A scaling-relation for disc galaxies: circular-velocity gradient versus central surface brightness
aa r X i v : . [ a s t r o - ph . C O ] A p r Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 2 June 2018 (MN L A TEX style file v2.2)
A scaling-relation for disc galaxies: circular-velocitygradient versus central surface brightness
Federico Lelli ⋆ , Filippo Fraternali , , Marc Verheijen Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV, Groningen, The Netherlands Department of Physics and Astronomy, University of Bologna, via Berti Pichat 6/2, 40127, Bologna, Italy
ABSTRACT
For disc galaxies, a close relation exists between the distribution of light and theshape of the rotation curve. We quantify this relation by measuring the inner circular-velocity gradient d R V (0) for spiral and irregular galaxies with high-quality rotationcurves. We find that d R V (0) correlates with the central surface brightness µ overmore than two orders of magnitude in d R V (0) and four orders of magnitudes in µ .This is a scaling-relation for disc galaxies. It shows that the central stellar density of agalaxy closely relates to the inner shape of the potential well, also for low-luminosityand low-surface-brightness galaxies that are expected to be dominated by dark matter. Key words: dark matter – galaxies: kinematics and dynamics – galaxies: structure
Scaling-relations are an ideal tool to investigate the struc-ture, the formation, and the evolution of galaxies. For discgalaxies, the Tully-Fisher (TF) relation (Tully & Fisher1977) is one of the best-studied scaling laws. It was originallyproposed as a correlation between the absolute magnitudeof a galaxy and the width of its global H I line profile. It isnow clear that the fundamental relation is between the totalbaryonic mass of the galaxy and the circular velocity alongthe flat part of the outer rotation curve ( V flat ), thought tobe set by the dark matter (DM) halo (e.g. McGaugh et al.2000; Verheijen 2001; Noordermeer & Verheijen 2007).While V flat is related to the total dynamical mass of agalaxy, the inner shape of the rotation curve provides in-formation on the steepness of the potential well. For discgalaxies (Sb and later types), the rotation curve is generallydescribed by an inner rising part (nearly solid-body) andan outer flat part (e.g. Bosma 1981; Begeman 1987; Swa-ters et al. 2009). For bulge-dominated galaxies, instead, therotation curve shows a very fast rise in the center, often fol-lowed by a decline and the flattening in the outer parts (e.g.Casertano & van Gorkom 1991; Noordermeer et al. 2007).The relation between the optical properties of a galaxyand the shape of its rotation curve has been debated formany years (e.g. Rubin et al. 1985; Corradi & Capaccioli1990; Persic & Salucci 1991). Several authors pointed outthat the shape of the luminosity profile and the shape ofthe rotation curve are closely related (Kent 1987; Caser-tano & van Gorkom 1991; Broeils 1992; Sancisi 2004; Swa-ters et al. 2009). In particular, de Blok & McGaugh (1996) ⋆ E-mail: [email protected] and Tully & Verheijen (1997) compared the properties oftwo disc galaxies on the same position of the TF relationbut with different central surface brightness, and found thathigh-surface-brightness (HSB) galaxies have steeply-risingrotation curves compared to low-surface-brightness (LSB)ones. Thus, for a given total luminosity (or V flat ), an ex-ponential light distribution with a shorter scale-length cor-responds to a steeper potential well (see also Amorisco &Bertin 2010). For HSB spirals, maximum-disc solutions canexplain the dynamics in the central regions with reasonablevalues of the stellar mass-to-light ratio M ∗ /L (e.g. van Al-bada & Sancisi 1986; Palunas & Williams 2000), suggestingthat either baryons dominate the gravitational potential ordark matter closely follows the distribution of light. Garridoet al. (2005) also found a clear trend between the inner slopeof the rotation curve and the central surface brightness of18 HSB spiral galaxies. For LSB galaxies, maximum-disc so-lutions can reproduce the inner parts of the rotation curves,but they often require high values of M ∗ /L that cannot beexplained by stellar population models (e.g. de Blok et al.2001; Swaters et al. 2011), leading to the interpretation thatLSB galaxies are dominated by DM at all radii. Finally, ga-laxies with a central “light excess” with respect to the ex-ponential disc (e.g. a bulge) show a corresponding “velocityexcess” in the inner parts of the rotation curve (e.g. M´arquez& Moles 1999; Swaters et al. 2009). This is usually referredto as the “Renzo’s rule” (Sancisi 2004): for any feature inthe luminosity profile of a galaxy there is a correspondingfeature in the rotation curve, and vice versa.In this Letter, we show that the inner circular-velocitygradient d R V (0) of a galaxy strongly correlates with thecentral surface brightness µ over more than two orders ofmagnitude in d R V (0) and four orders of magnitude in µ , c (cid:13) F. Lelli et al.
