A&A 458, 767-775 (2006)
DOI: 10.1051/0004-6361:20054453

Photometric determination of the [Fe/H] ratio for the globular cluster M 92

F. Smriglio¹ - A. K. Dasgupta¹ - R. P. Boyle²

1 - Phys. Dep., Univ. di Roma "La Sapienza'', P.le A. Moro, 2, 00185 Roma, Italy
2 - Vatican Observatory, 00120 Città del Vaticano

Received 1 November 2005 / Accepted 26 March 2006

Abstract
CCD observations of the northern globular cluster M 92 were obtained in the PXYVS colors of the Vilnius photometric system. The results for 230 stars in an area centered on the cluster and for 178 stars in an area west of the cluster down to V = 17.0 mag are presented. After a brief outline of the data pathway from observations to reduction via IRAF and a discussion of the photometric errors, the [Fe/H] ratio of the globular cluster M 92 is determined by multicolor PXYVS Vilnius photometry.

Key words: stars: fundamental parameters - stars: general - Galaxy: globular clusters: individual: M 92

1 Introduction

This paper is in continuation of a series of papers reporting CCD photometry in the Vilnius photometric system (hereafter VPS). Two fields in the direction of the very metal deficient globular cluster M 92 were investigated in five Vilnius colors PXYVS, with the purpose of producing CCD photometry for cluster members and of determining the [Fe/H] ratio of the cluster by a statistical approach.

Since the eighties, the CCD photometry technique progressively replaced the photoelectrically calibrated photographic plates, despite the small size of the field of view, relatively large pixels and surface imperfections, bringing a substantial improvement in sensitivity, linearity, and dynamic range. This "historical'' change produced a dramatic reduction in the internal errors of the observational data, and also made systematic differences evident in respect to the previous works.

At the same time adequate software has been developed. In particular, the point-spread function technique permitted an improved signal-to-noise ratio and the deconvolution of overlapping images. These new techniques have shown their effective potential for studying the typically very crowded globular cluster fields. In fact, a complete survey of a globular cluster area, with this considerably increased accuracy, has produced significant improvement in our knowledge of the morphology of the color-magnitude diagrams (CMD).

Having decided to make CCD photometric observations, we devoted our attention to the choice of a multicolor photometric system. After the international photographic and photovisual systems that appeared in the twenties, in the second half of the last century a number of photometric systems were developed, starting with the UBV system introduced by Johnson & Morgan (1953). With the RI system, proposed by Cousins (1976, 1980), an extension was set up, called the UBVRI broadband system. The limitations and the evolution of these systems are extensively described by Straizys (1992, 1999). Most CMDs for star clusters have been derived from the broad band UBV system and its extensions. But intermediate band systems (e.g. Strömgren and Vilnius) provide more accurate spectral information.

Hesser (1988) stressed the importance of using other multicolor photometric systems. He concluded that, despite the power of other systems to provide answers to many astrophysical problems, observational data on globular clusters obtained by multicolor photometric systems other than UBVRI are rather scarce in the literature. In general, this mainly depends on the necessary network of standards suitable for CCD photometry.

Bearing these arguments in mind, we started an observing program in the beginning of the nineties, in the VPS of regions in the direction of globular and open clusters. We had several aims in view: e.g., automated photometric classification of a wide variety of stars (Smriglio et al. 1986, 1988, 1990), CCD photometry in the Vilnius photometric system (Boyle et al. 1990a,b; Boyle et al. 1994; Dodd et al. 1996; Boyle et al. 1998; Smriglio et al. 1998; Smriglio et al. 2002), determination of interstellar extinction (Smriglio et al. 1991; Boyle et al. 1992; Janulis & Smriglio 1992), and the study of the cloudy structure of interstellar dust and the gas-to-dust ratio (Smriglio et al. 1994; Smriglio et al. 1995; Smriglio et al. 1996). The observations of globular clusters give the opportunity of calibrating photometric systems in terms of temperature, luminosity, and metallicity, since these parameters, and the interstellar reddening of the cluster stars can be measured with rather good accuracy. Moreover, the globular cluster population contains horizontal branch stars of different temperatures and metallicities, which are rather infrequent in the solar vicinity. Besides this, the morphology of the CMD is also very important for contributing to the knowledge of the age-metallicity relation.

In this paper we present CCD photometry in the VPS in the field of the globular cluster M 92, which is thought to be typical of very metal-deficient clusters. Furthermore we determine the [Fe/H] ratio for this cluster in a purely photometric way by use of a number of metallicity-sensitive color indices (Bartkevicius & Sperauskas 1983).

   
2 Observations

The observations were obtained using five Vilnius filters (P, X, Y, V, S), during two different observing runs in six nights in May and July 1996 with the 150 cm telescope of Bologna University situated at an altitude of about 3000 feet in Loiano (Northern Italy Apennines). A concise description of the VPS properties and the response functions of the Vilnius passbands are shown in a recent review of standard photometric systems (Bessel 2005). The U filter of the Vilnius system was not used, due to the very low ultraviolet efficiency of the presently available CCD camera; on the other hand, the U magnitude is not necessary for the photometric determination of metallicity using the technique described in the present paper. Also the observations in the Z band have been omitted because of the mediocre quality of the Z standard magnitudes available at that time.

