Our Method for Obtaining a Normalized Distibution
Following the initialization process, a binary population of 10,000 stars was evolved, via the Binary Evolution Algorithm (BSE), to an age of 7 Gyr (the age of NGC 188). This process was repeated 1,000 times to account for statistical fluctuations in the number of BSs produced. For each of the 1,000 simulations, the BSs generated were selected photometrically by setting limits on their luminosities and B-V (blue minus visual) color. Effectively, photometric selection of BSs is done by overlaying the region of the observed CMD where BSs are identified on top of a simulated CMD. Any stars that lie within the observed region created by the simulation, are then identified as BSs. As seen below, any star above the main sequence and blueward of the dashed line in the observed CMD is selected as a BSs in the synthetic CMD.
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 CMD Provided by Geller et al. 2008
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Next, the period and eccentricity of the selected BSs were pulled from an output file generated by the code. The final period and eccentricity distributions were examined by creating respective histograms for all BSs. The height of the bins in the histograms represents the mean for the particular eccentricity or period range over all 1,000 runs. The error bars represent the standard fluctuation of that mean, likewise. Shown below are the resulting histograms for the averaged final period and averaged final eccentricity accumulated over 1,000 simulation.
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The above left histogram represents the final averaged distribution for the periods of the binary BSs selected from the CMD, and the right histogram similarly represents the eccentricty. For the eccentricity histogram, notice the large number of BSs that have eccentricity of or near zero. This illustrates that whatever the formation pathway that produced the BSs, a majority of them tends to have tidally circularized orbits as opposed to elliptical orbits. Interstingly, this same results is also observed (Geller et al. 2008). In terms of the final average period distribution of the BSs, the apparent peak at p = 1,000 days proves to be a most interesting result due to the strong correlation with observation (see below).
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Do the Averaged Distributions Agree with Observation?
As opposed to plotting individual histograms for the period and eccentricity of the observed BSs, one can combine the two by plotting the eccentricity as a function of the orbital periods (see the plot at right). The feature of notice in this plot is that the majority of the observed BSs in binaries have a period of approximately 1,000 days which is rather reminiscent of the simulated period distribution.
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 Observational Data Provided by Geller et al. 2008
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The quite remarkable results of our simulation are shown in the diagram below. While a K-S will ultimately establish the degree of correlation between the two sets of data, the initial conclusion is that the two populations correlate strongly. In fact, the correlation appears so strong that we can make the following claims:
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If a binary has been evolving for 7 Gyr according to binary evolution, and assuming our comprehensive model is accurate, then the only way that its eccentricity can be any value other than zero is if its period is near 1,000 days.
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The period of the binary plays a greater role in BSs formation than does its eccentricity.
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In support of the above claim, Geller et al. (2008) finds 21 BSs in NGC 188, 16 of which are in binaries. In other words, 76% of all the observed BSs in NGC 188 are found in binaries.In our simulation, we found that on average, the simulation produced 18 ( 3) BSs, 10 of which were in binaries, giving 57% of the BSs to be in binaries. The immediate conclusion is that for NGC 188, the quantity of BSs are within limits of BSEs production capability.
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For the sake of curiosity, the simulation was ran with Hurley's initial conditions, that is the flat period and thermal eccentricity distribution for the parent population of binaries. While the quantity of BSs produced was in fact greater than the amount produced by the observed parent distributions, only 25% of the BSs produced were in binaries, and this percentage disagrees with the observed 76%.
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