Radial alignment of elliptical galaxies by the tidal force of a cluster of galaxies

Abstract

Unlike the random radial orientation distribution of field elliptical galaxies, galaxies in a cluster are expected to point preferentially towards the centre of the cluster, as a result of the cluster's tidal force on its member galaxies. In this work, an analytic model is formulated to simulate this effect. The deformation time-scale of a galaxy in a cluster is usually much shorter than the time-scale of change of the tidal force; the dynamical process of tidal interaction within the galaxy can thus be ignored. The equilibrium shape of a galaxy is then assumed to be the surface of equipotential that is the sum of the self-gravitational potential of the galaxy and the tidal potential of the cluster at this location. We use a Monte Carlo method to calculate the radial orientation distribution of cluster galaxies, by assuming a Navarro–Frenk–White mass profile for the cluster and the initial ellipticity of field galaxies. The radial angles show a single-peak distribution centred at zero. The Monte Carlo simulations also show that a shift of the reference centre from the real cluster centre weakens the anisotropy of the radial angle distribution. Therefore, the expected radial alignment cannot be revealed if the distribution of spatial position angle is used instead of that of radial angle. The observed radial orientations of elliptical galaxies in cluster Abell 2744 are consistent with the simulated distribution.

galaxies: clusters: general, galaxies: clusters: individual: Abell 2744, galaxies: kinematics and dynamics, galaxies: structure

1 INTRODUCTION

The total mass of a cluster of galaxies is dominated by dark matter, which couples with ordinary matter only through gravity. Therefore, to map the density profile of a cluster, one needs to find how luminous matter in a cluster is related to its gravitational potential. Each way of doing so leads to a method of cluster mass mapping. Assuming some sort of dynamical balance is established, one can model how the spatial distribution and velocity dispersion of galaxies respond to cluster gravitational potential and then develop a technique to determine the mass distribution (Carlberg et al. 1997; Kent & Gunn 1982). If hydrostatic equilibrium is supposed, the density and temperature profiles of hot gas are related to the shape of the gravitational potential well (Fabricant, Lecar & Gorenstein 1980; Xue & Wu 2002). Finding the manner in which light rays are bent by gravity gives birth to the method of gravitational lensing (Kaiser & Squires 1993; Umetsu & Broadhurst 2008). Although many approaches have been made, the results from these are not always in good agreement (Wu et al. 1998; Bartelmann & Steinmetz 1996). Therefore, it is still useful to find other independent probes.

An extended object in a non-uniform gravitational field feels a distorting tidal force. The tidal force from the mass of a cluster can change the shape of its galaxies in such a way that the radial orientation angles (see the upper panel of Fig. 1 for an illustration) of elliptical galaxies in a cluster show an anisotropic distribution towards the cluster centre that is different from the isotropic distribution of field galaxies; this phenomenon is referred to as radial alignment (RA). If RA really exists and is caused by the tidal effect of cluster mass, then a new link between an observable, i.e. the radial angle distribution (RAD) of elliptical galaxies, and the gravitational potential of the cluster can be found.

$Upper panel: illustration of the radial orientation angle. R0 = L0/rs is the dimensionless offset distance, where L0 is the distance between the real mass centre of the cluster and the reference centre used in observations, φ is the angle between the direction of the projected elliptical galaxy's long axis and the vector pointing from the galaxy centre to the reference centre. Lower panel: definition of the coordinate system. $\boldsymbol {V}_{\rm c}$ is the unit vector pointing in the direction of the centre of the cluster, with an angle θc between it and the z axis. lt is the distance from the mass element to vector $\boldsymbol {V}_{\rm c}$; lr is the projection of the mass element's position vector on to $\boldsymbol {V}_{\rm c}$.$

Figure 1.

Upper panel: illustration of the radial orientation angle. R₀ = L₀/r_s is the dimensionless offset distance, where L₀ is the distance between the real mass centre of the cluster and the reference centre used in observations, φ is the angle between the direction of the projected elliptical galaxy's long axis and the vector pointing from the galaxy centre to the reference centre. Lower panel: definition of the coordinate system. |$\boldsymbol {V}_{\rm c}$| is the unit vector pointing in the direction of the centre of the cluster, with an angle θ_c between it and the z axis. l_t is the distance from the mass element to vector |$\boldsymbol {V}_{\rm c}$|⁠; l_r is the projection of the mass element's position vector on to |$\boldsymbol {V}_{\rm c}$|⁠.

