© ESO, 2016

1. Introduction

For years, mass estimation of groups and clusters of galaxies has been one of the classical tasks of astronomers (see, for example: Zwicky 1933; Rood et al. 1970; Gott & Turner 1977; Rood & Dickel 1978; Rood 1981; Huchra & Geller 1982; Geller 1984; Karachentsev 2005). Galactic kinematics, observations of X-rays from hot intra-cluster gas or gravitational lensing are some of the techniques used to infer the mass of these systems.

From the redshifts and positions of galaxies, the virial mass is the classical mass estimator. Bahcall & Tremaine (1981) studied the reliability of the masses derived from the virial estimator and proposed another estimator based on the projected mass. Heisler et al. (1985) extended the work of Bahcall & Tremaine by exploring alternative mass estimators to the virial theorem for self-gravitating systems.

The virial theorem was introduced into astronomy by Zwicky (1933) in an approximate form to estimate the mass of the Coma cluster. It was applied to open star clusters by Chandrasekhar (1942), and to globular clusters by Schwarzschild (1954).

The virial theorem is derived by multiplying the steady Jeans equations by coordinate vectors and integrating them over a volume. If magnitudes are not null at the outer surface of this volume, a surface term appears. Mass estimation derived from this equation, when the surface term is neglected, is commonly known as the virial mass. This is because the virial theorem is usually applied without considering surface effects.

In the continuous model, when spherical symmetry and steady and static conditions are assumed, the virial theorem corresponds to the volume integral of the radial Jeans equation multiplied by the radial distance, r. The volume integral of the radial Jeans equation multiplied by r² has also been used by some authors (see, for example, Heisler et al. 1985). In fact, the mass estimation, which is derived when surface term is neglected, is known as the projected mass.

Equations equivalent to both volume integrals can be proposed to deal with a sample of N galaxies. From these equations, the virial mass and the projected mass are also derived when their surface terms are not taken into account. If a system extends beyond the sample field, both masses could differ appreciably; in which case, the surface term contributions would be important.

In this work, we explore how the use of both equations allows the mass that is contained in the volume occupied by the sample to be estimated without calculating surface terms. Cosmological constant terms are also included in the equations. For isolated systems, surface terms are positive or null; hence, the sample mass must be smaller than, or equal to, both virial and projected masses; i.e., if surface terms are neglected, the mass is overestimated.

To estimate the mass of a sample, distances and radial velocities are required with sufficient precision for all its component galaxies (H I rotational velocity must also be known to derive indicative masses of galaxies). The Catalog of Neighboring Galaxies (Karachentsev et al. 2004) contains quality observational data within 10 Mpc. In fact, these data were used by Karachentsev (2005) to estimate virial and projected masses for galaxy groups within 5 Mpc. Our study will be applied to these groups.

In Sect. 2, the two equations that come from the volume integrals of the radial Jeans equation multiplied by r (virial theorem) and by r² are presented in a continuous model for an isolated bound spherical system of galaxies. They are evaluated at a volume contained inside the whole system and thus, surface terms appear. The equation derived from them, which allows the mass inside the volume to be estimated without requiring surface term calculations, is also shown. Equations are also expressed as a function of projected quantities.

In Sect. 3, two other equations, equivalent to those of the continuous model, are proposed for dealing with a sample of N galaxies. In these equations, cosmological constant terms are also included. Sections 3.1 and 3.2 are devoted to virial mass and projected mass estimators, respectively, derived from neglecting surface terms in these equations. Section 3.3 presents a third equation, which allows the mass inside the volume of the sample to be estimated without having to calculate surface terms.

Section 4 is devoted to applying the previously mentioned equations to neighbouring groups of galaxies to estimate virial, projected, and sample masses.

Section 5 deals with galaxy complexes composed of the galaxy groups addressed in this work. In Sect. 5.1, virialized masses of these complexes are calculated from measured turn-around radii. In Sect. 5.2, these virialized masses are compared with the sum of sample masses of the groups estimated in Sect. 4.

Finally, our conclusions are presented in Sect. 6.

2. Mass estimators for isolated bound spherical systems of galaxies: continuous model

In the continuous model, a system of galaxies is described by a phase space density f_g(r,v,t), where r, v, and t are the spatial coordinate, velocity, and time. The mass density is then given by ${\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}{r}\mathit{,t}\mathrm{\right)}\mathrm{=}{}^{\mathrm{\int }}\mathit{f}_{\mathrm{g}}\mathrm{\left(}{r}\mathit{,}{v}\mathit{,t}\mathrm{\right)}\mathrm{d}{v}$; so, the average in velocities of any quantity, η(r,v,t), is calculated by $\overline{\mathit{\eta }}\mathrm{\left(}{r}\mathit{,t}\mathrm{\right)}\mathrm{=}{\mathrm{[}}^{\mathrm{\int }}\mathit{\eta }\mathrm{\left(}{r}\mathit{,}{v}\mathit{,t}\mathrm{\right)}{\mathit{f}}_{\mathrm{g}}\mathrm{\left(}{r}\mathit{,}{v}\mathit{,t}\mathrm{\right)}\mathrm{d}{v}\mathrm{]}\mathit{/}{\mathit{\rho }}_{\mathrm{g}}\mathrm{(}{r}\mathrm{)}$. Let us assume that the evolution of the phase space density is described by the collisionless Boltzmann equation. The integration over all velocities of the product of the Boltzmann equation with the velocity vector v leads to the Jeans equations.

The radial Jeans equation, assuming spherical symmetry and steady and static conditions ($\overline{{v}}\mathrm{=}\mathrm{0}$), is given by (see, for example, Binney & Tremaine 1987, p. 198) $\frac{\mathrm{d}}{\mathrm{d}\mathit{r}}\left[{\mathit{\rho }}_{\mathrm{g}}\overline{{\mathit{v}}_{\mathit{r}}^{\mathrm{2}}}\right]\mathrm{+}\frac{{\mathit{\rho }}_{\mathrm{g}}}{\mathit{r}}\left[\mathrm{2}\overline{{\mathit{v}}_{\mathit{r}}^{\mathrm{2}}}\mathrm{-}\mathrm{(}\overline{{\mathit{v}}_{\mathit{\theta }}^{\mathrm{2}}}\mathrm{+}\overline{{\mathit{v}}_{\mathit{\varphi }}^{\mathrm{2}}}\right]\mathrm{=}\mathrm{-}{\mathit{\rho }}_{\mathrm{g}}\frac{\mathrm{d}\mathit{\phi }}{\mathrm{d}\mathit{r}}\mathit{,}$(1)where φ is the gravitational potential generated by matter and dark energy. In Eq. (1), subindexes r, θ, and ϕ indicate radial, polar, and azimuthal components. As $\overline{{\mathit{v}}_{\mathit{r}}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{\theta }}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{\varphi }}}\mathrm{=}\mathrm{0}$, velocity dispersions of galaxies fulfill ${\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{r}}^{\mathrm{2}}}$, ${\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{\theta }}^{\mathrm{2}}}$ and ${\mathit{\sigma }}_{\mathit{\varphi }}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{\varphi }}^{\mathrm{2}}}$; and hence, ${\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathit{r}}^{\mathrm{2}}}\mathrm{+}\overline{{\mathit{v}}_{\mathit{\theta }}^{\mathrm{2}}}\mathrm{+}\overline{{\mathit{v}}_{\mathit{\varphi }}^{\mathrm{2}}}$. In the following, we use ${\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}$, ${\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}$, and ${\mathit{\sigma }}_{\mathit{\varphi }}^{\mathrm{2}}$ instead of $\overline{{\mathit{v}}_{\mathit{r}}^{\mathrm{2}}}$, $\overline{{\mathit{v}}_{\mathit{\theta }}^{\mathrm{2}}}$ and $\overline{{\mathit{v}}_{\mathit{\varphi }}^{\mathrm{2}}}$.

