Surface Brightness and Intrinsic Luminosity of Ellipticals
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 15 August 2018 (MN L A TEX style file v2.2)
Surface Brightness and Intrinsic Luminosity of Ellipticals
Barun Kumar Dhar ∗ , Liliya L.R. Williams School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455 USA
Accepted 12th December, 2011; in original form 2nd June, 2011.
ABSTRACT
We show that the surface brightness profiles of elliptical galaxies can be parametrized usinga linear superposition of two or three components, each of which is described by functionsdeveloped in Dhar & Williams as the 2D projections of a 3D Einasto density profile. Fora sample of 23 ellipticals in and around the Virgo Cluster with total absolute V -magnitude −
Gerard de Vaucouleurs (1948) showed that a remarkably simpleparametrization of the radial surface brightness profile of galaxiesexists for a wide range of ellipticals. He proposed a two parameterfitting function of the form of equation (1.1) with m = q = effective ∗ E-mail:[email protected] or half-light radius , R E , contains half the total projected light. Σ( R ) = Σ R E exp ( − q "(cid:18) RR E (cid:19) m − (1.1)where, Σ( R ) is the 2D surface brightness at a plane of sky projectedradial distance R , Σ R E =Σ( R E ) , q = q ( m ) and Σ S (0)=Σ R E e q .Observing that our Galaxy is made up of multiple subsystemsEinasto (1965) proposed a modification, equation (1.2), of the 2Dde Vaucouleurs law to model the intrinsic (3D) baryonic mass den-sity ρ ( r ) of each subsystem by allowing the shape parameter n to c (cid:13) Barun Kumar Dhar and Liliya L.R. Williams be a free parameter, and b = b ( n ) . ρ ( r ) = ρ s exp ( − b "(cid:18) rr s (cid:19) n − (1.2)where, ρ s is the density(3D) at a scale radius r s and ρ (0)= ρ s e b .Around the same time Sersic (1968) observed that m in equa-tion (1.1), characterized by m = 4 in de Vaucouleurs law, was notthe same for all galaxies and rendering it a free-parameter providedmuch better fits to the surface brightness profiles (hereafter, SB).Equation (1.1) is the current standard paradigm for describing theglobal structural properties of galaxies.Over the past forty years, the pioneering works of Caon, Ca-paccioli, Einasto, Ferrarese, Graham, Kormendy, Lauer and theircollaborators have shown that no single commonly used three pa-rameter fitting function could model the SB over the entire dynamicradial range. They showed that the SB profiles of all galaxies re-veal an inherently multi-component structure such that the outer re-gions can be described with a Sersic or a Sersic+exponential mod-els while the central regions can be described with power-laws.Additionally, for some nearby spirals and the giant ellipticalM87 (NGC4486), Einasto and collaborators (Rummel, Haud andTenjes) have shown that if spectroscopic and kinematic constraintsare used in addition to the SB data, then the intrinsic 3D mass den-sity including their central regions can be described with a multi-component Einasto model. While the power-law+Sersic models are widely accepted as pro-viding an accurate description of the SB profiles, the fit residualsof these models are often larger than measurement errors. Further,these functions do not have the flexibility to model deviations frompower-laws within the central regions. Since these fits are oftenused to draw inferences on galaxy structure, formation and evo-lution, it is important to address two crucial issues when modellinggalaxy structure:(i) Models of the SB profiles must be consistent with measure-ment errors over the entire available radial range; and(ii) Model fitting functions must allow one to easily infer theintrinsic 3D luminosity density from the 2D SB profiles.Our goal in this paper is to address the above issues with anew function derived in Dhar & Williams (2010) (DW10) to modelthe projected surface mass density profile of Einasto-like systems.We show that a multi-component parametrization with this function(hereafter, the DW-function) provides excellent fits, consistent withmeasurement errors, to the SB of ellipticals over a large dynamicradial range down to the HST resolution limit.This suggests that the 3D density distribution of light in galax-ies can be described with a multi-component Einasto model.Such an interpretation is similar to that of Einasto and collab-orators, except that a) our work extends to a much larger sample ofshallow and steep cusp ellipticals; and b) we deduce the intrinsic3D Einasto model properties by fitting the 2D SB using a multi-component DW-function.For the rest of this paper, we draw a distinction between com-ponents deduced from fitting and physically distinct kinematic sys-tems or stellar populations. We shall refer to a component as a sin-gle spherically symmetric closed form fitting function described byat most three parameters – a scale length, r α , a shape parameter, α ,and some normalization, Σ or Σ α . Our multi-component fits con-sist of the minimum number of such functions (i.e. DW-functions) required to model the entire dynamic radial range down to the res-olution limit of the HST. The minimum number depends on thequality of data, the available degrees of freedom, and the amplitudeof systematic patterns in the residuals.One should hence use caution in interpreting these fit compo-nents as kinematically distinct systems or stellar populations sincethe physical properties of the components, and even their presencewill depend on the choice of parametrization. It is however possiblethat some of our fitted components do coincide with kinematicallyidentified systems or stellar populations, or combinations thereof,which shall then facilitate interesting interpretations.Also note that neither fit components nor physically distinctsystems have to correspond to structure in the total gravitational po-tential. Only where ρ baryons ≫ ρ DM , or ρ baryons ∝ ρ DM featuresobserved in the SB profiles will trace the total mass density, andhence will provide information about the gravitational potential.Further, since no galaxy has a truly flat density core, we refrainfrom using the terms ’core-’ and ’cusp-’ galaxies and instead referto them as shallow-cusp and steep-cusp galaxies, respectively. In section 2 we provide a brief history and basis for believing thatgalaxies have a multi-component structure. This is followed by anoverview, in section 3, of the most flexible fitting functions triedto date, and the motivation for this work. The data we use are de-scribed in section 4 and important features of our 2D parametriza-tion and the fitting procedure are discussed in section 5. Section6 discusses results of our 2D fits to a sample of 23 ellipticals inand around the Virgo cluster spanning absolute V-magnitudes inthe range -24 Navarro et al. (2004) (N+04) showed that -parameter fitting func-tions, especially those with a power-law logarithmic slope like theEinasto profile, are able to describe the 3D mass density distri- c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams bution of spherically averaged Λ CDM N-body haloes better thanany of the -parameter fitting functions tried to date. Subsequentsimulations by Merritt et al. (2006) (M+06), Prada et al. (2006),Gao et al. (2008), Hayashi & White (2008), Stadel et al. (2009),Navarro et al. (2010) have confirmed the need for a -parameterfunction and for over 30 such dwarf, galaxy and cluster size N-body haloes the Einasto profile seems to be the best performingfunction in comparison to other -parameter fitting functions. TheEinasto index deduced from these simulations are typically in therange 5 . n . 8. Stadel et al. (2009) proposed a -parameter func-tion that provides fractionally better fit in terms of rms than theEinasto profile for two haloes they simulated. However, even forthese cases the Einasto profile has comparable residuals.Note that current resolution of N-body simulations does not al-low one to probe the very central regions of galaxies where baryonsreveal a multi-component structure. One may thus conclude thatwithin the resolved and converged domain of N-body simulations,the dark matter distribution can be described as a single componentsystem. Noting the similarity between the functional form of the Einastoand the Sersic profiles, Merritt et al. (2005) obtained sphericallyaveraged non-parametric estimates of the 3D intrinsic and 2D pro-jected mass densities of the N-body haloes described in N+04. Theyfound that the same fitting function, the Sersic profile, which is usedto describe the SB of ellipticals also describes the surface densitiesof Λ CDM N-body haloes, whose intrinsic 3D density is best de-scribed by a function of similar form - the Einasto profile.Dhar & Williams (2010) show that while it is possible to findlimited radial ranges over which a Sersic profile can approximate aprojected Einasto profile, over a large radial range a Sersic profile isnot a good representation of a projected Einasto profile, and usingsuch fits can lead to a misinterpretation of the best fit parameters.DW10 point out that the fitted Sersic profile parameters dependstrongly on the radial range of a projected Einasto profile. In otherwords, fits with a Sersic profile to a single projected Einasto profileimplies the existence of a variable Sersic shape parameter. DW10provide an accurate analytical approximation for the 2D projectionof an Einasto profile in terms of the 3D Einasto profile parameters,thereby allowing one to explore the intrinsic properties of systemsbelieved to be Einasto-like from 2D observations of those systems. With the ability to resolve the central regions of galaxies withthe HST, it was found that a single -parameter fitting function,like the Sersic profile, is not able to model the SB profile over alarge dynamic radial range down to the HST limit; less so with -parameter functions, for example, the de Vaucouleurs and the Jaffeprofiles. Ferrarese et al. (1994) thus proposed the -parameter dou-ble power-law while Lauer et al. (1995) proposed the -parameterNuker profile - which is a modified double power-law with an addi-tional parameter to control the sharpness of transition between thepower-laws.Lauer et al. (1995) pointed out that the more flexible -parameter Nuker profile is designed to model only the small cen-tral regions, ∼ -parameter Sersic profile.This is because power-laws assign a fixed logarithmic slope to thedensity distribution while the light of galaxies at large R exhibit avariable slope. However, even with 5 parameters modelling a smallradial range it can be seen that the central-most regions show resid-uals larger than measurement errors. This implies that a total of8 parameters are required for a near complete description exclud-ing the very central inner regions. Moreover, in the central regionsmany galaxies exhibit a sharp change in slope, i.e. a transition ra-dius. Even though the presence of the transition is unambiguous,the domain over which one should fit the Nuker profile to obtainrobust parameters is not always obvious.Graham et al. (2003) showed that the best-fitting Nuker pa-rameters are extremely sensitive to the chosen domain of fit. Hence,observing that the central regions can be modelled as power-lawswhile the outer regions require a Sersic profile, they proposed the -parameter Core-Sersic profile which has an inner power-law cou-pled to an outer Sersic profile along with a parameter to controlthe sharpness of transition. Trujillo et al. (2004) suggested a mod-ification of the Core-Sersic profile by allowing an infinitely sharptransition along with a step-function that reduces the Core-Sersicprofile to a -parameter function, but as expected produces an un-physically sharp break in the profile quite unlike the much smoothertransition observed in galaxies. Even then, the large fit residuals inthe central region continue to exist.Ferrarese et al. (2006) found that the light excess in the cen-tral regions (often referred to as ’nuclei’) of the steep-cusp galaxiesin their sample could be best described with a -parameter Kingmodel. Such ’nuclei’ are not as rare as was originally believed;but may not always be very prominent. Cote et al. (2006) showedthat in the ACS Virgo Cluster survey (Cote et al. 2004), hereafterACSVCS, 66-82 per cent of the galaxies have such a central fea-ture. It thus appears that one needs a King model for the nuclearregion, and either a Sersic, a Core-Sersic or a Nuker+Sersic profileto model the rest of the galaxy, i.e., a total of 6-11 parameters.While the overall rms of the fits with 6-11 parameters appearsmall, the residuals in many radial sections remain large, & & mag arcsec − ; considerably larger than the0.01-0.05 mag arcsec − uncertainty of HST quality data. For il-lustration we refer the reader to the fits in Ferrarese et al. (2006)using a combination of Sersic, Core-Sersic and King models wherelarge, and sometimes divergent, residuals can be found in someregions. A comparison of fits with Core-Sersic, King and Nukerprofiles in the central regions of some galaxies are also shown inLauer et al. (2007), clearly revealing the lack of a good fit in thecentral-most regions. More detailed discussion is provided in § R → . While this may seem to be of academicinterest, since the density at R = R = All of the above power-law and Core-Sersic parametrizations arewell-guessed but ad-hoc empirical fitting functions in 2D, in thesense that they are not a result of well established theoretical mod-els of galaxy formation. However, despite the existence of residualslarger than measurement errors, the parameters of the fits are usedto draw inferences on galaxy structure and evolution, which are in- c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals trinsically 3D phenomena. This could have been meaningful if the2D models were deduced from an underlying physically motivated3D distribution. In order to draw such inferences from functionsthat are merely fitting functions, one needs, at the very least, tohave residual profiles consistently comparable to measurement er-rors over a large dynamic radial range, and not just a low rms .Kormendy et al. (2009), hereafter KFCB09, addresses this is-sue partly by fitting a single modified exponential function, the Ser-sic profile, over a rigorously tested range where the function pro-duces residuals comparable to measurement errors. However, thisrange is chosen to ensure that the Sersic profile produces a good fitand specifically excludes the entire domain within the break radius.Since the Sersic profile is an ad-hoc fitting function, parameters de-duced from fits within a limited radial range can lead to misleadinginterpretations of the physics involved and in estimating propertiesof the region inside the break radius.Since the Sersic profile produces good fits over a large radialrange and lenticulars have been modelled with a Sersic + exponen-tial profile, a plausible alternative is to use a double-Sersic profileto model the entire galaxy. This approach had not received muchattention until recently when Gonzalez, Zabludoff & Zaritsky(2003) showed that a double-Sersic profile provides much betterfits to twelve Brightest Cluster Galaxies (BCG). An example ofsuch a fit for the BCG in Abell 2984 is shown in their figure 2.Cote et al. (2007) also arrive at a similar conclusion for some galax-ies in the ACSVCS and Fornax cluster surveys. A comprehensivestudy of fits with a double-Sersic profile is provided in Hopkins etal. (2009a,b) where they obtain low residuals over a large dynamicradial range. While they do not provide a residual profile, the rms of their fits are often larger than the measurement errors.From a mathematical stand point, the Sersic profile presentsanother difficulty: its deprojection, that can provide insightsin to the 3D structure of the galaxy, is not very well an-alytically tractable. Asymptotic limits of deprojection can befound in Ciotti (1991) and approximate expressions are givenin Prugniel & Simien (1997) and Lima Neto, Gerbal & Marquez(1999) (hereafter, PS97 and LGM99). However, the PS97 andLGM99 approximations are not accurate at small R − R E , andthe 3D density becomes undefined as r → for Sersic indices m> (Ciotti 1991), while all galaxies observed to date have m> .Baes & Gentile (2011) (BG11) provide an exact analyticalexpression for the deprojection of Sersic profiles for all m in termsof the Fox H function, which is extremely difficult, if not impossi-ble to compute, even numerically. However, for rational values of m they show that the deprojection can be expressed in terms of theMeijer G function. Rational m requirement is not too stringent be-cause for practical purposes any m can be well approximated by arational number. The singularity in the deprojected central densityfor m > is, however, inherent to the form of the Sersic function.We also note that the deprojections discussed above are assumed toextend to infinite 3D radius. However, just because a deprojectionis analytically difficult, does not, by any means, suggest that thetrue 2D light distribution of galaxies is not described by a Sersicprofile, and the 3D distribution is not a deprojected Sersic profile.What might suggest that the Sersic profile is not an optimal functionover very large radial ranges is that even a -component model, i.e.a double Sersic (as in Hopkins et al. (2009a,b)), often yield residu-als larger than measurement errors.Motivated by the finding that the Einasto profile provides bet-ter fits to pure dark matter high resolution N-body simulations ofdwarf, galaxy and cluster sized haloes, Dhar & Williams (2010)presented, for the first time an extremely accurate – fractional de- viations of ∼ − to − – analytical approximation to the sur-face mass density of a 3D Einasto profile. This function is validfor n & . (see section (5) below), and is expressed in terms ofthe 3D Einasto profile parameters, ρ s , r s , and n . Given the issuesdescribed above with the existing forms of the fitting function, inthis paper we explore the quality of fits to the surface brightnessprofiles of ellipticals with a multi-component DW-function, whichhas the interesting property that the intrinsic 3D luminosity densityis a multi-component Einasto profile. Since our primary goal is to explore how well a multi-componentDW-function describes the surface brightness profiles of ellipticals,we restrict ourselves to ellipticals for which a large dynamic ra-dial range of high resolution data is available. We hence looked atthe well studied Virgo Cluster, for which KFCB09 provides, forthe first time, an excellent composite compilation of many groundand space based observations, spanning up to five decades, i.e. thelargest available radial range.In order to be able to probe the central regions, ∼ µ H-band data for its central regions (trans-formed to V-magnitudes) to account for dust absorption in V ; b)the 1.6 µ images in Quillen, Bower & Stritzinger (2000) show thatdust absorption in near-IR is weak and restricted to the very cen-tral regions, and the V-H colour image shows that any nuclear pointsource is shielded by the dust; c) it has a shallow cusp and the ef-fects of a psf are not as strong as in steep-cusp galaxies; and d)during fitting with a multi-component DW-function, discussed insections 5.1 and 6, we varied the lower end of the fit range from0.07 to 0.3 arcsec (4 × per cent.We thus obtain a sample of 23 ellipticals in and around Virgo,comprising of 22 ellipticals from the dataset presented in KFCB09and a composite profile of NGC4494 from Napolitano et al. (2009). Critical to our modelling of multiple components is the requirementthat the fit-residuals be consistent with measurement errors. It istherefore important to discuss the various uncertainties reported byKFCB09 for their dataset.Zero-points in KFCB09 are reported to have systematic un-certainties mag arcsec − , and random errors of ∼ c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams mag arcsec − . While the authors do not provide quantitative val-ues for profile measurement errors (either for every data point or an rms for each galaxy) they do state that fits with a Sersic functionare considered good when the resulting rms of fit is comparable toprofile measurement errors, which are of the order of a “few hun-dredths of a mag arcsec − ” . KFCB09 also report that the median rms of fits with a Sersic profile, over a restricted radial range, is0.04 mag arcsec − with a dispersion of 0.02 mag arcsec − . Wethus conclude that individual galaxies in the sample can have pro-file measurement errors in the range of (0.02-0.06) mag arcsec − ,with an rms of random errors ∼ mag arcsec − .For the central most regions Lauer et al. (1998) report psf de-convolution errors of around 0.07 mag arcsec − , which alongwith the random errors of 0.03 mag arcsec − imply that thecentral-most data points, at . ∼ mag arcsec − . KFCB09 also reportuncertainties in sky subtraction and errors due to matching profilesat large radii are around 0.1 mag arcsec − . In Dhar & Williams (2010) the authors derive an extremely accu-rate analytical approximation for the line-of-sight 2D projection ofthe 3D Einasto profile, equation (1.2). In terms of a scaled radius X = Rr s the surface density at a projected radial distance R is Σ DW ( R ) = Σ Γ( n + 1) ( n Γ h n, b ( ζ X ) n i + b n r nb X ( − n ) γ (cid:20) , ζ b n X n (cid:21) e − bX n (5.1) − δb n Xe − b (cid:16) √ ǫ X (cid:17) n ) where, Σ = Σ(0) = 2 e b r s ρ s n Γ( n ) b n , (5.2)with Γ( n ) as the Gamma function, Γ[ n, x ] and γ [ n, x ] as the upperand lower incomplete gamma functions, respectively, and b = b ( n ) is related to r s . The parameters ζ , ǫ , δ , and µ are functions of n ,as derived in Dhar & Williams (2010) (assuming b =2 n and ζ = ζ = 1 . . n − . n (5.3) ζ = 1 (5.4) ǫ = ζ + ζ (5.5) δ ( X ) = ( ζ − ζ ) { − exp [ − X µ ] } (5.6)with, µ = 1 . n + 0 . n − . n . (5.7)As described in Paper III (in preparation), generalizing the δ term(5.6) allows equations (5.1)-(5.7) to describe the surface density ofa projected Einasto profile for any choice of scale radius r s and anassociated b ( n ) . However, these equations simplify considerablyfor the N+04 parametrization of b =2 n (DW10). This parametriza-tion also has another nice feature that in a log-log plot of the surface density, the slope of the profile is − at R ≈ r − which is easilyidentified visually as the point where the profile begins to flatten outin log-space. In this paper we therefore adopt b = 2 n and r s = r − and use the parametrizations derived in Paper III to infer the ( r E , valid for n & 2D half-light radius ( R E , valid for n & r E = r − (cid:18) . − . n + 0 . n . (cid:19) n (5.8) R E = r − (cid:18) . − . n + 0 . n . (cid:19) n . (5.9)It should be evident from geometry that R E 10. The fractional residuals with re-spect to a numerically projected Einasto profile were very small, ∼ − to − . In order to model the light of ellipticals, we com-pared equation (5.1) to the numerical projection of (1.2) for a widerrange of n . For 10 n 50 the residuals are ∼ − , and ∼ − for 0.5 n 1. The function works very well even for 0.2 n< r s , but with anoverall rms n inthis latter range. We also found that for the entire range 0.2 n 50, the uncertainties in recovering the 3D Einasto profile parame-ters from 2D fits with (5.1) to a numerical projection of an Einastoprofile are better than − .The tests were conducted over a large dynamic range in radii,corresponding to the domain within which the surface density dropsfrom Σ to ∼ − Σ , which translates to a difference in magni-tude ∆ µ = mag arcsec − . The set of equations (5.1)-(5.7) canthus be used to estimate the 3D parameters of a projected Einastoprofile for a very wide range in radii, and shape parameter 0.2 n 50, corresponding to 0.02 ( α =1 /n ) As with other -parameter functions discussed in section (2), a sin-gle DW-function could not fit the SB profiles over the entire dy-namic radial range of our galaxies. Hence, we start with the as-sumption that a minimum of two DW-functions, each with threeparameters, are required to describe the light of ellipticals.The decision to add a third component must be based on thelevel of measurement errors in the data (section 4.1), as well ason the available degrees of freedom, i.e. the addition of the thirdcomponent must be statistically justifiable.We decide whether to fit a -component model after consider-ing the following factors.(i) Overall rms of residuals: The random errors in zero-pointsof the KFCB09 data are ∼± mag arcsec − , and result primar-ily from matching profiles of different filter magnitude systems. Ifthe rms of residuals are much less than this level, it may indicateover-fitting. In that case -component models are not considered.On the other hand, if a -component model has a larger rms , weexplore a -component model. c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals (ii) Examination of the fit residuals: A low rms of residu-als does not necessarily imply a good or reliable model. One canobtain a low rms due to very small residuals over a large radialrange, and large residuals over a small range. We therefore exam-ine -component model residuals over the entire radial range andconsider the fit to be good if it has consistently low residuals ( . mag arcsec − ), except possibly at very large R , and at thesmallest R . ≈ HST-WFPC2 pixels, where residualsup to 0.1 mag arcsec − are considered acceptable (section 4.1).If the overall rms is low, but there are systematically high residu-als in some regions, other than at very large and very small R , weexplore a -component model.(iii) F-test: Having fit a -component model, we employ theF-test to ensure that the reduction in rms is statistically significantat > σ , or . per cent, and is not merely due to an increase inthe number of parameters.Note that a failure of the F-test does not indicate that an extracomponent is certainly not present. It just means that the number ofdegrees of freedom do not justify a statistically significant detec-tion of the extra component. Better resolution, observations with adifferent filter and an increase in the number of independent datapoints may lead to a significant detection of an additional compo-nent from the surface brightness profile.The resulting surface brightness is then given as: Σ DW ( R ) = N X i =1 Σ i ( R ) (5.10)with N = 2 or indicating the number of components and Σ i ( R ) as the density of the i th component given by equation (5.1)-(5.7), with each component uniquely characterized by the set { Σ i , r si = r − i , n i } . As indicated later in section 6.1, some galaxies mayhave N> but we do not explore this option in this paper.For a -component model we shall refer to a central compo-nent, which typically describes the region within the break radius,and an outer component, which typically identifies the main bodyof the galaxy. For a -component model there is an additional in-termediate component, which describes a transition region betweenthe central and outer components, except in two cases, NGC4621and NGC4434, where this component indicates the presence of aweak system embedded within the outer component.It should be noted that in our models of a galaxy as asuperposition of components, the central and intermediate DW-components are in excess to an inward extrapolation of the outerDW-component. Figures 1-21 show fits to the SB profiles of the Virgo ellipti-cals with a multi-component DW-function along with the best-fitresiduals. For all galaxies we present a -component model. Forfourteen galaxies a -component model could be justified, while a -component model is sufficient for the other nine galaxies. Thefigure captions identify whether the 2- or -component model isstatistically significant. The results of fits are summarized in Ta-ble 1.The residuals of our models are consistent with measurementerrors (section 4.1) over large dynamic ranges ∼ in radius forthe largest shallow-cusp galaxies down to the resolution limit ofthe HST and ∼ in surface brightness for the smaller steep-cuspgalaxies. The rms is often as low as ∼ mag arcsec − , witha median sample rms of 0.032 mag arcsec − . The multi-component fits were carried out through a non-linear least squares Levenberg-Marquardt minimization using GNUPLOT . During the fitting process all components were allowedthe entire dynamic radial range and no pre-defined restricted rangein R was imposed. While convergence does depend on a reasonableinitial guess in any non-linear fitting, we did not find any strong de-generacies between the fit parameters especially for models whereresiduals are consistent with measurement errors.As noted in section 5.1 we seek a -component model whenthe rms of our -component model is greater than the 0.03 mag arcsec − rms of random errors in our sample. We then per-form an F-test to either accept or reject the -component model at > σ , or . per cent. However, in the figures we present twocases where we do not rely on the F-test alone, and consider otherfactors.(i) In NGC4649 (Fig.3) an F-test indicates there is a 27 percent chance that the reduction in rms due to a -component modelis not statistically significant; hence for the rest of the paper weuse the -component model. However, from our understanding oferrors in the central regions (section 4.1) it can be seen that the -component model certainly improves the fit near the centre and isacceptable at . σ . Hence, a failure of the F-test does not neces-sarily mean that a physically distinct system is not present. Its ex-istence can be verified with more information, say spectroscopic,about the central region. In section 10.1 we will show that the -component model may be necessary to infer the intrinsic 3D den-sity. (ii) We show a -component model for NGC4636 (Fig.9) asan illustration where an F-test does not reject the -componentmodel (at > σ ), but we do. This galaxy has a large dynamicradial range and a -component model may well be admissi-ble. We however reject it since our first criteria to admit a -component model is that the -component model must have an rms > mag arcsec − while its -component rms is 0.029 mag arcsec − . Hence, we do not feel confident in accepting this -component model and emphasize that reliance on statistics andphysical interpretations of models must be made with respect tothe level of measurement errors in data. In this section we detail peculiarities observed while modellingsome of the galaxies and in section 6.2 we discuss three galax-ies where interpretation of their components may require detailedmodelling using additional (for example, spectroscopic) informa-tion. a) For some of the larger galaxies like NGC4486 (M87,Fig.2), even a -component model leaves systematic patterns ∼ mag arcsec − in the fit residuals. Hence, although the fits lookvery good and rms is better than what is typically achieved in theliterature for such a large dynamic radial range, we believe that thiscould be an indication of an underlying fourth component. We donot explore four components in this paper.b) NGC4459 (Fig.13) has well known embedded dust featureswhich are clearly evident in the SB profiles. Its outer n of the -component model is fairly robust with respect to whether we in-clude or exclude the radial range affected by the dust. This is, how-ever, not true of its inner n , which changes appreciably based onthe inclusion or exclusion of the dusty region. We adopt the fit thatincludes the dust region.c) NGC4473 (Fig.14) is quite an interesting case. The SB isfit extremely well with an rms of 0.031 mag arcsec − using only c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams Table 1. Multi-component DW model properties of Virgo EllipticalsComponent Parameters Effective Radii RMS (mag)Name Type D Scale n c n int n o r Ec r Eint r Eo r E R E µ q c A v V T M V T L V T Multi- Double-(Mpc) (pc) (pc) (kpc) (kpc) (kpc) (kpc) ( mag (cid:3) ) (mag) (mag) ( × L ⊙ ) DW Sersic(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)NGC4472 E2 17.14 83.10 0.717 1.860 5.359 382.75 2.09 29.87 25.59 18.88 15.92 0.806 0.072 7.88 -23.29 176.5 0.031 0.07NGC4486 E1 17.22 83.48 1.090 2.619 6.451 30.81 3.96 75.26 50.98 37.29 16.21 0.722 0.072 8.04 -23.14 153.9 0.050 0.09NGC4649 E2 17.30 83.87 0.997 —– 5.444 951.50 —– 14.96 13.78 10.20 15.64 0.828 0.086 8.46 -22.73 106.2 0.032 0.08NGC4406 E3 16.83 81.59 0.895 1.488 3.084 240.85 32.58 4.14 24.61 17.96 15.66 0.709 0.096 8.39 -22.74 106.9 0.049 0.13NGC4365 E3 23.33 113.11 0.970 3.720 7.233 549.05 5.07 43.98 28.07 20.70 15.58 0.717 0.068 9.09 -22.75 107.5 0.036 0.09NGC4261 E2 31.60 153.20 0.896 1.446 5.023 533.37 2.35 22.45 16.29 11.92 16.11 0.794 0.059 9.90 -22.60 93.45 0.024 0.08NGC4382 E2 17.86 86.59 1.545 0.903 3.330 186.71 0.70 12.21 11.14 8.25 14.65 0.761 0.101 8.78 -22.48 83.74 0.090 0.11NGC4636 E3 14.70 71.27 1.282 —– 6.064 632.50 —– 26.25 25.44 18.85 16.41 0.760 0.090 8.54 -22.30 70.82 0.029 0.04NGC4552 E1 15.85 76.84 0.754 3.306 6.528 89.56 1.09 20.34 12.51 9.14 14.48 0.873 0.133 9.29 -21.71 41.22 0.049 0.09NGC4621 E4 14.93 72.38 3.714 0.985 9.561 16.13 4.67 8.77 7.62 5.72 10.21 0.742 0.107 9.29 -21.58 36.75 0.027 0.05 † NGC4494 E1 15.85 76.84 1.701 2.057 4.171 10.95 0.19 5.54 5.15 3.81 12.61 0.838 0.067 9.90 -21.10 23.50 0.025 —–NGC4459 E2 16.07 77.91 4.091 —– 3.835 257.37 —– 4.84 4.44 3.29 13.16 0.804 0.149 10.09 -20.94 20.36 0.050 0.05NGC4473 E4 15.28 74.08 2.300 —– 5.649 532.50 —– 5.48 4.26 3.13 14.44 0.607 0.092 10.00 -20.92 20.03 0.031 0.05NGC4478 E2 16.98 82.32 0.941 1.217 2.641 4.25 0.07 1.49 1.45 1.08 13.55 0.822 0.080 11.37 -19.78 6.99 0.035 0.10NGC4434 E0 22.39 108.55 1.238 0.576 5.532 15.50 2.78 1.47 1.73 1.27 13.61 0.928 0.072 12.18 -19.57 5.74 0.032 0.07NGC4387 E4 17.95 87.02 3.562 —– 2.621 82.31 —– 1.68 1.65 1.23 14.18 0.633 0.107 12.14 -19.13 3.83 0.037 —–NGC4551 E3 16.14 78.25 1.634 1.214 2.446 23.12 0.15 1.67 1.59 1.18 14.47 0.734 0.125 11.95 -19.09 3.69 0.026 0.05NGC4458 E1 16.37 79.36 1.502 3.078 3.149 23.05 0.12 2.07 1.74 1.28 12.93 0.879 0.077 12.12 -18.95 3.24 0.026 0.06NGC4464 E3 15.85 76.84 1.994 1.222 3.094 13.66 0.08 0.78 0.69 0.51 12.74 0.749 0.071 12.59 -18.41 1.97 0.021 0.06NGC4467 E3 16.53 80.14 2.673 —– 2.402 31.67 —– 0.54 0.51 0.38 14.38 0.813 0.074 14.18 -16.92 0.499 0.020 0.02VCC1440 E0 16.00 77.57 1.378 —– 4.976 8.79 —– 0.91 0.90 0.67 14.36 0.965 0.088 14.14 -16.88 0.484 0.032 0.04VCC1627 E0 15.63 75.78 2.002 —– 2.907 17.47 —– 0.38 0.37 0.27 14.63 0.928 0.127 14.53 -16.44 0.324 0.037 0.04VCC1199 E1 16.53 80.14 1.896 —– 2.389 13.49 —– 0.23 0.21 0.16 14.22 0.869 0.071 15.55 -15.54 0.140 0.023 0.02NOTES.