Transcript PPV poster - Astrophysics

A New Technique for Fitting Colour-Magnitude Diagrams
Tim Naylor School of Physics, University of Exeter
R. D. Jeffries Astrophysics Group, Keele University
Abstract We present a new method for fitting colour magnitude diagrams of clusters and associations.
The method allows the model to include binaries, and gives robust parameter uncertainties.
WHAT’S THE PROBLEM? Despite the acquisition of many excellent colour-magnitude
diagrams for young clusters and associations, and the calculation of good pre-main-sequence
models, fitting these models to the data is still largely done “by eye”. There is a good reason for
this, even though it is not clearly elucidated in the literature. Were the data drawn from a single star
sequence the problem would become one of fitting an arbitrary line to data points with
uncertainties in two dimensions. Even this is not a straightforward problem, but was largely solved
by Flannery & Johnson (1982; ApJ 263 166). However, this solution remains little used for the
obvious reason that our data are not drawn from a single-star sequence, but from a population
which contains a large fraction of binary stars. These binary stars lie above the (pre-)mainsequence, resulting in a two dimensional distribution of objects in the colour magnitude plane (see
Figure 1). Faced with this, most galactic astronomers have taken the “by eye” approach, fitting the
single-star model sequences to the lower envelope of the data.
There have been more determined attempts to solve the problem in the extragalactic context, where
the distributions are spread even further from single isochrones because star formation continues
over large periods of time. Dolphin (2002, MNRAS 332 91, and references therein) bin the data in
two dimensions, but in doing so blur out our hard-won photometric precision. This is especially
serious in the case of clusters where the differences in the position of the isochrone with age are
typically rather small. Tolstoy & Saha (1996; ApJ 462 672) suggest making a two-dimensional
simulation of the data, but only using a similar number of points to that in the original dataset. Thus
some of the precision of the data is lost in the graininess of the model, and it is unclear how one
could determine uncertainties in parameters.
FIGURE 1 The grey scale is the best fit model isochrone to
the X-ray selected members of NGC2547 (circles). The data
points have been dereddened and then shifted by the best fit
distance modulus.
AN INTUITIVE SOLUTION Our solution to this problem can be envisaged in the following way.
Figure 1 shows a grey scale model which includes binaries, where the intensity of the grey scale is the
probability of finding a object at that colour and magnitude. Imagine moving the data points in Figure
1 over the grey scale, and collecting the values of the probability at the position of each data point.
The product of all these values is clearly a goodness-of-fit statistic, and is maximised when the data of
are placed correctly in colour and magnitude over the model. This method can be refined to include
the (two dimensional) uncertainties of each data point (see below), at which point we call our statistic
2. It can be formally derived from maximum likelihood theory, and as such can be viewed as either a
Bayesian or perfectly respectable Frequentist method. We have found that if the model is a single
sequence with uncertainties in one dimension 2 is identical to 2, i.e. 2 is a special case of 2. One
can derive uncertainties in the fitted parameters in a similar way to a 2 analysis, and we show in
Figure 2 the 2 space for fitting the data of Figure 1. The expected correlation between distance
modulus and age is clearly visible.
FORMAL DEFINITION The formal definition of our statistic is given by
FIGURE 2 A grey scale plot of 2 as a function of age and
distance modulus for the model and data of Figure 1. The
white contour is the 67% confidence level, whose structure is
probably caused by the clipping procedure for data points
lying outside the model.
where Pi, the probability for a single data point at (ci,mi) is given by the integral of the model 
multiplied by PD the probability distribution due to the uncertainties for that data point. We can show
this reduces to 2 if  is a line and PD a one-dimensional Gaussian. Then the product is only non-zero
where the two intersect, and has a value proportional to the value of the Gaussian at that point. Thus
the integral reduces to exp((c-ci)2/i2), leading to the normal form for 2.
OTHER PARAMETERS There is a large range of possible parameters one
could fit, but for NGC2547 we have been experimenting with binary fraction.
Figure 3 shows the distribution of 2 from fitting the data and that expected from
theoretical considerations. Clearly the data has too many points at high 2,
which corresponds to too many data points in the region of low (but non-zero)
probability in Figure 1.
We have experimented in increasing the binary
fraction, which increases the expected number of stars in this region of the
CMD, which cures the problem, but does not significantly change the best-fit
parameters for age and distance.
CONCLUSIONS 2 appears to be a very powerful technique for extracting
robust parameters with uncertainties from colour magnitude diagrams. Although
our own immediate interest is such datasets, it appears the method is very
general, and should have many applications to sparse datasets, and datasets with
uncertainties in two (or more) dimensions.
FIGURE 3 The expected distribution of 2 (curve) compared
with that obtained for the fit in Figure 1 (histogram).