Surface mass density of the Einasto family of dark matter haloes: Are they Sersic-like?
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 9 November 2018 (MN L A TEX style file v2.2)
Surface mass density of the Einasto family of dark matter haloes:Are they Sersic-like?
Barun Kumar Dhar ⋆ , Liliya L.R. Williams † School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455 USA
Accepted 29th January, 2010; in original form 30th October 2009
ABSTRACT
Recent advances in N-body simulations of dark matter haloes have shown that three-parametermodels, in particular the Einasto profile characterized by d ln ρ ( r ) /d ln r ∝ r α with a shapeparameter α . . , are able to produce better fits to the 3D spatial density profiles thantwo-parameter models like the Navarro, Frenk and White (NFW), and Moore et al. profiles.In this paper, we present for the first time an analytically motivated form for the 2Dsurface mass density of the Einasto family of dark matter haloes, in terms of the 3D spatialdensity parameters for a wide range of the shape parameter . α . Our model describesa projected (2D) Einasto profile remarkably well between and (3 − r , with errors lessthan . per cent for α . and less than per cent for α as large as 1. This model (in 2D) canthus be used to fit strong and weak lensing observations of galaxies and clusters whose totalspatial(3D) density distributions are believed to be Einasto-like. Further, given the dependenceof our model on the 3D parameters, one can reliably estimate structural parameters of thespatial (3D) density from 2D observations.We also consider a Sersic-like parametrization for the above family of projected Einastoprofiles and observe that fits with a Sersic profile are sensitive to whether one fits the pro-jected density in linear scale or logarithmic scale and yield widely varying results. Structuralparameters of Einasto-like systems, inferred from fits with a Sersic profile, should be usedwith caution. Key words: gravitational lensing – galaxies: clusters: general – galaxies: fundamental pa-rameters – galaxies: haloes – galaxies: structure – dark matter.
Gravitational lensing signatures are a response to the projected sur-face mass column density of matter Σ( ~R ) along the line of sightin galaxies and clusters. Upon a suitable deprojection and circularaveraging, an estimate of Σ( R ) can, in principle, be used to tracethe spherically averaged 3D density profile ρ ( r ) .In the past few years, N-body simulations have shown[Power et al. (2003), Navarro et al. (2004) (Nav04), Merritt et al.(2006) (M06), Stadel et.al. (2009)(S09)] that three-parametermodels, especially the Einasto (Einasto 1965) profile and thePrugniel and Simien (1997) de-projected Sersic profiles (PS97), areable to produce better fits to the 3D density profiles of galaxyand cluster-sized dark matter haloes than two-parameter models(Navarro, Frenk and White (1997) & (1996) (NFW), Moore et al.(1999)). While the PS97 profile has a well known 2D sky projectedform - the Sersic (Sersic 1968) profile, there has been no such ana-lytical counterpart for the Einasto profile.In this paper, we present a very good approximation for the 2D ⋆ E-mail:[email protected] † E-mail:[email protected] projection of the Einasto family of 3D profiles. Thus, if the 3D totalmass density is believed to be Einasto-like, our model can be usedto parametrically describe the projected 2D surface mass densitiesof galaxies and clusters in the weak and strong lensing regimes.However, note that even upon radial averaging to smooth out thesubstructure, not all haloes subscribe to an Einasto profile. For therest of this paper, we will limit the discussion to the 2D projectionof the special case where the 3D profile is Einasto-like.In 3D, the functional form of the Einasto (Einasto 1965) pro-file is given by: ln[ ρ ( r ) ρ s ] = − b [( rr s ) n − (1.1)where, ρ ( r ) is the 3D (spatial) density at r , n (or α = n ) is theshape parameter, b is a function of n , ρ s the