4 The value of R

Table 4 indicates that the clusters Be 62, NGC 637 and NGC 663 show a smaller value whereas the clusters NGC 869 and NGC 436 show a higher value for X. The remaining 8 clusters show a normal value for X. The data given in Tables 2, 3 and 4 are used to estimate the weighted (according to associated errors) mean value of the colour excess ratios $E(\lambda -V)/E(B-V)$ and the ratios are given in Table 5.

We define a parameter r which is the ratio of $[E(\lambda-V)/ E(B-V)]_{\rm cluster}$ (the ratio of colour excesses in the cluster region) and $[E(\lambda-V)/E(B-V)]_{\rm normal}$ (the ratio of colour excesses for the normal reddening law). The values of $E(\lambda -V)/E(B-V)$ given in Table 5 are used to obtain the ratio r for the cluster region and resultant value of r is given in Table 6. For the clusters where the data from U band to Kband are available, r is plotted as a function of $\lambda ^{-1}$ in Fig. 12.

 

Table 5: Weighted mean value of the colour excess ratios $E(\lambda -V)/E(B-V).$

Cluster	$\frac{E(U-V)}{E(B-V)}$	$\frac{E(R-V)}{E(B-V)}$	$\frac{E(I-V)}{E(B-V)}$	$\frac{E(J-V)}{E(B-V)}$	$\frac{E(H-V)}{E(B-V)}$	$\frac{E(K-V)}{E(B-V)}$
NGC 654	$1.72\pm0.07$	$-0.65\pm0.03$	$-1.35\pm0.07$	$-2.18\pm0.25$	$-2.42\pm0.24$	$-2.60\pm0.26$
NGC 663	$1.60\pm0.07$	$-0.55\pm0.02$	$-1.28\pm0.04$	$-2.43\pm0.29$	$-2.84\pm0.33$	$-3.08\pm0.36$
NGC 869	$1.95\pm0.07$	$-0.64\pm0.03$	$-1.28\pm0.02$	$-2.16\pm0.16$	$-2.54\pm0.15$	$-2.65\pm0.17$
NGC 884	$1.72\pm0.07$	$-0.66\pm0.03$	$-1.21\pm0.02$	$-2.29\pm0.09$	$-2.61\pm0.09$	$-2.77\pm0.11$
NGC 1502	$1.76\pm0.07$			$-1.97\pm0.19$	$-2.14\pm0.24$	$-2.20\pm0.24$
IC 1805	$1.72\pm0.07$		$-1.25\pm0.05$	$-2.55\pm0.22$	$-2.86\pm0.23$	$-3.07\pm0.24$
Normal	1.72	-0.60	-1.25	-2.30	-2.58	-2.78

Figure 12 indicates that the extinction in most of the cluster regions seems to be normal at $\lambda > \lambda_I$, except for the clusters NGC 1502 (where the colour excess ratios for $\lambda > \lambda_I$ are less than the normal one) and NGC 1805 (where the colour excess ratio at $\lambda \geq \lambda_{J}$ are higher than the normal one). Recent studies support a universality of the extinction curves for $\lambda > \lambda_I$(see e.g. Cardelli et al. 1989; He et al. 1995). It is suggested that the normalization should be done using the E(V-K) instead of E(B-V)(Tapia et al. 1991) because the E(V-K) does not depend on properties like chemical composition, shape, structure, degree of alignment of interstellar dust (cf. Mathis 1990 and references therein).

Cardelli et al. (1989) found that the mean R dependent extinction law can be represented by the following relation

$$\frac{A_\lambda}{A_V}=a_\lambda + \frac{b_\lambda}{R}$$ (1)

where $a_\lambda$ and $b_\lambda$ can be obtained from the relations given by Cardelli et al. (1989). The above relation can be written, in terms of $\frac{E(\lambda-V)}{E(B-V)}$, as

$$\frac{E(\lambda-V)}{E(B-V)}=R(a_\lambda-1) + b_\lambda.$$ (2)

Figure 11: CCDs for the clusters Be 62, NGC 436 and NGC 637. The continuous curve represents the MS shifted along the reddening vector given in Table 4 and dashed curve shows the MS shifted along a normal reddening vector.

The ratio of total-to-selective extinction towards the cluster direction " $R_{\rm cluster}$'' is derived using the Eq. (2). The value of $\frac{E(\lambda-V)}{E(B-V)}$ ${(\lambda \geq \lambda _J)}$ given in Table 5 is used to estimate the value of $R_{\rm cluster}$ and the results estimate for $R_{\rm cluster}$ are given Table 6. The clusters that have a broad spectrum of data are discussed below.

 

Table 6: Mean value of the ratio r as a function of wavelength.

$\lambda^{-1} (\mu$m-1)	2.90	1.56	1.25	0.80	0.61	0.45	$R_{\rm cluster}$
Cluster	U	$R_{\rm C}$	$I_{\rm C}$	J	H	K
NGC 654	$1.00\pm0.04$	$1.08\pm0.05$	$1.08\pm0.06$	$1.07\pm0.14$	$0.98\pm0.09$	$0.95\pm0.03$	$2.97\pm0.30$
NGC 663	$0.93\pm0.04$	$0.92\pm0.04$	$1.02\pm0.03$	$1.07\pm0.11$	$1.09\pm0.11$	$1.10\pm0.11$	$3.50\pm0.40$
NGC 869	$1.19\pm0.02$	$1.07\pm0.05$	$1.02\pm0.02$	$0.98\pm0.05$	$1.02\pm0.05$	$1.00\pm0.04$	$3.04\pm0.20$
NGC 884	$1.00\pm0.04$	$1.02\pm0.05$	$0.97\pm0.02$	$0.97\pm0.03$	$0.99\pm0.02$	$0.98\pm0.03$	$3.19\pm0.12$
NGC 1502	$1.02\pm0.03$			$0.85\pm0.05$	$0.83\pm0.05$	$0.81\pm0.04$	$2.57\pm0.27$
IC 1805	$1.00\pm0.03$		$1.00\pm0.02$	$1.12\pm0.09$	$1.14\pm0.08$	$1.16\pm0.07$	$3.56\pm0.29$

NGC 654

Sagar & Yu (1989) concluded that at wavelengths greater than 5500 Å, the extinction is normal. The presence of unusually well aligned interstellar grains indicated by the polarization measurements seems to increase the extinction in the U and B bands slightly (Sagar & Yu 1989). In the present work we find a rather normal extinction law.The value of $R_{\rm cluster}$ is $\sim$ $2.97 \pm 0.30 (\sigma)$ which, within the error, is close to the normal value of 3.1.