— Galaxy properties from the multi-component DW models best-fitting the surface brightness (SB) profiles compiled from KFCB09, except that of NGC4494 (marked with a † ) whose data is fromNapolitano et al. (2009); we have, however, transformed their intermediate-axis profile to major-axis so as to have uniformity with the KFCB09 sample. Columns contain: (2) Galaxy type as defined in KFCB09.(3) Distance in Mpc from KFCB09 and references therein. (4) shows the physical size of arcsec in pc at the distance given in column 3. (5-10) Einasto shape parameter n and the intrinsic (3D) effective orhalf-light radius r E of the central-, intermediate- (when applicable) and outer components, deduced from the multi-component DW fits to the SB profile. r E of the components have been computed from therespective best fitting r − using equation (5.8). (11-12) Total effective or half-light radii of the galaxy, intrinsic ( r E ) and projected ( R E ), deduced by numerically integrating the best fitting multi-component DWmodel to infinity. (13) V-band Central ( r =0 ) surface brightness in mag arcsec − deduced from the best fitting multi-component DW model and corrected for Galactic extinction (column 15). (14) characteristicaxis-ratio of the galaxy (see section 6.3). (16-17) Galactic extinction corrected total V-band apparent ( V T ) and absolute ( M V T ) magnitude. (18) V-band extinction corrected luminosity assuming M V ⊙ = rms of residuals between the best-fitting multi-component DW models (this paper) and double-Sersic models in Hopkins et al. (2009a,b), for the same SB dataset of KFCB09. a -component DW-function. The outer component has an Einastoshape parameter n = M V T = -20.92, L V T =2 × L ⊙ and rather elliptical (E4/E5) isophotes,typical of the steep-cusp galaxies. The KFCB09 composite dataextends to 300 arcsec (22 kpc), an extent that is larger than thatof all steep-cusp ellipticals (except the unusual NGC4621, section6.2), but smaller than that of the smallest shallow-cusp ellipticalNGC4552 (Fig.10). It has a Galactic extinction corrected centralsurface brightness of . mag arcsec − which is also found inboth families of galaxies (Table1. However, its intrinsic 3D centraldensity as well as the overall intrinsic density profile appears to besimilar to that of shallow-cusp galaxies.d) NGC4387 (Fig.14) is well fit with a -component model butmay well have a third component. Since the -component rms isalready reasonably low at 0.0367, we do not explore a -componentmodel. However the SB profile is very similar to that of NGC4551(Fig.17) which clearly shows a -component structure. The centralcomponent for NGC4387 is also spatially more extended than thatof NGC4551 and has a larger n compared to its outer component; afeature that is different from most galaxies in the sample. The n and r − of the outer component for NGC4387 and NGC4551 are alsovery similar. This suggests that the central component of NGC4387could well be a sum of two components, which a SB analysis withthe present data is not able to distinguish.Note that the fading light of the giant elliptical NGC4406(M86) beyond 575 arcsec (Fig.4 and section 6.2), affects the en-tire region of NGC4387. The centres of these two galaxies are sep-arated by 668 arcsec. Their SB profiles extend through 800 and93 arcsec for NGC4406 and NGC4387, respectively. The high SB central regions of NGC4387 are unlikely to be affected by the lowsignal ( > mag arcsec − ) from NGC4406, but the outer regionsbeyond ∼ arcsec could be. However, the SB of NGC4387 showno detectable features in its outer profile.e) VCC1440 (Fig.20) is unusual in that despite being a fairlylow luminosity steep-cusp galaxy ( M V T = − . , L V T = × L ⊙ ) it has a fairly large n = The analysis of the SB profiles of NGC4406, NGC4382 andNGC4621 do not lead us to an unambiguous conclusion of whichcomponent characterizes the main body of the galaxy. We hence ex-clude these galaxies from studies involving specific aspects of indi-vidual components, for example, trends involving shape parametersand luminosity of components. The figure captions identify them as exceptions . The details are explained below:a) The SB of NGC4406 (Fig.4) is fit well with a -componentmodel with an rms of 0.049 mag arcsec − . However, it is not clearwhat constitutes the main body of the galaxy. The n = n = arcsec and continues through arcsec. It is hence intrinsic to NGC4406 and not a feature dueto incorrect subtraction of the light of NGC4387 and NGC4374.This galaxy is streaming into Virgo at 1400 km s − and itis possible that it has also gone through a recent interaction or a c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals merger. Chandra images and the presence of large plumes of HIgas around this galaxy do indicate such a possibility. Therefore itis not surprising that the n of its outer component is different fromthe typical values of other galaxies. Same is true of the intermediatecomponent.Our -component fit over four decades in radius (with an up-per limit of 153 arcsec, as in KFCB09) is not a good fit, although itsouter n is consistent with that of other large shallow-cusp galaxies.However, as mentioned earlier estimating fit parameters and struc-tural properties from a limited radial range can be misleading. Forexample, the half-light estimated with the -component model is31 kpc, much larger than the -component half-light of 18 kpc.In Fig.4 and Figs.(22,23 and 24) we identify the componentwith n = n = -component model for NGC4382 (Fig.7), the outercomponent contains 94 per cent of the total light. However, therather abnormal bump in the profile around n of its -component model may not be repre-sentative of well relaxed systems, although its central and interme-diate components are similar to those of other shallow-cusp galax-ies analyzed in this paper. In fact, if the bump around arcsecis excluded (Fig.8), we find an outer n of a -component modelmore consistent with those of other shallow-cusp galaxies. How-ever, we do not want to draw inferences on components obtainedafter excluding some radial ranges from the fit, especially in the ab-sence of a theoretical motivation for doing so. We hence adopt the -component model in Fig.7.c) NGC4621 (Fig.12) is an interesting case. With M V T = -21.58, it is the most luminous steep-cusp galaxy in our sample. Ingeneral, galaxies with M V T < -21.5 appear to have shallow-cusps,and are spatially more extended, while those with M V T > -21.5have much smaller spatial extent and usually have steep cusps.However, NGC4621 has the steepest central cusp and has the high-est central surface brightness of all galaxies in our sample and isphysically almost as large, and as luminous as the smallest shallow-cusp galaxy NGC4552 (Fig.10).The residuals of a -component fit and an F-test justifies a -component model, which also gives a far more reasonable n = n = -component model. However, the outer n of its -component modelis much larger than that of any galaxy in our sample, including theshallow-cusp galaxies that generally have large outer n . This couldbe due to the embedded intermediate component with n = n values. At the same time this galaxycould just be a special case, as mentioned above.NGC4434 (Fig.16) may be a similar system. Its intermediatecomponent, which is nearly gaussian, with n = n is larger than that of all other steep-cusp galaxies, and is are more consistent with that of the shallow-cusp galaxies.However, like NGC4621, the embedded component may have al-tered the shape of the outer component. Its outer n may thereforehave large uncertainties. In order to calculate the total luminosity, the cumulative luminositywithin R and the half-light radii one has to, in principle, account forthe varying ellipticity ǫ =1 − q ; where q = b/a is the axis ratio. Nu-merically, this can be done by expressing the area element dA ( R ) in terms of an axis ratio, q ( R ) , as dA ( R ) = 2 πq ( R ) R + πR dqdR (6.1)and the projected luminosity is then given by L p ( R a ) = Z R a Σ( R ) dA ( R ) (6.2)where R and R a are along the major axis. However, this involvestaking derivatives of axis ratios which show large and some timesabrupt variations. Further, in addition to real variation in q therecould also exist artificial variations due to the limitations of theellipse-fitting process.To avoid the above difficulties, during the ellipse-fitting pro-cess one can add up the projected light non-parametrically in el-liptical isophotes, L NPp ( R a ) , through the last data point, R a , anddefine a characteristic axis ratio, q c , such that q c = L NPp ( R a )2 π R R a Σ AN ( R ) RdR , (6.3)where Σ AN ( R ) is the spherically symmetric analytical function;here a multi-component DW-function.To estimate the total luminosity, we use this value to integratethrough infinity, even though this ellipticity seldom represents theaxis ratio of the outer most isophotes. Doing so can be justified forprofiles with large dynamic radial range, as in this paper, since bydefinition (equation (6.3)) it is a weighted axis ratio and the ex-tremely low luminosity regions extending through infinity are un-likely to change this weight. Further, since we do not know the trueaxis ratio beyond the last data point we would not like to assumethat the axis ratio through ∞ is equal to that of the last data point.The total luminosity is then given by L pT = 2 πq c Z ∞ Σ AN ( R ) RdR (6.4)and the projected half-light radius, R E , can be estimated from L pT = 4 πq c Z R E Σ AN ( R ) RdR. (6.5)Note that the characteristic axis ratio defined above, (6.3), isneither the mean nor a luminosity weighted axis ratio in the usualsense. Nevertheless, it is a useful definition that reduces the error inestimating the total light and half-light radii, and also ensures thatthey are weakly dependent on the specific choice of parametrizationof the SB profile and assumptions of ellipticity; provided of coursethat the fit parameters have been deduced after modelling the entiredynamic radial range and not from a limited range.Analytical expressions for the integrals in equation (6.3) and(6.4) may not exist. However, for the projection of an Einasto pro-file, analytical forms are given in Paper III, which accurately mod-els a numerical integration with the DW-function as well. c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.1-0.05 0 0.05 0.1 0.1 1 10 100 1000 µ − µ D W R (arcsec) 1" ≈ 83 pc rms=0.0497 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02"-1052.0"]NGC4472 VCC1226r s =9.124, n=2.286r s =49.727, n=5.101 2-component DW Fit 0.1 1 10 100 1000 R (arcsec) 1" ≈ 83 pc rms=0.0309Fit range [0.02"-1052.0"]NGC4472 VCC1226r s =3.864, n=0.717r s =13.240, n=1.860r s =45.709, n=5.359 3-component DW Fit Figure 1. Multi-component DW models of the surface brightness (SB) profiles of galaxies in the Virgo Cluster. For each galaxy, the figure keysshow the best-fitting Einasto index n and scale radius r s = r − of the components. The rms of residual is shown in the adjoining residual profile.The fit range is listed in the figure panels and in a few cases when some points are excluded from the fit, the excluded region is also marked onthe SB profile. A -component model is always shown, while a -component model is shown when it is found to be statistically significant throughan F-test (section 5.1). The model accepted is marked ’ adopted ’ in the caption. Circles indicate data from KFCB09, solid (red) line shows the totalmulti-component DW model, dot-dashed (blue) marks the central component, short-dashed (black) the outer component or the component resemblingthe main body of the galaxy and long-dashed (magenta) shows an intermediate or embedded component in some galaxies. Note that the central andintermediate DW-components are in excess to an inward extrapolation of the outer DW-component in those regions. Above NGC4472 (VCC1226) Left: -component, Right: -component (adopted). The large residuals around 1.5 arcsec in the -component modelindicates that a -component model may be necessary, which not only has a much lower rms of 0.0309 but also has consistent low and non-divergentresiduals . ∼ . Also note the occurrence of an Einasto index of n . In this paper, we use the apparent magnitudes estimated non-parametrically by KCFB09 through the last data point to calculatethe numerator in equation (6.3). As a cross-check one can verifythat the characteristic axis ratio estimated through equation (6.3)is usually consistent with the E-type of the galaxies listed in Ta-ble 1 and the ellipticity values in the literature. The V -band total(integrated through infinity) magnitudes are listed as V T and thecorresponding total absolute magnitudes are listed under M V T . c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -0.1-0.05 0 0.05 0.1 0.1 1 10 100 1000 µ − µ D W R (arcsec) 1" ≈ 83 pc rms=0.0585 16 18 20 22 24 26 28 µ [ m a g a r c s ec - ] Fit range [0.017"-1779"] NGC4486 VCC1316r s =19.877, n=2.168r s =40.621, n=7.892 2-component DW Fit 0.1 1 10 100 1000 R (arcsec) 1" ≈ 83 pc rms=0.0497Fit range [0.017"-1779"] NGC4486 VCC1316r s =0.266, n=1.090r s =18.347, n=2.619r s =73.616, n=6.451 3-component DW Fit Figure 2. NGC4486 (VCC1316 or M87) Left: -component; Right: -component (adopted). The -component model shows an overall improvement and areasonable rms =0.0497 with non-divergent residuals over an extremely large radial range ∼ . However, the presence of ( ∼ ∼ -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 83 pc rms=0.0320 14 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.03"-502"] NGC4649 VCC1978r s =8.273, n=1.085r s =22.750, n=5.348 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 83 pc rms=0.0312Fit range [0.03"-502"] NGC4649 VCC1978r s =0.066, n=2.422r s =8.147, n=1.151r s =23.486, n=5.271 3-component DW Fit Figure 3. NGC4649 (VCC1978) Left: -component (adopted); Right: -component (significant at 1.2 σ ). The -component model could be used to comparewith non-parametric deprojections (section10.1). Refer to caption of Fig.1 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.2-0.1 0 0.1 0.2 0.1 1 10 100 1000 µ − µ D W R (arcsec) 1" ≈ 81 pc rms=0.0776 14.5 16.5 18.5 20.5 22.5 24.5 26.5 µ [ m a g a r c s ec - ] Fit range [0.016"-153.3"] NGC4406 VCC881r s =3.009, n=1.358r s =31.624, n=5.029 2-component DW Fit 0.1 1 10 100 1000 R (arcsec) 1" ≈ 81 pc rms=0.0487Fit range [0.016"-800"] NGC4406 VCC881r s =2.302, n=0.895r s =244.462, n=1.488r s =16.242, n=3.084 3-component DW Fit Figure 4. NGC4406 (VCC881) Left: -component fit upto 153.3 arcsec - the upper-limit of fit with a Sersic profile shown in KFCB09; Right: -component(adopted) fit over the entire radial range through 800 arcsec. However, which component forms the main body of the galaxy is not clear and hence this galaxyis an exception (section 6.2). Refer to caption of Fig.1 and section 6.2. -0.2-0.1 0 0.1 0.2 0.1 1 10 100 1000 µ − µ D W R (arcsec) 1" ≈ 113 pc rms=0.0638 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02"-657"]NGC4365 VCC731r s =5.245, n=2.204r s =20.571, n=6.331 2-component DW Fit 0.1 1 10 100 1000 R (arcsec) 1" ≈ 113 pc rms=0.0361Fit range [0.02"-657"]NGC4365 VCC731r s =3.671, n=0.970r s =11.085, n=3.720r s =23.123, n=7.233 3-component DW Fit Figure 5. NGC4365 (VCC731) Left: -component; Right: -component (adopted). Refer to caption of Fig.1 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -0.2-0.1 0 0.1 0.2 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 153 pc rms=0.0579 17 19 21 23 25 µ [ m a g a r c s ec - ] Fit range [0.07"-240"]NGC4261 VCC345r s =4.919, n=2.256r s =23.332, n=4.652 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 153 pc rms=0.0240Fit range [0.07"-240"]NGC4261 VCC345r s =2.714, n=0.896r s =9.548, n=1.446r s =21.360, n=5.023 3-component DW Fit Figure 6. NGC4261 (VCC345) Left: -component; Right: -component (adopted). The improved fit with a -component model, around the transition radius,and the near vanishing of large scale systematic patterns along with a significant reduction in rms is clearly visible. Refer to caption of Fig.1 for details. -0.2-0.1 0 0.1 0.2 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 87 pc rms=0.0980 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02-606"] NGC4382 VCC798r s =2.648, n=2.678r s =41.073, n=3.334 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 87 pc rms=0.0901Fit range [0.02-606"] NGC4382 VCC798r s =1.290, n=1.545r s =6.252, n=0.903r s =40.871, n=3.330 3-component DW Fit Figure 7. NGC4382 (VCC798) Left: -component; Right: -component (adopted). Although this galaxy shows a visibly shallow-cusp, note the lower outer-nand an inner n = 1 . similar to that of some of the less massive steep-cusp galaxies. The -component fit is very good, except in a region around arcsec .Hence the best-fitting parameters of its outer component may not represent the true values. The outer regions of this galaxy also show indications that it maynot have relaxed from a recent