2 The method

We detect clusters with maximum likelihood method in a way similar to that by P96, using models of surface density and apparent magnitude distribution of cluster galaxies and field galaxies. The cluster model has two free parameters, namely, its redshift and richness.

2.1 Models

2.1.1 Cluster

We assume spherical symmetry for simplifying the cluster model. The radial distribution of the galaxies in the model cluster matches the King model with $c \equiv \log (r_{\rm tidal}/r_{\rm core}) = 2.25$ (King 1966; Ichikawa 1986) and $r_{\rm core} = 170~{\rm h}^{-1}{\rm kpc}$ (Girardi et al. 1995), where $r_{\rm core}$ and $r_{\rm tidal}$ are core radius and tidal radius, respectively. The luminosity function of the galaxies obeys the Schechter function (Schechter 1976) with $\alpha = -1.25$ and M_B^* = -20.4 (converted from $M_{B_{\rm J}}^* = -20.12$ by Colless 1989, rescaling H₀ and using $M_{B_{\rm J}} = M_B - 0.18$ by Yoshii & Takahara 1988). Morphological type mixture, luminosity segregation, substructure, and influences by cD galaxies are not considered. To compute apparent features of the model cluster, two more parameters, namely redshift $z_{\rm fil}$ (hereafter we call it "filter redshift'' after the manner of P96) and richness N, must be assigned. Obviously, if $z_{\rm fil}$ is larger, angular extension of the model clusters becomes smaller and member galaxies become fainter. Dimming by the K-correction effect is assumed to be $K_B(z_{\rm fil}) = 4.7 \times z_{\rm fil}$ (for 0.6), which is the value for elliptical galaxies (Fukugita et al. 1995). We define N to be the number of all galaxies brighter than (M^*+5). The parameter N roughly represents the population of bright, giant galaxies in a cluster, ignoring dwarf galaxies whose natures such as spatial distribution or luminosity function are still unclear.

2.1.2 Field

We assume that field galaxies are randomly distributed on the sky, namely, angular two-point correlation function is not considered. For simulations in Sect. 3, we adopt deep galaxy number count data by Metcalfe et al. (1995) as the model of apparent magnitude distribution of field (foreground and background) galaxies. For the actual galaxy data discussed in Sect. 4, we use the magnitude distribution of all galaxies in the survey area to search clusters as if it were that of pure field galaxies. This causes an overestimate of the number of field galaxies, especially when the survey area is small and there is a cluster covering a bulk portion of the area by chance. Hence we have to deal with an enough large area so that clusters or even a large-scale structure will not seriously affect the estimate of the number of field galaxies in the area. Yet sometimes iteration would be needed. For that case, we mark conspicuous "cluster'' regions by referring to the first-time result and then execute the second calculation using the "more accurate'' field galaxy sample in the area except for the "cluster'' regions.

2.2 Algorithm

What we need at the beginning is just a usual catalog of galaxies containing projected positions and apparent magnitudes. Figure 1 shows an example galaxy distribution. The galaxies are generated by a Monte Carlo simulation based on the model described in Sect. 2.1. A cluster with (z,N) = (0.20,1000), roughly equal to an Abell richness class 0-1 cluster, is located at the center.

  

Figure 1: Galaxy distribution in an area containing an artificial cluster with (z, N) = (0.20, 1000) at the center. The symbol size changes with the apparent magnitude. The largest and smallest symbols correspond to mB = 16.0 and 23.5, respectively

Next we compute the likelihood ${\cal L}$ that a cluster is present at a particular point. We consider $n_{\theta}$ concentric annular regions centered on the position. Their angular inner radii and widths are $\theta _i$ and $\Delta \theta _i (1\leq i\leq n_{\theta})$, respectively. By counting the number of galaxies which fall in n_m magnitude bins ($m_j\leq m<m_j+\Delta m_j\; (1\leq j\leq n_m)$) for each annular region, we obtain an array $O_{ij}\; (1\leq i\leq n_{\theta}, 1\leq j\leq n_m)$ consisting of $n_{\theta} \times n_m$ galaxy numbers.

On the other hand, we can calculate an equivalent array M_ij for the model galaxies. M_ij is described as

$$M_{ij} = N\; C_{ij} + F_{ij},$$ (1)

where C_ij is an array for member galaxies in a normalized model cluster located at the center of $n_{\theta}$ concentric annular regions, and F_ij is an array for field galaxies. C_ij is written as

$$C_{ij} = 2\pi \int _{\theta _i}^{\theta _i+\Delta \theta _i}... ...ta \; \int _{m_j}^{m_j+\Delta m_j} \phi _{\rm c}(m)\; {\rm d}m,$$ (2)

where $\sigma _{\rm c}(\theta )$ is surface density profile and $\phi _{\rm c }(m)$ is differential luminosity function of cluster galaxies. Both $\sigma _{\rm c}(\theta )$ and $\phi _{\rm c }(m)$ depend on filter redshift and are normalized as

$$2\pi \int _0^{\theta _{\rm tidal}} \theta \sigma _{\rm c}(\theta )\; {\rm d}\theta = 1$$ (3)

and

$$\int _{-\infty }^{m^*+5} \phi _{\rm c}(m)\; {\rm d}m = 1.$$ (4)

F_ij is written as

$$F_{ij} = 2\pi \sigma _{\rm f} \int _{\theta _i}^{\theta _i+\... ...a \; \int _{m_j}^{m_j+\Delta m_j} \phi _{\rm f}(m)\; {\rm d}m,$$ (5)

where $\sigma _{\rm f}$ is surface density and $\phi _{\rm f}(m)$ is differential luminosity distribution of field galaxies.

