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

3. The virial mass estimator

In this section we test the virial theorem as a mass estimator. We consider its discrete and continuous form in order to understand and clarify some matters regarding its use that have recently appeared in the literature (e.g. Carlberg et al. 1996). We first outline the deduction of the scalar virial theorem for the sake of completeness and future reference (see e.g. HTB).

3.1. Discrete virial theorem

Consider a system of N equal-mass particles interacting gravitationally. First, we differentiate twice the moment of inertia of the system with respect to time:

The second term on the right-hand side is twice the total kinetic energy of the system, and the first term is the potential energy

where .

To derive the common form of the virial theorem we require that the time average of the second derivative of the moment of inertia vanishes (e.g. Goldstein 1980). In this case we have, after averaging over all possible orientations and assuming an isotropic velocity distribution (Limber & Mathews 1960), that

where is the projected separation among pairs of particles. This is what we explicitly mean by the virial mass estimator, VME. An important observational advantage of the VME in determining the total mass of a system is that it has virtually no dependence on velocity anisotropy for near-spherical systems (e.g. The & White 1986).

One may also derive a virial estimator by using the distances of test particles to the centre of the system (Bahcall & Tremaine 1981). Nonetheless, such form of the virial theorem is of no interest here since we do not consider galaxies in clusters to be properly represented by test particles.

3.1.1. Numerical test of the VME

We applied the VME to the numerical models described in Sect. 2. We proceeded as in the testing of the PME using different aperture radii, where we applied the VME to the particles inside the corresponding aperture. The mass derived using the VME, , is shown also in Fig. 2 for the Dehnen's model. Note that the VME follows more closely the projected mass than the PME over most radii, especially for .

Nonetheless, significant deviations occur at . In Fig. 3 we show for the Dehnen's model the difference between the VME estimate and the physical mass at different radii (short-dashed ), , and that of the non-projected virial estimator with respect to (solid ), . The maximum differences are % and %. For the King profile we obtained a somewhat higher %, this since it has a shallower profile than the Dehnen's one. For the Hernquist's model we obtained lower values in general: % and %. Also shown in Fig. 3 is the boundary term (15) that appeared in the PME (dot-dashed ).

Fig. 3. Differences in virial mass at different radii, both projected and non-projected, with respect to the true mass of the Dehnen's model are shown. Also, the boundary term from the PME is shown.

The application of the VME to our simulated N-body system does not yield any underestimate of the true mass. On the contrary, overestimates for the mass at the inner regions of the numerical systems were found. These results are contrary to those found by TC in their application of the VME, but in the same direction as noted by CYE. We will now turn to explain the reason why we have such discrepancies in the result from our recent application of the VME and at different radii.

Here we recall that the basic idea behind the use of the virial theorem as a mass estimator is to relate the gravitational potential energy and the kinetic energy of the system. If the complete system is in virial equilibrium any subsystem has to be also in equilibrium. Computing of the kinetic energy poses no problem, but the gravitational energy, i.e. the term involving , has not been correctly considered earlier when we computed .

Unlike force, the potential energy at a particular point inside a sphere depends on the particles outside this radius. In other words, the virial theorem holds at all radii if the total potential energy of the particles inside this is accounted for. As said before, the kinetic part of the virial theorem does not suffers from this inconvenient. All this is important to take into account when mass profiles of clusters are to be computed and compared e.g. to mass profiles derived from X-ray observations or gravitational lensing.

Therefore, applying the VME to clusters, for which we probably do not have a fair value of its true boundary, can yield an overestimate of the mass because one ignores the contribution to the potential energy of the outer parts of the `total' cluster which may be dark. The neglect of this extra potential energy is transformed apparently as a mass excess inside the sampled cluster, since the system requires this extra mass to be in equilibrium in order to equate the corresponding kinetic energy. This is also equivalent to having particles of different masses as we increase the aperture in the system (heavier particles inside and lighter ones outside). We need to consider a surface pressure term in the VME (17) if e.g. the system were confined by an external force, but this is not the case here. Note that we do not take into account here the probable effect of external tidal fields on the mass determination of clusters of galaxies.

