Total mass distributions of Sersic galaxies from photometry $&$ cent\ ral velocity dispersion
aa r X i v : . [ a s t r o - ph . I M ] F e b Astronomy&Astrophysicsmanuscript no. ms˙chakrabarty˙jackson˙printer c (cid:13)
ESO 2018October 6, 2018
Total mass distributions of Sersic galaxies fromphotometry & central velocity dispersion Dalia Chakrabarty and Brendan Jackson School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, U.K. e-mail: dalia.chakrabarty @ nottingham.ac.uk Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK e-mail: [email protected]
October 6, 2018
ABSTRACT
Aims.
We develop a novel way of finding total mass density profiles in Sersic ellipticals, to about 3 timesthe major axis e ff ective radius, using no other information other than what is typically available for distantgalaxies, namely the observed surface brightness distribution and the central velocity dispersion σ . Methods.
The luminosity density profile of the observed galaxy is extracted by deprojecting the measuredbrightness distribution and scaling it by a fiduciary, step-function shaped, raw mass-to-light ratio profile( M / L ). The resulting raw, discontinuous, total, 3-D mass density profile is then smoothed according to aproposed smoothing prescription. The parameters of this raw M / L are characterised by implementing theobservables in a model-based study. Results.
The complete characterisation of the formalism is provided as a function of the measurements ofthe brightness distribution and σ . The formalism, thus specified, is demonstrated to yield the mass densityprofiles of a suite of test galaxies and is successfully applied to extract the gravitational mass distribution inNGC 3379 and NGC 4499, out to about 3 e ff ective radii. Key words.
Methods: analytical – Galaxies: fundamental parameters (masses)
1. Introduction
Any evaluation of the total mass in distant galaxiesis a struggle against the paucity of available obser-vational evidence. Photometry is hardly enough toindicate the content of both the luminous as wellas the dark matter, unless the functional dependencebetween luminosity content and dark mass is ac-cessible. This is of course not the case; the exis-tence of such a relation is itself uncertain, espe-cially in early type galaxies. While kinematic in-formation of tracers has often been advanced as in-dicators of mass distributions in galaxies, the im-plementation of such information is tricky, pri-marily because of the mass-anisotropy degeneracy.Thus, one often resorts to cleverly designed ob-servational techniques and / or algorithms and for-malisms which thrive even in light of the lim-ited measurements. Examples of these include thePlanetary Nebula Spectrograph (Romanowsky et al. et al. Send o ff print requests to : Dalia Chakrabarty The X-ray emission from X-ray active systemscan be analysed to o ff er insight into the gravitationalmass distribution in the galaxy, under the assumptionof hydrostatic equilibrium (Humphrey et al. 2008;Lemze et al. 2008; Mahdavi et al. 2008; Zhang et al.2007; Fukazawa et al. 2006; O’Sullivan & Ponman2004, to cite some recent work). However, whatmakes this method potentially unreliable is the lackof information about the distribution of the frac-tion of hot gas that is in hydrostatic equilibrium(Churazov et al. 2008; Diehl & Statler 2007).Comparatively, a more stable route to mass dis-tribution determination is via lensing measurements.However, the biggest shortcoming of mass determi-nation from lensing measurements alone is the un-availability of the full three-dimensional mass dis-tribution. To improve upon this, lensing data is of-ten supplemented by dynamically obtained massestimates (Czoske et al. 2008; Bolton et al. 2008;Gavazzi et al. 2007; Koopmans & Treu 2003).However, there are questionable implementa-tional problems involved in (parametric) dynamicalmass determination, the chief of which are typicallythe mass-anisotropy degeneracy, binning-triggeredinstability of scant velocity dispersion data, reliance Dalia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ on the modelling of the stellar mass density andan even more fundamental worry caused by the as-sumption of one smooth parametric representationof the phase-space distribution function of the usedtracer and by its relation to the phase-space densityof the whole galaxy. This is of course in additionto the uncertainties in the mass distribution resultingfrom substituting the real geometry of the system bysphericity, as is typically done with all mass determi-nation procedures. Above all, using tracer kinemat-ics for mass determination (de Lorenzi et al. 2008;Douglas et. al 2007, etc.) is limited in applicabilitygiven the reliance on the size of the tracer kinematicdata! Large data sets are of course hard to attain insystems that are not close by. Moreover, this methodis unsuitable for fast evaluation of the mass distri-bution of individual galaxies that are members of alarge sample, as for example, a galaxy obseved in alarge survey.On the contrary, it would be highly beneficialto design a method that is comparatively less data-intensive in that it demands only what is easily avail-able from observations. We advance a methodologythat provides total gravitational mass density distri-butions to about 3 e ff ective radii, as compared to only1 e ff ective radius (Cappellari et al. 2006), in a fastand easy-to-implement fashion.This advanced formalism is inspired by a trickthat was reported in Chakrabarty (2007) (hereafter,Paper I). This trick involves the exploitation of onlyphotometry and the central velocity dispersion mea-sure ( σ ) in a galaxy, in order to generate the to-tal local mass-to-light ratio ( M / L ) profile to a dis-tance that is about thrice the semi-major axis e ff ec-tive radius. This cuto ff distance is described in detailsbelow. The prescription for constructing this profilewas provided in Paper I, though only for a certainclass of power-law galaxies. However, the exact na-ture of this prescription is very much a function of thephotometric class that the galaxy belongs to. Thus,the formula reported in Paper I cannot be invoked toshed light on the mass distribution in galaxies thatbetray a di ff erent (and more ubiquitous) photometricclass, eg. ellipticals, the surface brightness of whichcan be fit by a Sersic profile (Sersic 1968). This isprecisely what is reported in this paper.This paper is arranged as follows: the basicframework of the suggested formalism is discussedin Section 2, followed by a note on the modelsthat we use. The method used to obtain the soughtfunctional forms is briefly mentioned in Section 4.Results obtained from our work are subsequentlydiscussed in Section 5. Tests of the method are de-scribed in Section 6 while Section 7 deals with appli-cations to real galaxies NGC 3379 and NGC 4494.The paper is rounded o ff with a section devoted todiscussions of relevant points.
