ACCURATE GRAVITIES OF F, G, AND K STARS FROM HIGH RESOLUTION SPECTRA WITHOUT EXTERNAL CONSTRAINTS

We obtained 43 spectra of 42 Kepler asteroseismic targets using the Keck HIRES spectrograph in the red configuration at a resolution of $R\approx 70,000$. The signal-to-noise ratio (S/N) was typically $\gt 200$ per pixel column in the region around the Mg i b triplet for all but four fainter targets that had S/N of $\approx 120$. All spectra were reduced using the standard pipeline of the California Planet Search team.

The asteroseismic surface gravities of main sequence stars depend only weakly on the effective temperature and metallicity of the star (Gai et al. 2011) with their importance increasing slightly as the star evolves. We obtained temperature and metallicity from our spectral analysis, then used those values as initial inputs in the asteroseismic analysis. We then iterated once, fixing the gravity in our spectral analysis to the value returned from asteroseismology and used the newly derived ${{T}_{{\rm eff}}}$ and [Fe/H] values in the asteroseismic analysis. The iteration resulted in changes of less than 0.05 dex in the final gravities from those initially determined from the asteroseismic analysis.

As a baseline, we began by analyzing the spectra using the line list of VF05 with the same version of SME they used. We then updated the line list over the same wavelength region and updated the version of SME to v439,⁴ which uses an updated radiative transfer code and updated atmosphere interpolation algorithm. These changes reduced the differences in spectroscopic gravities relative to our asteroseismic reference values by about half; however, trends in these differences as a function of both metallicity and temperature remained. Between 5000 and 6500 K, Δ ${\rm log} {\mkern 1mu} g$ (spectroscopic minus asteroseismic ${\rm log} {\mkern 1mu} g$) spanned 0.5 dex.

We then included additional spectral intervals and decreased the number of simultaneously free parameters in our model to decouple the fitting of global parameters from individual abundances. These changes dramatically improved the accuracy in our derived ${\rm log} {\mkern 1mu} g$ values, reducing the rms scatter in Δ ${\rm log} {\mkern 1mu} g$ to 0.1 dex and removing the trend with respect to ${{T}_{{\rm eff}}}$. Further improvements were achieved by iterating the fitting procedure using the derived abundance pattern and allowing the abundances of alpha elements to be free while fitting the global parameters. This approach removed the trend in derived gravity with respect to [M/H] and reduced the rms scatter in Δ ${\rm log} {\mkern 1mu} g$ to only 0.05 dex.

The line list now includes more than 350 Å in 20 segments between 5160 and 7800 Å and includes nearly 7500 atomic and molecular lines (Table 1). The lines were obtained from the VALD-3 database (Kupka et al. 2011) and then tuned to better match a high resolution disk-integrated solar atlas (Wallace et al. 2011). The VALD-3 data contains many astrophysically tuned line parameters, but we found that about 15% of the VALD-3 lines still needed adjustment in one or more of line position, ${\rm log} (gf)$, or neutral perturber broadening coefficients in order to match the solar spectrum. Where available, we also used VALD-3 values for the temperature dependent broadening prescription of Barklem & O'Mara (1998), which gives better fits to the line profiles than the traditional van der Waals formulation.

Table 1.  Wavelength Ranges of Spectral Regions Used in This Analysis. Those With a Check in the VF05 Column Cover the Same Wavelengths as Segments from VF05

			Decreasing ${\rm log} {\mkern 1mu} g$		Decreasing ${{T}_{{\rm eff}}}$
Segment	${{\lambda }_{{\rm start}}}$	${{\lambda }_{{\rm end}}}$	Lines Grow	Lines Weaken	Lines Grow	Lines Weaken	${\rm VF}05$
a	5164	5190	0.17%	67.07%	48.14%	0.00%	x
j	5190	5207	0.63%	10.20%	16.17%	0.00%	⋯
k	5232	5262	1.93%	5.77%	10.78%	0.01%	⋯
b	6000	6015	0.08%	0.33%	1.68%	0.02%	x
c	6015	6030	0.02%	0.78%	2.20%	0.00%	x
d	6030	6050	0.29%	0.03%	0.35%	0.03%	x
e	6050	6070	0.10%	0.35%	1.08%	0.02%	x
f	6100	6120	0.21%	0.65%	1.88%	0.01%	x
g	6121	6140	0.30%	2.92%	3.89%	0.00%	x
h	6143	6160	0.60%	0.14%	1.11%	0.04%	x
i	6160	6180	0.04%	4.09%	5.42%	0.00%	x
l	6295	6305	0.11%	0.67%	0.78%	0.00%	⋯
m	6311	6320	0.02%	0.11%	1.34%	0.01%	⋯
n	6579	6599	0.16%	0.09%	1.13%	0.04%	⋯
o	6688	6702	0.00%	0.01%	0.22%	0.00%	⋯
p	6703	6711	0.00%	0.00%	0.43%	0.00%	⋯
q	6711	6718	0.01%	0.09%	0.63%	0.00%	⋯
r	7440	7470	0.16%	0.55%	1.16%	0.01%	⋯
s	7697	7702	0.00%	0.30%	0.20%	0.00%	⋯
t	7769	7799	0.64%	0.36%	0.53%	0.68%	⋯

