Intrinsic Shapes of Very Flat Elliptical Galaxies
aa r X i v : . [ a s t r o - ph . C O ] O c t Mon. Not. R. Astron. Soc. , 1–7 (2010) Printed 14 May 2017 (MN L A TEX style file v2.2)
Intrinsic Shapes of Very Flat Elliptical Galaxies
D. K. Chakraborty ⋆ , A. K. Diwakar † and S. K. Pandey ‡ School of Studies in Physics & Astrophysics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh 492010, India
Accepted xx. Received xx; in original form xx
ABSTRACT
Photometric data from the literature is combined with triaxial mass models to de-rive variation in the intrinsic shapes of the light distribution of elliptical galaxies
N GC , . . Key words: galaxies : photometry - galaxies : structure
Intrinsic shapes of the individual elliptical galaxies havebeen investigated by Binney (1985), Tenjes et al. (1993),Statler (1994a,b), Bak and Statler (2000), Statler (2001),and Statler et al. (2004). These authors have used the kine-matical data and the photometric data, and have used thetriaxial models with the density distribution ρ ( m ), where m = x + y /p + z /q with axial ratios p and q . Here,( x, y, z ) are the usual Cartesian co-ordinates, oriented suchthat x -axis ( z -axis) lies along the longest (the shortest) axisof the model. It was shown analytically that the projecteddensity of such a distribution ρ ( m ) with constant ( p, q )is stratified on similar and co-aligned ellipses (Stark 1977;Binney 1985). Statler (1994a) uses (apart from the kinemat-ical data) a constant value of ellipticity, which is an averageover a suitably chosen range of radial distance, for the shapeestimates. The shape estimates are robust, and are describedby a pair of the shape parameters, namely the short to longaxial ratio c L of the light distribution and the triaxiality T M of the mass distribution.A complementary problem was attempted by(Chakraborty et al. 2008, hereafter C ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
These models are fixed by assigning the values of axialratios ( p , q ) and ( p ∞ , q ∞ ) at small and at large radii,respectively. These axial ratios are related to triaxialities T and T ∞ , respectively, at small and large radii. We useBayesian statistics, and obtain the variation in the shape,following the methodology described in Statler (1994a).We find that the marginal posterior density ( MP D ) islikelihood dominated, so that it relatively insensitive to theunknown prior density. We use a flat prior. We use a largeensemble of models, so that the shape estimates may bemodel independent.The basic ingredients of our method are the same as inStatler (1994a), and we adopt all the necessary alterationsdescribed in C
08. We use ( q , T , q ∞ , T ∞ ) as the shape pa-rameters and use the ellipticities ǫ in , ǫ out and the positionangle difference Θ out − Θ in at two suitably chosen points R in and R out from the profiles of the photometric data ofthe galaxies. We find that the best constrained shape pa-rameters are q , q ∞ and the absolute value of the triaxialitydifference T d , defined as | T d | = | T ∞ − T | . C
08 have estimated the shapes of 10 elliptical galaxieswhich are comparatively rounder, with ellipticities . NGC , . .
5. We use triaxial models which are very flat. Wetake models with the lower limit of ( q , q ∞ ) ∼ . c (cid:13) D. K. Chakraborty, A. K. Diwakar and S. K. Pandey
Appendix A), and are not employed in the present shapeestimates.Determination of the intrinsic shape using photometryis important because the number of galaxies with good pho-tometric is many more than those with good kinematics.Besides, the results obtained by alternative models and tech-niques can be used for a comparison. Photometry constrainsthe flattening ( q , q ∞ ) but can not constrain ( T , T ∞ ). Thus,our work is complementary and not contradictory to that ofStatler and his coworkers.Sect. 2 presents the models. The necessity for the choiceof small values of the lower limits of ( q , q ∞ ), and the intrin-sic shapes of the galaxies are presented in sect. 3. Sect. 4 isdevoted to results and a discussion. We use models, which are triaxial generalizations of thespherical γ models of Dehnen (1993), with density ρ givenby ρ ( r ) = M (3 − γ ) b π r − γ ( b + r ) − γ , (1)where M is the mass of the model, r is the radial coordinate,0 γ < b is the scale length. The models have cusp atthe centre, and the density decreases as r − at large radii.Dehnen’s models are the generalization of the well studiedmodels of Jaffe (1983) and Hernquist (1990), correspondingto γ = 2 and γ = 1, respectively. The projected surface den-sity of the model of Dehnen, corresponding to γ = 1 .