In Fig. 4 we show the results of a consistent calculation of the potential energy in the VME (Eq. 17) that supports our previous qualitative explanation. We consider here only the projected form of this virial estimator owed to its use in observational astronomy; an almost perfect agreement was found when we used its non-projected form. In dotted line we show the mass estimated using the virial theorem by considering particles only within a certain `aperture' radius ; i.e. the summation of inter-particle separations in Eq. (17), for each i-th particle, considers only the j-th particles that their physical position is . The projected mass of the Dehnen's model is shown as a solid line, and the mass estimated using what the theory of the virial theorem indicates, , is shown in open squares; i.e. the potential energy of the i-th particle inside this is computed by adding the contributions of all other particles to , including those that have .

Fig. 4. Projected mass profile derived from the virial theorem (Eq. 17) applied to an N-body equilibrium and isotropic Dehnen system. assumes that the summation of interparticle distances is constrained to particles inside a particular radius . The solid line indicates the corresponding projected mass, and considers correctly that the particles outside contribute also to the potential energy of particles inside .

As observed from Fig. 4, the two above procedures yield different results, except for small discrepancies which we attribute to numerical artifacts in the construction of the N-body system. The error bars in Fig. 4 correspond to Poisson fluctuations , and the maximum discrepancy is which consistent with the numerical noise. Similar differences were observed for the other models considered here. For systems with less particles the behaviour is the same, but of course the errors are larger.

The physical reason for the overestimation of the mass in the application of the PME is also now clear: it does not account for the long-range nature of gravity. The product in the PME only considers the distance of a galaxy to the centre of the mass distribution and does not account for its interaction with other particles, as the summation involving in the VME does.

The behaviour shown in Fig. 4 supports the idea that the VME is an excellent mass estimator. In real applications, we can expect the error due only to the VME method to be within % for realistic mass profiles if sampling is made around the total effective radius of the cluster. On other hand, taking ( in Fig. 4) as a good estimate of the mass profile its difference with may indicate us how far into the equilibrium part of a gravitational system are we sampling, and thus lower the previous error bound. It might also tell us something about its velocity anisotropy.

In astronomical applications mass estimates are somewhat unreliable due to the fact that clusters may not be in virial equilibrium and/or due to the presence of interlopers. Also, the determination of the physical extent of a cluster is problematic. Theoretically speaking, it is important to recall that the VME is not very sensitive to anisotropies (e.g. Aceves & Perea 1998) so its output can be more indicative of the equilibrium state of the system than of its true mass. On other hand, the use of gravitational lensing methods, which are not hindered by the requirement of the cluster to be in equilibrium, can in conjunction with VME results assess the reliability of the latter. However mass determination by lensing methods are particularly affected if the cluster under investigation is elongated along the line-of-sight.

A comment on the TC results for the VME is also pertinent here. Their projected mass estimator is again only suited when particles (e.g. galaxies) can be treated as test particles. If the latter is not true, the application of underestimates the actual mass. In fact, when such estimator was applied e.g. to our Hernquist N-body model we obtained , a factor 5 lower than the true mass which is in agreement with the behaviour observed by TC. We now turn to consider the continuous form of the virial theorem.

3.2. Continuous virial theorem

To derive the scalar virial theorem in its continuous form we multiply the Jeans hydrostatic equilibrium equation by , and then integrate over the volume of the gravitational system (Binney & Tremaine 1987). Using Eq. (2) for an isotropic velocity system, we have:

Integrating by parts the left hand side and expressing the right hand side in terms of the mass interior to r, we have:

The surface pressure term is (e.g. CYE). For future reference, we may estimate the position of the maximum of this term for the Dehnen's model, : , yielding a maximum value of .

It is readily verified that the Jeans equation holds at every radius, giving e.g. for the Dehnen's model at each side of Eq. (9): . Thus, neglecting the term, which comes from an integration by parts, would obviously produce an error in the mass estimate when using Eq. (19) (e.g. The & White 1986). Now, since the value of the mass is inside an integral we cannot simply subtract the term point by point, or its maximum value, to an estimate of the mass using the VME (17) in order to obtain the correct value; but see below when this term is expressed in mass units (e.g. Girardi et al. 1998). This is because the term is local while the integrals in Eq. (19) and the mass are cumulative quantities.