Note. The letters correspond to those at the tops of the plots in Figure 1. The columns for decreasing ${\rm log} {\mkern 1mu} g$ and ${{T}_{{\rm eff}}}$ quantify the total amount of information contributed to the ${{\chi }^{2}}$ determination for each segment as plotted in the figure.

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We added wavelength regions to increase the number and quality of gravity and temperature dependent lines and to allow abundance determinations for additional elements. SME calculates ${{\chi }^{2}}$ from the difference between observed and model spectra. To evaluate the influence that the new segments had on fitting for ${\rm log} {\mkern 1mu} g$ and ${{T}_{{\rm eff}}}$ we created a model spectrum with solar parameters, two models with ${\rm log} {\mkern 1mu} g$ adjusted by ±0.2 dex, and two adjusted by ±200 K in ${{T}_{{\rm eff}}}$. We differenced pairs of models and used the squared difference at each unmasked spectrum point as a proxy for the information contributed to ${{\chi }^{2}}$ when fitting to spectra. We then plotted the cumulative distribution of these squared differences to examine this gravity and temperature information as a function of the increasing wavelength point in our unmasked spectrum (Figure 1). The information content of each segment is also detailed in Table 1. The expanded line list added 28% more gravity information and 51% more temperature information. Additionally, although deep gravity sensitive lines tend to get stronger with increasing ${\rm log} {\mkern 1mu} g$ and temperature sensitive lines stronger with decreasing ${{T}_{{\rm eff}}}$, there are some lines which display the opposite behavior. As noted by Gray (2008), weak lines of ions or atoms of an element that is predominantly found in the same ionization state (e.g., weak Fe ii lines) provide important gravity information because these lines become stronger with decreasing gravity. The opposing line growth of these lines is especially helpful in constraining ${{T}_{{\rm eff}}}$ and ${\rm log} {\mkern 1mu} g$ and we more than doubled the number of these spectral lines. Since Fe i is in the next lower energy state than the dominant Fe ii, weak lines of Fe i will not be very pressure sensitive and so ionization equilibrium provides an additional gravity constraint to the Mg i b wings, for lines where non-LTE effects do not significantly affect ionization equilibrium.

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In preparing the spectra for analysis we use the same continuum normalization procedure as VF05 but also mask out telluric lines as our spectral range now includes regions with significant telluric contamination. We perform cross correlation with the solar atlas (Wallace et al. 2011) to find an approximate radial velocity and before shifting the spectrum to observatory wavelengths, we apply a telluric mask based on telluric lines found in the Wallace et al. (2011) atlas. This masking had a beneficial effect on continuum normalization in some segments for spectra where telluric lines fell on continuum features near the ends of the segment.

Initial stellar parameters are all set to solar values except for the temperature, determined by $V-K\ {\rm or}\ B-V$ color relation (Boyajian et al. 2013), and gravity which is arbitrarily set to 4.5. In the first step ("Step 1"), we allow the global parameters ${{T}_{{\rm eff}}}$, ${\rm log} {\mkern 1mu} g$, [M/H], and macroturbulence (${{v}_{{\rm mac}}}$) to be free along with individual abundances for Ca, Si, and Ti. These alpha elements represent the largest number of non-iron peak lines in the spectral regions we are analyzing. They also have relatively high abundances that can differ greatly from the solar abundance pattern especially for low metallicity stars. In our initial fits, overall metallicity is strongly correlated with [Fe/H] due to the preponderance of iron lines. By letting these alpha elements be independent of overall metallicity, we partially account for non-solar abundance patterns in fitting the global parameters. VF05 showed that microturbulence, important in equivalent width analysis, seems not to play a large role in forward modeling and is degenerate with [M/H] so we fixed it to our adopted solar value of 0.85 km s⁻¹. After fitting, we perturb the temperature ±100 K in the resulting model and re-fit, using these new initial parameters to encourage the nonlinear least squares solver to converge to a nearby minimum. The new analysis provides more consistent results with median standard deviations between the three models of 11 K, 0.01 dex, and 0.005 dex for ${{T}_{{\rm eff}}}$ ${\rm log} {\mkern 1mu} g$, and [M/H] for the stars in our sample.