2. Formalism
The only Sersic model that was considered in Paper Idid actually indicate that the mass estimation tricksuggested for the power-law systems might be pos-sible for Sersic galaxies too. Following this lead, asin Paper I, we first invoke a raw two-stepped M / L profile of the Sersic galaxy at hand, where it is thedistribution of M / L with the major axis coordinate x ,that is relevant. This raw M / L distribution is subse-quently smoothed, (according to the smoothing pre-scription provided in Paper I and discussed below) toprovide the real M / L distribution of the system, to adistance that is by definition, 3 times the major-axise ff ective radii for a Sersic galaxy with sersic index n = ff ec-tive radius for all other values of n . The formal def-inition of this distance is given in Equation 1 whilethe justification for our choice of this length scaleis delineated in Section 8.4. Figure 1 represents aschematic diagram of this raw M / L profile against x . As can be appreciated from Figure 1, the unpro-cessed M / L profile is a two-stepped function that ischaracterised by three free parameters, including theposition of the step, or the jump radius x in and the M / L amplitudes inside and outside x in . The object ofthe current exercise is to fully characterise this un-processed M / L profile, so that upon smoothing, thefinal, total, local M / L distribution over x is retrieved.This final form of the M / L distribution is signifi-cantly di ff erent from the discontinuous, 2-stepped,raw M / L that we initially choose to work with. Infact, the final form of the profile is smooth and the M / L values vary a lot with x , sometimes abruptly, de-pending on the details of the model galaxy. Figure 4in Paper I represents a comparison of one such raw M / L profile and the final smoothed M / L distributionthat is advanced as the representative M / L profile forthe test system at hand. In fact, later in Figure 6, a raw M / L (exemplified in Figure 1) is compared to the true M / L distribution of the test galaxy under considera-tion - the di ff erence between the raw and final formsof the M / L profiles is clear in that figure.As in Paper I, x in is set equal to 3 X e , where X e = − . m ! (1)and m is the slope of the straight line that is fit tothe plot of core-removed log (I) against x / (asin Paper I), i.e. X e is the major axis equivalent ofthe e ff ective radius, if the Sersic index is 4. For allother values of the Sersic index, X e is at most an ap-proximation to the major axis e ff ective radius. Theadvantage of using X e over the exact definition ofthe major-axis e ff ective radius is described below inSection 8.4.The smoothing prescription used here is the sameas in Paper I - we smooth the raw total mass density alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 1.
Schematic view of a raw M / L distributionalong the major axis coordinate x , for a model galaxy.The dashed line marks the position of the jump ra-dius x in which as defined in Equation 1, is 3 timesthe major axis e ff ective radius of the model galaxy,if the galaxy is described by a Sersic index of 4; oth-erwise, x in is an approximation for the major-axis ef-fective radius. The amplitudes inside and outside X in have been marked as Υ in and Υ out . While the mea-sured central velocity dispersion is used to estimatethe range of Υ in values allowed, Υ out is constrainedvia the sought analytical relation that connects Υ out to Υ in . The point of this paper is to seek such rela-tions, in order to fully characterise the raw M / L pro-file; once the raw M / L distribution is known, it issmoothed to obtain the final, total M / L ratio profileof the galaxy, which is significantly di ff erent fromthis unprocessed M / L shown above.profile by two successive applications of a box filterof size corresponding to X e .The amplitude of M / L ( x ) for x ≤ x in is referredto as Υ in and that for x > x in it is Υ out . Now that x in has been pinned down by construction, we hope toget a constraint on the available choices for Υ in fromthe central velocity dispersion ( σ ) and also hopeto identify a functional dependence of Υ out on Υ in .The exact form of such a constraint or function isyet unknown but we begin by expecting these to bedefined in terms of the photometric parameters thatdescribe the surface brightness profile of the Sersicgalaxy under consideration, namely, the Sersic index n and X e that we have defined above (Equation 1).The central brightness is not a free parameter sincewe normalise all luminosity density profiles to a cen- tral value of 1000 L ⊙ pc − . Thus, we want to find thefunction f ( n , X e , Υ in ), where Υ out = f ( n , X e , Υ in ) (2)The inspiration for the hypothesis that f dependson the photometric parameters is discussed below.Given that by construction, Υ in is the uniformamplitude of the raw M / L profile for x < x in , weexpect it to be related to the “central” mass-to-lightratio of the galaxy, where by “central” is impliedthe distance at which the measurement of the cen-tral velocity dispersion ( σ ) is obtained (at x = x ,with x typically less than x in ). However, in a realsystem, the true central M / L cannot be securely de-termined from the measurement of σ alone, ow-ing to uncertainties about the validity of the assump-tions that are invoked, in order to translate knowl-edge of σ to that of mass enclosed within x (us-ing virial theorem). Such uncertainties basically stemfrom the presence of anisotropy in phase-space. Inthe mass modelling trick advanced in Paper I, roomis allowed for the accommodation of such uncertain-ties, as long as the deviations from the assumptionsused in the virial estimate of mass are not atypicallymore than what has been observed with real ellipti-cals (Padmanabhan et al. M / L ,from σ was parametrised by α . It was found thatfor a measured σ , as long as Υ in lies within a rangeof values (the details of this range correspond to thegiven α ), compatibility between the predicted andknown (model) mass density distributions is ensured.In other words, for α calculated from a given σ , Υ in can be safely chosen to belong to a range of M / L values: Υ minin to Υ maxout . Such positioning of Υ in can bechecked by comparing the mass distributions recov-ered with Υ in = Υ minin and Υ in = Υ maxin , for consis-tency. Here Υ minin = α while Υ maxin is an unknownfunction of the photometric parameters and α - say g ( n , X e , α ). Υ maxin = g ( n , X e , α ) , (3)where g ( n , X e , α ) is unknown and the choice of itsdependence on the photometric parameters is the fol-lowing. In Paper I we had success upon choosing thefunctions f and g as dependent on the photometricdetails of the system; this was the case for a suite ofmodel power-law galaxies. Such “success” is qual-ified in terms of the identification of the hypothe-sised dependence on the photometric properties ofthe model galaxies. Motivated by this, we endeavourto find forms of f and g .Thus, there are two unknown functions that wewish to constrain: f ( n , X e , Υ in ) and g ( n , X e , α ). Thesefunctions, when known, will provide Υ out from Υ in which will be known from α , (i.e. σ ) and the ob-served brightness distribution of the galaxy. Oncewe know Υ out from Υ in , we would then have fully Dalia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ characterised the raw M / L distribution over x . Thesmoothed out version of this raw M / L profile willthen be advanced as the true M / L profile of the sys-tem at hand. This in conjunction with the luminositydensity distribution will allow knowledge of the massdensity distribution.We hope to recover the analytical forms of thesetwo unknown functions for Sersic galaxies to 3 X e ,through an analysis of a sample of model Sersicgalaxy surface brightness profiles. Analytical fits aresought to the data that is collated from the suite ofmodels that we work with, in order to recognise pat-terns, if any, that may show up in the relations be-tween the various quantities, in particular, Υ out - Υ in and Υ in − α . The variation in these relations withchanging models is then explored to unravel the re-liance of these relations on the photometric parame-ters.
