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Astron. Astrophys. 343, 760-774 (1999)

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4. Dynamical analysis

With the assumption that the X-ray emitting gas is isothermal and in hydrostatic equilibrium with dark matter, the dynamical mass was estimated as a function of distance to the cluster center by Pislar et al. (1997) from X-ray (ROSAT PSPC) data (see their Fig. 13). Within the X-ray image, which is limited to a distance of 1.3 Mpc around the cluster center, the total mass estimated is about [FORMULA] [FORMULA].

The basic physical picture behind this determination is that if the cluster ABCG 85 is in equilibrium, then the velocities of any subpopulation should reflect the mass distribution. We have calculated the ellipticity of the galaxy distribution using the momentum method. This provided us with a direction of the major axis and the ratio [FORMULA] of the small to large axes. We have then changed the coordinates of the position of each galaxy by an anamorphosis along the major axis: [FORMULA]. The galaxy distribution then appears spherical. This defines a new radius R which will be used in the following equations. This transformation assumes that the cluster major axis is parallel to the plane of the sky, in the prolate as well as the oblate cases. We may then infer the enclosed mass from the measured velocities of a tracer population. Another possible method which is commonly used is to perform counts in concentric circular rings. Our method is more accurate since the galaxy count estimate is adapted to the geometry, but Eq. (2) assumes spherical symmetry and is not fully correct. The other method cumulates both defects.

Let us investigate the properties of the cumulative mass profile derived from the different tracers when isotropy is assumed. This is achieved via parametric and non parametric modeling. The Appendix gives a more detailed description of the non-parametric methods involved.

4.1. Data structure

4.1.1. Binning procedure

Optimal inversion techniques should avoid binning while relying on techniques such as Kernel interpolation. We found here that for such a sample and when assessing quantities which are two derivatives away from the data, the Kernel introduces spurious high frequency features in the recovered mass profile. Binning the projected quantities on the other hand allows us to control visually the quality of the fit. We use floating binning which is defined as follows: for each galaxy we find its p-nearest neighbors, and define a ring which encompasses them exactly; the estimator for the density, [FORMULA], would be defined as p divided by the area of that ring. For the projected velocity dispersion squared, [FORMULA], we could sum over the velocity squared (measured with respect to the mean velocity of the cluster) of the p neighbors and divide by p; in practice a better estimator, [FORMULA], accounting for velocity measurement errors is given by


where [FORMULA] is the sum of the error on the measured variance, [FORMULA], and of the square of the measured error on the velocity [FORMULA]. Bootstrapping is applied to estimate [FORMULA] while first neglecting these measurement errors. An estimate of the projected energy density is given by [FORMULA]. In practice, binning over 10 to 15 neighboring galaxies is applied, yielding estimates of the Poisson noise induced by sampling.

4.1.2. Bias and incompleteness

The sample is truncated in projected radius R. Since generically, truncation and deprojection will not commute, the estimation of the cumulative mass profile arising from a truncated sample in projected radius will be biased. In physical terms, this follows because we cannot distinguish between projected galaxies which are truly within a sphere of radius R, and those which are beyond but happen to fall along the line of sight. Considerations about the physical properties of the tracer may help reduce the confusion, but a bias remains in the estimated mass when the sample is truncated in projected radii. Extrapolation provides some means of correction. Note that extrapolation has a different meaning depending on what the true profile is. Specifically the boundary conditions (exponential splines, edge spline, truncation at two or five times the last measured radius) will make a difference in the recovered profile. Since the completeness of our redshift catalogue decreases with increasing radius, we restrict our analysis to the inner region of the cluster within 1000 arcsec (1.62 Mpc at the cluster redshift), where this catalogue is fairly complete (92% for R[FORMULA]). In practice we check that all mass estimates converge to the same total mass within the error bars.

4.2. Method

4.2.1. Jeans equation

The equilibrium of an isotropic stationary spherical galactic cluster obeys Jeans' equation:


where [FORMULA] is the gravitational potential generated by all the types of matter, i.e. stellar matter, X-ray emitting plasma and unseen-matter, [FORMULA] the density of galaxies in the cluster and [FORMULA] the radial velocity dispersion. Eq. (2) can be applied locally to assess the cumulative dynamical mass profile.

