Testing a simple recipe for estimating galaxy masses from minimal observational data

Abstract

1 INTRODUCTION

The accurate determination of galaxy masses is a crucial issue for galaxy formation and evolution models. Disentangling dark matter and baryonic matter of a galaxy permits testing the predictions of Λ cold dark matter cosmology and probing the mass function. An algorithm for deriving the mass of a spiral galaxy is straightforward – one just need to measure a rotation curve from gas or stars that can be safely assumed to be on circular orbits. For elliptical galaxies the situation is less simple. There is no ‘perfect’ (in terms of accuracy) tracer to measure the total gravitational potential. The main problem is the degeneracy between the anisotropy of stellar orbits and the mass. The shape of stellar orbits is not known a priori and different combinations of orbits may give the same distribution of light. Several different approaches for mass determination were proposed and successfully implemented, such as strong and weak lensing (e.g. Mandelbaum et al. 2006; Gavazzi et al. 2007), modelling of X-ray emission of hot gas in galaxies (e.g. Humphrey et al. 2006; Churazov et al. 2008), Schwarzschild modelling of stellar orbits, etc. Accurate data on the projected line-of-sight velocity distribution with information on higher order moments enable an accurate determination of the mass distribution for nearby ellipticals (e.g. Gerhard et al. 1998; Thomas et al. 2011). However, in the case of minimal available data, detailed modelling is often not possible. Therefore, it is important to find a method to measure galaxy masses with reasonable accuracy which gives an unbiased estimate when averaged over a large number of galaxies. In particular, it can be extremely useful while analysing large surveys, especially at high redshifts when detailed observational data of each individual galaxies are often not available.

The simplest way of estimating the mass of a galaxy is based on the projected velocity dispersion in a fixed aperture (e.g. Cappellari et al. 2006). A slightly more complicated approach is described in Churazov et al. (2010). To estimate the mass, the only information required is the light profile and either the dispersion profile measurement or at least a reliable dispersion measurement at some radius. Testing this particular method on a sample of simulated galaxies is the subject of this paper. The main questions that we want to address are as follows. (i) What is the accuracy of this method? (ii) Does it give an unbiased result? (iii) What are the restrictions for application of this method?

The structure of this paper is as follows. In Section 2, we provide a brief description of the method. In Section 3, we describe the sample of simulated galaxies which is used to test the method. The analysis of the accuracy of the method is presented in Section 4 where we also discuss alternative methods for determining the circular velocity. A summary on the bias and accuracy of the various methods is given in Section 5 with conclusions in Section 6.

2 DESCRIPTION OF THE METHOD

The main idea of the method is described in Churazov et al. (2010). Here we just provide a brief summary.

The method is based on the stationary non-streaming spherical Jeans equation:

1

where j(r)1 is the stellar luminosity density, is the radial component of the velocity dispersion tensor (weighted by luminosity), is the stellar anisotropy parameter (σ_θ=σ_φ because of the assumed spherical symmetry) and Φ(r) is the gravitational potential of a galaxy.

While the stellar luminosity density j(r) and radial dispersion cannot be observed directly, they contribute to the 2D surface brightness I(R) and the velocity dispersion σ(R) profiles:

2

3

Assuming we note that β= 0 for systems where the distribution of stellar orbits is isotropic, β= 1 if all stellar orbits are radial and β→−∞ if the orbits are circular.

Assuming the logarithmic form of the gravitational potential and using local properties of given I(R) and σ(R), one can calculate a circular velocity V_c for three different types of stellar orbits: isotropic (⁠⁠, β= 0), radial (σ_φ=σ_θ= 0, β= 1) and circular (⁠⁠, β→−∞). These relations are given by

4

where

5

In the case of noisy data on the dispersion velocity profile the subdominant terms γ and δ can be neglected, i.e. the dispersion profile is assumed to be flat, and equations (4) are simplified to

6

Let us call a sweet spot the radius at which all three curves and are very close to each other. One can hope that at the sweet spot the sensitivity of the method to the stellar anisotropy parameter β is minimal and the estimation of the circular speed at this particular point is reasonable. For example, from equations (6) it is clear that in the case of the power-law surface brightness profile with α= 2 and β= const, the relation between the circular speed and the projected velocity dispersion does not depend on the anisotropy parameter (e.g. Gerhard 1993). While the derivation of equations (4) and (6) relies on the assumption about a flat circular velocity profile, tests on model galaxies with non-logarithmic potentials, non-power-law behaviour of the surface brightness and line-of-sight velocity dispersion profiles and with the anisotropy parameter β varying with radius (Churazov et al. 2010) have shown that the circular speed can still be recovered to a reasonable accuracy. Now we extend these tests to a sample of simulated elliptical galaxies.

This method for evaluating the circular speed is not only simple and fast in implementation but it also does not require any assumptions on the radial distribution of anisotropy β(r) and mass M(r).

The mathematical derivation of equations (4)–(6) can be found in Churazov et al. (2010). A similar approach and analytic formulae for kinematic deprojection and mass inversion can also be found in Wolf et al. (2010) and Mamon & Boué (2010).