3. Models
We identify the relation between Υ in and Υ out inSersic galaxies, by recognising (and then quantify-ing) the patterns that emerge between these quanti-ties, as diverse model galaxies are scanned. To rem-inisce, Sersic galaxies are those, the surface bright-ness profile I ( R ) of which can be approximated as: I ( R ) = I exp h − b n ( R / R e ) / n i , (4)where I is the central surface brightness, R e is theprojected e ff ective half-light radius, n is the Sersicindex that determines the curvature of the bright-ness profile and b n is a function of n : for n > . , b n ≈ n − / + . / n (Prugniel & Simien1997; Lima Neto et al. ρ L ( s ) = ρ (cid:18) R e s (cid:19) p e − b n ( s / R e ) / n where (5) p = . − . n + . n for n ∈ (0 . , . Here, the equation for p is due to Lima Neto et al. (1999) and the ellipsoidal coordinate s is defined as s ≡ x a + y b + z c . (6)for the axial ratios of a:b:c for the ellipsoidal systemat hand. We use X e in place of R e (see Section 7.5). Itis to be noted that Equation 5 is an approximation andother forms have been used by Trujillo et al. (2002). As in Paper I, in our models, we assumeoblateness and an inclination of 90 ◦ , with a uni-form projected axial ratio of 0.7; the work ofPadilla & Strauss (2008) corroborates such consider-ations of geometry and ellipticity. Also, the modelsystems are assumed to be viewed in the ACS z -band, at a distance of 17 Mpc. Though we recoverthe sought functions for these chosen configurations,generalisations to other systems will be suggested inSection 5.3 and Section 5.4.Actually Terzi´c & Sprague (2007) gives the moregeneral 5-parameter mass density model which de-scribes Sersic galaxies with a core inside a givenbreak radius. However, we work with the simplercase of Sersic galaxies that can be qualified by a 3-parameter model. In fact, we constrict this further,by normalising the central luminosity density ρ to1000 M ⊙ pc − . Also, we look for the luminosity dis-tribution along the major axis. Thus, our models aredistinguished only by the Sersic index n and the X e defined in equation 1.Models were formulated for Sersic indices n = X e of 726 pc, 1000 pc, 1400pc, 2000 pc. While these models were extensivelyexplored to achieve the sought functions, we inves-tigated selected models with n lying in this rangeand higher X e values. In particular, we are inter-ested in models with Sersic index ≤
4, with larger X e . Our viable models include galaxies with n = X e ≤ n = X e ≤ n . Additionally, models with other n were alsofound to work to X e =
10 kpc. This choice of modelsis supported by the results that – in the nearby Virgo cluster, the ACSVCS survey(Ferrarese et al. 2006) reports that nearly 90%of the targetted early-type systems (Sersic andcored-Sersic included) fall in the range of R e . z -band (which is the waveband di-rectly comparable to our models). All the pro-gramme galaxies with n ∈ [3,4] were found tohave R e ≤ n ∈ (4,5.3] and R e > ff ective radius R e isreally the geometric mean of the extent along thephotometric semi-axes. Thus, the extent alongthe semi-major axis is greater (by about a fac-tor of about 1.2, for an axial ratio of 0.7) than thereported R e . In other words, the surrogate for X e is about 1.2 times the values of R e quoted earlierin this paragraph. Even when this factor in takeninto account, the range of our models covers theprogramme galaxies of the ACSVCS. – measurement of e ff ective radius is waveband de-pendent (Temi et al. 2008; Ko & Im 2005), sothat for n = X e val-ues in the z -band, are compatible with observa- alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 2.
Right panel: trends in Υ out with Υ in , with changing Sersic index n , from model galaxies with X e =
726 pc. Thecolour coding used for the di ff erent n values is: yellow for n =
3, magenta for n =
4, black for n = n =
6, blue for n =
7, green for n = n =
9. In the left panel, Υ minin is plotted against α in broken lines; this plot is universallygiven by Υ minin = α . Υ maxin is plotted against α for the di ff erent model galaxies used for the figure on the right, with thesame colour coding. For a model at hand, at a given α , Υ in can be chosen from the range defined by the values of Υ minin and Υ maxin . tions (Ferrarese et al. 2006; Trujillo et al. 2001;La Barbera et al. 2005). – at high redshifts, systems display evolution to-wards higher compactness (Buitrago et al. 2008).Though this evolution is marked for systemsat z > ≤ z ≤ X e .Sersic indices 2 or lower appeared not to be vi-able for the formalism to function, implying that thisformalism is suitable only for elliptical systems (seeSection 3). Sersic indices greater than 9 were not ex-amined since such systems are very rare.These luminosity density models, described byEquation 5 and the used values of n and X e , wereembedded in an NFW-type dark halo (Navarro et al. ρ dark , to give a total mass den-sity of: ρ t ( x ) = ρ dark ( x ) + αρ L ( x ) , (7)where by choice, ρ dark ( x ) = M s π x ( x + r s ) . (8) Here M s and r s are mass and length scales of thehalo, respectively. Our results are valid for halo pa-rameters that correspond to the points in the greenquadrilateral in the M s − r s space that is depicted inthe right panel of Figure 10.We ascribe ± σ errors of about 10% to the lu-minosity density distributions that we generate andsearch for compatibility between the model massdensity distribution and the predicted one, withinthese error bars.