Any light leaks outside the desired band were less than 1% of the transmission across a spectral region from 0.3 to 1.2 micron. A cryogenically-cooled CCD camera was used consisting of a Thomson TH 7896A CDA chip with $1024 \times 1024~ \rm pixels$, each $19 \mu~ \rm square$, operated at -130 $^{\circ}$C. The chip size corresponds to $9.6 \times 9.6~ \rm arcmin$ on the sky at the f/8 Cassegrain focus of the telescope.

The observations were performed in two fields partially overlapping each other, and this overlap will be very valuable for estimating the accuracy of the photometry. The first one (hereafter M92C) was centered on the cluster at the coordinates $\rm RA=17^{h}17^{m}00^{s}$, $\rm Dec= 43^{\circ} 08{'} 00{''}$ and the second one (hereafter M92W) was at $\rm RA= 17^{h}16^{m}5^{s}$, $\rm Dec= 43^{\circ}08{'}46{''}$ (epoch 1950). The common region has an extension of about $8.96 \times 4.1~ \rm arcmin$.

The exposure times for M92C were 30 min in P, 15 min in X, and 10 min in the rest of the filters. Similarly, the exposure times for M92W were 15 min in P, 7 min in X, and 5 min in the rest of the filters. The finding charts of both regions are shown in Figs. 1 and 2, respectively.

3 Reductions

The data were partially reduced at Wellington Victoria University (New Zealand) by one of us (A.K.D.) and at Rome University "La Sapienza'', using the crowded field photometry package DAOPHOT from IRAF in the point-spread function mode (Tody 1986, 1993; Stetson 1987).

The flatfielding was done using domeflat exposures. Flatfielded exposures in each filter were aligned by task imalign and then added in order to improve the signal-to-noise ratio (Newberry 1994). In the imcombine task we set "combine'' = average and "reject'' = crreject. In this rejection set-up the low pixels are ignored and only high pixels rejected to ensure that any star falling on a bad pixel or trap is rejected. This will also remove any star affected by cosmic rays.

We performed photometry on the combined images from each filter using tasks: phot, psf, nstar, substar, and allstar in this sequential order. The psf stars (photoelectric standards, as well as many bright and isolated stars) were selected using the pstselect procedure. The task psf was run interactively using the output list from pstselect and then accepting or rejecting a candidate as a psf star after inspecting the profile and the contour of the star. A good candidate psf star must have no neighbours within fitrad pixels and also must be free of cosmetic blemishes. The fitrad ${\approx}$ FWHM was set for each combined image per filter. The FWHM and sky ${\sigma}$ were obtained from each combined image per filter, using task imexamine. These procedures were repeated until we had a good list of those psf stars that do not contain any non-stellar, multiple, or cosmetic blemishes. From this procedure the instrumental magnitudes for all the band-passes were obtained.

The instrumental magnitudes ( $m_{\rm ccd}$) for each band-pass were converted to the standard ( $m_{\rm st}$) ones of the VPS using a color equation (CE) of the form:

 

$$m_{\rm ccd} = m_{\rm st} + m_{0} + m_{1} (Y-V)_{\rm st}$$ (1)

where (Y-V) is the color index most sensitive to temperature. The coefficient m₀ is the zero point shift and m₁ is the first-order color coefficient (slope). The values of these coefficients were obtained using task fitparams as described in IRAF.

Eight photoelectric standards from Zdanavicius (1992) were identified in the M92C observed area and other four standards in the M92W area, but only the stars listed in Table 1 were used for calibration. Unfortunately, this is due to the rather low (and in some cases very low) accuracy of 50% of these standards, as indicated by the author.

The first, second, and third columns of Table 1 list the identification numbers of the standards taken from Zdanavicius (1992), Sandage & Walker (1966) and this paper, respectively, and Cols. 4 to 7 show the V magnitude and color indices (P-X), (X-Y), (V-S) observed photoelectrically by Zdanavicius.

Table 1: The list of Vilnius photoelectric standard stars observed in the studied fields and used for calibration.

As already said in Sect. 2, an area of $8.96 \times 4.1~ \rm arcmin$. is common to both the observed fields, and 64 stars were identified in this area. For these stars two different sets of magnitudes and colors, obtained in two different runs, are available and their corresponding identification numbers, Nc for the M92C and Nw for M92W, are listed in Table A.3 (see Appendix).

Table 2: The fitting results in each pass-band for the stars common to both the observed fields.

These common stars give the opportunity for estimating the accuracy of our photometry. A comparison between the corresponding VPS magnitudes of these stars was performed by first degree polynomial fittings which give the results presented in Table 2. The fitting was performed with the stars in the range of magnitude indicated in Col. 2 for each pass-band. The slope and the zero-point with the corresponding errors are also shown. As an example, the result of the fitting for the Vilnius V magnitudes of these common stars up to V=14.5 is represented in Fig. 3.

	Figure 3: V magnitudes (Vw) of M92W versus V magnitudes (Vc) of M92C for stars that are common to both areas and brighter than V=14.5.
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From Table 2 we can see that for each pass-band the slope is systematically equal to 1 within a 1% rms error, while the errors for the zero-point are larger, due to the already-mentioned mediocre accuracy of the photoelectric standards. The standard deviation indicates a scatter within 2% for V, P, Y, and S colors and reaches 3% for X. For the magnitudes between V=15to 17, the scatter is larger than 3%, mainly due to the decreasing signal-to-noise ratio with increasing magnitudes by rule of photon statistics. Bearing in mind that the M92C area is affected by a background due to faint undetected stars and that it is relatively crowded compared to the western area, these results may be considered quite satisfactory for our purposes, as seen later.