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The idea that the cluster tidal force can cause RA was first proposed by Thompson (1976), in order to explain ‘a possible indication that the galaxies are preferentially aligned along the radius vector to the centre of the cluster’ for the Coma cluster (Hawley & Peebles 1975). In recent years, more observational evidence has been found to support this idea (Pereira & Kuhn 2005; Agustsson & Brainerd 2006; Faltenbacher et al. 2007). Theoretically, Ciotti & Dutta (1994) studied the dynamical process of shape changing of an elliptical galaxy under the cluster's tidal field using N-body simulations and showed an alignment tendency between the galaxy major axes and the radial direction. Pereira, Bryan & Gill (2008) found strong RA of dark matter substructures in a cluster halo in N-body simulations. Usami & Fujimoto (1997) studied a gaseous ellipsoid of uniform density orbiting in a logarithmic potential and discussed the application of their result to the alignment of galaxies in rich clusters. They concluded that, inside a critical radius, galaxies would be disrupted by the tidal force of the cluster, whereas beyond the critical radius the major axes of elliptical galaxies are locked in the radial direction. None of the previous works takes the intrinsic ellipticity distribution of galaxies into consideration. In Usami & Fujimoto (1997), all elliptical galaxy intrinsic shapes were set to spherical, which is unlikely in real cases. In our work, the ellipticity distribution of field elliptical galaxies, which can be considered as unaffected by environment, is used as the intrinsic ellipticity distribution.

This work aims to formulate an analytic model to simulate RA quantitatively. In Section 2, we verify qualitatively that the shape-changing time-scale for most galaxies in a cluster is short enough compared with the Keplerian motion period that we can ignore the dynamical process of tidal distortion and assume an equilibrium shape for the galaxy. In Section 3, we calculate the equilibrium shape of an elliptical galaxy analytically. Then projection effects are included and a RAD is generated by a Monte Carlo method. Finally, the observed RADs of the cluster Abell 2744 are compared.

2 QUALITATIVE DISCUSSION

Consider a galaxy in a cluster, where the vector from the galaxy centre to the cluster centre is L. The enclosed mass within radius L is denoted by M(L) and its potential field is Φ_c. The tidal force acting on a mass element of the galaxy is approximately

\begin{equation} \ddot{r}_i\sim -\frac{{\mathrm{\partial} }}{{\mathrm{\partial} } {\hat{r}}_j}\frac{{\mathrm{\partial} } \Phi _{\rm {c}}}{{\mathrm{\partial} } {\hat{r}}_i}\mid _{\hat{r}=L}r_j, \end{equation}

(1)

where r and |$\hat{r}$| are the vectors from the mass element to the galaxy centre and to the centre of the cluster, respectively, i = 1, 2, 3 denote the three Cartesian coordinate components of a vector and the Einstein summation convention is used in this work. We define a tidal potential corresponding to the tidal force, which is

\begin{equation} \Phi _{\rm {T}}=\frac{{\mathrm{\partial} } \Phi _{\rm {c}}}{{\mathrm{\partial} } {\hat{r}}_i}\mid _{\hat{r}=L}r_i. \end{equation}

(2)

The mass distribution of the cluster is treated as spherically symmetric, so its gravity acting on a mass element can be written as

\begin{equation} \Phi _{\rm {c}}=-\frac{GM(\hat{r})}{|\hat{r}|}+\Phi _{\rm {M}}(\hat{r}), \end{equation}

(3)

where |$M(\hat{r})$| denotes the enclosed mass in radius |$\hat{r}$| and |$\Phi _{\rm {M}}(\hat{r})$| denotes the gravity potential caused by the mass gradient. Here, since |$-\nabla \Phi _{\rm {c}}=[({GM(\hat{r})})/{|\hat{r}|^3}]\hat{r}$| (Binney & Tremaine 1987), we obtain

\begin{equation} \frac{G\cdot \nabla M(\hat{r})}{|\hat{r}|}-\nabla \Phi _{\rm {M}}(\hat{r})=0. \end{equation}

(4)

Substituting equations (3) and (4) into equation (2), and neglecting the variation of |$M(\hat{r})$| on a small scale, we obtain

\begin{equation} \Phi _{\rm {T}}=\frac{GM(L)}{L^3}(\frac{1}{2}l_{\rm {t}}^2-l_{\rm {r}}^2), \end{equation}

(5)

where l_r is the component of r along L and l_t is the perpendicular component, both of which are illustrated in Fig. 1. Therefore the tidal force acting on a unit mass is obtained by |$\ddot{r}_{\rm {T}}=-\nabla \Phi _{\rm {T}}$|⁠.

A unit mass element is also subjected to the self-gravity from the galaxy,

\begin{eqnarray} \ddot{r}_{\rm {grav}}&=&\ddot{r}_{\rm {e}}+\ddot{r}_{\rm {dm}}\nonumber\\ &\simeq & \frac{GM_{\rm {G}}}{R_{\rm {G}}^2}+\frac{GM_{\rm {dm}}}{R_{\rm {G}}^2}\nonumber\\ &\simeq & G\pi \rho _{\rm {G}}R_{\rm {G}}+\frac{GM_{\rm {dm}}}{R_{\rm {G}}^2}, \end{eqnarray}

(6)

where |$\ddot{r}_{\rm {e}}$| and |$\ddot{r}_{\rm {dm}}$| are the gravity acting on the unit mass from the luminous and dark matter at the effective radius of a galaxy; M_G and R_G are the mass and effective radius of luminous matter in the galaxy. The luminous matter is assumed to be homogeneous, with density ρ_G ∼ 10⁸ M_⊙. M_dm is the enclosed mass of dark matter within the effective radius. In this work, we only consider tidal distortion within the effective radius. The density profile of the dark matter in an elliptical galaxy is (Robles & Matos 2012)

\begin{equation} \rho _{\rm {dm}}=\frac{\rho _{\rm {c}}}{1+(r/r_{\rm {c}})^2}, \end{equation}