spatial density at ascale radius r s and ρ (0) = ρ s e b .In 2D, the Sersic (Sersic 1968) profile, which has been usedto describe the projected surface brightness profiles of galaxies, issimilar in form to the Einasto profile in 3D (1.1) and is given by: ln[ Σ S ( R )Σ R E ] = − q [( RR E ) m − (1.2)such that Σ S (0)=Σ R E e q where, R is the projected distance in c (cid:13) Barun Kumar Dhar and Liliya L.R. Williams the plane of the sky, Σ R E is the line of sight projected surfacebrightness at a projected scale radius R E , which can be definedto be the half-light radius of a Sersic profile under the condition: q = 2 m − . . /m (PS97) with m (or λ = m ) char-acterizing the shape of the Sersic profile.The parameter α = n in (1.1) defines the shape of the Einastoprofile. In the first ever fits to N-body haloes with a Einasto profile,Nav04 found an average value of . ± . for a wide rangeof halo masses from dwarfs to clusters. For galaxy-sized haloesPrada et al. (2006) found . α . . Hayashi and White(2008) observed an evolution of α with mass and redshift ( z ) in theMillennium Simulation ( MS ) of Springel et.al. (2005) and found α ∼ . for galaxy and ∼ . for cluster-sized haloes. A simi-lar trend is supported by Gao et al. (2008) where α ∼ . for themost massive clusters in MS . Hence, although in this paper we dis-cuss our approximation to a projected Einasto profile for . α ,particular attention is drawn to the domain α . . ; where as weshall show in §
3, the errors due to our approximation out to r − ( ∼ (3 − r ) are < . .The parameters r s , b and n in the Einasto profile are not in-dependent. It can be seen, that in terms of a dimensionless length X = rr s , the logarithmic slope of the density profile is given by: β = d ln( ρ/ρ s ) d ln X = − bn X /n (1.3)Nav04 chooses to define r s such that b/n = 2 (the isothermal valueof β = − ) at r = r s and hence label r s as r − and ρ s as ρ − .Another approach is to use the convention of M06, requiring r s to include half the total mass. They quote a numerical estimate of b =3 n − .
333 + 0 . /n . In this paper we will follow the Nav04parametrization.Since the Sersic profile describes the 2D surface brightnessprofiles of galaxies reasonably well, a natural question is: doesthe 2D surface mass density of Einasto-like 3D dark matter haloesalso follow a Sersic-like description? For the Nav04 simulations,Merritt et al. (2005) (M05) have shown that a Sersic function doesproduce fits with acceptable errors in the range of r conv to r ,where r conv is the minimum radius of convergence and r is thevirial radius in the N-body simulations of P03 and Nav04.In this paper, we focus on the analytical description of the pro-jected Einasto profile. In order to see how well a projected Einastoprofile is described by a Sersic profile, we numerically project (1.1)and find that a Sersic profile produces acceptable fits only in a lim-ited range of the projected radius R . We shall show in § R .The choice of Sersic fit (log or linear) may, for example, havepossibly strong implications in the strong and weak lensing regimesrespectively, yielding incorrect results. It is with this perspective,and the observation that the projection of an Einasto profile (i.e.integral of a Sersic-like function) is not a Sersic-like function butrather a Gamma-like function, we present a derivation of the sur-face mass density ( § r , r s (or r − ), ρ s (or ρ − ), b and n (or α = 1 /n ))and 2D pro-jected parameters as R , R E , q and m (or λ = 1 /m ). We will alsorefer to the Sersic profile as Σ S , our approximation to a projected2D Einasto profile as Σ E and a numerically projected Einasto pro-file as Σ N . Further, for this paper, r conv and r have no physical meaning as such. We note from the Nav04 simulations, that on anaverage r conv ∼ per cent of r − , and r ∼ − per cent of r − for galaxies and ∼ per cent of r − for clusters. In this paperwe will, for the sake of discussion, refer to r conv ∼ . r − and r as ∼ r − .In § Σ E , whichfor the Nav04 parametrization of b = 2 n , simplifies to (2.11). In § Σ E , followed by a discussion of errors in our approximation.We then present a comparison between Σ N and the best-fitting Ser-sic profile Σ S . In § The Einasto profile has generated considerable interest of late andhas been used in recent studies of dark matter haloes (Nav04,Pr06, M06, G08, HW08, S09). For the N-body haloes in Nav04,Merritt et al. (2005) have shown that within the limited radial rangeof r conv r r and shape parameter . α . ,a deprojected Sersic profile also fits the 3D distributions almostas well as the Einasto profile, and a Sersic profile fits the non-parametrically estimated 2D surface densities of dark matter haloeswith acceptable errors ( ∼ per cent).In the following discussion, we argue that if a 3D distributionis Einasto-like, the 2D distribution need not be Sersic-like and pro-vide an analytically motivated functional form for the 2D projec-tion, which can be used to describe the surface mass density of theEinasto family of dark matter haloes subscribing to a wide range ofthe shape parameter ( n or . α ) and over a wideradial (projected) range R r . In terms of the line of sight distance ( z ), the surface mass density Σ (R) can be estimated from: Σ( R ) = 2 Z ∞ ρ ( p z + R ) dz (2.1)While an exact analytical expression for the integral in (2.1) forthe Einasto profile (1.1) has so far eluded us, we derive below anexcellent semi-analytical approximation.To intuitively motivate the functional form of Σ E ( R ) for theEinasto profile (1.1), observe that at R = 0 , the integral (2.1)presents us with an exact solution: Σ E (0) = 2 e b r s nρ s b n Γ[ n ] (2.2)where, Γ[ n ] is the complete Gamma Function.Hence, for R >
0, it is reasonable to expect the integral to depend onterms involving incomplete gamma functions. In fact, for the sakeof discussion, one can make a very crude assumption that most ofthe contribution to the integral in (2.1) at a given R (especially for R < r − ) comes from the region z > R . Integrating, from some ζ R to ∞ (where ζ > ), one gets: Σ( R ) = 2 e b r s nρ s b n Γ[ n, b ( ζRr s ) n ] (2.3) c (cid:13) , 000–000 urface mass density of the Einasto family of dark matter haloes: Are they Sersic-like? Similarly, the integral for
R > r − will have dominant contribu-tions from terms involving γ [ a, x ] ; where Γ[ a, x ] and γ [ a, x ] arethe upper-incomplete and lower-incomplete gamma functions re-spectively.This is quite unlike a Sersic (1.2) function. Hence, although aSersic profile may fit a projected Einasto profile in a limited rangeof R it need not be a very good fit for all R . Σ E (R): An analytical approximation of projectedEinasto profile With the 3D spatial distance r = √ z + R , where R is the 2Dprojected distance in the plane of the sky, and z the line of sightdistance from the object to the observer, at any given R , one candefine 3 regions for the integral in (2.1) for the Einasto profile(1.1):Region I: z < R , integrating from z = 0 to ζ R , with ζ Region II: z > R , integrating from z= ζ R to ∞ , with ζ > , andRegion III: z ∼ R , in a neighborhood δ between ζ R and ζ R In Region I: z < R , the first term on the right hand side(RHS) of (1.1) can be written as: − b (cid:18) z + R r s (cid:19) n = − b (cid:18) Rr s (cid:19) n (cid:20) (cid:16) zR (cid:17) (cid:21) n (2.4)Neglecting 4th and higher order terms in ( z/R ), in the binomialexpansion of (2.4), the integral of (2.1) (from z = 0 to ζ R) has ananalytical approximation: ρ s r s exp " − b (cid:20) Rr s (cid:21) n − ! nb (cid:18) Rr s (cid:19) ( − n ) × γ " , ζ b n (cid:18) Rr s (cid:19) n (2.5)In Region II: z > R , a binomial expansion of the first term of theRHS of (1.1) gives us: − b ( zr s ) n (1 + ( Rz ) ) n = − b ( zr s ) n − b ( Rr s ) n (cid:20) n ( Rz ) − n + 14 n ( 12 n − Rz ) − n + ... (cid:21) (2.6)Fits to N-body simulations with the Einasto profile have so far indi-cated an n> . ( α . . ). Hence, observing that the leading