## 2. The projected mass estimatorThis estimator uses the average of the quantity over the whole tracer population (e.g. galaxies in a cluster) relating it to the total mass by (HTB, Perea et al. 1990) where is the observed line-of-sight velocity of a galaxy relative to the cluster mean, and is its projected radius from the centre of the distribution. The factor is related to the orbital distribution of galaxies and therefore a function of the anisotropy parameter (see below), being equal respectively to and for the cases of radial () and isotropic () 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) where is the mass density,
the gravitational potential,
the radial velocity dispersion, and
the anisotropy function;
is the tangential velocity
dispersion. Multiplying Jeans equation by
, and integrating by parts up to a
radius On the other hand, the average in Eq. (1) up to a particular radius is estimated as (e.g. HTB): where is the velocity dispersion along the line-of-sight, , and is the phase space density, defined such that is the mass contained in the phase space volume element . Evaluating the integral in Eq. (4), by using Eq. (3) and
Since and we have, after rearranging terms: This function 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 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: . Therefore, in using Eq. (1) for an isotropic system one must
remember that this form of holds
only when the total extent of the stellar system has been sampled;
otherwise, the correction term has
to be applied. As will be shown below, in astronomical situations the
use of can yield a significant
overestimate of the system's mass especially if we sample only around
its ## 2.1. Boundary term for spherical systemsIn this subsection we estimate theoretically the boundary term for four different spherical models: (1) de Vaucouleurs' or (de Vaucouleurs 1948), (2) Hernquist's (1990), (3) Dehnen's () (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
, with Here the units adopted for the numerical calculations are such that
the total mass is unity, , the total
system's ## 2.1.1. De Vaucouleurs modelThe surface brightness of the de Vaucouleurs' model is: where is the The mass density 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: where , and . The product can be obtained from the Jeans equation by imposing the condition that as . Since and are non-negative, and always positive, we have: The mass follows from integrating the mass density, and the boundary term can then be computed numerically. ## 2.1.2. Hernquist modelThis model is similar to an model, but has the virtue of being expressed in terms of simple functions. Some of the quantities of interest here are: where Using Hernquist's expressions and integrating several times by parts Eq. (9), we find that: ## 2.1.3. Dehnen modelDehnen (1993) has provided a family of potential-density models for spherical systems. We use his model since it provides a core resembling King models and allows simple analytic calculations (see also Tremaine et al. 1994, model ). The density and mass are given respectively by: The velocity dispersion is
. ## 2.1.4. King modified modelThe 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: where , and
, the being the Gamma function. Hereafter we will consider only the value of , which is close to fittings to the Coma cluster profile (e.g. Perea et al. 1990) and to the spectroscopic value of a -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 termIn Fig. 1 we show the function 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 The maximum of occurs at , or for our chosen units, and has a value of . 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 . The radius at which these three maxima occur are at 1.07, 0.66, and 0.43, respectively; i.e. all occur at .
A trend for the maximum of 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 ,
, and
is constant, from where we obtain
that . Therefore
grows as the `isothermal' region of
the model grows. Since the boundary term is positive and linear in
## 2.2. Numerical test of the PMEHere we evaluate as a mass
estimator by using A Monte Carlo realization of particles, each of mass , 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 units, with softening parameter and tolerance parameter . 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 reaching
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. . After this test
was conducted, we considered our initial To compute we proceeded as one
would do observationally to compute the mass at different radii of
clusters. We divided the In Fig. 2 we plot as a function
of projected radius for our Dehnen numerical cluster. Also shown are:
the virial mass (), as discussed
below, the true projected mass , and
the real 3-D mass within radius
## 2.3. Intrinsic `error' in the PMEFrom 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 % of its true value if the sampling is made only to 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 Before we leave this section we address the results of Thomas &
Couchman (1992) on the PME. Examining their formulae
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. , would
help solving the discrepancy they mention, although it is not clear
that one can characterize by a constant anisotropy their © European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |