A New Technique for Fitting ColourMagnitude Diagrams

1
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.
WHATS 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-)main-sequence, 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.
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.
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.