4. Method
Combinations of the parameters α , Υ in and Υ out thatshowed compatability with the known (model) massdensity profiles, up to 3 X e , at the aforementionedfour separate ( M s , r s ) coordinates, were searched forby our smoothing formalism that is automated.We record the list of Υ in values that imply com-patibility for a chosen α (chosen typically in therange of 2 to 10). The starting value of Υ in wastypically about α while the upper value was set as3 α , which was always su ffi cient to find the wholerange of Υ in values that correspond to compatability.Similarly, we record the Υ out value corresponding toa given Υ in .The plots of Υ minin and Υ maxin , as functions of α aremonitored, with the aim of recognising the analyticalform of these functional dependences. Similarly, theplot of Υ out as a function of Υ in is analysed at di ff er-ent n and X e , to identify f ( n , X e , Υ in ). Dalia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ
5. Results
With X e held at 726pc, as n is varied, the plots of Υ in against α are shown in the left panel of Figure 2 whilethe relation between Υ out against Υ in is depicted forthese models, in the right panel of this figure. Υ in & α When the data collated from the di ff erent modelsis plotted it is noted that while Υ minin = α , thelog( Υ maxin ) − log( α ) relation is well fit by a quadraticfunction of log( α ). This latter functional form hasonly a weak bearing on the Sersic index n , though Υ maxin is found to increase slightly as n is increasedfrom 5.3 to 9. Thus, by denying this increased ampli-tude of log Υ maxin with higher values of n , we merelyconstrict the range of values from which Υ in canbe chosen. Thus, our attempt at simplification ofthe sought functional form preempts a small reduc-tion in the applicability of our formalism to moreanisotropic galaxies than what is e ff ectively allowed.In Figure 3, we present the values of log( Υ maxin )at distinct values of log( α ), for di ff erent X e , and asingle Sersic index of 5.3. The functional form ofthe relation is recovered from our analysis, and thisis over-plotted on the data, in solid lines. This func-tional form is given by: Υ maxin = A ( X e ) + A ( X e ) log α + A ( X e )(log α ) (9) A ( X e ) = − . + . X e − . × − X e A ( X e ) = . − . × − X e + . × − X e A ( X e ) = . − . × − X e − . × − X e . Υ out & Υ in The data from the assorted models indicate that Υ out is noted to fall exponentially with Υ in ; in fact, a goodfit to this relation is given by the following equation: Υ out = ∆ ( n , X e ) + A ( n , X e ) exp[ − Υ in /τ ( n , X e )] , (10)where ∆ , A and τ are functions of X e and n . The re-lations between n and ∆ , A and τ are shown in theleft, middle and right panels of Figure 4, for the fourdi ff erent X e values that we use. From our models, weseek the functional form of these relations; these re-covered functions are over-plotted in solid lines, onthe data in the three panels in Figure 4. This compar-ison indicates that the known trends in log ∆ , ln A and τ ( n ), with increasing log( n ) and n , for the as-sorted X e values, are well replicated by the predictedfunctional forms of the relevant quantities. Fig. 3.
The values of log Υ maxin obtained at distinctvalues of log α , from our models, for n = X e ; in black filled circles for X e =
726 pc, redfor X e = X e = X e = Υ maxin on log α (Eqn. 9) is plotted in solid lines, for the 4di ff erent values of X e that we use, in correspondingcolours. This functional form holds approximatelyfor other values of n as well ( n ≥ Υ maxin only increases slightly with n .We advance the following formulae for ∆ , A and τ :log[ ∆ ( n , X e )] = C ( X e ) + C ( X e ) log( n ) where (11) C ( X e ) = . − . × − X e − . × − X e and C ( X e ) = . + . X e − . × − X e ln[ A ( n , X e )] = D ( X e ) + D ( X e ) n + D ( X e ) n (12) + D ( X e ) n where D ( X e ) = . − . X e + . X e − . × − X e , D ( X e ) = − . + . X e − . X e + . × − X e , D ( X e ) = . − . X e + . × − X e − . × − X e and D ( X e ) = − . + . X e − . × − X e alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 4.