The surface density of galaxies is related to the density via an Abel transform:


where [FORMULA] is the projected galaxy density and R the projected radius as measured on the sky. Similarly the line of sight velocity dispersion [FORMULA] is related to the intrinsic radial velocity dispersion, [FORMULA], via the same Abel transform (or projection)


Note that [FORMULA] is the projected kinetic energy density divided by three (corresponding to one degree of freedom) and [FORMULA] the kinetic energy density divided by three. Inverting Eqs. (3)-(4) into Eq. (2) yields:


Therefore, assuming we have access to estimators for [FORMULA] and [FORMULA], the cumulative mass distribution follows.

4.2.2. Algebraic dynamical mass estimators

The Bahcall & Tremaine (1981) mass estimator for test particles around a point mass assumes completeness and isotropy and is given by:


4.3. Parametric modelling

In a nutshell, given that formally the inverse of Eq. (4) is


it is straightforward to construct parametric pairs of projected and deprojected fields.

In order to describe the density profile of galaxies, and/or the energy density, we have used various kinds of parametric forms: 

  1. A [FORMULA]-model:


    for the spatial profile, and


    for the projected profile.
  2. A Sersic profile


    for the projected profile, to which corresponds the spatial profile: 


    where [FORMULA] (Gerbal et al. 1997).
  3. A power-law for both the spatial and surface energy density since the inverse Abel transform of a power law is a power law with a power index decreased by one.

4.4. Non parametric analysis

The non parametric inversion problem is concerned with finding the best solution to Eq. (5) for the cumulative mass profile when only discretized and noisy measurements of [FORMULA] and [FORMULA] are available (Gebhardt et al. 1996, Merritt 1996, Pichon & Thiébaut 1998 and references therein). In order to achieve this goal these functions are written in some fairly general form involving generically many more parameters than constraints and such that each parameter controls only locally the shape of the function. The corresponding inversion problem is known to be ill-conditioned: a small departure in the measured data (due to noise) may produce drastically different solutions since these solutions are dominated by artifacts due to the amplification of noise. Some kind of trade off must therefore be found between the level of smoothness imposed on the solution in order to deal with these artefacts on the one hand, and the level of fluctuations consistent with the amount of information in the signal on the other hand. Finding such a balance is called the "regularization" of the inversion problem.

In practice the regularization can be imposed either directly upon the projected model in data space or upon the unprojected model. The latter (the non parametric inversion) is preferable for Abel transforms such as Eq. (7) since the projection on the sky of a given galaxy distribution is bound to be smoother than the galaxy distribution itself. Moreover, physical constraints such as positivity of the galaxy distribution are also more stringent (and better physically motivated) in model space. Nevertheless it is sometimes more straightforward to carry the regularization in model space (a non parametric fit) and then carry the inversion numerically when an explicit inversion formula such as Eq. (7) is available.

Here, we apply both techniques to the recovery of the mass profile of ABCG 85.

4.4.1. Non parametric fit

We fit a regularized spline to [FORMULA] and [FORMULA] as a function of [FORMULA] with a linear penalty function on the second derivative (i.e. which leaves invariant linear functions of [FORMULA] which are power laws of R). We then make explicit use of Eq. (7) to compute numerically [FORMULA] and [FORMULA] together with their derivative. Note that this procedure is a non parametric fit rather than a non parametric inversion, and the regularization parameter needs to be boosted to account for the fact that the fit is then inverted to yield supposedly smooth deprojected quantities. In practice we use the regularization parameter [FORMULA] where [FORMULA] is given by General Cross Validation as defined in the Appendix.

