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Astron. Astrophys. 345, 439-447 (1999)

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2. The projected mass estimator

This estimator uses the average of the quantity [FORMULA] over the whole tracer population (e.g. galaxies in a cluster) relating it to the total mass by (HTB, Perea et al. 1990)

[EQUATION]

where [FORMULA] is the observed line-of-sight velocity of a galaxy relative to the cluster mean, and [FORMULA] is its projected radius from the centre of the distribution. The factor [FORMULA] is related to the orbital distribution of galaxies and therefore a function of the anisotropy parameter [FORMULA] (see below), being equal respectively to [FORMULA] and [FORMULA] for the cases of radial ([FORMULA]) and isotropic ([FORMULA]) orbits.

Eq. (1) is obtained as follows. On the one hand, for a spherically symmetric system under steady-state conditions and no rotation the Jeans equation holds (Binney & Tremaine 1987)

[EQUATION]

where [FORMULA] is the mass density, [FORMULA] the gravitational potential, [FORMULA] the radial velocity dispersion, and [FORMULA] the anisotropy function; [FORMULA] is the tangential velocity dispersion. Multiplying Jeans equation by [FORMULA], and integrating by parts up to a radius r we have:

[EQUATION]

On the other hand, the average [FORMULA] in Eq. (1) up to a particular radius [FORMULA] is estimated as (e.g. HTB):

[EQUATION]

where [FORMULA] is the velocity dispersion along the line-of-sight, [FORMULA], and [FORMULA] is the phase space density, defined such that [FORMULA] is the mass contained in the phase space volume element [FORMULA].

Evaluating the integral in Eq. (4), by using Eq. (3) and assuming isotropy ([FORMULA]), we have:

[EQUATION]

Since [FORMULA] and [FORMULA] we have, after rearranging terms:

[EQUATION]

This function [FORMULA] gives us directly the effect of neglecting the boundary term on the estimation of the mass via the PME when partial sampling is done. Now, since one usually fits a particular projected model to a set of data, the quantities to evaluate [FORMULA] follow from the corresponding deprojected model. One may use e.g. Eq. (9) to evaluate the radial velocity dispersion in case this does not exists explicitly beforehand for the adopted model. Integrating over the whole system we recover the standard formula of the projected mass for an isotropic stellar system: [FORMULA].

Therefore, in using Eq. (1) for an isotropic system one must remember that this form of [FORMULA] holds only when the total extent of the stellar system has been sampled; otherwise, the correction term [FORMULA] has to be applied. As will be shown below, in astronomical situations the use of [FORMULA] can yield a significant overestimate of the system's mass especially if we sample only around its effective radius. Furthermore, since the boundary of an astronomical system is somewhat ambiguous, particularly if large amounts of dark matter exist at the outer optical edges, it is important to quantify the error one might commit when using the PME. We proceed to estimate such error in the next subsection.

2.1. Boundary term for spherical systems

In this subsection we estimate theoretically the boundary term [FORMULA] for four different spherical models: (1) de Vaucouleurs' or [FORMULA] (de Vaucouleurs 1948), (2) Hernquist's (1990), (3) Dehnen's ([FORMULA]) (1993), and (4) a King's modified profile. We consider these models to bracket most of the possible profiles that can be fitted to a cluster of galaxies, which will be here the astronomical object of interest. Evidently, these profiles can be applied to other spherical systems, as well as the mass estimators.

Navarro et al. (1996), hereafter NFW, have proposed a profile model appropriate for clusters of galaxies of the form [FORMULA], with a being a scale radius. Carlberg et al. (1997b) have shown that the average mass density profile of clusters is well described by the NFW profile though they mention that the Hernquist's model is statistically acceptable. However discordant results exist in the literature on the subject of cluster profiles (e.g. Adami et al. 1998). We have not used the NFW profile here due to its intrinsic difficulty in defining formally a total mass, hence a half-mass or effective radius, unlike the other profiles. Indeed, in the NFW profile [FORMULA] as [FORMULA] and the total mass diverges. We do not expect much difference in the boundary term [FORMULA] when the NFW and Hernquist profiles are fitted to real clusters. Thus, for convenience, we deal here only with the Hernquist's model.

Here the units adopted for the numerical calculations are such that the total mass is unity, [FORMULA], the total system's effective radius [FORMULA], and the gravitational constant [FORMULA]. Following, for completeness, we write some pertinent quantities to compute [FORMULA] for the different models to be considered.