3 THE SAMPLE OF SIMULATED GALAXIES

3.1 Description of the sample

Simulations provide a useful opportunity to test different methods and procedures as all intrinsic properties of a system at hand are known. The main drawback of simulated objects is that they may not include all physical processes that take place in reality and thus may not reflect all complexity of nature. To test the procedure under consideration, we have used a sample of 65 cosmological zoom simulations partly presented in Oser et al. (2010). These smoothed particle hydrodynamics (SPH) simulations include feedback from type II supernovae, a uniform ultraviolet-background radiation field, star formation and radiative hydrogen and helium cooling but do not include ejective feedback in the form of supernovae-driven winds. Present-day stellar masses of simulated galaxies range from 2.18 × 10¹⁰ to 28.68 × 10¹⁰ M_⊙h⁻¹ inside 30 kpc. The softening length used in simulations is about R_soft= 400 pch⁻¹, h= 0.72. Typically the softening can affect profiles up to ⁠, which is ≃1.7 kpc in our case. We have followed a conservative approach and restricted the analysis to radii larger than 3 kpc. It should be noted that low-mass simulated galaxies may have no real counterparts possibly due to the lack of important physical processes (e.g. significant winds) in simulations. However, it has been demonstrated in Oser et al. (2011) that the massive simulated galaxies have properties very similar to observed early-type galaxies (see also Fig. 4), i.e. they follow the observed scaling relations and their evolution with redshift. For detailed description of simulations and included physics see Oser et al. (2010).

Figure 4

relation. The blue solid line is the linear fit to data points from the simulations. The green dashed line is the observed mass–size relation from Auger et al. (2010).

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To effectively increase the number of galaxies, we have considered three independent projections of each galaxy. Therefore, the whole sample of simulated galaxies consists of 195 objects.2

3.2 Isothermality of potentials in massive galaxies

First of all, we have found that massive galaxies in the sample have almost isothermal rotation curves over broad range of radii. To demonstrate this statement (Fig. 1), we have selected galaxies with a projected velocity dispersion at the effective radius σ(R_eff) (procedure of computation R_eff is described in Section 3.3) greater than 200 km s⁻¹ and plotted their circular velocity curves as a function of r/R_eff. G is the gravitational constant, M(<r) is the mass enclosed within r and R_eff is the effective radius of the galaxy. The circular velocity curves were normalized to the value of V_c averaged over r∈ [0.5R_eff, 3R_eff]. Three circular velocity curves that make the most significant contribution to the rms actually correspond to galaxies with the effective radius R_eff < 6 kpc. Therefore, in these, galaxies, the circular velocity curves may be affected by the gravitational softening especially near 0.5R_eff.

Figure 1

Circular velocity curves of massive galaxies [σ(R_eff) > 200 km s⁻¹] as a function of radius r. Individual rotation curves normalized to the speed averaged over [0.5R_eff, 3R_eff] are shown in black, green dashed lines indicate the interval [1− rms, 1+ rms ], where rms = 4.9 per cent, and the red thick line represents the overall trend ⁠.

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3.3 Analysis procedure

The analysis of each galaxy consists of several steps as described below.

Step 1: excluding satellites from the galaxy image. Usually, an image of a simulated galaxy (the distribution of stars projected on to a plane) contains many satellite objects and needs to be cleaned. Exclusion of satellites makes the surface brightness and the line-of-sight velocity dispersion profiles smoother and reduces the Poisson noise associated with satellites. The algorithm we used for removing satellites is as follows: first, for each star a quantity w characterizing the local density of stars (⁠⁠) and analogous to the SPH smoothing length was calculated and the array of these values was sorted. Then the (0.4N_stars) th term of the sorted w-array was chosen as a reference value w_o. N_stars is the total number of stars in a galaxy, and a factor in front of N_stars is some arbitrary parameter (the value 0.4 was chosen by a trial-and-error method). Stars with the 3D radius r > 10 kpc and w < w_o are considered as members of a satellite. After projecting stars on to the plane perpendicular to the line of sight, we have excluded all satellites together with an adjacent area of 1.5 kpc in size. The initial and final images of some arbitrarily chosen galaxy (the virial halo mass is ≃1.7 × 10¹³ M_⊙h⁻¹) are shown in Fig. 2.

Figure 2

Excluding the satellites (150 kpc × 150 kpc). Left: initial galaxy image. Right: cleaned galaxy image.

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Step 2: evaluating I(R) and σ(R). All radial profiles have been computed in a set of logarithmic concentric annuli around the halo centre. To calculate the surface brightness profile, corrected for the contamination from the satellites, we have first counted the number of stars in each annulus, excising the regions around satellites. The surface area of each annuli has also been calculated, excluding the same regions. The ratio of these quantities gives us the desired ‘cleaned’ surface brightness profile. The average line-of-sight velocity of stars and the projected velocity dispersion have been calculated similarly.

Importance of the ‘cleaning’ procedure and the resulting profiles of I(R) and σ(R) are shown in Fig. 3. The surface brightness data [open circles correspond to the initial (‘uncleaned’) image and red solid circles to the ‘cleaned’ image] and the smoothed curves (the calculation of these curves is described in Step 3) are shown in the upper panel, and the projected velocity dispersion profiles are shown in the middle panel. The true circular velocity (black solid curve) and the circular velocity [recovered from the initial data (blue dashed line) and from ‘cleaned’ data (blue solid line)] for the isotropic distribution of stellar orbits (the first equation in equations 4) are shown in the bottom panel. The last curve is in better agreement with the true velocity profile. All results and figures in this paper are restricted to the region R > 3.0 kpc.