4.4.2. Non parametric inversion

We fit the projection of a B-spline family which is sampled logarithmically in radius with a linear penalty function on the second derivative as discussed in the Appendix. The coefficients of the fit yield directly [FORMULA] and [FORMULA] which together with Eq. (2) lead to the cumulative mass profile of ABCG 85. As expected, the error bars on the corresponding mean profile are larger for the non parametric inversion since this method imposes the weakest prejudice on the expected mass profile.

4.5. Results and discussion

We have obtained various dynamical integrated mass profiles, some of which are displayed in Fig. 14, superimposed on those obtained from X-ray data (Pislar et al., 1997). In order to avoid having too many curves on this figure, we omitted the three curves corresponding to the following cases: 2D non parametric fit, galaxy density and pressure following [FORMULA]-models, and galaxy density following a Sersic distribution and pressure following a [FORMULA]-model; these three cases are almost indistinguishable from the power law case. Notice that the limiting radius for the mass obtained from optical data (`optical' dynamical mass) - 1000 arcsec - is smaller than the limiting radius - 1300 arcsec - for the mass obtained from X-ray data (`X-ray' dynamical mass); this is due to the lack of completeness of the galaxy velocity catalogue in a larger region.

[FIGURE] Fig. 14. Dynamical mass as a function of radius derived with various methods. Full lines: mass derived from X-ray data assuming a [FORMULA]-model and a Sersic model for the gas distribution; small dotted line: complete non parametric inversion (i.e. 3D model projected in data space); the mean profile and the error bars are estimated while varying the binning from 11 to 18 neighbours; dot-dashed line: galaxy density and pressure following a power law. The large square is the dynamical mass estimated with the Bahcall & Tremaine method.

We give in Fig. 15 an example of a non-parametric fit of the observed pressure (bottom panel) and observed profile (middle panel). The observed points for the pressure are the result of the product of the numerical profile with the velocity dispersion shown in the top panel. This profile corresponds to the velocity distribution as a function of projected distance to the cluster center displayed in Fig. 16.

[FIGURE] Fig. 15. Top panel: velocity dispersion profile; middle panel: density profile; bottom panel: pressure profile. In the two bottom panels , the observed points are indicated with crosses and typical fits with lines. All these data were obtained with a floating binning mean method. The horizontal lines show the corresponding bins.

[FIGURE] Fig. 16. Velocity as a function of projected distance to the cluster center for galaxies in the cluster velocity range.

One may notice that the various `optical' dynamical mass es are very close to one another, the distances between the curves being smaller than the error bars (see figure); the dynamical mass estimated with the Bahcall & Tremaine method is totally consistent with our estimate.

4.5.1. `Optical' dynamical masses versus `X-ray' dynamical mass es

`Optical' dynamical masses are larger than `X-ray' dynamical mass es. At a distance of R=1000 arcsec, `optical' dynamical mass es are [FORMULA] [FORMULA], while `X-ray' dynamical mass es are [FORMULA] [FORMULA].

A simple explanation would be the following: from the spectral capability of the ROSAT PSPC, Pislar et al. (1997) have derived an isothermal plasma temperature of about 4 keV. However Markevitch et al. (1998) using ASCA have shown that in the centre of ABCG 85 (where our analysis is performed) the temperature is about 8 keV. Such a discrepancy between ROSAT and ASCA determined temperatures is not uncommon, since the energy range of ROSAT is lower than that of ASCA, and therefore not well suited to measure cluster temperatures. Since `X-ray' dynamical mass es are proportional to the temperature, the use of the Markevitch et al. temperature would lead to a dynamical mass [FORMULA] [FORMULA] at R=1000 arcsec, equal to the `optical' dynamical mass es. It is then tempting to conclude that the Markevitch et al. high plasma temperature is confirmed by the comparison of `optical' and `X-ray' dynamical mass es.

A careful analysis of the ABCG 85 temperature map provided by ASCA (Markevitch et al. 1998) raises the question of the actual gas temperature; the observed value of 8 keV is only valid for a region of about 500 arcsec radius.

