3.3 Analysis
3.3.2 Effective temperature from FeH lines
We also investigated the possibility to determine𝑇effusing FeH molecular lines, which are very sensitive to temperature. Our approach is to find𝑇effat which the synthetic spectra best reproduce the strengths of FeH lines. An advantage of this approach is that the𝑇effcan be homogeneously determined by the same technique for all the IRD samples, which is suitable for comparing the abundances between individual objects.
The so-called Wing-Ford band of FeH lines at∼990–1020 nm (Wing & Ford,1969) is one of the most prominent molecular features observed in the near-infrared spectra of M dwarfs. We selected 57 well-isolated FeH lines in the band to measure EW as indicators of the𝑇eff. The selected line list is provided in Table3.2. Since there is no information on the van der Waals damping parameters for these lines, we calculated the pressure broadening by applying the hydrogenic approximation ofUnsöld(1955).
Table 3.2: FeH molecular line list Species 𝜆(nm)∗ 𝐸low(eV)† log𝑔 𝑓‡
FeH 990.0494 0.225 -0.663
FeH 990.4994 0.134 -0.770
FeH 990.5874 0.177 -0.712
FeH 990.7749 0.201 -0.686
FeH 991.2003 0.153 -0.791
FeH 991.4707 0.086 -0.954
FeH 991.6444 0.061 -1.075
FeH 992.4867 0.084 -1.111
FeH 992.8899 0.137 -0.891
FeH 993.3278 0.176 -0.760
Table3.2(Continued)
Species 𝜆(nm)∗ 𝐸low(eV)† log𝑔 𝑓‡
FeH 993.5211 0.056 -1.695
FeH 993.5984 0.066 -0.928
FeH 994.1595 0.199 -0.732
FeH 994.2386 0.343 -0.584
FeH 994.4552 0.139 -0.891
FeH 994.5821 0.173 -0.816
FeH 994.7144 0.053 -0.983
FeH 995.0860 0.407 -0.552
FeH 995.3069 0.156 -0.852
FeH 995.6651 0.376 -0.568
FeH 995.7322 0.194 -0.783
FeH 996.2875 0.175 -0.816
FeH 996.5181 0.442 -0.537
FeH 997.0824 0.216 -0.754
FeH 997.3036 0.410 -0.552
FeH 997.3811 0.196 -0.783
FeH 997.4490 0.108 -1.162
FeH 999.4835 0.051 -1.588
FeH 1000.3446 0.266 -0.700
FeH 1000.7714 0.073 -1.724
FeH 1001.0072 0.334 -0.617
FeH 1001.0404 0.177 -0.883
FeH 1001.9538 0.066 -1.446
FeH 1003.3067 0.427 -0.565
FeH 1003.6239 0.089 -1.775
FeH 1003.9977 0.043 -1.407
FeH 1004.3975 0.322 -0.654
FeH 1005.2318 0.066 -1.544
FeH 1005.5334 0.097 -1.640
FeH 1005.7509 0.239 -0.778
56 Chapter 3. Chemical analysis of IRD planet survey
Table3.2(Continued)
Species 𝜆(nm)∗ 𝐸low(eV)† log𝑔 𝑓‡
FeH 1005.8284 0.051 -1.262
FeH 1006.1634 0.115 -1.684
FeH 1006.5967 0.084 -1.198
FeH 1007.1895 0.262 -0.749
FeH 1008.8971 0.602 -0.488
FeH 1009.4644 0.073 -1.080
FeH 1009.5954 0.288 -0.722
FeH 1009.9352 0.468 -0.550
FeH 1009.9628 0.108 -1.254
FeH 1010.7984 0.106 -1.050
FeH 1011.0859 0.538 -0.523
FeH 1011.5008 0.086 -1.015
FeH 1011.5689 1.043 -0.633
FeH 1011.7964 0.315 -0.697
FeH 1011.9702 0.416 -0.597
FeH 1012.0335 0.108 -1.050
FeH 1013.5384 0.121 -0.993
∗Wavelength in vacuum †Lower excitation potential
‡Oscillator strength
Synthetic spectra based on the line list by Plez reproduce the individual lines in the band well. We confirmed the FeH lines are exclusively sensitive to𝑇eff rather than [Fe/H] or log𝑔. Figure3.4shows the EW of FeH lines calculated using model atmospheres normalized to those for𝑇eff =3000K as a function of𝑇eff. The figure demonstrates that the EWs of the FeH lines are sensitive to temperature, though the sensitivity is generally dependent on the line strengths mainly due to saturation effect.
