4 Selecting cluster member candidates

4.1 Considering only CCD photometry

Following the same procedure as in Foster et al. ([1997]) and Rolleston & Byrne ([1997]), candidate cluster members were selected using the theoretical pre-main sequence isochrones of D'Antona & Mazzitelli ([1994]), and in particular those resulting from computations using opacities from Alexander et al. ([1989]) and Rodgers & Iglesias ([1992]) and the mixing model of Canuto & Mazzitelli ([1991]).

Isochrones are presented in terms of stellar effective temperature and luminosity, and need to be transformed to the observational domain of magnitude and colour. Unfortunately, these calibrations are not yet well defined for the M-type dwarfs. Stauffer et al. ([1995]) have made a comparison of the magnitudes and colours of known Pleiads with combinations of isochrones and transformations, viz. models provided by Vandenberg and Swenson (private communication) in addition to those of D'Antona & Mazzitelli ([1994]). They found that the best agreement between the theoretical tracks and the observed photometry was achieved using the D'Antona & Mazzitelli models mentioned above, combined with an ad hoc "tuned'' temperature scale. However, of the observationally defined transformations, the temperature scales from Kirkpatrick et al. ([1993]) with bolometric corrections from Bessell ([1991]) provided the closest fit to the data. Thus, we have adopted the temperature scales from Kirkpatrick et al. and bolometric corrections from a more recent paper by Bessell ([1995]) for the range $T_{\rm eff}\le 3500$ K, and the temperature scale of Bessell ([1991]) for $4000~{\rm K}\ge T_{\rm eff}>3500~{\rm K}$. For $T_{\rm eff}>4000$ K, we have used the temperature scales and bolometric corrections by Wood & Bessell (private communication) which are available via anonymous ftp from mso.anu.edu.au.

The selection regions for cluster membership in the B,B-V and R,R-I colour-magnitude diagrams were initially defined by the 80 and 120 Myr isochrones. This region was subsequently broadened to allow for an uncertainty of 0.2 mag in the distance modulus (Stock [1956]; Piskunov [1980]), an uncertainty of 0.01 mag in the reddening (Krzeminski & Serkowski [1967]), the photometric errors as listed in Table 2, and the effects of binarity. In the case of reddening, only the redder isochrone, i.e. the 80 Myr one, is shifted redwards, the bluer one not. The effect of binarity on the location of stars with respect to the isochrones depends on the frequency of binaries and the distribution of their mass ratios. However, Dabrowski & Beardsley ([1977]) have shown that the maximum increase in brightness would correspond to $\sim$ 0.8 magnitudes, and hence this has been incorporated in our bright selection limit.

Only stars fullfilling the selection criteria in both colour-magnitude diagrams were considered as candidate cluster members, and 118 stars with magnitudes in the range 14.2 < V < 20.4 were identified as such. The colour-magnitude diagrams are shown in Figs. 1 and 2.

In our previous paper, Foster et al. ([1997]) estimated the contamination due to background stars for IC 2602 using CCD observations of an offset field, and as a second approach, by comparison with previously determined star densities for the Pleiades. No offset frame has been observed in the case of Stock 2, and only the latter method can be applied here. This comparison is justified given the similar age and same richness class (Lynga [1987]) of Stock 2 and the Pleiades. In an equivalent area imaged by our CCD frames, one would expect to find 52 Pleiads. Hence, our list of CCD photometrically selected candidate members may be contaminated by background stars by up to 50%.

Figure 1: B vs. B-V colour-magnitude diagram from CCD photometry. The solid lines represent isochrones for cluster ages of 80, 100, 120, and 200 Myr, respectively. The region between the dashed lines defines the location of possible cluster members, as described in detail in Sect. 4.1. Diamonds in both colour-magnitude diagrams indicate cluster member candidates

Figure 2: Same as Fig. 1, but showing the R vs. R-I colour-magnitude diagram

 

 

Table 4: The final solutions to the proper-motion distribution model by magnitude bin

