangles. We also show the bulk rotation velocity corrected with i BTF derived from the Baryonic Tully − Fisher (BTF)relation. • Mass density profile − The derived mass density profile. The dashed and solid lines show the best fits of theNFW halo model and the pseudo-isothermal halo model to the rotation curve, respectively. The measured inner slope α is shown in the panel. • Mass modeling results − Disk-halo decomposition of the bulk rotation curve (asymmetric drift corrected whereneeded) is made under various Υ ⋆ assumptions (Υ . ⋆ , maximum disk, minimum disk + gas and minimum disk). ForM81dwB, the asymmetric drift corrected hermite h rotation curve is used.8 Oh et al. Fig. A.1.—
Data:
Total intensity maps and velocity fields of IC 2574. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m,optical R and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 600 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 8 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from −
10 km s − to110 km s − with a spacing of 20 km s − . ark and luminous matter in THINGS dwarf galaxies 19 Fig. A.2.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of IC 2574. The open gray circles in allpanels indicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity fieldafter fixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotationcurve to the inclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in thebottom-middle panel. We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resultingrotation curves are indicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Harmonicanalysis:
Harmonic expansion of the velocity fields for IC 2574. The black dots and gray open circles indicate the results from the bulkand hermite h velocity fields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of thepotential measured using the bulk and hermite h velocity fields. Fig. A.3.—
Mass models of baryons:
Mass models for the gas and stellar components of IC 2574. (a):
Azimuthally averaged surfacebrightness profiles in the 3.6 µ m, R , V and B bands (top to bottom). (b)(f): The stellar mass-to-light values in the K and 3.6 µ m bandsderived from stellar population synthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellarcomponent. (e):
Optical colors. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotationvelocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves derived using different types ofvelocity fields (i.e., bulk, IWM, hermite h , single Gaussian and peak velocity fields as denoted in the panel) for IC 2574. This figure isthe same as the panel (a) of Fig. 2. Mass density profile:
The derived mass density profile of IC 2574. The open circles and tripod-likesymbols represent the mass density profiles derived from the bulk and hermite h rotation curves assuming minimum disk, respectively.The inner density slopes α are measured by least squares fits (dotted and dot-dashed lines) to the data points indicated by gray dots andlarger tripod-like symbols, and shown in the panel. ark and luminous matter in THINGS dwarf galaxies 21 Fig. A.4.—
Mass modeling results:
Disk-halo decomposition of the IC 2574 rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the bulk rotation curve, and the short and long dashedlines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermalhalo models (long dash-dotted lines) are denoted on each panel. TABLE 2Parameters of dark halo models for IC 2574
NFW halo (entire region) NFW halo ( < ⋆ assumption c V χ red. c V χ red. (1) (2) (3) (4) (5) (6) (7)Min. disk < . ) 674.6 ± ± ) 2.88 ( ) < . ± ... 3.39Min. disk+gas < . ) 524.3 ± ± ) 1.65 ( ) < . ± ... 2.32Max. disk < . ) 634.4 ± ... ( ± ) 2.33 ( ) < . ± ... 1.63Model Υ . ∗ disk < . ) 873.9 ± ... ( ± ) 1.81 ( ) < . ± ... 1.96Pseudo-isothermal halo (entire region) Pseudo-isothermal halo ( < ⋆ assumption R C ρ χ red. R C ρ χ red. (8) (9) (10) (11) (12) (13) (14)Min. disk 5.77 ± ± ± ± ± ± ± ± ± ± ... ± . ∗ disk 7.23 ± ± ± ± Note. − (1)(8): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk” uses the values derived from the populationsynthesis models in Section 4.1. (2)(5): Concentration parameter c of NFW halo model (NFW 1996, 1997). We also fit the NFWmodel to the rotation curves with only V as a free parameter after fixing c to 9. The corresponding best-fit V and χ red valuesare given in the brackets in (3) and (4), respectively. (3)(6): The rotation velocity (km s − ) at radius R where the densityconstrast exceeds 200 (Navarro et al. 1996). (4)(7)(11)(14): Reduced χ value. (9)(12): Fitted core-radius of pseudo-isothermalhalo model (kpc). (10)(13):
Fitted core-density of pseudo-isothermal halo model (10 − M ⊙ pc − ). ( ... ): blank due to unphysicallylarge value or not well-constrained uncertainties. ark and luminous matter in THINGS dwarf galaxies 23 Fig. A.5.—