interaction and we therefore mark this galaxy as an exception (see section 6.2). Refer to caption of Fig.1 for further details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.2-0.1 0 0.1 0.2 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 87 pc rms=0.0417 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02-26" and 221-488"] NGC4382 VCC798r s =2.269, n=2.425r s =17.252, n=5.688 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 87 pc rms=0.0215Fit range [0.02-26" and 221-488"] NGC4382 VCC798r s =0.887, n=0.915r s =3.712, n=1.680r s =16.649, n=5.841 3-component DW Fit Figure 8. NGC4382 (VCC798) Left: -component; Right: -component. As noted in Fig. 7, this galaxy is an exception . Here we show a fit by excluding aregion around the bump from − arcsec, as in KFCB09. Note the larger outer-n and the n ∼ for the inner component, similar to the pattern seen inother massive shallow-cusp galaxies. However, we adopt the -component fit in Fig. 7 since excluding domains of fit with ad-hoc fitting functions, may notreveal the true structure of these regions. Refer to caption of Fig.1 for further details. -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 71 pc rms=0.0288 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.02-657"] NGC4636 VCC1939r s =5.916, n=1.282r s =35.231, n=6.064 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 71 pc rms=0.0234Fit range [0.02-657"] NGC4636 VCC1939r s =6.375, n=1.457r s =127.382, n=0.404r s =35.782, n=6.244 3-component DW Fit Figure 9. NGC4636 (VCC1939) Left: -component (adopted); Right: -component. The -component model is shown as an illustration of a case where anF-test does not reject it but we do, since the -component model has a rms ∼ (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -0.3-0.2-0.1 0 0.1 0.2 0.3 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 77 pc rms=0.0735 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.045-495"] NGC4552 VCC1632r s =2.965, n=3.786r s =16.964, n=6.954 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 77 pc rms=0.0488Fit range [0.045-495"] NGC4552 VCC1632r s =0.963, n=0.754r s =4.133, n=3.306r s =20.950, n=6.528 3-component DW Fit Figure 10. NGC4552 (VCC1632) Left: -component; Right: -component (adopted). At M V T = − . , the smallest shallow-cusp elliptical in Virgo is moreluminous than all other steep-cusp ellipticals. As in most shallow-cusp ellipticals, it also has an n< for its central component which is not seen in any of thesteep-cusp ellipticals. HST observations (Renzini et al. (1995), Cappellari et al. (1999)) reveal variable UV-flare activity in the centre which is interpreted toarise from a low-level AGN. Contributions from such a point source to the central-most data point is excluded from the fit. See caption of Fig.1 for details. -0.15-0.1-0.05 0 0.05 0.1 0.15 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 77 pc rms=0.0656 12 14 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.02-250"]NGC4494 Coma I cloudr s =1.671, n=9.985r s =41.140, n=1.603 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 77 pc rms=0.0251Fit range [0.02-250"]NGC4494 Coma I cloudr s =0.080, n=1.701r s =1.231, n=2.057r s =14.857, n=4.171 3-component DW Fit Figure 11. NGC4494. Left: -component; Right: -component(adopted). This galaxy is in the Coma-I cloud around Virgo and comparable to the moreluminous steep-cusp ( > × L V ⊙ ) Virgo ellipticals. The composite data from Napolitano et al. (2009) is plotted in terms of the major axis radius, as inother galaxies from the KFCB09 sample. This is the only galaxy that is not in the KFCB09 sample. Refer to caption of Fig.1 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 72 pc rms=0.0477 11 13 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02-512"] NGC4621 VCC1903r s =0.067, n=8.788r s =5.698, n=7.424 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 72 pc rms=0.0272Fit range [0.02-512"] NGC4621 VCC1903r s =0.055, n=3.714r s =48.550, n=0.985r s =2.803, n=9.561 3-component DW Fit Figure 12. NGC4621 (VCC1903) Left: -component; Right: -component (adopted). The most luminous ( M V T = -21.58, L V T = × L V ⊙ ) steep-cusp elliptical in Virgo. The -component model improves the fit in the region from 9-130 arcsec and also gives a more reasonable central-n. However, italso has a much larger outer-n. The SB has sharp pointed isophotes and it is not clear how much the embedded component and its ellipticity influences ourdetermination of outer-n. We hence mark this galaxy as an exception (see section 6.2). Also see caption of Fig.1. -0.2-0.1 0 0.1 0.2 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 78 pc rms=0.0498 13 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.019-263"] NGC4459 VCC1154r s =0.703, n=4.091r s =14.661, n=3.835 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 78 pc rms=0.0403Fit range [0.019-1.24" and 9.6-263"]NGC4459 VCC1154r s =1.021, n=5.626r s =17.271, n=3.612 2-component DW Fit Figure 13. NGC4459 (VCC1154) Left: -component(adopted); Right: -component excluding the region affected by dust from . − . arcsec, as inKFCB09. It is one of three galaxies in our sample that has an n central > n outer . Refer to caption of Fig.1 for details. c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 74 pc rms=0.0308 14 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.02-311"]NGC4473 VCC1231r s =3.164, n=2.300r s =8.362, n=5.649 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 87 pc rms=0.0367Fit range [0.02-93"] NGC4387 VCC828r s =0.249, n=3.562r s =7.470, n=2.621 2-component DW Fit Figure 14. Left NGC4473 (VCC1231): Its -component model has an rms of only 0.031 over a dynamic radial range ∼ and mags in SB. It is the leastluminous of ∼ L V ⊙ ellipticals in Virgo with properties similar to both the steep and shallow cusp families (section 6.1). However, unlike the massiveshallow cusps that have central- n . , it has n = n central >n outer and has a luminosity of 3.83 × L V ⊙ . Its profile could beaffected by the light of NGC4406 (section 6.1). Note the similarity of this profile with that of NGC4551 (Fig. 17) where -components could be justified. -0.2-0.1 0 0.1 0.2 0.01 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 82 pc rms=0.0837 13 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.01-106"] NGC4478 VCC1279r s =1.133, n=6.546r s =9.864, n=1.902 2-component DW Fit 0.01 0.1 1 10 100 R (arcsec) 1" ≈ 82 pc rms=0.0355Fit range [0.01-106"]NGC4478 VCC1279r s =0.040, n=0.941r s =0.617, n=1.217r s =6.950, n=2.641 3-component DW Fit Figure 15. NGC4478 (VCC1279) Left: -component; Right: -component (adopted). As in NGC4494, three components are clearly visible. It is the mostluminous of ∼ L V ⊙ ellipticals in Virgo. Along with NGC4434, they are the only two steep-cusp ellipticals with a central-n ∼ . See Fig.1 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.2-0.1 0 0.1 0.2 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 109 pc rms=0.0554 14 16 18 20 22 24 26 28 µ [ m a g a r c s ec - ] Fit range [0.022-115"] NGC4434 VCC1025r s =0.922, n=6.802r s =11.209, n=1.674 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 109 pc rms=0.0316Fit range [0.022-115"] NGC4434 VCC1025r s =0.097, n=1.238r s =22.778, n=0.576r s =1.609, n=5.532 3-component DW Fit Figure 16. NGC4434 (VCC1025) Left: -component; Right: -component (adopted). A -component model is clearly needed. However, it has an unusuallylarge outer-n typical of the more massive shallow-cusp galaxies. Refer to caption of Fig.1 for details. -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 78 pc rms=0.0434 13 15 17 19 21 23 25 27 µ [ m a g a r c s ec - ] Fit range [0.02-110"] NGC4551 VCC1630r s =0.493, n=5.404r s =9.160, n=2.412 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 78 pc rms=0.0263Fit range [0.02-110"] NGC4551 VCC1630r s =0.170, n=1.634r s =1.314, n=1.214r s =8.842, n=2.446 3-component DW Fit Figure 17. NGC4551 (VCC1630) Left: -component; Right: -component (adopted). Refer to caption of Fig.1 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 80 pc rms=0.0506 14 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.02-95"] NGC4458 VCC1146r s =0.232, n=3.003r s =6.457, n=3.594 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 80 pc rms=0.0265Fit range [0.02-95"] NGC4458 VCC1146r s =0.177, n=1.502r s =0.505, n=3.078r s =8.153, n=3.149 3-component DW Fit Figure 18. NGC4458 (VCC1146) Left: -component; Right: -component (adopted). Refer to caption of Fig.1 for details. -0.1-0.05 0 0.05 0.1 0.1 1 10 100 µ − µ D W R (arcsec) 1" ≈ 77 pc rms=0.0406 12 14 16 18 20 22 24 26 28 µ [ m a g a r c s ec - ] Fit range [0.02-69"] NGC4464 VCC1178r s =0.290, n=6.664r s =3.911, n=2.822 2-component DW Fit 0.1 1 10 100 R (arcsec) 1" ≈ 77 pc rms=0.0214Fit range [0.02-69"] NGC4464 VCC1178r s =0.089, n=1.994r s =0.718, n=1.222r s =3.225, n=3.094 3-component DW Fit Figure 19. NGC4464 (VCC1178) Left: -component; Right: -component (adopted). It is the least luminous of ∼ L V ⊙ ellipticals in Virgo. Refer tocaption of Fig.1 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.1-0.05 0 0.05 0.1 0.1 1 10 µ − µ D W R (arcsec) 1" ≈ 80 pc rms=0.0205 14 16 18 20 22 24 26 µ [ m a g a r c s ec - ] Fit range [0.02-24"] NGC4467 VCC1192r s =0.150, n=2.673r s =2.822, n=2.402 2-component DW Fit 0.1 1 10 R (arcsec) 1" ≈ 78 pc rms=0.0320Fit range [0.02-43"] VCC1440 IC798r s =0.073, n=1.378r s =1.749, n=4.976 2-component DW Fit Figure 20. Left NGC4467 (VCC1192); Right: VCC1440. Low luminosity ellipticals in Virgo ( ∼ L V ⊙ ) bordering the dwarf elliptical population. Bothgalaxies show a steep-cusp, but note the large outer-n for VCC1440 typical of the more massive shallow cusps. Refer to caption of Fig.1 for details. -0.1-0.05 0 0.05 0.1 0.1 1 10 µ − µ D W R (arcsec) 1" ≈ 76 pc rms=0.0370 14 16 18 20 22 24 26 28 µ [ m a g a r c s ec - ] Fit range [0.02-29"] VCC1627r s =0.115, n=2.002r s =1.739, n=2.907 2-component DW Fit 0.1 1 10 R (arcsec) 1" ≈ 80 pc rms=0.0231Fit range [0.02-14"] VCC1199r s =0.087, n=1.896r s =1.205, n=2.389 2-component DW Fit Figure 21. Left: VCC1627; Right: VCC1199. The least luminous ellipticals in Virgo ( ∼ L V ⊙ ). Refer to caption of Fig.1 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals The goal of this paper is to investigate how well can a multi-component DW-function model the SB and consequently whetherthe intrinsic 3D density can be described with a multi-componentEinasto model. A detailed comparison with other parametrizationsfor every galaxy is beyond the scope of this paper. However, theseparametrizations often show residuals larger than measurement er-rors (sections 3.1 and 3.2). Since the literature has a few examplesof fits with other parametrizations, for the galaxies we model here,we present a comparison in sections 7.1, 7.2 and 7.3. All galaxies in our sample have been fit with a combination ofpsf-convolved Sersic, Core-Sersic (hereafter, CS) and King mod-els to the ACSVCS Sloan g- and z-band profiles in Ferrarese et al.(2006). While the residuals are difficult to ascertain from their fig-ures, fit residuals of King+Sersic, Core-Sersic and Nuker models inthe central . − arcsec , or ∼ radial decades, are presentedin Lauer et al. (2007) (L+07); see their fig. 9 and 10. These are thesame psf-deconvolved profiles that were used in KFCB09 and thepresent paper where the data in the central regions are from psf-deconvolved WFPC V-band and . µ NICMOS images of Laueret.al. (1992, 1995 & 2005). Note that L+07 fit the same models usedin F+06 (for a given galaxy) to psf-deconvolved WFPC/NICMOSimages and then compare the F+06 models to fits with a Nukermodel. L+07 also notes that differences in choice of psf, cameraand observing band are not significant to prevent a comparison.The Nuker fits usually exclude fitting ’nuclei’. These regionsare fit by F+06 with a King and a Sersic ( -parameters) or CS pro-file ( -parameters); the CS+King galaxies of F+06 are not shownin L+07. Further, the fits in L+07 are compared within the cen-tral − arcsec domain of validity of the Nuker profile. Allother parametrizations have been fit over larger radial ranges. Fitsin this paper, which use KFCB09 data, use the largest range. The galaxies that overlap with our sample are discussed below:(i) Steep-cusp with a King model for ’nuclei’: For most steep-cusp galaxies F+06 required a King profile for the central regionand identified them as ’nuclei’. Note that King models have atruncation radius and a flat core in 3D, and hence in 2D as well.The Nuker fits exclude these regions. For galaxies that we have incommon with F+06 and L+07 – NGC4387, NGC4467, NGC4551,NGC4458, VCC1199, VCC1440 and VCC1627 – we show thatthey can be well fit with DW-functions, often with just two com-ponents over the entire dynamic radial range, and with much betterresiduals than with either the King+Sersic or the Nuker profiles.(ii) Steep-cusp with a single Sersic model: For some galaxies,F+06 showed that the entire profile can be described with a sin-gle component Sersic model. The comparison plots in L+07 showthat a single Sersic profile can not fit the central-most regions ofNGC4478, NGC4473, NGC4621, NGC4434 and NGC4464 wherethe Nuker performs better. We show that it is possible to easilyquantify these regions using the DW-function with better residualsthan with the Nuker profile.(iii) Shallow-cusp: NGC4365, NGC4382, NGC4406,NGC4552, NGC4649, NGC4472 and NGC4486 have been fitwith the Core-Sersic and Nuker profiles. Although the residualsare smaller than those in the case of the steep-cusp galaxies, weshow that fits with a multi-component DW-function produce evensmaller residuals. It should also be noted that the Nuker fits apply to a limited range, and as shown by Graham et al. (2003) theparameters depend strongly on the selected radial domain of fit. KFCB09, whose composite SB profiles we use in this paper,presents fits to the SB with a single Sersic -parameter function foreach galaxy. Since such a 1-component Sersic profile cannot ade-quately fit the full radial range of 4-5 decades of KFCB09 profiles,the authors estimate the largest radial range over which a singleSersic profile produces robust fits and residuals comparable to mea-surement errors. This range is typically 1-2.5 radial decades, andexcludes regions interior to the transition radius, where the slopechanges rather abruptly.KFCB09 also had to invoke certain constraints to estimate thetotal luminosity and half-light radii, for example, a limiting magni-tude up to which to integrate the light. This was usually done forgalaxies where the Sersic profile failed to model the SB profile overa large radial range. While as noted in KFCB09 the existence of aphysically justified limiting magnitude is possible – for example,a tidal truncation radius or incorrectly subtracted light of a neigh-bour – such limits should not exist due to the failure of an ad-hocfitting function to model the entire SB profile. Further, since the fitswere obtained from a limited range, they give a biased estimate ofthe Sersic index (for example, in the case of NGC4406 and othermassive galaxies in table 1 of KFCB09).We agree with KFCB09 that fits must be consistent with themeasurement errors, and show that this can be achieved using amulti-component DW-function, which also avoids systematic de-viations between the data and the fits. Our estimates of structuralproperties are therefore likely to be more meaningful, and can beused to obtain direct estimates of the intrinsic 3D structural proper-ties of galaxies. Hopkins et al. (2009a,b) fit a -component Sersic profile to allgalaxies in the KFCB09 sample that we use here. While they donot show the residual profiles, we note that the rms of their fitswith a -component Sersic profile is usually per cent larger thanthe rms of our -component DW models, and nearly − per cent larger than the rms of our -component DW models. InTable 1 we list the rms of fits of our best-fitting DW models andthat of the double-Sersic models in Hopkins et al., for comparison.From their double Sersic models Hopkins et.al. conclude thatsteep-cusp and shallow-cusp galaxies are two disjoint populations,and that the outer Sersic index m does not depend on the mass orluminosity of these galaxies.Our fits lead us to a different conclusion. We observe that theouter Einasto index n does increase with luminosity (and conse-quently size, r E - the 3D half-light radius) in a seemingly con-tinuous manner. A similar trend was also noted by Graham et al.