Bin	f	$\mu_{x_{{\rm c}}}$	$\mu_{y_{{\rm c}}}$	$\mu_{x_{{\rm f}}}$	$\mu_{y_{{\rm f}}}$	$\sigma$	$\Sigma_{x}$	$\Sigma_{y}$	N
$12<B\le 14$	0.875	17.140	-15.534	-0.017	-1.143	4.531	9.441	7.740	160
$14<B\le 16$	0.922	16.752	-12.951	0.208	-1.078	4.651	7.424	6.793	1306
$16<B\le 18$	0.916	15.216	-12.876	-0.245	-0.720	5.449	6.136	5.690	2241
$18<B\le 20$	0.872	14.600	-9.263	-0.675	-0.302	6.663	6.323	5.954	1756

4.2 Considering proper motion data

The first step in the analysis was to remove as many of the field stars as possible from consideration, using the plate photometry. A photometric selection was made from the photographic B,B-R colour-magnitude diagram using a 100 Myr isochrone from D'Antona & Mazzitelli ([1994]), using the temperature scale and bolometric corrections as described above, and including the transformation to the photographic photometry system as described in Sect. 3.1. Stars lying within a band 0.8 magnitudes fainter and 1.8 magnitudes brighter were selected as possible cluster members. This reduced the number of stars to be considered from more than 300 000 to $\sim7500$. The band around the isochrone was chosen rather broad because of the errors in plate photometry calibration and the position dependent colour-shift mentioned in Sect. 3.2. Using a 170 Myr isochrone instead of 100 Myr, thus following the age determination of Robichon et al. ([1997] and [1999]), would not change the result in a significant way.

The proper motions of the selected stars were analysed using the method described by Sanders ([1971]). The distribution of stars in the VPD is modeled as the sum of two bivariate Gaussians, a circular distribution for the cluster stars and an elliptical distribution for the field stars:

$$\rho\left(\mu_{x_{i}},\mu_{y_{i}}\right) = \phi_{\rm c}\left(... ...i }}\right) + \phi_{\rm f}\left(\mu_{x_{i}},\mu_{y_{i}}\right)$$ (5)

where

$$\phi_{{\rm f}} \frac{fN}{2\pi\Sigma_{x}\Sigma_{y}}$$ =

$$\exp \left\{ -\frac{1}{2} \left[ \left( \frac{\mu_{x_{i}}-\mu_{x_... ...( \frac{\mu_{y_{i}}-\mu_{y_{{\rm f}}}}{\Sigma_{y}} \right)^{2} \right] \right\}$$ (6)

is the field star distribution,

$$\phi_{{\rm c}} \frac{\left(1-f\right)N}{2\pi\sigma^{2}}$$ =

$$\exp \left\{ -\frac{1}{2} \left[ \left( \frac{\mu_{x_{i}}-\mu_{x_... ...left( \frac{\mu_{y_{i}}-\mu_{y_{{\rm c}}}}{\sigma} \right)^{2} \right] \right\}$$ (7)

the cluster star distribution, N is the total number of stars in the model, f is the fraction of field stars, $\Sigma_{x}$ & $\Sigma_{y}$ are the standard distribution of the field star positions in x and y, $\sigma$ is the standard deviation of the cluster population, $(\mu_{x_{{\rm f}}},\mu_{y_{{\rm f}}})$ & $(\mu_{x_{{\rm c}}},\mu_{y_{{\rm c}}})$ the field and cluster centres in the VPD, and $(\mu_{x_{i}},\mu_{y_{i}})$ the proper motion of the ith star in the fit. Thus, there were a total of 8 parameters to be determined. The model as formulated above contains no dependence on the spatial distribution of cluster and field stars. In their proper motion studies of Praesepe, Jones & Stauffer ([1991]) and Hambly et al. ([1995]) use a model with a uniform surface density of field stars, and a radial exponential decrease in cluster members (the characteristic radius being another free parameter) from the cluster centre. Their formulation is unsuitable for fitting the central regions of the cluster, as they were primarily interested in searching for members in the outer regions of Praesepe. We felt that an a priori assumption of the spatial distribution would discriminate against cluster members deviating from such a form and that the spatial aspect of the cluster distribution would be better analysed after a selection on the basis of proper motion.

The solution was determined iteratively using the method of maximum likelihood (see Sanders [1971] for further details), and employing the method of bisection (Press et al. [1992]). Our model was adapted from a code written by Hambly (private communication), to solve for the root of each equation in turn until all equations could be satisfied with a single set of the 8 parameters. The code was found to be insensitive to the initial estimates of the parameters, but somewhat sensitive to the adopted data points. Points lying far from the centre of the VPD caused the code to either converge on an unrealistic model of the data containing a large fraction ($\sim50\%$) of cluster members located close to the centre of the VPD, or crash as the result of a division-by-zero error (resulting from the extremely low density of field stars located far from the centre of the distribution). Thus for the purposes of determining the model, the data were restricted to stars with proper motions in the range $\left\vert\mu_x,\mu_y\right\vert\leq30$ mas/yr. The model parameters were independently determined for each of the 4 magnitude bins spanning the range $12\le B\le 20$, and the fitted parameters are shown in Table 4.

 

 

Table 5: Coordinates, photographic photometry and proper motions of 634 Stock 2 stars with a membership probability $\ge 50\%$ based on the proper motion analysis, ordered by magnitude. Only the first 10 stars are shown here. The complete table is available by anonymous ftp from the Centre de Données Stellaire, Strasbourg, or from the Armagh Observatory WWW server (ftp://www.arm.ac.uk/pub/ath/stock2/)

$\alpha$ (J2000.0) $\delta$		B	B-R	$\mu_{x}$	$\mu_{y}$
2:18:57.63	58:59:50.4	12.84	1.25	28.6	-20.5
2:16:00.47	59:28:31.2	13.16	1.12	13.0	-13.8
2:18:29.65	60:29:59.7	13.22	1.25	25.0	-17.9
2:13:44.80	59:00:50.7	13.31	1.10	13.8	-24.3
2:14:34.65	59:29:10.3	13.31	1.34	26.2	-24.4
2:19:22.58	60:21:34.1	13.36	1.26	32.3	-17.0
2:17:14.74	59:31:34.5	13.40	1.33	22.9	-13.1
2:18:00.67	60:21:11.8	13.42	1.38	23.1	-35.2
2:15:08.48	59:39:16.1	13.44	1.21	18.0	-9.8
2:16:33.93	59:24:38.2	13.55	1.16	25.0	-24.5

Membership probabilities were calculated for each star based on the model using the ratio of the densities of cluster stars ($\phi$) to cluster plus field stars ($\rho$):

$$P\left(\mu_{x_{i}},\mu_{y_{i}}\right)=\frac{\phi_{{\rm c}}\le... ...y_{i} }\right)}{\rho\left(\mu_{x_{i}},\mu_{y_{i}}\right)}\cdot$$ (8)

Stars with membership probabilities greater than 50% were considered candidate cluster members (634 in total) and are listed in Table 5.

A VPD is shown in Fig. 3 in which the cluster is clearly visible.

Figure 3: Vector-point diagram after photometric preselection. Heavy dots denote stars with a membership probability greater than 50%. The cluster is clearly separated from the background field stars. However, formal cluster members in the outer region of the distribution may not be members due to some shortcomings in the general method. See text for details. Note also that coordinates are plate coordinates transformed into milliarcseconds, but not right ascension and declination

At this stage, some shortcomings of the method in selecting cluster members become evident, especially in cases like ours where the contamination of the sample with background field stars is very high. In this context, it is interesting to note from Table 4 that the width of the Gaussian distribution fitting the cluster stars ($\sigma$) increases with increasing magnitude. We believe this to be for the following reason. With increasing magnitude, stellar images on the plates become less distinct, and their positions, measured as intensity weighted centroids, become less accurate. As a consequence, proper motions for faint stars also become less accurate. This results in a wider distribution in the VPD. The fitting procedure, however, tends to regard all stars in the outer region in the VPD which are closer to the cluster centre (in our case stars in the lower right region of the VPD) as cluster members. This also has the effect that the width of the field star distribution ( $\Sigma_{x}$ and $\Sigma_{y}$) does not increase with increasing magnitude. Another consequence is that in the combined VPD in Fig. 3, where all magnitudes are plotted together, some non-members appear to be surrounded by members in regions away from the centre of the cluster distribution. Those non-members are brighter stars for which the cluster distribution is narrower than for the faint stars.

Consequently, the process of determining membership probabilities can lead to some stars having high membership probabilities, yet lying quite far from the centre of the cluster centre in the VPD. For example, the star at $\left(-5.8, -34.7\right)$in Fig. 3 has a formal membership probability of 95%, but is extremely unlikely to be a cluster member.

The sharp cut-off dividing the "candidate members'' from the "non-members'' is another consequence of our method used for determining membership probabilities. Considering Fig. 3 and Eq. (8), stars toward the lower right region of the VPD tend to be members because of divisions by small values of $\rho$, and the sharp cut-off represents the boundary where the membership probability becomes less than 50% due to divisions by high values of $\rho$. The list of stars with membership probabilities will probably contain some non-members and the level of this contamination has been estimated by applying the fitted distribution to "altered'' datasets. The fit for the real dataset of $\left(\mu_{x},\mu_{y}\right)$was applied to the three "fake'' datasets $\left(-\mu_{x},\mu_{y}\right)$, $\left(-\mu_{x},-\mu_{y}\right)$ and $\left(\mu_{x},-\mu_{y}\right)$, and the number of stars with membership probabilities greater than 50% was determined in each case. These numbers are shown in Table 6, along with the estimated percentage of field stars calculated from the mean of the three "fake'' selections. The number of stars in each bin, $N_{\rm b}$, is slightly larger than that used to determine the model, N, since no restriction on the stellar proper motions was imposed. For comparison, if no photometric pre-selection was made, the contamination in the $16<B\le 18$ bin has been estimated to be greater than 75%. This clearly demonstrates the importance of the combined approach used here. The list of non-members will certainly contain stars that are cluster members, but whose position in the VPD places them too close to the central distribution for them to have a membership probability greater than 50%.

Thus, when compiling a list of stars, a final membership list must take into account the location of the star in the proper motion VPD as well as its membership probability.

This becomes evident once more when we plot the colour-magnitude diagram of the plate photometry (Fig. 4). The cluster members are highlighted therein as heavy dots. Given the estimated error in the photometry of $\approx 0.15$ mag both in $B_{\rm J}$ and $R_{\rm 63F}$, we cannot expect the cluster members to lie nicely on an isochrone. In order to reduce these discrepancies, it would be desireable to obtain more extensive CCD photometry covering a larger area, and to correct for the position dependent effect mentioned in Sect. 3.2. Most notably, however, the diagram suggests that pushing the faint border used to preselect member candidates (lower dotted line in Fig. 4) to an even fainter limit will result in a selection of more cluster members. However, this is merely the result of increasing contamination of the sample with field stars. The field star distribution in the VPD soon becomes broader and grows so high that it dominates the total distribution. The large majority of field stars then makes it impossible to fit a bivariate Gaussian, which reduces the effectiveness of this method for distinguishing cluster members.

In summary, our analysis has certainly freed the sample from most of the field stars. However, if one wants to obtain a sample more free of field stars, spectroscopic methods have to be used to classify spectral types and distinguish background giants from cluster stars, or to identify the cluster stars as a group with different radial velocities than the field stars.

Figure 4: Photographic colour-magnitude diagram of all stars. Stars with a membership probability greater than 50% are plotted as heavy dots. The solid line denotes the 100 Myr isochrone, the dashed lines enclose the area out of which cluster member candidates have been preselected (see text for details, Sect. 4.2)

Figure 5: The spatial positions of the stars. Upper left: membership probability $P \ge 50\%$ and photographic magnitude $B\le 16$; upper right: $P \ge 50\%$ and B > 16; lower left $P < 50\%$ and $B\le 16$; lower right: $P < 50\%$ and B > 16

 

 

Table 6: The number of stars with membership probabilities greater than 50% for the dataset (Col. 3) and the three "fake'' datasets (Cols. 4-6) used in extimating the contamination due to background objects

Mag.	$N_{\rm b}$	$\left(\mu_{x},\mu_{y}\right)$	$\left(-\mu_{x},\mu_{y}\right)$	$\left(-\mu_{x},-\mu_{y}\right)$	$\left(\mu_{x},-\mu_{y}\right)$	Cont.
$12<B\le 14$	173	28	4	3	4	14%
$14<B\le 16$	1363	119	31	19	21	20%
$16<B\le 18$	2321	228	68	50	51	25%
$18<B\le 20$	1838	259	95	82	148	42%

 

 

Table 7: Candidate members of Stock 2 selected using CCD photometry and proper motions, sorted by V-magnitude. Stars that satisfy both the CCD photometric and proper motion selection criteria are labelled "Y'' in the last column (22 stars). Five more stars which were judged as members from CCD photometry, possess proper motions that formally do not fulfill the selection criteria, but which are located close to members in the VPD diagram, and hence are judged to be possible members and are added with a "N?''. In addition we include 13 of the CCD selected candidates for which is was not possible to derive proper motions, labelled with a "?'', as they could be members

No.	$\alpha$ (J2000.0) $\delta$		V	B-V	V-R	R-I	$\mu_{x}$	$\mu_{y}$	member?
1	2:14:24.80	59:25:39.1	14.28	1.30	0.61	0.68	11.8	-22.9	Y
2	2:14:00.51	59:19:55.8	14.65	1.22	0.73	0.75	--	--	?
3	2:15:34.06	59:17:05.6	14.67	1.22	0.72	0.73	9.4	-12.7	N?
4	2:15:14.98	59:18:51.5	14.69	1.20	0.73	0.71	17.6	-19.0	Y
5	2:13:59.86	59:16:45.6	14.85	1.28	0.76	0.77	14.7	-5.0	N?
6	2:15:42.66	59:31:23.4	14.86	1.19	0.63	0.71	--	--	?
7	2:15:55.42	59:16:28.7	14.89	1.25	0.75	0.83	--	--	?
8	2:13:36.99	59:22:54.5	15.17	1.40	0.80	0.73	26.6	-13.8	Y
9	2:15:46.29	59:15:25.0	15.23	1.26	0.71	0.77	13.8	-9.2	Y
10	2:15:55.36	59:17:45.5	15.26	1.40	0.81	0.78	18.7	-14.2	Y
11	2:15:40.28	59:11:34.6	15.32	1.31	0.77	0.85	--	--	?
12	2:16:01.51	59:14:06.9	15.41	1.43	0.82	0.85	21.1	-11.9	Y
13	2:15:04.19	59:15:43.3	15.46	1.54	0.80	0.76	13.8	-13.7	Y
14	2:14:38.74	59:22:16.3	15.73	1.42	0.73	0.80	16.3	-7.5	Y
15	2:14:17.63	59:25:12.0	15.88	1.46	0.75	0.83	10.0	-11.9	N?
16	2:14:42.58	59:12:48.9	15.92	1.56	0.99	0.84	--	--	?
17	2:14:19.26	59:15:53.7	16.15	1.46	0.94	0.82	--	--	?
18	2:14:12.06	59:30:22.1	16.38	1.65	0.94	0.84	14.8	-15.2	Y
19	2:15:01.19	59:08:25.1	16.46	1.75	1.13	1.10	12.8	-12.9	Y
20	2:14:58.08	59:16:08.1	16.49	1.56	0.97	0.90	13.1	-15.6	Y
21	2:15:42.01	59:31:26.7	16.49	1.57	0.94	0.98	--	--	?
22	2:13:45.01	59:20:43.9	16.55	1.61	0.93	0.92	18.7	-15.4	Y
23	2:15:06.70	59:22:37.9	16.70	1.71	0.85	1.27	16.6	-16.2	Y
24	2:14:10.64	59:23:40.5	17.16	1.79	1.07	1.04	15.3	-9.3	Y
25	2:15:13.90	59:23:51.6	17.27	1.78	0.89	1.29	12.8	-12.8	Y
26	2:15:44.26	59:20:50.0	17.41	1.73	1.09	1.10	--	--	?
27	2:15:04.66	59:26:22.7	17.55	1.79	0.91	1.31	14.6	-6.3	N?
28	2:14:06.11	59:21:31.1	17.57	1.78	1.04	1.05	10.1	-15.7	Y
29	2:14:57.79	59:13:07.8	17.83	1.83	1.29	1.21	--	--	?
30	2:15:09.72	59:19:19.3	17.95	1.87	1.25	1.20	15.6	-13.3	Y
31	2:13:43.84	59:17:28.0	18.21	1.77	1.20	1.39	18.9	-2.5	N?
32	2:14:23.83	59:15:37.2	18.26	1.80	1.28	1.26	20.3	-9.4	Y
33	2:14:12.03	59:13:26.9	18.29	1.85	1.26	1.23	--	--	?
34	2:13:47.03	59:31:37.2	18.48	1.83	1.37	1.48	28.5	-16.9	Y
35	2:13:46.62	59:16:03.7	18.90	1.82	1.22	1.53	--	--	?
36	2:13:44.81	59:20:09.2	18.93	1.81	1.26	1.40	26.4	-13.6	Y
37	2:14:59.51	59:17:05.3	18.96	1.94	1.37	1.48	--	--	?
38	2:13:55.33	59:22:18.1	19.08	1.87	1.28	1.34	18.6	-15.9	Y
39	2:14:19.69	59:23:57.6	19.30	1.91	1.16	1.58	13.2	-10.6	Y
40	2:15:54.71	59:23:20.0	19.82	1.91	1.45	1.54	--	--	?

