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5 Field-star contamination and clusterparameters

In order to maximise the cluster members and reduce field-star contamination in the sample to be used for determination of cluster parameters, we have considered stars within a radius of 200 CCD pixel ( $\sim 2\hbox{$.\mkern-4mu^\prime$ }5$) from the cluster center. In the sample, we have also included the stars identified as proper motion members in the cluster database by Mermilliod (1995). As these stars are generally bright (V < 14 mag), they are very important for the determination of cluster age and for the study of stellar evolutionary status of the Cepheids and other evolved stars. Before deriving cluster parameters from this sample, we shall quantify the amount of field-star contamination and the same is done below.

5.1 Field-star contamination

As the cluster location near the Perseus arm contributes a good number of background and foreground stars, it is difficult to separate the field-stars from the cluster members, only on the basis of their closeness to the MS in the CMD and colour-colour diagrams of the cluster (see Romeo et al. 1989, for a detailed discussion). The probability of cluster membership is small for stars located well away from the MS. To know the actual number of cluster members from the remaining stars, their kinematic information like proper motions and radial velocities are required. Due to lack of such information for stars fainter than $V \sim 13$ mag, it is difficult to establish firmly the cluster membership of these stars. In order to get an idea of the field-star contamination in the cluster region, Pedreros et al. (1984) measured stars in the adjacent field (see their Fig. 1) which is located at $\sim 8'$ (i.e about $\sim $ 2.2 cluster radius) away from the cluster center. So, we also consider it as the field region. Frequency distribution of the stars in different parts of the V, B-V diagram in the cluster and field regions normalized for the difference in their areas is listed in Table 4. In order to avoid the effect of relatively large data incompleteness, the analysis is restricted to the brightness level which is $\sim $ 1 mag above the limiting magnitude of Pedreros et al. (1984) observations. To derive the frequency distribution of stars, the V, (B-V) diagram is divided into seven magnitude bins from V = 12 to 19 and three colour bins called blueward, near and redward of MS. We find that the number of MS stars is generally more than that in adjacent field area and that the differences are statistically significant, while the differences in the numbers of stars, which are blueward of MS and redward of MS, of the two regions are generally not statistically significant. The table indicates that the degree of field-star contamination in the MS of our sample is increasing with faintness. The cluster parameters are derived using the sample stars assuming that field-star contamination may not change the results derived below significantly.

Table 4: Frequency distribution in the V, (B-V) diagrams of the cluster region and adjacent field region (taken from Pedreros et al. 1984) are presented. The number of stars in the cluster and field regions are normalised to the cluster area. $N_{\rm BC}$, $N_{\rm MC}$ and $N_{\rm RC}$ denote the number of stars in the cluster region located blueward, near and redward of MS respectively. The corresponding numbers for the field region are $N_{\rm BF}$, $N_{\rm MF}$ and $N_{\rm RF}$ respectively
V $N_{\rm BC}$ $N_{\rm BF}$ $N_{\rm MC}$ $N_{\rm MF}$ $N_{\rm RC}$ $N_{\rm RF}$
12.0-13.0 0 0 4 4 1 4
13.0-14.0 0 0 13 0 4 11
14.0-15.0 0 0 14 0 12 7
15.0-16.0 0 0 39 4 17 14
16.0-17.0 0 0 63 25 38 21
17.0-18.0 5 18 75 39 32 21
18.0-19.0 16 18 117 32 35 25

5.2 Interstellar extinction in the direction of cluster

In order to estimate the interstellar extinction to the cluster, we plot apparent (U-B) versus (B-V) diagram in Fig. 5 for the sample stars. Adopting the slope of reddening line E(U-B)/E(B-V) as 0.72, we fitted the intrinsic zero-age main-sequence (ZAMS) given by Schmidt-Kaler (1982) to the MS stars of spectral type earlier than A0 in Fig. 5. This yields a mean value of E(B-V) = 0.51 mag with an uncertainty of $\sim $ 0.03 mag for NGC 7790. The observed cluster sequence in Fig. 5 is well defined for hotter stars, indicating that interstellar extinction is uniform across the cluster region in agreement with the conclusions made by Sandage (1958); Pedreros et al. (1984) and Romeo et al. (1989). Our reddening estimate agrees fairly well with most of the earlier estimates (see Table 2 in Romeo et al. 1989), except in the case of the photographic determination of Pedreros et al. (1984) which has a value of 0.64$\,\pm\,$0.04 mag.