(7)

where |$\rho _{\rm {c}}\sim 0.1\,\mathrm{M}_{{{\odot }}}\,\rm {kpc}^{-3}$| is the central density and r_c ∼ 5 kpc is the scale radius. Therefore the dark mass enclosed within r is

\begin{equation*} M_{\rm {dm}}=4\pi \rho _{\rm {c}}r_{\rm {c}}^3 \left[\frac{r}{r_{\rm {c}}}-{\rm {arctan}}\left(\frac{r}{r_{\rm {c}}}\right)\right]. \end{equation*}

At the effective radius R_G ∼ 10 kpc, |$\ddot{r}_{\rm {e}}\simeq G\pi \rho _{\rm {G}}R_{\rm {G}}\sim 10^9$|⁠, |$\ddot{r}_{\rm {dm}}\simeq GM_{\rm {dm}}/R_{\rm {G}}^2\sim 10^8$|⁠, where πG = 1 is used. Therefore |$\ddot{r}_{\rm {dm}}\ll \ddot{r}_{\rm {e}}$|⁠, |$\ddot{r}_{\rm {grav}}\simeq \ddot{r}_{\rm {e}}$|⁠, implying that the effect of dark matter in an elliptical galaxy can be neglected within the effective radius. This has been comfirmed by de Paolis, Ingrosso & Strafella (1995), who found that dark matter inside the effective radius is negligible with respect to luminous matter.

Define β as the ratio of the tidal force and self-gravity:

\begin{equation} \beta \equiv \frac{|\ddot{r}_{\rm {T}}|}{|\ddot{r}_{\rm {grav}}|}\simeq \frac{|\nabla \Phi _{\rm {T}}|}{|\ddot{r}_{\rm {grav}}|}\le \frac{GM(L)/L^3}{G\pi \rho _{\rm {G}}R_{\rm {G}}}\frac{{\mathrm{\partial} } (r^2)}{{\mathrm{\partial} } r}\mid _{r=R_{\rm {G}}}\simeq \frac{M(L)}{\pi \rho _{\rm {G}}L^3}, \end{equation}

(8)

which reflects the significance of the tidal force compared with self-gravity. In the case of cluster A2744, the mass distribution can be modelled by a Navarro–Frenk–White (NFW) profile with parameter r_s ≈ 160 kpc (Zhang et al. 2006). We take r₂₀₀, i.e. the radius within which the mean density should be 200ρ_c, as the boundary of the cluster and set |$\rho _{\rm {G}}=10^8\,\mathrm{M}_{{{\odot }}}\,\rm {kpc}^{-3}$|⁠, the same order of magnitude as the average density of the Milky Way. We find that galaxies with β > 1 only comprise 0.1 per cent of the volume of the cluster. Thus, for a rich cluster like A2744, the condition β ≪ 1 is satisfied for most galaxies, which means that the tidal force is weak compared with the self-gravity of these galaxies.

Carter & Luminet (1983) calculated the response of a star under a tidal force that is weak compared with its self-gravity. Analogous to a star, a galaxy will have a trace-free quadrupole oscillation, with frequency |$\omega _{\rm {osc}}\simeq \sqrt{\pi G\rho _{\rm {G}}}$|⁠. Therefore the deformation time-scale of a galaxy should be |$\tau _{\rm {D}}\simeq 1/\omega _{\rm {osc}}=1/\sqrt{G\pi \rho _{\rm {G}}}$|⁠. Using equation (8) to substitute πρ_G, we obtain |$\tau _{\rm {D}}\simeq 1/\sqrt{GM(L)/\beta L^3}=\sqrt{\beta }\tau _{\rm {K}}$|⁠, where |$\tau _{\rm {K}}\simeq 1/\sqrt{GM(L)/L^3}$| is the Keplerian time-scale characterizing the rate of change of the tidal force. Since β ≪ 1, we have τ_D ≪ τ_K. It follows that the galaxy will remain close to the stationary equilibrium state determined by the instantaneous value of the tidal potential (Carter & Luminet 1983), where the equilibrium shape of the galaxy will coincide with the isopotential surface of the equilibrium state Φ_eff = Φ_grav + Φ_T.