In the next step ("Step 2") we fix the global parameters to the ${{\chi }^{2}}$ weighted average of the three models from Step 1. We then allow all of our elemental abundances to be free including the three alpha elements which were free in Step 1. We have a total of 15 free elemental abundances (C, N, O, Na, Mg, Al, Si, Ca, Ti, V, Cr, Mn, Fe, Ni, Y), which were selected based on our interest in the element, the number of lines available in our spectral range, the sensitivity of those lines to small changes in abundances, and the reliability of recovering the solar abundances in our asteroid spectra (see below). A few of those elements, carbon, nitrogen, oxygen, and magnesium are also components of molecular species with lines in our spectral range and our inclusion of molecules in the spectral synthesis model provide additional constraints on these elemental abundances. The resultant model now has the full set of global parameters and individual abundances.

We then iterate, repeating Step 1 and Step 2 using the abundance pattern derived from the output of the first iteration instead of solar values. The differences in the models between the first and second iterations are small but serve to erase a subtle trend in Δ ${\rm log} {\mkern 1mu} g$ as a function of metallicity that was seen at the end of the first iteration. The standard deviation of the differences between the first and second iteration is 0.056 dex in ${\rm log} {\mkern 1mu} g$ and only 0.01 dex for [M/H].

To evaluate the accuracy of our returned parameters, we analyzed 20 spectra of reflected sunlight from four different asteroids taken on six different epochs. These spectra were obtained in the same manner as our other targets. Because we calibrated the atomic line data to the high resolution NSO solar spectrum, we expect that we should recover the solar parameters when we fit the reflected solar spectra in the asteroid observations. We found that our procedure recovers the solar parameters for the asteroid spectra (Figure 2) with ${{\chi }^{2}}$ weighted mean offsets of only 1 K in ${{T}_{{\rm eff}}}$, 0.02 dex in ${\rm log} {\mkern 1mu} g$, and 0.02 dex in [Fe/H] and rms scatter of 5 K, 0.006 dex, and 0.003 dex respectively.

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Our analysis is one of differential solar measurements made after tuning our line list against a solar atlas. In doing so, we adopted a microturbulence value of 0.85 km s⁻¹ for the Sun and left that fixed in the analysis of other stars. After fitting using our new procedure, we analyzed the effects of this decision to follow VF05 in fixing the microturbulence to the value we adopted for the Sun with two tests. The first fixed ${{v}_{{\rm mic}}}$ using the final parameters from our analysis in the empirical formula of Ramirez et al. (2013)

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{v}_{{\rm mic}}} & = & 1.163+7.808\times {{10}^{-4}}\times (T{\rm eff}-5800) \\ {} & {} & -0.494\times ({\rm log} {\mkern 1mu} g-4.30)-0.05\times [{\rm Fe}/{\rm H}] \\ \end{array}\end{eqnarray} \tag{ 1 }$$

and then ran our analysis as before. For the asteroid spectra ${{v}_{{\rm mic}}}$ = 1.07 was 0.2 km s⁻¹ higher than our value of 0.85 and resulted in models on average 30 K hotter and 0.03 dex higher in ${\rm log} {\mkern 1mu} g$. For the asteroseismic stars, the resultant gravities were discrepant from asteroseismic values by up to 0.35 dex and there were clear trends in ${\Delta }{\rm log} {\mkern 1mu} g$ with both ${{T}_{{\rm eff}}}$ and [M/H]. Temperatures also increased for hotter stars by up to 200 K. Allowing for the possibility that the offset in ${{v}_{{\rm mic}}}$ at solar from our adopted value could be responsible for the differences, we re-ran this test after subtracting a constant 0.22 km s⁻¹ from the ${{v}_{{\rm mic}}}$ values. By construction, this then returned the solar parameters for the asteroid spectra but the trend in ${\Delta }{\rm log} {\mkern 1mu} g$ with ${{T}_{{\rm eff}}}$ remained with a slightly shallower slope (${\Delta }{\rm log} {\mkern 1mu} g$ up to 0.23 dex).

