1. Proper-Motion Membership Determinations in Star Clusters Dana I. Dinescu (Yale U.)

2. Ebbighausen 1939 – NGC 752King et al 1998 – NGC 6397 The Goal

3. How To Do It 1) From a set of photographic plates or CCD frames taken over a relevant time span, calculate relative proper motions. 2) Assign a membership probability to a kinematical group, taking into account proper-motion uncertainties. 3) Use membership probabilities to separate the cluster from the field population and then study other physical properties (CMD, LF, light profile) of the populations separately.

4. Relative Proper-motion Determination The method most commonly used is the iterative central-plate overlap technique (Eichhorn & Jefferys 1971, see also Girard et al. 1989). All plate measures are transformed to a standard-plate coordinate system. The transformation has the general form: X s +  x  t = a 1 + a 2 X + a 3 Y + a 4 X 2 + a 5 XY + a 6 Y 2 + a 7 (B-V) + possible h.o.t Y s +  y  t = b 1 + b 2 Y + b 3 Y + b 4 X 2 + b 5 XY + b 6 Y 2 + b 7 (B-V) + possible h.o.t Mean positions and proper motions are estimated by a least-squares fit to the positions as a function of epoch over all plates on which a star appears. New proper motions are calculated, and the process is iterated until it converges to the final proper motion values. Proper-motion uncertainties are determined from the scatter about the best-fit line. Typically, the reference stars used to determine the plate coefficients are cluster stars. VERY IMPORTANT: Photographic plate positions are affected by a number of systematics, of which the most notable is magnitude equation. For the appropriate treatment of these systematics see e.g., Kozhurina-Platais et al. 1995.

5. Proper-motion Membership References: Vasilevskis et al. 1958, Sanders 1971, De Graeve 1979, Girard et al. 1989, Kozhurina-Platais et al. 1995, Dinescu et al. 1996, Cabrera-Cano & Alfaro (1990), Galadi-Enriquez 1998, and references therein Parametric a) Fit observed proper-motion distributions in each coordinate with Gaussian functions; this is the “classical/traditional method” (Vasilevskis et al. 1958). b) Maximum likelihood applied to the observed proper- motion distribution. Assumes Gaussian functions for the cluster and the field distributions (Sanders 1971). Non-parametric No functional form is assumed when the proper-motion distribution is made (Cabrera-Cano & Alfaro 1990).

6. With this approach, the proper-motion distribution is smoothed by the individual errors, which is advantageous for a large range in the proper-motion error. Observed Proper-motion Distributions - The proper motion axes are rotated so as to align them with the major and minor axis of the field distribution; this ensures that the proper-motion distribution in one coordinate is independent of the one in the other coordinate (most important for the field proper-motion distribution). - The observed proper-motion distribution function must be constructed from a set of discreet proper-motion measurements. This generally requires binning, or in some other way, smoothing the data. Thus, one-dimensional marginal distributions are constructed by taking into account individual proper-motion uncertainties. The frequency of stars per unit of proper motion (in x and y) is given by: Parametric, Conventional Method

7. Dinescu et al. 1996

8. Proper-motion Membership Probability The observed proper-motion distribution in each axis is fit with a model distribution consisting of the sum of two Gaussians: the cluster and the field. The free parameters determined from the fit are: the number of cluster stars (N c ), the center and dispersion of the cluster and field distributions along each axis  c,f;x,y,  c,f;x,y )  The frequency distributions are (e.g, for the cluster): The proper-motion cluster membership probability is defined as: In reality, the proper motion of an individual star is not precisely known. So, integrate over the proper-motion error ellipse for star i:

9. Difficulties with the conventional method (De Graeve 1979) The computed probabilities (high values) will be significant only when the peak of f c is much higher than the corresponding ordinate of f f.. This happens if:  there is a big difference in the location of the two peaks  there is a high proportion of cluster stars  there is a high-precision proper-motion set If none of these conditions are met, the computed probabilities will rapidly loose significance.

10. More difficulties with the conventional method (e.g., Girard et al. 1989, Galadi-Enriquez et al. 1998)  The cluster distribution can differ from a Gaussian: for samples where the proper- motion error varies significantly as a function of magnitude, the sum of Gaussians of different  is not a Gaussian.  The field distribution is not a Gaussian; physically, it is determined by the combined Solar peculiar motion and Galactic rotation.