Table 1.
Galaxy sample. The entire table is published in the on-line version of the journal. The last column provides references for thedistance, the surface photometry, and the rotation curves: a) Jacobs et al. (2009); b) Swaters & Balcells (2002); c) Swaters et al. (2009);d) Saha et al. (2006); e) Kent (1987); f) Begeman (1987); g) Tully et al. (2009); h) Kent (1987); i) de Blok et al. (2008); j) Tully (1988);k) Noordermeer & van der Hulst (2007); l) Noordermeer et al. (2007). See Sect. 2 for details.Name Type Dist Method i µ ,R V max d R V (0) R m χ ν Ref.Mpc ◦ mag/ ′′ km/s km/s/kpc kpc(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)UGC 7559 IBm 5.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
72 1.7 3 0.57 g, h, iUGC 11670 S0/a 14.2 ± ± ± ± ± thereby extending and firmly establishing the correlationhinted at by Fig. 8 of Garrido et al. (2005). This is a scaling-relation for disc galaxies. We discuss the implications of thisrelation for the stellar and DM properties of galaxies. We define the circular-velocity gradient d R V (0) as the in-ner slope of a galaxy rotation curve, i.e. dV /dR for R → d R V (0) can be estimated if the rising part of the rotationcurve is well-sampled, but this requires high-quality dataand a careful modelling of the gas kinematics in the innerparts. To minimize the uncertainties, we use four samplesof galaxies with high-quality rotation curves: Swaters et al.(2009) (hereafter S09), de Blok et al. (2008) (hereafter dB08or THINGS), Verheijen & Sancisi (2001) (hereafter VS01),and Begeman (1987) (hereafter B87). The rotation curveswere derived using interferometric H I observations and cor-rected for beam-smearing effects. We select only galaxiesviewed at inclination angles i between 40 ◦ and 80 ◦ , as therotation velocities of face-on discs require a large correctionfor i , while the observed rotation curves of edge-on discs maybe affected by unseen holes in the central H I distribution.We projected each derived rotation curve onto the corre-sponding position-velocity diagram to verify that they havebeen properly determined. We exclude the galaxies from S09and VS01 with low-quality data ( q >
2, see S09). The spiralNGC 3521 (from dB08) is close to edge-on in the inner re-gions and we neglect the two innermost velocity points. Fivegalaxies are in common between B87 and dB08. We use thenew rotation curves from THINGS except for two galaxies(NGC 2903 and NGC 3198), as the inner parts of their rota-tion curves are better traced by B87, who applied a carefulbeam-smearing correction (see Figs. 9 and 12 of dB08). Thesample of S09 has one object (NGC 2366) in common withdB08 and another one (UGC 6446) in common with VS01;we use the rotation curves from S09. In all these cases thedifferences in d R V (0) are, however, within a factor two. Thefinal sample comprises 63 galaxies with morphological typesranging from Sab to Sd/Im.To this high-quality sample, we add 11 rotation curvesof S0/Sa galaxies (with 40 ◦ i ◦ ) from