Figure 3

Influence of satellites on the surface brightness (upper panel) and the projected velocity dispersion profiles (middle panel). Open black circles correspond to the initial galaxy image and solid red circles correspond to the galaxy image without satellites. The black dashed curve is the smoothed curve for the initial data and the black solid curve is for the cleaned data. The bottom panel shows the true circular velocity (black thick line) and recovered circular velocity for the isotropic distribution of stellar orbits (in blue) for initial data (dashed) and cleaned data (solid). It is clear that removing satellites reduces the scatter in the line-of-sight velocity dispersion data and makes the profile smoother.

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Step 3: taking derivatives. To take derivatives, we follow the procedure described in appendix B of Churazov et al. (2010). The main idea is that all data points participate in calculating the derivative but with different weights. The weight function is given by

7

where R₀ is the radius at which the derivative is being calculated and the parameter Δ is the width of the weight function.

Both observed and simulated surface brightness profiles are typically quite smooth, so we have used Δ_I= 0.3 to calculate the logarithmic derivative dln I(R)/dln R. For the line-of-sight velocity dispersion data we have used Δ_σ= 0.5. With the assumed values of Δ, the local perturbations are smoothed out but the global trend of the profiles is not affected. Changing values Δ_I and Δ_σ in the range [0.3, 0.5] does not significantly influence our final result.3 The difference (in terms of circular velocity) is less than 1 per cent. As an example, the smoothed curves for the I(R) and σ(R) data in Fig. 3 are calculated using this procedure.

We have also tested the influence of parameters of the presented smoothing algorithm. As long as the smoothed curve describes data reasonably well, neither the functional form of the weight function nor other parameters [like higher order terms in expansion ln I(R) =a(ln R)²+bln R+c or σ(R) =a(ln R)²+bln R+c] significantly affect the final result.

Step 4: estimating the circular velocity. Applying equations (4) or (6) to the smoothed I(R) and σ(R), we have calculated V_c profiles assuming isotropic, radial and circular orbits of stars. Then we have found a radius (a sweet point R_sweet) at which the quantity ⁠, where ⁠, is minimal. The value of the isotropic velocity profile at this particular point is the estimation of the circular velocity speed we are looking for. We take as an estimate of the V_c(R) (rather than or ⁠) for two reasons. First, at around one effective radius the dominant anisotropy for most elliptical galaxies is (Cappellari et al. 2007). The spherically averaged anisotropy is therefore only moderate (see also Gerhard et al. 2001, fig. 4). Massive elliptical galaxies are the most isotropic. Thus, an isotropic orbit distribution is a much better approximation than purely radial or circular orbits. Secondly, the value of is less prone to spurious wiggles in I(R) and σ(R).

The effective radius R_eff is calculated as a radius of the circle which contains half of the projected stellar mass, taking into account effects of cleaning. We found that in the simulated data set the value of the effective radius depends on the maximal radius used to calculate the total number of stars in a galaxy. The problem is especially severe for the most massive galaxies as they have an almost power-law 3D stellar density distribution with a≃ 3. In our analysis (in contrast to Oser et al. 2011), we have not introduced any artificial cut-off and used all stars in the smooth stellar component (excluding substructure) of the main galaxies out to their virial radii for the calculation of the effective radius. The resulting effective radii as a function of total stellar mass (in logarithmic scale) are shown in Fig. 4. The slope and the normalization of the relation are close to the fit of Sloan Lens ACS Survey (SLACS) data by Auger et al. (2010).

The axis ratio q of each projection of a galaxy is calculated as a square root of eigenvalues of the diagonalized inertia tensor. The inertia tensor is computed within the effective radius without excluding substructures. We have found that q is not sensitive to our cleaning procedure as normally there are almost no satellites within R_eff.

4 ANALYSIS OF THE SAMPLE

4.1 At a sweet point

For each galaxy in the sample, we have performed all steps described above and selected the radius at which the circular velocity curves for isotropic, circular and radial orbits (equations 4) intersect or lie close to each other. Then we have calculated the value of the isotropic speed at this radius. To measure the accuracy of our estimates, let us introduce a deviation from the true circular speed ⁠, where and should be taken at the sweet spot R_sweet. The subscript ‘opt’ (optical) is used to distinguish this method (based on optical data) from circular speed calculations (based on X-ray data). We have plotted the number of galaxies (normalized to the total number of galaxies and expressed in per cent) versus the deviation in the form of a histogram. To have an idea whether the method under consideration gives reasonable accuracy, histograms for deviations at R_eff, 0.5R_eff and 2R_eff are also shown. The whole sample (‘subsample A’) is presented in Fig. 5. The sample-averaged value of the deviation is slightly less than zero in all cases. For example, at the sweet point per cent while the rms =8.6 per cent.4

Figure 5

The fraction of galaxies (in per cent) as a function of deviation evaluated via equations (4) at different radii: R_sweet (panel A), R_eff (panel B), 0.5R_eff (panel C) and 2R_eff (panel D).

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Large deviations (∼30–40 per cent) are seen only in galaxies with ongoing merger activity. The influence of mergers appears as ‘waves’ in the projected velocity dispersion profile. The example of such a system is shown in Fig. 6 (right-hand panel). The presence of such ‘waves’ indicates that the circular speed could be significantly overestimated (by a factor of ∼1.2–1.5), which is not surprising as the method is based on the spherical Jeans equations and the assumption about dynamical equilibrium is violated. When the profiles I(R) and σ(R) are smooth and monotonic, the circular speed can be recovered with much higher accuracy (Fig. 6, left-hand panel).