\end{figure} Figure 5: The (U-B), (B-V) diagram for the sample stars in NGC 7790. The continuous straight line represents the slope (0.72) and direction of the reddening vector. The dotted curve represents the locus of Schmidt-Kaler's (1982) ZAMS, shifted in the direction of reddening vector for the values of E(B-V) and E(U-B) indicated in the diagram

For determining the nature of interstellar extinction law in the direction of the cluster, we used the stars earlier than A0 selected from their location in the (U-B) versus (B-V) diagram (Fig. 5) which reveals that stars with (B-V) < 0.65 mag are the desired objects. For them, the (B-V)0, E(B-V) and E(U-B) values have been determined using the spectral type (available only for 10 stars) taken from the cluster database and the UBV photometric Q method (cf. Johnson & Morgan 1953; Sagar & Joshi 1979), and the calibration given by Schmidt-Kaler (1982). For calculating E(V-R) and E(V-I) values, we used the apparent (V-R) and (V-I) measurements; Sagar &Cannon's (1994) calibration between (B-V)0 and (V-R)0 and Walker's (1985) calibration between (B-V)0 and (V-I)0. The mean values of the colour-excess ratios derived in this way are listed in Table 5. They are in fair agreement with the normal values. We have therefore considered normal and uniform interstellar extinction in the direction of the cluster with E(B-V) = 0.51 mag in our further analyses.
Table 5: A comparison of the colour-excess ratios with E(B-V) for NGC 7790 with the corresponding values for the normal interstellar extinction law given by Schmidt-Kaler (1982) for E(U-B)/E(B-V); by Alcalá & Ferro (1988) for E(V-R)/E(B-V) and by Dean et al. (1978) for E(V-I)/E(B-V)
Object E(B-V) E(U-B)/E(B-V) E(V-R)/E(B-V) E(V-I)/E(B-V)
Normal interstellar   0.72 0.65 1.25
NGC 7790 (spectroscopic) 0.51 $\pm$ 0.04 0.71 $\pm$ 0.04 0.65 $\pm$ 0.04 1.32 $\pm$ 0.05
NGC 7790 (photometric) 0.51 $\pm$ 0.06 0.69 $\pm$ 0.05 0.66 $\pm$ 0.05 1.38 $\pm$ 0.07

5.3 The CMDs and distance to the cluster

In order to determine the distance modulus of the cluster, we plot intrinsic V0, (U-V)0; V0, (B-V)0; V0, (V-R)0 and V0, (V-I)0 diagrams in Fig. 6 for the sample stars of NGC 7790. For this, we convert apparent V magnitude and (B-V), (U-B), (V-R) and (V-I) colours into intrinsic ones using the value of E(B-V) = 0.51 mag and following relations for E(U-B); E(V-R); Av and E(V-I) (see Sagar & Cannon 1994 and references therein)

\begin{displaymath}\frac{E(U-B)}{E(B-V)} = X + 0.05 E(B-V) \end{displaymath}

where X = 0.62 - 0.3 (B-V)0 for (B-V)0 < -0.09
and X = 0.66 + 0.08 (B-V)0 for (B-V)0 > -0.09;

\begin{displaymath}\frac{E(V-R)}{E(B-V)} = E1 + E2 \times E(B-V) \end{displaymath}

where E1 = 0.6316 + 0.0713 (B-V)0 and
E2 = 0.0362 + 0.0078 (B-V)0;

\begin{displaymath}\frac{A_v}{E(B-V)} = 3.06 + 0.25 (B-V)_0 + 0.05 E(B-V) \end{displaymath}


\begin{displaymath}\frac{E(V-I)}{E(B-V)} = 1.25[1+0.06(B-V)_0+0.014E(B-V)] .\end{displaymath}