Noordermeeret al. (2007) (hereafter N07). Since early-type galaxies usu-ally lack H I emission in their central regions, the rotationcurves were derived combining H α long-slit spectroscopy (for the inner parts) with H I observations (for the outer parts).We exclude UGC 12043 as the H α observations of this galaxyhave very low velocity resolution. The values of d R V (0) forearly-type galaxies are more uncertain than those for late-type galaxies.We derive d R V (0) by fitting the inner rotation curvewith a polynomial function of the form V ( R ) = m X n=1 a n × R n (1)and consider the linear term a = lim R → dV /dR = d R V (0).The fit is error-weighted and constrained to pass through V = 0 at R = 0. The value of a depends on i) the radialrange used in the fit, and ii) the order of the polynomial m .We define R as the radius where the rotation curve reaches90% of its maximum velocity, and fit only the points within R . This choice allows us to maximize the number of pointsalong the rising part of the rotation curve without includingpoints along the flat part. Rotation curves with less then 3points within R are excluded, as they are not well-resolvedin the inner parts. 16 galaxies from the high-quality sam-ple and 3 galaxies from N07 are excluded by this criterion,thus the high-quality and total samples reduce to 47 and 55objects, respectively. For a pure exponential disc with scale-length R d , R ≃ . R d . Thus, the first fitted point of the ro-tation curve is typically at R . . R d and the derived valueof d R V (0) is representative of the innermost galaxy regionsthat are accessible by the available rotation curves. To derivethe best-fit order of the polynomial, we proceed as follows.We start with a linear fit ( m = 1) and progressively increase m until the reduced- χ ( χ ν ) approaches 1. In practice, weminimize the function P χ ( χ ; ν ) − .
5, where P χ ( χ ; ν ) is theintegral probability of χ and ν is the number of degrees offreedom; the procedure is halted in case χ ν would drop below1. Visual inspection showed that this method works betterthan the F-test (e.g. Bevington & Robinson 2003), that insome cases returns high values of m and thus increases thenumber of free parameters in the fit.We test our automatic procedure on a set of model ro-tation curves, calculated by summing the contributions ofa disc, a bulge, and a DM halo. We add typical errors tothe velocity points ( ∼ − ) and try several spatial sam-plings. We find that, even if the rotation curve is poorly-sampled ( ∼ R ), the actual value of d R V (0)can be recovered with a error of ∼ d R V (0) may be under-estimated by a factor ∼ c (cid:13) , 000–000 scaling-relation for disc galaxies C i r c u l a r V e l o c i t y ( k m / s ) Radius (kpc)
UGC 7559 (m = 1) C i r c u l a r V e l o c i t y ( k m / s ) Radius (kpc)
NGC 3198 (m = 2) C i r c u l a r V e l o c i t y ( k m / s ) Radius (kpc)
NGC 5055 (m = 3) C i r c u l a r V e l o c i t y ( k m / s ) Radius (kpc)
UGC 11670 (m = 5)
Figure 1.
Results of the polynomial-fit for four representative ga-laxies. Filled circles show the points of the rotation curve within R , while open circles show the points excluded in the fit. Thesolid, red line shows the fitted polynomial function, while thedashed, blue line shows its linear term. The order m of the poly-nomial is indicated. See Sect. 2.1 for details. laxies that require polynomial fits of different orders. Late-type galaxies (Sb to Im) are generally well-fitted by poly-nomials with m = 1 (e.g. UGC 7559) or m = 2 (e.g.NGC 3198), but several cases do require m > m >
4, e.g. UGC 11670), as their ro-tation curves may have complex shapes characterized by asteeply-rising part followed by a decline and a second rise.For some bulge-dominated galaxies from N07, the value of d R V (0) is rather uncertain, since there may be no datapoints in the inner radial range where the linear term a is representative of the rotation curve (see Fig. 1, bottom-right). Table 1 provides the fit results for the galaxies inFig. 1. The results for the entire galaxy sample are providedin the on-line version of this Table.The error δ d R V (0) on d R V (0) is estimated as δ d R V (0) = s δ a + (cid:18) d R V (0) δ i tan( i ) (cid:19) + (cid:18) d R V (0) δ D D (cid:19) (2)where δ a is the nominal error on the fitted linear term a , δ i is the error on the disc inclination i , and δ D is the erroron the galaxy distance D . δ D typically gives a negligiblecontribution for galaxies with distances derived using the tipof the red giant branch (TRGB) and/or Cepheids (Ceph),whereas it can dominate the error budget for galaxies withdistances estimated from the TF relation or the Hubble flow. For the high-quality sample of disc-dominated galaxies, weconsider two ways to estimate the central surface brightness:i) the disc central surface brightness µ d , obtained from anexponential fit to the outer parts of the luminosity profile; and ii) the observed central surface brightness µ , obtainedfrom a linear extrapolation of the luminosity profile in theinner few arcseconds to R = 0 (see Swaters & Balcells 2002). µ takes into account possible deviations from a pure expo-nential disc. This may carry valuable information on themass distribution, e.g. if a pseudo-bulge/bar is present, butmay also reflect variations in the stellar populations and/orin the internal extinction, e.g. if the star-formation activ-ity is enhanced in the central parts. We use the observedcentral surface brightness µ . Since we are considering disc-dominated galaxies (Sb and later types), we correct µ forinclination; we assume an optically-thin disc. Given the am-biguity in using either µ or µ d , we include the difference∆ µ = µ d − µ in the error δ µ . This is estimated as δ µ = p (∆ µ/ + [2 . e ) tan( i ) δ i ] . (3)For the galaxies from S09, we use the values listed inTable A.5 of Swaters & Balcells (2002) (Harris R -band). Forthe galaxies from VS01, we use the surface photometry fromTully et al. (1996) (Cousins R -band). For the galaxies fromdB08 and B87, we use the surface photometry from threedifferent sources (in order of preference): Swaters & Balcells(2002) (Harris R -band), Kent (1987) ( r -band), and Mu˜noz-Mateos et al. (2009) (Harris R -band or r ′ -band). The opticalfilters are comparable, but there can be systematic differ-ences of ∼ R -band photometry available andhave been excluded, reducing the total sample to 52 objects.The S0/Sa galaxies from N07 require a different ap-proach, because i) the surface brightness rapidly increasesin the central regions due to the presence of a dominantbulge; and ii) several galaxies are at large distances ( & &
150 kpc). For these galaxies, Noordermeer& van der Hulst (2007) provide the R -band disc central sur-face brightness µ d , extrapolated from an exponential fit andcorrected for i , and the bulge central surface brightness µ b ,extrapolated from a Sersic fit to the inner parts after sub-tracting the disc contribution. We estimate µ by summingthe contributions of µ d and µ b ; the latter value is not cor-rected for i as the bulge is assumed to be spherical. Theerrors are given by Eq. 3, where ∆ µ is now the differencebetween µ and the innermost value of µ observed. d R V (0) − µ SCALING RELATION
In Fig. 2 (left), we plot µ against d R V (0) for the totalsample of 52 galaxies. There is a clear, striking relation. Alinear, error-weighted fit to the data yieldslog[ d R V (0)] = ( − . ± . µ + (6 . ± . . (4)As discussed in Sect. 2, the values of d R V (0) for the S0/Sagalaxies from N07 are uncertain. However, it is clear thatthese bulge-dominated galaxies follow the same trend de-fined by disc-dominated ones. Fig. 2 (left) also shows a linearfit excluding the objects from N07 (dashed line). This givesonly slightly different values of the slope ( − . ± .