3 QUANTITATIVE CALCULATION OF THE EFFECT OF TIDAL FORCE ON RAD OF GALAXIES IN A CLUSTER

3.1 Coordinate system

Undoubtedly, an elliptical galaxy with a central luminous ellipsoid and dark matter halo component is a very complex entity and therefore we need to assume some necessary simplifications and assumptions in order to be able to study mathematically the deformation of such a complicated system under the tidal force from a cluster. Since we only consider the tidal distortion within an effective radius r ∼ 10 kpc, hereafter the effect of dark matter is neglected (see Section 2 for details). We assume that the surface of the luminous matter is an oblate ellipsoid (Lambas, Maddox & Loveday 1992) at the effective radius, with ellipticity ϵ before tidal distortion, hence the equation for the shape can be written as

\begin{equation} \frac{x^2+y^2}{\mu ^2}+\frac{z^2}{\nu ^2}=1, \end{equation}

(9)

where x, y and z axes are the principal axes of the ellipsoid and ν and μ are related by the following equation:

\begin{equation} \nu ^2=(1-\epsilon ^2)\mu ^2. \end{equation}

(10)

We require that the vector pointing to the centre of the cluster falls in the plane x–z; thus the coordinate system can be uniquely defined, as long as the centre of the cluster is not on the z axis.

The direction of the cluster centre is denoted as

\begin{equation} \boldsymbol {V}_{\rm {c}}=(\sin \theta _{\rm {c}},0,\cos \theta _{\rm {c}}), \end{equation}

(11)

where θ_c is the angle between the direction of the cluster centre and the z axis. The coordinate system and vector |$\boldsymbol {V}_{\rm {c}}$| are illustrated in the lower panel of Fig. 1.

3.2 Deformation of a single elliptical galaxy

In order to simplify the calculation, according to the homoeoid theorem (Binney & Tremaine 1987), we assume that the isopotential surface of the gravity potential of luminous matter at the effective radius coincides with the local surface of the luminous ellipsoid. Then the self-gravity potential of an elliptical galaxy at the effective radius before tidal distortion can be written as

\begin{equation} \Phi _{\rm {grav}}=A_1(x^2+y^2)+A_3z^2, \end{equation}

(12)

where A₁ = (1 − ϵ²)A₃. It is easy to find A₃ ≃ Gπρ_G.

The contribution of tidal force is equivalent to adding a tidal potential Φ_T to the self-gravity potential Φ_grav; here we take

\begin{equation} \Phi _{\rm {eff}}=\Phi _{\rm {grav}}+\Phi _{\rm {T}}. \end{equation}

(13)

A little calculation gives

\begin{eqnarray} \Phi _{\rm {T}}&=&-\frac{B}{4}\Bigl [(1-3\cos 2\theta _{\rm {c}})x^2-2y^2\nonumber \\ &&{}+(1+3\cos 2\theta _{\rm {c}})z^2+6xz\sin 2\theta _{\rm {c}}\Bigr ], \end{eqnarray}

(14)

where B = GM(L)/L³. Hence an explicit expression for Φ_eff is given by

\begin{equation} \Phi _{\rm {eff}}=ax^2+by^2+cz^2+dxy+exz+fyz, \end{equation}

(15)

where

\begin{eqnarray*} a&=&A_1-\frac{B}{4}(1-3\cos 2\theta _{\rm c}),\\ b&=&A_1+\frac{B}{2},\\ c&=&A_3-\frac{B}{4}(1+3\cos 2\theta _{\rm c}),\\ e&=&-\frac{3B}{2}\sin 2\theta _{\rm c} \end{eqnarray*}

and

\begin{eqnarray*} d=&f=0. \end{eqnarray*}

Therefore, the final shape of the distorted galaxy is fully described, given the initial ϵ, |$\boldsymbol {V}_{\rm {c}}$| and ρ_G, together with the cluster mass profile M(L), which is taken as the NFW profile (Navarro, Frenk & White 1996):

\begin{equation} M(r)=4\pi \rho _{\rm {0}}r^3_{\rm {s}}\left[\ln (1+r/r_{\rm {s}})-\frac{r/r_{\rm {s}}}{1+r/r_{\rm {s}}}\right], \end{equation}

(16)

where ρ₀ and the ‘scale radius’ r_s are parameters of the distribution. Define δ₀ ≡ ρ₀/ρ_c, where ρ_c is the critical density, defined as (3H²)/(8πG) where H is the Hubble constant and G is the gravitational constant. Then

\begin{equation} \delta _0=\frac{200}{3}\frac{c^3}{\ln (1+c)-c/(1+c)}, \end{equation}

(17)

where c is the so-called ‘concentration parameter’, defined as c = r₂₀₀/r_s where r₂₀₀ is the radius within which the mean density should be 200 × ρ_c (Navarro et al. 1996). Substituting equation (16) into the expression for B, we obtain

\begin{equation} B=4\pi G\rho _0r^{\prime -3}\left[\ln (1+r^{\prime })-\frac{r^{\prime }}{1+r^{\prime }}\right], \end{equation}

(18)

where r^′ = r/r_s is the dimensionless radius. Given c and r^′, B can be obtained from equations (17) and (18). Analogously, A₁, A₃ and all the parameters from a to f of the ellipsoid can be calculated.