Figure 6

Left: example of the galaxy that perfectly suits for the analysis. The surface brightness and the projected velocity dispersion profiles are shown in panels (A) and (B) correspondingly. Data are represented as red points, and smoothed curves that were used to compute derivatives (α, γ, δ) are represented as black solid lines. The auxiliary coefficients α, γ, −δ and α+γ are shown in panel (C) in red solid, blue dotted, green dash–dotted and black dashed lines, respectively. Circular velocity profiles for isotropic orbits of stars (blue solid line), pure radial (green dash–dotted) and pure circular (magenta dashed) orbits as well as the true circular speed (black thick curve) are presented in panel (D) for the full version of the analysis (equations 4). And the same curves for the simplified analysis (equations 6) are shown in panel (E). Right: example of the galaxy with large deviation due to merger activity. The crest in the projected velocity dispersion profile at R≃ 20 kpc leads to the significantly overestimated value of the circular speed.

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The sample includes galaxies with different values of ellipticity. The axis ratio q (computed from the diagonalized inertia tensor within R_eff) ranges from 0.19 to 0.99. To test the possible influence of the ellipticity on the accuracy of estimates, we have selected galaxies with axis ratio q < 0.6. The resulting distribution as a function of the circular speed deviations is almost symmetric, unbiased, with per cent (Fig. 7). On the other hand, if we consider the same galaxies seen in a projection with the maximum value of the axis ratio q, we get the distribution appreciably biased towards negative values of the deviation (⁠ per cent). The reason for this bias is rotation. When observing a galaxy along its rotation axis, the projected velocity dispersion is appreciably smaller than for perpendicular directions. To further test this statement we have rotated each galaxy so that the principal axes of the galaxy (A≥B≥C) coincide with the coordinate system (x, y and z, correspondingly) and analysed velocity maps for each projection. As a criteria for rotation we have used the anisotropy parameter ⁠, where v is the average rotation velocity of stars, is the mean velocity dispersion and q is the axis ratio (Binney 1978; Bender & Nieto 1990). If (v/σ)* > 1.0, then the object is assumed to be rotating. We have found that the most massive simulated galaxies usually do not rotate or rotate slowly and show signs of triaxiality, while less massive galaxies rotate faster and show signs of axisymmetry. This statement is in agreement with observational studies (e.g. Cappellari et al. 2007 and references therein). Moreover, the majority of rotating galaxies appears to be oblate, rotating around the short axis. Hence, for the oblate galaxies observed along the rotation axis (and as a consequence seen in a projection with the axis ratio q close to unity), the method gives underestimated values of the circular speed. It should be noted that when observing the rotating galaxies along long axes the circular speed estimate is slightly biased towards overestimation (Thomas et al. 2007 reached the similar conclusion). The average deviation for the subsample of oblate galaxies seen perpendicular to the rotation axis is biased high by per cent with rms = 6.3 per cent.

Figure 7

Left: shown in cyan is the histogram for deviations for galaxies with the axis ratio q < 0.6, in black is the histogram for the same galaxies but seen in a projection with the axis ratio q close to unity (seen along the rotation axis). Right: the histogram for deviations for the sample when merging and oblate galaxies seen along the rotation axis are excluded (subsample ‘G’). The average deviation per cent and rms = 6.8 per cent.

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To investigate possible projection effects on the results of our analysis, we have picked one rotating galaxy (the virial halo mass is ≃2.2 × 10¹² M_⊙h⁻¹) and calculated the surface brightness and the velocity dispersion profiles for different lines of sight. While the light profiles are quite similar, the velocity dispersion profiles may differ significantly when the line of sight is parallel to the rotation axis and perpendicular to it. We have calculated the average value of the circular speed estimates taking into account the probability of observing the galaxy at different angles. For the selected galaxy, the average deviation from the true V_c is about −4.9 per cent and the maximum deviation (when observing along the rotation axis) is about −25 per cent.

It should be mentioned that the method under consideration was designed for recovering the circular speed in massive elliptical galaxies and it does not pretend to give accurate results for low-mass galaxies. In addition, not so many elliptical galaxies with σ < 150–200 km s⁻¹ are observed (e.g. Bernardi et al. 2010).

It is convenient to distinguish low- and high-mass simulated galaxies by the value of the projected velocity dispersion at the effective radii. Let us call ‘massive’ galaxies with σ(R_eff) > 150 km s⁻¹. If we apply our analysis to the subsample of massive galaxies and exclude merging and oblate galaxies seen along the rotation axis (the subsample ‘MG’), we get an unbiased distribution with rms = 5.4 per cent. The resulting histogram is shown in Fig. 8, left-hand image, panel (A). Estimations at other radii give slightly more biased and slightly less accurate results (Fig. 8, left-hand image, panels B–D).

Figure 8

Left: distribution of galaxies from the subsample ‘MG’ [massive galaxies with σ(R_eff) > 150 km s⁻¹ when merging and oblate galaxies observed along the rotation axis are excluded] according to their deviations. Deviations are calculated at R_sweet (panel A), R_eff (panel B), 0.5R_eff (panel C) and 2R_eff (panel D). Right: the same histograms but for the simplified version of the analysis (equations 6).