In the absence of other photometric observations of the globular cluster M 92 in the VPS, we can try to get some information about the accuracy of our photometry by comparing our Vilnius V magnitude with the broad-band Johnson V magnitudes available in the two observed regions. For this purpose we identified 22 stars in M92C and an other 20 stars in M92W coincident with just as many standard stars observed by Sandage. They are listed in Tables A.4 and A.5, respectively (see Appendix). In these

Tables our identification numbers and the corresponding Sandage numbers are shown in Cols. 1 and 2. The V magnitudes are listed in Cols. 3 and 4. In Cols. 5 and 6 the (B-V) color indices calculated from Vilnius colors and those measured by Sandage in the UBV broad-band Johnson system are shown. The (B-V) in Col. (5) of each table has been calculated using the following relations between Vilnius and the UBV system (Straizys 1977, 1992): (B-V) = 0.50 [(X-V) + (Y-V)]

for $[(X-V) + (Y-V)]\leq 2.26$, or (B-V) = 0.48 (X-V) + 0.02

for [(X-Y) + (Y-V)] >2.26
where (X-V) and (Y-V) are the Vilnius color indices. These transformations can be applied both for reddened and unreddened stars.

The (B-V) index was also corrected for a chemical composition effect by using the empirical relations between (B-V) and (Y-V) color indices for metal poor stars of different metallicities, studied by Bartkevicius & Lazauskaité (1992). But, with this sample and the present accuracy, we did not obtain significant differences, except for a slight second order effect for the brightest stars in the M92C field. On the contrary, a slightly larger, but also inconsequential, scatter of the data was observed in the color-magnitude diagram. This effect is certainly due to the errors introduced by these equations, which, according to the authors, can reach a value of 3%.

The comparison of our V magnitudes and (B-V) color indices with those determined photoelectrically by Sandage & Walker (1966), listed in Tables A.4 and A.5 in the Appendix, was done by using a first degree least-square solution technique. The result for the V magnitude in the M92C area is given by the equation $V_{\rm c}=(1.02 \pm 0.01) V_{\rm s} - (0.27 \pm 0.22)$. For the M92W area we obtain $(V)_{\rm w}=(1.00 \pm 0.01)V_{\rm s}- (0.24\pm 0.13)$, where the suffix "s'' indicates Sandage's values. In both the areas the standard deviation is <0.03. Moreover, the result for the (B-V) color index is given by the equation $(B-V)_{\rm c}=(0.97 \pm 0.02) (B-V)_{\rm s} - (0.01 \pm 0.01)$ in the M92C area and $(B-V)_{\rm w}=(1.03 \pm 0.02) (B-V)_{\rm s} - (0.01 \pm 0.01)$ in the M92W area. The standard deviation is 0.02 and 0.03, respectively.

On the basis of these results, it appears that the accuracy of this photometry remains within 2-3%, which in general cannot be considered as very accurate photometry. But considering the poor quality of the standards and of the high surface density of stars around the globular cluster, these results are acceptable. On the other hand, as we have already suggested in a previous paper (Smriglio et al. 1998), it is not excluded that the available broad band photometry can contain systematic errors. For testing this hypothesis, we identified a number of stars common to two different photoelectric observations in this region (see Table A.6 in Appendix), published by Sandage & Walker (1966) and Sandage (1969). The comparison between the V magnitude from Sandage & Walker (1966) (VS66) and the V magnitude from Sandage (1969) (VS69) gives the equation $V_{S69}=V_{S66}(1.06 \pm 0.03)- (0.93 \pm 0.46)$ with a standard deviation $\rm SD=0.04$. For this fitting the stars XI-11 and XII-6, rejected by the authors, were removed from the sample. These values confirm a probable presence of large scatter and also systematic differences in the UBV photoelectric photometry of globular cluster stars.

By applying the same technique, also the (B-V) index ((B-V)S66) from Sandage & Walker (1966) and the corresponding index ((B-V)S69) from Sandage (1969) (Table A.6 in Appendix) where compared. The fitting gives the equation $(B-V)_{S69}=(B-V)_{S66}(1.05 \pm 0.08)-(0.01 \pm 0.01)$. This result shows a large rms error in the slope and, plotting the data in Fig. 4, we can also notice a systematically larger scatter for stars bluer than $(B-V)\simeq 0.25$.

In conclusion, the broad-band photoelectric UBV observations, available in the M 92 region, do not allow any estimation of the external errors of our CCD photometry, because of the considerable scatter and the systematic differences existing in the published data.

4 Results

The Vilnius intermediate-band CCD photometry presented in the previous Sections has produced a catalogue of 344 stars brighter than V=17.5 mag, the vast majority of them belongs to globular cluster M 92. For determining the metallicity of the cluster by a purely photometric technique, the stars brighter than V=14.8, for reasons described later, were selected. In this section the statistical procedure, followed for determining the [Fe/H] ratio of the studied cluster, and all the results are presented.

4.1 The catalogue

The final catalogue is presented in Table A.1 for the stars observed in the M92C field and in Table A.2 for the stars in the M92W field (see Appendix). In Table A.1, a list of 230éstars is complete in V magnitude from V=11.76 (the brightest value in the field) to V=16.83. Similarly, the list in Table A.2 contains 178 stars from V=11.72 to V=17.47 mag. In both tables, eight columns consecutively give (1) the sequential running number, (2) and (3) the coordinates $X_{\rm c}$ and $Y_{\rm c}$, respectively, in pixels of the center of each star image, (4) the CCD V magnitude in the VPS, and (5) to (9) the Vilnius color indices Y-V, P-V, X-V, V-S, and Johnson B-V color index. This last index was calculated from the Vilnius color indices (X-V) and (Y-V), as indicated in the previous section.

