The study of embedded clusters can provide the key to a better understanding of a number of fundamental problems in astrophysics, e.g. distances, masses, ages, Metalicity. Moreover, the analysis of the complex gravitational interactions of their individual members contributes to the study of stellar dynamics.
Several of these clusters have been investigated using only photometric techniques to determine the members. As open cluster members share similar photometric and kinematic properties, it is useful to complete the study of the stars of the cluster region by employing astrometric methods. Recently, Corti and Orellana  and Orellana et al.  combine astrometric and photometric membership results in order to reduce the uncertainty introduced by each technique individually.
In this work we improve the membership probability of the stars in the regions of van den Bergh open clusters vdB92, vdB146 and vdB150.
Open cluster vdB92: has equatorial coordinates and (Bonatto and Bica  ) and is placed in Canis Major. It was first identified by van den Bergh  and contains the stars BD-11 1763 (UCAC4 393-021223) and BD-11 1761 (UCAC4 393-021212). Racine  performed photometry and spectroscopy for 92b (UCAC4 393-021223) and photometry for 92a (UCAC4 393-021212) and for 92c (UCAC4 393-021222). Soares and Bica  employ 2MASS photometry to study the brighter sequences and derive the cluster age of 5-7 Myr and a distance of 1.5 kpc. Froebrich et al.  list the cluster as FSR 1188 and derive a core radius of 1.1', a tidal radius of 24' and 297 stars as members based on H photometry. A more extensive analysis up to 2MASS fainter magnitudes is done by Bonatto and Bica  . They derive the fundamental parameters from a colour-magnitude diagram characterized by a poorly populated main sequence and a dominant fraction of pre-MS stars. They obtain , , giving a distance of kpc, a value of , for the central coordinates and for the cluster radius. Dias et al.  employ the UCAC4 catalogue to determine the mean proper motion and stellar membership in a region centred in , containing 36 stars up to magnitude 17. They find a value of ( , ) mas/yr, a radius of 2.5' and 26 cluster members.
Open cluster vdB146: is situated in Cepheus and its coordinates are and (Dias et al.  ). When van den Bergh  catalogues the reflection nebula, he remarks that stars BD + 65 1637 (UCAC4 781-037984) and BD + 65 1638 (UCAC4 781-037989) are in a small clustering. The reflection nebula vdB-RN146 is also named as NGC 7129 in the literature. In 1968 Racine identifies two stars more as cluster members: LkHa234 (UCAC4 781-037993) and vdB146d (UCAC4 781-037976). Aveni and Hunter  perform photometry and spectropscopy of 30 stars in the cluster as a part of their observational studies relating to star formation. They determine 9 cluster members, a distance modulus 10.5 and a total mass of 120 Me.
More recently, an important investigation of vdB146 and the dust cloud where it is embedded is made by Straižys et al.  using medium-band seven color photometry of 159 stars in the region and determining the spectral and luminosity classes for half of them. They locate the cloud and the cluster at kpc, and calculate the cluster age between 3.1 and 0.20 Myr employing six cluster members of spectral clases B3 to A1.
Open cluster vdB150: discovered in 1966 by van den Bergh, it is placed in Cepheus and its coordinates are and (Dias et al.  ). van den Bergh  finds a reflection nebula associated with the star BD +72 1020 (UCAC4 817-026634), to which Racine  performs photometry and spectroscopy to determine a distance modulus of 8.5. Aveni and Hunter  analyse a sample of 19 stars in the area and report the possible existence of a widespread group at a distance modulus of 7.5.
The images of the optical environments of the three open clusters are taken from the Digital Sky Survey (DSS) (Figure 1).
Figure 1. Images of the optical environments of open clusters vdB92 (a); vdB146 (b) and vdB150 (c). Right ascension and declination are given in the abscissa and ordinate, respectively. Orientation: north to the top and east to the left. We thank the Leicester Database and Archive Service (LEDAS) http://www.ledas.ac.uk/DSSimage/aboutdss and the Space Telescope Science Institute (STScI) for the use of their images.
2. The Data
We perform our investigation taken the stellar position and proper motions from the UCAC4 Catalogue, which has 113 million stars and 105 million of them with proper motions. The catalogue is complete to about magnitude R = 16 having an average density over 2000 stars/(')2.
Positional errors are about 15 to 20 mas for stars in the 10 to 14 mag range. Errors in proper motions of the bright stars (to ) run from about 1 to 3 mas/yr, typical errors of the fainter stars using SPM and NPM data are at the level of 2 to 6 mas/yr.