If dark energy is treated from a hydrodynamical point of view and the cosmological constant model is assumed, φ can be derived from ${\mathrm{\nabla }}^{\mathrm{2}}\mathit{\phi }\mathrm{=}\mathrm{4}\mathit{\pi G}\left[\mathit{\rho }\mathrm{+}\frac{\mathrm{1}}{{\mathit{c}}^{\mathrm{2}}}\mathrm{(}{\mathit{\rho }}_{\mathrm{\Lambda }}\mathrm{+}\mathrm{3}{\mathit{p}}_{\mathrm{\Lambda }}\mathrm{)}\right]\mathit{.}$(2)In Eq. (2), ρ is the mass density of matter and ρ_Λ and p_Λ, the energy density and pressure of the dark energy (${\mathit{\rho }}_{\mathrm{\Lambda }}\mathrm{=}\frac{{\mathit{c}}^{\mathrm{4}}}{\mathrm{4}\mathit{\pi G}}\mathrm{\Lambda }\mathrm{=}\mathrm{-}{\mathit{p}}_{\mathrm{\Lambda }}$, Λ being the cosmological constant). Thus, taking into account the spherical symmetry of the system, $\frac{\mathrm{d}\mathit{\phi }}{\mathrm{d}\mathit{r}}\mathrm{=}\frac{\mathit{Gm}}{{\mathit{r}}^{\mathrm{2}}}\mathrm{-}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}\mathit{r,}$(3)where m(r) is the mass of matter enclosed up to r, i.e. $\mathit{m}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{\mathit{r}}\mathit{\rho }{\mathit{r}}^{\mathrm{\prime }\mathrm{2}}\mathrm{d}{\mathit{r}}^{\mathrm{\prime }}\mathit{.}$(4)

2.1. The virial theorem

The virial theorem is derived by multiplying Eq. (1) by 4πr³ and integrating from r = 0 up to some r = R. Thus, $\begin{array}{ccc}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{3}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}\mathit{R}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{R}\mathrm{\right)}& \mathrm{=}& {\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\\ & & \mathrm{-}\mathit{G}{\mathit{M}}_{\mathrm{g}}{⟨\frac{\mathit{m}}{\mathit{r}}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}\end{array}$(5)In Eq. (5), M_g is the mass of galaxies contained in the sphere of radius R; and for any quantity η(r), ${⟨\mathit{\eta }⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{M}}_{\mathrm{g}}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{R}}{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{2}\mathit{\pi }}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{\pi }}{\mathit{\rho }}_{\mathrm{g}}\mathit{\eta }{\mathit{r}}^{\mathrm{2}}\sin \mathit{\theta }\mathrm{d}\mathit{\theta }\mathrm{d}\mathit{\varphi }\mathrm{d}\mathit{r}\mathit{.}$(6)When we deal with a galaxy system, its mass is defined by the spatial extension of the sample. Hence, if the system extends beyond the observed field and the surface term in Eq. (5) is neglected, the mass obtained up to its observed extension is overestimated (the pressure at the outer surface of the observed field causes the mass needed to keep the system in equilibrium to be smaller).

One way of dealing with the surface term of Eq. (5) was developed by The & White (1986). For the Coma cluster, they estimated the maximum possible value of the surface term by determining the largest value of ${\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}$ at $\mathit{R}\mathrm{=}\mathrm{5.4}{\mathit{h}}_{\mathrm{50}}^{-1}$ Mpc (h₅₀ = H₀/ 50 km s^-1 Mpc^-1; H₀ being the Hubble function at present), consistent with the observational data. They concluded that the surface term should be smaller than 42% of the kinetic term, and assumed a value of 0.21 times the kinetic term. Rines et al. (2003) also calculated the overestimation of the virial mass when the system does not lie entirely within the virial radius for the clusters treated in the CAIRNS (Cluster and Infall Region Nearby Survey) project.

An estimation of an upper limit for the surface term can also be derived assuming a singular isothermal sphere. The fact that bound gravitational systems show decreasing velocity dispersions with radius means that isothermal spheres have a greater surface term. If galaxies are assumed to trace the mass, the mass density of galaxies, ρ_g, and matter, ρ, would be given by $\begin{array}{ccc}{\mathit{\rho }}_{\mathrm{g}}& \mathrm{=}& \frac{{\mathit{M}}_{\mathrm{g}}}{\mathrm{4}\mathit{\pi R}{\mathit{r}}^{\mathrm{2}}}\mathit{,}\\ \mathit{\rho }& \mathrm{=}& \frac{\mathit{M}}{\mathrm{4}\mathit{\pi R}{\mathit{r}}^{\mathrm{2}}}\mathit{,}\end{array}$M_g and M being the mass of galaxies and the mass of matter enclosed up to R. When dark energy is not taken into account, Eqs. (7)–(8) are solutions of the Jeans Eq. (1) if $\mathit{M}\mathrm{=}\frac{{\mathit{\sigma }}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}{\mathit{G}}\mathit{R}\mathit{.}$(9)This result is also derived from Eq. (5). However, when the surface term in (5) is neglected, the mass derived does not correspond to that shown in Eq. (9); in fact, it is overestimated by a factor ${\mathit{\sigma }}^{\mathrm{2}}\mathit{/}\mathrm{(}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{)}$. Hence, the surface term contributes appreciably to the mass. For isotropic orbits, Eq. (9) tells us that $\mathit{M}\mathrm{=}\mathrm{(}\mathrm{2}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathit{/}\mathit{G}\mathrm{)}\mathit{R}$. In that case, ${\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathit{/}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{=}\mathrm{1}\mathit{/}\mathrm{3}$, which can be compared with the assumption of The & White (1986) for the surface term.

When cosmological constant effects are taken into account, ρ_g and ρ given by Eqs. (7) and (8) do not fulfill the Jeans Eq. (1). In any case, assuming these mass densities, an estimation for the mass inside a sphere of radius R can be derived from Eq. (5). This equation leads to $\mathit{M}\mathrm{=}\frac{{\mathit{\sigma }}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}{\mathit{G}}\mathit{R}\mathrm{+}\frac{\mathrm{8}\mathit{\pi }{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{9}{\mathit{c}}^{\mathrm{2}}}{\mathit{R}}^{\mathrm{3}}\mathit{.}$(10)Equation (10) was used by Membrado & Aguerri (2004) and Membrado & Pacheco (2014) in the study of the collapse of spherical mass shells in a cosmological background of matter and cosmological constant. This equation allowed the virial radius of a shell to be estimated from its turn-around radius.

2.2. A way of avoiding surface terms

When the Jeans Eq. (1) is multiplied by r⁴ and integrated from r = 0 up to r = R, the equation obtained is ${\mathit{R}}^{\mathrm{4}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}\mathit{R}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{R}\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{R}}{\mathit{r}}^{\mathrm{3}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{[}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{]}}\mathrm{d}\mathit{r}\mathrm{-}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{R}}{\mathit{r}}^{\mathrm{4}}{\mathit{\rho }}_{\mathrm{g}}\frac{\mathrm{d}\mathit{\phi }}{\mathrm{d}\mathit{r}}\mathrm{d}\mathit{r}\mathit{.}$(11)Then, using Eqs. (3) and (6) in Eq. (11), we obtain $\begin{array}{ccc}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{4}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}\mathit{R}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{R}\mathrm{\right)}& \mathrm{=}& {\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}\mathit{r}{\mathrm{[}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{]}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\\ & & \mathrm{-}\mathit{G}{\mathit{M}}_{\mathrm{g}}{⟨\mathit{m}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{3}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}\end{array}$(12)In the case of assuming singular isothermal spheres (see Eqs. (7) and (8)), Eq. (9) is also derived from (12) when dark energy is not taken into account. If the surface term is neglected, the mass is overestimated by a factor $\mathrm{(}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{)}\mathit{/}\mathrm{(}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{)}$. We should also point out that the contribution of the cosmological constant term is 3 / 2 times greater than that shown in Eq. (10). This is because Eqs. (7) and (8) do not fulfill the Jeans Eq. (1).

The bonus of Eq. (12) is that its surface term can be eliminated by using the virial theorem given by Eq. (5). Thus, $\begin{array}{ccc}\mathit{R}\left[{\mathrm{⟨}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{-}\mathit{G}{⟨\frac{\mathit{m}}{\mathit{r}}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\right]\mathrm{=}& & \end{array}$${\mathrm{⟨}\mathit{r}{\mathrm{[}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{]}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{-}\mathit{G}{⟨\mathit{m}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{3}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(13)Following with the example of singular isothermal spheres, the mass given by Eq. (9) is again derived from (13) when dark energy terms are not considered. When those terms are not neglected, they lead to a contribution which is half that shown in Eq. (10).

2.3. Using projected quantities

From observations of galaxies, we have access to their velocities in the line of sight, to their projected separations and, in some cases, to their distances. Therefore, it is useful to express Eqs. (5) and (12) in a manner in which we can include observational measurements. This can be done by using projected quantities.

Let us consider a point that has a position vector r with respect to the mass centre, and a position vector R with respect to the observer. Taking the line of sight of the mass centre as the z-direction, we can define the polar angle with respect to the mass centre, θ, and the polar angle with respect to the observer, Θ; these angles fulfill and . At r, we have a distribution of velocities given by the distribution function f(r,v).