The stars of both samples are shown in order of increasing V magnitudes and, following a criterion already used in previous papers of the same series, can be easily identified in the finding charts in Figs. 1 and 2 by their coordinates $X_{\rm c}$ and $Y_{\rm c}$, respectively.

4.2 Interstellar extinction

The interstellar reddening in the direction of the globular cluster M 92 has been estimated in the course of time by several authors using a wide variety of methods. A concise, but in practice complete, list of color excess determination methods has been published by Webbink (1985). He grouped all the techniques available in the literature in six echelons, also giving an estimate of the accuracy of each method. The first-echelon technique relies on the intrinsic colors of the BHB stars, which are nearly independent of metallicity. With accurate photometry these measures can be accurate to $\pm$ $0\hbox{$.\!\!^{\rm m}$ }02$. The second and third-echelon measures rely on the intrinsic colors of the RR Lyrae stars and on the intrinsic flux distributions of individual red giants stars. The results are probably not very inferior to those of the first-echelon. The author relegates the asymptotic reddenings of foreground field stars technique in the fourth position only, because these estimates are essentially statistical in nature and are normally significantly more uncertain than the higher-echelon measures. In fact, the measurement uncertainty in E_B-V increases in proportion to the reddening itself and, in the case of small-sized fields, it is typically $0.15\times E_{B-V}$. The results are quite satisfactory in general for high-latitude clusters. For the sake of completeness we say that the fifth-echelon measures employ the blue and red edges of the RR Lyrae instability strip and the sixth-echelon can be obtained by comparing of integrated photometry of the cluster in question with the intrinsic colors of clusters of a similar spectral type. Their accuracy is comparable to that of second and third-echelons. In the absence of a color-excess determination by one of the above methods, one can resort to the cosecant-law, but the accuracy cannot be better than $\pm$ $0.4 \times E_{B-V}$.The color-excesses E_B-V for the cluster M 92 are diffusely quoted in the literature and an extended selection, used in this paper, is listed in Table 3.

Table 3: A selection of the color-excess E_B-V of M 92 from several authors in the literature (in order of publication).

The measure by Sandage (1969) was performed by using the two-color diagram of blue horizontal branch stars, which is a first-echelon technique for determining the reddening of a globular cluster (Webbink 1985). According to this author, in the case of very accurate photometry, the error could be better than $0\hbox{$.\!\!^{\rm m}$ }02$, but, after our photometric test (see Sect. 3, Fig. 4), this condition could not be achieved sufficiently. Both Harris & Racine (1979) and Zinn (1980a) have taken weighted means of the available reddening estimates in the literature and, while these averages are identical, they found differences of up to 0.03 mag for many clusters on their list. The reddening has been measured recently by Vilnius photometry (Smriglio et al. 1998) using seven suspected non-member foreground stars (Sandage & Walker 1966) of solar metallicity. The metal abundance of these stars is also confirmed by the color-color diagrams in the VPS.

	Figure 4: Comparison between the (B-V) color-index by Sandage (1969) and the same color-index observed by Sandage & Walker (1966) for 17 stars.
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From the available data, it is evident that E_B-V=0.02 is the most probable value for the reddening of M 92, but we also think that it is comparable to, or smaller than, the photometric errors obtainable by all these techniques. Consequently, we have decided not to apply any reddening correction to the observed color indices.

4.3 The color-magnitude diagram

Plotting the stars of one of the two fields (e.g. M92W) in the color-magnitude diagram [V, B-V], (CMD, shown in Fig. 5), we can select a sample of stars able to be used for the photometric determination of the [Fe/H] ratio of M 92.

	Figure 5: The color-magnitude diagram for the stars observed in the field M92W: the empty circles indicate the stars brighter than V= 14.8; the empty squares the stars fainter than this value; filled circles and filled squares are photoelectric standards by Sandage & Walker (1966) and Sandage (1970), respectively. The measured [Fe/H] ratio for three stars is indicated (see text)
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For reasons that will be justified in the next paragraphs, the stars brighter than about 14.8 V-magnitude have been selected. By chance, they are the brightest of the catalogue and then affected by smaller errors. These objects are marked in Fig. 5 by open circles. The stars fainter than this limit are indicated by empty squares. On the whole, two samples of about 40 stars in M92W and 80 stars in M92C are available.

Moreover, the CMD diagram gives the opportunity of making an immediate and simple comparison of our photometry with that of Sandage & Walker (1966). In fact, the filled circles indicate 37 photoelectric standards, observed in the region of M 92 and identified by these authors as members of this cluster. Likewise, the filled squares indicate six other photoelectric standards, fainter than V=16 mag, later observed by Sandage (1970). It is evident that, within the 2-3% accuracy of this photometry, the distribution of CCD stars does not show any significant systematic effect with respect to the UBV photometry. In any case, the presence of some small systematic effect in the (B-V) index, calculated from VPS color indices, would not influence the determination of metallicity, which is obtained directly from Vilnius color indices.

4.4 The photometric determination of [Fe/H] ratio

The efficiency of photometry in the VPS for distinguishing metal deficient stars (MD) and for determining their abundances and luminosities has been extensively shown by V. Straizys and his group in the seventies and eighties. A new technique for determining all these parameters and, in particular, the [Fe/H] ratio for MD stars was published by Bartkevicius & Sperauskas (1983) and is applied in the present work.