As the interstellar extinction seems to be uniform across the cluster region (see last section), we have used the same value of E(B-V) for all sample stars. The overall morphology of the CMDs confirms those obtained by earlier studies (Sandage 1958; Pedreros et al. 1984; Romeo et al. 1989; Phelps & Janes 1994). In all the CMDs a well populated cluster MS down to V0 = 18 mag is clearly seen. The V0, (B-V)0 and V0, (V-I)0 diagrams show the faintest part of the MS. Evolutionary effects are clearly visible in the upper part of the cluster MS. The stars seem to be distributed in a clumpy fashion along the MS, giving rise to gaps. The most prominent one amongst them is located near turn-off point (see Fig. 6). It has a width of $\sim $0.25 mag. Following Hawarden (1971), the $\chi^2$ value of this gap is estimated and found to be 0.012% indicating that this is a genuine gap. This gap is similar to the those observed on the rising branches of the evolving part of MS (cf. Sagar & Joshi 1978 and references therein). In V0, (U-V)0 and V0, (B-V)0 diagrams, we fitted the ZAMS given by Schmidt-Kaler (1982) while in V0, (V-R)0 and V0, (V-I)0 diagrams, the ZAMS given by Walker (1985) was fitted. The (V-R)0 colour for the ZAMS on the present photometric system was taken from Sagar & Cannon (1994). After accounting for the colour dispersion expected from the error in observations, the visual fit of the ZAMS to the bluest envelope of the CM diagrams gives $(m-M)_0=12.6\pm0.1$. The visual fit has been done for stars in the unevolved part of the MS (V0 > 14 mag).
\end{figure} Figure 6: The V0, (U-V)0; V0, (B-V)0; V0, (V-R)0 and V0, (V-I)0 diagrams for the sample stars in NGC 7790. Continuous curves are the ZAMS fitted to the unevolved part of the cluster MS for the values indicated in the diagram. The mean value of true distance modulus (m-M)0 to the cluster is 12.6 mag

The mean value of (m-M)0 is $12.6\,\pm\,0.15$ mag for NGC 7790. The uncertainty in the value is estimated from the errors in R, E(B-V) and the errors in fitting the ZAMS. The distance modulus yields a distance of $3300\pm230$ pc to NGC 7790. Present determination of distance modulus agrees with those recently determined by Romeo et al. (1989) and Phelps & Janes (1994). However, it is larger than the values of 12.3, 12.15 and 11.98 mag determined by Pedreros et al. (1984), Balona & Shobbrook (1985) and Schmidt (1981) respectively. Present distance determination should be considered as more reliable because they have been derived by fitting the ZAMS over a wide range of the unevolved part of the cluster MS. As the cluster contains three Cepheid variables, we have used the period-luminosity relation (PLR) given by Sandage et al. (1999) for the Galactic Cepheids for determining their distance modulus. For this, the average $<\!\!\!V\!\!\!>$ values and observed period given in Table 6 are used. This yields a value of $12.63\pm0.1$ mag for the true distance modulus of the Cepheids assuming present detemination of E(B-V) and the value of R given above. Thus, the agreement between two independent estimates of distance is excellent. The present distance determination also agrees very well with the value of $3230\pm160$ pc determined for the Cepheid CF Cas by Matthews et al. (1995) using the PLR.