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Thereby we have marked out four subsamples – the whole sample without exceptions (‘A’– all), the sample without merging or oblate galaxies seen along the rotation axis (‘G’– good), the subsample of massive galaxies (‘M’– massive) with σ(R_eff) > 150 km s⁻¹ and, finally, the subsample of massive galaxies when merging and oblate galaxies observed along the rotation axis are excluded (‘MG’– massive and good).

In the case of missing or unreliable data on the line-of-sight velocity dispersion profile, Churazov et al. (2010) suggest to apply a simplified version of the aforementioned analysis (equations 6). By neglecting terms γ and δ, we assume that the projected velocity dispersion profile is flat. Then the radius at which I(R) ∝ R⁻² is the sweet point. The resulting histograms for the subsample ‘MG’ are shown in Fig. 8, right-hand panel. It can be seen that data on the projected velocity dispersion play noticeable role in the analysis if the required accuracy is of the order of several per cent. Neglecting its derivatives leads to a bias towards underestimated values of V_c (⁠ per cent at the sweet point) and broader wings/tails (rms = 6.4 per cent at R_sweet) compared to Fig. 8, left-hand panel. Nonetheless, if only the surface brightness profile and some data on the projected velocity dispersion are available, the simplified version of the method seems to be a good choice.

4.2 Simulated galaxies at high redshifts

We have also tested the same procedure for galaxies at higher redshifts, namely at z= 1 and 2. The fraction of merging galaxies in the sample is larger at high redshift than at z= 0 and the number of stars in each halo is considerably smaller. Nevertheless, results are quite encouraging. For the subsample ‘MG’, the average deviation of the circular speed for the isotropic distribution of orbits at the sweet point (estimated via equations 4) from the true one is close to zero and the scatter is modest. At redshift z= 1 the average deviation is per cent and rms = 6.0 per cent, and at per cent and rms = 8.0 per cent (see Discussion).

4.3 Mass from integrated properties

Assuming the logarithmic form of the gravitational potential ⁠, we can estimate the potential Φ over some range of radii up to a constant. To calculate the potential one needs to know the circular velocity profile. If we assume that this profile roughly coincides with the isotropic profile over some range of radii (let us choose R∈ [0.5R_eff, 3R_eff] as a range of radii easily available for observations), we can define

8

where R∈ (0.5R_eff, 3R_eff) and can be found from the full version of the analysis (the first equation in equations 4) or from the simplified version (the first equation in equations 6). As the true potential is known, we can write ⁠. In the ideal case, ⁠. The accuracy of such an approach is illustrated in Fig. 9. In cyan is shown the distribution of subsample ‘MG’ of galaxies as a function of Δ_Φ= (1 −κ) × 100 per cent. Just to remind, this subsample consists of the massive galaxies with σ(R_eff) > 150 km s⁻¹ and merging galaxies as well as oblate galaxies seen along their rotation axes are excluded. In the case of the full analysis, the distribution is almost unbiased (the average value of κ is 1.02 ± 0.02) with rms = 10.9 per cent. For the simplified formula of we see some offset and rms = 11.8 per cent. In approximation of small deviations, the rms defined for the potential calculations is twice as large as the rms for the circular velocity calculations because the potential Φ scales as ⁠. To compare this approach with previous results, let us introduce the deviation estimated at the sweet point R_sweet. The resulting distribution for the same subsample is shown in black in Fig. 9. As expected, the width of this distribution is nearly two times larger than the distribution of circular velocity estimates (Fig. 8).

Figure 9

Accuracy of the derived potential of massive galaxies (merging and oblate objects seen along the rotation axis are excluded). The histogram shown in cyan is for the quantity Δ_Φ= (1 −κ) × 100 per cent, where ⁠. The histogram shown in black is for the deviation of the one estimated at the sweet point from the true one ⁠. Left: histograms for the full version of the analysis (equations 4). The average value of κ is and rms = 10.3 per cent. The average value of the deviation is −0.2 ± 1.9 per cent and rms = 11.3 per cent. Right: histograms for the simplified version of the analysis (equations 6). and rms =11.8 per cent, and per cent and rms = 12.7 per cent.

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As we see the gravitational potential can be estimated via with reasonable accuracy. This fact is in agreement with the aforementioned statement that most massive galaxies in the sample have almost flat circular velocity profiles in broad range of radii.

4.4 Circular speed derived from the projected dispersion in a fixed aperture

When data on the velocity dispersion are available only in the form of aperture dispersions one can estimate the circular speed using the simple relation

9

where is the velocity dispersion measured within some aperture R. To test this way of evaluating the circular speed, we have computed the luminosity-weighted dispersion within the aperture of radius R (excluding the region R<2R_soft= 3 kpc where softening may affect results of the analysis),

10

and calculated the deviation from the true circular speed at different radii: at the sweet point R_sweet for the full version of the analysis (equations 4), at R= 0.5R_eff, R_eff and 2R_eff. The resulting histograms for the subsample ‘MG’ are presented in Fig. 10. Comparing with circular speed estimations at a single radius (in particular, at the sweet point) this method gives a biased result per cent and noticeably larger rms (at R_eff rms = 7.8 per cent).

Figure 10

Distribution of galaxies from the subsample ‘MG’ (galaxies with σ(R_eff) > 150 km s⁻¹ when merging and oblate galaxies seen along the rotation axis are excluded).