These data are supplemented by 2MASS photometric data for about 110 million stars and 5-band (B, V, g, r, i) photometry from the APASS (AAVSO Photometric All-Sky Survey) for over 50 million stars. A detailed description of the construction of UCAC4 catalogue can be found in Zacharias et al.  .
Data extraction has been performed using the Vizier1.
3. Astrometric Analysis
3.1. Cluster Centre Coordinates, Mean Proper Motion and Membership Determination
Astrometrically, the existence of an open cluster is pointed out by an overdensity in the sky as well as in the Vector Point Diagram (VPD). The identification of members of the cluster with respect to field stars is done by evaluating both overdensities simultaneously.
The projected stellar density of each cluster is analysed in a region whose size and magnitude limit depend on local characteristics. We consider the faintest magnitude up to which an overdensity can be distinguished. Up to the 14 magnitude, we examine a region sized of for vdB92. Up to the 15 magnitude, we examine another of for vdB146. A region of is chosen for vdB150 analysis up to magnitude 13.
The observed local density is evaluated at the nodes of a grid of a given size by adding the stars within a circle whose radius is weighted by a smoothing parameter following Stock and Abad’s rule (Stock and Abad  ). The node with the highest density is adopted as the centre of the cluster with coordinates ( , ). It represents an approximate value of the cluster centre because it is determined only considering the spatial over-density. The grid and radius sizes depend on the local cluster regions, as shown in Table 1.
The analysis is continued by adopting the proper motion distribution of Vasilevskis-Sanders (Vasilevskis et al.  -Sanders  ) for cluster and field stars. The inclusion of an exponential function describing the areal stellar density for cluster stars and a constant one for field stars (Jones and Walker  ) improves the model.
The proper motion distribution for the i-th star takes the form:
Table 1. Sizes of the grid and radius selected to evaluate the local density in the region of the clusters and the approximate cluster centre coordinate ( , ).
where , , are components of proper motion in x and y and the distance from the approximate cluster’s centre for the i-th star, is the central cluster stellar density, is the characteristic radius, is the constant areal stellar density for field stars, , are the field mean proper motion; , the elliptical dispersions for field stars, , the cluster mean proper motion, and its circular dispersion for cluster stars. These parameters are obtained after applying the method of maximum likelihood to Equation (1).
The probability for the i-th star is calculated as
A cluster member is found when .
Open cluster vdB92:
We choose the stars in a circular region of radius 30' centered at the given in Table 1 which contains 569 stars. The proper motion data are analysed in an elliptical subregion of the VPD containing 523 stars.
The parameters , and adjust the function to the radial stellar density profile.
After Equation (1) is resolved, the cluster parameters are , , and the field parameters , , and .
60 stars are found to be astrometric members of the cluster. Their coordinates lead to calculate the equatorial coordinates of the centre of the cluster , and an amount of 17.11' for the radius, in good agreement with Bonatto and Bica’s value.
The location of the members in the spatial distribution and in the VPD is shown with black circles in Figure 2(a) and Figure 3(a), respectively. Table 2 gives our numbering system, the UCAC4 number, the equatorial coordinates with their errors, the components of proper motion with their errors and the membership probability ( ).
Figure 2. Stellar positions in the region of vdB92 (a); vdB146 (b) and vdB150 (c). The black circles represent the astrometric cluster members and crosses the rest of the stars in the ellipse.
Table 2. vdB92 astrometric members.
Figure 3. Vector point diagram in the region of vdB92 (a); vdB146 (b) and vdB150 (c), proper motions are in units of mas/yr. The black circles represent the astrometric cluster members and crosses the rest of the stars in the ellipse.
Dias et al.  published an astrometric research using UCAC4 catalogue. They analysed a small region of diameter 3' centred at , and detected an open cluster whose components of mean proper motion are and having 26 members. The area considered in their investigation does not include the cluster members observed by Racine  and the ones mentioned by Bonatto and Bica  .
Open cluster vdB146:
We take 63 stars in a circular region of radius 10' centered at , . The elliptical subregion of the VPD contains 59 stars.
The parameters , and fit the radial stellar density profile.
The cluster parameters take the values , , and the field parameters , , and .
We find five astrometric cluster members and then calculate the equatorial coordinates of the centre of the cluster , and the value of 4.31' for the radius. The location of the members in the spatial distribution and in the VPD is shown with black circles in Figure 2(b) and Figure 3(b), respectively. Table 3 gives our numbering system, the UCAC4 number, the equatorial coordinates with their errors, the components of proper motion with their errors and the membership probability ( ).
Open cluster vdB150:
The model is applied in a circular region of radius 30' and centered at , with 84 stars. The elliptical subregion of the VPD contains 77 stars.
The parameters , and adjust the radial stellar density profile.
The cluster parameters take the values , , and the field parameters , , and .
Seven stars are found to be the astrometric members of the cluster, three of them are fainter than 13 mag. Therefore, the equatorial coordinates of the centre of the cluster and radius take the values , and 7.25' respectively.
The location of the members in the spatial distribution and in the VPD is shown with black circles in Figure 2(c) and Figure 3(c), respectively. Table 4 gives our numbering system, the UCAC4 number, the equatorial coordinates with their errors, the components of proper motion with their errors and the membership probability (Pci).
In summary, the centre coordinates and components of the mean proper motion of each cluster are shown in Table 5.
3.3. Proper Motion Errors
We analyse the incidence of stellar proper motions errors in the determination of the cluster parameters and stellar membership. We follow the method proposed
Table 3. vdB146 astrometric members.
Table 4. vdB150 astrometric members.
Table 5. Centre coordinates, mean proper motion, number of members N and diameters D of the clusters.
by Chen et al.  . We have generated 25 simulations by adding an increment to the real values of each star in the region of each cluster. After this, we apply the procedure presented in Subsection 1 to each simulated sample and obtain the cluster parameters and the astrometric members. We then calculate the mean value of and the mean number of members, as shown in Table 6.
Comparing these results to those obtained in Subsection 3.2 (see Table 5), it is easy to see that the error does not significantly change the kinematic parameters of the studied clusters.
In addition, the effect of the proper motion errors on the determination of cluster members for each cluster is examined. Therefore, we count the number
Table 6. Influence of observational errors: means of the mean proper motion and of the number of members N from the 25 simulated samples for each cluster.
of times in which a cluster member obtained in Subsection 3.2 keeps its membership condition throughout the 25 simulations.
Figure 4 shows these results by a histogram for each cluster, where the abscissa shows our numbering system for vdB92, vdB146 and vdB150 respectively.
It is easy to see that between 80% and 90% of the members maintain their condition, despite the proper motion errors.
We compare the stellar astrometric membership condition in the regions of vdB92, vdB146 and vdB150 with the photometric one given in the literature.
Open cluster vdB92:
In 2010, Bonatto and Bica made an infrared photometric analysis using 2MASS data and derived fundamental parameters. We take these parameters to plot the CMD of the 60 astrometric members (Figure 5). Up to 13 magnitude, we compare their distribution with the one of the photometric members drawn in Figure 6 right of Bonatto and Bica  . We find that three astrometric members shown with a triangle in Figure 5 are not photometric members.
In their investigation, Bonatto and Bica  calculate the average spectroscopic distance of five photometric members using SIMBAD data, in excellent agreement with the photometric value. It is important to remark that four of these stars-HRW14 (UCAC4 393-021237), BD-11 1763 (UCAC4 393-021223), HIP 34133 (UCAC4 393-021497) and NSV 3364 (UCAC4 393-021345)-are astrometric members.
Open cluster vdB146:
We analyse the astrometric membership condition of the nine cluster members obtained by Aveni and Hunter  . Only five members fulfil both astrometric and photometric condition, they are stars No. 1 (UCAC4 781-037989), No. 2 (UCAC4 781-037993), No. 4 (UCAC4 781-037984), No. 7 (UCAC4 781-038012) and No. 10 (UCAC4 781-037976). The other ones are not astrometric members for different reasons: star No. 3 (UCAC4 781-037991) cannot be evaluated by our astrometric analysis as it has no proper motion components, stars No. 23 (UCAC4 782-037014) and No. 24 (UCAC4 780-039381) are located outside the cluster radius and finally star No. 9 (UCAC4 781-037982) does not fulfil the condition due to the value of its proper motion. These results are shown in Table 7.
Figure 4. Histogram showing the probability in which a cluster member previously determined keeps its membership condition throughout the 25 simulations for vdB92, vdB146 and vdB150.
Figure 5. Colour-magnitude diagram of vdB92 astrometric members using Bonatto and Bica (2010) parameters. The ones symbolized by a triangle are not photometric members according to Figure 6 right of the mentioned article.
