Transcript Slide 1

Proper-Motion Membership
Determinations in Star Clusters
Dana I. Dinescu (Yale U.)
The Goal
Ebbighausen 1939 – NGC 752
King et al 1998 – NGC 6397
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:
Xs+ mxDt = a1 + a2X + a3Y + a4X2 + a5XY + a6Y2 + a7(B-V) + possible h.o.t
Ys+ myDt = b1 + b2Y + b3Y + b4X2 + b5XY + b6Y2 + b7(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.
Proper-motion Membership
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 propermotion 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).
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, Conventional Method
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:
 (m )  
i
 ( m  mi ) 2
1
exp
2

2 i
2  i





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.
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 (Nc), the center and dispersion of the cluster and field
distributions along each axis (mc,f;x,y, sc,f;x,y). The frequency distributions
are (e.g, for the cluster):
f c (m x , m y )  f c, x (m x ) f c, y (m y )
f c, x (m x ) 
Nc
2 s c , x
 ( m x  mc , x ) 2 

exp 
2


2
s
c, x


The proper-motion cluster membership probability is defined as:
fc
P
fc  f f
In reality, the proper motion of an individual star is not precisely known. So,
integrate over the proper-motion error ellipse for star i:
( m x  m x ,i ) 2 ( m y  m y ,i ) 2
fc
P( mi )   
exp{ 

}dm x dm y
2
2
2 x ,i
2 y ,i
   ( f c  f f ) 2  x ,i  y ,i
 
Difficulties with the conventional method (De Graeve 1979)
The computed probabilities (high values) will be significant only when the peak of
fc is much higher than the corresponding ordinate of ff.. 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 s 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
Overcoming some of the difficulties
Proper-motion and Spatial Membership Probability
Include the spatial information - De Graeve (1979). A spatial frequency
distribution can be constructed for the cluster and the field stars (Sc, Sf ). The
form of this function, for open clusters, is taken to be an exponential (~ exp(-r/r0),
where r0 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:
P
f c Sc
f c Sc  f f S f
NOTE: No inference on the spatial distribution of the cluster stars can be made
when these probabilities are used !
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:
s obs 2  s int 2  s meas 2  s smooth 2
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, ssmooth  smeas.
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:
f c, x (m x ) 
Nc
2 s c , x
 ( m x  mc, x ) 2 
 * Ec , x ( m x )
exp 
2


2s c , x


where
Nc
Ec , x  
i
 ( m x  D x ,i ) 2 

exp 
2

2

2  x ,i N c
x ,i


1
The observed proper motion of star i is offset by Dx,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:

i
 ln  (m x,i , m y ,i )
p j
0
pj ; j = 1..9 - 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. 2004 – NGC 1817
Galadi-Enriquez et al. 1998 – NGC 1750 and NGC 1758
Galadi-Enriquez et al. 1998
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 propermotion errors.
Applications
Deriving physical parameters of the cluster from a cleaned CMD
Galadi-Enriquez et al. 1998
CMD morphology, stars of special interest
NGC 188 – Platais et al. 2003
Constraining stellar evolution models: core convective overshoot
Sandquist 2004 – M 67 CMD cleaned with propermotion 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 wellseparated 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.