At the point r, the component of the velocity in the line of sight of the mass centre, v_z, as a function of the radial, polar, and azimuthal components of the velocity is ${\mathit{v}}_{\mathrm{z}}\mathrm{=}\mathrm{-}{\mathit{v}}_{\mathit{\theta }}\sin \mathit{\theta }\mathrm{+}{\mathit{v}}_{\mathit{r}}\cos \mathit{\theta }\mathit{.}$(14)Hence, as ${\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathrm{z}}^{\mathrm{2}}}\mathrm{-}\mathrm{(}\overline{{\mathit{v}}_{\mathrm{z}}}{\mathrm{)}}^{\mathrm{2}}$ and $\overline{{\mathit{v}}_{\mathrm{z}}}\mathrm{=}\mathrm{0}$, ${\mathrm{⟨}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{3}}{\mathrm{⟨}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{2}}{\mathrm{3}}{\mathrm{⟨}{{\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(15)For the sake of simplicity, we could assume ${\mathit{\sigma }}_{\mathit{\theta }}\mathrm{=}{\mathit{\sigma }}_{\mathit{\varphi }}\mathit{,}$(16)and a constant anisotropy parameter β, given by $\mathit{\beta }\mathrm{=}\mathrm{1}\mathrm{-}\frac{{\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}\mathrm{\left(}\mathit{r}\mathrm{\right)}}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{r}\mathrm{\right)}}\mathrm{·}$(17)Hence, ${\mathrm{⟨}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{3}}{\mathrm{⟨}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\left[\mathrm{1}\mathrm{-}\frac{\mathrm{2}\mathit{\beta }}{\mathrm{3}}\right]{\mathrm{⟨}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{·}$(18)With respect to the component of the velocity in the line of sight, v_los, it is easy to realize that v_los = −v_θsin(θ − Θ) + v_rcos(θ − Θ). When isotropy is assumed (i.e. β = 0) and as $\overline{{\mathit{v}}_{\mathrm{los}}}\mathrm{=}\mathrm{0}$, ${\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{=}\overline{{\mathit{v}}_{\mathrm{los}}^{\mathrm{2}}}\mathrm{=}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma }}_{\mathit{\varphi }}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}$. Thus, ${\mathrm{⟨}{{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}{\mathrm{⟨}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{3}}{\mathrm{⟨}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(19)Now, let us have a look at the second term on the right hand side of Eq. (5). Taking into account that ${\mathit{r}}_{\mathrm{\perp }}\mathrm{=}\mathit{r}\sin \mathit{\theta ,}$(20)then, ${⟨\frac{\mathit{m}}{{\mathit{r}}_{\mathrm{\perp }}}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}}{⟨\frac{\mathit{m}}{\mathit{r}}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(21)With respect to the cosmological constant term of Eq. (5), the use of Eq. (20) leads to ${\mathrm{⟨}{{\mathit{r}}_{\mathrm{\perp }}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{2}}{\mathrm{3}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{2}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(22)Hence, using Eqs. (18), (21), and (22) in Eq. (5), the virial theorem for a steady and static gravitational system, assuming spherical symmetry and a constant anisotropy parameter, reads as $\begin{array}{ccc}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{3}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}\mathit{R}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{R}\mathrm{\right)}& \mathrm{=}& \mathrm{3}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\\ & & \mathrm{-}\frac{\mathrm{2}}{\mathit{\pi }}\mathit{G}{\mathit{M}}_{\mathrm{g}}{⟨\frac{\mathit{m}}{{\mathit{r}}_{\mathrm{\perp }}}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{{\mathit{r}}_{\mathrm{\perp }}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}\end{array}$(23)If isotropy is assumed, Eq. (19) can be used instead of (18).

With respect to Eq. (12), the first and third terms on the right hand side of the equation can also be expressed as functions of projected quantities. From Eqs. (14) and (20), ${\mathrm{⟨}{\mathit{r}}_{\mathrm{\perp }}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{16}}{\mathrm{⟨}\mathit{r}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{3}\mathit{\pi }}{\mathrm{16}}{\mathrm{⟨}\mathit{r}{{\mathit{\sigma }}_{\mathit{\theta }}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(24)Thus, if a constant anisotropy parameter given by Eqs. (16) and (17) is assumed, ${\mathrm{⟨}{\mathit{r}}_{\mathrm{\perp }}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{32}}\frac{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{⟨}\mathit{r}{\mathrm{[}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{]}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{;}$(25)and in the case of isotropy (β = 0 and ${\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}$), ${\mathrm{⟨}{\mathit{r}}_{\mathrm{\perp }}{{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{16}}{\mathrm{⟨}\mathit{r}{\mathrm{[}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{+}{{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}}^{\mathrm{]}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(26)With respect to the third term of Eq. (12), from Eq. (20), ${\mathrm{⟨}{{\mathit{r}}_{\mathrm{\perp }}^{\mathrm{3}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{=}\frac{\mathrm{3}\mathit{\pi }}{\mathrm{16}}{\mathrm{⟨}{\mathit{r}}^{\mathrm{3}}\mathrm{⟩}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}$(27)Now, using Eqs. (25) and (27), Eq. (12) reads as $\begin{array}{ccc}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{4}}{\mathit{\rho }}_{\mathrm{g}}\mathrm{\left(}\mathit{R}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{\left(}\mathit{R}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{32}}{\mathit{\pi }}\frac{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{\mathit{r}}_{\mathrm{\perp }}{{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\\ & & \mathrm{-}\mathit{G}{\mathit{M}}_{\mathrm{g}}{⟨\mathit{m}⟩}_{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{128}\mathit{G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{9}{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}{\mathrm{⟨}{{\mathit{r}}_{\mathrm{\perp }}^{\mathrm{3}}}^{\mathrm{⟩}}}_{{\mathit{M}}_{\mathrm{g}}}\mathit{.}\end{array}$(28)

3. Mass estimators for isolated bound spherical systems of galaxies: discrete model

Let us suppose now that an isolated galaxy system extends beyond the field occupied by a sample of N galaxies. The masses, positions, and velocities of galaxies are denoted by [m_g]_i, R_i, and V_i. The total mass of the N galaxies is ${\mathit{M}}_{\mathrm{g}}\mathrm{=}{\sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}$; and the mass of matter inside the extension of the sample, M. In the intergalactic medium, the distribution of dark matter is unknown; therefore, the mass distribution in galaxy systems is also unknown. In this work, for the sake of simplicity, we assume that galaxies trace the mass.

For the sample of galaxies, the position and velocity of the centre of mass are ${⟨{{R}}_{\mathit{i}}⟩}_{\mathrm{M}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{M}}_{\mathrm{g}}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}{{R}}_{\mathit{i}}\mathit{,} {⟨{{V}}_{\mathit{i}}⟩}_{\mathrm{M}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{M}}_{\mathrm{g}}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}{{V}}_{\mathit{i}}\mathit{.}$(29)Hence, the positions and velocities of galaxies relative to the mass centre are ${{r}}_{\mathit{i}}\mathrm{=}{{R}}_{\mathit{i}}\mathrm{-}{⟨{{R}}_{\mathit{i}}⟩}_{\mathrm{M}}\mathit{,} {{v}}_{\mathit{i}}\mathrm{=}{{V}}_{\mathit{i}}\mathrm{-}{⟨{{V}}_{\mathit{i}}⟩}_{\mathrm{M}}\mathit{.}$(30)The component of r_i perpendicular to the line of sight of the mass center, ẑ ≡ ⟨R_i⟩_M/ | ⟨R_i⟩_M |, is denoted by [r_i]_⊥; and the projection of v_i on the line of sight of the mass centre, by [v_i]_z.

3.1. Virial mass estimator

We will assume that the system shows spherical symmetry and a constant anisotropy parameter. In this case, an expression for the virial theorem for the sample of the N galaxies, similar to that given for a continuous model by Eq. (23), is: $\begin{array}{ccc}& & {⟨{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}⟩}_{\mathit{t}}\mathrm{=}\mathrm{3}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{v}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{z}}^{\mathrm{2}}⟩}_{\mathit{t}}\\ & & \mathrm{-}\frac{\mathrm{2}}{\mathit{\pi }}\mathit{G}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\frac{\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}}}{\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}}⟩}_{\mathit{t}}\mathrm{+}\frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{{\mathit{c}}^{\mathrm{2}}}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}^{\mathrm{2}}⟩}_{\mathit{t}}\mathit{.}\end{array}$(31)In Eq. (31), R_s is the radius of the sphere containing the N galaxies. S_vir(R_s) is a surface term (see surface term of Eq. (23)) evaluated at a distance R_s. [ℳ]_i is the mass covered by a spherical surface of radius r_i. Finally, the brackets stand for time averages, which are taken to obtain steady magnitudes. It is, of course, not possible to obtain time averages from observations. However, the sums of individual contributions at present on the right hand side of Eq. (31) would introduce a kind of averaging. It is expected that the magnitudes at the present epoch fulfill the virial theorem (31) reasonably well.