A 3-D temperature profile must be defined by at least three quantities, i.e. a slope (or equivalently an adiabatic index), a length scale and a value of the temperature at a given radius. In the case of ABCG 85, the data are too poor to recover the three above values (Markevitch, private communication). However we have estimated the dynamical mass assuming a "mean" value for the temperature slope compatible with the whole sample of the Markevitch et al. data and using the hydrostatic equation. Notice first that the asymptotic behaviour for the integrated dynamical mass is (obviously) no longer the isothermal one (i.e. [FORMULA], with [FORMULA]). As a consequence the amount of dark matter at large scale would be largely reduced compared to the isothermal behaviour. However, we adress the question of the dynamical mass in a region of only 1000 arcsec. Using a value for the scale length of order 500 kpc we are able to derive a [FORMULA] profile; the obtained mass compared to the isothermal one is displayed in Fig. 17. As it is the case for some clusters analyzed by Markevitch et al. (private communication and poster at the Paris Texas meeting) the non-isothermal mass is higher for small radii but intersects the isothermal profile at a radius of [FORMULA] arcsec. The resulting profile is clearly located in the region covered with error bars as indicated in Fig. 17.

[FIGURE] Fig. 17. `X-ray' dynamical mass es estimated in two cases: dotted line: isothermal gas, full line: non-isothermal gas; the error bars are the same as in Fig. 14 for `optical' dynamical mass es.

Our conclusion is that a non-isothermal analysis is not currently possible due to the weakness of the temperature analysis at least for ABCG 85, but is certainly a promising possibility in the future.

One may notice that `optical' dynamical mass es show the same rate of growth (i.e. [FORMULA]) as the `X-ray' dynamical mass es in the isothermal regime; the corresponding densities vary as [FORMULA]. We have calculated the 3-D galaxy velocity dispersion profile (defined as the pressure to density ratio). The mean velocity dispersion is 1072 km s-1 and the residual of this profile compared to this mean value is displayed in Fig. 18; the variation is [FORMULA] indicating that the dispersion profile is constant with radius.

[FIGURE] Fig. 18. Residual of the deprojected velocity dispersion profile compared to a mean velocity dispersion of 1072 km s-1.

Therefore with similar hypotheses - isothermality for the X-ray emitting plasma, isovelocity for the galaxies - `X-ray' dynamical mass es and `optical' dynamical mass es are equal. This coherence validates the two independent techniques; notice however that the static hypothesis is common for the two methods.

It is interesting to note that the dependence of the mass of the different components with radius (between [FORMULA]100 and 1000 arcsec) is: [FORMULA] for galaxies, [FORMULA] for the X-ray gas and [FORMULA] for dynamical matter, as noted above in the isothermal case.

4.5.2. Additional comments

The various modelling techniques implemented onto the masss profile of ABCG 85 have led to very similar featureless powerlaw profiles for the enclosed mass (Fig. 14). This property follows because even with a sample of about 300 galaxies (all the galaxies in the cluster velocity interval were included, i.e. 272 NoELGs and 33 ELGs) the inversion of Eq. (4) or (3) is dominated by shot noise and requires stringent regularisation. As mentioned previously by Merritt & Tremblay (1994), we find here that a truly non parametric inversion would require that nature provides many more galaxies per cluster.

We remind the reader that we have assumed an isotropic velocity dispersion for the galaxies; the agreement found above shows that this hypothesis is reasonable. It is easily accounted for by the fact that the dominance of radial orbits is probably due to the continuous infall of galaxies and small groups, as shown in particular for ABCG 85 (Durret et al. 1998b), which occurs mostly in the outskirts of the cluster. It is only in the outer regions that the hydrostatic hypothesis is also questionable.

A possible caveat is the following: only the velocity dispersion is observed, while it is the pressure which is the important physical quantity. The estimate of the pressure as the numerical density multiplied by the velocity dispersion is correct if there is no equipartition between small and large galaxy masses, or expressed differently if the velocity dispersion does not depend on the galaxy mass. We have checked in a central region of 750 arcsec radius that the velocity dispersion does not depend on the magnitude; assuming a constant mass to light ratio for all galaxies, this implies that the velocity dispersion does not depend on galaxy mass.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999