Trends in the three di ff erent parameters that define the exponential fall-o ff of Υ out with Υ in , withchanges in n and X e ; marked in filled circles of colours that are distinctive of the X e value used - black for X e =
726 pc, red for X e = X e = X e = X e .These functional dependences were recovered using model galaxies with { n = , . , , , , e ≤ } . + . × − X e .τ ( n , X e ) = τ ( X e ) + τ ( X e ) n + τ ( X e ) n + (13) τ ( X e ) n + τ ( X e ) n where τ ( X e ) = − . + . X e − . X e + . × − X e ,τ ( X e ) = . − . X e + . X e − . × − X e ,τ ( X e ) = − . + . X e − . X e + . × − X e and τ ( X e ) = . − . X e + . × − X e − . × − X e . These relations are valid for all model systemswith n ≥ n =
3, only systems with X e ≤ n = X e ≤ It is always possible that the observed photometryis presented in a wave-band di ff erent from the ACS z -band for which we predict the above relations be-tween the properties of the raw M / L , though the z -band is in general a better choice for high redshiftsystems than a bluer waveband. If the available pho-tometry is in a band di ff erent from z , (say the w -band), then the factor by which every value of Υ in and Υ out should be changed, is obtained from the fol-lowing considerations. We realise that if w is suchthat a model galaxy is brighter in z than in the w -band, then the M / L ratio by which the inner and outerparts of the brightness profile in z need to be scaled,are smaller than the same by which the profile in w needs to be scaled. In fact, the value of Υ out corre-sponding to observations in w should be scaled bya factor of 10 . L ⊙ w − L ⊙ z ) . Here L ⊙ w is the solar ab-solute magnitude in the w -band. In the ACS z -band L ⊙ z = w -band is really an approximation since we assume allthe way that X e is the same over wavebands. This isnot true (Temi et al. 2008; Ko & Im 2005). It is important to enquire about the stability of theposited forms of Υ out and Υ in when the distance to agalaxy is di ff erent from what has been used in themodels, namely 17 Mpc (approximate distance toVirgo). The only influence of the distance D (in Mpc)to the observed galaxy is in a ff ecting the luminos-ity density distribution that is obtained be deproject-ing the observed surface brightness profile, through aterm that is linear in D − . It is this luminosity densityprofile that is scaled by Υ in and Υ out in the inner andouter parts of the galaxy. Thus, if the galaxy is in re-ality further than 17 Mpc, the implementation of thesuggested M / L values would amount to an overesti-mation of the luminosity density. To compensate forthis, we need to modulate the relevant Υ out and Υ in values by the factor D /
6. Tests
In this section we discuss the testing of our advancedparametric forms of Υ out = f ( n , X e , Υ in ) (Equation9 with inputs from 10, 11 and 12) and Υ maxin = g ( n , X e , α ) (Equation 8). The scheme is tested onmodels that were excluded from the fitting exercise Dalia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ that resulted in our identification of the forms of thefunctions f and g .Figure 5 shows a comparison of the predictedand model mass density profiles along the x -axis forthe model galaxy that has a Sersic index of 3.5 and X e = α is set to 5 = ⇒ σ ≈
250 kms − , av-eraged over a radius of about X e / =
375 pc. For thissystem, we extract Υ maxin ≈ Υ out ≈ Υ out − Υ in values are used to scale the luminos-ity density distribution for this model galaxy whichis then smoothed to o ff er a final mass density distri-bution along the major axis. These Υ out − Υ in valuesimply a predicted mass density distribution (in red inFigure 5) that tallies favourably with the model (ingreen in Figure 5).The predicted mass density distribution whencompared to the luminosity density distribution of-fers the final or smoothed M / L which is shown inFigure 6 to be significantly di ff erent from the raw M / L profile (in this black line).However, this was the example of one givenmodel galaxy, characterised by a given central ve-locity dispersion. Tests were undertaken to validatethe relations predicted between the properties of theraw M / L and the photometric parameters of modelgalaxies, across the full range of σ . The tests werecarried out with model galaxies with Sersic indexof 4 and X e of 726 pc and 1400 pc. In Figure 7,we see the calculated Υ out value corresponding to agiven Υ in , for which compatibility is noted betweenthe known (model) and predicted total mass densitydistributions, to 3 X e . Such Υ out values are shown inthe two panels of this figure, in black dots.It is to be noted that these calculated values of Υ out very closely straddle (within errors of ± Υ out and Υ in that ispredicted in Equations 9, 10, 11 and 12, for given X e and n ; in this case for n = X e =
726 pc (left) and n = X e = ff er confidencein the formalism that we suggest.
7. Applications
In this section, we check out the e ffi cacy of the ad-vanced scheme in recovering gravitational mass den-sity distribution of two elliptical galaxies, NGC 4494and NGC 3379, to 3 X e . Our predicted mass distribu-tions are compared to independent dynamical massmodels for these systems. NGC 3379 was reported by Romanowsky et al. (2004) and Douglas et. al (2007) to contain very lit-tle dark matter on the basis of a Jeans equationanalysis of the kinematics of around 200 plane-tary nebulae (PNe) that reside in the dark halo ofthis galaxy. An independent estimation of the dis-tribution of the total mass density of this galaxy
Fig. 5.
Figure to compare the predicted total massdensity distribution with x (in red) with the known(model) mass density profile (in green), for the modeldescribed by n = X e = α of 5,this model yields Υ maxin ≈ Υ out ≈ M / L distribution thatis defined by these “inner” and “outer” amplitudesand subsequently smoothing the discontinuous massdensity profile thus obtained, (using the smoothingprescription mentioned in the text).was performed by Chakrabarty, 2009 (submitted to AJ ), by implementing these PNe velocities in theBayesian algorithm CHASSIS (Chakrabarty & Saha2001; Chakrabarty & Portegies Zwart 2005). As ac-knowledged by Chakrabarty (2009), these massdistributions from CHASSIS indicate somewhathigher masses than the estimates of Douglas et. al(2007), owing to the assumption of isotropy withinCHASSIS. While details of such mass estimationtechniques are irrelevant to the current work, herewe present a comparison between the mass densityprofile obtained from our formalism with the sameobtained from CHASSIS. We also present a compar-ison between the cumulative mass result M ( r ) via thequantity defined as v c = √ GM ( r ) / r , where v c is re-ferred to as the circular velocity.Our predictions are based on the analysis ofthe photometry of NGC 3379 in the B -band(Capaccioli et al. 1990) and the data for pro-jected central velocity dispersion, as given byStatler & Smecker-Hane (1999). Using this σ p dataalong the major axis of the galaxy, (Table 1A ofStatler & Smecker-Hane 1999), we get an α of about7.5. The photometry suggests an X e of 2.2 kpc.Additionally, we deproject the B -band surface bright-ness profile using the Bayesian deprojection algo- alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 7.
Comparison of model Υ out , as obtained from our calculations (black filed circles) and predicted Υ out (red line), for given Υ in , corresponding to model systems with Sersic index = X e =
726 pc (left) and X e = n and X e for the model testsystem, in Equations 9, 10, 11 & 12. The calculated Υ out values result from our analysis. Fig. 8.
The radial distribution of the recovered total mass density is depicted in black, on the left. On thisis superimposed the mass density profile obtained by implementing the kinematic information of a sampleof planetary nebulae in NGC 3379 (the same sample as used by Douglas et. al, 2007), in the Bayesianalgorithm CHASSIS (Chakrabarty 2009, communicated to AJ ). The quantity v c , as recovered from thesemass distributions are shown in corresponding colours on the right, with the v c profile from Douglas et. al(2007) overplotted in red.rithm DOPING (Chakrabarty & Ferraese 2008). Weconsider NGC 3379 to be at a distance of 11 kpc,as in Douglas et. al (2007). Our predicted values of(upper limit on) Υ in and Υ out , as modulated by di ff er-ences in the wave-band of the available photometryand distance to the system, are about 9.3 and 5.9.The resulting mass density distribution that weadvance for NGC 3379 within 3 X e , is depicted inblack on the left of Figure 8. This is compared to the mass model identified by CHASSIS. v c derived fromour calculated mass distribution, from the dynami-cal mass modelling by CHASSIS and that reportedby Douglas et. al (2007) are represented in the rightpanel of Figure 8.The comparison of the total mass density pro-file indicates the clear trend for our predictedmass distribution to be on the lower side com-pared to the gravitational matter density provided σ Fig. 6.