The second term on the right hand side of Eq. (31) comes from the trace of the Chandrasekhar gravitational potential energy tensor in Eq. (5) (see, for example, Binney & Tremaine 1987, p. 67). However, it should be noted that the gravitational energy is more commonly used. Proof of the virial theorem in the discrete model using gravitational energy and projected magnitudes can be seen, for example, in Heisler et al. (1985). Here, the surface term is not taken into account and the cosmological constant effects are neglected. When those contributions are considered, the virial theorem can be expressed as $\begin{array}{ccc}{⟨{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}⟩}_{\mathit{t}}& \mathrm{=}& \mathrm{3}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}{[}{\mathit{v}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{z}}^{\mathrm{2}}⟩}_{\mathit{t}}\\ & & \mathrm{-}\frac{\mathrm{2}}{\mathit{\pi }}\frac{\mathit{GM}}{{\mathit{M}}_{\mathrm{g}}}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\sum _{\mathit{j}\mathit{<}\mathit{i}}\frac{\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{j}}}{\mathrm{[}{\mathit{r}}_{\mathit{ij}}{\mathrm{]}}_{\mathrm{\perp }}}⟩}_{\mathit{t}}\\ & & \mathrm{+}\frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{{\mathit{c}}^{\mathrm{2}}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathrm{⟨}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{r}}_{\mathit{i}}{{\mathrm{]}}_{\mathrm{\perp }}^{\mathrm{2}}}^{\mathrm{⟩}}}_{\mathit{t}}\mathit{,}\end{array}$(32)where ${\mathit{r}}_{\mathit{ij}}\mathrm{=}\mathrm{\left|}{{r}}_{\mathit{ij}}\mathrm{\right|}\mathrm{=}\mathrm{|}{{r}}_{\mathit{i}}\mathrm{-}{{r}}_{\mathit{i}}\mathrm{|}\mathrm{=}\mathrm{|}{{R}}_{\mathit{i}}\mathrm{-}{{R}}_{\mathit{j}}\mathrm{|}\mathit{.}$(33)Some authors use [r_ij]_⊥ as the component of r_ij perpendicular to the line of sight of the ith galaxy; others, as the component of r_ij perpendicular to the line of sight of the mass center.

Equations (31) or (32) tells us that galaxy contributions should be weighted by galaxy masses. In this respect, indicative masses can be derived from HI rotation velocities. Some authors take the same mass for all galaxies. This would introduce another kind of averaging in the equations. In this sense, the N-body simulations developed by Heisler et al. (1985) indicated that their estimators worked well, even if galaxies had a range of masses.

Neglecting the surface term and the contribution of the cosmological constant term, mass estimators derived from different expressions for the virial theorem have been extensively used (see, for example: Limber & Mathews 1960; Huchra & Geller 1982; Carlberg et al. 1996; Karachentsev 2005). Chernin et al. (2012) included the cosmological constant term in mass estimators. They concluded that the cosmological constant term must be taken into account in groups of galaxies, but its contribution is negligible in galaxy clusters.

The aim of this work is to estimate the effects of surface terms on mass estimators when applied to a galaxy system. To develop this, accurate distances and radial velocities are required for all galaxies in the system. Moreover, to estimate the mass of each galaxy, the H I rotational velocity must be known. All these data are available for galaxy systems within 10 Mpc from the Catalog of Neighboring Galaxies (Karachentsev et al. 2004). In fact, the quantity and quality of these observational data allowed Karachentsev (2005) to determine masses for galaxy groups within 5 Mpc (Milky Way, M 31, M 81, NGC 2403, Centaurus A, M 83, IC 342, Maffei, Sculptor Filament and Canes Venatici I Cloud). In this paper, we also deal with these neighbouring galaxy systems.

Each one of these groups has a massive galaxy, which contains nearly all the mass of galaxies of the system. In this work, we assume that the mass centre of a group coincides with the centre of its most massive galaxy, denoted by i = 1 galaxy.

For a sample of N galaxies with a main galaxy that dominates, the virial theorem given by Eq. (31) is applied to the rest of the N − 1 galaxies (note that r₁ = 0 and [ℳ]₁ = 0). To estimate (31), we assume that such N − 1 galaxies have the same mass; i.e.

$\begin{array}{ccc}& & \mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}\mathrm{=}\mathrm{1}}\mathrm{\gg }\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}\mathit{,}\\ & & \mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}\mathrm{=}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathit{/}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}\mathit{,}\\ & & {\mathit{M}}_{\mathrm{g}}\mathrm{=}\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}\mathrm{+}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathit{.}\end{array}$

Thus, the virial theorem reads as ${\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}\mathrm{=}\mathrm{3}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}\mathrm{-}\frac{\mathit{GM}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{\mathrm{\left[}{\mathit{R}}_{\mathrm{H}}\mathrm{\right]}}\mathrm{+}\frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{[}{{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}}^{\mathrm{]}}\mathit{.}$(37)The kinetic term of Eq. (31) has been approximated by the kinetic term of Eq. (37); in this term, we take $\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}\mathrm{\approx }\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}$ (both quantities that coincide in the case of isotropy; see Eq. (19)), where $\mathrm{[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}{\mathrm{]}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ is the dispersion of line of sight velocities with respect to that of the sample centroid. In Eq. (37), [R_H] is the mean harmonic radius, and $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}$, the mean square projected radius. Thus,

$\begin{array}{ccc}& & \mathrm{\left[}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}\mathrm{\approx }\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\left[\mathrm{[}{\mathit{V}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{los}}\mathrm{-}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{V}}_{\mathit{j}}{\mathrm{]}}_{\mathrm{los}}\right]}^{\mathrm{2}}\mathit{,}\\ & & \mathrm{\left[}{\mathit{R}}_{\mathrm{H}}\mathrm{\right]}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}}{\left[\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\frac{\mathrm{1}}{\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}}\left(\frac{\mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}}}{\mathit{M}}\right)\right]}^{-1}\mathit{,}\\ & & \mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}^{\mathrm{2}}\mathit{.}\end{array}$

To deal with [ℳ]_i in Eq. (39), we explore three models: model PM takes a central point mass; model MG considers that the companions to the main galaxy have the same mass (see Eq. (35)); and, model EM assumes equal mass for all galaxies. These models are used for reasons of simplicity and because they introduce another type of averaging in the calculations.

In Sect. 4, neighbouring galaxy groups are dealt with. The results in that section seem to indicate that [R_H] shows a dependence on galaxy positions, which is stronger than that of the mass contribution of the companions to the main galaxy. Thus, some groups with a massive main galaxy are better represented by a harmonic radius that is calculated assuming equal mass galaxies. In contrast, other groups with a main galaxy, but where companions contribute appreciably to the mass, are better represented by a harmonic radius calculated assuming central point mass.

All models [ℳ]_i fulfill $\mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathrm{=}\mathrm{1}}\mathrm{=}\mathrm{0.}$(41)When galaxies are numbered according to their distances from the main galaxy, [ℳ]_{i> 1}’s from models PM, MG, and EM are given by $\begin{array}{ccc}\mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}& \mathrm{\equiv }& \mathrm{[}{\mathrm{ℳ}}_{\mathrm{PM}}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}\mathrm{=}\mathit{M,}\\ \mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}& \mathrm{\equiv }& \mathrm{[}{\mathrm{ℳ}}_{\mathrm{MG}}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}\mathrm{=}\frac{\mathit{M}}{{\mathit{M}}_{\mathrm{g}}}\left[\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}\mathrm{+}\mathrm{(}\mathit{i}\mathrm{-}\mathrm{2}\mathrm{)}\frac{{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\right]\mathit{,}\\ \mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}& \mathrm{\equiv }& \mathrm{[}{\mathrm{ℳ}}_{\mathrm{EM}}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}\mathrm{=}\frac{\mathit{M}}{{\mathit{M}}_{\mathrm{g}}}\left[\mathrm{(}\mathit{i}\mathrm{-}\mathrm{1}\mathrm{)}\frac{{\mathit{M}}_{\mathrm{g}}}{\mathit{N}}\right]\mathrm{·}\end{array}$It can be seen that when ${\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\ll }\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}$ (main galaxy dominates the mass), Eq. (43) tends to Eq. (42); and, when $\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}\mathrm{=}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathit{/}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}$ (all galaxies have the same mass), Eq. (43) coincides with Eq. (44).

Equations (43) and (44) are not directly used in Eq. (39). To introduce another kind of averaging in the discrete model, these equations are averaged in the (N − 1) companions to the main galaxy. Thus, for models MG and EM, we take $\begin{array}{ccc}\mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}& \mathrm{\equiv }& \mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{MG}}\mathrm{\right]}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathrm{ℳ}}_{\mathrm{MG}}{\mathrm{]}}_{\mathit{i}}\\ & \mathrm{=}& \mathit{M}\left[\frac{\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}}{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{2}\mathrm{)}}{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\frac{{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{{\mathit{M}}_{\mathrm{g}}}\right]\mathit{,}\\ \mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}\mathit{>}\mathrm{1}}& \mathrm{\equiv }& \mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{EM}}\mathrm{\right]}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathrm{ℳ}}_{\mathrm{EM}}{\mathrm{]}}_{\mathit{i}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\mathit{M}\mathit{.}\end{array}$When using Eq. (42) in Eq. (39), the harmonic radius for model PM is given by $\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{PM}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}}{\left[\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\frac{\mathrm{1}}{\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}}\right]}^{-1}\mathrm{·}$(47)This is the case for galaxies orbiting around a point mass. The virial theorem for this system, neglecting the cosmological constant term and surface term, was addressed, for example, by Bahcall & Tremaine (1981).

The harmonic radius for model MG can be derived using Eqs. (45) in (39). Thus $\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{MG}}\mathrm{=}{\left[\frac{\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}}{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{2}\mathrm{)}}{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\frac{{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{{\mathit{M}}_{\mathrm{g}}}\right]}^{-1}\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{PM}}\mathit{.}$(48)The greater the difference between [R_H]_MG and [R_H]_PM, the greater the mass contributions of the companions to the main galaxy.

And, when Eq. (46) is used in Eq. (39), $\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{EM}}\mathrm{=}\mathrm{2}\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{PM}}$(49)(note that when all galaxies have the same mass, [R_H]_EM = [R_H]_MG). Some authors (see, for example, Karachentsev 2005), instead of using [R_H]_EM, use $\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{K}}\mathrm{=}\frac{\mathrm{2}\mathit{N}}{\mathit{N}\mathrm{-}\mathrm{1}}\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{PM}}\mathit{.}$(50)Equation (50) can be deduced from Eq. (39), taking for [ℳ]_i, the average of [ℳ_EM]_i (see Eq. (44)) in the N galaxies.

Hereafter, [M] will be used to represent mass estimations which do not take into account cosmological constant terms. Estimations of mass obtained considering cosmological constant terms will be indicated by {M}.

In the case of neglecting the surface term of Eq. (37), an estimator for the mass inside the sphere of radius R_s is $\begin{array}{ccc}\mathrm{\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\}}\mathrm{=}\mathrm{\left[}{\mathit{M}}_{\mathrm{vir}}\mathrm{\right]}\left(\mathrm{1}\mathrm{+}\mathrm{\left[}{\mathit{f}}_{\mathrm{vir}}\mathrm{\right]}\right)\mathit{,}& & \end{array}$(51)where $\begin{array}{ccc}\left[{\mathit{M}}_{\mathrm{vir}}\right]& \mathrm{=}& \frac{\mathrm{3}}{\mathit{G}}\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}\mathrm{\left[}{\mathit{R}}_{\mathrm{H}}\mathrm{\right]}\mathit{,}\\ \left[{\mathit{f}}_{\mathrm{vir}}\right]& \mathrm{=}& \frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}}{\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{\Lambda }}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}}{\mathrm{\left[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}}\mathrm{·}\end{array}$In Eq. (53), H₀ is the Hubble function at present (${\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}\mathrm{=}\mathrm{\left[}\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{0}}\mathrm{\right]}\mathit{/}\mathrm{\left[}\mathrm{3}{\mathit{c}}^{\mathrm{2}}\mathrm{\right]}$; ρ₀ being the energy density of the present Universe), and Ω_Λ is the fraction of dark energy in the Universe at present (Ω_Λ = ρ_Λ/ρ₀). Values for Ω_Λ and H₀ can be taken from Spergel et al. (2003); in this work we assume H₀ = 71 km s^-1 Mpc^-1 and Ω_Λ = 0.73.

Equation (51) can be compared with the following virial mass estimator proposed by Chernin et al. (2012) which includes a cosmological constant term: $\mathrm{\left\{}{\mathit{M}}_{\mathrm{C}}\mathrm{\right\}}\mathrm{=}\frac{{\mathit{v}}^{\mathrm{2}}\mathit{r}}{\mathit{G}}\left[\mathrm{1}\mathrm{+}\frac{\mathrm{8}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{3}{\mathit{c}}^{\mathrm{2}}}{\left(\frac{\mathit{r}}{\mathit{v}}\right)}^{\mathrm{2}}\right]\mathrm{=}\frac{{\mathit{v}}^{\mathrm{2}}\mathit{r}}{\mathit{G}}\left(\mathrm{1}\mathrm{+}{\mathit{f}}_{\mathrm{C}}\right)\mathit{.}$(54)In Eq. (54), r and v are the characteristic size and velocity of the system. For groups of galaxies, these authors took r ≈ 1 Mpc and v ≈ 70 km s^-1, so f_C ≈ 0.75. For clusters of galaxies, they considered r ≈ 5 Mpc and v ≈ 1000 km s^-1 leading to f_C ≈ 0.09. Hence, according to Chernin et al. (2012), the contribution of the cosmological constant term is negligible in the estimation of virial masses of galaxy clusters, but considerable in galaxy groups.

It is obvious that when observational measurements are used in mass estimators that do not include Λ-terms, the masses derived are not those in a Λ = 0 Universe. Hence, the contribution of Λ-terms in mass estimators should not be confused with the effect of the cosmological constant. To estimate the actual effect of Λ, it would be necessary to know positions and velocities of galaxies in a Λ = 0 Universe (such data cannot be obtained from observational measurements, and only simulations of structure formation or infall models in a Λ = 0 Universe could provide that information).

We evaluated the surface term S(R_s) at the distance of the farthest galaxy of the sample from the centre of the massive dominant galaxy; i.e. at ${\mathit{R}}_{\mathrm{s}}\mathrm{=}\begin{array}{c}\max \end{array}\left\{\mathrm{\left|}{{r}}_{\mathrm{1}}\mathrm{\right|}\mathit{,}\mathit{...}\mathit{,}\mathrm{\left|}{{r}}_{\mathit{N}}\mathrm{\right|}\right]\mathit{.}$(55)Equation (55) has been used, for example, by The & White (1986) to evaluate the surface term used in the estimation of the virial mass of the Coma cluster.

It should be noted that R_s is not the radius of the sphere containing the virialized mass of the sample. An estimation of this type of radius would require knowing the galaxy positions of the sample in time; thus, steady magnitudes could be obtained by a time average. As this is not possible, distances averaged in the sample at present, such as the harmonic radius or the averaged distance between the central galaxy and its companions, would give information about this quantity.

3.2. Projected mass estimator

For the N galaxies of the sample addressed in this Sect. 3, the equivalent equation to (28), derived from the continuous model, is $\begin{array}{ccc}{⟨{\mathit{R}}_{\mathrm{s}}{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}⟩}_{\mathit{t}}& \mathrm{=}& \frac{\mathrm{32}}{\mathit{\pi }}\frac{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}\mathrm{[}{\mathit{v}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{z}}^{\mathrm{2}}⟩}_{\mathit{t}}\\ & & \mathrm{-}\mathit{G}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}}⟩}_{\mathit{t}}\mathrm{+}\frac{\mathrm{128}\mathit{G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{9}{\mathit{c}}^{\mathrm{2}}}{⟨\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathit{i}}\mathrm{\right[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}^{\mathrm{3}}⟩}_{\mathit{t}}\mathit{.}\end{array}$(56)Thus, by assuming Eqs. (34)–(36), and taking into account that r_{i = 1} = 0 and [ℳ]₁ = 0, Eq. (56) can be read as $\begin{array}{ccc}{\mathit{R}}_{\mathrm{s}}{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{32}}{\mathit{\pi }}\frac{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}\\ & & \mathrm{-}\mathit{GM}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\left[}\mathit{\alpha }\mathrm{\right]}\mathrm{+}\frac{\mathrm{128}\mathit{G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{9}{\mathit{c}}^{\mathrm{2}}}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\mathrm{\right]}\mathit{,}\end{array}$(57)where $\begin{array}{ccc}\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}& \mathrm{\approx }& \mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]} \\ & \mathrm{=}& \frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}{\left[\mathrm{[}{\mathit{V}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{los}}\mathrm{-}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{V}}_{\mathit{j}}{\mathrm{]}}_{\mathrm{los}}\right]}^{\mathrm{2}}\mathit{,}\\ \left[\mathit{\alpha }\right]& \mathrm{=}& \frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\frac{\mathrm{[}\mathrm{ℳ}{\mathrm{]}}_{\mathit{i}}}{\mathit{M}}\mathit{,}\\ \left[{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\right]& \mathrm{=}& \frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}^{\mathrm{3}}\mathit{.}\end{array}$As can be seen, the kinetic term of Eq. (56) has been approximated by the kinetic term of Eq. (57). Besides, in Eq. (57) we have assumed $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}\mathrm{\approx }\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}$; it should be noted that both quantities are not equal, but coincide in the isotropy case (see Eqs. (25) and (26)).