5.4 Age of the cluster members

The stars brighter than Mv = 0.0 mag show evolutionary effects and most of them are proper motion cluster members. In order to derive the cluster age, we converted apparent V, (B-V) diagram of the sample stars into intrinsic ones. We used this diagram instead of other CMDs as the other colours are not available for all bright stars. Figure 7 shows the Mv, (B-V)0 diagram for NGC 7790. We have estimated age by fitting stellar evolutionary isochrones given by Bertelli et al. (1994) in Fig. 7. The isochrones include the effects of mass loss and overshooting of the convective core in the theoretical calculations. The effects of the binaries have also been considered while estimating ages. In order to define upper limit of the effects of binarity in the CMDs, the isochrones which are derived from theoretical stellar evolutionary models for single stars have been brightened by 0.75 mag keeping the colour same. The isochrones fitted in this way explains the presence of stars around the MS and the red GB. This also indicates that some fraction of cluster members seems to be in the form of binaries. The population I (X=0.7, Y=0.28, Z=0.02) isochrone of age log t = 8.1 fits the brighter MS stars, except the two bluest and brightest stars (see Fig. 7). As they are located near ZAMS in the CMDs (see Fig. 6) and also have high probability of proper motion cluster membership, we consider them as blue straggler as Ahumada & Lapasset (1995) did. The positions of the Cepheid variables CEa Cas, CEb Cas and CF Cas are shown as crosses in Fig. 7. For this, we used the photometric data provided by Sandage (1958) for CF Cas and by Opal et al. (1987) for others. The extremes of their variabilty are also indicated. For the Cepheids, the blue loop of the solar metallicity isochrones does not reach the instability strip, which was also noticed by Romeo et al. (1989). At the same time, their spectroscopic chemical abundance determinations indicate lower metallicity of [Fe/H] = -0.2 (Fry & Carney 1997). In order to see whether the lower metallicity isochrones fit these variables, we fit the Z = 0.008 isochrones given by Bertelli et al. (1994). The single star isochrone fits the Cepheid variables well for an age of log t = 8.1 but for slightly different cluster parameters which can be accounted in terms of the colour difference expected at different chemical composition. This, therefore supports the lower metallicity determination of the Cepheids by Fry & Carney (1997) and also the fact that the Cepheids are known to be single stars. However, their observed positions (more or less in the middle of the blue loop) are not at the generally expected location which is near the bluest point of the blue loop, where the life time is relatively longer. This may indicate that either theoretical loop morphology is not completely correct as it depends on several variables (main being the mass loss rate and metallicity) of the Cepheids which are not accounted in the model or they are in extremely fast evolutionary stage with the brightest one passed the bluest point while other two are approaching towards it.
\end{figure} Figure 7: The Mv, (B-V)0 diagram of NGC 7790. It is plotted only for brighter stars ( (B-V)0 < 1.5 mag and Mv < 3.4 mag) so that various type of stars can be identified clearly. The two stars located above and blueward of the cluster turn-off point are blue stragglers. The three Cepheid variables have been shown with crosses denoting extremes of their variability. Isochrones of age log t = 8.1 from Bertelli et al. (1994) for both Pop. I (solid) and Z=0.008 (dashed) have been fitted to the bright cluster members and the MS for the marked values of distances. The extent of brightening in the corresponding isochrones due to binaries of equal mass are also shown. An age of $\sim $ 120 Myr is thus assigned to NGC 7790

In the light of the above discussions, we conclude that the cluster is $120\pm20$ Myr old. This agrees very well with the value of 100 Myr given by Romeo et al. (1989).
Table 6: Mean $<\!\!\!V\!\!\!>$ mag and Period P in days for the Cepheids in NGC 7790 are taken from Sandage (1958) and Opal et al. (1987). The $<\!\!M_v\!\!>$ values are derived assuming Cepheids as cluster member and using present determination of distance and reddening. The zero-points (C1) are determined using these $<\!\!M_v\!\!>$ in the period-luminosity relation given by Sandage et al. (1999)
Cepheid P $<\!\!V\!\!>$ $<\!\!M_v\!\!>$ C1
  (day) (mag) (mag) (mag)
CEa Cas 5.14087 10.90 -3.36 -1.35
CEb Cas 4.47928 11.02 -3.24 -1.40
CF Cas 4.87522 11.14 -3.12 -1.18
Average       $-1.31\pm0.1$

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