IMPROVED SPECTROSCOPIC PARAMETERS FOR TRANSITING PLANET HOSTS

In recent years, the number of extrasolar transiting planets has expanded considerably, with discoveries being made at a rapid pace both from the ground and increasingly also from space by the CoRoT and Kepler missions. With this large assembly of data, studies have begun to focus on examining patterns and correlations among the global properties of these planets and their parent stars, and what this can tell us about planet formation and evolution.

While the characteristics of some of these systems are very well known (e.g., HD 209458, HD 189733, TrES-1), those of others are much less well determined and have remained so since their discovery. Our knowledge of the planetary properties depends critically on understanding the parent stars. This is because, for transiting systems, the light curves only give information on the size of the planet relative to that of the star (R_p∝R_⋆), and spectroscopic observations only reveal the mass of the planet if we know the mass of the star (M_p∝M^2/3_⋆). The stellar mass and radius, in turn, depend on other properties that can be gleaned from the stars' spectra such as the effective temperature (T_eff), surface gravity (log g), and chemical composition (commonly represented by [Fe/H]).

For many of the known transiting planet systems, follow-up photometric observations have been undertaken after the initial discovery of the planet, usually for the purpose of measuring the times of mid-transit and seeking departures from strict periodicity (transit timing variations) that may indicate the presence of additional bodies in the system. These new transit light curve observations have also served to improve the radius determinations in many cases. However, it is much less common for known transiting systems to be re-observed spectroscopically. As a result, our knowledge of the stellar properties is often limited by whatever information was reported in the discovery papers, which is sometimes preliminary. Inaccuracies in the stellar T_eff, log g, and [Fe/H] propagate through to the determination of the planetary properties, and may obscure correlations with other quantities and prevent us from gaining valuable insight into the nature of planets. To make matters worse, the methods of determining T_eff, log g, and [Fe/H] in the literature are highly inhomogeneous, as they have been carried out by many groups using different assumptions and methodologies.

In one of the few studies to redetermine spectroscopic properties for transiting planet hosts in a uniform way, Ammler-von Eiff et al. (2009) obtained new temperatures, surface gravities, and metallicities for 13 systems based on new or existing spectra. They combined their determinations with those for 11 additional systems made by others using similar techniques, and compiled a list of 24 host stars with uniformly derived properties. Comparison with results for the same stars by other groups uncovered significant systematic differences in some cases, the causes of which are unknown.

One of the motivations for the present paper is to derive spectroscopic properties in a homogeneous manner for a much larger sample of more than 50 transiting planet hosts, and thereby reduce any dispersion in the stellar and planetary properties that is caused by the variety of methodologies used in the past. For this, we make use of the stellar parameter classification (SPC) technique introduced by Buchhave et al. (2012). We also wish to understand potential systematic errors in such determinations. In particular, it has been recognized for some time (e.g., Sozzetti et al. 2007; Holman et al. 2007; Winn et al. 2008) that surface gravities are often poorly constrained by spectral analyses. This is an unfortunate limitation, because the surface gravity would otherwise help to establish the luminosity of the star (and hence its radius) by placing it on the H-R diagram.

In cases for which the surface gravity is poorly constrained, the use of an external constraint on the luminosity becomes very important to allow accurate determinations of the mass and radius of the star. However, a detail that has usually been overlooked is that the determinations of other spectroscopic quantities such as the temperature and metallicity can also be affected by the poor constraint on gravity. This is because the uncertainties in T_eff and [Fe/H] are strongly correlated with log g in at least some of the commonly used analysis techniques. This can be a significant source of systematic error. Therefore, a second goal of our work is to compare spectroscopic determinations for a subset of the sample obtained using two additional procedures widely employed in previous studies, and to investigate and quantify systematic errors stemming from the weakly constrained gravities.

Ultimately, the stellar quantities that enter into the calculation of the planetary characteristics are the masses and radii inferred from T_eff, log g, and [Fe/H], as mentioned earlier. A third objective of the present paper is to study and quantify how errors in the spectroscopic properties propagate into the stellar masses and radii.

The paper is organized as follows. In Section 2, we report our spectroscopic observations, which consist of new and archival spectra obtained with three different telescopes. Our spectroscopic analysis techniques are described in Section 3. Our results with and without the application of external constraints on the surface gravity are presented in Section 4. Also presented there are the final results from this work. These are then compared with work by others in Section 5. In Section 6, we study the impact of the different assumptions regarding log g on the stellar masses and radii. We discuss our results in Section 7, and end with a summary of our conclusions in Section 8.