03) andthe intersect (5 . ± . c (cid:13) , 000–000 F. Lelli et al. C i r c u l a r- V e l o c i t y G r ad i en t ( k m s - k p c - ) Central Surface Brightness (R mag arcsec -2 )S09dB08VS01B87N07 101001000 812162024Central Surface Brightness (R mag arcsec -2 ) ∆ µ > 1.50.5 ≤ ∆ µ ≤ ∆ µ < 0.5 ∆ µ ≤ -0.5 Figure 2.
The circular-velocity gradient versus the central surface brightness. The solid and dashed lines show a liner fit to the data-points for the total and high-quality samples, respectively.
Left: galaxies coded by the reference for the rotation curve (S09: Swaters et al.2009, dB08: de Blok et al. 2008, VS01: Verheijen & Sancisi 2001, B87: Begeman 1987, N07: Noordermeer et al. 2007).
Right: galaxiescoded by the value of ∆ µ = µ d − µ (in R mag arcsec − ), that quantifies the “light excess” over an exponential profile. than two orders of magnitude in d R V (0) and four orders ofmagnitude in µ .The values of the slope and the intersect are likely moreuncertain than the formal errors, due to several effects inthe determination of d R V (0) and µ . Possible concerns arei) the different linear resolutions (in kpc) of the H I andoptical observations, and ii) the effects on µ of internalextinction, recent star-formation, and/or a LINER core. Weperformed several fits using different methods to estimate µ and d R V (0), such as calculating V /R at the innermost pointof the rotation curve. We obtained slopes always between − .
25 and − .
15, and we think that the actual slope mustbe constrained between these values.The scatter around the relation is largely due to obser-vational uncertainties on d R V (0). Major sources of uncer-tainties are i) the galaxy distance, ii) the inclination, andiii) the innermost points of the rotation curve (see Eq. 2).However, part of the scatter is likely to be intrinsic anddue to differences in the 3-dimensional (3D) distribution ofbaryons and in the structural component that defines µ (adisc, a bulge, a bar, or a nuclear star cluster).To investigate the role played by different structuralcomponents, in Fig. 2 (right) we plot the same data-pointscoding the galaxies by the value of ∆ µ = µ d − µ . Thisquantifies the deviation from an exponential law in the innerparts of the luminosity profile (in R mag arcsec − ). We dis-tinguish between four cases: i) galaxies dominated by a bulge(∆ µ > . . ∆ µ .
5) like a pseudo-bulge or a bar, iii)galaxies with an exponential disc ( − . < ∆ µ < . µ − . µ &
18 mag arcsec − )is populated by bulge-dominated galaxies. It is clear that,for these galaxies, the use of µ d instead of µ would shiftthem away from the relation, as µ d . − (the “Freeman value”, Freeman 1970). On the lower-left end of the relation, instead, one can find both pure ex-ponential discs and galaxies with central light concentra-tions/depressions. For these disc-dominated galaxies, the useof µ d instead of µ would still lead to a correlation, but thiswould have a steeper slope ( ∼ -0.25). For galaxies with si-milar values of µ , d R V (0) does not seem to depend on thedetailed shape of the luminosity profile (simple exponentialor with a central light depression/concentration). The correlation between the central surface brightness µ and the circular-velocity gradient d R V (0) implies that thereis a close link between the stellar density and the gravita-tional potential in the central parts of galaxies. This holdsfor both HSB and LSB objects, covering a wide range ofmasses and asymptotic velocities (20 . V flat .
300 km s − ).The relation between the distribution of light and thedistribution of mass has been extensively discussed in thepast (see Sancisi 2004 and references therein). However,only few attempts have been made to parametrize this re-lation, notably by Swaters et al. (2009) (S09). Fig. 10 ofS09 plots the logaritmic-slope between 1 and 2 disc scale-lenghts S , = log[ V (2 h ) /V ( h )] / log(2) versus the “light ex-cess” with respect to an exponential disc ∆ µ = µ d − µ . Itshows that a “light excess” (a bulge-like component) cor-responds to a “velocity excess” in the rotation curve withrespect to the expectations for the underlying exponentialdisc. The limitation of that parametrization is that it doesnot capture the dynamical difference between HSB and LSBdiscs, that are known to have steeply-rising and slowly-risingrotation curves, respectively (e.g. Tully & Verheijen 1997).In Fig. 10 of S09, indeed, both HSB and LSB exponentialdiscs have ∆ µ ≃ S , ≃ .
