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Astron. Astrophys. 357, 471-483 (2000)

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2. The data and data reduction

2.1. Photometry

CCD images of both clusters were taken with the 1.23 m telescope at Calar Alto Observatory on October 15, 1998, in photometric conditions. The seeing was of the order of [FORMULA]. The telescope was equipped with the [FORMULA] pix CCD chip TEK 7_12 with a pixel size of [FORMULA] and the WWFPP focal reducing system (Reif et al. 1995). This leads to a resolution of [FORMULA] and a field of view of [FORMULA]. Both clusters were observed in Johnson B and V filters, the exposure times were 1 s, 10 s, and 600 s in V, and 2 s, 20 s, and 900 s in B. Figs. 1 and 2 show CCD images of both clusters.

[FIGURE] Fig. 1. 600 s V CCD image of NGC 1960. The field of view is approximately [FORMULA], north is up, east to the left

[FIGURE] Fig. 2. 600 s V CCD image of NGC 2194. As in Fig. 1, the field of view is approximately [FORMULA] with north up and east to the left

The data were reduced with the DAOPHOT II software (Stetson 1991) running under IRAF. From the resulting files, we deleted all objects showing too high photometric errors as well as sharpness and [FORMULA] values. The limits were chosen individually for each image, typical values are [FORMULA] to [FORMULA] for the magnitudes, [FORMULA] to 1 for sharpness, and 2 to 4 for [FORMULA].

Resulting photometric errors of the calibrated magnitudes in different V ranges valid for both clusters as given by the PSF fitting routine are given in Table 1.


[TABLE]

Table 1. Typical photometric errors for stars in different magnitude ranges


The data were calibrated using 44 additional observations of a total of 27 Landolt (1992) standard stars. After correcting the instrumental magnitudes for atmospheric extinction and to exposure times of 1 s, we used the following equations for transformation from instrumental to apparent magnitudes:

[EQUATION]

where capital letters represent apparent and lower case letters (corrected as described above) instrumental magnitudes. The extinction coefficients [FORMULA] and [FORMULA], zero points [FORMULA] and [FORMULA] as well as the colour terms [FORMULA] and [FORMULA] were determined with the IRAF routine fitparams as:

[EQUATION]

We checked the quality of these parameters by reproducing the apparent magnitudes of the standard stars from the measurements. The standard deviations derived were [FORMULA] and [FORMULA].

Johnson & Morgan (1953) published photoelectic photometry of 50 stars in the region of NGC 1960. Their results coincide with ours with a standard deviation of approx. [FORMULA] in V and [FORMULA] in [FORMULA], respectively. There is only one exception, star 110 (Boden's (1951) star No. 46) for which we found [FORMULA], [FORMULA], which differs by [FORMULA] and [FORMULA] from the value of Johnson & Morgan (1953). In their photographic photometry, Barkhatova et al. (1985) found values for this star which coincide with ours. We therefore assume the difference most likely to be caused by a mis-identification of this star by Johnson & Morgan (1953).

All stars for which B and V magnitudes could be determined are listed in Tables 2 (NGC 1960, 864 stars) and 3 (NGC 2194, 2120 stars), respectively. We derived the CMDs of the two clusters which are shown in Figs. 3 and 4. A detailed discussion of the diagrams is given in Sects. 3 and 4.

[FIGURE] Fig. 3. Colour magnitude diagram of NGC 1960 and the surrounding field. This diagram contains all stars for which both B and V magnitudes were determined. Stars with too high photometric errors were excluded beforehand. This CMD is still contaminated with field stars. For a CMD which is field star corrected see Fig. 10

[FIGURE] Fig. 4. Colour magnitude diagram of all stars in the field of NGC 2194. For further remarks see Fig. 3. The star marked with a cross is claimed to be a blue straggler by Ahumada & Lapasset (1995). This statement is discussed in Sect. 4.2. Fig. 14 shows the field star corrected CMD of NGC 2194


[TABLE]

Table 2. List of the photometric data of all stars measured in the CCD field of NGC 1960. For cross-identification, the star numbers of Boden (1951) are given, too. Only the ten brightest stars for which both photometry and proper motions were determined are listed here, the complete table is available online at the CDS archive


2.2. Actual cluster sizes

Mass segregation might lead to a larger "true" cluster size than stated, e.g., in the Lyngå (1987) catalogue: While the high mass stars are concentrated within the inner part of the cluster, the lower mass stars might form a corona which can reach as far out as the tidal radius of the cluster (see, e.g., the recent work of Raboud & Mermilliod 1998). Therefore, the range of the cluster stars had to be checked. We applied star counts in concentric rings around the centre of the clusters.