For this work, we have relied in part on new spectra collected with the TRES instrument (Fűrész 2008) on the 1.5 m Tillinghast reflector at the F. L. Whipple Observatory (Mount Hopkins, AZ), and also on publicly available spectra obtained with the HIRES instrument (Vogt et al. 1994) on the Keck I telescope (Mauna Kea, HI) and with the FIES instrument (Djupvic & Andersen 2010) on the 2.5 m Nordic Optical Telescope (La Palma, Spain). The archival spectra have generally not been used previously for a determination of the stellar properties, and even when they have, we have redetermined those properties here using different methodologies for comparison purposes.

We obtained new TRES spectra for 22 stars (see Table 1). They cover the wavelength range ∼3860–9100 Å and were taken at a typical resolving power of R ≈ 48, 000 with the medium fiber of that instrument. The signal-to-noise ratios (S/Ns) for individual exposures range between 27 and 130 per resolution element of 6.2 km s⁻¹, and refer to the region of the Mg i b triplet (∼5200 Å). For stars with multiple exposures, the S/N reported is the average. All spectra were reduced using the procedures described by Buchhave (2010) and Buchhave et al. (2010).

The archival HIRES spectra (Table 2) were obtained with the typical setup employed for extrasolar planet searches with that instrument. These spectra cover the full optical range and were gathered at two different nominal resolving powers of R ≈ 68, 000 and R ≈ 51, 000. The S/Ns in the Mg b region range from about 40 to 360 per resolution element (4.4 km s⁻¹ and 5.9 km s⁻¹, respectively). Many of the spectra available from this instrument used an iodine cell in the beam to impose a dense set of molecular absorption lines that serve as a fiducial in determining precise Doppler shifts. For the present work, we analyzed only the spectra taken without the iodine cell (referred to as "templates"). They were reduced following standard procedures for this instrument.

The archival FIES spectra cover the wavelength range from about 3600 Å to 7400 Å, and were obtained with the medium fiber of that spectrograph at a typical resolving power of R ≈ 46, 000. The S/Ns at ∼5200 Å for individual exposures range from 21 to 220 per resolution element of 6.5 km s⁻¹. The list of stars is given in Table 3. Reductions were carried out as described by Buchhave (2010) and Buchhave et al. (2010) using nightly flat-field, bias, and dark frames, as well as thorium–argon exposures taken immediately before and after the science exposure. This instrument does not use an iodine cell, and relies instead on its intrinsic stability to enable the measurement of precise radial velocities of exoplanet hosts. As a result, there are usually many more spectra of each object available for spectroscopic analysis.

Sample spectra from each of the instruments are shown in Figure 1.

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Three different techniques were used to determine the spectroscopic parameters T_eff, log g, and [Fe/H]. One procedure was applied to all of our spectra, to provide a homogeneous data set. The other two techniques were applied to subsets of the spectra in order to evaluate systematic differences in the effective temperatures, surface gravities, and iron abundances. We describe them below. Two of the methods also give the projected rotational velocities, vsin i.

The technique that was applied to all of our TRES, HIRES, and FIES spectra is referred to as SPC, and is described fully by Buchhave et al. (2012). It is based on a cross-correlation of the observed spectrum against a library of synthetic spectra calculated from Kurucz model atmospheres (Castelli & Kurucz 2003, 2004), and determines the temperature, surface gravity, metallicity, and projected rotational velocity by seeking the maximum of the cross-correlation coefficient as a function of those parameters (see also Torres et al. 2002; Latham et al. 2002). The grid of synthetic spectra covers a wide range in the above four parameters, although it spans only a limited wavelength region between 5050 Å and 5360 Å (Buchhave et al. 2012). This allowed the use of five echelle orders in the FIES spectra, three in the TRES spectra, and two for HIRES. For simplicity, all synthetic spectra in this library were computed with a microturbulent velocity of ξ = 2 km s⁻¹ and a macroturbulent velocity of ζ_RT = 1 km s⁻¹. The metallicities derived with this method are not strictly [Fe/H], but represent instead an average abundance [M/H] of the elements producing absorption features in the spectral region under consideration. Although iron lines tend to dominate, the two indices could be different for a star with peculiar abundances, or more generally for metal-poor stars in which the α elements are often enhanced. However, for the present sample there is no evidence of such anomalies from detailed abundance studies by ourselves or others (e.g., McCullough et al. 2006; Sozzetti et al. 2006; Deleuil et al. 2008; Torres et al. 2008; Anderson 2010), nor are any of the stars particularly metal-poor so that we would expect [α/Fe] to be significantly different from zero. In the following, we have therefore assumed that [M/H] is equivalent to [Fe/H].

A second technique was applied to nearly all of the HIRES spectra, and relies on the widely used analysis package Spectroscopy Made Easy (SME; see Valenti & Piskunov 1996) with the atomic line database of Valenti & Fischer (2005). The procedure assumes local thermodynamical equilibrium (LTE) and plane parallel geometry, and synthesizes spectra adjusting T_eff, log g, and [M/H] to achieve the best match to the observed spectra by minimizing a χ² function. [M/H] is a global abundance parameter in SME that is used to interpolate in the grid of model atmospheres and to scale the solar abundance pattern (except for helium and a few other elements including iron) when calculating opacities. For the present work, we report instead the iron abundance, [Fe/H], which is a separate variable in the fits. We used eight wavelength segments of approximately 20 Å each, one including the gravity-sensitive Mg b triplet and the others spanning the range 6000–6180 Å. Following Valenti & Fischer (2005), we used a fixed value of ξ = 0.85 km s⁻¹ for the microturbulent velocity in SME, and the radial–tangential macroturbulent velocity was computed with the prescription given by the same authors.⁶ The performance of this method for deriving effective temperatures, surface gravities, metallicities, and rotational velocities is documented in detail in the same work.

Finally, for about half of the HIRES spectra and some of the TRES spectra we made use of a third method that follows a more classical curve-of-growth approach, as implemented in the spectral synthesis code MOOG⁷ (Sneden 1973). We used MOOG in conjunction with a grid of Kurucz ATLAS model atmospheres (Kurucz 1993). This technique determines spectroscopic properties under the assumption of LTE, imposing the conditions of excitation and ionization equilibrium. It relies on equivalent widths of selected Fe i and Fe ii lines, which we measured either manually or using the automated ARES⁸ tool (Souza et al. 2007). Between 100 and 200 relatively isolated lines were measured in each spectrum, depending on the S/N, with equivalent widths in the range from 2 to 120 mÅ. Microturbulence was determined by imposing the constraint that the Fe i abundance should not depend on the reduced equivalent width. For details of the procedure see, e.g., Sozzetti et al. (2006, 2007). For the TRES spectra, the wavelength region used is ∼4300–6750 Å (echelle orders 10 to 38), and for HIRES we used all orders between 4980 Å and 6420 Å.

4.1. Unconstrained determinations

In this section, we report spectroscopic determinations of the temperatures, surface gravities, metallicities and projected rotational velocities with SPC, SME, or MOOG in which we solved for those three parameters simultaneously, without making use of any external information about the stars. We refer to these as "unconstrained" results, to distinguish them from the results discussed in the next section in which we hold the surface gravities fixed. The unconstrained results are the most commonly reported in the exoplanet literature, although as we discuss later, the constrained values are generally preferable. For this reason, we defer a tabulation of the final parameters until Section 4.3. Tables 1–3 provide a listing of the method(s) applied to each spectrum. For stars with two or more spectra available from a given telescope, we have taken an average of the individual determinations.

A significant fraction of our stars have unconstrained determinations from two or more analysis methods. This allows us to compare results and check for systematic differences, which has seldom been done for transiting planet hosts. In Figure 2, we display the unconstrained values of T_eff, log g, and [Fe/H] from the different procedures against each other. For the purpose of this comparison, the new SME results obtained here have been augmented with other SME-based results from the literature, relying mostly on HIRES spectra obtained by the HATNet project (Bakos et al. 2004). Those spectra have also been reanalyzed with SPC. The average offsets from at least 30 stars in common between the methods are listed in Table 4, along with the uncertainty of the mean. The average differences are well below 100 K in temperature, and under 0.1 dex in both log g and [Fe/H]. In computing these differences we have adjusted the MOOG abundances to account for a small difference in the adopted solar iron abundance in our implementation of that technique (A(Fe) = 7.52)⁹ compared to SPC and SME (A(Fe) = 7.50). On average, MOOG is seen to give slightly hotter temperatures and higher surface gravities than both SPC and SME, but lower iron abundances. The SPC results are intermediate between the other two.