5. The latter result is due tothe fact that S , is, by definition, a scale-invariant quantity, c (cid:13) , 000–000 scaling-relation for disc galaxies that is not expected to depend on µ or V max . In contrast, d R V (0) measures the inner slope of the rotation curve inphysical units (km s − kpc − ) and is directly related to thecentral dynamical surface density (in M ⊙ pc − ), providinginsights in the underlying physics, as we now discuss.For a 3D distribution of mass, the rotation velocity V ofa test particle at radius R is given, to a first approximation,by V R = α GM dyn R (5)where G is Newton’s constant, M dyn = 4 / πR ρ dyn is thedynamical mass within R , and α is a factor that depends onthe detailed mass distribution (for a spherical distributionof mass α =1, while for a thin exponential disk α ≃ .
76 at R = 0 . R ). For R →
0, we have dVdR = VR = p βGρ dyn , = r βG ρ bar , f bar , (6)where β = 4 / πα , ρ dyn , and ρ bar , are, respectively, thecentral dynamical and baryonic mass densities, and f bar , = ρ bar , /ρ dyn , is the baryon fraction in the central regions.Note that f bar , may strongly differ from the “cosmic”baryon fraction, and can vary widely from galaxy to galaxy,depending on the formation and evolution history. Observa-tionally, we measure µ which is related to ρ bar , by µ = − . ρ bar , ∆ z ( M bar /L ) − ] (7)where ∆ z is the typical thickness of the stellar component(either a disk or a bulge) and M bar /L is the baryonic mass-to-light ratio, including molecules and other dark baryoniccomponents. Thus, we expect the following relationlog[ d R V (0)] = − . µ + 0 . (cid:18) βG M bar /L ∆ z f bar , (cid:19) . (8)In Sect. 3, we mentioned that the slope of our rela-tion is not well-determined due to several uncertainties inthe measurements of d R V (0) and µ . However, it is con-sistent with − . − . − .
25. Were the slope exactly − .
2, the second term ofEq. 8 would be a constant, implying a puzzling fine-tuningbetween the 3D distribution of baryons ( β and ∆ z ), thebaryonic mass-to-light ratio ( M bar /L ), and the dark mattercontent ( f bar , ).Despite the uncertain value of the slope, the results pre-sented here show a clear relation between the central stellardensity in a galaxy and the steepness of the potential well(see also Sancisi 2004; Swaters et al. 2011). This impliesa close link between the density of the baryons, regulatedby gas accretion, star-formation, and feedback mechanisms,and the central density of the DM halo, together shapingthe inner potential well. This may represent a challenge formodels of galaxy formation and evolution. Future observa-tional studies may help to better constrain the slope of therelation, while theoretical work should aim to understandits origin. We measured the circular-velocity gradient d R V (0) for asample of spiral and irregular galaxies with high-quality rotation curves. We found a linear relation betweenlog[ d R V (0)] and the central surface brightness µ witha slope of about − .
2. This is a scaling-relation for discgalaxies that holds for objects of very different morpholo-gies, luminosities, and sizes, ranging from dwarf irregularsto bulge-dominated spirals. This relation quantifies thecoupling between visible and dynamical mass in the centralparts of galaxies, and shows that the central stellar densityclosely relates to the inner shape of the potential well.
Acknowledgements:
We are grateful to Renzo Sancisifor stimulating discussions and insights. We also thankErwin de Blok and Rob Swaters for providing us with thehigh-quality rotation curves.
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