To quantify the effect of the term on the mass determination we should formally solve the integral Eq. (19). Fortunately, if one does not considers this surface pressure term the situation is simple. We have:

which we may solve readily, leading to an expression for the mass with no surface pressure term as follows:

From Eq. (21) we can see that the neglect of the term in (19) does not always overestimates the mass, but can even underestimate it in realistic cases due to the radial dependence of the velocity dispersion. Indeed, for the extreme case of an isoth ermal sphere, where is a constant, we have an overestimation since . But, e.g. for the Dehnen's model where at large radii the mass will cease to increase settling to a constant value at some finite radius. We may estimate the mass interior to , using the expression for in Sect. 2.1.1, giving ; i.e. an underestimation of the total mass by %. Therefore, in general, an overestimation occurs only at the region of the system that may be considered nearly isothermal, e.g. (see Fig. 5), while an underestimation occurs outside of it. The maximum overestimate is approximately given by the numerical value of .

Fig. 5. Effect of neglecting the surface pressure () term in the continuous form of the virial theorem. indicates that no term is considered in the solution of the integral equation (19). takes the referred term into consideration when actually solving the integral equation (19) for , and is the theoretical Dehnen mass profile. Neglecting the term yields an overestimate of mass at while an underestimate for . In the lower panel the difference is shown for a particular range of values.

Girardi et al. (1998) have introduced a correction term in Eq. (17) when partial sampling of a system is done based on an suggestion by CYE; namely, that an overestimate in mass occurs when applying the VME (Eq. 17) due to the neglect of the term in Eq. (19). Girardi et al. have proposed the following formula to correct for virial mass estimates when partial sampling is done up to a boundary radius b

where refers to the integrated three-dimensional velocity dispersion, which we calculate in our numerical models as

where refers to the total number of particles inside radius R and the summation considers only the particles inside this R. The correction term in (22) expresses the term in `mass units' by dividing it by a term related to the total kinetic energy of the system up to the radius b.

In Fig. 6 (upper panel ) we show the computed projected mass profile using the VME by apertures including the correction term (dashed line , ) as in Eq. (22) for our N-body equilibrium system. The values of , , and were taken here directly from the theoretical model, hence representing an ideal observational situation, while the integrated velocity dispersion was calculated as indicated above. For comparison, the mass using the VME when the outer particles are considered, , is also shown. In the lower panel the corresponding differences are displayed.

Fig. 6. Projected mass profile by the application of the VME by apertures, corrected by a surface pressure term for a Dehnen's model, . denotes the true projected mass for the model. In the lower panel, the differences between these quantities are shown. The difference cannot be accounted by Poisson error bars.

From Fig. 6 it follows that both the masses obtained from the consistent application of the VME (Eq. 17), and the correction due to the term introduced to aperture values of the VME (Eq. 22) provide good estimates of the mass profile . However the application of the VME to the N-body system yields a better result, within numerical errors, than Eq. (22). We emphasize that in astronomical practice the VME does not require to assume any particular form of the radial velocity dispersion of galaxies, which is needed when constructing the correction in (22). This being perhaps the best feature of the virial mass estimator.

Moreover, in an astronomical situation, in order to obtain the term one has to determine the number surface density of galaxies and then apply a deprojection integral to obtain the physical number density , or use the corresponding of the fitted model. Here, implicitly, some confidence is required that the equilibrium part of a cluster has been sufficiently sampled to obtain a reliable total virial mass M and integrated velocity profile; as recognized by Girardi et al. (1998). Hence, in this respect, the estimation of a mass profile using the correction term in (22) and that on the VME rest on the same working conditions, but the latter does not require any assumption on or its projected version. The PME correction term (6) is hindered by the same situation. Thus, we are lead to conclude that the VME is a reliable estimator of mass and mass profiles, having aside the virtue of being straightforward to apply.

As the PME, the VME can overestimate the luminous matter of a cluster if its extent is small in comparison to the extent of the mass. In this case one is led to conclude that the mass-to-light ratios of the visible parts of clusters may be importantly overestimated, say by %, if their outer parts are mainly dark. To conclude this section we recall that in deriving Eq. (17) the assumptions were the same as in the derivation of the PME, so these equations are valid in this context, as we have verified by numerical experiments that have correctly considered the potential energy term in Eq. 17. In a future work we will consider the estimation of mass profiles of several clusters and compare the results with those obtained by other authors, both in the optical and X-ray region of the spectra.

© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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