TÉCNICAS Y HERRAMIENTAS

My initial point of comparison between the two methods was a simple scatter plot. The dashed line in Figure 2.6 denotes y = x, and in the case where the two methods provided similar answers, I would expect a tightly correlated sequence centred on the line (within errors). However there are a preponderance of points lying towards the isochronal side of the line, suggesting that stellar model fitting tends to return ages that are older than those preferred by gyrochronological methods.

Another interesting facet of Figure 2.6 is the distribution of the points along both axes. More than half of the systems lie within a region defined by age_gyro < 4 Gyr and age_iso < 6 Gyr. This is not entirely surprising given the region of parameter space to which I have restricted the study. Rough estimates ofτ_MS for stars at the limit of my parameter space are τMS = 3.5 Gyr for an F7 star andτMS =11.4 Gyr for a G9 star (using masses from table B1

of Gray 2008). A drop-off after roughly 4 Gyr is consistent with this, as systems at the hotter end of the parameter range start to evolve off the MS, and are therefore no longer targeted by transit search programs.

In terms of the different methods, 75 percent of the gyrochronology estimates are less than 4 Gyr, with the youngest and oldest systems being 0.14 and 9.68 Gyr old respectively. For the stellar model fitting estimates, 68 percent are younger than 6 Gyr, with the estimates ranging from 1.45 to 13.4 Gyr. It therefore seems that gyrochronology tends to return stellar age estimates which occupy a slightly more narrow range, and which are more biased towards younger ages than the results from stellar model fitting.

The distributions for the different methods highlight the difference between the stellar model fitting and gyrochronology methods. Binning the data in 1 Gyr increments produced the distributions shown in Figure 2.7. Both show a similar overall structure, with a peak at the lower end of their age range followed by a tail towards older systems, but the peaks and median values of the two distributions differ by around 2 Gyr, with the gyrochronology distribution clearly peaking at a younger age. A 2D Kolmogorov-Smirnov (KS) test on the two datasets indicates that there is a less than 1 percent probability of the two having a common

Figure 2.6: Gyrochronology age, calculated using equation (2.11), as a function of stellar model fitting age, found using the Yonsei-Yale isochrones. The dashed line denotes y = x; systems clustered around this line show similar age values for different methods of calculation. The maximum age on both axes is set to the age of the Universe. Direct measurements of the stellar rotation period were available for systems marked in blue. For systems marked in black, P_rotwas inferred from vsinI and R_s. It appears that gyrochronology tends to return younger system ages than stellar model fitting.

parent distribution.

One drawback of the KS test is that it fails to account for the uncertainties in my age estimations. I therefore evaluated the χ2 goodness-of-fit of my data to the line age_Gyro = age_Iso,

χ2_{= Σ}(ageGyro−ageIso) 2 σ2 Gyro+σ 2 Iso , (2.13)

where σ_Gyro and σ_Iso are the average uncertainties in each value of the gyrochronological and isochrone fitting ages respectively. I found that χ2 = 698.2, with a reduced value of χ2

red=9.8, suggesting that my ages are a poor fit for the null hypothesis that the two methods

return the same age values. The P-value for this result is P(χ2)∼ 0, a strong indication of significance.

2.4.2

∆ageanalysis

To further investigate this bias, I calculated the differences between the two ages for each of the systems in my sample. I then binned the results in 0.1 Gyr increments and produced a cumulative probability distribution, Figure 2.8. If the two methods were producing broadly similar results, then I would expect that the distribution would pass through the intersection

Figure 2.7:Age distributions for the results that I obtained from stellar model fitting using the Yonsei-Yale isochrones, and from gyrochronology using equation (2.11). The gyrochronology (blue, open) distribution seems to peak at a younger age than the distribution for stellar model fitting (black, hashed). Thick, vertical lines denote the medians of the distributions, and show the same offset.

of the lines∆age=0 and probability=0.5. Given the preliminary results from the previous section this is unlikely to be the case, but the deviation from this ‘ideal case’ will be interesting to characterise.

Figure 2.8 shows an apparent offset towards positive ∆age, in line with the conclusion from the previous section that stellar model fitting is returning ages which are slightly older than those from gyrochronology. The distribution passes through probability=0.5 at 2.3 Gyr, which roughly corresponds to the offset between medians seen in Figure 2.7. The average upper and lower error bars on∆age are 4.2 and 2.1 Gyr respectively, so this is a 1.1σeffect. For comparison, all the other possible combinations of stellar models and gyrochronology equations also show positive offsets of between 2.2 and 4.1 Gyr, giving significance of between 1.2σand 2.7σfor the effect.

It seems that there might be a disagreement between the ages that are produced by gy- rochronology and stellar model fitting. Does this correlate with a physical parameter in the systems that I am studying? Is it that stellar model fitting is overestimating ages, or that gyrochronology is underestimating ages (or a combination of the two)?

Figure 2.8: A cumulative probability distribution for the difference between the age results obtained by stellar model fitting using the Yonsei-Yale isochonres, and by gyrochronology using equation (2.11). The dotted lines denote∆age = 0and probability= 0.5. If the two methods produced similar stellar age distributions, the plot would pass through the intersection of these lines. The x -axis range is±the age of the Universe. There is an apparent offset towards positive∆age, again suggesting that stellar model fitting is returning ages which are older than those from gyrochronology.