contri-bution comes from the 1st term b ( z/r s ) n , one can drop the remain-ing terms. This is especially true in our primary domain of interest( n , i.e. α . ). Even for n , although the approx-imation is not as good (as it is for n > ), it continues to providebetter fits than a Sersic profile in the entire range n . Withthis understanding, the integral from z = ζ R to ∞ , in Region IIcan be written as: e b r s nρ s b n Γ " n, b (cid:18) ζ Rr s (cid:19) n (2.7)In Region III: ζ R z ζ R , one can not make the approx-imations made in regions I and II. However, there exists a point ǫR between ζ R and ζ R , where the mean-value approximationwill be valid in a domain δ about ǫR . Since the density profile fallsrapidly for R << r − and gradually for R >> r − , the domainof applicability of the mean-value approximation will be such that δ < ( ζ − ζ ) for R < r − ( → as R → ) and tending to ζ − ζ for R >> r − . Further, it should be obvious that ζ , ζ and δ will depend on the shape parameter n as well. A function describing δ ,with such a property is: δ = ( ζ − ζ )[1 − exp( − ( R/r s ) µ )] (2.8)with µ = µ ( n ) The remaining (small) excluded region does not add significantlyto the integral (refer to error plots in § δ around R - from an application of the mean-valuetheorem at ǫR can be written as: δRρ ( z = ǫR ) = 2 δRρ s exp − b "(cid:18) √ ǫ Rr s (cid:19) n − (2.9)A few important observations are in order. First, neglectingterms in the integrand of region I (2.4) and II (2.6), leads to over-estimating the integrands in those regions. Second, ignoring thecontribution from region III and fitting only for ζ and ζ , producesgood fits (refer to discussion following (2.13)) but understandablywith a ζ greater than ζ with region III included. i.e. includingregion III, lowers ζ allowing it to be closer to 1, resulting in alarger contribution from the upper-incomplete gamma function. Asimultaneous fit of ζ , ζ , ǫ and µ (in δ ) accounts for these excesscontributions through a negative sign from region III.In region I, since th and higher order terms in z/R are ne-glected, one can fix ζ = 1 . Further, although ǫ should in principlebe estimated, we found it to be a reasonable approximation to fix ǫ = ζ + ζ . This reduces the number of parameters to fit to onlytwo - ζ and µ - which in turn, during the fitting process, compen-sates for the approximations made on ζ and ǫ .With these approximations, the surface mass density Σ E (R),for n with X = R/r s can be written as: Σ E ( R ) = Σ E 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 (2.10) − δb n Xe − b (cid:16) √ ǫ X (cid:17) n where, Σ E = Σ E (0) = 2 e b r s ρ s Γ( n + 1) b n For the Nav04 parametrization of b n = 1 and by choosing ζ = 1 , (2.10) simplifies to (2.11); where, after substituting factorslike b/ n with in (2.10), we leave b in the rest of the equation inorder to reduce clutter. This gives us: Σ E ( R ) = Σ E Γ( n + 1) " n Γ h n, b ( ζ X ) n i + b n X ( − n ) γ (cid:20) , X n (cid:21) e − bX n − δb n Xe − b (cid:16) √ ǫ X (cid:17) n (2.11)with, δ = ( ζ − ζ )[1 − e − ( R/r s ) µ ] ǫ = ζ + ζ c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams ζ n Best fit ζ (n) Figure 1. ζ ( n ) (2.12) for . α . ( . n . ). Each pointhas been obtained by fitting the numerically projected Einasto profile with(2.11) for b = 2 n and ζ = 1 . ζ = 1 b = 2 nX = R/r − and, ζ and µ are numerically estimated (Fig.1 and Fig.2) for thespecific conditions of b = 2 n and ζ = 1 to yield: ζ = 1 . . n − . n (2.12) µ = 1 . n + 0 . n − . n (2.13)We would also like to note that based on the accuracy of an approx-imation needed, one can neglect the contribution from region III(the δ term) and with ζ = 1 fit only for ζ . Although we have notexplored a functional relation of ζ ( n ) , we note that the best-fitting ζ varies weakly from . at n = 10 to . at n = 3 . to . at n = 1 . , with a maximum error within r − (usuallyreaching a peak around r ) of . at n = 10 , . at n = 3 and . at n = 1 . Hence, within the current domain of < n < from N-body simulations (corresponding to . < α < . ) onecan neglect the δ term and set ζ = 1 , and ζ = 1 . at the costof an error in the range ( . to . ).The parametrizations (2.12) and (2.13) worked very well evenat n=14 and n=0.95. However, we have not tested n > and n < values rigorously. If greater accuracy is needed, we recommendfitting for ζ and µ .