By far the largest limiting factor when determining stellar ages is the uncertainty in the mass of the star. To explore the extent of the effect of mass when calculating age, I selected five stars, varied their masses by 20%, and recalculated their ages using the method described in the previous section. As Table 8.3 shows, even a 20% error in the mass leads to an enormous difference in the ages of the stars, up to a factor of 10.
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Table. 8.3: Mass’ Effect on Age Determinations
Varying mass: Age corresponding to: HD 0.8×M M 1.2×M 0.8×M M 1.2×M
(M) (M) (M) (Gyr) (Gyr) (Gyr)
3651 0.63 0.79 0.95 60.0 24.0 6.6 19994 1.07 1.34 1.61 11.3 4.8 1.1 177830 0.92 1.15 1.38 22.5 10.0 5.1 190228 0.66 0.83 1.00 39.1 17.3 8.8 190360 0.77 0.96 1.15 35.8 14.1 3.9
Given the small errors in the exoplanet host stars’ Teff and the new stellar radii
measured here, it was possible to run a range of stellar models using the Dartmouth Stellar Evolution Web Server for a given star. The metallicity listed in Santos et al. (2004) remained the same while a variety of masses for each star was considered. The mass listed in the planet discovery paper was the starting point, and then a range of different masses were tested until the datapoint was fit. This procedure was performed for stars whose error in the linear radius was ≤10%, which amounted to 15 stars. The results are shown in Figures 8.2 through 8.15. AllTeff are from (Santos
et al. 2004) unless otherwise noted in Table 8.4.
The errors for these masses were estimated by moving the observed datapoint by 1σtowards an evolutionary track other than the best fit track. The new datapoint was then compared to the distance between the two evolutionary tracks and converted to a mass difference. For example, say two evolutionary tracks were for two masses of 1.0 Mand a 1.1Mwith the 1.0 Mtrack being the best fit for the observed datapoint. If when the datapoint was moved one-sigma towards the 1.1 M evolutionary track,
it covered half the distance between the tracks, then the error in the mass would be 0.05 M.
Figure. 8.2: HD 3651: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.3: HD 9826: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
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Figure. 8.4: HD 19994: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.5: HD 38529: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.6: HD 59686: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars. Allende Prieto
& Lambert (1999) claim a 1 K error on their Teff but the error bar printed here is
10 K so it is visible.
Figure. 8.7: HD 75732: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
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Figure. 8.8: HD 104985: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars. Teff is from
(Takeda et al. 2005), who quotes no error bars, so a 10-K error bar was assigned.
Figure. 8.9: HD 117176: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.10: HD 120136: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.11: HD 143761: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
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Figure. 8.12: HD 189733: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.13: HD 190360: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
Figure. 8.14: HD 196885: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars. Allende Prieto
& Lambert (1999) claim a 10 K error on their Teff but the error bar printed here is
10 K so it is visible.
Figure. 8.15: HD 217014: Determining stellar mass and age. The three lines represent evolutionary tracks for stars of the masses listed on the plot; ♦s represent 1-Gyr intervals; and + represents the measured Teff and R with error bars.
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Table. 8.4: New Stellar Mass and Age
HD Teff R Moriginal Mnew Ageoriginal Agenew EE Age
(K) (R) (M) (M) (Gyr) (Gyr) (Gyr)
3651 5173±35 0.94±0.03 0.79 0.82±0.01 24.0 ∼20 5.1 9826 6212±64 2.21±0.09 1.3 1.49±0.05 4.8 2-3 3.8 19994 6217±67 1.89±0.07 1.34 1.48±0.04 4.8 2-3 8.9 38529 5674±40 2.60±0.25 1.4 1.37±0.06 5.2 3-4 5.1 59686 4571±1 10.74±0.75 1.15 1.45±0.15 7.4 2-3 - 75732 5279±62 1.15±0.04 0.95 0.82±0.02 10.6 20-21 5.5 104985 4877 11.16±0.65 1.6 2.20±0.30 2.0 0-1 3.0 117176 5560±34 1.94±0.05 0.92 1.11±0.03 13.8 6-7 7.1 120136 6339±73 1.30±0.03 1.2 1.35±0.03 4.0 0-1 2.5 143761 5853±25 1.29±0.08 1.0 0.92±0.01 7.3 10-11 9 189733 5051±47 0.78±0.05 0.82 0.78±0.04 6.7 ∼11 >0.6 190360 5584±36 1.18±0.04 0.96 1.05±0.02 14.1 8-9 12.1 196885 6310 1.73±0.17 1.27 1.49±0.06 6.2 ∼2 8.4 217014 5804±36 1.23±0.05 1.0 1.12±0.02 11.3 5-6 4
Note. AllTeff from Santos et al. (2004) except the following: HD 34445, HD 59686, and HD 196885
are from Allende Prieto & Lambert (1999), HD 104985 is from Takeda et al. (2005), and HD 189733
is from Sousa et al. (2006).
Figure 8.16 plots the new mass versus the original mass from the planet discovery papers, and in almost all cases, the new mass is larger than the original one, which also increases the mass of the companion. The mean Mnew/Moriginal is 1.09, and the
standard deviation of that mean is 0.14, indicating a significant departure from the original mass estimate. These systematic effects may be an effect of the stellar model used.
The errors for the original masses were estimated using the following method: for each star, the mass corresponding to the subtypes surrounding the given spectral
classification was determined using Cox (2000). For example, if the star’s spectral classification was listed as a K2 V in the planet discovery paper, the masses for the K1 V and K3 V types were determined. Then the difference between the given mass and the mass for each surrounding subtype was calculated for each star and then averaged for the 15 stars seen in Figures 8.2 through 8.15. The average was ∼6%, so the original masses show 6% error bars in Figure 8.16. This is a rough estimate, as few planet discovery papers list error estimates for the stellar masses.
Figure. 8.16: New vs. old stellar masses. The solid line represents the 1:1 ratio for the masses. Note how almost all the new mass estimates are larger than the previous estimates. The largest outlier is for HD 104985.
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