In a few words, it is possible to observe the ultraviolet excess for MD stars by their position on color-color diagrams by plotting a color index that is sensitive to metal deficiency as a function of a color index that mainly depends on the temperature. In the VPS the (Y-V), (V-S), and (Y-S) indices are the best indicators of the effective temperature for F-M type MD stars, while (P-X), (X-Y), and (P-Y) are the indices most sensitive to the metallicity. The (U-P) index is indispensable for determining luminosities of all MD stars and for distinguishing peculiarities as carbon-rich stars (CH, barium, and R stars) and red horizontal branch (RHB) stars. From these color-color diagrams it is possible to define a metallicity index $\delta^{'}$(CI), that is nearly independent of temperature (Bond 1980), for each star and also, to calibrate it directly in terms of [Fe/H]. For estimating the amount of variation of the abundance-sensitive color indices as a function of the metallicity, the two-color diagram (P-X) vs. (Y-V), calibrated in metallicities (Straizys 1982), can be used. In this diagram the unreddened stars of normal chemical composition of spectral types G0-K2 and luminosities IV-III, and G5-K2 for supergiants, form a single sequence and $\delta (P-X)$ "is a function of the metallicity only''. This diagram is calibrated in [Fe/H] using values of metallicity measured by spectroscopy. The variation $\delta (P-X)/0.1 [$Fe/H] dex is on the order of 1% for G0-G5 and 1.5-1.8 % for K0-K2 types and does not show any luminosity effect. The amount of variation obtained by other combinations of filters appears to be on the same order of magnitude.

In our case the color-color diagrams (P-V), (Y-V) and (X-V), (Y-V) where plotted in Figs. 6a-d. In these figures the continuous line represents the main sequence, while the sequence of G-K giants of solar metallicity is shown by a dashed line. The arrow gives the slope of the interstellar reddening line and its length corresponds to E_Y-V=0.20.

Figure 6: The color-color diagrams (P-V) vs. (Y-V) and (X-V) vs. (Y-V) in both cluster and west fields. The continuous line represents the main sequence, and the dashed line the sequence of the normal G-K giants. The stars, observed in the fields M92C ( a) and  c)) and M92W ( d)), are plotted by dots. The encircled dots are the stars brighter than about V=15th magnitude. In c) only the stars brighter than this limit are represented (see text).

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Figure 6a shows the (P-V),(Y-V) diagram, plotted for the 230 stars observed in M92C field (Table A.1 in Appendix), as are the stars brighter than V=15 mag. The analogous diagram for the 178 stars observed in M92W field (Table A.2 in Appendix) is plotted in Fig. 6b.

From these diagrams, noticeable color excesses $\delta (P-V)$ and $\delta (X-V)$ due to differential line blanketing are evident. The distribution of dots in Figs. 6a-c show that the scatter depends essentially on the stars fainter than the 15th V-magnitude and also that it is obviously larger in the more crowded region M92C.

In particular, stars fainter than 15th V-magnitudes have been omitted in Fig. 6c to show the fairly good sequence of the MD stars more clearly. This sequence, as already shown by Smriglio et al. (1998), displays a good fit with the sequence of a sample of field stars, with same metallicity and luminosity and after dereddening of their color indices. Again in this figure, six suspected foreground stars of solar metallicity and two suspected RHB stars are present, as their measured metallicity afterwards confirmed.

4.5 Results

The method for statistically determining the [Fe/H] ratio of a globular cluster consists in measuring the metallicity of a properly selected sample of cluster members. An accurate statistical analysis of these data permits a mean value of [Fe/H] ratio to be determined with its rms error.

In the case of M 92, we decided to select the stars brighter than about 14.8 V-magnitudes. These stars are mainly red giants (RG). But a few stars of the red end of the horizontal branch (RHB), some stars of the asymptotic branch and some sub-giants are included in this sample. It is known that the RHB stars have either no ultraviolet excess or only a slight one relative to solar metallicity stars. The explanation is that the deblanketing due to low metal abundance nearly compensates for the gravity effect (e.g., Johnson & Morgan 1953; Eggen & Sandage 1964). Also some field star could appear in the same range of color and magnitude. But these objects show a relatively high value of metallicity (apparent or real) and then they could pollute the mean metallicity of the sample from a statistical point of view. For instance, in Fig. 5, the value of measured metallicity has been indicated for the stars scattered around the giant branch sequence. It is evident that they appear to be metal rich or slightly richer than the actual metal abundance of the cluster. On the other hand, we estimate that only a relatively small number of RHB stars can be present in this sample. Bearing in mind that this globular cluster is extremely metal-poor, we think that the usual $3\sigma$ statistical cleaning of the data is an efficient technique for excluding nearly all of these relatively metal-rich objects. In case of 1% photometry, that is to say, a more accurate measurement of the metallicity, an individual elimination could also be performed on the basis of their measured [Fe/H] value, followed finally by the statistical cleaning. In all cases, a more effective removal of these RHB stars from the sample will probably reduce the dispersion of the measured [Fe/H] ratios and/or some systematic effect.

The metallicity for each star was determined from the three color excesses $\delta (X-Y), \delta (P-Y)$ and  $\delta (P-X)$. On the whole, six different sets of values have been obtained. The final sample of stars, selected for our purpose, and the measured metallicity are shown in Tables A.7 and A.8 for the two fields M92C and M92W, respectively (see Appendix).