(1996) and F+06 based on their fits with Sersic profiles. An examination of the SB profiles show that the central regionshave distinct variations in slope. This is more true for the steep-cuspgalaxies than shallow-cusp ones. This deviation from pure power-laws are reflected in the large fit residuals in the central regions asdiscussed in section 7.1. c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams Fig. 1-21, when compared to the fits in the literature describedabove, demonstrate that the 2- or -component DW-function pro-vides a better fit, over a larger radial range, than other existing func-tional forms. Our overall rms are comparable to, or lower than thatof other models, and our residuals are consistently low over the 4-5radial decades of the available composite observations (KFCB09).Furthermore, the DW-function is a very accurate 2D projection ofthe 3D Einasto profile, and is expressed in terms of the 3D Einastoprofile parameters. This means that if the 2D fits are good, the in-trinsic 3D luminosity structure is that of superimposed Einasto pro-files, and can be inferred directly, with no further modelling.We hence propose that the light of ellipticals that was believedto be well fit with a Sersic profile in 2D, is instead better describedby a multi-component form of a similar function (the Einasto pro-file) in 3D, whose 2D projection is given by a multi-componentDW-function. n In this section we investigate the structural properties of the compo-nents deduced from the multi-component DW fits. We shall be re-ferring to the statistically significant best-fitting models only, listedin Table 1, and not all -component and -component fits shown inFigs.1-21. Nine galaxies are described with two DW-components(two shallow cusps and seven steep cusps), and fourteen galaxiesare described with three DW-components (seven each of shallowand steep cusp galaxies).In our modelling of a galaxy as a linear superposition of com-ponents, the central and intermediate DW-components described inthe following sections 8.1-8.3 are in excess to an inner extrapola-tion of the outer DW-component; and do not contain all of the lightin the central and intermediate regions. This is consistent with sim-ilar decompositions in the literature. Superposition of components,comprising of a Sersic profile for the central bulge superimposedon an underlying exponential model (a Sersic profile with m = 1 ),is often applied to the case of lenticulars and spirals. Cote et al.(2007) have shown that a similar decomposition using a double-Sersic profile as a fitting function can also be applied to ellipticalswith M B & − . . The resulting central Sersic-component is thenused to evaluate physical properties of the central region.For the case of shallow-cusp galaxies, as in the spectro-scopic and kinematic modelling of M87 in Tenjes, Einasto & Haud(1991) and the fitting function (double Sersic) based modelling of’core’ galaxies in Hopkins et al. (2009b), we also find that thesegalaxies can be modelled as a linear superposition of components.While such fit components need not correspond to real physi-cal systems, we show in section 8.4, three cases of shallow-cusp(’core’) galaxies whose central DW-components coincide very wellwith spectroscopically identified systems.In order to estimate luminosities of components, a knowledgeof a characteristic axis ratio (equation (6.3)) over the extent of eachcomponent is required. Since the components are superimposed, itis very difficult to isolate their characteristic axis ratios using SBanalysis; except maybe the outer-most dominant one. However, el-liptical galaxies have axis ratios of 0.3 (E7) q q the uncertainty in estimating the luminos-ity can at most be off by a factor of − . In our sample, except 3galaxies which are E4, the rest are between E1 and E3. Our worsterrors in estimating component luminosities are therefore less than a factor of . Additional spectroscopic or kinematic modelling maybe used to constrain the ellipticities of individual components.With this understanding, for the purpose of computing lumi-nosity and half-light radius of components, we assume that the axisratio of all components are the same as the characteristic axis ratio q c for the entire galaxy (equation (6.3)). Note that uncertainties ininterpreting component luminosities do not affect our estimate ofthe total luminosity of the galaxy. From the nine galaxies with two components and fourteen galaxieswith three components, we observe that:1) In all galaxies, the central component has a lower n thanthat of the outer component. Since Einasto functions of the form e − x ( n = → faster than e − x ( n = n implies alarger concentration. Hence, the central component is more con-centrated than the outer component.2) The central component of the shallow-cusp galaxies areone or two orders of magnitude more luminous than that of thesteep-cusp galaxies (Fig.22). Recall that the uncertainty in q is atworst a factor of while the luminosities differ by a factor of − . The shallow-cusp galaxies thus host an unambiguously largerluminosity in their central component. Note that more luminousgalaxies, M V T . − . mag, typically have a shallow cusp, whilefainter galaxies have steep central cusps.3) The 3D half-light radius r E of the central component ofshallow-cusp galaxies is typically an order of magnitude larger thanthat of steep-cusp galaxies (Fig.23). The same is also true for thescale radius r − (listed in the keys of Figs.1-21), which character-izes a radius inside which the logarithmic slope of the 3D profilefalls below − , and that of the SB profile falls below − .4) The shape parameter n of the inner component of shallow-cusp galaxies is always significantly smaller than that of their outercomponent, with the difference reduced for the steep-cusp galaxies(Fig.24). Exceptions, with n central ≈ n outer , are NGC4467 andNGC4459 whose central region is affected by a huge dust disk,and NGC4387 that has n central > n outer , whose light could beaffected by NGC4406 or, as discussed in section 6.1, could be a -component system being modelled with -components.5) All of the nine massive shallow-cusp galaxies( M V T − . ) have n . n = n > n ∼ n ∼ n implies a larger concentration. Hence, thecentral components of shallow-cusp galaxies appear to be moreconcentrated than those of the steep-cusp galaxies.The above observations indicate a new trend with regards tothe central components of galaxies: Even though steep-cusp galax-ies have a higher central density, the central component of theshallow-cusp galaxies is far more luminous and massive, spatiallymore extended, and more concentrated than that of the steep-cuspgalaxies.6) The large incidence of n ∼ m = c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals l og ( L u m i no s it y L ⊙ ) Total Absolute Magnitude M VT Total LuminosityCentral+IntermediateCentral+Interm(exception)CentralCentral (exception) Figure 22. Luminosity of components resulting from our multi-componentDW models as a function of total luminosity of the galaxy ( M V T ). Galax-ies whose component parameters are uncertain (see section 6.2) are markedas exceptions in the figure key. The central component of massive (gen-erally shallow-cusp) ellipticals are ∼ more luminous than that of theless massive (generally steep-cusp) ellipticals. This effect is more apparentfor the total luminosity in the central+intermediate component. Note thatluminosities of the central and intermediate DW-components are excess lu-minosities in those regions with respect to contributions from the outer DW-component. A colour version of this figure is available in the online edition. r outer exceptionsr central exceptions10 -2 -1 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 D H a l f- li gh t r a d i u s r E ( kp c ) Total Absolute Magnitude M VT r central exceptionsr intermediate exceptions Figure 23. The intrinsic (3D) half-light radii r E of all components appearto be generally increasing with galaxy luminosity. The r E of the massiveshallow cusps are generally a factor of larger than that of the steep-cuspgalaxies. This trend is stronger for the outer component. Note that, as shownin Table1, r Eo of the outer component is generally slightly larger than r E of the galaxy which is computed using the total light from all components.Also refer to caption of Fig. 22. A colour version of this figure is availablein the online edition. spectroscopically identified disks, and our -component – but not -component – fits show that they have n ∼ 1. We reiterate thatwhile disks usually have m = 1, not all m = . pc, they found a similar trend as mentionedabove; all steep-cusp and shallow-cusp galaxies show a light ex-cess in their central regions with respect to the inner extrapolationof an outer Sersic profile. They also note that, on an average, they n outer exceptionsn central exceptions 1 2 3 4 5 6 7 8 9-24 -23 -22 -21 -20 -19 -18 -17 -16 -15 D E i n a s t o i nd e x n Total Absolute Magnitude M VT n central exceptionsn intermediate exceptions Figure 24. Einasto index n of components. All massive shallow-cusp galax-ies have n . n & 2. The n of the outer component shows a distinct trend increas-ing with the total luminosity of the galaxy. Also note the large differencein n between the central and outer components of the shallow-cusp galax-ies, which diminishes with decreasing luminosity for the steep-cusp family.Refer to caption of Fig. 22 for exceptions . A colour version of this figure isavailable in the online edition. could recover the total light in the true central component of steep-cusp galaxies by fixing, without fitting, the Sersic index of the in-ner component to m = 1. However, whenever they fit for the Sersicindex m in the steep-cusp galaxies observed with the HST, theytypically obtain a wide range of 0.6 . m . m ∼ n weget for our -component DW models in cases where -componentDW models produce significantly better fits. Using three compo-nents in these cases, not only is the variance in the inner n reduced,we also see a tendency of n ∼ n ∼ m ∼ Our multi-component models indicate that of the galaxieshave an intermediate DW-component. These include both steepcusp and shallow cusp galaxies. Only in two out of the galax-ies (NGC4621 and NGC4434) it identifies features within the outerDW-component, but in the rest it is located between the inner andouter components, and thus forms a transition region. From Fig.22we also observe that within a factor of two, all galaxies containa similar fraction of the total light in their central+intermediateDW-components. There appears to be an indication that the mas- c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams sive shallow-cusp galaxies may contain a larger fraction, althoughthis is not very clear from our small sample.Without additional kinematic or spectroscopic data it is notpossible to ascertain the physical origin of the intermediate compo-nent, but based on the results of existing galaxy formation models,we speculate that at least three scenarios are possible:We suggest that, the stars making up the intermediate compo-nent – (i) may have formed as a result of local star formation (asin Mihos & Hernquist (1994), Hopkins et al. (2008) for the centralregions), (ii) scattered into this region during mergers by a cen-tral supermassive black hole (SMBH) (Begelman, Blandford, Rees(1980), Boylan-Kolchin, Ma & Quataert (2004)) , or (iii) in thecase of shallow-cusp galaxies only, stars could have been scatteredto these radii by the central SMBH, or coalescing binary SMBHswhich are believed to scour out few 100 pc regions in galaxy cen-tres (Gualandris & Merritt 2008).The physical interpretation of the central and outer regions ofa galaxy depends on how the overall SB profile is modelled. In ourmodels for shallow-cusp (’core’) galaxies, we observe an excess lu-minosity due to the central DW-component with respect to an outerDW-component. In section 8.4, we show that for three ’core’ galax-ies, the central DW-component does correspond to spectroscopi-cally identified real systems. Hence, the central DW-component isnot necessarily a mere mathematical construct. Alternatively, as hasbeen suggested in the literature, the mechanisms mentioned in theprevious paragaraph are believed to have caused a deficit of massin the central regions, with respect to an inward extrapolation ofan outer Sersic profile which is fitted to the SB at radii beyond thetransition, or break radius. In other words, the existence of a shal-low cusp may imply – (a) a deficit of mass in the central regions or(b) an excess of mass with small n , which is superimposed on tothe outer DW-component, as in this paper. The dynamical interpre-tation of the central region will probably be different depending onwhether (a) or (b) are mostly correct, but either case is consistentwith SMBH scouring out mass. However, SB analysis alone cannotresolve this issue; more data and dynamical modelling are required.Further discussion on mass deficits is provided in section 11. The multi-component DW models reveal a huge dominant outercomponent that usually contains a much larger fraction of the totallight than any of the other components. Fig. 24 indicates that theouter n increases with luminosity. From the SB profiles (Fig. 1-21) we note that for galaxies with outer n & 5, the outer componentmakes a significant contribution to the density in the central region.This is usually the case for the massive shallow-cusp galaxies, butis also seen in smaller, less luminous galaxies, like NGC4473 andNGC4434, where the SB appears to indicate a shallow inner slope.On the other hand, the outer components of the steep-cusp galaxiesdo not contribute fractionally as much light to the central regions.From the above discussion of components in sections 8.1-8.3we observe that all shallow-cusp galaxies have a sharp transitionto a shallow inner slope. These SB profiles are well modelled by acombination of – (i) a central n ∼ r E (Table 2) is greater than about ten times thatof steep-cusp galaxies, and (ii) a non-negligible contribution to thedensity in the central region from a n & n ∼ n ’s, and when projected it produces a step-function likesharper transition in a log-log plot of density versus radius. The components obtained through fitting ad-hoc functions may notcorrespond to physically distinct kinematic systems or stellar pop-ulations, unless the form of our fitting function happens to be cor-rect. While a detailed analysis of components and systems for everygalaxy is beyond the scope of this paper, we present three interest-ing connections between the components we deduce from fittingthe V-band SB profiles, and spectroscopically identified systems.Jaffe et al. (1993) and Ford et al. (1994) reported the earli-est detections of nuclear disks around supermassive black holes(SMBHs) in the centres of galaxies. We discuss structural simi-larities between such spectroscopic detections of systems and thecomponents deduced from our multi-component DW models forthree galaxies in our sample.1) Images of M87 (NGC4486) show a prominent nuclear disk.The central DW-component of our -component model (Fig.2)from the broadband I-band (WFPC F785LP) images, suitablyscaled to V-magnitudes in KFCB09, has a best fit Einasto shapeparameter n = r = per cent; a size consistent with thespectroscopic observations of Harms et al. (1994). Tsvetanov et al.