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4.5 Circular speed from X-ray data

Another way to measure the mass of galaxies comes from the analysis of the hot X-ray gas in galaxies. By measuring the gas number density n and the temperature T profiles from X-ray observations, one can find the total mass assuming that the gas is in the hydrostatic equilibrium (HE) (e.g. Mathews 1978; Forman, Jones & Tucker 1985). Assuming spherical symmetry, the equation of HE can be written as

11

where P=nkT is the gas pressure (n is the gas number density), is the gas density (m_p is the proton mass), Φ is the gravitational potential, V_c is the circular velocity and M is the total mass of the galaxy. In simulations, the mean atomic weight μ is assumed to be equal to 0.58. Strictly speaking, assuming the HE one neglects possible non-thermal contribution to the pressure, which can be due to the presence of (i) turbulence in the thermal gas and (ii) cosmic rays and magnetic fields (e.g. Churazov et al. 2008).

To estimate deviations from HE, the so-called mass bias, we took a subsample of the most massive galaxies with M > 6.5 × 10¹² M_⊙. X-ray properties of low-mass galaxies in the sample are influenced by gravitational softening in the central 3–4 kpc and are strongly dominated by cold and dense clumps in the centre. Moreover, we know from observations that only the most massive galaxies have massive X-ray atmospheres (e.g. O’Sullivan, Forbes & Ponman 2001).

The typical profiles of the gas density and temperature extracted from simulations are shown in Fig. 11. We used the median value of the electron density n_e and T determined in each spherical shell, so that we are free of cold dense clumps contamination (Zhuravleva, Churazov & Kravtsov 2011). Calculated pressure and circular velocity (equation 11) are also shown in Fig. 11. The spurious feature of simulations is that in the central 3–4 kpc cold and dense clumps are strongly dominating. Even using the median value does not remove these clumps, causing strong increase of density and drop of temperature in the centre. These clumps are moving ballistically and are not in the HE.

Figure 11

Profiles of hot gas electron number density, temperature, pressure and circular velocity of simulated galaxy. Dotted vertical curves show the upper and lower limits on R. V_c plot: solid and dashed curves show mass from HE and total mass from simulations, respectively.

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Deviations from HE, ⁠, were calculated at R_eff (cyan histogram in Fig. 12) and 2R_eff (black histogram in Fig. 12). The average over the subsample value of the deviation at R_eff is per cent and rms = 4.4 per cent. At 2 per cent and rms = 3.8 per cent. The average value of over R_eff < R < 2R_eff is 6.8 per cent.

Figure 12

Distribution of galaxies according to their deviations of estimated circular speed from the true value at R_eff (shown in cyan) and at 2R_eff (shown in black). Circular speed is derived using the HE equation for the hot gas. The sample consists of 12 galaxies.

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To calculate the average ratio of kinetic energy to thermal energy E_kin/E_therm in R_eff < R < 2R_eff, one should exclude cold dense clumps since their contribution to the kinetic energy can be significant. The procedure to exclude clumps is described in Zhuravleva et al. (2011). In brief, in each radial shell, we exclude particles with density exceeding the median value by more that 2 standard deviations. An example of initial and diffuse projected densities is shown in Fig. 13. The calculated mean ratio of E_kin/E_therm for diffuse component is 4.4 per cent, which is close to the bias in mass from HE.

Figure 13

Projected number density of hot gas in simulated galaxy. Left: initial density; right: density with excluded clumps.

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5 DISCUSSION

In Table 1, we summarize the bias and accuracy of all methods discussed above. The sample of simulated galaxies was divided into four subsamples: the whole sample without exceptions (‘A’), the subsample ‘G’ (‘good’) for which merging and oblate galaxies observed along the rotation axis are excluded, the subsample ‘M’ of massive galaxies with σ(R_eff) > 150 km s⁻¹, and the subsample ‘MG’ of massive galaxies when merging and oblate galaxies seen along the rotation axis are excluded. For estimations of the potential, the bias and the rms are nearly twice as large as those for the circular speed estimations. To avoid possible confusion, all values in the table are associated with V_c estimations.

Table 1

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Summary of the methods discussed.

	A		G		M		MG
	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)
Full analysis, Rsweet	−1.8	8.6	−1.2	6.8	0.2	7.7	0.0	5.4
Full analysis, Reff	−2.0	8.6	−2.4	5.9	−0.6	8.5	−1.0	5.1
Simplified analysis, Rsweet	−5.9	9.6	−5.8	7.4	−3.3	9.2	−4.0	6.4
Simplified analysis, Reff	−4.3	8.9	−4.6	6.6	−2.7	8.7	−3.0	5.8
⁠, equations (4)	3.7	8.7	2.9	7.1	1.2	6.7	1.2	5.1
⁠, equations (6)	7.5	10.1	6.7	8.2	4.4	7.8	4.5	5.9
Aperture dispersions, Reff	−1.4	10.3	−1.5	9.2	1.1	9.2	1.0	7.8
	Reff				2Reff
	(per cent)		rms (per cent)		(per cent)		rms (per cent)
X-ray	−3.0		4.4		−4.0		3.8

	A		G		M		MG
	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)
Full analysis, Rsweet	−1.8	8.6	−1.2	6.8	0.2	7.7	0.0	5.4
Full analysis, Reff	−2.0	8.6	−2.4	5.9	−0.6	8.5	−1.0	5.1
Simplified analysis, Rsweet	−5.9	9.6	−5.8	7.4	−3.3	9.2	−4.0	6.4
Simplified analysis, Reff	−4.3	8.9	−4.6	6.6	−2.7	8.7	−3.0	5.8
⁠, equations (4)	3.7	8.7	2.9	7.1	1.2	6.7	1.2	5.1
⁠, equations (6)	7.5	10.1	6.7	8.2	4.4	7.8	4.5	5.9
Aperture dispersions, Reff	−1.4	10.3	−1.5	9.2	1.1	9.2	1.0	7.8
	Reff				2Reff
	(per cent)		rms (per cent)		(per cent)		rms (per cent)
X-ray	−3.0		4.4		−4.0		3.8

Table 1

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Summary of the methods discussed.