At this point, it is important to notice that the values of metallicity reported in these tables can only be used for statistical analysis. In fact, a single value could show a large difference compared to the mean, and it would be meaningless to use it as a measurement of metallicity for a single star. Obviously, the information about the metallicity of the cluster is contained in the whole set of measurements and must be extracted by statistical treatment. The [Fe/H] values, listed in Tables A.7 and A.8, have been plotted in histograms and the resulting six histograms are shown in Fig. 7. The bell-shaped distribution of data can be studied with a Gaussian fit and, for each set of data, the mean value $\overline{[\rm Fe/H]}$with its rms error  $\sigma_{i}$ is obtained. The results are listed in Table 4.

	Figure 7: The histograms of the six sets of [Fe/H] ratios measured in both fields cluster (in  a),  c),  e)) and west (in  b),  d),  f)), obtained from the three color excesses $\delta (X-Y), \delta (P-Y)$ and $\delta (P-X)$. The Gaussian fittings for each distribution are also shown.
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This statistical treatment requires some justification: for a normal distribution of a set of measurements, the probability that a single value is affected by a given error must be absolutely random, which is to say that the error must be independent of the measurement itself. In the case of astronomical photometry, this may not be completely true, because of the dependence of the photon statistical errors on the apparent magnitudes. This effect gives a contribution to the final observational error, making it dependent on the magnitude, and its influence by degrees significantly increases with the magnitude itself. But in our case all the stars of the selected samples are brighter than $V \simeq 14.5$ mag and the signal-to-noise ratio errors remain confined within $0.5\%$ up to $V \simeq 14$ mag and within $1\%$ for stars fainter than $V\simeq 14{-}14.5$ mag. Then, remembering that the estimated final errors, in this range of magnitudes, are on the order of $2{-}3\%$, we can assume that this systematic effect is actually negligible.

Considering that the distributions shown in Fig. 7 at first sight appear to follow a normal distribution, with a more or less good approximation, some statistical parameters were determined for testing the normal distribution. The features to be considered are the median, the skewness (the degree of symmetry of a distribution, hereafter SKN), and the kurtosis-excess (the degree of peakedness, hereafter KRX).

In Table 5, the SKN and the KRX are shown in Cols. 3 and 5, respectively, for each set of data in both regions M92C and M92W. The values given in Cols. 4 and 6 are obtained by multiplying the standard deviations of the SKN and the KRX by 2.6. This is known to be a rapid test for estimating the symmetry and the shape of a Gaussian distribution: it can be shown that, when they are smaller than their correspondent limit, the SKN and the KRX are significantly near to zero.

Table 4: The [Fe/H] ratios measured for each color excess in both fields M92C and M92W with the corresponding rms error and the number of stars involved in computing each value.

Table 5: The results of the statistical analysis. The values of median, skewness, kurtosis-excess are indicated for each set of metallicity measurements, in each field (C for M92C and W for M92W). The values listed in Cols. 4 and 6 give the limit for the skewness and the kurtosis-excess to be considered significantly equal to zero (see the text).

From Table 5, it is evident that the values of SKN and the KRX indicate a symmetric and mesokurtic normal distribution for each set of data, with only the exception of the $[{\rm Fe/H}]{\it cluster}PY$. In fact, both the SKN and the KRX are positive for this set, and larger than their own limits, that is, the normal distribution is slightly asymmetric on the right side with more concentration around its mean value (leptokurtic). This effect is also indicated in Fig. 7 and is more evident in Fig. 7c. On the other hand, the symmetry of these normal distributions is also confirmed by the fact that the mean and the median are coincident within the same bin (Tables 4 and 5). In conclusion, the statistical approach by a Gaussian fitting to the determination of metallicity seems to be correct and fairly justified.

The final value of metallicity $\overline{[\rm Fe/H]}$ is obtained by calculating the weighted mean of the six mean values in Table 4; that is, $\overline{[{\rm Fe/H}]}=\sum_{i}^{}p_{i}[{\rm Fe/H}]_{i}/\!\sum_{i}^{}p_{i}$and assuming as weight $p_{i}=1/\sigma_{i}^{2}$, where $\sigma_{i}$, with i from 1 to 6, is the rms error of each [Fe/H] determination in Table 4. The rms error of the mean is given by the known formula: $\sigma_{f} ~=~1/\!\sqrt{\sum_{i}^{}(1/\sigma_{i}^{2})}$. This statistical procedure provided the value $\overline{[{\rm Fe/H}]}=-2.11\pm 0.14$. It is interesting that the rms error, obtained with the present 2-3% photometry, is lower than (0.15-0.20) dex, which is the value of the accuracy obtained, with a 1% Vilnius photometry, for the [Fe/H] measurement of a single star (Straizys 1999).

5 Discussion and conclusions

The main aim of the present work is the statistical determination of the [Fe/H] ratio of the globular cluster M 92 by Vilnius photometry and an estimate of the accuracy of the method. Multicolor stellar photometry in the band-passes PXYVS of the Vilnius system has been carried out in two fields, $9.6~ \times~ 9.6$ arcmin each, in the direction of the globular cluster M 92. The two fields overlapp each other and the common area has an extension of about $9 \times 4$ arcmin. The U Vilnius filter band-pass was not used, due to the very low UV efficiency of the CCD camera then available at the Loiano 152 cm telescope. Also, the observations in the Z band have been omitted because of the mediocre quality of the Z standard magnitudes available in the literature. However, the band-passes U and Z are not required for a photometric determination of metallicity by using the quoted method. Moreover, no classification of stars is necessary, because the reddening in the direction of M 92 is already known and is negligible within the present accuracy.

