5.2 Turn-Off detection methods
5.2.3 Uncertainties in the turn-off detection
M´endez et al. (2002) and Tabur et al. (2009) use the bootstrap method (Babu & Feigelson 1996) to estimate uncertainties in the magnitude of the Red Giant Branch Tip, because it is robust and widely documented. Since our task is similar to the TRGB detection, we decided to use the same method to study the errors.
A bootstrap resampling simulates the act of making the same observation multiple times. This allows realistic estimates of the standard deviation in the mean. Consider a stellar population with temperature function Φ. From this function we determine the turn-off Teff,TO(Φ) with one of the methods described above. We then randomly resample,
with replacement – i.e. a star may appear more than once or not at all –, all the stars of the stellar population and we consider only a randomly selected sample containing the 80% of the resample. We repeat the procedure to detect the TO, this time using the temperature function from the latter subsample Φ∗
, and we obtain T∗
eff,TO(Φ
∗ ).
By bootstrap resampling 500 times we take the standard deviation of the distribution T∗
eff,TO(Φ
∗
) as the 1σ error in the measurement of Teff,TO(Φ). We consider the uncertainty
in our detection as 3σ, which corresponds to 99% of probability for the star being at the turn-off. The uncertainties for our sample with [Fe/H] = -1.95 are plotted in Fig. 5.3 with red dashed lines. Comparing the values obtained for errors of both methods, we can see that the TO-detection obtained with the Sobel Kernel filter is more accurate than that obtained with the Cubic Fit.
Effect of errors in metallicity measurements
The effect of the metallicity measurement accuracies of about 0.25 dex in the final TO- detection were analyzed with Monte Carlo simulations. For a star with metallicity value [Fe/H]0 and error σ[Fe/H]0 we gave a random value distributed as a Gaussian in the range of [[Fe/H]0−σ[Fe/H]0,[Fe/H]0+σ[Fe/H]0]. This was done for each star and then we looked for the turn-off in this new sample. As in the previous case, we repeated this process 500 times and calculated the standard deviation of the TO-determination.
Monte Carlo simulation moves the stars in a perpendicular direction to that of where the TO is detected. Therefore it is expected that an error in the metallicity determination does not affect the temperature distribution (and the determination of the TO) significantly. We found, however, a significant effect on the TO-detection errors due to the metallicity bin size used for the temperature distribution. This happens if the metallicity bin size is smaller than the averaged 2σ[Fe/H]0, because a star can move from one bin to the other, which makes the temperature distribution change. For example, a star located on the second metallicity bin can well be located on the third bin during the next resample. In the same way, a star that was in the first sample at the third bin, can be during the next resample at the second one. These two stars have different temperature values
and therefore the temperature distributions will be different in both resamples. Like this an effect in the TO-detection is produced, which is similar to the bootstrapping method explained above. In this case we exclude stars from each metallicity bin due to movements of the Monte Carlo simulations and in the previous case we excluded them by considering randomly 80% of the stars from the resample, meaning that this effect becomes important for smaller stellar samples. In the left panel of Fig. 5.4 we have plotted the error due to temperature bootstrapping (“Boot”) with the dashed line and the error due to Monte Carlo simulations of metallicity measurements with the solid line. The binning in the metallicity for the temperature distribution was 0.2 dex, which is smaller than the 2σ[Fe/H]0 accuracies of the metallicity measurements of 0.25 dex. We can see how both curves behave similarly, where the error in the TO-detection becomes larger at the metal-rich side. This happens because at high metallicities there are less stars (discussed in detail below), meaning that a resampling of them will affect more the shape of the temperature distribution.
Figure 5.4: Left panel: errors in the TO-determination due to bootstrapping method in temperature (Boot, dashed line) and due to Monte Carlo simulations of metallicity mea- surement (MC, [Fe/H], solid line) considering a metallicity bin size of 0.2 dex. Middle panel: errors in the TO-determination with Monte Carlo simulations using different metallicity bin size. Right panel: as left panel, but considering the bin size of 0.8 dex for MC.
In case that the binning is much larger than the averaged 2σ[Fe/H]0 errors of the metal-
licity measurements, there is less probability that a star is moved from one bin to the other one. The temperature distribution of the stars at each metallicity bin in the different Monte Carlo simulations remains equal and the TO-detection variates less. This is shown in the middle panel of Fig. 5.4, where different curves represent the standard deviation of the TO-determination using different sizes in the metallicity binning. We can see how the errors become smaller with larger binning size, even at high metallicities, where we do not have so many stars. When the binning gets too large, the TO-detection becomes less accurate at the metal-poor border. For a last comparison we have plotted at the right panel of Fig. 5.4 again with the dashed line the error due to temperature bootstrapping
5.2 Turn-Off detection methods 61 (“Boot”) and with the solid line the error due to Monte Carlo simulations of metallicity measurements. In this case the binning in the metallicity for the temperature distribution was of 0.8 dex, which is much larger than the 2σ[Fe/H]0 errors of the metallicity measure-
ments. It can be seen how the errors due to metallicity measurements become negligible when compared with the bootstrapping ones.
We must take into account that the cut-off at the turn-off becomes noisy if the metallic- ity bin is too large. We need to find an agreement for the optimal size of the binning so that the errors due to metallicity are small but without loosing accuracy in the bootstrapping method. A in intermediate binning size of ∼0.4 dex would still give an accurate turn-off determination, where the errors due to metallicity measurements would still smaller than those obtained with bootstrapping.
Note that the “Boot” curve has its minimum at [Fe/H] =−1.7, which is related with the maximum amount of stars in the sample, as will be seen in Sect. 5.2.3. Allende Prieto et al. (2006) studied the metallicity distribution of F and G SDSS stars. As discussed in detail in Chap. 3, their atmosphere parameters have an offset of ∼0.3 dex with respect to ours. The peak in their metallicity distribution for halo stars is at [Fe/H]= -1.4, which is exactly 0.3 dex higher than our peak in the metallicity distribution at [Fe/H]= -1.7. In addition, Carollo et al. (2007) have found a peak in the metallicity distribution for the inner halo at [Fe/H]= -1.6.