From Eq. (59), the quantity M [α] is the average of [ℳ]_i in the N − 1 companions to the main galaxy. Thus, for models PM, MG, and EM, we obtain (see Eqs. (42)–(46)) $\begin{array}{ccc}\left[\mathit{\alpha }\right]& \mathrm{\equiv }& \mathrm{[}\mathit{\alpha }{\mathrm{]}}_{\mathrm{PM}}\mathrm{=}\mathrm{1}\mathit{,}\\ \left[\mathit{\alpha }\right]& \mathrm{\equiv }& \mathrm{[}\mathit{\alpha }{\mathrm{]}}_{\mathrm{MG}}\mathrm{=}\frac{\mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{MG}}\mathrm{\right]}}{\mathit{M}}\mathrm{=}\left[\frac{\mathrm{[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}}{{\mathit{M}}_{\mathrm{g}}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{2}\mathrm{)}}{\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\frac{{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{{\mathit{M}}_{\mathrm{g}}}\right]\mathit{,}\\ \left[\mathit{\alpha }\right]& \mathrm{\equiv }& \mathrm{[}\mathit{\alpha }{\mathrm{]}}_{\mathrm{EM}}\mathrm{=}\frac{\mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{EM}}\mathrm{\right]}}{\mathit{M}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{·}\end{array}$As can be seen, when the main galaxy dominates the group mass, [α]_MG tends to [α]_PM; and if all galaxies have the same mass, [α]_MG = [α]_EM. When Eq. (61) is used, the gravitational term of Eq. (57) is that derived for N − 1 galaxies orbiting around a point mass. In the case of using Eq. (63), this gravitational term corresponds to N galaxies with the same mass.

If the surface term of Eq. (57) is neglected, the projected mass estimator is obtained: $\mathrm{\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{=}\mathrm{[}{\mathit{M}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{)}\mathit{,}$(64)where $\begin{array}{ccc}[{\mathit{M}}_{\mathrm{proj}}\mathrm{]}\beta & \mathrm{=}& \frac{\mathrm{32}}{\mathit{\pi G}}\frac{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}}{\mathrm{\left[}\mathit{\alpha }\mathrm{\right]}}\mathit{,}\\ [{\mathit{f}}_{\mathrm{proj}}\mathrm{]}\beta & \mathrm{=}& \frac{\mathrm{4}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{\Lambda }}}{\mathrm{9}{\mathit{c}}^{\mathrm{2}}}\frac{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\mathrm{\right]}}{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{6}}{\mathit{H}}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Omega }}_{\mathrm{\Lambda }}\frac{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\mathrm{\right]}}{\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}}\mathrm{·}\end{array}$Projected mass estimations grow with the anisotropic parameter β, while the contributions of cosmological constant terms decrease with β. This can be seen from Eqs. (64)–(66); i.e. $\begin{array}{ccc}\{{\mathit{M}}_{\mathrm{proj}}\mathrm{\}}\beta & \mathrm{=}& \left[\mathrm{3}\frac{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }\mathrm{=}\mathrm{-}\mathrm{\infty }}\right]\left[\frac{\mathrm{\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }\mathrm{=}\mathrm{-}\mathrm{\infty }}}{\mathrm{1}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }\mathrm{=}\mathrm{-}\mathrm{\infty }}}\right]\mathit{,}\\ [{\mathit{f}}_{\mathrm{proj}}\mathrm{]}\beta & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{3}}\frac{\mathrm{(}\mathrm{4}\mathrm{-}\mathrm{3}\mathit{\beta }\mathrm{)}}{\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }\mathrm{)}}\mathrm{[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }\mathrm{=}\mathrm{-}\mathrm{\infty }}\mathit{.}\end{array}$The leader term of the mass estimator (64), [M_proj]_β, depends on $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{z}}^{\mathrm{2}}\mathrm{\right]}$. This kind of dependence was proposed by Page (1952) for estimating the mass of binary galaxies. The Page estimator was based on the mean value of the projected mass and on the assumption of circular orbits (β → −∞) for galaxies. Wolf & Bahcall (1972) also derived a similar expression assuming straight-line orbits (β = 1). Bahcall & Tremaine (1981) derived another mass estimator also based on projected mass that depended on the mean value of the square of the Keplerian orbit eccentricities, e; their estimator, which does not take into account the cosmological constant contribution, is that of Eq. (65) with ${}^{\mathrm{⟨}}\mathit{e}^{\mathrm{2}}{}^{\mathrm{⟩}}\mathrm{=}\mathrm{(}\mathrm{2}\mathrm{-}\mathit{\beta }{\mathrm{)}}^{-1}$. The mass estimator by Bahcall & Tremaine (1981) has been used, as an example, by Karachentsev (2005) to estimate the mass of the nearest groups of galaxies; he adopted ${{}^{\mathrm{⟨}}\mathit{e}^{\mathrm{2}}{}^{\mathrm{⟩}}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{=}\mathrm{0.7}$ as the average eccentricity. This value is close to that for isotropic orbits, β = 0. Heisler et al. (1985) showed that when the distribution of orbit eccentricities is unknown, the assumption of isotropic orbits resulted in a good agreement with the numerical experiments that they carried out.

3.3. Sample mass estimator

Using Eqs. (51)–(53) in (37), and Eqs. (64)–(66) in (57), the equations for the mass contained in the sample field or sample mass, M ≡ {M_s}_β, can be derived. Thus, $\begin{array}{ccc}{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}& \mathrm{=}& \frac{\mathit{G}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{\mathrm{\left[}{\mathit{R}}_{\mathrm{H}}\mathrm{\right]}}\mathrm{[}\mathrm{\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\}}\mathrm{-}\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{]}\mathit{,}\\ {\mathit{R}}_{\mathrm{s}}{\mathit{S}}_{\mathrm{vir}}\mathrm{\left(}{\mathit{R}}_{\mathrm{s}}\mathrm{\right)}& \mathrm{=}& \mathit{G}{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}\mathrm{\left[}\mathit{\alpha }\mathrm{\right]}\mathrm{[}\mathrm{\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{-}\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{]}\mathit{.}\end{array}$As in this work, galaxy groups are assumed to be isolated, the surface terms of Eqs. (69) and (70) must be greater than or equal to zero. Hence, $\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{\le }\begin{array}{c}\min \end{array}\mathrm{\left[}\mathrm{\right\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\left\}}\mathit{,}\mathrm{\right\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{]}\mathit{.}$(71)Besides, the sample mass must be greater than or equal to the total mass of the N galaxies; i.e. ${\mathit{M}}_{\mathrm{g}}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathit{.}$(72)The mass estimator for the sample, {M_s}_β, can be obtained from Eqs. (69) and (70). Thus, $\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{=}\frac{\{{\mathit{M}}_{\mathrm{vir}}\mathrm{\}}\mathrm{-}\mathrm{[}\mathit{\gamma }\mathrm{\left]}\mathrm{\right\{}{\mathit{M}}_{\mathrm{proj}}\mathrm{\}}\beta }{\mathrm{1}\mathrm{-}\mathrm{\left[}\mathit{\gamma }\mathrm{\right]}}\mathit{,}$(73)where $\mathrm{\left[}\mathit{\gamma }\mathrm{\right]}\mathrm{=}\mathrm{\left[}\mathit{\alpha }\mathrm{\right]}\frac{\mathrm{\left[}{\mathit{R}}_{\mathrm{H}}\mathrm{\right]}}{{\mathit{R}}_{\mathrm{s}}}\mathrm{·}$(74)When [γ] < 1, the smallest value of {M_s}_β is derived for β = 1 (the greatest value of {M_proj}_β is reached for radial orbits; see Eq. (67)). If M_g ≤ {M_s}_{β = 1}, the upper limit for the anisotropy parameter is β_M = 1. But when M_g> {M_s}_{β = 1}, the upper limit β_M< 1 is imposed by M_g = {M_s}_βM.