In a previous paper, Smriglio et al. (2002) have shown that, with a 1.5-m class telescope, the rms error of the CCD photometry is generally less than 2% up to 16.5 V-magnitude. The rms error for the present photometry ranges between 2% and 3%, which is not very accurate. But, in spite of this mediocre accuracy, it has been possible to well determine the metallicity for this cluster fairly, due to the measurement of the [Fe/H] ratio for a large number of cluster members and to a statistical analysis of these data.

In fact, taking the 64 stars common to both fields into account, a catalogue of 344 stars on the whole has been obtained up to about 17.5 V-magnitude. Of these objects, a sample of about 120 stars brighter than $V\sim 14.8$ mag has been selected.

The final result obtained by the statistical method presented in this paper gives the value $\overline{[{\rm Fe/H}]}=-2.11\pm 0.14$ for the metallicity of the globular cluster M 92. In order to compare this result with those measured by other authors, a short random selection of M 92 metal abundances, published in different times, is reported here without any claim to completeness.

The main techniques followed for investigating the metal abundances of globular clusters are based on spectroscopic or photoelectric photometry observations and are applied both to integrated and individual cluster members. Spectroscopic observations in the blue region of the integrated spectra by Morgan (1956, 1959), Kinman (1959) and van den Bergh (1969) have shown rather good efficiency in ranking the globular cluster by metal abundance. In the seventies a number of observers obtained similar results by observations of individual cluster stars (e.g., Osborn 1973; Butler 1975; Canterna 1975; Hesser et al. 1977; Cooley et al. 1978; Searly & Zinn 1978; Cohen 1978, 1979), but much less sensitivity in distinguishing small metallicity differences among very metal-poor globular clusters (i.e., $\rm [Fe/H]<-1.5$) was pointed out by Zinn (1980a). Photoelectric photometry observations in the same spectral region were also successfully used by several authors (e.g., van den Bergh & Henry 1962; Gascoigne & Koelher 1963; McClure & van den Bergh 1968; Johnson & McNamara 1969; Faber 1973; Harris & Canterna 1977).

A number of metallicity determinations, measured or averaged from the results of different authors, and the respective accuracy (when available) are listed in Table 6. The Harris & Racine (1979) [Fe/H] value in Table 6 is based on the average of abundance estimates from individual cluster giants, published in the seventies (for related references see their paper). The typical internal errors in [Fe/H] estimated by these authors, are $\pm$0.2 dex. The Zinn (1980a,b) [Fe/H] value in Table 6 is obtained by photoelectric photometry of the cluster's integrated colors. These measurements and the measurements by other investigators have yelded estimates of [Fe/H] with an average precision of $\pm$0.1 dex for a total of 84 clusters.

Table 6: Some [Fe/H] determinations of the globular cluster M 92 measured by different authors in different epochs.

Later on, Zinn & West (1984) derived the metallicity of M 92 as a weighted mean of the following determinations: $\rm [Fe/H]_{IR}=-2.01$ and $[{\rm Fe/H}]_{T_{\rm e}}=-2.14$, both from the effective temperature $T_{\rm e}$ of the giant branch (Frogel et al. 1983); $[{\rm Fe/H}]_{\Delta S}=-2.34$, from the $\Delta S$ parameter, which relies on strength of Ca II K line in the spectra of RR Lyrae variables; $\rm [Fe/H]_{BG}=-2.4$, by Bell & Gustafsson (1983) from comparison of scans of red giants with synthetic spectra, $\rm [Fe/H]_{DDO}=-2.27$, from Hesser et al. (1977) DDO photometry of red giants; and $\rm [Fe/H]_{Q_{39}}=-2.12$, from the photometric integrated light index Q₃₉. The standard error of each measurement, as assumed by the authors, is $\pm$0.2 dex, except for $\rm [Fe/H]_{Q_{39}}$ whose limiting accuracy has been estimated to be $\pm$0.15 dex. The resulting weighted average gives the value $\rm [Fe/H]=(-2.24 \pm 0.08~dex)$. This standard deviation was calculated by propagating the ${\sigma}$s of the individual measurements; but, as pointed out by the authors themselves, this value represents the precision with which the clusters are ranked on Cohen's metallicity scale and not the accuracy, in absolute terms, of the value of [Fe/H]. Moreover, this value of metallicity depends on a particular procedure for determining the mean. In fact, the straight mean of the data gives a value of metallicity $\rm\overline{[Fe/H]}=(-2.21\pm 0.15)~dex$, where the error indicates the standard deviation obtained from the scatter of the measurements themselves.

On the basis of high-resolution, high signal-to-noise spectra of bright red giants, a value of $\rm [Fe/H]=-2.16$ has been measured for the M 92 metallicity by Gratton et al. (1997). They show that the availability of a homogeneus set of high-quality abundances provides a standard error of $\sim$0.07 dex and also pointed out that Carretta & Gratton abundances (1997) for metal-poor and metal-rich clusters agree quite well with Zinn & West values (1984). Moreover, a direct estimate of the accuracy in abundance measurements by using multicolor photometry in comparison with high dispersion analysis has been made by these authors and the errors, derived from the rms scatter of differences between Strömgren intermediate band uvby photometry and high dispersion metallicity, are on the order of 0.13 dex.