[Refer to the discussion following (2.9), on whythe δ term is negative]. This is a useful result because the surfacemass density is expressed entirely as a function of the 3D spatialdensity parameters, which is not the case for any existing projectedfitting function. Equation (2.10) or (2.11) thus serves two purposes:One, given a 3D Einasto profile, (2.10) gives a good approximationto its 2D surface mass density. And two, a good fit to some 2Dobservations with (2.10) for example surface mass density fromlensing, will give us the 3D spatial density parameters of a Einasto-like profile.This is also quite unlike fitting 2D Einasto profiles with Sersic-like functions, where one first needs to fit for the 2-D and then de-project to fit for the 3D shape parameters (M05), which usually aredifferent without any known existing functional relation betweenthem. µ n Best fit µ (n) Figure 2. µ ( n ) (2.13) . α . ( . n . ). Each pointhas been obtained by fitting the numerically projected Einasto profile with(2.11) for b = 2 n and ζ = 1 . ζ and µ for Σ E We numerically integrate (2.1) for the profile in (1.1) and obtain Σ N ( R ) in the domain R : (0 − r − for 90 profiles with ashape parameter in the range . α . A resolution in R of . r − ( ∼ . r conv ), allows us to quantify errors due to ourapproximation in a domain R << r conv , and we report compari-son of errors up to r − or up to a R where Σ N ( R ) ∼ − Σ(0) whichever is earlier.With b = 2 n and ζ = 1 , we use a non-linear least squaresLevenberg-Marquardt algorithm to estimate the best-fitting valuesfor ζ and µ by fitting (2.11) to each of the numerically generated Σ N profiles. We find that ζ (2.12) and µ (2.13) are best describedby a second and third degree polynomial respectively, in α = 1 /n (Fig.1 and Fig.2). Σ E approximation Fig. 3 and Fig. 4 describe the fractional error profile between ourmodel Σ E and the numerically projected Einasto profile Σ N (R)for a wide range of the shape parameter α = 1 /n . Fig. 3 is morerelevant to current N-body simulations, where in α seems to be inthe range 0.1 to 0.25. It is worth noting here, that for α as high as0.25 ( n = 4 . ), the largest errors are < . in the range (0 to30) r − .In case the range n becomes relevant in the future,where N-body haloes for α & . have not yet been found, we alsopresent in Fig. 4 a comparison between Σ E and Σ N . This is alsothe domain where our assumptions in the z > R region are weaker.Nevertheless, the accuracy of the approximation is striking with theworst error < within R < r − . In this section we discuss results of fits to Σ N with a Sersic func-tion and superimpose the Σ E model (black solid lines) (2.11) for arelative comparison between Σ S and Σ E (Fig. 5 and Fig. 6). Thefits presented here were obtained using log scale of density. Onecan also obtain fits with density in linear scale; a comparison of c (cid:13) , 000–000 urface mass density of the Einasto family of dark matter haloes: Are they Sersic-like? -0.0035-0.003-0.0025-0.002-0.0015-0.001-0.0005 0 0.0005 0.001 0.01 0.1 1 10 ∆ Σ E / Σ N R/r -2 Σ E n=10.0, α =0.10 Σ E n=7.14, α =0.14 Σ E n=5.88, α =0.17 Σ E n=4.76, α =0.21 Σ E n=4.00, α =0.25 Figure 3.
Fractional Error between Σ E and Σ N for . α . , n . The x-axis plotted in log scale is expressed as a ratio of the2D projected radius R and the 3D scale radius r − of the correspondingEinasto profile. r ∼ r − while r conv ∼ . r − . -0.02-0.015-0.01-0.005 0 0.005 0.01 0.1 1 10 ∆ Σ E / Σ N R/r -2 Σ E n=3.03, α =0.33 Σ E n=2.50, α =0.40 Σ E n=2.00, α =0.50 Σ E n=1.50, α =0.67 Σ E n=1.25, α =0.80 Σ E n=1.00, α =1.00 Figure 4.