Data:
Total intensity maps and velocity fields of NGC 2366. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m,optical V and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 1000 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 8 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from 30 km s − to180 km s − with a spacing of 20 km s − . Fig. A.6.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of NGC 2366. The open gray circles inall panels indicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocityfield after fixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotationcurve to the inclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in theright-middle panel. We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resultingrotation curves are indicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Harmonicanalysis:
Harmonic expansion of the velocity fields for NGC 2366. The black dots and gray open circles indicate the results from the bulkand hermite h velocity fields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of thepotential measured using the bulk and hermite h velocity fields. ark and luminous matter in THINGS dwarf galaxies 25 Fig. A.7.—
Mass models of baryons:
Mass models for the gas and stellar components of NGC 2366. (a):
The azimuthally averaged3.6 µ m surface brightness profile. (b)(f): The stellar mass-to-light values in the K and 3.6 µ m bands derived from stellar populationsynthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellar component. (e):
Optical color. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotation velocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves for NGC 2366. See Section 3.4 for a detailed discussion. Theserotation curves have also been discussed in detail Oh et al. (2008). Mass density profile:
The derived mass density profile of NGC 2366.The open circles represent the mass density profile derived from the bulk rotation curve assuming minimum disk. The inner density slope α is measured by a least squares fit (dotted line) to the data points indicated by gray dots, and shown in the panel. Fig. A.8.—
Mass modeling results:
Disk-halo decomposition of the NGC 2366 rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the bulk rotation curve, and the short and long dashedlines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermalhalo models (long dash-dotted lines) are denoted on each panel. ark and luminous matter in THINGS dwarf galaxies 27 TABLE 3Parameters of dark halo models for NGC 2366
NFW halo (entire region) NFW halo ( < ⋆ assumption c V χ red. c V χ red. (1) (2) (3) (4) (5) (6) (7)Min. disk < . ) 901.5 ± ± ) 1.72 ( ) < . ± ... < . ) 727.8 ± ... ( ± ) 1.08 ( ) < . ± ... < . ) 936.1 ± ... ( ± ) 0.89 ( ) < . ± ... . ∗ disk < . ) 630.7 ± ... ( ± ) 0.98 ( ) < . ± ... < ⋆ assumption R C ρ χ red. R C ρ χ red. (8) (9) (10) (11) (12) (13) (14)Min. disk 1.47 ± ± ± ± ± ± ± ± ± ± ± ± . ∗ disk 1.36 ± ± ± ± Note. − (1)(8): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk” uses the values derived from the populationsynthesis models in Section 4.1. (2)(5): Concentration parameter c of NFW halo model (NFW 1996, 1997). We also fit theNFW model to the rotation curves with only V as a free parameter after fixing c to 9. The corresponding best-fit V and χ red values are given in the brackets in (3) and (4), respectively. (3)(6): The rotation velocity (km s − ) at radius R wherethe density constrast exceeds 200 (Navarro et al. 1996). (4)(7)(11)(14): Reduced χ value. (9)(12): Fitted core-radius ofpseudo-isothermal halo model (kpc). (10)(13):
Fitted core-density of pseudo-isothermal halo model (10 − M ⊙ pc − ). ( ... ): blank due to unphysically large value or not well-constrained uncertainties. Fig. A.9.—
Data:
Total intensity maps and velocity fields of Ho I. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m, optical R and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 400 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 3 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from 120 km s − to160 km s − with a spacing of 10 km s − . ark and luminous matter in THINGS dwarf galaxies 29 Fig. A.10.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of Ho I. The open gray circles in all panelsindicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity field afterfixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotation curve to theinclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in the right-middle panel.We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resulting rotation curves areindicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Asymmetric drift correction:a:
Gray filled dots indicate the derived radial velocity correction for the asymmetric drift σ D . Black open and filled dots represent theuncorrected and corrected curves for the asymmetric drift, respectively. b: Azimuthally averaged H i velocity dispersion. c: Azimuthallyaveraged H i surface density. d: The dashed line indicates a fit to Σ σ with an analytical function. Harmonic analysis:
Harmonicexpansion of the velocity fields for Ho I. The black dots and gray open circles indicate the results from the bulk and hermite h velocityfields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of the potential measured usingthe bulk and hermite h velocity fields. Fig. A.11.—
Mass models of baryons:
Mass models for the gas and stellar components of Ho I. (a):
Azimuthally averaged surfacebrightness profiles in the 3.6 µ m, R , V and B bands (top to bottom). (b)(f): The stellar mass-to-light values in the K and 3.6 µ m bandsderived from stellar population synthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellarcomponent. (e):
Optical color. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotationvelocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves derived using different types ofvelocity fields (i.e., bulk, IWM, hermite h , single Gaussian and peak velocity fields as denoted in the panel) for Ho I. See Section 3.4 formore information. Mass density profile:
The derived mass density profile of Ho I. The open circles represent the mass density profilederived from the bulk rotation curve assuming minimum disk. The inner density slope α is measured by a least squares fit (dotted line) tothe data points indicated by gray dots, and shown in the panel. ark and luminous matter in THINGS dwarf galaxies 31 Fig. A.12.—
Mass modeling results:
Disk-halo decomposition of the Ho I rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the bulk rotation curve, and the short and long dashedlines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermalhalo models (long dash-dotted lines) are denoted on each panel. TABLE 4Parameters of dark halo models for Ho I
NFW haloΥ ⋆ assumption c V χ red. (1) (2) (3) (4)Min. disk 12.6 ± ) 28.2 ± ± ) 4.05 ( )Min. disk+gas 19.0 ± ) 18.6 ± ± ) 3.66 ( )Max. disk 34.9 ± ) 11.4 ± ± ) 8.03 ( )Model Υ . ∗ disk 20.8 ± ) 17.2 ± ± ) 4.00 ( )Pseudo-isothermal haloΥ ⋆ assumption R C ρ χ red. (5) (6) (7) (8)Min. disk 0.44 ± ± ± ± ± ± . ∗ disk 0.22 ± ± Note. − (1)(5): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk”uses the values derived from the population synthesis models in Section 4.1. (2): Con-centration parameter c of NFW halo model (NFW 1996, 1997). We also fit the NFWmodel to the rotation curves with only V as a free parameter after fixing c to 9.The corresponding best-fit V and χ red values are given in the brackets in (3) and(4), respectively. (3): The rotation velocity (km s − ) at radius R where the densityconstrast exceeds 200 (Navarro et al. 1996). (4)(8): Reduced χ value. (6): Fittedcore-radius of pseudo-isothermal halo model (kpc). (7):
Fitted core-density of pseudo-isothermal halo model (10 − M ⊙ pc − ). ( ... ): blank due to unphysically large value ornot well-constrained uncertainties. ark and luminous matter in THINGS dwarf galaxies 33 Fig. A.13.—
Data:
Total intensity maps and velocity fields of Ho II. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m,optical R and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 600 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 5 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from 100 km s − to200 km s − with a spacing of 15 km s − . Fig. A.14.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of Ho II. The open gray circles in all panelsindicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity field afterfixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotation curve to theinclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in the right-middle panel.We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resulting rotation curves areindicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Asymmetric drift correction:a:
Gray filled dots indicate the derived radial velocity correction for the asymmetric drift σ D . Black open and filled dots represent theuncorrected and corrected curves for the asymmetric drift, respectively. b: Azimuthally averaged H i velocity dispersion. c: Azimuthallyaveraged H i surface density. d: The dashed line indicates a fit to Σ σ with an analytical function. Harmonic analysis:
Harmonicexpansion of the velocity fields for Ho II. The black dots and gray open circles indicate the results from the bulk and hermite h velocityfields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of the potential measured usingthe bulk and hermite h velocity fields. ark and luminous matter in THINGS dwarf galaxies 35 Fig. A.15.—