3.3 Projection effects

From the view of an observer, the ellipsoidal shape of a galaxy is projected to become an ellipse. The next step is to determine the major axis of the projected ellipse and evaluate the angle between that and the direction towards the cluster centre. The observer is assumed to be in the direction |$\boldsymbol {V}_{\rm {o}}$|⁠, which can be expressed as

\begin{equation} \boldsymbol {V}_{\rm {o}}=(\sin \theta _{\rm {o}}\cos \phi _{\rm {o}},\sin \theta _{\rm {o}}\sin \phi _{\rm {o}},\cos \theta _{\rm {o}}). \end{equation}

(19)

The problem is then reduced to a constrained extremum value problem: to find the farthest point |$\boldsymbol {X}$| from the vector |$\boldsymbol {V}_{\rm {o}}$| on the surface described by

\begin{equation} {\rm Const}=ax^{{2}}+by^{{2}}+cz^{{2}}+dxy+exz+fyz, \end{equation}

(20)

the projection of |$\boldsymbol {X}$| on the celestial sphere, |$\boldsymbol {X}_{\rm {p}}$|⁠, is then the direction of the major axis. Defining |$\boldsymbol {V}_{\rm {cp}}$| as the projection of vector |$\boldsymbol {V}_{\rm {c}}$| on the celestial sphere, we obtain the radial orientation angle

\begin{equation} \varphi =\arccos \frac{\boldsymbol {V}_{\rm {cp}}\cdot \boldsymbol {X}_{\rm {p}}}{\mid \boldsymbol {V}_{\rm {cp}}\mid \mid \boldsymbol {X}_{\rm {p}}\mid }. \end{equation}

(21)

The details of solving the constrained extremum value problem are presented in the Appendix.

3.4 RAD of elliptical galaxies

In order to derive the RAD of elliptical galaxies in a cluster, we use Monte Carlo simulations. In each run, a set of elliptical galaxies is generated, each of which is given a set of parameters sampled with certain probability distributions, as discussed below. With M(r) given, each galaxy from the set results in a φ_i (i = 1, 2...N). c = 10 is used to simulate a cluster with high concentration like Abell 2744 (see section 4.1); note that c = 10 is also the concentration of the Coma cluster (Rines & Diaferio 2006), where RA was found for the first time. The effective density of each galaxy is set to be 10⁸ times ρ_c. In this run, 100 000 galaxies are generated, with the intrinsic ellipticity distribution shown in Fig. 2, which displays the ellipticity distribution of field galaxies (Fasano & Vio 1991). Here, the ellipticity of a generated ellipsoid is defined as ϵ ≡ (l_max − l_min)/l_max, where l_max and l_min are the major and minor axes of the ellipsoid, respectively. The dimensionless projected distance of galaxies is sampled with a probability inversely proportional to |$r^{\prime 2}_{\rm p}$|⁠, from |$r^{\prime }_{\rm p}=2.5$| to |$r^{\prime }_{\rm p}=5$|⁠, corresponding to an approximation to the projected galaxy number density distribution in Baier (1976). Under such parameters, the maximum value of β is 0.0057, which is far smaller than unity. For each galaxy, the direction towards the cluster centre |$\boldsymbol {V}_{\rm {c}}$| and the direction towards the observer |$\boldsymbol {V}_{\rm {o}}$| in the principal axes coordinate system are sampled with an isotropic probability, corresponding to an isotropic position distribution, which is assumed as the initial configuration of galaxies in the cluster.

Figure 2.

The intrinsic (dashed line) and distorted (solid line) ellipticity distributions of galaxies.

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Fig. 3 shows RADs for the different centre offset distances R₀, where R₀ ≡ L₀/r_s is defined as the dimensionless offset distance and L₀ is the distance between the real mass centre of the cluster and the reference centre used in observations. When R₀ = 0, the upper left panel in Fig. 3 shows a single peak. Therefore, if the reference centre is the real cluster centre, the major axes of elliptical galaxies tend to point to the cluster centre, i.e. we can see an obvious radial alignment of elliptical galaxies. However, if the reference centre has an offset distance from the real cluster centre, since the major axes tend to point to the real centre rather than the reference centre, two peaks are found in RADs. As the centre deviation distance increases, the two peaks depart further from each other with increasing R₀. Then the two peaks merge at −90/90° as R₀ keeps increasing. Finally, the RAD becomes a uniform distribution if R₀ ≫ 0; at this moment, the radial angle distribution becomes the position angle distribution.

RADs for different centre offset distances R0.

Figure 3.

RADs for different centre offset distances R₀.

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We also plot the distorted ellipticity distribution of ellipsoids after the simulation in Fig. 2. The width of the ellipticity distribution is broadened by about Δw = w₂ − w₁ ≃ 0.025, where w₁ and w₂ are the full widths at half-maximum (FWHM) of the ellipticity distributions before and after the simulation, respectively; meanwhile, the ellipticity distribution is shifted to higher values compared with the intrinsic one.

