Tau-squared Example 1

## Example 1 - High-mass stars.

This example is the fit to the high-mass stars presented in Jeffries et al (2007). The output files are given in the download in a directory called Example1, but you should create a clean directory to try it in. The precise inputs to the programs are given in a series of files with the suffix .in also in the Example1 directory.

First we run monte to create the 2D isochrone. We will need a mass range of 1 to 16 solar masses. We will use the Geneva isochrones with the colours supplied with the isochrones (option 0), specificcally V vs B-V. We require a binary fraction of 0.5. Since we only require one age, we set the age "range" to go from log10(age) is 7 to 7.

Monte tends to take a couple of minutes to run (on my PC at least), but at the end produces a file called V_B-V_07.000.fit. This is a genuine FITS file, so you can use your favourite FITS reader to look at it, and convince yourself that its right.

Now you need to run grid. The input data catalogue is called ubv.cat (which you can copy from CMDfit/Example1). This has U-B as well as B-V in it, so you'll be prompted for which colours you want (numbers 1 and 2 on the list). Next you can add an extra uncertainty to the magnitudes and colours of each star. This is added in quadrature to the uncertainties given in the data file. In this case, though, you should give the extra uncertainty to be added as zero. Next come the details of the isochrones you are using. The log10(age) range is 7 to 7, and you should search a distance modulus range of 9.7 to 10.5, with 100 points in the "grid". Finally, E(B-V)=0.20. You are now asked for the maximum range of tau-squared which is to be allowed (i.e. between the best fitting and worst fitting points). This should be set to a negative number, i.e. we don't want any clipping. It could be set to 20, and any data point which would normally exceed this limit is, as explained in the paper, "soft clipped" to this value.

There are a host of useful output files. The tau-squared space in this case is a simple plot of tau-squared against distance, and is given in grid.fit. Normally this is a two-dimensional space (say age and distance), and so is given as a FITS file (hence its extension). In this case, though, as its just one dimensional there is also a simple ASCII file (grid.fit.asc). The best fitting CMD, best_model.fit is also a FITS file. Obviously you want to see the data overlaid on this. We use the Starlink program gaia to look at images, and so provide you with the program for_gaia, which will take your data file (ubv.cat) and an image (in this case best_model.fit) to give you a regions definition file (ubv.ard), which the regions sub-menu in gaia allows you to plot the data over the model. For final publication plots you may wish to cut down the size of the FITS image, and bin it up in magnitude, before using for_gaia. Your favourite data reduction package will no doubt do this, but we also provide binup and subset to do it.

The output from grid should tell you that the best fitting value of parameter 2 (the distance modulus) was about 10.12, for a tau-squared of 381 (though your answers may deviate slightly due to the the random process which generates the models). As explained in the paper, the tau-squared alone is meaningless, you need the reduced tau-squared. You do this using the program tau2. You are first prompted for the best fitting model (best_model.fit), then you need the remaining data points after clipping, which was written for you by grid into unclipped.cat (though this should be identical to ubv.cat, provided you have not added an extra uncertainty in grid). Next you will be prompted for the number of the magnitude (1) and colour (2) being fitted from this file and the number of free parameters (1). This program tells you probability that the model is a good description of the data (Pr(tau-squared). You will find that the chance of exceeding the 381 grid gave us, is around 0.03. Thus this fit is on the margins of being acceptable, but we probably cannot justify clipping points out.

Finally, if you want to work out a confidence interval you need to run uncer. It automatically reads the file which is the tau-squared grid from grid (grid.fit) and writes a file called uncer.out, which is the tau-squared levels for given percentage confidence levels. It prints the 68 percent confidence level to the screen, which it should say is a tau-squared of 382. If you look back at grid.fit.asc you find this gives you a 68 percent confidence interval in distance modulus of 10.06 to 10.18, but again these numbers are also printed to the screen. These values deviate slightly, but not significantly from those in Jeffries et al. partly because of randomness, but partly due to improvements in the technique and the software since that paper was published.

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