Mass models of baryons:
Mass models for the gas and stellar components of Ho II. (a):
Azimuthally averaged surfacebrightness profiles in the 3.6 µ m, R , V and B bands (top to bottom). (b)(f): The stellar mass-to-light values in the K and 3.6 µ m bandsderived from stellar population synthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellarcomponent. (e):
Optical colors. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotationvelocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves derived from different types ofvelocity fields (i.e., bulk, IWM, hermite h , single Gaussian and peak velocity fields as denoted in the panel) and literature for Ho II. SeeSection 3.4 for more information. Mass density profile:
The derived mass density profile of Ho II. The open circles represent the massdensity profile derived from the bulk rotation curve assuming minimum disk. The inner density slope α is measured by a least squares fit(dotted line) to the data points indicated by gray dots, and shown in the panel. Fig. A.16.—
Mass modeling results:
Disk-halo decomposition of the Ho II rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the bulk rotation curve, and the short and long dashedlines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermalhalo models (long dash-dotted lines) are denoted on each panel. ark and luminous matter in THINGS dwarf galaxies 37 TABLE 5Parameters of dark halo models for Ho II
NFW haloΥ ⋆ assumption c V χ red. (1) (2) (3) (4)Min. disk 6.4 ± ) 36.7 ± ± ) 0.15 ( )Min. disk+gas 12.7 ± ) 20.4 ± ± ) 0.32 ( )Max. disk 48.3 ± ) 3.7 ± ± ) 0.30 ( )Model Υ . ∗ disk 11.5 ± ) 17.6 ± ± ) 0.22 ( )Pseudo-isothermal haloΥ ⋆ assumption R C ρ χ red. (5) (6) (7) (8)Min. disk 0.95 ± ± ± ± ± ... ± ... . ∗ disk 0.33 ± ± Note. − (1)(5): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk”uses the values derived from the population synthesis models in Section 4.1. (2): Con-centration parameter c of NFW halo model (NFW 1996, 1997). We also fit the NFWmodel to the rotation curves with only V as a free parameter after fixing c to 9.The corresponding best-fit V and χ red values are given in the brackets in (3) and(4), respectively. (3): The rotation velocity (km s − ) at radius R where the densityconstrast exceeds 200 (Navarro et al. 1996). (4)(8): Reduced χ value. (6): Fittedcore-radius of pseudo-isothermal halo model (kpc). (7):
Fitted core-density of pseudo-isothermal halo model (10 − M ⊙ pc − ). ( ... ): blank due to unphysically large value ornot well-constrained uncertainties. Fig. A.17.—
Data:
Total intensity maps and velocity fields of DDO 53. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m,optical R and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 350 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 3 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from −
10 km s − to40 km s − with a spacing of 15 km s − . ark and luminous matter in THINGS dwarf galaxies 39 Fig. A.18.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of DDO 53. The open gray circles in all panelsindicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity field afterfixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotation curve to theinclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in the right-middle panel.We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resulting rotation curves areindicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Asymmetric drift correction:a:
Gray filled dots indicate the derived radial velocity correction for the asymmetric drift σ D . Black open and filled dots represent theuncorrected and corrected curves for the asymmetric drift, respectively. b: Azimuthally averaged H i velocity dispersion. c: Azimuthallyaveraged H i surface density. d: The dashed line indicates a fit to Σ σ with an analytical function. Harmonic analysis:
Harmonicexpansion of the velocity fields for DDO 53. The black dots and gray open circles indicate the results from the bulk and hermite h velocityfields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of the potential measured usingthe bulk and hermite h velocity fields. Fig. A.19.—
Mass models of baryons:
Mass models for the gas and stellar components of DDO 53. (a):
Azimuthally averagedsurface brightness profiles in the 3.6 µ m, V and B bands (top to bottom). (b)(f): The stellar mass-to-light values in the K and 3.6 µ mbands derived from stellar population synthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellarcomponent. (e):