5, mostclosely resembles to the de Vaucouleurs R / law. Presently,we concentrate to γ = 1 . C
08. The modelis the density distribution of the same form as (1) with r replaced by M , where M = x + y P + z Q , (2)with varying axial ratios P − ( M ) = βb p − + M p − ∞ βb + M , (3)and Q − ( M ) = βb q − + M q − ∞ βb + M . (4)The axial ratios ( P, Q ) reduce to ( p , q ) at small radii andto ( p ∞ , q ∞ ) at large radii. β > p , q , p ∞ , q ∞ ) alters P and Q in the inter-mediate region. The models are fixed, once the axial ratios( p , q , p ∞ , q ∞ ) are chosen. The triaxialities T and T ∞ arerelated to the axial ratios at small and at large radii by T = 1 − p − q . T ∞ = 1 − p ∞ − q ∞ . (5)To fix up the scale length b of the triaxial models, we con-sider γ = 1 . R e = 1 . b of the spherical model. The effective radius ofthe triaxial models depends on the axial ratios, as well ason the viewing angles. However, such changes are small for γ models (de Zeeuw and Carollo 1996), and are neglected. The constant ρ surfaces are coaxial ellipsoids. Projec-tion of these models on a plane perpendicular to a line ofsight, and therefore, the calculation of ellipticity and po-sition angle are performed numerically. We refer to thesemodels as M models.Another form of triaxial generalization of (1) is inves-tigated by de Zeeuw and Carollo (1996), where two moreterms are added to equation (1), each one of these is a suit-able radial function multiplied by spherical harmonics of loworder. The models provide simple analytical representationof the observed surface brightness of triaxial elliptical galax-ies. However, for large values of flattening models becomepeanut shaped and are not used in the present investiga-tion. Very flat de Zeeuw - Carollo models are discussed inAppendix A. The galaxies chosen here are very flat. The morphologicalclassification of
NGC E /E , E /E E /E RC MP D ) P of the Bayesian estimate is obtained by integrating the pos-terior density over all viewing angles. To gain some insightinto the possible values of the intrinsic shape, which willbe obtained by Bayesian method, we perform the followingnumerical experiments. The objective of these experimentsis to find suitable limits of the axial ratios ( q , q ∞ ) for theplots of P . In the plots of P in C q and q ∞ extend from0 . .
0. Statler (1994a) chooses 0 . c L . N of viewingangles ( θ ′ , φ ′ ) and the axial ratio q which gives ellipticity0 . ǫ .
55 (plot 1A) and 0 . ǫ .
17 (plot 1B).The number of viewing angles is counted between 0 o . o . o .
0, both for θ ′ and φ ′ . The totalnumber of viewing angles in this numerical experiment is8100. The axial ratio p is taken as 0 .
9. Here, we use Starkmodel. We find that a higher values of ellipticity is producedby flatter models and a lower values of ellipticity is producedby rounder models, over a larger number of viewing angles.Therefore, the Bayesian estimate should pick up a flat modelto represent shape of the galaxies of our present investiga-tion. It is interesting to note that the plots (1 A &1 B ) showa maxima, which lies at q ∼ .
44 for the plot 1 A and at q ∼ .
82 for the plot 1 B The fig. 1 and its inferences are based on applying Starkmodel, which has constant values of the axial ratio ( p, q ).However, in our shape estimates, we use models with varyingaxial ratios. So, we re-examine the results of these plots byconsidering M models.Fig. 2 presents the marginal posterior density P as afunction of ( q , q ∞ ), summed over various values of ( T , T ∞ )for NGC q and q ∞ from theregion 0 . ( q , q ∞ ) .