Following with the case [γ] < 1, the greatest {M_s}_β appears for β = −∞ (circular orbits lead to the smallest value of {M_proj}_β; see Eq. (67)). When {M_s}_{β = −∞} ≤ min^[ {M_vir} , {M_proj}_{β = −∞}^], β_m = −∞ is the lower limit for β. However, if it happens that {M_s}_{β = −∞}> min^[ {M_vir} , {M_proj}_{β = −∞}^], the surface terms of Eqs. (69) and (70) are negative; then, β_m is calculated from the condition {M_s}_βm = {M_vir} = {M_proj}_βm.

Hence, if [γ] < 1 is fulfilled by a galaxy group, ${\mathit{M}}_{\mathrm{g}}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\}}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }}\mathit{,} {\mathit{\beta }}_{\mathrm{m}}\mathrm{\le }\mathit{\beta }\mathrm{\le }{\mathit{\beta }}_{\mathrm{M}}\mathit{.}$(75)In the case of [γ] > 1, β_M< 1 appears when {M_s}_{β = 1}> min^[ {M_vir} , {M_proj}_{β = 1}^] (the greater value of {M_s}_β is now for radial orbits): then, β_M derives from the condition {M_s}_βM = {M_vir} = {M_proj}_βM. This means that for the allowed values of β, {M_vir} ≥ {M_proj}_β. Finally, β_m ≠ −∞ if {M_s}_{β = −∞}<M_g (circular orbits lead to the smaller value of {M_s}_β); so, M_g = {M_s}_βm imposes the lower limit for β. Hence, for a [γ] > 1 galaxy group, ${\mathit{M}}_{\mathrm{g}}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{\le }\mathrm{\{}{\mathit{M}}_{\mathrm{proj}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{\le }\mathrm{\left\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\right\}}\mathit{,} {\mathit{\beta }}_{\mathrm{m}}\mathrm{\le }\mathit{\beta }\mathrm{\le }{\mathit{\beta }}_{\mathrm{M}}\mathit{.}$(76)We know that the extension of the sample coincides with that of the isolated system when the surface terms are zero. Hence, this occurs for γ< 1 galaxy groups whose anisotropy parameter is β_m ≠ −∞. Such a situation also appears in a galaxy group fulfilling γ> 1 if β = β_M< 1.

Finally, the contribution of cosmological constant terms can be determined from $\mathrm{\{}{\mathit{M}}_{\mathrm{s}}{\mathrm{\}}}_{\mathit{\beta }}\mathrm{=}\mathrm{[}{\mathit{M}}_{\mathrm{s}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{s}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{)}\mathit{,}$(77)where [M_s]_β is the sample mass neglecting cosmological constant terms. [M_s]_β fulfills a similar equation to (73), but where {M_vir} and {M_proj}_β are substituted by [M_vir] and [M_proj]_β. From Eqs. (73), (77), (51), and (64), [f_s]_β can be expressed as $\mathrm{[}{\mathit{f}}_{\mathrm{s}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{=}\frac{\mathrm{\left[}{\mathit{f}}_{\mathrm{vir}}\mathrm{\right]}\mathrm{-}\mathrm{\left[}\mathit{\eta }{\mathrm{]}}_{\mathit{\beta }}\mathrm{\right[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }}}{\mathrm{1}\mathrm{-}\mathrm{[}\mathit{\eta }{\mathrm{]}}_{\mathit{\beta }}}\mathit{,}$(78)where $\begin{array}{ccc}\mathrm{[}\mathit{\eta }{\mathrm{]}}_{\mathit{\beta }}\mathrm{=}\mathrm{[}\mathit{\gamma }\mathrm{]}\frac{\{{\mathit{M}}_{\mathrm{proj}}\mathrm{\}}\beta }{\mathrm{\{}{\mathit{M}}_{\mathrm{vir}}\mathrm{\}}}\frac{\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{vir}}\mathrm{\left]}\mathrm{\right)}}{\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{[}{\mathit{f}}_{\mathrm{proj}}{\mathrm{]}}_{\mathit{\beta }}\mathrm{)}}\mathrm{·}& & \end{array}$(79)

4. Mass estimations of some neighbouring groups of galaxies

In this section, the estimators of mass presented in Sect. 3 are applied to the galaxy groups studied by Karachentsev (2005): the Milky Way (MW), M 31, M 81, NGC 2403, Centaurus A, M 83, IC 342, Maffei, Sculptor Filament & Canes Venatici I Cloud (physical quantities have been taken from Karachentsev et al. (2004). All these groups have a main galaxy; this allows us to use the mass estimators when taking the main galaxy as the centre of the group. Hereafter, indicative masses of galaxies appearing in Karachentsev et al. (2004) are taken as galaxy masses.

Different sample mass models are explored. Each one of these models is characterized by two parameters: [R_H] (appearing in the virial mass estimator; see Eq. (51)), and [α] (used in projected mass estimations; see Eq. (64)). In the following, we deal with [R_H]_PM, [R_H]_MG, [R_H]_EM and [R_H]_K given by Eqs. (47)–(50); and with [α]_MG and [α]_EM from Eqs. (62), (63).

Some characteristics of the groups are shown in Table 1. The following data are presented in the rows in the table:

• 1.

  Number of members in the group, N. All members fulfill that their Keplerian cyclic periods are smaller than the timescale of the Universe at present (see Karachentsev 2005).

• 2.

  Sum of indicative masses of galaxies in the group, ${\mathit{M}}_{\mathrm{g}}^{\mathit{\star }}$. Each galaxy indicative mass derives from values of HI rotation velocities and major linear diameters (see Karachentsev et al. 2004). Hereafter, upper index “⋆” is used to indicate indicative mass.

• 3.

  Indicative mass of the main galaxy, $\mathrm{[}{\mathit{m}}_{\mathrm{g}}^{\mathit{\star }}{\mathrm{]}}_{\mathrm{1}}$. As can be seen, contributions of $\mathrm{[}{\mathit{m}}_{\mathrm{g}}^{\mathit{\star }}{\mathrm{]}}_{\mathrm{1}}$ to ${\mathit{M}}_{\mathrm{g}}^{\mathit{\star }}$ are 63% for CVn I, and around 80% for Maffei and Sculptor; from 88% to 93% for Centaurus A, NGC 2403, M 81, and M 81; while for the Milky Way and M 83, the mass contributions of main galaxies are greater than 96%.

• 4.

  Dispersion of the line-of-sight velocities in the group, $\mathrm{[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}{\mathrm{]}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ (see Eq. (38)). See the small value taken by this magnitude in NGC 2403.

• 5.

  Distance of the farthest galaxy from the central massive galaxy of the group, R_s.

• 6.

  Mean separation among the dominant galaxy and companions to it, [R]. This is a measure of the size of the group calculated from the mean projected separation of galaxies from the principal galaxy, [R_⊥], i.e.$\mathrm{\left[}\mathit{R}\mathrm{\right]}\mathrm{=}\frac{\mathrm{4}}{\mathit{\pi }}\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}\mathrm{\right]}\mathit{,}$(80)where $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}\mathrm{\right]}\mathrm{=}\left[\frac{\mathrm{1}}{\mathit{N}\mathrm{-}\mathrm{1}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{r}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{\perp }}\right]\mathit{.}$(81)(Equation (80) has been derived from the continuous model.)

• 7.

  Mean projected separation of galaxies from the principal galaxy, [R_⊥] (see Eq. (81)).

• 8.

  Mean square projected separation of galaxies from the principal galaxy, $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}$ (see Eq. (40)).

• 9.

  Mean cube projected separation of galaxies from the principal galaxy, $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\mathrm{\right]}$ (see Eq. (60)).

• 10.

  $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}$ (see Eq. (58)). We note that this quantity is much smaller for NGC 2403 and Maffei than for the rest of the galaxy groups.

• 11.

  Mean harmonic radius assuming that N − 1 galaxies orbit around a point mass, [R_H]_PM (see Eq. (47)).

• 12.