	A		G		M		MG
	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)
Full analysis, Rsweet	−1.8	8.6	−1.2	6.8	0.2	7.7	0.0	5.4
Full analysis, Reff	−2.0	8.6	−2.4	5.9	−0.6	8.5	−1.0	5.1
Simplified analysis, Rsweet	−5.9	9.6	−5.8	7.4	−3.3	9.2	−4.0	6.4
Simplified analysis, Reff	−4.3	8.9	−4.6	6.6	−2.7	8.7	−3.0	5.8
⁠, equations (4)	3.7	8.7	2.9	7.1	1.2	6.7	1.2	5.1
⁠, equations (6)	7.5	10.1	6.7	8.2	4.4	7.8	4.5	5.9
Aperture dispersions, Reff	−1.4	10.3	−1.5	9.2	1.1	9.2	1.0	7.8
	Reff				2Reff
	(per cent)		rms (per cent)		(per cent)		rms (per cent)
X-ray	−3.0		4.4		−4.0		3.8

	A		G		M		MG
	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)	(per cent)	rms (per cent)
Full analysis, Rsweet	−1.8	8.6	−1.2	6.8	0.2	7.7	0.0	5.4
Full analysis, Reff	−2.0	8.6	−2.4	5.9	−0.6	8.5	−1.0	5.1
Simplified analysis, Rsweet	−5.9	9.6	−5.8	7.4	−3.3	9.2	−4.0	6.4
Simplified analysis, Reff	−4.3	8.9	−4.6	6.6	−2.7	8.7	−3.0	5.8
⁠, equations (4)	3.7	8.7	2.9	7.1	1.2	6.7	1.2	5.1
⁠, equations (6)	7.5	10.1	6.7	8.2	4.4	7.8	4.5	5.9
Aperture dispersions, Reff	−1.4	10.3	−1.5	9.2	1.1	9.2	1.0	7.8
	Reff				2Reff
	(per cent)		rms (per cent)		(per cent)		rms (per cent)
X-ray	−3.0		4.4		−4.0		3.8

In the case of the subsample ‘MG’, the estimation of the circular speed at the sweet point with the help of equations (4) gives the unbiased result (⁠ per cent) and reasonable accuracy (⁠ per cent). To test whether the unbiased average is not just a coincidence, we have performed a ‘jackknife’ test. The resulting average for randomly chosen subsamples is less than 1 per cent. The subsample ‘MG’ consists of 106 objects and the statistical uncertainty in this case is about 0.9 per cent.

For the subsample ‘M’ of massive galaxies [127 objects, 26 of them (13.3 per cent) are oblate, three of them (2.4 per cent) are with ongoing merger activity] we also got almost the unbiased average (⁠ per cent). From an observational point of view merging objects can be easily excluded, while information on the ‘oblateness’ of galaxies may not be available. If we exclude merging galaxies from the subsample ‘M’, we get the average value of the deviation per cent and rms = 5.9 per cent. Therefore, the result is almost unbiased. If we run the ‘jackknife’ tests we get on average slightly underestimated values of the circular speed with less than 1.5 per cent.

The method is not restricted to nearby galaxies, it also allows us to recover the circular speed for high-redshift ellipticals. The circular speed estimate averaged over the subsample of massive and slowly or non-rotating simulated objects (mergers are excluded) at z= 1 is per cent with rms = 6.0 per cent, and at z= 2 the average deviation is per cent with rms = 8.0 per cent (see Fig. 14). Therefore, the V_c estimates are also almost unbiased with modest scatter of 6–8 per cent as in the case of subsample ‘MG’ at z= 0.

Figure 14

Distribution of high-redshift galaxies from the subsample ‘MG’ [massive galaxies with σ(R_eff) > 150 km s⁻¹ when merging and oblate galaxies observed along the rotation axis are excluded] according to their circular speed deviations. Deviations are calculated at R_sweet (panels A and C) and at R_eff (panels B and D).

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While derivation of equations (4) and (6) is based on the assumption of the logarithmic form of the gravitational potential, we have shown that the circular speed estimate at the sweet point is still reasonable even if true circular velocity is not flat.

The case of slowly changing V_c with radius can be illustrated by the following example. If we assume that V_c varies with radius as a power law along with other quantities we end up with the following relation between V_c and (from Jeans equation):

12

where is the gamma function. This relation is insensitive to the anisotropy parameter β when α+γ= 2. Therefore, one can hope that for slopes slowly varying with radius the sweet point will be located at the radius where this condition is met. Substituting α+γ= 2 in equation (12) yields the relation between V_c and which coincides with equation (5) for isotropic orbits for α= 2. Deviations of α from 2 by 10 per cent cause modest ∼3 per cent variations in V_c.