NGC 663

Using the colour excesses E(V-K) and E(B-V) Tapia et al. (1991) found weak evidence for an anomalous reddening law with a value of $R_{\rm cluster} = 2.73 \pm 0.20$, which is marginally lower than the normal value of 3.1. However, they felt that the scatter in their data is too large to conclude about the value of $R_{\rm cluster}$. Yadav & Sagar (2001) reported values for $E(\lambda -V)/E(B-V)$ ( $\lambda \geq \lambda_{J}$) which are significantly smaller than the normal ones.

The (weighted) mean value of $R_{\rm cluster} = 3.50 \pm 0.40 (\sigma)$suggests a marginally anomalous reddening law in the NGC 663 cluster region but in the opposite sense to that reported by Tapia et al. (1991) and Yadav & Sagar (2001). More near-IR data is needed to determine the $R_{\rm cluster}$ in the NGC 663 cluster region. Here it is interesting to mention that the behaviour of the extinction curve towards UV also deviates from the normal one. Figure 12 indicates a lower value for the E(U-V)/E(B-V) ratio, whereas Yadav & Sagar (2001) reported a normal value for this ratio. They supplemented their data with the photometric spectral types which are based on the Q method, where they adopted E(U-B)/E(B-V)=0.72. Presumably a dominance of photometric spectral determination forced the ratio of E(U-B)/E(B-V) to a normal value.

NGC 869 and 884 (h and $\chi$ Persei)

From the extinction curve analysis Johnson (1965) found a value of $R_{\rm cluster} = 3.0$ in the NGC 869 and NGC 884 cluster region. Tapia et al. (1984) also reported a normal reddening law in the cluster region. Recently Yadav & Sagar (2001) found that the $E(\lambda -V)/E(B-V)$ ratios for $\lambda \geq \lambda_{J}$ are smaller than the normal ones.

The colour excess diagrams (method "A'') of NGC 869 indicates somewhat lower values for the ratios E(J-V)/E(B-V) and E(K-V)/E(B-V) but the errors are large, whereas in the case of NGC 884 the errors in the estimation of colour excess ratios are too large. The reason for the large errors is a small range in the $E(B-V) (\sim$0.3 mag). Because of the large errors we have not used the colour excess ratios obtained from the CEDs of NGC 869 in the further analysis. The TCDs (method "B'') indicate that the ratio of colour excesses $E(\lambda -V)/E(B-V)$ in both the clusters for $\lambda \geq \lambda_I$ are perfectly normal. It is interesting to note that the CED for NGC 869 yields $E(U-V)/E(B-V) = 1.90 \pm 0.13$. The V/(B-V) and V/(U-B) CMDs also seem to support the ratio of E(U-V)/E(B-V) = 1.9. In a recent study Keller et al. (2001) have adopted E(U-B)/E(B-V) = 0.72 to fit the ZAMS to the stellar distribution on the V/(U-B) CMD of the stars in the NGC 869 and NGC 884 cluster region. We find that only in the case of NGC 884 the V/(U-B) CMD supports a normal value for the reddening vector X in the cluster region.

NGC 1502

Tapia et al. (1991) found the ratio E(V-K)/E(B-V) = 2.20 corresponding to $R_{\rm cluster} = 2.42\pm0.09$, which is significantly lower than the reddening value. Yadav & Sagar (2001) also found that the colour excess ratios $E(\lambda -V)/E(B-V)$ for $\lambda \geq \lambda_{J}$ are significantly smaller than the normal ones. The colour excess ratios for $\lambda \geq \lambda_{J}$ obtained in the present study are in good agreement with those reported by Tapia et al. (1991) and Yadav & Sagar (2001). The value of $R_{\rm cluster}$ is estimated as $2.57\pm0.27$ which is in good agreement with that obtained by Tapia et al. (1991).

Figure 12: The ratio r as a function of $\lambda ^{-1}$.

IC 1805

Various studies have been carried out to estimate the value of Rin the cluster region of IC 1805, but the results are not conclusive. Johnson (1968), Ishida (1969) and Kwon & Lee (1983) reported an anomalous reddening law in the cluster region with values of $R_{\rm cluster} \sim 5.7$, $\sim$3.8 and $\sim$3.44 respectively. Kwon & Lee also reported a regional variation in the value of $R_{\rm cluster}$, with a maximum value of $R_{\rm cluster} = 3.82 \pm 0.15$ for stars located in the outer region and the minimum of $3.06 \pm 0.05$ for stars located in the central region. Sagar & Yu (1990) found that the interstellar extinction law in the direction of most of the cluster members is normal. The colour excess ratios $E(\lambda -V)/E(B-V)$ for $\lambda \geq \lambda_{J}$ obtained in the present work indicate an anomalous reddening law in the cluster region of IC 1805. The value of $R_{\rm cluster}$ is estimated as $3.56\pm0.29$, which also indicates an anomalous reddening behaviour in the cluster region.