The logarithmic likelihood is given by

$$\ln {\cal L} = \sum _{i, j} \ln \left\{ \frac{M_{ij}^{O_{ij}}\; {\rm e}^{-M_{ij}}}{O_{ij}!} \right\}$$

 

$$= \sum _{i, j} \left\{ O_{ij}\ln (N\, C_{ij}\!+\!F_{ij})\!-\!(N\, C_{ij}\!+\! F_{ij})-\ln (O_{ij}!) \right\}.$$ (6)

Here we assume that O_ij obeys Poisson statistics since their values amount to about 10 or less for typical clusters at 0.2 with our choice of values for $\Delta \theta _i$ and $\Delta m_j$ (see the next section). If we assume Gaussian distribution for O_ij, Eq. (6) becomes simply equivalent to $-\chi ^2$ (P96 did so, expecting that there would be enough background galaxies. See Eq. (12) of their paper). However, this assumption leads us to overestimating N about 20% of the true value for the case of (z,N) = (0.20, 1000). This is because Poissonian distribution is not symmetrical and has a longer tail toward larger value. The use of Poisson statistics helps to reduce the possible systematic error in the estimation of the redshift and the richness.

Equation (6) is a function of both filter redshift $z_{\rm fil}$ and richness N. In order to simplify calculations, we first fix $z_{\rm fil}$ to a certain value and maximize ${\cal L}$ by optimizing only N. The partial derivative of the logarithmic likelihood (6) with respect to N is  

$$\frac{\partial }{\partial N}\ln {\cal L} = \sum _{i, j} \left\{ \frac{O_{ij}C_{ij}}{N\; C_{ij}+F_{ij}} - C_{ij} \right\}.$$ (7)

Equation (7) is apparently a monotonically decreasing function of N. If we find a certain richness value $N_{\rm p}$ for which Eq. (7) becomes zero, ${\cal L}$ has a peak value ${\cal L}_{\rm p}$ at $N=N_{\rm p}$. Computing ${\cal L}_{\rm p}$ and $N_{\rm p}$ at every point in the whole image, we obtain a "likelihood image'' ${\cal L}_{\rm p} (x,y)$ and a "richness image'' $N_{\rm p} (x,y)$ for the fixed filter redshift.

  

Figure 2: "Likelihood image'' (upper panel a)) and "Richness image'' (lower panel b)) of the artificial cluster in Fig. 1 for $z_{\rm fil}$ = 0.20

Figures 2a and 2b, respectively, show the "likelihood image'' and the "richness image'' for $z_{\rm fil}=0.20$ generated from the galaxy distribution shown in Fig. 1. We can recognize the existence of a cluster by a peak in both images. However, appearances of the peaks are quite different. While the peak in the "richness image'' is simple and very prominent, there exists a ring-like region of slightly lower likelihood around a weak peak in the center of the "likelihood image'', and ${\cal L}_{\rm p}$ increases again toward further out of the ring. This is because it is difficult to discriminate a cluster with very small N from "field''. Though only "likelihood image'' is theoretically needed to detect clusters, peaks in "likelihood image'' corresponding to clusters are often very obscure as seen in Fig. 2a. We therefore find a peak in "richness image'' at first, and then check if there is also a peak in corresponding "likelihood image''. If a peak exists at nearly the same point in both images, we regard it as a cluster candidate.

We obtain several pairs of "likelihood image'' and "richness image'' and find peaks in both images for other filter redshifts. Then we plot the peak ${\cal L}_{\rm p}$ and the peak $N_{\rm p}$ as functions of filter redshift for each cluster candidate. An example is shown in Figs. 3a and 3b, respectively. If we find a peak in ${\cal L}_{\rm p}-z_{\rm fil}$ plot (Fig. 3a), $z_{\rm fil}$ at the peak is the redshift estimate of the cluster candidate (hereafter $z_{\rm est}$). Once $z_{\rm est}$ is obtained, $N_{\rm p}$ for that $z_{\rm est}$ (hereafter $N_{\rm est}$) can also be found as shown in Fig. 3b. Some cluster candidates do not show any remarkable single peak of ${\cal L}_{\rm p}$ in the ${\cal L}_{\rm p}-z_{\rm fil}$ plot. Such candidates may be spurious.

  

Figure 3: Upper panel a) displays peak logarithmic likelihood as a function of filter redshift for the artificial cluster in Fig. 1. Lower panel b) shows peak richness as a function of filter redshift for the same data