Fractional Error between Σ E and Σ N for . α . , n . The x-axis plotted in log scale is expressed as a ratio of the2D projected radius R and the 3D scale radius r − of the correspondingEinasto profile. r ∼ r − while r conv ∼ . r − . residuals for the case α = 0 . is presented in Fig. 7. The bestfit parameters and consequently the error profile are quite differ-ent and is an indication that the best-fitting Sersic profile does notprovide an adequate representation of a projected Einasto profile.Not evident in the plots (Fig.5-7) are the extremely large errorsin the central density Σ S (0) when fitting using log density. For α =0 . (Fig. 5) the relative error at R = 0 is , for α = 0 . (Fig.5and 7), the relative error is and for α = 1 . However, weobserved empirically that fits in log density happen to reflect thesignificance of R E as enclosing half the total mass of Σ N (it is tobe noted that R E by definition encloses half the total mass of Σ S ).This is because the domain over which Σ S (with fits obtained in logdensity) overestimates the density is a relatively small contributorto the total mass and fits with log density in the region R > r − are good. Since, we can reliably estimate one of the Sersic profileparameters ( R E ) through fits with log density (as opposed to nonein linear scale), we have presented results of fits in log density inFig. 5 and Fig.6.The large errors in the central density also have serious con- sequences for strong lensing. Typically the strong lensing regimeextends up to ∼ . r − . In strong lensing, image positions cor-respond to extremum (minima, maxima and saddle) points of thetime delay surface. The j th image θ ij for the i th source β i is givenby: ~θ ij = ~β i + 1 π Z ( ~θ ij − ~θ ′ ) κ ( ~θ ′ ) | ~θ ij − ~θ ′ | d θ ′ (3.1)where κ ( ~θ ) is the normalized surface mass density at an angularposition ~θ .Consequently contributions to the integral in (3.1) from un-usually large density near the center (as a result of fits with a Sersicprofile in log density), will produce image separations larger thanwhat one can expect from a numerical projection of the 3D Einastoprofile.Sersic profile fits to model a projected Einasto profile shouldthus be avoided especially if the central region ( R< . r − ) is be-ing excluded. This is because, as shown in Fig.7, one can get areasonably good fit (relative error within ) over a large rangeof R & . r − (fitting in log density) giving an indication that theSersic profile is a good representation of projected Einasto profile,but doing so will lead to even larger errors in the excluded centralregion ( R< . r − ). One should thus use caution in interpretingthe other structural parameters, the shape parameter m or λ and thecentral density Σ S (0) .Fits in linear density present a different problem. Even thoughthe central errors are much better (relative error ∼ for α =0 . ,Fig.7) than fits using log density, the best-fitting R E does not en-close half the total mass of Σ N and the errors for large R keepincreasing with R.Although the Sersic profile is not a good representation of aprojected Einasto profile, one can fit Sersic profiles in limited do-mains of R and obtain an estimate of the shape of the projectedEinasto profiles in those domains only, but be careful to not use theresulting best-fitting values of R E and Σ S (0) as a true represen-tation of the half-mass radius and central density of the projectedEinasto. In Fig.8, we present two such relations between the 2DSersic index ( λ = m ) and the 3D Einasto index α = n in two do-mains R< r − and R> r − . In these 2 regions, λ ( α ) can be de-scribed as power laws. For the domain R< r − we find: λ ( α ) = 1 . α . (3.2)and for R > r − defined as in § λ ( α ) = 1 . α . (3.3)Not only is the shape ( λ ) different in the two domains, their evolu-tion with α is also different. Nevertheless, this result can be usefulin obtaining an estimate of the shape of projected Einasto profile inthese two domains demarcated by r − .The Σ E model, with errors < . does not face any of theabove issues. Further, unlike the Sersic profile, the Σ E model canpredict the central density with almost errors due to the exis-tence of an analytical solution. Thus, if the underlying 3D distri-bution is Einasto-like, the 2D distribution should be modeled with Σ E . Σ E model The Σ E model is not just a good description of the projectedEinasto profile but is also expressed in terms of the 3D Einasto c (cid:13) , 000–000 Barun Kumar Dhar and Liliya L.R. Williams -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.01 0.1 1 10 ∆ Σ S / Σ N R/r -2 Σ E n=10.0, α =0.10 Σ E n=5.88, α =0.17 Σ E n=4.76, α =0.21 Σ E n=4.00, α =0.25 Σ S n=10.0, α =0.10 Σ S n=5.88, α =0.17 Σ S n=4.76, α =0.21 Σ S n=4.00, α =0.25 Figure 5.