9. We use the M models with β = 1 .
0. The probability of the shape is plotted in dark greyshade : darker is the shade, higher is the probability. Thewhite contour encloses the region of 68% highest posteriordensity (HPD), which may be interpreted as 1 σ error bar. c (cid:13) , 1–7 ntrinsic Shapes of Very Flat Elliptical Galaxies Figure 1.
Plot between the number of viewing angles and theaxial ratio q , which would reproduce ellipticity in a chosen inter-val. Figure 1A is drawn for the ellipticity 0 . ǫ .
55 while1B is drawn for the ellipticity 0 . ǫ . This figure indicates that higher probability region is con-fined, approximately between 0 . . q , q ∞ ). Hence, itis more appropriate to choose the lower and the upper limitsof both q and q ∞ as 0 . . q , q ∞ ), fig. 3shows shape P ( q , q ∞ ) of a rounder galaxy NGC ǫ in = 0 . , ǫ out = 0 .
133 at R in = 15 ′′ . R out = 49 ′′ .
3. The effective radius R e of NGC ′′ .
5. We find that the
HP D region is confinedbetween( q , q ∞ ) > . q , q ∞ ) al-lowed in this plot. Figure 2.
Plot of marginal posterior density ( P ) as a functionof q , q ∞ (= q ), summed over various values of ( T , T ∞ ), for NGC720 using the limits 0 . .
9, both for q and q ∞ . Figure 3.
Same as Fig 2, for NGC 3379.
The observed data of
NGC ǫ increases monotonically from 0 .
315 at R in = 8 . .
442 at R out = 51 . o .
5. We consider the uncertainty in the ellip-ticity as 0 .
02 and in the position angle is 1 o .
0, both at R in and at R out . These are the typical errors in observations(de Carvalho et al. 1991; Penereiro et al. 1994). The effec-tive radius of the galaxy is 52 . c (cid:13) , 1–7 D. K. Chakraborty, A. K. Diwakar and S. K. Pandey
Figure 4.
Plot of unweighted sum of MPD ( P ) as a functionof q , q ∞ (= q ), for NGC 720 using the limits 0 . . q and q ∞ . The sum is taken over the M models with β =5 . , . , . , . .
2. Plus marks the location of the maximumprobability.
Figure 5.
Same as Fig 4, for NGC 720 using the limits 0 . .
8, both for q and q ∞ . of models, as described in sect. 2, with β = 5 . , . , . , . .
2. Taking the sum of the marginal posterior densityover all possible values of T and T ∞ , and taking the un-weighted sum over all the models, we obtain shape estimate P as a function of ( q , q ∞ ).Fig. 4 presents the shape estimate P ( q , q ∞ ) of NGC . . q and q ∞ . We find that the 1 σ region is very narrow Figure 6.
Three dimensional plot of the unweighted sum of MPD( P ) as a function of q , q ∞ , | T d | , for NGC 720. The sum is takenover the M models with β = 5 . , . , . , . .
2. Values of | T d | are constant in each section. In each section, q goes from theleft to right hand side from 0.25 to 0.75, and q ∞ runs betweenthe same values from the bottom to the top. which should be the consequence of the choice of the limitsof q and q ∞ . Examining this limit in fig. 1, we find that thischoice falls in the region where high values of the ellipticitywill not be reproduced. Therefore, we need to go to smallervalues of ( q , q ∞ ) to obtain higher ellipticities, which maybe close to observed ellipticities of NGC P ( q , q ∞ ) wherein we have al-lowed the limits 0 . . q and q ∞ . 1 σ region iswider now (but narrow enough to satisfy the requirementof the likelihood dominated shape estimate). Although, itis the plot of MP D ( P ) as a function of shape parameters,which constitute the Bayesian estimate of the shape, somestatistical summary of the shape is very convenient for itsdescription. The expectation values < q >, < q ∞ > and lo-cation of the peak values q P , q ∞ P are such quantities. Table2 provides such a summary. The expectation values of theflattening at small and at large radii are < q > = 0 .
64 and < q ∞ > = 0 .