11. Examples of proper-motion distributions

12. Include the spatial information - De Graeve (1979). A spatial frequency distribution can be constructed for the cluster and the field stars (S c, S f ). The form of this function, for open clusters, is taken to be an exponential (~ exp(-r/r 0 ), where r 0 is the half-light radius (van den Bergh & Sher 1960). For the field, the function is assumed to be a constant. The combined membership probability is: Proper-motion and Spatial Membership Probability NOTE: No inference on the spatial distribution of the cluster stars can be made when these probabilities are used ! Overcoming some of the difficulties

13. The Cluster Proper-motion Dispersion: What to Use When Estimating Probabilities ? The cluster proper-motion dispersion obtained from the fit of the sum of two Gaussians to the observed proper-motion distribution, consists of the following terms: The intrinsic proper-motion dispersion which is given by e.g., internal motions in a cluster, is generally very small. For an open cluster, the velocity dispersion is ~ 1 km/s, corresponding to a proper-motion dispersion of 0.2 mas/yr for a cluster at a distance of 1 kpc from the Sun. The measurement proper-motion dispersion is the dominating value, and it is given by the “mean”, collective proper-motion error of individual stars in the sample. This proper-motion error varies over a large range: 0.2 to 2-3 mas/yr. When building the observed proper-motion distribution – by smoothing with individual errors of each star – the total dispersion is increased by the smoothing process. If proper-motion errors are estimated correctly,  smooth  meas. When estimating probabilities, use the individual proper-motion errors, rather than the dispersion obtained from the fit (Dinescu et al. 1996); this is especially good for accurate probabilities of bright, well-measured stars.

14. Better Modeling the Cluster Proper-motion Distribution When proper-motions are of high quality, and proper-motion errors are accurate but vary with magnitude, one can build a better model to incorporate the errors into the proper-motion distribution (Girard et al. 1989). The modeled proper-motion distribution is convolved with an error function E: The observed proper motion of star i is offset by  x,i, which is drawn from a normal error distribution of dispersion  x,i.

16. Parametric, Maximum likelihood (Sanders 1971, Slovak 1977) Assumes that the cluster and field proper-motion distributions are Gaussian; the 9 parameters (number of cluster stars, cluster and field centers and dispersions in x and y) are determined simultaneously, in an iterative procedure from the equations of condition: p j ; j = 1..9 - the parameters Membership probabilities follow from the modeled proper-motion frequency distributions.

17. The Non-Parametric Method (Cabrera-Cano & Alfaro 1990, Galadi-Enriquez et al. 1998) The parametric methods work only when there are two proper-motion groups (cluster and field stars) distributed according to normal bivariate function. The most common departure from these assumptions is the non-Gaussian shape of the field proper-motion distribution (Sun’s peculiar motion + Galactic rotation). Combined with low “signal-to-noise” of the cluster, the traditional approach can fail to produce reliable results. In the non-parametric method, the proper-motion distribution function (PDF) is determined empirically. Basically, for a sample of N points distributed in a 2D space, it is possible to tabulate the frequency function by evaluating the observed local density at each node of a given grid. A kernel is used to estimate the local density around any given point (typically a circular Gaussian kernel). The field PDF is constructed from a region (of the physical space) where a negligible number of cluster stars are contributing. Then, the cluster empirical PDF is determined as a difference between the total PDF and that of the field (e.g., Galadi-Enriquez et al. 1998, Balaguer-Nunez et al. 2004).

18. Balaguer-Nunez et al. 2004 – NGC 1817

19. Galadi-Enriquez et al. 1998 – NGC 1750 and NGC 1758

20. Galadi-Enriquez et al. 1998 NGC 1750 NGC 1758

21. Membership Probabilities: The Concept and the Real World In the real world there are only cluster stars and field stars; how about P ~ 50% ? Intermediate values show our inability to separate the two populations due to proper- motion errors.

22. Applications Galadi-Enriquez et al. 1998 Deriving physical parameters of the cluster from a cleaned CMD

23. NGC 188 – Platais et al. 2003 CMD morphology, stars of special interest

24. Constraining stellar evolution models: core convective overshoot Sandquist 2004 – M 67 CMD cleaned with proper- motion memberships from Girard et al 1989

25. Other properties of the cluster: internal dynamics, mass function  Mass segregation  Surface density profile; tidal radius  Velocity dispersions and velocity anisotropy – dynamical mass of the cluster  Luminosity function to mass function For globular clusters: see Drukier et al. papers and Anderson, King et al. papers

26. Concluding Remarks To obtain reliable membership probabilities, a high-quality set of proper motions and a realistic description of the proper-motion errors are required. The classical method works well when the cluster dominates and/or is well- separated from the field. According to specific scientific interest, additional information (spatial, CMD, radial velocities), can be included separately from the proper-motion analysis, or combined with it.