(1999) study the morphology of the disk in detail with the narrowband F658N filter and observe that a significant light excess is de-tected inside 0.5 arcsec. We note that at 0.5 arcsec the intensity ofthe central DW-component falls to per cent of maximum.2) NGC4261 also has large spectroscopically confirmed cen-tral systems (Jaffe et al. (1993), Ferrarese, Ford & Jaffe (1996)).FFJ96 fit a double exponential model with scale lengths of 1.83and 8.73 arcsec. Our best fit -component model (Fig.6) show thatthe central and intermediate components have nearly exponentialprofiles. The central component has a shape parameter n = r − = n = r − = arcsec. Our -component DW model (Fig.14) has a central component with n =2 . and r − = arcsec contributes ∼ Given that a multi-component DW-function fits the SB profiles ex-tremely well, our next major goal is to explore the conditions underwhich the 3D intrinsic luminosity density profiles can be describedwith a multi-component Einasto model. An attractive feature of c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals such an interpretation is that the Einasto profile can be a likelydescriptor of both the baryonic, and Λ CDM N-body dark matterhaloes, revealing an universality in their functional form.Deprojections, however, are generally not unique. In this sec-tion we review some of the limitations in obtaining unique depro-jections of surface density profiles that have been taken into consid-eration while providing the intrinsic 3D luminosity density profilesin the next section. Rybicki (1987) showed that based on the angle of inclination withthe line of sight i of an axisymmetric system, there exists a cone ofignorance ( θ =90 ◦ − i ), such that a family of densities – called konusdensities (Gerhard & Binney 1996) – that is non-zero only within θ , can project to yield zero SB unless the system is seen edge-on( i =90 ◦ ). The range of possible deprojections increases with de-creasing i . For triaxial systems this non-uniqueness increases dra-matically (Gerhard & Binney 1996). Kochanek & Rybicki (1996)further extend the range of possible functions through semi-konusdensities and their linear combinations. van den Bosch (1997) (hereafter, vdB97) highlight physical admis-sibility conditions that limit the range of possible semi-konus den-sities. Through a generalized set of semi-konus densities, he inves-tigated how they may (or may not) have an effect on the dynamicaland photometric properties of galaxies. We summarize here someof the key results from his work:1) For intrinsic axis ratio, q & . the effects of semi-konusdensities are negligible for almost all inclination angles. This issimilar to saying that spherically symmetric systems have uniquedeprojections.2) For oblate spheroids with a constant core, the presenceof konus densities manifests as detectable wiggles along the mi-nor axis, but not along the major axis, an effect seldom seen inreal galaxies. Although real galaxies seldom have perfect constantdensity cores, this can be an important distinguishing feature forshallow-cusp galaxies.3) Using a more representative, double power-law (Qian et al.1995) parametrization of the intrinsic 3D density of the central re-gions of real galaxies, vdB97 showed that even for an intrinsic axisratio of q = 0 . , the maximum amount of semi-konus density thatcan be added is negligible for inclination angles i & ◦ . This isseen for a wide range of cusp steepness, − α inside thecore radius, or break radius of the double power-law, say R b . Forsmaller inclination angles, i . ◦ , the maximum semi-konus den-sity that can be added increases as the ratio of the scale-length ofthe konus density (say r k ) to the core radius of the galaxy decreases(refer to his fig. 8).A consequence of this effect of decreasing r k /R b is that thesemi-konus density does not add significantly to the mass, or light.Furthermore, as r → the mass of the central supermassive blackhole will completely overwhelm any contribution of the konus den-sity. It should also be noted that the amplitude shown in his fig. 8is the ratio of maximum konus-density at r = 0 to the power-lawgalaxy density at r = R b . Since galaxy densities continue to risefor r < R b , this means the relative strength of konus to galaxydensity is even smaller at comparable r .4) The generalized konus densities are not power-laws. How-ever, to characterize their effect in terms of the slopes of the konus densities (not to be confused with slope of the galaxy SB profile),vdB97 approximates them as power-laws. He showed that if onecan approximate ρ konus ( r ) ∝ r − α , then α must be . i.e., thekonus densities themselves can not be too cuspy. The discussion in the previous section applies to axisymmetric sys-tems. For triaxial systems the range of possible deprojections in-creases. However, this increase in non-uniqueness will also dependon the degree of triaxiality.Triaxiality as well as axisymmetry can leave an imprint onthe kinematic structure of galaxies. Emsellem et al. (2004) providesdetailed kinematic maps of 48 E/S0 galaxies from the SAURONsurvey (de Zeeuw et al. 2002); eight of these galaxies are part ofour sample. Using an estimate of their specific angular momen-tum λ R , within one effective radius R E , Emsellem et al. (2007)characterized galaxies as fast ( λ R > λ R < M BH ∼ . M gal ) are not able to sustain a triaxial shape.This indicates that the steep-cusp galaxies which are typically fastrotators can be very well approximated as axisymmetric systems.For galaxies with shallow cusps which typically rotateslowly, theoretically, triaxiality can not be excluded. However,Cappellari et al. (2006) observe that significant triaxiality will leadto an increase in the otherwise relatively small scatter in the M/L- σ relation of the SAURON sample deduced using axisymmetricmodels. It thus appears that the shallow cusps may not be stronglytriaxial. In our sample the shallow-cusp galaxies also appear nearlyspherically symmetric in projection and hence we model them asoblate axisymmetric systems as well. In addition to the above issues pertaining to konus densities,Merritt & Tremblay (1994) and Gebhardt et al. (1996) highlight theimportance of non-parametric deprojections. This should be the ap-proach of choice to reveal the range of possibilities in 3D sincesmall deviations from fits to the SB with ad-hoc fitting functionscan translate into larger deviations in the 3D distribution.However, with very high precision data as used in this paper,large deviations, or features in 3D will manifest themselves as de-tectable, but possibly small features in the 2D SB profiles. Uponprojection, smaller features in 3D may remain hidden within mea-surement errors. Hence if the goal is to extract smaller, local fea-tures in the 3D distribution, then a non-parametric inversion mustbe performed.Our goal in this paper is, however, to investigate whether thegross properties of the 2D and 3D distribution of light can be de-scribed with a multi-component Einasto model. Further, in section10.1 we show that for two galaxies in our sample that have non-parametric deprojections in the literature, our parametric estimatesare in good agreement with the non-parametric estimates. We hencedo not take a non-parametric option and note that our inferred in-trinsic profile is likely to fall within the confidence intervals of anon-parametric deprojection. c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams 10 Intrinsic 3D luminosity density profiles In this section we discuss the luminosity density profiles of galaxies in our sample where non-uniqueness of deprojection canbe reasonably minimized. Based on the discussion in section 9.3we model all galaxies under the assumption of oblate axisymmetry.For oblate axisymmetric systems with intrinsic axis-ratio q and with a minor axis inclined at an angle i with the line of sight( i = 90 ◦ is edge-on), it can be shown that the surface brightness isgiven by: Σ( R ) = 2 q p cos ( i ) + q sin ( i ) Z ∞ ρ ( r ) dζ (10.1)where, R is a coordinate along the projected major axis, which foroblate axisymmetric systems is the same as the true major axis; ζ = z/q is the reduced coordinate along the line of sight and r = p R + ζ .It can be further shown that the observed axis-ratio, q ′ , is givenby: q ′ = q cos ( i ) + q sin ( i ) (10.2)When the SB is described in terms of the semi-major axisusing a spherically symmetric function, generally from equation(10.1), it can be seen that the inferred intrinsic central density ρ ∝ Σ ( q ′ /q ) . An estimate of the light enclosed within an intrin-sic radius r , measured along the semi-major axis, when expressedin terms of Σ , does not require knowledge of the true axis ratio q and inclination angle i , and is given by L ( r ) ∝ Σ q ′ .For the Einasto profile, equation (1.2), using (5.1) and (5.2)this leads to: ρ = Σ b n r s Γ( n + 1) (cid:18) q ′ q (cid:19) , (10.3)and, L ( r ) = 2 π (cid:18) r s b n (cid:19) γ (cid:16) n, b [ rr s ] n (cid:17) Γ( n ) Σ q ′ (10.4)The ρ ( r ) and L ( r ) profiles in Figures 25-38 have been obtainedusing equation (10.3) in (1.2) and (10.4).Table 2 lists the inclination angles (and the corresponding ref-erences) used to obtain the intrinsic luminosity density profilesand the cumulative light enclosed as a function of intrinsic 3Dradius r in Figures 25-38. For galaxies with observed axis ratio & . for which we could not find inclination angles in the lit-erature, we infer their 3D intrinsic luminosity density by assum-ing an arbitrary inclination angle i = 85 ◦ , to distinguish themfrom galaxies with i = 90 ◦ estimated from modelling. Giventhat the residuals of our fits to the 2D surface brightness (median rms = 0 . mag arcsec − ), are similar to the rms of randomerrors of the data ∼ . mag arcsec − , we can be fairly con-fident that our parametric description of the 3D distribution as amulti-component Einasto profile will fall within the wider confi-dence limits of non-parametric inversion. This is especially true atlarge R and for galaxies with uniformly low residuals over the en-tire dynamic radial range. This is however not true for cases likeNGC4382 where our -component model has systematic residualssignificantly larger than measurement errors. We also do not pro-vide luminosity density profiles for NGC4406 which has a strongcentral dip in its SB profile, as well as galaxies with large inclina-tion angles, for example, NGC4261. Table 2. Luminosity density sampleName q ′ DW q ′ ref i Reference(deg)NGC4472 0.806 0.83 90 van der Marel, Binney & Davies (1990)NGC4486 0.722 0.96 90 Cappellari et al. (2007)NGC4649 0.828 0.90 90 Shen & Gebhardt (2010)NGC4365 0.717 0.75 68 van den Bosch et al. (2008)NGC4636 0.760 —– 85 Arbitrary assumed i NGC4552 0.873 0.96 90 Cappellari et al. (2007)NGC4621 0.742 0.66 90 Cappellari et al. (2007)NGC4478 0.822 —– 85 Arbitrary assumed i NGC4434 0.928 —– 85 Arbitrary assumed i NGC4473 0.607 0.61 73 Cappellari et al. (2007)NGC4458 0.879 0.88 90 Cappellari et al. (2007)NGC4467 0.813 —– 85 Arbitrary assumed i VCC1627 0.928 —– 85 Arbitrary assumed i VCC1199 0.869 —– 85 Arbitrary assumed i NOTES.— Sample of galaxies whose intrinsic (3D) luminosity profiles are shownin Figs.25-38. Refer to sections 9 and 10. In the columns above: Galaxy type arefrom KFCB09; q ′ DW is the characteristic axis ratio deduced using equation (6.3); q ′ ref and i are the axis-ratio and inclination angle of the minor-axis with respect tothe line-of-sight, from the reference listed in the last column. For some galaxies thatare fairly round but for which we could not find an inclination angle in the literature,we have assumed an arbitrary inclination angle of i = 85 ◦ . Some of the galaxies for which we present the intrinsic lumi-nosity density, were also deprojected by other authors. The -component model for NGC4649, Fig. 27, can be compared to thenon-parameteric inversion of the same data set (from KFCB09) pre-sented in Shen & Gebhardt (2010) (SG10). Note that in this pa-per we use values for observed axis ratio q and inclination angle i quoted by SG10. The overall profiles are consistent once a correc-tion is made for the distance to the galaxy; SG10 adopt . Mpc,while we use . Mpc. The central luminosity density is steeper inSG10. However, one must note that the central-most points in thesurface brightness have larger errors, arising primarily from psf-deconvolution.Also as mentioned in section 6, including a third componentimproves the fit to the SB in the central regions (Fig. 3), making itconsistent with the ∼ . mag arcsec − errors in the central-mostdata points. We do not use the -component model because the F-test rejects it at . σ . One can thus use the -component modelpurely as a fitting function (without making strong inferences onthe resulting components) and in doing so we find very good agree-ment with the non-parametric deprojection in SG10. This indicatesthat a subdued third component could well exist, as shown in Fig. 3.A similar comparison of our -component parametric model(Fig. 2) with a non-parametric deprojection of NGC4486 (M87) inGebhardt & Thomas (2009) also shows that they are consistent. Most of the galaxies for which we present luminosity density pro-files are fairly round: ten are E0-E2, and four are E3/E4. Twelvegalaxies have large, ∼ ◦ inclination angles, and two are at i ∼ ◦ . None have any wiggles or dips in the central regions oftheir SB profiles. These considerations ensure that the contribu-tion of semi-konus densities to the density profile is small, or non-existent. Further, our multi-component models have residuals con-sistent with measurement errors over a large dynamic radial rangethat allows for a parametric deprojection. c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals -7 -6 -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ DW =0.81, i vM =90 o q int =0.81r =25.59R =18.88 NGC 4472 total Einasto densityr =0.382, n=0.717r =2.09, n=1.860r =29.836, n=5.359 0.01 0.1 1 10 100 Intrinsic major-axis radius (r) in kpc q’ SN =0.96, i SN =90 o q int =0.96r =50.98R =37.29 NGC 4486 total Einasto densityr =0.031, n=1.090r =3.957, n=2.619r =75.18, n=6.451 Figure 25. Multi-component Einasto models of the intrinsic (3D) volume luminosity density profiles for galaxies in our sample (see Table 2 andsection 10) assuming oblate-axisymmetry. Only the statistically significant best-fitting models (Table 1) are shown. The component profiles have beencomputed using equation (10.3) in (1.2). Colours and line types are as in Fig.1. The figure keys list the Einasto shape parameter n and the intrinsic (3D)effective or half-light radius of the components (in kpc), estimated from the best-fitting values of r − , using equation 5.8. The total half-light radii inkpc, intrinsic ( r E ) and projected ( R E ), are also shown separately in the figure panel. Also listed are the observed axis ratio q ′ , the inclination angle i of the minor axis to the line-of-sight with suffixes labelling the references(see Table 2) – DW (this paper), vM (van der Marel et al. 1990), SN standsfor SAURON (Cappellari et al. 2007), SG(Shen & Gebhardt 2010) and vB(van den Bosch et al. 2008). For some galaxies that are fairly round and forwhich we could not find an inclination angle in the literature, we have assumed an arbitrary i = 85 ◦ . These cases are labelled as i arb . Generally weuse q ′ from the same reference that contains the i listed in Table 2. However, if q ′ DW ≈ q ′ of the reference, we use q ′ DW . The intrinsic axis ratio q int is computed using equation 10.2. The horizontal axis showing the intrinsic (3D) radius (in kpc) is up to ∼ . × the projected radius of available data. Above NGC4472 (left) and NGC4486 (right). (Colour versions of these figures are available in the online edition.) We conclude that for the galaxies presented in this section, the 3D intrinsic density profile can be described with a multi-componentEinasto model. These galaxies span a wide range of luminosities − < M V T < − and belong to both the steep-cusp andshallow-cusp families. It is therefore likely that the intrinsic 3D baryonic density of other ellipticals can also be described with amulti-component Einasto model. Here we note a few important factors that may affect our interpretation of the intrinsic luminosity profiles.i) We have assumed a constant ellipticity while galaxies seldom have constant ellipticity. A varying ellipticity can be incorporated inour multi-component models for a more accurate deprojection.ii) We show luminosity density profiles of galaxies which we could justify as oblate axisymmetric systems. However, some galaxiesare prolate while some are triaxial and some have a combination of axisymmetric and triaxial regions. For such galaxies, the 3Dintrinsic density should not be inferred from a SB analysis alone. One may assume a multi-component Einasto profile for the intrinsicdensity and use additional kinematic information to include triaxiality in the kinematics to constrain the Einasto profile parameterseither in 3D or in 2D through the DW-function.iii) The luminosity density profiles are subject to uncertainties in estimating the absolute magnitude (which are subject to uncertaintiesin distance measurements) and Galactic extinction. These uncertainties however do not affect the overall shape of the density profile. c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.81r =25.59R =18.88NGC 4472 total 3D luminosityr =0.382, n=0.717r =2.088, n=1.860r =29.84, n=5.359 0.01 0.1 1 10 100 Intrinsic major-axis radius (r) in kpc q’ DW =0.72r =50.98R =37.29NGC 4486 total 3D luminosityr =0.031, n=1.090r =3.957, n=2.619r =75.18, n=6.451 Figure 26. Cumulative light enclosed within an intrinsic (3D) radius r ( kpc ) for the galaxies for which we infer the intrinsic luminosity density (see Table 2and section 10). The component contributions have been estimated using (10.4). Profiles are shown only for the statistically significant best-fitting models(Table 1). The observed axis ratio q ′ and the total half-light radii, intrinsic r E and projected R E , are also shown separately within the figure panels. Coloursand line types are as in Fig.1 and details on figure keys and labels are as in Fig.25. Above NGC4472 (left) and NGC4486 (right). (Colour versions of thesefigures are available in the online edition.) -6 -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ SG =0.90, i SG =90 o q int =0.90r =13.78R =10.20NGC 4649 total Einasto densityr =0.9504, n=0.997r =14.94, n=5.444 0.01 0.1 1 10 100 Intrinsic major-axis radius (r) in kpc q’ DW =0.72, i vB =68 o q int =0.66r =28.07R =20.70 NGC 4365 total Einasto densityr =0.548, n=0.970r =5.065, n=3.720r =43.94, n=7.233 Figure 27. Intrinsic (3D) luminosity density for NGC4649 (left) and NGC4365 (right). Refer to caption of Fig.25 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.83r =13.78R =10.20NGC 4649 total 3D luminosityr =0.950, n=0.997r =14.94, n=5.444 0.01 0.1 1 10 100 Intrinsic major-axis radius (r) in kpc q’ DW =0.72r =28.07R =20.70NGC 4365 total 3D luminosityr =0.548, n=0.970r =5.065, n=3.720r =43.94, n=7.233 Figure 28. Cumulative volume luminosity profile for NGC4649 (left) and NGC4365 (right). Refer to caption of Fig.26 for details. -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ DW =0.76, i arb =85 o q int =0.76r =25.44R =18.85 NGC4636 total Einasto densityr =0.6325, n=1.282r =26.25, n=6.064 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ SN =0.96, i SN =90 o q int =0.96r =12.51R =9.14 NGC 4552 total Einasto densityr =0.089, n=0.754r =1.084, n=3.306r =20.31, n=6.528 Figure 29. Intrinsic (3D) luminosity density for NGC4636 (left) and NGC4552 (right). The vertical marker on the profile for NGC4552 shows the inner radiusbeyond which the luminosity density profile shown, can be trusted. This is because the best fitting -component DW model for this galaxy (Fig.10) is not agood representation of the surface brightness profile within . arcsec where the light is affected by the variable UV flare activity interpreted to arise froma low-level AGN (Renzini et al. (1995), Cappellari et al. (1999)). Also refer to caption of Fig.25 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.76r =25.44R =18.85NGC 4636 total 3D luminosityr =0.633, n=1.282r =26.25, n=6.064 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ SN =0.96r =12.51R =9.14NGC 4552 total 3D luminosityr =0.089, n=0.754r =1.084, n=3.306r =20.31, n=6.528 Figure 30. Cumulative volume luminosity profile for NGC4636(left) and NGC4552(right). The vertical marker on the profile for NGC4552 shows the in-ner radius beyond which the profile shown, can be trusted. This is because the best fitting -component DW model for this galaxy (Fig.10) is not a goodrepresentation of the surface brightness profile within . arcsec . Refer to caption of Fig.26 and Fig.29 for details. -6 -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ SN =0.66, i SN =90 o q int =0.66r =7.62R =5.72 NGC 4621 total Einasto densityr =0.0161, n=3.714r =4.669, n=0.985r =8.76, n=9.561 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ DW =0.82, i arb =85 o q int =0.82r =1.45R =1.08 NGC 4478 total Einasto densityr =0.0042, n=0.941r =0.074, n=1.217r =1.49, n=2.641 Figure 31. Intrinsic (3D) luminosity density for NGC4621 (left) and NGC4478 (right). Refer to caption of Fig.25 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.74r =7.62R =5.72NGC 4621 total 3D luminosityr =0.016, n=3.714r =4.669, n=0.985r =8.76, n=9.561 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ DW =0.82r =1.45R =1.08NGC 4478 total 3D luminosityr =0.004, n=0.941r =0.074, n=1.217r =1.49, n=2.641 Figure 32. Cumulative volume luminosity profile for NGC4621 (left) and NGC4478 (right). Refer to caption of Fig.26 for details. -6 -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ DW =0.93, i arb =85 o q int =0.93r =1.73R =1.27 NGC 4434 total Einasto densityr =0.015, n=1.238r =2.78, n=0.576r =1.471, n=5.532 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ DW =0.61, i SN =73 o q int =0.56r =4.26R =3.13 NGC 4473 total Einasto densityr =0.5319, n=2.300r =5.47, n=5.649 Figure 33. Intrinsic (3D) luminosity density for NGC4434 (left) and NGC4473 (right). Refer to caption of Fig.25 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.93r =1.73R =1.27NGC 4434 total 3D luminosityr =0.015, n=1.238r =2.776, n=0.576r =1.47, n=5.532 0.01 0.1 1 10 Intrinsic major-axis radius (r) in kpc q’ DW =0.61r =4.26R =3.13NGC 4473 total 3D luminosityr =0.532, n=2.300r =5.47, n=5.649 Figure 34. Cumulative volume luminosity profile for NGC4434 (left) and NGC4473 (right). Refer to caption of Fig.26 for details. -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ DW =0.88, i SN =90 o q int =0.88r =1.74R =1.28NGC 4458 total Einasto densityr =0.023, n=1.502r =0.12, n=3.078r =2.073, n=3.149 0.01 0.1 1 Intrinsic major-axis radius (r) in kpc q’ DW =0.81, i arb =85 o q int =0.81r =0.51R =0.38NGC 4467 total Einasto densityr =0.0317, n=2.673r =0.54, n=2.402 Figure 35. Intrinsic (3D) luminosity density for NGC4458 (left) and NGC4467 (right). Refer to caption of Fig.25 for details.c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.88r =1.74R =1.28NGC 4458 total 3D luminosityr =0.023, n=1.502r =0.125, n=3.078r =2.07, n=3.149 0.01 0.1 1 Intrinsic major-axis radius (r) in kpc q’ DW =0.81r =0.51R =0.38 NGC 4467 total 3D luminosityr =0.032, n=2.673r =0.54, n=2.402 Figure 36. Cumulative volume luminosity profile for NGC4458 (left) and NGC4467 (right). Refer to caption of Fig.26 for details. -5 -4 -3 -2 -1 V o l u m e l u m i no s it y d e n s it y ρ (r) ( L ⊙ p c - ) Intrinsic major-axis radius (r) in kpc q’ DW =0.93, i arb =85 o q int =0.93r =0.37R =0.27VCC 1627 total Einasto densityr =0.0175, n=2.002r =0.38, n=2.907 0.01 0.1 1 Intrinsic major-axis radius (r) in kpc q’ DW =0.87, i arb =85 o q int =0.87r =0.21R =0.16VCC 1199 total Einasto densityr =0.0135, n=1.896r =0.23, n=2.389 Figure 37. Intrinsic (3D) luminosity density for VCC1627 (left) and VCC1199 (right). Refer to caption of Fig.25 for details.c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams V o l u m e l u m i no s it y L (r) ( L ⊙ ) Intrinsic major-axis radius (r) in kpc q’ DW =0.93r =0.37R =0.27VCC 1627 total 3D luminosityr =0.017, n=2.002r =0.38, n=2.907 0.01 0.1 1 Intrinsic major-axis radius (r) in kpc q’ DW =0.87r =0.21R =0.16VCC 1199 total 3D luminosityr =0.013, n=1.896r =0.23, n=2.389 Figure 38. Cumulative volume luminosity profile for VCC1627 (left) and VCC1199 (right). Refer to caption of Fig.26 for details. 11 Mass deficit in massive ellipticals Peebles (1972) predicted that a compact massive object, of mass M , can alter the density structure around its radius of influence, r inf = GM/σ ∗ (11.1)such that for r < r inf a stellar density cusp (in 3D) ∝ r − / forms with a velocity dispersion σ ( r ) ∝ r − / ; and a core, of con-stant density and velocity dispersion, forms within a core-radius r c >>r inf . Here σ ∗ is the velocity dispersion at r>>r inf .Begelman, Blandford, Rees (1980) discuss the formation,evolution, eventual coalesence and possible recoil, of a binarySMBH, as a result of galaxy mergers. Since then, N-body simu-lations have shown that the evolution of the binary leads to a co-evolution of the density profile of the galaxy, in a region around theradius of influence, due to ejection of an amount of mass propor-tional to the mass of the binary.Our work, describing the structure of ellipticals as a superpo-sition of DW-profiles, implies the presence of excess light in thecentre of all ellipticals with respect to an inner extrapolation of theouter DW-components. We emphasize that our models describe thecurrent structure, post any modification by the binary, and can beconsistent with mass ejection by SMBHs.In this section we discuss a related concept of mass deficit inmassive ellipticals, which has its genesis in -i) the identification of massive ellipticals exhibiting a shal-lower surface brightness (SB) profile in their central regions, in-wards of a projected break radius, with respect to the trend of SBoutside the break radius - i.e., a light deficit. This is in contrast toless massive ellipticals that seem to contain steeper light profilesinside the break radius - i.e., a light excess;ii) the assumption that evolution of the binary SMBH is thesole factor responsible for the observed break radius; and iii) the assumption that in the absence of the binary SMBH,the trend (functional form) of SB outside the break radius wouldhave continued its form all the way to the centre of the galaxy.In section 11.3 we show that estimates of ’observed’ deficitin real galaxies, made with respect to an inner extrapolation of theSB profile outside the break radius, not only have large variationwith respect to predictions from N-body simulations but also varywidely between different researchers for the same galaxy. We alsocompare signatures of mass-ejection by the binary SMBH with theobserved profiles of real massive ellipticals and find that currentpredictions from N-body simulations are not able to account forsome of the largest ’cores’ in Virgo galaxies. Prior to such discus-sion, in section 11.1 we briefly describe the various phases of evo-lution of the binary SMBH and in section 11.2 we review the pre-dictions from simulations about the amount of mass ejected, spa-tial extent and resulting slope of density profiles, in the context of’dry’ dissipationless (gas-free) mergers, believed to be the forma-tion mechanism for the massive ’core’ ellipticals. The discussion isthus directed towards systems with the most massive & M ⊙ SMBHs, as is the case for the ’core’ ellipticals in Virgo. The significant phases in the evolution of the binary SMBH are: Phase 1 : Subsequent to a galaxy-galaxy merger, a SMBH bi-nary is said to form when the SMBH separation reduces to a dis-tance r inf - the radius of influence of the more massive SMBH. Phase 2 : Their separation shrinks as the binary loses energydue to dynamical friction and by scattering of stars through thegravitational sling-shot mechanism. A ’hard-binary’ is eventually c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals defined to form when the separation reduces to a size, a hard = Gµ σ ∗ = q q ) GM σ ∗ (11.2)where, µ = M M /M is the reduced mass of the binary with q = M /M M = M + M is the combined total mass ofthe binary. a hard is thus a fraction of r inf (equation (11.1))At this stage, due to depletion of stars, the decrease inbinary separation can stall unless additional mechanisms cancontinue removing energy from the binary to drive it to-wards Phase 3. Berczik et al. (2006), Hoffman & Loeb (2007),Berentzen et al. (2009), Khan, Just & Merritt (2011) show mech-anisms through which stalling may easily be avoided. Also, seeDotti, Sesana & Decarli (2011) for a review and references therein. Phase 3 : The binary then continues to harden until its semi-major axis reduces to a size a gw where energy loss due to emissionof gravitational waves begins to dominate energy losses due to scat-tering of stars, driving the binary towards coalesence. Phase 4 : During the last few stable orbits before coalesence,the centre of mass recoils with a recoil or kick velocity that sen-sitively depends on the binary mass-ratio and spins, relative spinallignement and orientation of spin-axis with the angular momen-tum vector. If the kick velocity V kick > V esc , the escape velocity,the coalesced SMBH can be completely removed from the galaxy.For lower kick velocities, the SMBH performs damped oscillatorymotion in the central regions and eventually settles down to a Brow-nian motion about the centre. Volonteri, Haardt & Madau (2003) (V+03a) andVolonteri, Madau & Hardt (2003) (V+03b) have shown that,in galaxy mergers:i) SMBHs of mass ratio q = z ∼ . z . q< q ∼ ρ ∝ r − pro-files and mergers (accompanied by coalesence of the binary) notonly erase the cusps completely to form a constant density corebut also preserve such cores, they estimate that at the end of a suc-cession of mergers the resulting mass deficit, M def , is given by M def = ± M BH (equation 15 of V+03b). Note that most’core’ galaxies have a shallow cusp rather than a constant densitycore. The above estimate should therefore be considered a limitingvalue. At the same time, mass deficit due to the scouring effect ofSMBHs in Phase 4 is not accounted for in this study.iii) Even when the entire r − cusp is erased and a constantdensity core forms, the resulting core size is only ∼ 60 pc for a M ⊙ halo (fig.1 and 2 of V+03b) – a mass resembling themost massive ’core’ ellipticals in Virgo. For example, assuming M halo /L V = 50 the halo mass of M87, the giant ’core’ galaxyin Virgo with L V T = × M V ⊙ , is ∼ × M ⊙ . Thebreak radius for this galaxy is however ∼ 600 pc >> 60 pc coreformed by the merging binaries; a point we will return to in thenext section. V+03b also find that such core sizes are typically ∼ GM BH /σ ∗ =3 r inf . M def before coalesence Using Dehnen (1993) models – a power-law of the form ρ ( r ) ∝ r − γ at small r – to describe the intial density profiles, Merritt (2006) conducted 39 N-body simulations to study the evolutionupto the hard-binary stage (Phase 2), for the case of ’dry’ merg-ers in non-rotating spherically symmetric systems, including nineremergers. He showed that, after each merger:i) M def = q . M . For initial density profiles as Dehnen γ = q = (0.1,0.25) (V+03a), M def = (0.33,0.46) M (table 1, Merritt06).ii) there is a lowering of the density profile in a region aboutthe size of the radius of influence. This can be seen by comparing r ′ h from his table 1 with the radius at which reduction in densityoccurs after the first merger in his figs. 6a,6b and 6c.Merritt, Mikkola & Szell (2007), MMS07, study the evolutionin Phase 3. For intial Dehnen density profiles with γ = q = M =10 M ⊙ , they obtain M def = M . Whenadded to the mass deficit upto Phase 2, this yields M def ∼ M .Further, as can be seen from their figure 21, the net deficit at theend of Phase 3 is also within a region ∼ r h - the region of influencedefined as the radius containing a stellar mass equal to twice thetotal black hole mass. Note that r h is typically much less than theobserved break radius (’core’) of massive ellipticals. M def after coalesence Assuming that the binary coalesces and recoils with a kickvelocity (Phase 4), Boylan-Kolchin, Ma & Quataert (2004) andMerritt et al. (2004) investigate the resulting effects on the den-sity structure. They find that for V kick > (0.25-035) V esc , cores ofsize r h can form as a result of the kicks; with larger deficits for V kick In this section we investigate whether the signatures of mass deficitin N-body simulations are in congruence with the estimates of massdeficit from SB profiles of galaxies and highlight limitations of ex-isting models of galaxy structure - a single sersic, a core-sersic anda Nuker profile - in estimating such mass deficits as well as in con-firming predictions from simulations.In real galaxies, estimates of the amount of mass deficit andthe extent of break radius vary widely for the same galaxy. Thiscan be seen by comparing the mass deficits computed in Graham(2004) (their table 1), with data from Rest et al.(2001).For instance, for NGC4168, Graham04 estimated a projectedbreak radius, R b , of 0.72 arcsec (108 pc) and a M def = M BH using a CS profile, while estimates with a Nuker profile yield R b = M def = M BH ; for the later,the prescription in Milosavljevic & Merritt (2001) assuming thatthe unscoured intrinsic (3D) profile within the break radius was ρ ( r ) ∝ r − , has been used. However, Milosavljevic et al. (2002)(MMRvB02) estimate an R b = M def = M BH . Not only dothese estimates differ by a factor of for the same galaxy, eventhe most extreme estimates of M def from N-body simulations arenot able to account for deficits as large as (20-50) M BH .For NGC2986, MMRvB02 obtain a lesser estimate of M def = M BH from a larger break radius of 400 pccompared to the (Nuker,CS) profile estimates (Graham04) of M def = (26.7,7.02) M BH over a smaller R b = (174,94) pc.Assuming r − initial profiles, MMRvB02 show that on av-erage M def =10 M BH for the case of ellipticals. This estimate issimilar to the predictions of V+03a ( ∼ M BH ) and the averagevalue of 10 M BH in KFCB09 for Virgo ellipticals. However theydo not agree with the (2.4 ± M BH estimates of Ferrarese et al.(2006) (F+06) for the Virgo ellipticals, the (2.1 ± M BH es-timate of Graham04, the 2 M BH estimate of Hoffman & Loeb(2007) (HL07), including Virgo ellipticals, and the (2.29 ± M BH and (1.24 ± M BH estimates of Hyde et al. (2008).For the giant ’core’ elliptical in Virgo, M87(NGC4486),HL07 estimate 2.5 M BH , MMRvB02 estimate 8.7 M BH andKormendy & Bender (2009) (KB09) estimate 14.1 M BH , allscaled to M BH = × M ⊙ . Despite such large deficits, the SBprofile of M87 shows a distinct rising trend; apart from the fact thatit has an unsually large break radius. NGC4649 also has a fairlylarge break radius. While KB09 estimate 5 M BH , HL07 estimateonly 1 M BH , both for M BH = × M ⊙ . Another Virgo galaxywith an apparent ’core’ is NGC4382. KB09 estimate a deficit of 13 M BH , however Gultekin et al. (2011) find that this galaxy has anunusually small black hole mass, consistent with no black hole, yetobtain a mass deficit of 45.6 M BH .Estimates of M def /M BH do depend on a number of factorslike uncertainties in estimates of M BH , the assumed form of M/L etc. However, the large differences are generally due to discrep-ancy in – estimation of R b ; estimation of the inner power-law in-dex; and the huge uncertainty in our assumptions about what mighthave been the shape of the density profile prior to the action of theSMBHs. For instance, KFCB09 defines the central region of lightdeficit or ’extra-light’ based on the region over which a Sersic fitstheir estimate of the non-central region. F06 on the other hand fits a Core-Sersic model which defines the central region as the regioninside R b of the core-sersic model. Consequently the two meth-ods yield widely varying estimates of what the ’central region’ ofdeficit is. This leads to a large discrepancy in the estimated M def .Also see fig.2 of MMRvB02.Hopkins & Hernquist (2010)(HH10) suggest a new non-parametric method to obtain mass deficit and find that M def varieswith radius such that at ∼ pc, M def /M BH ’ assymptotes toa maximum of 0.5-2 ’ and for the largest galaxies at core-radius of ∼ kpc, M def = (2-4) M BH . While, this is similar to the estimatesof F+06 and Graham04, it is at odds with the KFCB09 estimateswhere the authors reasoned that M def ∼ M BH accounts for thenet mass deficit through Phase 4 (results from MMS07 and GM08)and in agreement with V+03a and MMRvB02.As discussed in section 11.2.3, M def = (10-20) M BH are hardto explain using the results of N-body simulations. However, evenif such were to be true, the resulting region over which KFCB09and MMRvB02 estimates such deficit is much larger than the re-gion of influence of the SMBH over which simulations predict massdeficits; a point also noted in MMRvB02. This is also true for esti-mates of R b using a Core-Sersic or Nuker model; especially for thelargest ’cores’ as in NGC4486 and NGC4649. The shallow cusps(’core’) are thus not entirely due to mass ejection by the SMBHs. Disagreements in the literature are limited not only to the amount,but also the sign of the deficit. In MMRvB02, NGC4478 is shownto have a mass deficit of 15.85 M BH . However, its SB profileshows an apparent ’cusp’ (see Fig.15, of this paper), due to whichF+06, Cote et al.