Optical color. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotationvelocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves derived using different types ofvelocity fields (i.e., bulk, IWM, hermite h , single Gaussian and peak velocity fields as denoted in the panel) for DDO 53. See Section 3.4for more information. Mass density profile:
The derived mass density profile of DDO 53. The open circles represent the mass densityprofile derived from the bulk rotation curve assuming minimum disk. The inner density slope α is measured by a least squares fit (dottedline) to the data points indicated by gray dots, and shown in the panel. ark and luminous matter in THINGS dwarf galaxies 41 Fig. A.20.—
Mass modeling results:
Disk-halo decomposition of the DDO 53 rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the bulk rotation curve, and the short and long dashedlines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermalhalo models (long dash-dotted lines) are denoted on each panel. TABLE 6Parameters of dark halo models for DDO 53
NFW halo (entire region) NFW halo ( < ⋆ assumption c V χ red. c V χ red. (1) (2) (3) (4) (5) (6) (7)Min. disk < . ) 852.2 ± ... ( ± ) 0.59 ( ) < . ± ... < . ) 576.6 ± ... ( ± ) 0.70 ( ) < . ± ... < . ) 276.4 ± ... ( ± ... ) 1.04 ( ) < . ± ... . ∗ disk < . ) 472.1 ± ... ( ± ) 0.75 ( ) < . ± ... < ⋆ assumption R C ρ χ red. R C ρ χ red. (8) (9) (10) (11) (12) (13) (14)Min. disk 0.85 ± ± ± ± ± ± ... ± ± ± ... ± . ∗ disk 2.11 ± ± ... ± Note. − (1)(8): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk” uses the values derived from the populationsynthesis models in Section 4.1. (2)(5): Concentration parameter c of NFW halo model (NFW 1996, 1997). We also fit theNFW model to the rotation curves with only V as a free parameter after fixing c to 9. The corresponding best-fit V and χ red values are given in the brackets in (3) and (4), respectively. (3)(6): The rotation velocity (km s − ) at radius R wherethe density constrast exceeds 200 (Navarro et al. 1996). (4)(7)(11)(14): Reduced χ value. (9)(12): Fitted core-radius ofpseudo-isothermal halo model (kpc). (10)(13):
Fitted core-density of pseudo-isothermal halo model (10 − M ⊙ pc − ). ( ... ): blank due to unphysically large value or not well-constrained uncertainties. ark and luminous matter in THINGS dwarf galaxies 43 Fig. A.21.—
Data:
Total intensity maps and velocity fields of M81dwB. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m,optical R and B bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 500 mJy beam − km s − . (e): Velocitydispersion map (moment 2). Velocity contours run from 0 to 25 km s − with a spacing of 5 km s − . (f): Position-velocity diagram takenalong the average position angle of the major axis as listed in Table 1. Contours start at +2 σ in steps of 3 σ . The dashed lines indicatethe systemic velocity and position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for theaverage inclination from the tilted-ring analysis as listed in Table 1. (g)(h)(i)(j)(k)(l): Velocity fields. Contours run from 320 km s − to380 km s − with a spacing of 5 km s − . Fig. A.22.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of M81dwB. The open gray circles in allpanels indicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity fieldafter fixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotation curveto the inclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in the right-middle panel. We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resulting rotationcurves are indicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Asymmetric driftcorrection: a:
Gray filled dots indicate the derived radial velocity correction for the asymmetric drift σ D . Black open and filled dotsrepresent the uncorrected and corrected curves for the asymmetric drift, respectively. b: Azimuthally averaged H i velocity dispersion. c: Azimuthally averaged H i surface density. d: The dashed line indicates a fit to Σ σ with an analytical function. Harmonic analysis:
Harmonic expansion of the velocity fields for M81dwB. The black dots and gray open circles indicate the results from the hermite h andIWM velocity fields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of the potentialmeasured using the hermite h and IWM velocity fields. ark and luminous matter in THINGS dwarf galaxies 45 Fig. A.23.—
Mass models of baryons:
Mass models for the gas and stellar components of M81dwB. (a):
Azimuthally averagedsurface brightness profiles in the 3.6 µ m, R , V and B bands (top to bottom). (b)(f): The stellar mass-to-light values in the K and 3.6 µ mbands derived from stellar population synthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellarcomponent. (e):
Optical colors. (g)(h):
The mass surface density (scaled by 1.4 to account for He and metals) and the resulting rotationvelocity for the gas component.