A hundred degree change in the assumed𝑇effchanges the EWs by∼0.1 dex. In contrast, the variation in the EWs along with the change in the overall metallicity [M/H] of 0.1 dex is less than 0.01 dex. The change in log𝑔of 0.1 dex also does not affect the EWs by more than 0.05 dex.
Ishikawa̲phd
0
Figure 3.4: Sensitivity of EWs of the selected FeH lines in model spectra to the assumed 𝑇eff. The vertical axis shows the logarithmic EW normalized by the value at𝑇eff = 3000 K (EW0).
58 Chapter 3. Chemical analysis of IRD planet survey
250 200 150 100 50 0 50 100
Deviation of T
effof each line from the mean (K) 0
25 50 75 100 125 150 175
Number
Figure 3.5: The histogram of the deviation of𝑇eff derived from each FeH line from the mean of𝑇eff values from all the FeH lines of corresponding object. The data of the 11 objects with 2900<𝑇eff <3200 is stacked.
We measured the EWs of the FeH lines by fitting the Gaussian profiles to observa- tional data. The Gaussian profiles can fit to the line profiles satisfactorily because the damping wings of the FeH lines are not significant. We obtained𝑇eff from individual FeH lines as the observed EW is reproduced by calculation. The average of all the values is taken as the𝑇eff-FeHfor a certain object. The log𝑔= 5.0,𝜉 = 0.5 km s−1, and [M/H] = 0.0 are assumed throughout the above𝑇eff-FeHestimation. Figure3.5shows the histogram of the result from individual FeH lines. The low-temperature side has a gently extended tail, while the right side shows a steeper drop. The Shapiro-Wilk test results in a p-value far less than 0.05, which means that it is not a normal distribution.
The histogram for each object shows a similar shape, suggesting that a particular set of absorption lines might be producing outliers regardless of the object. Other possible causes include contamination by unidentified weak absorption lines, incomplete line parameters, or misestimation of the continuum level in a particular wavelength range. A more stringent selection of absorption lines by examining the𝑇efffrom each
2900 2950 3000 3050 3100 3150 3200 3250 TFeH
2600 2700 2800 2900 3000 3100 3200 3300
T J1119 J1306 J1523J1652J1703J1802 J1816 J1909J2026J2043 J2053J2343 GJ699GJ406
FeH (1 vs 1) IRD Input (V J) MG
TIC (GBPGRP) Interferometry
Figure 3.6: Comparison of different estimates of𝑇eff. The horizontal axis shows the estimates from the FeH lines in the IRD spectra and the black dashed line shows the unity slope. The green, purple, orange, and red corresponds the𝑇eff-𝑉−𝐽,𝑇eff-𝐺,𝑇eff-TIC, and𝑇eff-interf in the text, respectively. The orange dashed line shows a linear fit on 𝑇eff-TIC.
absorption line and its reasons will improve the accuracy of temperature estimation by the FeH lines.
We compare the𝑇eff-FeHwith different estimates of𝑇eff, including𝑇eff-TICthat we adopted for the abundance analysis, in Figure3.6. The𝑇eff-TICare represented by the orange circles. The green circles are the𝑇eff values in the IRD input catalog, estimated by the empirical relationship between𝑉 −𝐽 color and𝑇eff provided by Man15 (𝑇eff-𝑉−𝐽).
In order to apply the estimation to all the IRD target candidates, the𝑉 magnitudes were estimated from the𝐵and𝑅magnitudes of USNO-B Catalog (Monet et al.,2003) and the 𝐽 magnitudes were drawn from the Two Micron All Sky Survey (2MASS;
Skrutskie et al.,2006) for all the objects. The purple circles are the estimates by the empirical relation between the absolute𝐺-band magnitude and𝑇eff calibrated byRabus et al.(2019). The absolute𝐺-band magnitude is calculated from𝐺-band photometry and parallaxes measured by Gaia.