Figure to compare the final total (local) M / L distribution with the initially chosen raw M / L profile(in broken lines), for the test galaxy the total mass ofwhich is presented in Figure 5. The final M / L is pre-sented within 3 X e . The raw M / L profile is the sameas that shown in Figure 1; as is apparent from thisfigure, it is significantly di ff erent from the final M / L profile.by CHASSIS. This is only to be expected sinceCHASSIS in its current form, assumes isotropy,which a ff ects CHASSIS idiosyncratically to spuri-ously enhance mass density, as acknowledged byChakrabarty (2009). Consequently, the v c profile ad-vanced by our work is also on the lower side ofthe profile that follows from CHASSIS, though ourresult compares better with the result advanced byDouglas et. al (2007). NGC 4494 is a nearby elliptical, the mass dis-tribution of which to 7 e ff ective radii has re-cently been presented by Napolitano et al. (2008).The distance to this galaxy is given as 15.8 Mpcby Napolitano et al. (2008). The V -band photom-etry of this galaxy is presented in Table A1 ofNapolitano et al. (2008). This surface brightness pro-file indicates an X e of 3.37 kpc and is deprojected,given the radial variation of the projected eccen-tricity (assuming an oblate geometry and edge-onviewing). The projection of such recovered luminos-ity density distribution is shown in the left panel ofFigure 9, compared to the surface brightness data ofthis galaxy (in red). Napolitano et al. (2008) also citethe central velocity dispersion of this system as 150kms − (Paturel et al. 2003). Considering this σ p to be the dispersion averaged over R e /
8, where R e = α of 2. Our predictedvalues of Υ in and Υ out are about 2 and 0.121. Theseare used to characterise the raw M / L , which whensmoothed, gives rise to the total mass distributionthat is shown in the middle panel of Figure 9.The total M / L ratio of NGC 4494, in the V -bandis shown on the right panel of Figure 9 while theenclosed gravitational mass distribution is depictedin the middle panel. This is very similar to the ra-dial mass distribution of NGC 4494, within the in-ner 3 X e , as reported by Napolitano et al. (2008) (theirFigure 13).
8. Discussions and Summary
We have presented a novel mechanism for estimatingtotal mass density profiles of elliptical galaxies thatcan be described by a Sersic-type surface brightnessdistribution. This formalism uses nothing in excessof what is typically available in the observational do-main, namely, photometry and the central velocitydispersion profile. This allows the implementation ofthis scheme even for systems at high redshifts. (Suchan implementation of this scheme will be presentedin a future contribution - Chakrabarty & Conselice,under preparation). The advanced scheme uses themethodology presented in Paper I, in the context ofSersic galaxies.To begin with, we generate a sample of luminos-ity density distributions that project to Sersic bright-ness profiles, of assorted values of n and X e . Thesetoy galaxies are then assigned various values of thelocal central mass-to-light ratio (parametrised by α ).For each such configuration, we monitor the allowedrange of Υ in and the value of Υ out that corresponds toany such Υ in . This pair of Υ in − Υ out values define theraw or unsmoothed two-stepped local mass-to-lightratio distribution with x . The local mass density dis-tribution in the system (to 3 X e ) is judged by scalingthe luminosity density profile by the raw M / L pro-file and then smoothing the result by the prescribedsmoothing routine (two successive applications of abox filter of size corresponding to X e ).The dependence of Υ out on Υ in is found to be wellapproximated with an exponential decline that is de-scribed by three free parameters, which are actuallyall functions of the model characteristics, namely, n and X e . The dependence of these three parameters on n and X e are extracted from our model-based studyand analytical fits to these trends are presented. Suchfits are noted to be favourably represented by poly-nomials.The reliance of the band of the allowed values of Υ in on α is also quantified as ranging from α to Υ maxin ,where the form of the function Υ maxin ( n , X e , α ) is alsofound to be polynomial in nature. alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 9.