In order to reveal further the effect that occurs when the centre of the cluster is mis-identified observationally, we define the ‘amplitude’ of the RAD as |$A_{\rm {rad}}=({N_{\rm {max}}-N_{\rm {min}}})/{\bar{N}}$|⁠, where N_max, N_min and |$\bar{N}$| are the highest point, lowest point and mean value of the RAD. A_rad intuitively suggests the significance of a peak-type distribution deviating from uniform distribution, i.e. A_rad ≃ 0 when the distribution is similar to a uniform distribution, while A_rad ≫ 0 if there are significant peaks in the distribution. Fig. 4 shows A_rad as a function of the offset distance from the reference centre to the true centre of the cluster mass distribution. On the whole, A_rad decreases with increasing R₀. For R₀ ≫ 1, radial angle is equivalent to position angle. The result in Fig. 4 explains why it is difficult to find RA with only the position angle distribution of elliptical galaxies in a cluster.

Figure 4.

Amplitude of RAD versus R₀.

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4 OBSERVED RAD OF ABELL 2744

4.1 Selection of elliptical galaxies

Abell 2744 is one of the richest clusters of galaxies at intermediate redshift z ∼ 0.308, known as a gravitational lens (Merten et al. 2011). Its parameters are estimated as r_s = 160 kpc (Zhang et al. 2006), r₂₀₀ = 2.506 Mpc (Demarco et al. 2003), hence the concentration parameter c = r₂₀₀/r_s ≃ 15.7. Such a high mass concentration is expected to show some signature in its RAD. We thus choose this cluster for comparison with our theoretical work. The coordinate system is centred at |$\alpha \, ({\rm {J2000.0}}) = 00^{\rm {h}}14^{\rm {m}}21 {.\!\!\!\!^{{\mathrm {s}}}}04$|⁠, |$\delta \, ({\rm {J2000.0}}) = -30^{\circ }23^{\rm {m}}52 {.\!\!\!\!^{{\mathrm{s}}}}4$| (Boschin et al. 2012). However, note that this cluster is an actively merging cluster (Owers et al. 2011); therefore, the cluster central position cannot be determined unambiguously (Zhang et al. 2006), which may bias the RAD.

We retrieve Hubble Space Telescope (HST) Advanced Camera for Surveys/Wide-Field Camera (ACS/WFC) images in the F606W and F814W bands of two fields in the immediate vicinities of the cluster centre (date: 2009 October 27, PID: 11689, PI: Dupke). The associated images of A2744 are fully processed and drizzled. Photometry for objects is carried out using the SExtractor package (Bertin & Arnouts 1996). The configuration parameters are listed in Table 1 and the magnitudes of the sources in the cluster are calibrated by the MAG_ZEROPOINT parameter in the AB system. The derived source catalogues in the two bands are matched to obtain 3005 true objects. The parameter Class/Star >0.9 is used to get rid of stars and then the removed stars are inspected visually in the images. Arcs and arclets, spurious objects, sources in the margins of images and sources inside other giant bright sources are identified visually and removed. The optical magnitudes of targets are corrected for foreground extinction from the Galaxy according to the Schlegel–Finkbeiner–Davis Galactic reddening map (Schlafly & Finkbeiner 2011).

Table 1.

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SExtractor configuration parameters for Abell 2744.

Parameters	Values
DETECT_MINAREA	32
DETECT_THRESH	1.3
DEBLEND_NTHRESH	64
DEBLEND_MINCONT	0.005
CLEAN_PARAM	1.2
BACK_SIZE	40
BACK_FILTERSIZE	3
BACKPHOTO_THICK	24

Table 1.

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SExtractor configuration parameters for Abell 2744.

Parameters	Values
DETECT_MINAREA	32
DETECT_THRESH	1.3
DEBLEND_NTHRESH	64
DEBLEND_MINCONT	0.005
CLEAN_PARAM	1.2
BACK_SIZE	40
BACK_FILTERSIZE	3
BACKPHOTO_THICK	24

Limited by the ACS field size, the measurement reaches out to a distance of 1.28 Mpc from the cluster centre, which is about eight times r_s. These objects include both foreground and background galaxies. In order to select the real cluster members, a colour–magnitude diagram (CMD) is plotted in Fig. 5 using the magnitudes in the two filter bands, i.e. colour = m₆₀₆ − m₈₁₄, mag = m₆₀₆. Since the colours of elliptical galaxies are redder than those of spiral galaxies (Baldry & Glazebrook 2005) and the Butcher–Oemler condition is satisfied, i.e. the blue galaxies are those at least 0.2 mag bluer than the cluster ridgeline (Butcher & Oemler 1984), a linear fitting, i.e.

\begin{equation} {colour}=k\times mag+d, \end{equation}

(22)

where k and d are the slope and intercept respectively, is applied to obtain the slope of the red sequence (Brow, Lucey & Ellis 1992a,b) in the CMD, which is known as the collection of E/S0 galaxies. Here k ≃ −0.037, d ≃ 1.610 are obtained; therefore we select the E/S0 galaxies in the colour range |${colour}\sim -0.037m_{814}+1.610_{-0.2}^{+0.2}\,\rm {mag}$|⁠.A faint end at about m₈₁₄ ∼ 23.0 mag is used to divide the member elliptical galaxies from the background galaxies (Romano et al. 2010); this is approximately consistent with a rest-frame r-band absolute magnitude of M_r ∼ −17 mag. Finally, 199 elliptical galaxies are obtained. These selected elliptical galaxies are then inspected visually in the images.