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Table 4. Comparison of Unconstrained Results for T_eff, log g, and [Fe/H] from Different Analysis Techniques

Methods	ΔTeff	Δlog g	Δ[Fe/H]	N
	(K)	(dex)	(dex)
SPC−SME	+32 ± 12	+0.014 ± 0.017	−0.023 ± 0.015	49
SPC−MOOG	−37 ± 18	−0.049 ± 0.020	+0.015 ± 0.018a	37
SME−MOOG	−78 ± 18	−0.093 ± 0.030	+0.068 ± 0.014a	31

Note. ^aMOOG metallicities have been adjusted to the same solar iron abundance of A(Fe) = 7.50 adopted in SPC and SME.

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A comparison of the projected rotational velocities from SPC and SME is shown in Figure 3. For vsin i values larger than about 10 km s⁻¹, the SPC values tend to be systematically larger than those from SME. Part of this may be related to differences in the continuum-fitting algorithms used in the two techniques (see, e.g., Behr 2003; Royer et al. 2002a, 2002b; Glazunova et al. 2008; Hensberge et al. 2000).

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4.2. Constrained Determinations

One of the most important uses of the spectroscopic parameters for transiting planet hosts is to infer the mass and radius of the star (M_⋆, R_⋆), which are needed in turn to establish the mass and radius of the planet. The stellar dimensions are typically obtained by comparing T_eff, log g, and [Fe/H] with stellar evolution models, or by using empirical calibrations (Torres et al. 2010a; Enoch et al. 2010). The spectroscopic parameter that most directly affects the determination of the stellar radius is log g, which is a proxy for the luminosity (typically unknown for these stars since the parallaxes have generally not been measured). However, surface gravity has a rather subtle effect on the spectral line profiles, and is difficult to measure accurately. It has been advocated (see Sozzetti et al. 2007; Holman et al. 2007; Torres et al. 2008, and others) that a much better constraint on the luminosity of transiting planet hosts can be obtained from the normalized semimajor axis a/R_⋆ that is directly measurable from the transit light curve when the eccentricity is known (Seager & Mallén-Ornelas 2003). The quantity a/R_⋆ is closely related to the mean stellar density, ρ_⋆. The approach that is now most common in the field for inferring M_⋆ and R_⋆ is to compare T_eff, [Fe/H], and ρ_⋆ with stellar evolution models, or to use those three quantities as inputs to empirical calibrations. Once the mass and radius are known, a more accurate value of log g follows trivially. We refer to this as a density-based surface gravity, which is of course model or calibration dependent.

The log g values that emerge from this process do not always agree with the spectroscopic estimates. This inconsistency has usually been attributed to shortcomings in the spectral synthesis techniques. Because the surface gravity is typically correlated with metallicity and temperature in some of the most commonly used methods of analysis, these differences have motivated some authors to repeat the spectroscopic determination of T_eff and [Fe/H] holding log g fixed at the external density-based values, in order to avoid systematic errors that could propagate into the stellar mass and radius (see, e.g., Kovács et al. 2007; McCullough et al. 2008). We have taken the same approach here, using the best available estimates of the density-based log g for each system from published photometric analyses. New values of T_eff, [Fe/H] and vsin i have been derived with each of the three methodologies, and are presented in Table 5.¹⁰ The external values of log g are given later in Section 4.3, with our final results.

Table 5. Spectroscopic Results Using the External Constraint on log g from Photometry (Constrained Analysis)

Star	SPC Analysis			SME Analysis			MOOG Analysis		Tela
	Teff	[Fe/H]	vsin i	Teff	[Fe/H]	vsin i	Teff	[Fe/H]
	(K)	(dex)	(km s−1)	(K)	(dex)	(km s−1)	(K)	(dex)
CoRoT-1	6280 ± 50	−0.04 ± 0.08	4.3 ± 0.5	6312 ± 44	+0.08 ± 0.04	4.9 ± 0.5	6300 ± 75	+0.05 ± 0.11	H
CoRoT-2	5552 ± 50	−0.15 ± 0.08	10.6 ± 0.5	5602 ± 44	−0.01 ± 0.04	10.0 ± 0.5	5550 ± 75	−0.10 ± 0.11	H
CoRoT-7	5392 ± 50	+0.08 ± 0.08	0.5 ± 0.5	5274 ± 44	+0.02 ± 0.04	0.5 ± 0.5	5250 ± 75	0.00 ± 0.08	H
HAT-P-3	5201 ± 50	+0.45 ± 0.08	1.8 ± 0.5	⋅⋅⋅	⋅⋅⋅	0.5 ± 0.5	⋅⋅⋅	⋅⋅⋅	H
HAT-P-3	5247 ± 50	+0.36 ± 0.08	2.2 ± 0.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	T