Comparison of rotation curves:
Comparison of the H i rotation curves derived using different types ofvelocity fields (i.e., bulk, IWM, hermite h , single Gaussian and peak velocity fields as denoted in the panel) for M81dwB. See Section 3.4for more information. Mass density profile:
The derived mass density profile of M81dwB. The open circles represent the mass densityprofile derived from the hermite h rotation curve assuming minimum disk. The inner density slope α is measured by a least squares fit(dotted line) to the data points indicated by gray dots, and shown in the panel. Fig. A.24.—
Mass modeling results:
Disk-halo decomposition of the M81dwB rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the hermite h rotation curve, and the short and longdashed lines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermal halo models (long dash-dotted lines) are denoted on each panel. ark and luminous matter in THINGS dwarf galaxies 47 TABLE 7Parameters of dark halo models for M81 dwB
NFW haloΥ ⋆ assumption c V χ red. (1) (2) (3) (4)Min. disk 0.83 ± ) 1003.0 ± ... ( ± ) 0.36 ( )Min. disk+gas 0.29 ± ) 1411.0 ± ... ( ± ) 0.25 ( )Max. disk < . ) 283.7 ± ... ( ± ) 0.10 ( )Model Υ . ∗ disk 0.52 ± ) 1001.3 ± ... ( ± ) 0.19 ( )Pseudo-isothermal haloΥ ⋆ assumption R C ρ χ red. (5) (6) (7) (8)Min. disk 0.31 ± ± ± ± ... ± . ∗ disk 0.30 ± ± Note. − (1)(5): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk” usesthe values derived from the population synthesis models in Section 4.1. (2): Concentra-tion parameter c of NFW halo model (NFW 1996, 1997). We also fit the NFW model tothe rotation curves with only V as a free parameter after fixing c to 9. The correspond-ing best-fit V and χ red values are given in the brackets in (3) and (4), respectively. (3): The rotation velocity (km s − ) at radius R where the density constrast exceeds200 (Navarro et al. 1996). (4)(8): Reduced χ value. (6): Fitted core-radius of pseudo-isothermal halo model (kpc). (7):
Fitted core-density of pseudo-isothermal halo model(10 − M ⊙ pc − ). ( ... ):):
Fitted core-density of pseudo-isothermal halo model(10 − M ⊙ pc − ). ( ... ):): blank due to unphysically large value or not well-constraineduncertainties. Fig. A.25.—
Data:
Total intensity maps and velocity fields of DDO 154. (a)(b)(c): : Total intensity maps in
Spitzer
IRAC 3.6 µ m, K and J bands. (d): Integrated H i map (moment 0). The gray-scale levels run from 0 to 450 mJy beam − km s − . (e): Velocity dispersion map(moment 2). Velocity contours run from 0 to 15 km s − with a spacing of 5 km s − . (f): Position-velocity diagram taken along the averageposition angle of the major axis as listed in Table 1. Contours start at +3 σ in steps of 5 σ . The dashed lines indicate the systemic velocityand position of the kinematic center derived in this paper. Overplotted is the bulk rotation curve corrected for the average inclination fromthe tilted-ring analysis as listed in Table 1. (g)(h)(i)(j): Velocity fields. Contours run from 320 km s − to 440 km s − with a spacing of20 km s − . ark and luminous matter in THINGS dwarf galaxies 49 Fig. A.26.—
Rotation curves:
The tilted ring model derived from the bulk velocity field of DDO 154. The open gray circles in allpanels indicate the fit made with all ring parameters free. The gray dots in the VROT panel were derived using the entire velocity fieldafter fixing other ring parameters to the values (black solid lines) as shown in the panels. To examine the sensitivity of the rotation curve tothe inclination, we vary the inclination by +10 and − ◦ as indicated by the gray solid and dashed lines, respectively, in the right-middlepanel. We derive the rotation curves using these inclinations while keeping other ring parameters the same. The resulting rotation curvesare indicated by gray solid (for +10 ◦ inclination) and dashed (for − ◦ inclination) lines in the VROT panel. Harmonic analysis:
Harmonic expansion of the velocity fields for DDO 154. The black dots and gray open circles indicate the results from the hermite h andIWM velocity fields, respectively. In the bottom-rightmost panel, the solid and dashed lines indicate global elongations of the potentialmeasured using the hermite h and IWM velocity fields. Fig. A.27.—
Mass models of baryons:
Mass models for the gas and stellar components of DDO 154. (a):
The azimuthally averaged3.6 µ m surface brightness profile. (b)(f): The stellar mass-to-light values in the K and 3.6 µ m bands derived from stellar populationsynthesis models. (c)(d): The mass surface density and the resulting rotation velocity for the stellar component. (g)(h):
The masssurface density (scaled by 1.4 to account for He and metals) and the resulting rotation velocity for the gas component.
Comparison ofrotation curves:
Comparison of the H i rotation curves for DDO 154. See Section 3.4 for a detailed discussion. These rotation curves havealso been discussed in detail de Blok et al. (2008). Mass density profile:
The derived mass density profile of DDO 154. The open circlesrepresent the mass density profile derived from the bulk rotation curve assuming minimum disk. The inner density slope α is measured bya least squares fit (dotted line) to the data points indicated by gray dots, and shown in the panel. ark and luminous matter in THINGS dwarf galaxies 51 Fig. A.28.—
Mass modeling results:
Disk-halo decomposition of the DDO 154 rotation curve under various Υ ⋆ assumptions (Υ . ⋆ ,maximum disk, minimumdisk + gas and minimum disks). The black dots indicate the hermite h rotation curve, and the short and longdashed lines show the rotation velocities of the stellar and gas components, respectively. The fitted parameters of NFW and pseudo-isothermal halo models (long dash-dotted lines) are denoted on each panel. See de Blok et al. (2008) for more details. TABLE 8Parameters of dark halo models for DDO 154
NFW haloΥ ⋆ assumption c V χ red. (1) (2) (3) (4)Min. disk 5.3 ± ) 58.3 ± ± ) 1.48 ( )Min. disk+gas 5.2 ± ) 51.9 ± ± ) 1.01 ( )Max. disk < . ) 655.1 ± ... ( ± ) 1.52 ( )Model Υ . ∗ disk 4.4 ± ) 58.7 ± ± ) 0.82 ( )Pseudo-isothermal haloΥ ⋆ assumption R C ρ χ red. (5) (6) (7) (8)Min. disk 1.30 ± ± ± ± ± ± . ∗ disk 1.33 ± ± Note. − (1)(5): The stellar mass-to-light ratio Υ ⋆ assumptions. “Model Υ . ⋆ disk” usesthe values derived from the population synthesis models in Section 4.1. (2): Concentra-tion parameter c of NFW halo model (NFW 1996, 1997). We also fit the NFW model tothe rotation curves with only V as a free parameter after fixing c to 9. The correspond-ing best-fit V and χ red values are given in the brackets in (3) and (4), respectively. (3): The rotation velocity (km s − ) at radius R where the density constrast exceeds200 (Navarro et al. 1996). (4)(8): Reduced χ value. (6): Fitted core-radius of pseudo-isothermal halo model (kpc). (7):
Fitted core-density of pseudo-isothermal halo model(10 − M ⊙ pc − ). ( ... ):):