Fitting isochrones to open cluster photometric data

Monteiro et al
A&A 2010

Monteiro solved for the parameters of an open cluster using the Cross Entropy method and analyzed the errors using the bootstrap method, instead of direct computation of the direct tau-squared space. This allows for a much larger grid. Right now, I limit my space to those near the published values.

The cross entropy method first randomly draws a certain number of parameters, and evaluates some function S(x) to be minimized at each point The points are ordered and the best N are selected out. These N define a mean and standard deviation around which to normally draw the next set of points. Weighting the mean and standard deviation with those of the previous round allows the process to converge less quickly and avoid local minima.

The function S(x) used to evaluate the quality of a point was the -log likelihood. Like Flannery & Johnson 1982, they use a nearest point estimator, only paying attention to the point which is closest. I believe that the nearby points should be considered as well and the probabilities added. This makes it more favorable for a star to exist in a area with a larger possible mass range, an important factor.

When the compute the probability for one point, they use the same Gaussian distribution I have been using, but also factor in fitting U-B as well, using all available data.

One of the most important topics in the paper dealt with determining which stars to exclude and giving stars weighting factors. The triangle of cluttered background stars near the bottom was completely cut out. The exact point of cutting did not effect the end result significantly. A cluster radius was computed using density given by positions in the image. Background stars outside this radius were excluded.

To compute the weighting factor, each gained a box with 3 sigma on each side. A Gaussian was computed using the mean and standard deviation for B, B-V, and U-B of stars in that box. It also included a term for its radius from the cluster center. This put importance on stars in the densest of the diagram and in the cluster center. The weight of each star was a factor in its probability.

After setting up the process, they performed multiple checks to assess the quality of the program. They created multiple synthetic clusters and checked if they could regain the same values. They took well-studied clusters and cross-referenced their values with those published. They studied the effect that different IMF's and binary % had on the results. IMF did not have an effect, but binary ratio could effect the distance up to 10%, though age remained the same. The technique was largely successful.

They used standard photometric errors (not those intrinsic to the data) as published by Mointinho 2001. They worked of Padova isochrones like the ones I'm using. They also fitted for E(B-V) in the beginning using a color-color diagram and only allowed it to vary 10% from that value. They also isolated data from only one source, and chose those with U-B data as well. They took 100% binary ratio with companions pulled from the same IMF.