4.3 Spatial distribution of candidate members

The spatial distribution of the photometrically preselected stars along with the subset that have membership probabilities greater than 50% are shown in Fig. 5. In the upper two panels of this figure, only stars with a membership probability of 50% and more are plotted, and stars with a lower membership probability are shown in the bottom two panels. Stars brighter than or equal to B=16 are plotted to the left, stars fainter than that to the right. This figure clearly shows the possible problem resulting from the lack of calibrating stars across the entire field of interest, namely, the distribution of photometrically selected stars is highly uneven, and is magnitude dependent. As was argued in Sect. 3.2, since it is uncertain to what extent this is a real effect as opposed to just incorrect calibration, no ad hoc changes to the photometric selection have been made.

Figure 6: The difference between the star density distributions of members and non-members. The diamonds indicate the position of previously determined cluster members taken from Krzeminski & Serkowski ([1967])

Figure 7: Proper motions for stars with CCD photometry. Diamonds denote those stars which fulfill both selection criteria in the (B,B-V) and (R,R-I) CMDs. Filled symbols represent those stars with a membership probability greater than 50% based on the proper motion membership criteria. There are 22 such stars (see Table 7), but only 21 appear in this plot because two stars have almost identical proper motions

Figure 5 also indicates that some of the structure seen in the distribution of the candidate members may be the result of the background structure. A clearer indication of the cluster can be seen when the positional distributions are converted into star densities, and the photometrically selected star density is subtracted from the candidate member density, as shown in Fig. 6. The cluster is clearly visible as a density enhancement. Given that the plate scale is 67.14 arcsec/mm, this would imply that the cluster has a spatial diameter of approximately 30 arcmin. This is much smaller than the value of 60 arcmin quoted by Lynga ([1987]) based on the distribution of early-type cluster members. Figure 6 also shows the positions of 34 stars previously determined to be cluster members by Krzeminski & Serkowski ([1967]). These positions agree with our density distribution. The small diameter of the peak in the density may be indicative of the cluster core rather than the entire cluster. Certainly, the list of stars with membership probabilities less than 50% contains some cluster members, and so the spatial density of the "non-members'' will be enhanced, which may explain why only the cluster core is evident.

4.4 Analysis of members selected from CCD photometry

The proper motions of all the stars with CCD photometry are plotted in Fig. 7, showing the positions of the candidate members as open diamonds. Combining the results of the proper motion study, we find 22 stars with membership probabilities greater than 50%. This would imply that the list of candidate members as based on CCD photometry alone is contaminated by up to 80% with non-members. The level of contamination in the sample can be estimated from the number of stars located at a similar distance from the origin in a different quadrant of the plot. It is clear that the final candidate list is likely to contain at most one or two non-members.

Figure 7 shows one of the problems with the determination of the membership probabilities as discussed in the previous section. There are roughly 6 stars with proper motions that lie close in the VPD to the sharp cut-off boundary illustrated in Fig. 3. These stars have membership probabilities that are slightly less than 50% and hence have been formally deemed "non-members'' in the proper motion selection process. For this reason, these stars are listed in Table 7 with membership "N?''.