Simulated galaxies are of course more complicated than the above example. For our sample we have investigated possible correlations between the deviation of the estimated V_c from the true one and local (at ⁠) slopes of the velocity ⁠, surface brightness and velocity dispersion profiles. There is no obvious correlation between and α or γ. We do see a weak linear trend in and ⁠, although it is much smaller than the scatter in ⁠. Most of the galaxies in the sample V_c(R) slowly declines with radius near R_sweet (see Fig. 1). However, even after subtracting this trend, the rms scatter in is reduced from 5.4 to 5.0 per cent, i.e. only by 0.4 per cent.

Comparable results are obtained using over [0.5R_eff, 3R_eff] as an estimator of the gravitational potential.

The simplified version of the analysis (equations 6) at the sweet point gives almost the same result as at the effective radius. Therefore, if one has not enough data to calculate all necessary derivatives for applying equations (4) it makes sense to derive from the first formula of equations (6) and use as an estimation of the circular speed. The quality of such an approach depends on the ‘quality’ of the sample. In the case of non-interacting and almost spherical galaxies, the rms is about 7 per cent and the bias is about −4 ± 1.1 per cent. Assuming a flat projected velocity dispersion profile leads to the underestimation of the circular speed. If data on the line-of-sight velocity dispersion allow us to estimate the overall trend Δσ/ΔR, it may reduce the bias.

In general, we can expect the sweet point to be not far from the radius R₂ where ⁠. Indeed for a smooth surface brightness profile which gradually steepens with radius an integral diverges at low or high limits for greater than or lower than 2, respectively. Therefore, one can hope that at the radius R₂ the contributions to the integral of R < R₂ and R > R₂ to be comparable and R₂∼R_eff. Thus, R_sweet∼R_eff. For example, for a Sérsic model with index n (Graham & Driver 2005)

and it can be easily seen that the sweet point for the circular speed estimation is of the order of the effective radius. Moreover, as it was shown in Churazov et al. (2010) (table 4) for Sérsic models, the stellar anisotropy is close to minimal at about 0.5R_eff and this radius can be used as the sweet point for the circular speed determination.

We have tested the statement that R_sweet∼R₂∼R_eff on the sample of the simulated objects. If the slope of the surface brightness profile is close to −2 over some range of radii or is not monotonic, then there is an ambiguity in selecting R_sweet and R₂. To avoid this ambiguity, we have smoothed I(R) and σ(R) using the width of the window function Δ_I=Δ_σ= 1.0. As a result, α(R) has become monotonic for the majority of objects and newly determined ⁠, follow the relationship ⁠. However, a significant smoothing of data leads to a bias in estimating the circular speed per cent at both and ⁠.

6 CONCLUSIONS

Being an important issue, the total mass estimation for elliptical galaxies is often quite difficult, especially for galaxies at high redshift. We used a large sample of cosmological zoom simulations of individual galaxies to test a simple and robust procedure (see equations 4 and 6) based on the surface brightness and velocity dispersion profiles to estimate the circular speed and therefore the total mass of a massive galaxy. The method is very simple and does not require any assumptions on the stellar anisotropy profile. For massive ellipticals without significant rotation at redshifts z= 0-2, it gives an unbiased estimate of the circular speed [the bias is less than 1 per cent] with 5–6 per cent scatter. Therefore, this method is suitable for the analysis of large samples of galaxies with limited observational data at low and high redshifts. The method works best for the most massive ellipticals [σ(R_eff) > 200 km s⁻¹], which in the present simulations have almost isothermal circular velocity profiles over broad range of radii.

The method should be applied with caution to merging galaxies where the circular speed can be significantly overestimated. For rotating galaxies seen along the rotation axis the procedure gives substantially underestimated V_c.

The best estimate of the circular speed is obtained at a sweet point R_sweet where the sensitivity of the recovered circular speed to the stellar anisotropy is expected to be minimal (see Section 2). The R_sweet is expected to be not far from the projected radius where the surface brightness declines approximately as I ∝ R⁻². This radius is in turn close (within a factor of 2) to the effective radius R_eff of the galaxy. Our tests have shown that the accuracy (rms scatter) of the circular speed estimates at is 5–7 per cent for the most massive ellipticals.

An even simpler method – based on the aperture velocity dispersion (equations 9 and 10) – is found to be less accurate, although the results are still reasonable. For example, for massive galaxies without significant rotation the sample-averaged deviation of the circular speed at the effective radius is per cent with per cent. Other flavours of the circular speed estimates are described in Section 4.4.

Using the same simulated set, we have also tested the accuracy of the circular speed estimate from the HE equation for the hot gas in massive ellipticals. We found a negative bias at the level of 3–4 per cent and the scatter of ≃5 per cent. The presence of bias is caused by the residual gas motions.

Given the simplicity of the described method (Section 2), the low bias and modest scatter in the recovered value of the circular speed, it is suitable for the analysis of large samples of massive elliptical galaxies at low as well as high redshifts.

We are grateful to the referee for very useful comments and suggestions. NL is grateful to the International Max Planck Research School on Astrophysics (IMPRS) for financial support. TN acknowledges support from the DFG Excellence Cluster ‘Origin and Structure of the Universe’. The work was supported in part by the Division of Physical Sciences of the RAS (the programme ‘Extended Objects in the Universe’, OFN-16).

Footnotes

REFERENCES