(2007) (C+07), KFCB09 infer this galaxy to havea light (and mass) excess. Similarly in MMRvB02, NGC4473 hasthe highest M def = M BH amongst the Virgo ellipticals. F+06and C+07 also conclude that this galaxy has a mass deficit. ButKFCB09, argue that this galaxy has ’extra-light’ and not a deficit.Further, in this paper and in Hopkins et al. (2009 a,b), the authorsargue that all galaxies can be modelled with extra light above aninner extrapolation of the outer components.Thus, whether the SB profile of a galaxy ’exhibits’ an excessor deficit also depends on ones methodology. Given the large disagreements in estimates of M def ; the region af-fected by the binary SMBHs; and which profile best describes thelight of ellipticals; it can not be said that robust estimates of massdeficits, in real galaxies, have been made. Consequently, validat-ing predictions of N-body simulations and especially the stage(s)of SMBH evolution responsible for the observed deficit turn out tobe largely uncertain.Further, extrapolating the profile from the break radius, R b ,presents a number of conceptual difficulties:i) When the SB profiles of the massive ’core’ galaxies aremodelled with the Core-Sersic (F+06) or single Sersic (KFCB09)profiles, the non-central regions, beyond R b , generally have Ser-sic index m > ; sometimes as large as and . Larger the m ,steeper the density as R → and consequently larger the extrap-olated density. Since estimates of M def are correlated with an es-timate of R b and in most cases R b is much larger than the regionof influence of the SMBHs (example: NGC4486, NGC4649), ex-trapolating profiles with large Sersic indices will invariably implylarger, but incorrect, mass deficits (also see HH10). c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals ii) The SMBHs are believed to affect only the central regions,around the radius of influence, and not the global profile of thegalaxy. While it is believed that the mass deficit is due to ero-sion of the steep cusps, it remains to be explained what forms thelarge m ∼ R b . Further, baryonic effects are usually moredominant (over dark matter) in the central regions of all galaxies -core and cuspy alike - leading to relatively more concentrated cen-tral, than outer, regions (or components). This is precisely what ourmodelling indicates (section 8.1 and Fig.24). If this is true, it willnot be meaningful to extrapolate the outer component inwards toestimate the ’mass deficit’ and neither would comparing possiblyfundamentally different profiles of ’cuspy’ and ’core’ galaxies, toget non-parametric estimates of mass deficits, as in HH10.iii) If the massive ellipticals have formed through multiplemergers, it is likely that their outer profile has also been built up as acumulative effect. Consequently their Einasto or Sersic index oughtto have evolved with their merger histories. Recall that the large in-dices, for the massive galaxies, are due to the gradual fall-off oftheir extended stellar light well outside the central regions. We areinclined to believe that this index was different in the past and hasevolved over the merger history of these galaxies. The large spatialextent of these galaxies could also be indicative of a larger numberof mergers, compared to the much smaller steep cusp galaxies; apoint noted by KFCB09 as well. If this is true, and mergers alterthe shape (Sersic index) of the outer profile, then extrapolating thecurrent observed outer Sersic profile (index) to estimate the initialdensity profile will give us an incorrect estimate of the mass deficit.In this paper, we have shown that all galaxies, shallow andsteep cusps, can be modelled with very high precision over a largedynamic range, as a superposition of DW-components and that theyall have a ’light excess’ in their central regions with respect to an in-ner extrapolation of their outer components. As shown in section7,not only do the DW-models fit the SB profiles better than the Core-Sersic and Nuker profiles, at least for three galaxies the centralcomponent correlates with a real physical system and is not a meremathematical construction of the fitting process.Mass ejection and consequently some deficit due to core-scouring binary SMBHs could well have occurred and these arelikely to have shaped the central components of our DW modelsas well. For instance, in this work, we find that the central compo-nent of all ’core’ galaxies have an Einasto index n ∼ 1. It will beinstructive to see if such components can be robustly isolated inSPH + N-body simulations and how well can they be fit with n ∼ 12 The outer n of ellipticals In this section, we present two empirical speculations about thestructure and formation of elliptical galaxies.(1) Our multi-component models reveal an interesting prop-erty of the most luminous, & L V ⊙ galaxies in our sample. TheEinasto shape parameter n of their outer component is very similarto the n of pure dark matter haloes; 5 . n . 8, as shown in Fig. 39.We remind the reader that because dark matter haloes are well fitwith Einasto profiles (N+04, M+06), a direct comparison with the n of our galaxies is possible. † The plus symbols in Fig. 39 represent 26 galaxy-size dark mat-ter haloes compiled from the high resolution Λ CDM N-body simu-lations of Diemand, Moore & Stadel (2004), Navarro et al. (2004),Merritt et al. (2006), Prada et al. (2006), Stadel et al. (2009) andNavarro et al. (2010). We assume that M of these haloes is per cent of M gal , the combined mass of the dark matter andbaryons. To translate the stellar mass of our Virgo galaxies to M gal we have assumed (i) a stellar mass-light ratio M ⋆ /L V T = per cent of M gal .Observe that the masses of the galaxies hosted by these dark matterhaloes, ∼ - M ⊙ , are comparable to those of the low andhigh luminosity ellipticals in our sample, − < M V T < − .If we interpret the dark matter simulation results to meanthat Einasto shape parameter 5 . n . n outer ∼ n N − body , was also shaped by collisionless processes.This conclusion is broadly consistent with the prevailing notion thatmassive ellipticals form through dry, dissipationless mergers.(2) Ferreras, Saha & Williams (2005) and Ferreras et al.(2007) have shown using a combination of strong lensing and stel-lar population synthesis models that the outer regions & R E ofmassive galaxies are more dark matter dominated than those ofsmaller galaxies. Cappellari et al. (2006) arrive at a similar conclu-sion that fast-rotating galaxies, which are generally low-luminositywith steep cusps, have relatively lower dark matter content thanthe slow-rotating galaxies which are usually massive with shal-low cusps. Also, Auger et al. (2010) observe that the mean darkmatter fraction within R E / increases with galaxy size and mass.These observations are consistent with Planetary Nebula observa-tions (Douglas et al. (2007), Napolitano et al. (2007,2009,2011),Tortora et al. (2009)) that the outer regions of small and intermedi-ate mass ellipticals have varying degree of low dark matter content,while dark matter is quite dominant in the outer regions of massiveellipticals. These observations indicate that all massive ellipticalsare dark matter dominated. † Papers describing fits to the haloes in N-body simulations use the recip-rocal of our shape parameter, α =1 /n , and typical α values hover around0.17 which corresponds to n = (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams E i n a s t o i nd e x n log (Total Mass (dark+baryons) M ⊙ ) N-body haloesGalaxies with L VT > 10 L ⊙ Galaxies with L VT < 10 L ⊙ Figure 39. Einasto index n of Λ CDM N-body haloes and that of the outercomponent of the Virgo galaxies in this paper. The outer component ofthe most luminous (massive) galaxies have an n similar to that of the N-body haloes. This is not true for most of the lower luminosity ellipticals.The solid box at n = n = n may be uncertain (section 6.2). The other solidbox at n = Since all the massive ellipticals in our sample have outer 5 . n . 8, this could indicate that such galaxies are likely to be darkmatter dominated. There might even be a relation between the outer n of the galaxy’s SB profile and its dark matter content. 13 Summary and Discussion Critical to our understanding of galaxies, their formation and evo-lution, is our ability to accurately quantify the galaxies’ structuralproperties and their variation with mass. The Sersic profile is thesingle most commonly used function for describing galaxy lightdistribution. The works of Caon, Capaccioli, Einasto, Ferrarese,Graham, Kormendy, Lauer and their collaborators have revolution-ized our understanding of galaxy structure. These authors haveshown us that a single Sersic profile does not fit the surface bright-ness distribution of ellipticals consistent with measurement errorsover a radial range larger than - decades. To extend the radialrange of the fit, especially in to the central regions, the core-Sersic( parameters) and the Nuker ( parameters for the central - arcsec) profiles have been introduced, with a further addition of aKing model ( more parameters) for the nuclear region.But even with these flexible and multi-parameter models, thefit residuals often exceed the measurement errors in some radialranges, in spite of the overall rms of residuals being low due to thelarge regions over which the Sersic profile fits well. Double Sersicmodels (Hopkins et al. 2009 a,b) provide an improvement, but stillthe residuals remain larger than measurement errors.In addition to fit residuals exceeding measurement errors, theSersic profile has another drawback. The galaxy structure, dynam-ics and evolution exist in 3D, so it is more meaningful to describegalaxies using 3D functions. The Sersic profile is an intrinsically2D distribution whose deprojections preassumes an infinite 3D ex-tent. If, for example, a galaxy has a truncation radius in 3D, a Sersicprofile will not be able to model that. The parametric forms for de- projecting a Sersic profile (and power-laws, for that matter) are notwell behaved near the centre. Further, these analytic functions areeither reasonable but not extremely accurate (PS97, LGM99), orare accurate but extremely complicated (BG11).To overcome the above limitations, we propose that the 3D lu-minosity density may be described with a multi-component Einastoprofile whose parameters can be estimated by modelling the surfacebrightness using a multi-component DW-function. This is similar tothe observations of Einasto and collaborators with a small sampleof spiral (and the giant elliptical M87), but extends the idea to amuch larger and diverse sample of shallow and steep cusp ellipti-cals and also allows for a direct parametric description of the 2Dsurface brightness profiles.We model the surface brightness (SB) profiles of ellipti-cals in and around the Virgo cluster with a multi-component DW-function and summarize our observations as follows:1. Multi-component DW-function fits the SB profiles of el-lipticals with residuals consistently comparable to measurement er-rors over large dynamic ranges ∼ in radii, and ∼ in SB,with a median sample rms of . mag arcsec − . Nine galax-ies are well described with a -component (central and outer) DWmodel, while for fourteen galaxies a third component was required,and confirmed through an F-test. The third component acts as an in-termediate component between the central and outer components,except in NGC4621 and NGC4434 where it appears to be embed-ded within the outer component.2. All steep-cusp and shallow-cusp galaxies reveal a centralcomponent that is in excess to an inward extrapolation of the outercomponent. Its shape parameter n is usually less than that of theouter component, which implies, as expected, that the central com-ponent is more concentrated than the outer component. Exceptionsare NGC4459, NGC4387 and NGC4467 (section 8.3).3. The central component of all massive shallow-cusp galax-ies have n . , while those of steep-cusp galaxies generally have n > , although there are some cases with . n < (section 8.3).This indicates that the central components of shallow-cusp galaxiesa) are more concentrated than that of the steep-cusp galaxieseven though the later are more denser in their central regions; andb) could signal the presence of disk-like systems; however,this must be verified spectroscopically.4. The central component of the shallow-cusp galaxies is farmore luminous (and massive) and spatially more extended (large r − or r E ) than that of the steep-cusp galaxies. Further, withina factor of two, all galaxies appear to host a similar fraction oftotal light in their central+intermediate components, with a weakindication that massive galaxies may be hosting a larger fraction.The last point is inconclusive due to the small number of massiveellipticals in our sample.5. In most of the shallow-cusp galaxies the outer compo-nent makes a comparable contribution to the density in the centralregions with respect to that of the central component, while in thesteep-cusp galaxies the central component is dominant.From the modelling point of view (section 8.3), the shallow-cusp feature usually seen in massive galaxies is due to a combina-tion of a larger r − , a low n . implying a more concentratedcomponent, and the non-negligible contribution to the density inthe central regions from the n & outer component.6. Galaxy formation models indicate that in massive ellipti-cals, the formation, evolution and subsequent coalesence of binary c (cid:13) , 000–000 urface Brightness and Luminosity of Ellipticals SMBHs can remove nearly (2-3) M BH of stellar material from thecentral regions, in a typical merger (refer to iii, section 11.2.3),leading to mass deficits. Our observation that the central compo-nent in shallow-cusp galaxies are already massive, can be used toconstrain galaxy formation models by accounting for the amountof mass that must have assembled in these galaxies prior to massejection by the binary SMBH’s.It is possible that the current shapes of the central as well asthe intermediate DW-components, within the central kiloparsec orso, have been influenced by the evolution of the binary SMBHs;and consequently some deficit could well have occured. However,the amount and sign of such deficits depend on estimating a breakradius R b , and assuming that the functional form of the densityprofile beyond R b can be extrapolated all the way to the centre.This may not be meaningful and has led to large disagreements inthe literature – both for a given galaxy as well as for averages over asample. We note that core-scouring by SMBHs is unlikely to be thesole mechanism for producing some of the largest ’cores’ (shallow-cusps), and other processes in galaxy formation and evolution arelikely to have played their role as well in forming this feature. Insuch a case extrapolating the density profile inwards from R b , to estimate the mass deficit, will yield misleading results.7. For of the galaxies we could describe the intrin-sic 3D luminosity density distribution fairly uniquely with a multi-component Einasto model (section 10). Since these galaxies span awide range of luminosity − 8. Further, PNe and stronglensing observations indicate that massive ellipticals are more darkmatter dominated than less massive ellipticals. This indicates thatour result – the outer component of the surface brightness profilesof massive galaxies has 5 . n . n for their outer (major) component are dark matter dominated.10. In section 11, we have shown that the shapes of cen-tral and global profiles of shallow- and steep- cusp galaxies differmarkedly and could be a result of differing formation pathways.In section 8.3 we have shown that the shallow-cusps can be typ-ically described as systems with - i) central DW-components of low n ( . ) and large scale radius r − or r E , and ii) outer DW-component of large n ( & ) and central density comparable to thatof the central DW-component. And in section 12 we have shownthat large outer n systems are likely to be dark matter dominated.Hence, if dark matter has played a role in the formation andevolution of massive shallow-cusp galaxies, it will be instructive toexplore its role, if any, in shaping the central regions and conse-quently in forming the shallow cusps.Finally we note that the galaxies modelled in this paper are allin and around the Virgo Cluster. 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