Figure to bring out the results for the galaxy NGC 4494. The left panel depicts a projection of theluminosity density distribution that we estimate for the galaxy (in black), compared to the V -band surfacebrightness data (in red). The total gravitational mass distribution of the galaxy is represented in the middlepanel while the mass-to-light ratio on the V -band is shown on the right. Even though the polynomial fits are found to do agood job for most configurations, for certain mod-els, the predictions appear incorrect (by no more than10%), eg. τ may be judged under-estimated for n = X e = Υ out for these Υ in values from our models, ratherthan a failure of our predictions. Typically, such er-rors emanate from lack of refinement in the step-sizesused in our work and is potentially amendable. . . . As delineated in Section 3, the formalism that we ad-vance here, is applicable only to galaxies with Sersicindices n ≥
3. It is only for such systems that theluminosity density profile is steep enough, i.e. fallsquickly enough to ensure that in the “outer” partsof the system the luminous matter density is com-pletely overwhelmed by the dark matter density, sothat the total matter density is e ff ectively a represen-tation of the dark matter distribution. Thus, changesin the central M / L or α do not a ff ect the M / L in these“outer” parts. Now, to qualify “outer”, we state thatthis region corresponds to x ∼ X e . This is anywaytrue, only if the luminous matter fraction is not toosmall, in which case, we need to settle for progres-sively lower M / L in the outer parts, as α increases. If however, the luminous density profile is too flat,(as for n < Υ out will need to increase withincreasing α . Then, the exponential fall-o ff of Υ out with Υ in will not be noted. Qualitatively speaking,this is the reason why our formalism will not be validfor galaxies with Sersic indices less than 3. However, this increased flattening of the luminositydensity profile can be attained even for n ≥
3, forlarge X e values. Thus, in such systems, for the samereason as in the last paragraph, Υ out will increasewith an increase in the inner M / L instead of fallingexponentially with increasing Υ in . Thus, based onthese considerations, on the basis of our experiments,we find that the following galaxies cannot be accom-modated within our suggested formalism: – n = X e > – n = X e > – n < However, it is important to keep in mind that theseranges have been ascertained on the basis of a dis-cretised scanning of the parameters n and X e , of ourtwo-parameter galaxy models. Thus, when we saythat models with n < α, , Υ in , Υ out ,what we really mean is that our experiments witha model with n = n = n and X e de-scribed above owes, to our dealing only with modelsascribed the aforementioned discrete values (valuesmentioned in Section 3). Consequently, the rangeslisted above are a conservative interpretation. σ The practical question to ask is, are these sug-gested ranges compatible with real galaxies? Asitemised in Section 3, our choice is supported bythe trends observed for the Virgo cluster, withinthe ACS Virgo Cluster Survey, in alliance withthe predicted evolution towards greater compact-ness in high-redshift systems (Ferreras et al. 2009).Furthermore, keeping in mind that a waveband de-pendence of e ff ective radius exists, our model X e val-ues in the z -band fall within limits indicated by ob-servations (Ferrarese et al. 2006; Trujillo et al. 2001;La Barbera et al. 2005).Additionally, we question the relevance of theupper limits on the e ff ective radii at n = n . As far as the correlation between theshape parameter n and log of e ff ective radius is con-cerned, it is well known that in general, half-light ra-dius decreases with decreasing n (Boselli et al. 2008;Cˆot´e et al. 2008; Naab & Trujillo 2006; Brown et al.2003; Trujillo et al. 2001). In fact, such a correlationis physical and is hinted at by the global relationsbetween structural parameters; it is not merely a re-flection of the fitting procedure (Trujillo et al. 2001). Of course, it could be perceived that such fate of themass distribution at x ∼ X e would be dictated bythe exact details of the underlying dark matter distri-bution that defines the model galaxy. Indeed, for thehigh values of n , the M / L at x ≥ X e ( Υ out ) is dictatedby the details of the dark matter distribution, but itis the smoothing of this raw M / L profile that givesus the end product, namely the total local M / L ratioprofile of the galaxy upto the benchmark x of 3 X e . Itis precisely this smoothing procedure that brings intothe final M / L structure, elements of the mass distri-bution in the inner 3 X e in the galaxy, as well as arelatively less significant signature of the same thatlies outside the radius upto which the distribution issought . The di ff erence that smoothing makes to theraw M / L profile, as distinguished from the final M / L profile, is clear in Figure 6.Thus, given such (1) implementation of oursmoothing procedure and that (2) the jump radius inthe raw M / L profile is itself 3 X e , we find that our ex-periments bear the fact that changing the mass scaleand length scale of the used NFW dark matter densitydistribution across the wide ranges do not a ff ect themass configuration for x . X e , though further out inthe galaxy, the influence of changing the DM densitydistribution picks up quickly. The consistency in the Υ out − Υ in relation, noted with changes in M s and r s is brought out in the left panel of Figure 10. The al-lowed ranges in M s and r s are provided in the rightpanel of Figure 10. The modelling of the dark matter distribution in earlytype galaxies is obviously indirect and consequentlydi ffi cult and unreliable. Ferreras et al. (2008) suggesta large scatter in the outer slope of the dark mat-ter distributions that they recover for their targetedgalaxies. Gavazzi et al. (2007) corroborate an NFWmodelling of the dark haloes of their sample galax-ies, suggesting that the total matter density tends toan overall isothermal form. Given this degree of un-certainty, it might be argued that there is not muchsense in splitting hair to decide between an NFW andisothermal dark matter distributions, as long as thedark haloes that we probe in our analysis are com-patible with observations or simulations.This is indeed the case, when we compareour ranges of halo characteristics (right panel ofFigure 10) to the suggestion by Kleinheinrich et al.(2006) that the recovered range of virial masses oftheir NFW model to be [3 . × , . × ] M ⊙ .Such a mass range is compatible with the sugges-tion by Hoekstra et al. (2004) for a fiduciary galaxyof luminosity L B = L B ⊙ that has an NFW pro-file ( M ∈ [10 , . × M ⊙ , as indicated byFigure 4 in Hoekstra et al. 2004). This mass rangecorresponds to a range of about 8 kpc to about 30kpc for r s . In fact, we cover these ranges in the runs,the results of which are presented in Figure 10, andalso scan haloes of lower masses. For various haloesdefined within the red and black quadrilaterals inFigure 10, compatible Υ out − Υ in relations were re-covered.This explains our choice of sticking with theNFW prescription for our dark halo model. For rea-sons similar to what we explain above, we suggestthat upon smoothing, haloes of varying shapes butsimilar masses within 3 X e are not expected to a ff ectthe mass distribution within 3 X e in the massive ellip-ticals that we deal with in this formalism.Thus, we see that our methodology is valid, irre-spective of selecting haloes defined by widely dif-ferent M s and r s values, (i.e. NFW haloes definedby points inside the black and red quadrilaterals inFigure 10). This motivates us to choose to work witha smaller subset of all the haloes explored in theseruns; in particular, we choose to work with haloes de-fined by the more massive half of the log( M s ) rangethat limits the red quadrilateral. This chosen subset inthe M s − r s space is bound by the green quadrilateralin Figure 10. In addition to details of the DM distribution, we ad-mit that the model structure does include the free pa-rameters that describe the Sersic model of densitydistributions in general triaxial galaxies, as it ide-ally should, though this work has full potential of alia Chakrabarty and Brendan Jackson: Mass Distributions Only from Photometry & σ Fig. 10.