Figure 6. VPD of the eight Aveni and Hunter’s photometric members for vdB146 plotted with crosses, while the five astrometric members are plotted with large circles.
Taking into account that cluster members must share similar kinematic features, the VPD of the photometric members (Figure 6) explains the reason why stars No. 9, No. 23 and No. 24 cannot be considered members of the cluster. The photometric members are plotted with crosses and the five astrometric members with large circles.
Straižys et al.  find that the young age of the cluster makes that only six cluster members are located close to the main sequence and the majority of stars are on the pre-main-sequence. These stars are BD + 65 1637 (UCAC4 781-037984), BD + 651638 (UCAC4 781-037989), LkHa234 (UCAC4 781-037993), No. 96 (UCAC4 781-037976), No. 105 (UCAC4 781-037982) and No. 154 (UCAC4 781-038012). Note that the former five stars are members for Aveni and Hunter  .
While comparing our results with these six cluster members, it can be seen that only star No. 105 (UCAC4 781-037982) does not fulfil the astrometric membership condition due to its proper motion. These results are shown in Table 7.
Open cluster vdB150:
Aveni and Hunter  examine a sampling of 19 stars in the area surrounding the reflection nebula vdB-RN150 in order to interpret the nature of the loose group of stars. They suggest that this group, except star BD +72 1018 (UCAC4 815-027807) which belongs to nebula vdB149, is a cluster centred about star No. 1 (UCAC4 817-026634) and is located at a distance modulus of 7.5.
Even if the Aveni and Hunter’s stellar membership criterion of the sampling indicates an uncertainty for all the 18 stars, our study shows that four of these stars are astrometic members, 10 do not fulfil this condition due to the values of their proper motions, two are excluded of the analysis as they have no proper motion and two are located further than 30' of the adopted centre. These results are shown in Table 8. Among the astrometric members, star No. 1 (UCAC 4817-026634) has a parallax of 3.31 mas (302.11 pc) as it belongs to the Hipparcos
Table 7. vdB146. Identification of 9 photometric members determined by Aveni and Hunter (1972) in the UCAC4 catalogue.
catalogue (HIP 109389). This distance is in good agreement with the location of the cluster at 316.2 pc determined by Aveni and Hunter  . As cluster members must have similar kinematic parameters, the VPD of the uncertain photometric members (Figure 7) explain the reason why some possible photometric members should not be considered members of the cluster. The possible photometric members are plotted with crosses and the four astrometric members with large circles.
Taking profit that open cluster members share similar photometric and kinematic properties, astrometric and photometric results are combined in this work to reduce the uncertainty introduced by each technique individually.
We report that vdB92 has , , , 60 astrometric members, , and a value of 34' for the diameter. In the case of vdB146, the obtained values of these parameters are , , , 5 members, equatorial centre coordinates , and a diameter of 9'. In the case of vdB150, the obtained values of these parameters are , , , 7 members, equatorial centre coordinates , and a diameter of 15'. It is worthy to mention that the astrometric parameters of vdB146 and vdB150 are determined for the first time using UCAC4 data.
The incidence of proper motion errors in the determination of the cluster parameters and of the stellar membership is evaluated by applying the model to 25 simulated samples of each cluster. We conclude that the observational errors do
Figure 7. VPD of the 16 Aveni and Hunter’s possible photometric members for vdB150 plotted with crosses, while the four astrometric members are plotted with large circles.
Table 8. vdB150. Identification of 18 possible photometric members determined by Aveni and Hunter (1972) in the UCAC4 catalogue.
not significantly change the mean proper motion and the number of members of the examined clusters.
Finally, we compare our membership results and the photometric ones given in the literature to lead to a successful membership determination. In the case of vdB92, we take Bonatto and Bica  fundamental parameters and realize that 57 stars are astro-photometric members. For vdB146, the comparison with the photometric members given by Aveni and Hunter  affirms that five stars are both astrometric and photometric members. For vdB150, our analysis shows that four stars considered as possible photometric members by Aveni and Hunter  are astrometric members. In addition, the parallax of one astrometric member is in good agreement with the location of the cluster determined by Aveni and Hunter  .
This work was supported by Consejo Nacional de Investigaciones Cientficas y Técnicas (CONICET), Argentina, and by Universidad Nacional de La Plata (UNLP) under grant no. 11/G114.