Notes. ^aTelescope/instrument combination: H = Keck/HIRES, F = NOT/FIES, T = FLWO/TRES.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In Figure 4, we compare the constrained results against the unconstrained values from the previous section, separately for the three methods. For SPC and SME, the temperature and the metallicity changes were found to be strongly correlated with changes in surface gravity, in the sense that T_eff and [Fe/H] both increase when a higher gravity is adopted. The slopes of these correlations (determined with simple linear regressions of Y on X using the SLOPES code of Feigelson & Babu 1992) are such that for an increase of 0.5 dex in log g the temperatures change systematically by about +310 K for SPC and +350 K for SME. For a similar increase in log g, the metallicities from SPC and SME both change by about +0.19 dex. These correlations are hardly significant for MOOG: the formal changes per 0.5 dex increase in surface gravity are only +70 K and +0.04 dex in T_eff and [Fe/H], respectively, which are of the order of the typical internal errors or smaller. We also note that there is some evidence that these correlations depend on temperature, in the sense that for stars cooler than the Sun they are roughly half as large.

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SPC and SME are somewhat similar techniques in the sense that they both optimize T_eff, log g, and [Fe/H] by seeking the best match between the observed spectrum and a synthetic spectrum. The correlations described above are not completely unexpected in these procedures, as one spectroscopic quantity can play against another to some extent and lead to nearly the same cross-correlation value or χ² value. For example, stronger lines produced by adopting a higher metallicity for the synthetic spectrum can be compensated for, to first order, by a suitable increase in the effective temperature. A similar degeneracy exists between surface gravity and temperature. In the case of MOOG, the effect of these correlations is evidently much weaker. A significant difference between MOOG and the other methods is the line lists. In particular, for this work MOOG uses only Fe i and Fe ii lines, whereas SME additionally includes the region of the Mg i b triplet, and SPC relies heavily on this same spectral region as well. Therefore, for the two latter methods the Mg i b lines contain by far the strongest information on log g. We speculate that any errors in the synthesis of the broad wings of these pressure-sensitive lines may cause errors in the SPC and SME determinations that would likely also impact the T_eff and [Fe/H] values because of the correlations described above; MOOG, on the other hand, would be unaffected. While we have no evidence of such errors at the present time, this could be investigated by using different atmospheric models. We note also that the surface gravity in the unconstrained MOOG analysis is determined implicitly by requiring that the iron abundance for Fe i and Fe ii from the measured equivalent widths be the same (ionization equilibrium). When fixing log g to a value determined externally this condition is generally no longer met, although the discrepancy for our sample (about 0.05 dex, on average) is small compared to the nominal uncertainty in the Fe i abundance. This lack of ionization equilibrium seems to have little effect on the other parameters, and could be a sign of real differences between Fe i and Fe ii (see, e.g., Schuler et al. 2010, and references therein), or deficiencies in the models (Yong et al. 2004). For all three methods, we find that the goodness of fit (as quantified by χ² or the average cross-correlation coefficient) is only slightly lower in the constrained fits compared to those in which log g is left free.

Differences between the results with unconstrained surface gravities and those using the density-based values were found to be up to 0.5 dex for SPC and SME, and up to 0.3 dex for MOOG. To the extent that the photometric constraints on ρ_⋆ (and therefore log g) are accurate, this suggests that all three spectroscopic procedures are vulnerable to systematics, though perhaps not to the same degree or for the same reasons.

Indirect evidence of these effects may perhaps be seen already in published results from the two surveys that have produced the majority of the transiting planet discoveries from the ground: the WASP project (Pollacco et al. 2006) and the HATNet project (Bakos et al. 2004). The HATNet group has generally used SME for the spectroscopic analysis of the parent stars, and in most cases the studies have been iterated as described above, using the constraint from the transit light curves to set log g. The WASP team has occasionally also used SME, although more recently they have relied on the UCLSYN package (Smalley et al. 2001) and in general their spectroscopic results are from an unconstrained analysis (log g free).

A comparison of the distribution of published metallicities for the host stars from these two groups is seen in Figure 5. The average metallicities differ by about 0.17 dex, and a Kolmogorov–Smirnov test suggests the distributions are statistically different, with a false alarm probability (FAP) of 0.14%. Both surveys are magnitude limited, with the WASP program being typically shallower, so a difference in the metallicity distributions is possible in principle. However, we find that the mean visual magnitude of the two star samples is essentially the same (V ≈ 11.5 for WASP and V ≈ 11.3 for HATNet), and a K-S test indicates the brightness distributions are indistinguishable (FAP = 0.259). It seems likely, therefore, that the difference in the average metallicities is mostly due to the analysis. It goes in the direction expected, as the constrained log g values adopted by the HATNet team are more often larger than the unconstrained values, which according to Figure 4 should lead to higher metallicities in that survey, just as observed.

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Figure 6 shows that for SPC and SME the differences between the constrained and unconstrained results for T_eff, log g, and [Fe/H] are a function of the temperature of the star. In the case of SME, the external surface gravities for stars up to roughly 6000 K tend to be lower than the unconstrained values, on average, and in turn lead to systematically lower temperatures and metallicities. For hotter stars, the trend reverses sharply, reaching maximum differences of 0.5 dex in log g, 400 K in T_eff, and 0.2 dex in metallicity compared to those with log g free. For SPC, the external log g values differ somewhat less from the unconstrained determinations, but again the hotter stars give higher temperatures and [Fe/H] values when imposing this constraint. No such dependence is apparent for the MOOG results. We also found no significant dependence of ΔT_eff, Δlog g, or Δ[Fe/H] on either metallicity or surface gravity, in any of the three methods. The projected rotational velocities from SPC and SME are insensitive to the change in log g.

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As mentioned above, the trends in SPC and SME with effective temperature are most likely related to the fact that the information on surface gravity in these methods comes almost exclusively from the pressure-broadened wings of the Mg i b lines at ∼5200 Å. For stars hotter than about 6000 K, the wings of these lines weaken considerably, and the methods lose sensitivity to log g. It is less clear why the bias in log g for such stars is toward smaller values, as opposed to being toward higher values or simply having a larger scatter. In any case, the positive correlation between log g, T_eff, and [Fe/H] explains the upturn in the last two quantities in Figure 6.

As a result of the temperature correlation for SPC and SME shown in Figure 6, the net effect of applying the log g constraint with those methods is to shift the stars in the H-R diagram toward a less evolved state (closer to the zero-age main sequence). The less evolved state also seems more likely a priori, because of the slower speed of evolution for unevolved stars. We illustrate this for SME in Figure 7. The trend has a significant impact on the inferred radii of the stars, which we discuss more quantitatively in Section 6.

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The constrained results from the different methods are compared in Figure 8 for stars in common. Our expectation was that the methods would show better agreement than the unconstrained results, since some of the biases in the determination of T_eff and [Fe/H] that are intrinsic to each of the techniques would be reduced by enforcing an external constraint on log g. This is indeed the case, although residual trends can still be seen in some of the panels of Figure 8, such as between MOOG and SME in [Fe/H], or between SPC and SME in temperature and also metallicity. These are most likely a reflection of strong correlations between the spectroscopic parameters that are present in at least two of the methods (SPC and SME), as already described, and the different degrees to which the three procedures respond to the imposition of an external log g constraint. Many details of the spectroscopic analysis are likely to influence the results in ways that are difficult to predict or quantify. This includes the accuracy of the continuum normalization of the spectra, the model atmospheres used, the adopted line lists, and the sensitivity to the S/N of the spectra, which varies from method to method. Nevertheless, average differences between the final values from SPC, SME, and MOOG remain quite small, as seen in Table 6.

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Table 6. Comparison of Constrained Results for T_eff and [Fe/H] from Different Analysis Techniques

Methods	ΔTeff	Δ[Fe/H]	N
	(K)	(dex)
SPC−SME	+27 ± 9	−0.020 ± 0.015	44
SPC−MOOG	+27 ± 23	+0.049 ± 0.019a	36
SME−MOOG	+8 ± 22	+0.081 ± 0.017a	29

Note. ^aMOOG metallicities have been adjusted to the same solar iron abundance of A(Fe) = 7.50 adopted in SPC and SME.

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4.3. Final Spectroscopic Results

Prior to this study, the largest effort to determine spectroscopic parameters for transiting planet hosts in a homogeneous way was that of Ammler-von Eiff et al. (2009), who used MOOG following the precepts of Santos et al. (2004, 2006). Our sample and theirs have 14 stars in common.

A comparison of our results from Table 7 with theirs is seen in Figure 9, where the differences in T_eff, [Fe/H], and log g for these 14 stars are plotted in the sense 〈Ammler-von Eiff minus Table 7〉. As indicated earlier, our adopted surface gravities are the density-based values, not the spectroscopic values. The temperatures on the horizontal axis of the figure are from our own determinations.