The left panel depicts the relation between Υ out and Υ in , obtained using the model with n = X e = ff erent runs that seek this relation, given NFW haloes characterised by ( M s , r s )values within the quadrilaterals in black (see right panel) and in red (see right panel). When the halo densityparameters are as contained within the black quadrilateral, the obtained Υ out − Υ in relation is shown in openblack circles in the left panel. When the halo density is characterised by M s and r s values as points insidethe red quadrilateral on the right, the Υ out − Υ in relation is shown in red filled circles in the left panel. Thepredicted Υ out − Υ in relation in shown on the left in solid black lines. The green quadrilateral represents thesubset in the M s − r s space, which defines the dark matter density distributions for which concurrent f and g functions were recovered in general runs done with assorted models, within our scheme.being extended into two-dimensions and include anaxisymmetric description of the galaxy. Now, the tri-axiality of the models needs to be defined in terms ofa chosen geometry and intrinsic eccentricities - it isof course not possible to constrain such characteris-tics from observations alone.In our models, these were fixed as oblatenesswith an axial ratio of 0.7. It is indeed possible thatthe unknown functions f and g that we attempt toconstrain, harbour dependence on these intrinsic ge-ometric factors. However, the preparatory assump-tions involved in the deprojection of an observedbrightness distribution, namely the underpinning ofintrinsic geometry and viewing angle, are essentiallyunconstrained, unless the system is favourably in-clined or flattened. In other words, the unknown inour modelling are the usual quantities that render de-projection non-unique.As for the specification of the ellipticity inour models, a typical value has been adopted -Padilla & Strauss (2008) suggest an axial ratio dis-tribution for a large sample of SDSS ellipticals, witha mode in the range of 0.6 to 0.8. Inspired by this, weuse an axial ratio of 0.7 in our models. Again, the un-certainty in intrinsic ellipticity, cannot be known fora general observed galaxy. The quantification of thedeprojection e ff ects on the formulae provided aboveis possible, at least in a statistical sense, and it is en-visaged that the same will be pursued in the future. The case of a central mass condensation in the sys-tem was dealt with in Paper I, in reference to the ex-ample of the galaxy M87 - the recovered mass dis-tribution of M87 was demonstrated to be consistentwith the same obtained from kinematical consider-ations. If independent measurements indicate an ob-served system to harbour a massive central mass con-densation, then the scheme delineated in Paper I willbe followed. X e insteadof R e ? The usage of X e instead of R e is preferred since X e asper its definition here, as well as in Paper I, is derivedsolely from the inputs to the methodology, namelythe surface brightness profile along the semi-majoraxis which we consider to be along the ˆx -axis. Thus,any changes to the shape of the brightness profile willbe directly reflected in a linear change in X e but notnecessarily so in the half-light radius that is estimatedfrom isophotal analysis. In fact, here we use the samedefinition for X e as in Paper I, (see Section 2).The usage of X e should not be cause for concernsince X e is merely one definition of the semi-majoraxis e ff ective radius and reduces to the conventionaldefinition of the major-axis e ff ective radius for n = .When an unknown galaxy is being analysed withinthis formalism, its half-light radius is just as much σ an unknown as is the X e that we define here. Thus,there is no loss of connection with observations bythe implementation of X e .. It merits mention that other choices for the smooth-ing prescription and x in may also work, but here weconcentrate on the above mentioned configurationand the specification of Υ in and Υ out are accordinglyunique to these choices. M / L In a similar context, it may be argued that the defi-nition of α that we use herein will leave an imprint.We equate α to the central local M / L , inspired bythis result that is achieved in PaperI. Here the totalgravitational mass found enclosed within the radius x is M ( x ) where M ( x ) is linked to the 3-D velocitydispersion at x via: σ = GM ( x ) / x . Then, accord-ing to our definition, α is given as the ratio of M ( x )and the enclosed light within x . For other definitionsof α , other ranges of Υ in will be valid (and thereforeother forms of dependences of Υ out on Υ in , for thesame galaxy). Thus, the pairs of f and g functionsthat we advance here, work for the used choice of thedefinition.We would like to emphasise that the extraction ofthe exact value of the mass enclosed within x (andtherefore of α ), from a measurement of σ , is notthe point of this exercise ; uncertainties in this extrac-tion do not undermine the advanced results either, aslong as the galaxy at hand is “not too” aspherical oranisotropic at x . Here we qualify “not too” as thoseconfigurations for which we obtain consistent massprofiles using the two extreme values of Υ in , that areallowed for the extracted value of α . Thus, when themethod fails, we know that it does. As long as thisaforementioned consistency is noted, choosing Υ in from anywhere within the range corresponding to thegiven α will lead to consistent total mass distribu-tions within 3 X e . Additionally, this range is neithertoo constricted nor too relaxed, as was discussed inPaper I. The presented device is based upon conclusions thatare drawn from a sample of model Sersic galaxies.This would naturally imply that the success of thisformalism is crucially dependant on the generalityof the models. In particular, we have discussed theranges of n and X e for which our advanced results are Semi-major axis e ff ective radius has been used before,for example by Naab & Trujillo (2006) true. We have also discussed the e ff ect of changingproperties of the dark halo that we use in the modelsand find the advanced scheme robust to such modelparameter variations.In fact, the formalism presented above is uniquein its scope and structure. Estimates of total massdistributions in distant elliptical systems are di ffi -cult and therefore rare; the formulae presented hereinare therefore advantageous and could be treated asguides to decipher the total mass distribution ofSersic galaxies in large surveys.Most importantly, the advanced methodology issuccessful within a severely constricted data domain,compared to any other scheme that aims to obtainmass distributions in elliptical galaxies. All that theadvanced method demands in terms of data is whatis typically available - surface brightness profile anda measure of central velocity dispersion. The unde-manding nature of our method renders it applicableeven at high redshifts. The simplicity of implementa-tion of the input data is advantageous in that it allowsfor the scheme to be used in an automated way, to ob-tain mass distributions for large samples of galaxies.Tricks such as this and Nipoti et al. (2008) exploit thebasic configuration within galaxies and o ff er novelways for characterisation of distant systems. Acknowledgements.
DC is funded by a Royal Society DorothyHodgkin Fellowship. BJ acknowledges the support of a Universityof Nottingham Summer Studentship. We thank Sebastian Foucaudfor useful discussions that helped enrich the paper.
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