4 Determination of proper motions and cluster membership

 

4.1 Proper motion determination

  In the proper motion determination we followed the same principles as in the proper motion study in the field around the globular cluster M3 (Scholz et al. 1997). The idea is to build up a time series of coordinates for each single object measured on n plates. After the plate matching of all comparison plates with respect to one reference plate (the POSS1 plate e1457), all matched objects are used in plate-to-plate solutions for transforming the measured coordinates on the comparison plates into the coordinate system of the reference plate. The plate-to-plate solutions are done with 3rd order polynomials and the method of stepwise regression described in Hirte et al. (1990). For the proper motion study we used only the Palomar plates measured with PDS measuring machines. Whereas for the blue POSS1 plate o1457 no independent astrometric measurements were available (see Sect. 2.2), we did not use the Tautenburg plates in the proper motion study since the epoch difference compared to the latest Palomar plate was only small and the Tautenburg plates did not go as deep as the Palomar ones.

Scholz et al. (1997) used exclusively Tautenburg Schmidt plates measured on the APM with a field of about $3\hbox{$^\circ$}\times 3\hbox{$^\circ$}$. For the absolute proper motion determination of the globular cluster Pal5, Scholz et al. (1998) combined Palomar, UKST and Tautenburg Schmidt plates measured on the APM using a smaller 1 square degree field. In both of these fields due to their high Galactic latitude large numbers of reference galaxies were available. Therefore, a zero point correction was applied by subtracting the mean coordinate difference of all galaxies measured on the comparison plate and on the reference plate before the proper motion determination.

In the low Galactic latitude field of IC348 ($b \sim -18\hbox{$^\circ$}$) with its large interstellar extinction no background galaxies are available so that we can only obtain relative proper motions. Therefore, we repeated the plate-to-plate solutions after a preliminary proper motion determination and selection of all objects with small proper motions ($\vert\mu\vert<15$ mas/yr). The relative proper motions of all objects with respect to the objects with small proper motions from the first run were then determined from the linear regression of the coordinates (x,y)_j over the epochs Ep_j, with $j=1...~n_{\rm pl}$ where the number of plates $n_{\rm pl}$ for a given object is mainly dependent on its magnitude.

After the plate-to-plate solutions by stepwise regression with 3rd order polynomials and before the proper motion solution by linear regression of the coordinates over the epochs, we investigated the coordinate differences between each comparison plate and the reference plate for systematic effects as a function of the position of the stars in the field. No significant systematic errors were found. We repeated this error investigation taking the POSS2 R plate as reference plate and did also not find significant systematic errors.

Different proper motion determinations by using different subsets of plates were carried out and the results compared with the proper motions of common bright stars from the catalogue of Fredrick (1956). Using 12 stars from Fredrick (1956) with V>11 (see Sect. 5.5) the dispersion of the proper motion differences decreased from about 4mas/yr with the proper motion determination from one pair of plates (POSS1 + POSS2 R plates) to about 3mas/yr with three POSS plates (POSS1 + POSS2 R plates + POSS Quick survey plate) and 2.5mas/yr with all four available POSS plates.

  

Figure 4: Proper motions in the Hipparcos system of all 1431 stars of our sample in the one square degree field around IC348. In the proper motion determination only the DSS frames of four POSS plates were used. pmx corresponds to $\mu_{\alpha}\cos{\delta}$, pmy corresponds to $\mu_{\delta}$. One can see a slightly elongated distribution in the direction of the standard antapex. The scale of the figure is the same as in Fig. 7 so that this proper motion diagram can be directly compared with those from other proper motion catalogues. The formal proper motion errors of our sample obtained from the linear regression of 4 (minimum 3) plate coordinates over the plate epochs are about 3 mas/yr in both components. This accuracy was confirmed by the comparison with the highly accurate proper motions from Fredrick (1956), see Sect. 5.5. The dependence of our proper motion errors on magnitude are shown in Fig. 3

Figure 4 shows the proper motion diagram of the complete sample of 1431 stars. Here and throughout the paper $(\mu_x,~ \mu_y)$ correspond to $(\mu_{\alpha}\cos{\delta},~ \mu_{\delta})$. The proper motions have been transformed to the Hipparcos system which is tied to the extragalactic reference system (Kovalevsky et al. 1997), in two steps (Sect. 5.5). First they were transformed to the system of Fredrick (1956) by Eq. (6) and then we applied the correction of the Fredrick system to the Hipparcos system given by Eq. (7).

The mean proper motion errors of all stars are $\sigma_{\mu_x}=3.0$mas/yr and $\sigma_{\mu_y}=3.3$mas/yr. Figure 3 shows how the proper motion errors change with the magnitudes of the stars. For all 820 stars with R < 17 the mean proper motion errors are $\sigma_{\mu_x}=2.4$mas/yr and $\sigma_{\mu_y}=2.7$mas/yr, respectively.

4.2 Cluster membership

  Cluster membership can be estimated from kinematical data (i.e. proper motions and/or radial velocities), from the number density of the stars around an assumed cluster centre and from the location of the stars in observed colour-magnitude diagrams (CMDs). Distance measurements of sufficient numbers of member candidates of a cluster are usually not available to be included in membership studies. Hipparcos (ESA 1997) only recently provided an all-sky catalogue of proper motions and parallaxes for nearly 120 000 stars with a limiting magnitude of $V \sim 12.4$, but completeness only to V < 8.5. We will discuss in Sect. 5.4 how these data can be used for the case of IC348.

Fredrick (1956) discussed the membership of 38 bright stars around IC348 for each star individually on the basis of its location in the proper motion diagram (see Fig. 7d), the position projected on the sky (see Fig. 1) and the location in the CMD (visual magnitudes versus (B-V) colour indices, not shown here). Depending on the selection criteria, between 17 possible and 8 highly probable cluster members were found among the 38 stars.

Significantly larger numbers of member candidates were recently obtained on the basis of a near-infrared imaging survey (Lada & Lada 1995), ROSAT X-ray observations (Preibisch et al. 1996) and UBVRI CCD photometry (Trullols & Jordi 1997). Lada & Lada (1995) estimated that 380 NIR sources, i.e. the majority of the stars observed in their field of investigation (see Fig. 1), are members of the cluster. Preibisch et al. (1996) found 56 stars to be probable new members, presumably weak line T Tauri stars, because of their X-ray properties. Trullols & Jordi (1997) classified 114 out of 123 stars investigated in the central region of IC348 (see Fig. 1) as cluster members on the basis of colour-colour and colour-magnitude diagrams.

For the determination of membership probabilities of much more distant but well concentrated (in projected positions) globular clusters as well as open clusters on the basis of Schmidt plate measurements we have discussed and applied different methods (see Scholz & Kharchenko 1994 for the globular cluster M3, Kharchenko et al. 1997 for M5 and Kharchenko & Schilbach 1995 for a study of open clusters). In the last investigation of M5 we determined membership probabilities of the globular cluster in a four-dimensional space of proper motions and coordinates using the maximum likelihood method. Different mean absolute proper motions and dispersions were found for three distribution functions of the cluster stars, near field stars and distant field stars. In addition to the four-dimensional distribution functions, we also investigated two-dimensional distribution functions including either only the proper motions or only the coordinates.

In the present case of IC348 we decided not to include the coordinates and to investigate only two-dimensional distribution functions in the proper motions. The reason for that was the relatively bad measurement of the central cluster region on the Schmidt plates due to strong image crowding of bright stars and nebulosities as well as the apparently weak concentration of the fainter stars around the assumed cluster centre. In all other aspects we followed the same principles of the determination of the parameters of three distribution functions (foreground field stars, cluster stars, background field stars) by the maximum likelihood method as described in Kharchenko et al. (1997).

  

Figure 5: Central part of proper motion diagram for different subsamples of stars used in the membership determination. Shown are the stars with proper motion errors < 4mas/yr, within a circular region around IC348 ($r_{\rm max}=30$arcmin) with different limiting magnitudes a) $R_{\rm max}=14.5$, b) $R_{\rm max}=15.5$, c) $R_{\rm max}=16.5$ and d) $R_{\rm max}=17.5$. The sample shown in b) was finally selected for the determination of the parameters of three distribution functions (foreground field stars, cluster stars, background field stars) by the maximum likelihood method

For the membership determination we selected from our catalogue of 1431 stars different subsamples of stars using the following criteria: We excluded stars with large proper motion errors $\sigma_{\mu}\gt\sigma_{\mu,\rm max}$, the faintest stars $R \gt R_{\rm max}$, those with very large proper motions ($\vert\mu\vert\gt\vert\mu\vert _{\rm max}$ in the original proper motion system, not yet transformed by Eqs. (6) and (7) (see Sect. 5.5) and stars outside a preselected cluster radius $r\gt r_{\rm max}$.

  

Table 4: Mean proper motions (with their dispersion given in parentheses) of three distributions derived from a subsample of 164 preselected stars. A - cluster stars, B - distant field stars, C - foreground stars

The parameters of three distributions in the original proper motion system from the solution with 164 stars selected by $R_{\rm max}=15.5, \sigma_{\mu,\rm max}=4$mas/yr, $\vert\mu\vert _{\rm max}=30$mas/yr and $r_{\rm max}=30$arcmin are shown in Table 4. This solution was rather stable, i.e. not very sensitive to changes in the selection criteria. Particularly, the separation of two distributions with small dispersion of the same order but different mean proper motion and a third distribution with much larger dispersion was a characteristic feature of all solutions with different subsamples. Figure 5 shows as an example, how the proper motion diagram of accurately measured stars ($\sigma_{\mu,\rm max}=4$mas/yr) within a circular field around IC348 ($r_{\rm max}=30$arcmin) changes with the limiting magnitude $R_{\rm max}$.