  Mean harmonic radius given by Eq. (48), [R_H]_MG. The difference between [R_H]_MG and [R_H]_PM for a galaxy group is a consequence of the influence of the companions to the main galaxy. The greatest difference is found in the CVn I Cloud: [R_H]_MG differs from [R_H]_PM by about 26%. For the Maffei group and for Sculptor Filament, it is 13%. For the rest of the groups, however, it is smaller than 7%. We note that NGC 2403 and CVn I show a [R_H]_MG greater than R_s, and that for the Sculptor group they are comparable.

• 13.

  Mean harmonic radius [R_H]_EM. This is twice [R_H]_PM (see Eq. (49)).

• 14.

  Mean harmonic radius in agreement with Karachentsev (2005), [R_H]_K (see Eq. (50)).

• 15.

  Adimensional quantity, [α]_MG, given by Eq. (62). This is used in the calculation of {M_proj}_β and {M_s}_β. This quantity is an average of the fraction of mass inside a sphere covering up to the location of each galaxy. When mass contributions of companions to the main galaxy are negligible, [α]_MG = 1.

• 16.

  Quotient between crossing time, t_cross and timescale of the Universe, 1 /H₀: $\mathrm{[}{\mathit{t}}_{\mathrm{cross}}\hspace{0.17em}{\mathit{H}}_{\mathrm{0}}\mathrm{]}\mathrm{\approx }\left(\frac{\mathrm{\left[}\mathit{R}\mathrm{\right]}}{\mathrm{1}\mathrm{Mpc}}\right){\left(\frac{\mathrm{[}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}{\mathrm{]}}^{\mathrm{1}\mathit{/}\mathrm{2}}}{\mathrm{1}\hspace{0.17em}{\mathrm{km}\hspace{0.17em}\mathrm{s}}^{-1}}\right)}^{-1}\left(\frac{{\mathit{H}}_{\mathrm{0}}}{\mathrm{1}\hspace{0.17em}{\mathrm{km}\hspace{0.17em}\mathrm{s}}^{-1}\hspace{0.17em}{\mathrm{Mpc}}^{-1}}\right)\mathrm{·}$(82)NGC 2403 presents a crossing time greater than the timescale of the Universe at present; this would mean that NGC 2403 is not virialized. The groups IC 342, Sculptor, and CVn I, show t_cross about half ${\mathit{H}}_{\mathrm{0}}^{-1}$; so, their virialization is not confirmed. The rest of the groups would be in a dynamic equilibrium.

For the Milky Way, to account for projection effects, [R_H]_PM, [R_⊥], $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}\mathrm{\right]}$, $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}\mathrm{\right]}$ and $\mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}\mathrm{\right]}$ have been calculated as follows: $\begin{array}{ccc}\mathrm{[}{\mathit{R}}_{\mathrm{H}}{\mathrm{]}}_{\mathrm{PM}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}}{\left[\frac{\mathrm{4}}{\mathit{\pi }\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\frac{\mathrm{1}}{\mathrm{\left[}{\mathit{D}}_{\mathit{i}}\mathrm{\right]}}\right]}^{-1}\mathit{,}& & \end{array}$(83)$\begin{array}{ccc}& & \mathrm{\left[}{\mathit{R}}_{\mathrm{\perp }}\mathrm{\right]}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{4}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{\left[}{\mathit{D}}_{\mathit{i}}\mathrm{\right]}\mathit{,}\\ & & \mathrm{[}{{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{2}}}^{\mathrm{]}}\mathrm{=}\frac{\mathrm{2}}{\mathrm{3}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{D}}_{\mathit{i}}{\mathrm{]}}^{\mathrm{2}}\mathit{,}\\ & & \mathrm{[}{{\mathit{R}}_{\mathrm{\perp }}^{\mathrm{3}}}^{\mathrm{]}}\mathrm{=}\frac{\mathrm{3}\mathit{\pi }}{\mathrm{16}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{[}{\mathit{D}}_{\mathit{i}}{\mathrm{]}}^{\mathrm{3}}\mathit{,}\\ & & \mathrm{[}{\mathit{R}}_{\mathrm{\perp }}{{\mathit{\sigma }}_{\mathrm{los}}^{\mathrm{2}}}^{\mathrm{]}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{4}\mathrm{(}\mathit{N}\mathrm{-}\mathrm{1}\mathrm{)}}\sum _{\mathit{i}\mathrm{=}\mathrm{2}}^{\mathit{N}}\mathrm{\left[}{\mathit{D}}_{\mathit{i}}\mathrm{\right]}{\left[\mathrm{[}{\mathit{V}}_{\mathit{i}}{\mathrm{]}}_{\mathrm{los}}\mathrm{-}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{[}{\mathit{V}}_{\mathit{j}}{\mathrm{]}}_{\mathrm{los}}\right]}^{\mathrm{2}}\mathit{,}\end{array}$where [D_i] is the distance to the ith galaxy.

4.1. Selection of harmonic radius model

In this subsection, we explore harmonic radius models. The differences between them come from the assumption made for the mass enclosed by the sphere covering up to the location of each galaxy, [ℳ]_i. One model takes a central point mass, (see [ℳ_PM]_i given by Eq. (42)). Another considers that companions to the main galaxy have the same mass (see [ℳ_MG]_i given by Eq. (43)). The third model assumes the same mass for all galaxies (see [ℳ_EM]_i given by Eq. (44)).

In the discrete model, to introduce another average, these masses have been averaged in the galaxy number of each sample. [ℳ_PM], [ℳ_MG] , and [ℳ_EM] come from averaging in the N − 1 companions to the main galaxy; and [ℳ_K], from averaging in the N galaxies of the sample. Results for harmonic radii, [R_H]_PM, [R_H]_MG, [R_H]_EM, and [R_H]_K (see Eqs. (47)–(50)) are shown in rows 11–14 of Table 1 for different galaxy groups.

In this subsection, we select just one harmonic radius model for each group. For this purpose, we come back to the continuous model that assumes power-like mass distributions. These mass distributions would represent [ℳ]_i in the continuous model. They have a power-exponent, p ≥ 0, and a distribution radius, R. The mass enclosed up to R is the sample mass. We impose that discrete and continuous models lead to the same harmonic radius and to the same mean separation between galaxies and the main galaxy. This allows us to fix p and R. The criterion we use is to choose the mass distribution of those mentioned, whose distribution radius, R, is the closest to the radius at which the farthest galaxy of the sample is located, R_s (see row 5 in Table 1).

For the equal mass galaxy model, we assume $\mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{EM}}\mathrm{\right]}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\mathit{M}{\left(\frac{\mathit{r}}{\mathit{R}}\right)}^{\mathit{p}}\mathrm{·}$(88)The point mass model is derived from Eq. (88), taking p = 0; i.e. $\mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{PM}}\mathrm{\right]}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\mathit{M}\mathit{.}$(89)For the case of companions to the main galaxy of equal mass, we use $\mathrm{\left[}{\mathrm{ℳ}}_{\mathrm{MG}}\mathrm{\right]}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\frac{\mathit{M}}{{\mathit{M}}_{\mathrm{g}}}\mathrm{\left[}\mathrm{\right[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}_{\mathrm{1}}\mathrm{+}\mathrm{\left[}{\mathit{m}}_{\mathrm{g}}{\mathrm{]}}^{\mathit{c}}\mathrm{\right(}\mathit{r}{\mathrm{\left)}}^{\mathrm{\right]}}\mathit{.}$(90)In Eq. (90), [m_g]₁ corresponds to the main galaxy mass of the sample; and [m_g] ^c(r) is the mass distribution of companions to it, given by

$\begin{array}{ccc}& & {\mathrm{[}{\mathit{m}}_{\mathrm{g}}\mathrm{]}}^{\mathit{c}}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\frac{{\mathit{M}}_{\mathrm{g}}^{\mathit{c}}}{{\mathit{R}}^{\mathit{p}}\mathrm{-}{\mathit{R}}_{\mathrm{1}}^{\mathit{p}}}\mathrm{(}{\mathit{r}}^{\mathit{p}}\mathrm{-}{{\mathit{R}}_{\mathrm{1}}^{\mathit{p}}}^{\mathrm{)}}\mathit{,} \mathit{r}\mathrm{\ge }{\mathit{R}}_{\mathrm{1}}\mathit{,}\\ & & {\mathrm{[}{\mathit{m}}_{\mathrm{g}}\mathrm{]}}^{\mathit{c}}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,} \mathrm{0}\mathrm{\le }\mathit{r}\mathit{<}{\mathit{R}}_{\mathrm{1}}\end{array}$(91)