Fractional Error between the best-fitting Σ S and Σ N withEinasto index . α . , n with fractional error forthe best-fitting Σ E (black solid lines ≈ ) superimposed. The x-axis plot-ted in log scale is expressed as a ratio of the 2D projected radius R and the3D scale radius r − of the corresponding Einasto profile. r ∼ r − while r conv ∼ . r − . -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.01 0.1 1 10 ∆ Σ S / Σ N R/r -2 Σ E n=3.03, α =0.33 Σ E n=2.00, α =0.50 Σ E n=1.33, α =0.75 Σ E n=1.00, α =1.00 Σ S n=3.03, α =0.33 Σ S n=2.00, α =0.50 Σ S n=1.33, α =0.75 Σ S n=1.00, α =1.00 Figure 6.
Fractional Error between the best-fitting Σ S and Σ N withEinasto index . α . , n with fractional error forbest-fitting Σ E (black solid lines ≈ ) superimposed. The x-axis plotted inlog scale is expressed as a ratio of the 2D projected radius R and the 3Dscale radius r − of the corresponding Einasto profile. r ∼ r − while r conv ∼ . r − . profile parameters. It should thus be possible to recover the 3D pa-rameters ( α, r − , ρ − ) from fits to 2D distributions that subscribeto an underlying 3D Einasto-like system.For the wide family of numerically projected Einasto profiles Σ N described in this paper, we could recover the 3D parametersfor all of them with an accuracy of ∼ − or better, by fitting Σ N with (2.11) and the parametrizations of (2.12) and (2.13) througha non-linear least squares Levenberg-Marquardt algorithm. Giventhat, as of now robust data (within virialized regions) from N-bodysimulations are in the domain r conv to r , the fits were performedin this domain. In passing, we note that our results are even betterif we fit from to r − . Such a high degree of accuracy indicatesthat if the 2D distribution is indeed a projected Einasto profile, the3D parameters can be recovered very well even from a limited ra-dial range of observations.We note that, for the entire range of . ( α = n ) the -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.01 0.1 1 10 ∆ Σ S / Σ N R/r -2 Σ S fits in linear scale for α =0.17 Σ S fits in log scale for α =0.17 Σ E n=5.88, α =0.17 Figure 7.
Fractional Error in Σ S from fits to Σ N (for α = 0 . ) with aSersic profile using log density (dot-dashed line) and a Sersic profile usinglinear density (dashed line). Not shown in the plots are the errors in cen-tral density ∆Σ S (0)Σ N (0) . The errors are for the fit with log density and for the fit with linear density. The fractional error in Σ E (black solidline ≈ ) is superimposed for comparison. The x-axis, plotted in log scale,is a ratio of the 2D projected radius R to the 3D scale radius r − of thecorresponding Einasto profile. r ∼ r − while r conv ∼ . r − . S e r s i c ( D ) s hape pa r a m e t e r λ = / m Einasto (3D) shape parameter α =1/n Sersic index λ ( α ) for R > r -2 Sersic index λ ( α ) for R < r -2 Figure 8.