Whereas the group C can easily be considered as near foreground stars, we have to decide which of the groups A and B represent the cluster IC348 and which the more distant field stars. The dispersions of both groups A and B are comparable to the proper motion errors in our sample. In Table 4 we have also included the parameters of the three distributions after transforming all proper motions by Eqs. (6) and (7) to the Hipparcos system. After this correction and taking into account the direction of the standard antapex, we interpret group A as a cluster at intermediate distance and group B as the more distant field stars. An additional strong argument supporting this interpretation is given by the different dependency of the two probabilities on the distance from the cluster centre (see Sect. 4.3, Fig. 6).

After the parameters of the distribution functions have been determined with a small subsample of stars, the probabilities for all stars to be a member of one of the obtained distributions were derived as:  

$$P = \exp{\left( \displaystyle -0.25\cdot\left( \left(\frac{\... ...erline{\mu_y}}{\sigma^{+1}_{\mu_y}(R)}\right)^2\right) \right)}$$ (4)

where $\overline{\mu_x}$ and $\overline{\mu_y}$ are the mean proper motions of the distribution from Table 4 and $\mu_x, \mu_y$ are the individual proper motion of a star. Instead of the dispersions given in Table 4 we preferred to use the magnitude dependent dispersions $\sigma^{+1}_{\mu_x}(R)$ and $\sigma^{+1}_{\mu_y}(R)$ obtained from the mean proper motion errors plus 1mas/yr (see Fig. 3). The magnitude dependent dispersions defined by this way vary between $\sim$3mas/yr and $\sim$4mas/yr, respectively for $\sigma^{+1}_{\mu_x}(R)$ and $\sigma^{+1}_{\mu_y}(R)$ in the magnitude interval 10.5 < R < 16.5 and rise up to $\sim$5mas/yr and $\sim$6mas/yr at R=18 (see Fig. 3).

Membership probabilities were determined for all 1431 stars of our proper motion sample, respectively determining the probability of each star to be a member of the cluster $P_{\rm cl}$ and of the distant field stars $P_{\rm df}$. For the foreground stars we have not determined a third membership probability criterion but defined them to have small values (e.g. < 5%) for both $P_{\rm cl}$ and $P_{\rm df}$.

4.3 Cluster radius

  With not including any information on the position of the stars in the membership determination (Sect. 4.2, Eq. (4)), it is straightforward to investigate the proper motion membership probabilities as a function of the position in the field. To do so, we divided the field into concentric rings around the assumed cluster centre of IC348 and computed the mean probabilities of all stars within a ring to be members of the cluster ($P_{\rm cl}$) or of the distant field stars ($P_{\rm df}$). If our interpretation of the two distributions A and B (Table 4), respectively as cluster stars and more distant field stars, is correct, we may expect no change of the mean probability $P_{\rm df}$ with the distance from the cluster centre. On the other hand we would expect a concentration of the cluster members in their positions, i.e. a decrease of the mean probability $P_{\rm cl}$with the distance from the cluster centre.

  

Figure 6: Mean membership probabilities $P_{\rm cl}$ (filled circles, dotted lines) and $P_{\rm df}$ (open circles, dashed lines) of all stars in concentric rings around the assumed centre of IC348 - a) for all stars with R < 16, b) for all stars with R > 16

Figure 6 shows the change of the mean probabilities $P_{\rm cl}$ (filled circles and dotted lines) and $P_{\rm df}$ (open circles and dashed lines) for the bright stars (R < 16) and faint stars (R > 16), respectively. As expected, the mean probability $P_{\rm df}$ does not show significant variation over the field. But for the mean cluster membership probabilities based only on the proper motion data and obtained without any positional parameters, we see a strong correlation with the distance from the cluster centre, if we look at the brighter stars (Fig. 6a). For the faint stars (Fig. 6b), there is only a small difference between the mean probabilities $P_{\rm cl}$ and $P_{\rm df}$ in dependence on the distance from IC348. This is due to the larger overlap of the two membership probabilities obtained by Eq. (4) and the use of the magnitude dependent dispersions $\sigma^{+1}_{\mu_x}(R)$ and $\sigma^{+1}_{\mu_y}(R)$ instead of the constant dispersion given in Table 4.

In addition to the support of our proper motion cluster membership we obtained from the result in Fig. 6a, we can also estimate the cluster radius as $\sim 10-15$arcmin. This is an important contribution to the discussion of the possible extension of the cluster IC348 (cf. Herbig 1998; Lada & Lada 1995).

The achieved accuracy of proper motions is not sufficient to substantiate a separation of the cluster into several subgroups as proposed by Lada & Lada (1995) from their NIR photometry (see also Sect. 8).

4.4 The catalogue

  The catalogue of positions $\alpha,\delta$(2000), proper motions and their errors (in the Hipparcos system), the original U, B, B_j, V, E and R magnitudes as obtained from each individual plate (without corrections of Eqs. (2) and (3)), membership probabilities $P_{\rm cl}$ of the cluster IC348 and $P_{\rm df}$ of the group of distant field stars, respectively are given for all 1431 stars. This table is only available in machine readable form from the CDS (cdsweb.u-strasbg.fr) via anonymous ftp.