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Astron. Astrophys. 361, 415-428 (2000)

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3. Influence of the lens parameters on depletion curves

For our first set of simulations, we used the analytical redshift distribution for the sources, allowing a fully analytical treatment of the simulations. The contamination by foreground objects with this distribution is weak (about [FORMULA] of foreground galaxies for [FORMULA]). In each case, we computed depletion curves, and then examined their behavior. Note that we limited our analysis outside the first 20" from the center where the signal cannot be constrained observationally (decrease in the observed area for each point, obscuration by the brightest cluster galaxies ...). For these reasons, this "forbidden" area will be shaded in each plot.

3.1. Influence of the velocity dispersion

We used a SIS model and varied the velocity dispersion [FORMULA] from 1000 to 1800 km s-1 (Fig. 1). As expected the radius of the minimum increases with [FORMULA]. Indeed for a SIS, the Einstein radius scales exactly as [FORMULA], so the minimum of the depletion curve, which roughly corresponds to the maximum Einstein radius (at [FORMULA] or more), also scales as [FORMULA].

[FIGURE] Fig. 1. Top: Depletion curves obtained for different velocity dispersions ranging from 1000 to 1800 km s-1. Bottom: Intensity of the minimum (solid line), position of the minimum in Mpc (dotted line) and half width at half minimum in Mpc (dashed line) of the depletion curves as a function of the velocity dispersion.

More surprisingly, the depth of the depletion at the position of the minimum is roughly constant and, at first order, does not depend on the velocity dispersion of the cluster or its total mass. Part of this effect is due to the density of foreground galaxies, but part of it is intrinsic to this SIS model, as other potential shapes do not show this property (see below). On the contrary there is a clear dependence on the half width at half minimum with [FORMULA] (Fig. 1).

3.2. Influence of the core radius

The introduction of a core radius in the model (Fig. 2) does not affect significantly the outer region of the depletion area and has little effect on the position of the minimum. As [FORMULA] increases, the inner width of the curve is enlarged and its slope is decreased. In fact, the study of this area for our purpose has no interest, because the spatial resolution of galaxy number counts variations is by far larger than this scale. In addition, in rich clusters of galaxies, the central density of galaxies is large enough so that empty areas are quite small between the envelopes of large and bright galaxies.

[FIGURE] Fig. 2. Depletion curves obtained with [FORMULA] km s-1 for different core radii ranging from 25 to 125 [FORMULA] kpc. The value of the Einstein radius at the cluster redshift is [FORMULA] 200 [FORMULA] kpc.

In practice, the only observable effect of a core radius is a deformation of the inner depletion curve with a significant departure from a symmetry with respect to the outer part. But we suspect that a quantitative estimate of [FORMULA] would be difficult to extract from the signal.

3.3. Influence of the slope of the mass profile

The simulations are done with the power-law density profile, characterised by its slope [FORMULA] (Fig. 3). The minimum position of the depletion area does not depend significantly of [FORMULA], because our scaling of the potential was fixed at the Einstein radius, nearly independent of [FORMULA]. On the contrary, the two other typical features of the depletion area (half width at half minimum and intensity of the minimum) strongly depend on this parameter. The width of the curve represents roughly the mass dependence with radius. In the case [FORMULA] for example, the shallower slope of the mass radial dependence creates a larger width of the depletion curve. The half width at half minimum of the curve decreases when [FORMULA] increases. The reason is the same as with the SIS, that is to say, when [FORMULA] increases, the mass is concentrated in the inner part of the cluster and the depletion effect is less extended to the outer regions.

[FIGURE] Fig. 3. Top: Depletion curves obtained with [FORMULA] M[FORMULA]/Mpc3 and for different slopes [FORMULA] ranging from 1.7 to 2.1. Bottom: Intensity of the minimum (solid line), position of the minimum in Mpc (dotted line) and half width at half minimum in Mpc (dashed line) of the depletion curves as a function of the slope [FORMULA].

3.4. Influence of the ellipticity of the potential

In these simulations, we use the singular isothermal ellipsoid potential, with [FORMULA] km s-1, and we study the depletion curves along the minor and major axis (Fig. 4). The increase of [FORMULA] does not affect the minimum value of the depletion area but leads to an increase/decrease of the half width at half minimum and of the minimum position along the major/minor axis as an homothetic transformation. The relative positions of the two minima of the depletion area along the main axis gives immediately the axis ratio and thus the ellipticity of the potential. An application of this differential effect along the two main axis is presented in Sect. 5.

[FIGURE] Fig. 4. Top: Variation of the depletion curve with ellipticity along the minor axis. Bottom: Idem along the major axis. The streching of the curves with ellipticity is characterised by an homothetic transformation.

3.5. Study of NFW profile

With the NFW model we varied separately the virial radius and the concentration parameter. We have chosen the values for these two parameters in the range of those found by Navarro et al. (1996). For the first set of simulations, we adopted [FORMULA] and varied [FORMULA] from 1800 to 3600 kpc (Fig. 5). An increase of the virial radius (more massive cluster) leads to an increase of the three characteristic features of the depletion area and to the appearance of a bump in the central region which grows with [FORMULA]. The increase of the half width at half minimum of the depletion curves with [FORMULA] can be explained by the fact that when [FORMULA] increases there is more mass in the outer regions. Consequently, the depletion area is more extended to the outer part of the cluster and reaches its asymptotic value less rapidly. The second set of simulations was done with [FORMULA] kpc and we varied c from 7 to 25 (Fig. 6). The variation of c affects only the inner part of the depletion curves. An increase of the concentration parameter (cluster with smaller characteristic radius) leads to an increase of the intensity and position of the minimum with a steeper slope for the small values of c. Contrary to [FORMULA], the variation of c does not affect significantly the half width at half minimum of the depletion area.

[FIGURE] Fig. 5. Top: Depletion curves obtained with [FORMULA] and for different virial radii. Bottom: Intensity of the minimum (solid line), position of the minimum in Mpc (dotted line) and half width at half minimum in Mpc (dashed line) of the depletion curves as a function of the virial radius.

[FIGURE] Fig. 6. Top: Depletion curves obtained with [FORMULA] kpc and for different concentration parameters. Bottom: Intensity of the minimum (solid line), position of the minimum in Mpc (dotted line) and half width at half minimum in Mpc (dashed line) of the depletion curves as a function of the concentration parameter.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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