43, respectively.Both in fig. 4 and 5, we choose the interval betweenhigher and lower limits of ( q , q ∞ ) as 0 .
5. This is basicallyto save the computer time, but maintaining the reliabilityof the results. Shape calculation requires a very large num-ber of projections, which need to be calculated numerically.We divide the parameter space of ( q , q ∞ ) in 48 ×
48 squarebins of equal size and calculate the likelihood at the centreof each bin. The bin size is small enough, so that the calcu-lated likelihood can be regarded as a continuum function of( q , q ∞ ), and at the same time, the number of bins is smallenough, so that the computer time is not unmanageable.Fig. 6 shows the 3-dimensional intrinsic shape of NGC
720 as a function of q , q ∞ and | T d | . We cut a totalof 16 sections, each perpendicular to | T d | axis, and arrangethese sections in a form of a two-dimensional array. The c (cid:13) , 1–7 ntrinsic Shapes of Very Flat Elliptical Galaxies Table 1.
Observational data of the galaxies.Galaxy R e R in R out ǫ in ǫ out Θ d NGC 720 52.0 8.5 51.8 0.315 0.442 -3.5NGC 2768 76.5 15.8 95.4 0.364 0.569 -1.4NGC 3605 22.5 5.9 20.4 0.305 0.418 -2.0
Table 2.
Statistical summary of the 2-dimensional shape esti-mates P ( q , q ∞ ) of the galaxies.Galaxy q p q ∞ p < q > < q ∞ >NGC
720 0.68 0.48 0.64 0.43
NGC
NGC value of | T d | is constant in each section, and is shown inthe plot. We find that the 1 σ region occupies larger areain the sections with smaller values of | T d | . Further, in eachsection of constant | T d | , 1 σ region occupies a small area of( q , q ∞ ) plane. We find that higher P is concentrated in sec-tions with | T d | between 0 .
28 to 0 .
47. The expectation valueof < | T d | > = 0 . The observational data used in the models of these galaxiesare presented in Table 1. Here also, the data is obtainedfrom R - band surface photometry of Peletier et al. (1990).Fig. 7 and 8 present the plot of P of NGC q , q ∞ ). The lower and upper limits of q and q ∞ are taken as 0 .
25 to 0 .
75. The HPD region showsthat these galaxies are very flat. The expectation values are < q > = 0 .
63 and < q ∞ > = 0 .
32 for
NGC < q > = 0 .
62 and < q ∞ > = 0 .
42 for
NGC P of NGC q , q ∞ and | T d | .Statistical summary of the intrinsic shapes of all thethree flat galaxies NGC , q and q ∞ as obtained from 2-dimensional estimates P ( q , q ∞ ) arereported in Table 2. These values are quite close but not ex-actly the same as those reported in Table 3. The differencesmay be attributed to ”resolution” : in 2-dimensional shapeestimates, we have divided the parameters space of ( q , q ∞ )in 48 ×
48 divisions, whereas in 3-dimensional shape esti-mate, space ( q , q ∞ ) is divided in 10 ×
10 divisions for each | T d | . Table 3.
Statistical summary of the 3-dimensional shape esti-mates P ( q , q ∞ , | T d | ) of the galaxies.Galaxy q p q ∞ p | T dp | < q > < q ∞ > < | T d | >NGC
720 0.68 0.38 0.41 0.56 0.40 0.41
NGC
NGC
Figure 7.
Same as Fig 4, for NGC 2768 using the limits 0 .
25 to0 .
75, both for q and q ∞ . Figure 8.
Same as Fig 4, for
NGC .
25 to0 .
75, both for q and q ∞ . We have presented the intrinsic shapes of 3 very flat galax-ies. A specific feature of these estimates is the choice of thelower limit of q and q ∞ . The lower limit was chosen as 0 . C
08 and as 0 . P for NGC q , q ∞ ) as their lower limits for very flat elliptical galaxies.We took the lower limit as either 0 .
25 or 0 . c (cid:13) , 1–7 D. K. Chakraborty, A. K. Diwakar and S. K. Pandey
Figure 9.