Star counts in the vicinity of NGC 2194 show no significant variations of the stellar density outside a circle with a diameter of [FORMULA] (corresponding to [FORMULA] pc at the distance of the object) around the centre of the cluster. For NGC 1960, this point is more difficult to verify, since its total stellar density is much lower than for NGC 2194, so that it is not as easy to see at which point a constant level is reached, and on the other hand, its smaller distance lets us reach fainter absolute magnitudes so that the effect of mass segregation might be more prominent within the reach of our photometry. However, our tests provided evidence, too, that the cluster diameter is no larger than [FORMULA]. It must be stressed that these figures can only provide lower limits for the real cluster sizes: Members fainter than the limiting magnitude of our photometry might reach further out from the centres of the clusters.

2.3. Proper motions

For our proper motion studies we used photographic plates which were taken with the Bonn Doppelrefraktor, a 30 cm refractor ([FORMULA], scale: [FORMULA]) which was located in Bonn from 1899 to 1965 and at the Hoher List Observatory of Bonn University thereafter. The 16 cm [FORMULA] 16 cm plates cover a region of [FORMULA]. They were completely digitized with [FORMULA] linear resolution with the Tautenburg Plate Scanner, TPS (Brunzendorf & Meusinger 1998, 1999). The positions of the objects detected on the photographic plates were determined using the software search and profil provided by the Astronomisches Institut Münster (Tucholke 1992).

In addition, we used the 1 s to 20 s Calar Alto exposures to improve data quality and - for NGC 2194 - to extend the maximum epoch difference. Furthermore, a total of 16 CCD frames of NGC 1960 which were taken with the 1 m Cassegrain telescope ([FORMULA] with a focal reducing system) of the Hoher List Observatory were included in the proper motion study. The latter observations cover a circular field of view of [FORMULA] in diameter which provides a sufficiently large area for the cluster itself and the surrounding field. The astrometric properties of this telescope/CCD camera system were proven to be suitable for this kind of work in Sanner et al. (1998). The stellar [FORMULA] positions were extracted from the CCD frames with DAOPHOT II routines (Stetson 1991). A list of the plates and Hoher List CCD images included in our study can be found in Table 4.


[TABLE]

Table 3. List of the photometric data of all stars measured in the CCD field of NGC 2194. As for Table 2, only the ten brightest stars for which we derived photometric data and proper motions are mentioned here. As a cross reference, the numbers from del Rio (1980) are added. The complete table is available at the CDS archive



[TABLE]

Table 4. Photographic plates form the Bonn Doppelrefraktor (prefix "R") and CCD frames of the 1m Cassegrain telescope of the Hoher List Observatory (prefix "hl") used to determine the proper motions of the stars in and around NGC 1960 and NGC 2194. For both clusters, the short ([FORMULA]) Calar Alto CCD photometric data (see Sect. 2.1 were included in the calculations, too


The fields of the photographic plates contain only a very limited number of HIPPARCOS stars (ESA 1997), as summarized in Table 5. Therefore, we decided to use the ACT catalogue (Urban et al. 1998) as the basis for the transformation of the plate coordinates [FORMULA] to celestial coordinates [FORMULA]. For NGC 2194 this decision is evident, for NGC 1960 we preferred the ACT data, too, as the brightest HIPPARCOS stars are overexposed on several plates, thus lowering the accuracy of positional measurements: It turned out that only three of the HIPPARCOS stars were measured well enough to properly derive their proper motions from our data. The celestial positions of the stars were computed using an astrometric software package developed by Geffert et al. (1997). We obtained good results using quadratic polynomials in x and y for transforming [FORMULA] to [FORMULA] for the photographic plates and cubic polynomials for the CCD images, respectively.