The second test allowed ${{v}_{{\rm mic}}}$ to be an additional free parameter in the global parameter step with its initial value set to 0.85 km s⁻¹ and our procedure was then run as before. There was a systematic offset of $\sim 0.2\;{\rm km}\;{{{\rm s}}^{-1}}$ lower in the returned ${{v}_{{\rm mic}}}$ with respect to the fixed test values. The asteroid spectra returned the solar parameters with only [M/H] increasing slightly by 0.01 dex. However, asteroseismic stars again showed clear trends in ${\Delta }{\rm log} {\mkern 1mu} g$ with both ${{T}_{{\rm eff}}}$ and [M/H] though slightly less (${\Delta }{\rm log} {\mkern 1mu} g$ up to 0.25 dex) than the fixed ${{v}_{{\rm mic}}}$ case.

The first step in our analysis was to empirically tune our line data against a solar atlas using fixed solar parameters including 0.85 km s⁻¹ for microturbulence. Using the formula of Ramirez et al. (2013) for microturbulence when fitting stellar spectra degraded our gravity accuracy because our lines had been tuned at this constant, lower value. It is possible that self-consistent use of the Ramirez et al. (2013) formula to tune the line data and fit stellar spectra might improve the accuracy of our abundance determinations without compromising the accuracy of the gravity.

In our models, microturbulence is partially degenerate with ${\rm log} {\mkern 1mu} g$, ${{T}_{{\rm eff}}}$ and [M/H]. Including it as an additional free parameter reduces our ability to recover accurate surface gravities. Solving for microturbulence while tuning the atomic line data might allow us to solve for microturbulence when fitting spectra without compromising the fit for solar-type stars. From these tests it is clear that we have to leave ${{v}_{{\rm mic}}}$ set at the value we adopted for the Sun since this is the value used in tuning our lines against the solar atlas.

The first of these is to compare our spectroscopic gravity with the surface gravity from the grid based asteroseismic method described in Section 3 as our fiducial. Our new analysis process results in surface gravities which match the asteroseismically determined values of the stars in our sample with an rms scatter of 0.05 dex and with virtually no offset (0.01 dex) in the zero-point. Improvements with respect to the prior analysis using the VF05 procedure can be seen in Figure 3. There are no significant trends in Δ ${\rm log} {\mkern 1mu} g$ between the asteroseimically and spectroscopically determined values with respect to ${{T}_{{\rm eff}}}$, ${\rm log} {\mkern 1mu} g$, or metallicity (Figure 4). Errors in these three parameters have been shown to be correlated in previous spectral synthesis modeling analyses using the VF05 line list and an older SME version (Torres et al. 2012). The extreme outlier at Δ ${\rm log} {\mkern 1mu} g$ = −0.2 dex (Figure 4) is a star with total rotational broadening ($v{\rm sin} i$ and ${{v}_{{\rm mac}}}$) greater than 25 km s⁻¹ which smooths away most of the information needed to determine gravity and temperature.

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A second comparison of our spectroscopic analysis comes from the gravity constraint of transiting exoplanets. We compared our spectroscopic ${\rm log} {\mkern 1mu} g$ values with those derived by Torres et al. (2012) using the $a/{{R}_{*}}$ density method (Sozzetti et al. 2007) for stars with transiting planets. Most of the spectra have S/N $\gt 100$ per wavelength bin, though a quarter of them have S/N below 100. Torres et al. (2012) found systematic trends between their gravities based on $a/{{R}_{*}}$ and SME gravities obtained using the VF05 procedure. In contrast, we find no systematic trends between Torres et al. (2012) gravities and our SME gravities obtained using more spectral segments and a two-stage fitting procedure (Figure 5). Our SME methodology works better than the VF05 procedure for the Torres et al. (2012) stars, which are generally warmer than the original VF05 sample. We find a constant offset of 0.06 dex in ${\rm log} {\mkern 1mu} g$, relative to Torres et al. (2012). Such a small offset could be a result of errors in our spectroscopic analysis or may be the result of small errors in the $a/{{R}_{*}}$ analysis that arise because of inaccuracies in the impact parameters or eccentricities (Huber et al. 2013).