$CMD of Abell 2744. The galaxies with ${colour}\sim -0.037m_{814}+1.610_{-0.2}^{+0.2}\,\rm {mag}$ and m814 < 23.0 mag are considered as E/S0 galaxies and denoted by a box (red in the online article).$

Figure 5.

CMD of Abell 2744. The galaxies with |${colour}\sim -0.037m_{814}+1.610_{-0.2}^{+0.2}\,\rm {mag}$| and m₈₁₄ < 23.0 mag are considered as E/S0 galaxies and denoted by a box (red in the online article).

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4.2 RAD of Abell 2744

With SExtractor, we are able to measure the orientations of the detected objects quantitatively. We define the radial angle as φ = 0 when the major axis of a projected ellipsoid points to the cluster centre and φ = ±90° when the major axis is oriented along the tangential direction. By separating the range −90° to +90° into 18 bins, we count the number of members in each bin. In order to obtain the RAD with high significance, we add together the number of galaxies in each bin of φ from the two filter bands. The RADs of all sources of Abell 2744, including foreground and background galaxies, and the RADs of the elliptical galaxies in Abell 2744 are presented in Fig. 6.

Figure 6.

Observed RADs of Abell 2744. The upper panel shows the RAD of all sources; the lower the RAD of elliptical galaxies. The RAD of elliptical galaxies is fitted by a double-Gaussian component plus a Lorentzian component.

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There are apparent tangential peaks centred at about ±90° in the upper panel of Fig. 6, which should be produced by gravitational lensing (e.g. Smail et al. 1997; Merten et al. 2011), since the shapes of background galaxies are stretched tangentially after lensing (Joachim 1997). Furthermore, since the selected elliptical galaxies are contaminated by some background galaxies, RADs in the bottom panel of Fig. 6 also show weak tangential peaks, which are fitted by a double-Gaussian component:

\begin{equation} N=\frac{N_{\rm {G}}}{\sqrt{2\pi }\sigma _{\rm {G}}} {\rm e}^{-(\varphi -\varphi _{\rm {G}})^2/2\sigma _{\rm {G}}^2}+\frac{N_{\rm {G}}}{\sqrt{2\pi }\sigma _{\rm {G}}} {\rm e}^{-(\varphi -\varphi _{\rm {G}}-180^{\circ })^2/2\sigma _{\rm {G}}^2}, \end{equation}

(23)

where φ_G = −90° is fixed. Additionally, a peak is located close to φ = 0, which is fitted by a Lorentzian component:

\begin{equation} N=\frac{N_{\rm {L}}\sigma _{\rm {L}}}{\pi [(\varphi -\varphi _{\rm {L}})^2+{\sigma _{\rm {L}}}^2]}. \end{equation}

(24)

As a comparison, we also fit the central peak with a constant:

\begin{equation} N=N_{\rm {C}}. \end{equation}

(25)

The two different fitting results are listed in Table 2. The relative error in N_L is 20 per cent, suggesting that the central peak really exists and thus the orientations of the elliptical galaxies are not random. The F-test is performed to obtain the probability p that the model DG+L is better than the model DG+C. Given the value of F, |$F=\chi _1^2 \times dof_2/\chi _2^2 \times dof_1$|⁠, where dof₁ = 13, dof₂ = 15 and |$\chi _1^2, \chi _2^2$| are the variances of the two models, respectively; thus we obtain p ∼ 83 per cent.

Table 2.

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Fitting results of the RAD of the elliptical galaxies. DG denotes a double-Gaussian component, L denotes a Lorentzian component and C denotes a constant component. All errors are at 1σ throughout this work.

DG+L

N_L

φ_L

σ_L

N_G

φ_G

σ_G

χ²/dof

8.3(1.7) × 10²

0.17(4.11)

24(7)

3.0(1.0) × 10²

−90(fixed)

22(8)

14.0/13

DG+C

N_C

N_G

φ_G

σ_G

χ²/dof

4.5(0.6)

5.3(5.4) × 10

−90(fixed)

6.8(9.5)

27.8/15

5 CONCLUSION AND DISCUSSION

We estimated qualitatively the shape-changing time-scale of a galaxy in a cluster, which is found to be much shorter than its Keplerian time-scale around the centre of the cluster for most member galaxies. Thus we ignored the dynamical process of shape changing and assumed an equilibrium shape of galaxies. The shape of an elliptical galaxy under the tidal force of a cluster is calculated by assuming that the density contour of the galaxy coincides with the equipotential surface of the sum of the tidal potential and self-gravitational potential of the galaxy. We then used the Monte Carlo method to simulate the RAD of galaxies in a cluster, taking a NFW mass profile of the cluster and the initial ellipticity distribution as that of field galaxies. Monte Carlo simulations also find that the elliptical galaxies in a cluster tend to be radially aligned if the real physical cluster centre is chosen as the reference centre. If the reference centre has an offset distance from the real centre, then the further the reference point away from the cluster centre, the weaker the anisotropy of the RAD. Therefore, in order to find significant RA, the reference centre should be set to the real physical centre of the cluster. In addition, a distribution of position angles equivalent to R₀ ≫ 0 cannot reflect RA. The observed RAD of cluster Abell 2744 is presented, using data from the HST. Comparing the observed RADs in Fig. 6 with the simulated ones in Fig. 3, we conclude that the observed non-uniform RADs are due to the tidal effects of these clusters.