Same as Fig 6, for
NGC
Figure 10.
Same as Fig 6, for
NGC
A summary of the intrinsic shapes of these very flatgalaxies is presented in Table 2 and Table 3. We find thatthese galaxies are little rounder inside (average value of ∼ . < q ∞ > ∼ . C
08, thesegalaxies may be termed as RF type.Intrinsic shapes of elliptical galaxies have implicationsfor their formation and evolution. As the galaxies studiedhere are very flat, we have given emphasis on the shape P ( q , q ∞ ), and on informations about the flattening at innerand at outer radii. ACKNOWLEDGEMENTS
DKC would like to thank the coordinator, IUCAA referencecentre, Pt. Ravishankar Shukla University, for the technicalsupport. AKD gratefully acknowledges the award of RajivGandhi National Fellowship grant No. F-16-71/2006 (SA-II), from UGC, New Delhi, India. We thank the reviewer forhis comments which helped us to improve the paper in itspresent form.
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APPENDIX A: VERY FLAT DE ZEEUW -CAROLLO MODELS
A simple family of triaxial models, with ellipticityvariation and position angle twist was presented byde Zeeuw and Carollo (1996) with density distribution ρ ( r, θ, φ ) = f ( r ) − g ( r ) Y ( θ ) + h ( r ) Y ( θ, φ ) , (A1)where f ( r ) is same as (1), g ( r ) and h ( r ) are two radial func-tions, and Y and Y are the usual spherical harmonics.Here, ( r, θ, φ ) are the standard polar co-ordinates. The pro-jected surface density of ( A
1) can be calculated easily, andoften analytically. g ( r ) and h ( r ) are fixed by assiging ax-ial ratios ( p , q ) and ( p ∞ , q ∞ ) respectively, at small large c (cid:13) , 1–7 ntrinsic Shapes of Very Flat Elliptical Galaxies Table A1.
Regions of negative ρ . p = p = p ∞ = 0 . , γ = 1 . , θ = 0 o . , φ = π . q = q = q ∞ r low b r high b > > ρ is positive at all r radii, where constant - ρ surfaces are approximately ellip-soidal. Numerical distribution function was shown to existfor prolate triaxials : ( p, q ) = (0 . , .
60) and for oblatetriaxials : ( p, q ) = (0 . , . q = q ∞ = q, p = p ∞ = p and γ = 1 . . C ρ surfacesbecome peanut shaped or dimpled for large values of flat-tening. Such models can not be a ”true” representation ofthe shape of an elliptical galaxy.In addition to above, we now find the appearance ofnarrow regions, where ρ is negative. Clearly, it is unphysicalto call such negative ρ as mass density. Such negative ρ appears in polar regions ( θ ∼ o . r extending from some r low to r high . Tables ( A
1) and ( A ρ on φ = π plane.Triaxials models of similar form as ( A
1) was proposedby Schwarzschild (1979) as a numerical model, where f ( r )is taken as the modified Hubble density distribution. Later,it was put into an analytical form by de Zeeuw and Merritt(1983). Projected properties of such triaxial modified Hub-ble model was studied by Chakraborty and Thakur (2000).We now find that sufficiently flat versions of triaxial modi-fied Hubble model also exhibit regions of negative ρ .We find that the appearance of negative ρ regions iscorrelated with the dimpleness of constant ρ surfaces. Weexamine an extended version of models ( A Y , Y and Y .Such models were studied by Chakraborty and Das (2003).It was found that the dimpleness reduces i.e., the modelsbecome more ellipsoidal - like. We now find that for thesame choice of the parameters as in Table ( A r high − r low ) of negative ρ decreases.It will be interesting to extend the studies of particle or-bit and the numerical distribution function of Thakur et al.(2007), to the models which exhibit regions of negative ρ .This paper has been typeset from a TEX/ L A TEX file preparedby the author.
Table A2.
Regions of negative ρ . p = p = p ∞ = 0 . , γ = 1 . , φ = π , q = 0 . θ r low b r high b o . o . o . o . > o . ρ is positive at all r c (cid:13)000