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Although the sample is small, there appears to be a significant systematic difference in the surface gravities that correlates with temperature (bottom panel), such that Ammler-von Eiff et al. (2009) measure a higher log g for the hotter stars. It is difficult to see how a bias in our log g values from Table 7 could cause this, as those determinations rest essentially on the a/R_⋆ measurements from the transit light curves. There may also be a trend of ΔT_eff with temperature (top panel), though it seems marginal. For the metallicities, there is no significant correlation with temperature, but we note that all of the Ammler-von Eiff et al. (2009) values of [Fe/H] are higher than ours, the average difference being +0.10 ± 0.03 dex.

In Figure 7 above, we presented a graphical illustration of how the inferred evolutionary state of the star can change considerably as a result of applying the external (photometric) constraint on log g in the spectroscopic analysis. As might be expected, this can also lead to significant changes in the mass and radius derived for the star, which are of more immediate interest for computing planetary properties in transiting planet systems.

To explore this more quantitatively, we have derived stellar masses and radii for the same stars shown in Figure 7 following the common procedure of comparing the spectroscopic quantities T_eff, log g, and [Fe/H] against stellar evolution models. For each star, we performed a Monte Carlo simulation in which we drew 10,000 values of the three spectroscopic parameters from Gaussian distributions characterized by the measured values and observational errors, assuming they are uncorrelated. We compared each set with isochrones from the Yonsei-Yale series by Yi et al. (2001), seeking the best match in a χ² sense. Ages were allowed to vary in steps of 0.1 Gyr over the range from 0.1 to 13.7 Gyr, and the α-element abundance was assumed to be solar for this test ([α/Fe] = 0.0). Stellar masses and radii were determined from the mode of the respective posterior probability distributions, and 1σ lower and upper confidence limits were defined by the 15.85% and 84.15% percentiles of the cumulative distribution. We carried out these calculations first using the unconstrained SME results, and then repeated the process with the constrained results.

A comparison between M_⋆ and R_⋆ from these two sets of parameters is shown in Figure 10 as a function of Δlog g, where the changes are displayed both in absolute units and as a percentage. In the most extreme cases, the unconstrained radii can be off by up to 100%. This occurs for the hotter stars, as can be seen in Figure 7. The impact on the mass is also not negligible, reaching ∼15% in some cases. Errors of this magnitude are almost always larger than other observational errors. Very similar results were obtained using the SPC determinations.

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The importance of this bias coming from the reliance on the weakly determined spectroscopic log g is now widely recognized in the community, and is largely avoided in current analyses of transiting planets by adopting instead the density-based log g to infer M_⋆ and R_⋆. However, many of those same studies still retain the temperatures and metallicities from unconstrained spectroscopic analyses in which the surface gravity was left free. Because T_eff and [Fe/H] typically suffer from strong correlations, as shown earlier in Figure 4, those values have a residual bias that is often not negligible and can propagate to the stellar masses and radii. This bias has been generally overlooked.

We have quantified this systematic effect by repeating the determination of M_⋆ and R_⋆ for the same sample above using the photometry-based log g, but deliberately adopting T_eff and [Fe/H] from the unconstrained spectroscopic analysis, to emulate the procedure often followed in published studies. Figure 11 compares these masses and radii with those based on our second iteration of SME, in which the temperature and metallicity were redetermined using the external log g constraint. There are clear differences in the stellar properties that mimic those seen in Figure 6. We conclude that even if a more accurate log g is adopted for inferring M_⋆ and R_⋆ from evolutionary models, using the unconstrained values of the temperatures and metallicities rather than those from a second iteration of the spectroscopic analysis can still lead to stellar masses that are up to 20% too large, and radii that are overestimated by as much as 10% for the hotter stars. Since planetary properties from Doppler and light curve analyses have dependencies M_p∝M^2/3_⋆ and R_p∝R_⋆, the above errors can translate into biases (overestimates) of 13% and 10% in the planetary masses and radii. For cooler stars the errors tend to be in the opposite direction and are smaller. Analogous results were obtained when using the determinations from SPC. In the case of MOOG, on the other hand, the effect on M_⋆ and R_⋆ from using unconstrained temperatures and metallicities is only marginally significant, consistent with the much smaller correlations seen in Figure 4.

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We defer a comprehensive examination and tabulation of stellar masses and radii for a future paper, since we are also compiling many new light curves for analysis, which may improve the external constraints.