2.1.1. De Vaucouleurs model

The surface brightness of the de Vaucouleurs' model is:

[EQUATION]

where [FORMULA] is the effective radius, a scale-radius where half of the total light is emitted, and [FORMULA]. We will consider here the effective radius to be the locus where half of the projected mass resides. The total mass is, assuming a mass-to-light ratio of unity, [FORMULA].

The mass density [FORMULA] can be obtained by deprojecting the surface brightness using the Abell integral equation (Binney & Tremaine 1987). The latter has been obtained in implicit form by Poveda et al. (1960) as:

[EQUATION]

where [FORMULA], and [FORMULA].

The product [FORMULA] can be obtained from the Jeans equation by imposing the condition that [FORMULA] as [FORMULA]. Since [FORMULA] and [FORMULA] are non-negative, and [FORMULA] always positive, we have:

[EQUATION]

The mass [FORMULA] follows from integrating the mass density, and the boundary term can then be computed numerically.

2.1.2. Hernquist model

This model is similar to an [FORMULA] model, but has the virtue of being expressed in terms of simple functions. Some of the quantities of interest here are:

[EQUATION]

where a is a scale-radius, and M the total mass. The effective radius here is [FORMULA].

Using Hernquist's expressions and integrating several times by parts Eq. (9), we find that:

[EQUATION]

2.1.3. Dehnen [FORMULA] model

Dehnen (1993) has provided a family of potential-density models for spherical systems. We use his [FORMULA] model since it provides a core resembling King models and allows simple analytic calculations (see also Tremaine et al. 1994, model [FORMULA]). The density and mass are given respectively by:

[EQUATION]

The velocity dispersion is [FORMULA]1 The effective radius in this model is [FORMULA].

2.1.4. King modified model

The simplicity of this empirical model, besides allowing larger cores, makes it well suited for fitting observational data. The projected and physical density, and projected mass are (e.g. Perea et al. 1990, Adami et al. 1998), respectively:

[EQUATION]

[EQUATION]

where [FORMULA], and [FORMULA], the structural length of the model, are fitting parameters. The effective radius, in terms of [FORMULA], is [FORMULA]. The total mass M may be estimated e.g. by the virial theorem or considered another fitting parameter. The following relations among the different parameters hold:

[EQUATION]

[FORMULA] being the Gamma function. Hereafter we will consider only the value of [FORMULA], which is close to fittings to the Coma cluster profile (e.g. Perea et al. 1990) and to the spectroscopic value of a [FORMULA]-model of X-ray emission in clusters (Sarazin 1988). We refer hereafter to this model as King.

2.1.5. Theoretical value of the boundary term

In Fig. 1 we show the function [FORMULA] for the above considered models. Of the models considered here the Dehnen's model lends itself to an easier computation of the maximum value of the boundary term. Indeed, the following expression for the boundary term is readily obtained

[EQUATION]

The maximum of [FORMULA] occurs at [FORMULA], or [FORMULA] for our chosen units, and has a value of [FORMULA]. For the de Vaucouleurs' model the maximum value of this boundary term is 0.166, for the Hernquist's one 0.176, and for the King's [FORMULA]. The radius at which these three maxima occur are at 1.07, 0.66, and 0.43, respectively; i.e. all occur at [FORMULA].

[FIGURE] Fig. 1. Boundary term in the PME as a function of radius for de Vaucouleurs', Hernquist's, Dehnen's, and King's models. The maximum values occur respectively at [FORMULA] and inside [FORMULA]. These values give an indication of the error that one may expect to make when estimating masses via the PME at different radii of e.g. a cluster of galaxies.

A trend for the maximum of [FORMULA] to increase as the system gets less concentrated is observed. The latter is explained if we consider the extreme case of an isothermal sphere, where [FORMULA], [FORMULA], and [FORMULA] is constant, from where we obtain that [FORMULA]. Therefore [FORMULA] grows as the `isothermal' region of the model grows. Since the boundary term is positive and linear in M we can be overestimating the total mass by about 20% by using the quantity [FORMULA] as in Eq. (1, [FORMULA]), if the sampling is done only up to regions near the effective radius of the gravitational system. The precise value of the error depends on the density profile of the system.