From the data reported in this sample, it seems reasonable to derive the following conclusions:

a)

the values of metallicity are spread out over a range of about 0.2 dex. This depends on the different techniques used for the measurements and on zero-point offsets of different scales. The latter is mainly due to different assumptions about the solar abundances in the high-dispersion analysis originally used in calibration. This is the typical value for zero-point offset, but, in some cases, a maximum zero-point difference of about $\leq$0.4 dex has been observed (Zinn & West 1984). This problem has now been solved by the ability of Kurucz model to reproduce most of the solar observations to the level of accuracy of the high dispersion spectral analysis (Gratton et al. 1997).

b)

Some nonlinearity may also be present on different scales. For example, the Zinn & West scale (1984) appears slightly nonlinear when calibrated against the best modern measurements of [Fe/H] (see Carretta & Gratton 1997; Rutledge et al. 1997), but this effect seems to be very small or absent for metal-poor clusters (which is the case of M 92).

c)

The accuracy of the measurements on a particular metallicity scale achieved a value of $\pm$0.08 dex (e.g. Zinn & West 1984) or, more recently, $\pm$0.07 dex (e.g. Gratton et al. 1997). On the other hand, in some cases (e.g. Zinn & West 1984), when the [Fe/H] ratio is obtained from a number of metal abundances measured by different techniques and/or different authors, it has been shown that the error is on the order of $\pm$0.15 dex. Moreover, a direct comparison between Strömgren multicolor photometry and high-dispersion spectroscopy measurements gives an rms error of 0.13 dex, and the accuracy with which the index Q₃₉ can predict [Fe/H] is limited to $\pm$0.15 dex.

In conclusion, on the understanding that the high dispersion spectroscopy of individual giants provides the most accurate results, we can conclude that the statistical method presented in this paper, based on CCD observations in the VPS and using crowded field photometry package DAOPHOT from IRAF in the point-spread function mode (Tody 1986, 1993; Stetson 1987), can achieve a precision that is typical of intermediate band photometry (e.g. uvby of Strömgren's system). Moreover, the accuracy seems to be better than achieved by using the photometric integrated light index Q₃₉.

We can say that this simple method can be useful for determining the [Fe/H] ratio for globular clusters from PXYV Vilnius photometry alone. We also think that better accuracy can be achieved by more accurate CCD photometry (i.e. using a more efficient CCD chip in the UV, establishing standards with accuracy better than 1%, using appropriate instrumental procedure to reduce effects due to scattered light on flat-fielding, etc.) and by improving the statistics, i.e. increasing the number of independent observations, as well as the number of cluster members involved in the statistical process.

Acknowledgements

The authors are grateful to Profs. B. Marano, and V. Zitelli for generous allocation of time at the 1.52 m telescope of Osservatorio Astronomico di Bologna, at Loiano (Italy). One of us A.K.D. would like to thank Father G.V. Coyne SJ for a very generous travel fund, Carter Observatory and Victoria University (New Zealand), for the hospitality provided during his visit as Visiting Research Fellow, and Cardiff University College (UK) for providing computing facilities. The authors are especially grateful to the IRAF Group for their enduring assistance and to Prof. A.G. Davis Philip for his careful reading of the manuscript. The useful suggestions by Prof. J.G. Cohen have been greatly appreciated.

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Online Material

	Figure 1: The finding chart of the M92C field. The central pixel ( $X_{\rm c}=1044$; $Y_{\rm c}=1022$) is at RA  $\rm 17^{h}17^{m}00\hbox{$.\!\!^{\rm s}$ }7$, Dec  $+43^{\circ }08'00''$ (1950.0). The X,Y pixel coordinates, with their respective origins in the East and South, correspond to those in Table 1A (see Appendix).
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	Figure 2: The finding chart of the M92W field. North is up, east to the left. The central pixel ( $X_{\rm c}: 1042$; $Y_{\rm c}: 1024$) is at RA  $\rm 17^{h}16^{m}25\hbox{$.\!\!^{\rm s}$ }7$, Dec  $+43^{\circ }08'46''$ (1950.0). The X,Y pixel coordinates, with respective origins in the East and South, correspond to those in Table 2A (see Appendix).
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Appendix A

Table A.1: CCD photometry in M92C.

Table A.2: CCD photometry in M92W.

Table A.3: The identification numbers Nc and Nw for the 64 stars common to both the observed regions M92C and M92W. The corresponding numbers are taken from Tables A.1 and A.2.

Table A.4: In Cols. 1 and 2 the identification numbers of M92C stars Nc and the corresponding Sandage & Walker numbers (1966) Ns. In Cols. 3 and 4 the Vc and Vs magnitudes and in Cols. 5 and 6 the (B-V) color indices. The color-index (B-V) in Col. 5 is determined by transformations of linear combinations of Vilnius color-indices (see text).

Table A.5: In Cols. 1 and 2 the identification numbers of M92W stars Nw and the corresponding Sandage & Walker numbers (1966) Ns. In Cols. 3 and 4 the Vw and Vs magnitudes and in Cols. 5 and 6 the B-V color indices. The color-index (B-V) in Col. 5 is determined by transformations of linear combinations of Vilnius color-indices (see text).

Table A.6: The V magnitudes and the (B-V) color-indices of 17 stars observed by Sandage & Walker (1966) and re-observed by Sandage (1969).

Table A.7: In Col. 1 are the id. numbers of the sample of stars selected in the M92C field. In Cols. 2-4 the measured values of [Fe/H] are shown.

Table A.8: In Col. 1 are the id. numbers of the sample of stars selected in the M92W field. In Cols. 2-4 the measured values of [Fe/H] are shown.