The𝑇eff-FeH, represented by the black dashed unity line in the figure, lies higher
60 Chapter 3. Chemical analysis of IRD planet survey than most of the other estimates based on the color. Notably, the relative temperatures between the individual objects are shared by𝑇eff-FeHand𝑇eff-TIC. This implies that these two independent estimates accurately reproduce the relative temperatures between the objects, probably thanks to the high𝑇eff dependence of the FeH lines and𝐺BP−𝐺RP color, the substantial number of FeH lines, and the high photometric accuracy of Gaia. However, the difference between𝑇eff-FeHand𝑇eff-TICtends to be larger at lower temperatures, as shown by the linear fit (orange dashed line in the figure). This can be because the𝑇eff-FeHare indeed precise but not accurate in absolute terms due to the imperfections in the modeling of FeH lines at such low temperatures, such as the treatment of pressure broadening. The deviation of the best-fit synthetic spectra from the observed one is not noticeable for the objects analyzed in this thesis. For cooler objects such as TRAPPIST-1 (𝑇eff ∼2400–2600 K), nonetheless, we confirmed that the FeH lines in our synthetic spectra have wider wings than observed ones. Thus the 𝑇eff-FeHmight be greatly overestimated at such low temperatures.
However, there is also an uncertainty in the empirical𝑇effestimation based on colors at the lower temperatures. The empirical relation based on the𝐺BP−𝐺RPcolor which was used for deriving𝑇eff-TICis calibrated by the objects in Man15, for which Man15 estimated the𝑇eff based on the BT-Settl-model fitting on the medium-resolution spectra. Further, this fitting method in Man15 was originally calibrated byMann et al.(2013b) (hereafter Man13), but their calibration sample does not include objects cooler than 3238 K. For the lower temperatures, it is undeniable that there still be uncalibrated systematic errors.
The calibration of Man13 is based on𝑇eff calculated by the Stefan-Boltzmann law from the radii directly measured by interferometers and bolometric luminosity.
𝑇eff estimation from interferometric measurements (𝑇eff-interf) is reliable because it is model-independent, but the faintness and small angular diameters of late-M dwarfs limit its application with sufficient accuracy to only a few objects very close to the Sun.
The coolest example is GJ 406 (𝑇eff-interf = 2657±20 K) reported byRabus et al.(2019).
The high-resolution near-infrared spectra of this object is available in the CARMENES GTO Data Archive, so we calculated𝑇eff-FeHand compared it to the𝑇eff-interf (the left red circle in Figure3.6). Adopting the linear relation between𝑇eff-FeHand𝑇eff-TIC(the orange dashed line), the provisional𝑇eff-TICestimated from our𝑇eff-FeHis 2741 K. Its difference from the𝑇eff-interf is well within the typical scatter. This suggests that the𝑇eff
estimation based on the EW of the FeH lines may retain to be useful even for such cool objects. The extension of the reach of interferometric measurements to cooler M dwarfs is expected to drive the further understanding of the accurate𝑇eff in this regime.
It should be noted that GJ 406 is magnetically active.Reiners et al.(2018) reported the significant emission of H𝛼 (EWH𝛼 ∼ −9.25 Å). We confirmed that the magnetically sensitive FeH lines are wider than magnetically insensitive lines due to the Zeeman effect. The FeH lines being clearly affected by the Zeeman effect are excluded from the 𝑇eff-FeHestimation of the object. The IRD-SSP sample does not include magnetically active M dwarfs, but if we want to go beyond that and analyze the magnetically active ones, we need to select the lines taking the Landé g-factors into consideration.
On the other hand, looking at the higher side of𝑇eff,𝑇eff-FeH,𝑇eff-TIC, and𝑇eff-interf are in good agreement. The rightmost objects in Figure3.6show the convergence of the estimates for GJ 699 at around 3250 K. Although it is not shown in the figure, the corresponding estimates for GJ 436 also well agree with each other within 40 K (3440 K, 3456 K, 3416 K, respectively). These two examples support the view that the𝑇eff discrepancy between the methods is due to problems specific to low temperatures.