2.2. Numerical test of the PME

Here we evaluate [FORMULA] as a mass estimator by using N-body models. We only present here for convenience the results for the Dehnen's model, but the same quantities and analogous results consistent with the theoretical expectations were found for the other three models of Sect. 2.1

A Monte Carlo realization of [FORMULA] particles, each of mass [FORMULA], was constructed. The numerical `cluster' was first evolved using a TreeCode (Barnes & Hut 1986) in order to test its stability before the PME is applied to the initial configuration. The total time of evolution lasted 160 time-steps, each of [FORMULA] units, with softening parameter [FORMULA] and tolerance parameter [FORMULA]. Quadrupole terms were included. Energy changed by less than 0.1% throughout the evolution.

The ratio of kinetic to potential energy, as provided by the code, was initially [FORMULA] reaching [FORMULA] at the end of the computation. The initial discrepancy of the numerical model from the ideal virial ratio is attributed to the initial positions and velocities not being in equilibrium with the code's potential which depends on the softening introduced: i.e. [FORMULA]. After this test was conducted, we considered our initial N-body system to be in stable equilibrium so we could apply to it the mass estimators considered in this work.

To compute [FORMULA] we proceeded as one would do observationally to compute the mass at different radii of clusters. We divided the N-body cluster into 40 projected concentric radii, spaced logarithmically, from [FORMULA]. Inside each radius the summation of [FORMULA], with [FORMULA], was calculated for all pertinent particles and then divided by their number [FORMULA]. In this way we estimated [FORMULA] or, equivalently, the observational projected mass [FORMULA].

In Fig. 2 we plot [FORMULA] as a function of projected radius for our Dehnen numerical cluster. Also shown are: the virial mass ([FORMULA]), as discussed below, the true projected mass [FORMULA], and the real 3-D mass within radius r, [FORMULA]. In the upper panel of Fig. 2, the difference [FORMULA] is shown as a long-dashed line, the difference [FORMULA] as a dotted line, and the theoretical value of [FORMULA] as in Eq. (15) is shown as a solid line. The difference between [FORMULA] and [FORMULA] can be said to be in perfect agreement, both in magnitude and in position of the maximum, with the theoretical expectations of Sect. 2.1. Poisson noise can explain the obtained differences as we have verified, although it is not shown in the figure to avoid overcrowding. The difference [FORMULA], although similar in shape to [FORMULA], is about 50% smaller in magnitude. A similar behaviour of these discrepancies was observed for the other models considered here.

[FIGURE] Fig. 2. Results of the application of the PME ([FORMULA]) and VME ([FORMULA]) as a function of radius to Dehnen's ([FORMULA]) numerical model of a cluster. Also, the theoretical projected mass profile [FORMULA] and physical mass [FORMULA] are shown for comparison. The upper panel shows the differences [FORMULA] (long-dashed ), [FORMULA] (dotted ), and the theoretical boundary term [FORMULA] is plotted as a solid line.

2.3. Intrinsic `error' in the PME

From the analysis of the boundary term in Eqs. (6,15) and the numerical calculations performed, we conclude that the PME can overestimates the mass by [FORMULA]% of its true value if the sampling is made only to [FORMULA] of the gravitational system. This maximum error is for the King's model, while lower values were obtained for more concentrated models like Hernquist's or de Vaucouleurs' in agreement with the theoretical expectations; see Fig. 1.

It is important to recall at this point the assumptions made in the derivation of the PME: spherical symmetry, isotropic velocity distribution and no mass spectrum; implicitly it also assumes that the particles we observe trace matter. If dark matter dominates the outer parts of galaxy clusters we expect that sampling the luminous part would yield an overestimate in the mass, not an underestimate, which can be of [FORMULA]%. This particular value depends, obviously, on the total mass profile, but can be considered typical for profiles suited for galaxy clusters.

Before we leave this section we address the results of Thomas & Couchman (1992) on the PME. Examining their formulae [FORMULA] we note that these equations correspond to the case of test particles orbiting a massive particle, as they acknowledge, but for non-test particles they would require an extra factor of two. It seems, from the numbers they give in their Table 5 that this extra factor of two and perhaps a higher constant anisotropy, e.g. [FORMULA], would help solving the discrepancy they mention, although it is not clear that one can characterize by a constant anisotropy their N-body clusters.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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