Sersic index( λ =1/m) of a projected (2D) Einasto profile as a func-tion of the 3D Einasto index ( α =1/n) in two domains R < r − (dotted) and R > r − (solid) indicating the absence of a unique Sersic index λ for all R.The power law relations for λ ( α ) in equations (3.2) and (3.3) are applicableonly in these domains. fits always converged for an intial guess of α < true α , r s > truer − and ρ − in the range ( . to ) ρ true − . An inital guess of verylow α ∼ . and a guess for r − ∼ r for the type of object(galaxy or cluster) being considered can be a reasonable startingvalue for the fit to converge. We also did not encounter any localminima. i.e. if the fit converges, it always converged to the true setof ( α , r − , ρ − ). Non-parameteric estimates of density profiles in N-body simula-tions (Nav04,M06) favour Einasto-like profiles, since they pro-vide better fits than the two-parameter NFW and Moore profiles.Merritt et al. (2006) have also shown that a de-projected Sersic pro- c (cid:13) , 000–000 urface mass density of the Einasto family of dark matter haloes: Are they Sersic-like? file fits the 3D halo mass distribution almost as well as the Einastoprofile, and a Sersic profile provides good fits to non-parametricestimates of surface mass densities (M05) of the Nav04 N-bodyhaloes.We have observed that fits with a Sersic function ( Σ S ) to anumerically projected Einasto profile ( Σ N ) are sensitive to whetherone fits using linear density Σ S (errors increasing for large R) orlog density ln (Σ S ) (errors increasing for small R) yielding widelyvarying results. Consequently, the Sersic profile does not give anadequate description of the projected Einasto profile.Sersic profile fits to the surface mass density of N-body haloes(M05), whose 3D spatial densities are well fit by Einasto profileswith . α . , have been obtained from the limited radialrange of r conv to r . For the haloes in M05 and Nav04, this rangeis generally less than two decades in radius. Hence, if the 3D dis-tribution is indeed Einasto-like, interpreting structural propertiesfrom fits with a Sersic profile, especially m in (1.2) as the shapeparameter, Σ S (0) as the central density and R E as the half-massradius, can be misleading.In this paper, we have provided an analytical approximation(2.11) to the projected surface mass density of Einasto-like 3Ddensity distributions. The fit errors are well contained to < forthe projected radial range R (10 − r − equivalent to (3 − r and shape parameter, n , or . α . Thismodel can therefore be used both as a fitting function for 2D ob-servations and also to extract the 3D parameters of Einasto-likeprofiles. Since Σ E fits a projected Einasto profile in a wide radialrange, it can be used for fitting strong and weak lensing observa-tions in systems whose total 3D density distribution is believed tobe Einasto-like. One can also numerically integrate (2.11) to getreliable estimates of the mass enclosed.Finally, we note that the form similarity of (1.1) and (1.2), i.e.fitting functions that describe the 3D mass density of dark matterhaloes and the 2D light distributions of galaxies, respectively, couldbe largely coincidental and should be used with caution when draw-ing conclusions about the similarity of dynamical evolution thatlead to the formation of the stellar components of ellipticals anddark matter haloes. ACKNOWLEDGMENTS
BKD and LLRW would like to acknowledge the support of NASAAstrophysics Theory Grant NNX07AG86G. We thank Jaan Einastoand Urmas Haud for pointing us to the original literature on theEinasto profile.
REFERENCES
Einasto J., 1965, Tartu Astron. Obser. Teated, No.17, 1Gao L., Navarro J.F., Cole S., Frenk C.S., White S.D.M., SpringelV., Jenkins A., Neto A.F., 2008, MNRAS, 387,536 (G08)Hayashi E., White S.D.M., 2008, MNRAS, 388,2 (HW08)Merritt D., Navarro J.F., Ludlow A., Jenkins A., 2005, ApJ,624:L85 (M05)Merritt D., Graham A.W., Moore B., Diemand J., Terzic B., 2006,AJ, 132:2685 (M06)Moore B., Quinn T., Governato F., Stadel J., Lake G., 1999, MN-RAS, 310, 1147 (Moore 99)Navarro J.F., Frenk C.S., White S.D.M., 1996, ApJ, 462, 563(NFW) Navarro J.F., Frenk C.S., White S.D.M., 1997, ApJ, 490, 493(NFW)Navarro J.F. et al., 2004, MNRAS, 349, 1039 (Nav04)Power C., Navarro J.F., Jenkins, A., Frenk C.S., White S.D.M.,Springel V., Stadel J., Quinn T., 2003, MNRAS, 338, 14 (P03)Prada F., Klypin A.A., Simonneau E., Betancort-Rijo J., Patiri S.,Gottlober S., Sanchez-Conde M.A., 2006, ApJ, 645:1001 (Pr06)Prugniel Ph. and Simien F., 1997, AA, 321, 111 (PS97)Sersic J.L. 1968, Atlas de Galaxias Australes, Observatorio As-tronomico de Cordoba, Cordoba, ArgentinaSpringel V. et.al. 2005, Nature, 435,629Stadel J., Potter D., Moore B., Diemand J., Madau P., Zemp M.,Kuhlen M., Quilis V., 2009, MNRAS, 398,1,L21 (S09) c (cid:13)000