[TABLE]

Table 5. Number of HIPPARCOS and ACT stars inside the field of view of the Doppelrefraktor plates used for the proper motion studies


Initial tests in the fields of both clusters revealed that the proper motions computed for some ten ACT stars disagreed with the ACT catalogue values. We assume that this is caused by the varying accuracy of the Astrographic Catalogue which was used as the first epoch material of the ACT proper motions or by unresolved binary stars (see Wielen et al. 1999). We eliminated these stars from our input catalogue.

The proper motions were computed iteratively from the individual positions: Starting with the ACT stars to provide a calibration for the absolute proper motions and using the resulting data as the input for the following step, we derived a stable solution after four iterations. Stars with less than two first and second epoch positions each or a too high error in the proper motions ([FORMULA] in [FORMULA] or [FORMULA]) were not taken into further account.

To determine the membership probabilities from the proper motions, we selected [FORMULA] wide areas around the centres of the clusters. This dimension well exceeds the proposed diameter of both clusters so that we can assume to cover all member stars for which proper motions were determined. Furthermore, this region covers the entire field of view of the photometric data. The membership probabilities were computed on the base of the proper motions using the method of Sanders (1971): We fitted a sharp (for the members) and a wider spread (for the field stars) Gaussian distribution to the distribution of the stars in the vector point plot diagram and computed the parameters of the two distributions with a maximum likelihood method. From the values of the distribution at the location of the stars in the diagram we derived the membership probabilities. The positions of the stars did not play any role in the derivation of the membership probabilities. In the following, we assumed stars to be cluster members in case their membership probability is 0.8 or higher.

2.4. Colour magnitude diagrams

Before analysing the CMDs in detail, we had to distinguish between field and cluster stars to eliminate CMD features which may result from the field star population(s). For the stars down to [FORMULA] (NGC 1960) and [FORMULA] (NGC 2194) we found after cross-identifying the stars in the photometric and astrometric measurements that our proper motion study is virtually complete. Therefore we used these magnitudes as the limits of our membership determination by proper motions. For the fainter stars we statistically subtracted the field stars:

We assumed a circular region with a diameter of 806 pixels or [FORMULA] to contain all cluster member stars. As seen in Sect. 2.2, this exceeds the diameters of the clusters. The additional advantage of this diameter of the "cluster" region is that this circle corresponds to exactly half of the area covered by the CCD images so that it was not necessary to put different weights on the star counts in the inner and outer regions. We compared the CMDs of the circular regions containing the clusters with the diagrams derived from the rest of the images to determine cluster CMDs without field stars. The method is described in more detail in, e.g., Dieball & Grebel (1998).

We fitted isochrones based on the models of Bono et al. (1997) and provided by Cassisi (private communication) to the cleaned CMDs. We assumed a Solar metallicity of [FORMULA] and varied the distance modulus, reddening, and ages of the isochrones. Comparison with the isochrones of other groups (Schaller et al. 1992, Bertelli et al. 1994) does not show any significant differences in the resulting parameters.

2.5. Mass function

For the IMF study it is important to correct the data for the incompleteness of our photometry. With artificial star experiments using the DAOPHOT II routine addstar we computed B-magnitude depending completeness factors for both clusters. The B photometry was favoured for these experiments since its completeness decreases earlier as a consequence of its brighter limiting magnitude. According to Sagar & Richtler (1991), the final completeness of the photometry after combining the B and V data is well represented by the least complete wavelength band, hence V completeness was not studied. The results, which are approximately the same for both NGC 1960 and NGC 2194, are plotted in Fig. 5: The sample is - except for a few stars which likely are missing due to crowding effects - complete down to [FORMULA], and for stars with [FORMULA], we still found more than 60% of the objects. In general, we found that the completeness in the cluster regions does not differ from the values in the outer parts of the CCD field. We therefore conclude that crowding is not a problem for our star counts, even at the faint magnitudes. However, crowding may lead to an increase in the photometric errors, especially in the region of NGC 2194, in which the stellar density is considerably higher than for NGC 1960.