In our model, the gravitational interaction between two nearby galaxies is neglected. We treated the galaxies as if they were moving under a smooth averaged field. In fact, when two galaxies get close, tidal effects from the nearby galaxy will hold weight over the entire cluster. This will increase the randomness of the RAD of the whole cluster, because encounter events are random and the effects are local. The non-spherical symmetry and substructures of the cluster are also not included, since we applied the isotropic NFW mass profile. Those effects will be included in our following works. Our simulations place galaxies in the region where the self-gravity of galaxies is far greater than the tidal force from the cluster, so tidal disruption of galaxies has not been taken into consideration.

In section 2, we assumed that a galaxy has a trace-free quadrupole oscillation under weak tidal distortion like a star (Carter & Luminet 1983), with a period τ_osc ∼ 0.1 Gyr. However, an elliptical galaxy is a stellar system rather than a fluid system and the oscillation can be quickly damped by Landau damping (Binney & Tremaine 1987; Weinberg 1994). Therefore, the time needed from the beginning of deformation to finally settling on the equilibrium state is indeed the damping time-scale τ_damp, if τ_damp > τ_osc. In a finite stellar system like an elliptical galaxy, the time-scale for damping the quadrupole oscillation (Weinberg 1994, l = 2 mode) is shorter than τ_osc/0.608 ≃ 1.6τ_osc (see Weinberg 1994 for details). Therefore an elliptical galaxy will remain in stationary equilibrium after 1.6τ_osc, which is also shorter than τ_K. Therefore Φ_eff = Φ_grav + Φ_T is a good approximation, where Φ_T is instantaneous.

In our Monte Carlo simulations, the initial ellipticity distribution and the effective density of the galaxies are chosen to be those of field galaxies. There is evidence showing that differences might exist between the densities of cluster galaxies and field galaxies (Pu & Han 2011); it is, however, possible that these differences are caused by the tidal effect we calculated here, which may change the density and ellipticity distributions of the member galaxies of a cluster. Although a different initial ellipticity distribution and effective density do not change our conclusions qualitatively, they will become important when analysing the RAD quantitatively. The effects of weak lensing are also not taken into consideration; these would cause images of some galaxies on the far and back side of the cluster stretched along the tangential direction and thus may weaken the observed RA resulting from the tidal force of the cluster.

Finally, in this work we aim to study the possibility of radial alignment of elliptical galaxies, the original orientations of which are random, under the tidal force in a cluster. There are certainly other possibilities for radial alignment; for example, the galaxies entering the cluster could be radially pre-aligned, since tidal forces are ubiquitous. However, the issue is beyond the scope of the present work and could be an interesting topic for future investigations. It is even possible to infer the original orientation and ellipticity distribution of galaxies by studying observed distributions using our analysis method, if the gravitational field of the cluster is determined independently using other methods. Once again, this could be an interesting topic for future investigations.

We appreciate the very insightful and constructive review report by the anonymous referee. We thank Professor Zu-Hui Fan for her help on galactic dynamics. Drs Yuan Liu and Jian Hu are thanked for their constructive suggestions and discussions. SNZ acknowledges partial funding support by 973 Program of China under grant 2014CB845802, the National Natural Science Foundation of China under grant Nos 11133002 and 11373036, the Qianren start-up grant 292012312D1117210 and by the Strategic Priority Research Program ‘The Emergence of Cosmological Structures’ of the Chinese Academy of Sciences, Grant No. XDB09000000. Part of this project was carried out under the support of the National Natural Science Foundation of China Nos 10878003, 10778752, 11003013, Shanghai Foundation No. 07dz22020 and the Leading Academic Discipline Project of Shanghai Normal University (08DZL805).

REFERENCES

Agustsson

Brainerd

T. G.

ApJ

2006

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L25

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Article Contents

Abstract

1 INTRODUCTION

2 QUALITATIVE DISCUSSION

3 QUANTITATIVE CALCULATION OF THE EFFECT OF TIDAL FORCE ON RAD OF GALAXIES IN A CLUSTER

3.1 Coordinate system

3.2 Deformation of a single elliptical galaxy

3.3 Projection effects

3.4 RAD of elliptical galaxies

4 OBSERVED RAD OF ABELL 2744

4.1 Selection of elliptical galaxies

4.2 RAD of Abell 2744

5 CONCLUSION AND DISCUSSION

REFERENCES

APPENDIX A: DETERMINING THE MAJOR AXIS OF AN ELLIPSOID PROJECTED ON TO THE CELESTIAL PLANE

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