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

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

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.

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

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).

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

Dinescu et al. 1996

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:

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.

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.

Examples of proper-motion distributions

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

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.

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.

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 = the parameters Membership probabilities follow from the modeled proper-motion frequency distributions.

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).

Balaguer-Nunez et al – NGC 1817

Galadi-Enriquez et al – NGC 1750 and NGC 1758

Galadi-Enriquez et al NGC 1750 NGC 1758

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.

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

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

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

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

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.