[FIGURE] Fig. 5. Completeness of the 900 s B exposures of NGC 1960 (solid line) and NGC 2194 (dotted line). Down to [FORMULA] both samples are at least 60% complete. Note that the IMF is computed on the base of the V magnitudes which alters the completeness function as we deal with main sequence stars with a colour of up to [FORMULA]

Several objects remained far red- or bluewards of the lower part of the main sequence after statistical field star subtraction. We assume that this results from the imperfect statistics of the sample. For a formal elimination of these stars we had to define a region of the CMD outside of which all objects can be considered to be non-members. This was achieved by shifting the fitted isochrones by two times the errors listed in Table 1 in V and [FORMULA] to the lower left and the upper right in the CMD (Since this procedure applies only to stars within the range of the statistical field star subtraction, we used the errors given for the faint stars in our photometry.). To take into account probable double or multiple stars we added another [FORMULA] to the shift to the upper right, and for NGC 2194 we allowed another [FORMULA] in the same direction as a consequence of the probably higher photometric errors due to crowding in the central part of the cluster. All stars outside the corridor defined in this way are not taken into account for our further considerations. The shifted isochrones are plotted as dotted lines in Figs. 10 and 14, respectively. It may be remarked that according to Iben (1965) we can exclude objects with a distance of several magnitudes in V or a few tenths of magnitudes in [FORMULA] from the isochrone to be pre-main sequence members of neither NGC 1960 nor NGC 2194.

We furthermore selected all objects below the turn-off point of the isochrones. For the remaining stars, we calculated their initial masses on the base of their V magnitudes. We used the mass-luminosity relation provided with the isochrone data. V was preferred for this purpose as the photometric errors are smaller in V compared to the corresponding B magnitudes. The mass-luminosity relation was approximated using [FORMULA] order polynomials

[EQUATION]

which resulted in an rms error of less than 0.01. Using [FORMULA] or lower order polynomials caused higher deviations especially in the low mass range. The values of the parameters [FORMULA] are listed in Table 6.


[TABLE]

Table 6. Parameters of the polynomial approximation of the mass-luminosity relation for the stars of the two clusters. See Eq. (6) for the definition of [FORMULA]


Taking into account the incompleteness of the data, we determined the luminosity and initial mass functions of the two clusters. The IMF slope was computed with a maximum likelihood technique. We preferred this method instead of the "traditional" way of a least square fit of the mass function to a histogram, because those histogram fits are not invariant to size and location of the bins: Experiments with shifting the location and size of the bins resulted in differences of the exponent of more than [FORMULA]. Fig. 6 shows the results of such an experiment with the NGC 1960 data. The fitted IMFs show an average [FORMULA] value of around -1.2 with individual slopes ranging from -1.1 down to -1.4. This can be explained by the very small number of stars in the higher mass bins which contain only between one and ten stars. In case only one member is mis-interpreted as a non-member or vice versa, the corresponding bin height might be affected by up to [FORMULA] in the worst case which will heavily alter the corresponding IMF slope. In addition, all bins of the histogram obtain the same weight in the standard least square fit, no matter how many stars are included. For very populous or older objects (globular or older open star clusters, see, e.g., the IMF of NGC 2194) this effect plays a minor role, because in these cases the number of stars per bin is much higher. On the other hand, the maximum likelihood method does not lose information (as is done by binning), and each star obtains the same weight in the IMF computation.

[FIGURE] Fig. 6. Results of an experiment with part of the NGC 1960 data. We computed histograms with an equal bin width of [FORMULA] (as in Fig. 12), but varying location of the bins, expressed by the [FORMULA] value of the left corner of the leftmost bin. This diagram shows the resulting IMF slope [FORMULA] and the corresponding error of the fit

Nevertheless, for reasons of illustration we sketch the IMF of our two clusters with an underlying histogram in Figs. 12 and 16.

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Online publication: June 5, 2000
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