While most transiting planet investigations use stellar evolution models to infer stellar masses and radii, others use empirical relations for M_⋆ and R_⋆ as a function of temperature, metallicity, and mean density. The biases described above can affect both. The Kepler mission is increasingly making use of asteroseismology as an alternative way of deriving the mean stellar density, based on oscillation frequencies measured directly from the light curves in favorable cases. Deriving the stellar mass or radius with this technique still requires stellar models, as well as accurate measurements of T_eff and [Fe/H], so there is still a danger of systematic errors from the use of unconstrained spectroscopic determinations.

Numerous investigations have addressed the persistent problem of the anomalously "inflated" radii of some of the Jovian planets—objects that can exceed 60% of the size of Jupiter in extreme cases—which has been with us since the discovery of transits in HD 209458, challenging our understanding of planet formation and evolution. A variety of mechanisms have been proposed that may play a role in "puffing up" the planets, but no universal process seems to account for all of these anomalies (see, e.g., Burrows et al. 2007; Miller et al. 2009; Fortney & Nettelmann 2010; Demory & Seager 2011, and references therein). Our results in the preceding section suggest that a portion of the discrepancy in R_p may have to do with systematic errors in the stellar radii in some cases, which can amount to ∼10%, as illustrated in Figure 11.

One of the notorious examples of inflated planets for which we have derived new spectroscopic parameters is WASP-12 b, with a radius of R_p = 1.79 ± 0.09 R_Jup (Hebb et al. 2009).¹¹ This study of WASP-12 b adopted spectroscopic properties derived from an unconstrained SME analysis yielding T_eff = 6290 K (rounded off by the authors to 6300 K, and assigned errors of ⁺²⁰⁰₋₁₀₀ K) and [M/H] = +0.30^+0.05_−0.15. These are both somewhat higher than we derive here from our constrained spectroscopic analyses: T_eff = 6118 ± 64 K and [Fe/H] = +0.07 ± 0.07. We note that their spectroscopic surface gravity (log g = 4.38) is also larger than the density-based (photometrically constrained) value they reported, log g = 4.17 ± 0.03. These differences are all consistent with the magnitude and sign of correlations shown in Figure 4, and leave open the possibility that the stellar mass and radius of Hebb et al. (2009) may be biased. To test this we repeated their isochrone analysis using the Yonsei–Yale stellar evolution models of Yi et al. (2001), first with the Hebb et al. (2009) temperature and metallicity, and then adopting ours. The surface gravity was held fixed at the density-based value they determined. The stellar radius we obtained with our revised spectroscopic parameters is 5% smaller, implying a planetary radius also 5% smaller, all else being equal. While this correction is far from what would be needed to solve the puzzle of the inflated radius of WASP-12 b, it does go in the right direction.

Accurate knowledge of the properties of the host stars in transiting exoplanet systems is essential to derive accurate characteristics for the planets. Considerable efforts have been devoted to improving the light curves of newly discovered as well as previously known transiting systems, but relatively little attention has been paid to refining the spectroscopic properties of the stars.

Here, we have derived new effective temperatures, metallicities, and projected rotational velocities for 56 transiting planet systems in a homogeneous manner, using the SPC technique. These determinations bring a needed measure of uniformity to the growing collection of stellar and planetary properties that should facilitate the discovery of patterns and correlations that may provide valuable insight into the nature of planets.

A key aspect of our spectral analysis is the application of an external constraint on the surface gravity, based on accurate knowledge of the mean stellar density of the star, which comes directly from the light curve modeling. Because log g is usually weakly constrained by the spectra, fixing it as we have done here prevents errors in log g from biasing the temperatures and metallicities, and from affecting the inferred stellar masses and radii. Those biases come mainly from strong correlations between T_eff, [Fe/H], and log g that are present in unconstrained determinations (log g free), not only when applying SPC, but also in the widely used SME procedure as implemented by Valenti & Fischer (2005). Both of these methods are based on spectral synthesis. We find that the correlations are much smaller with MOOG, which uses a more classical curve-of-growth approach.

We investigate the interagreement among the three spectroscopic techniques by applying SME and MOOG to subsets of our stars, and we show that the temperatures and metallicities are generally in good accord after application of the log g constraint (the mean differences being well under 50 K and 0.1 dex, respectively). We do, however, detect some remaining systematic trends as a function of temperature and metallicity that are occasionally larger in some regimes.

Virtually all current studies of transiting planets make use of the mean stellar density as a luminosity indicator to derive the stellar properties, either through a comparison with model isochrones, or using empirical relations. We demonstrate that not using ρ_⋆ can incur errors in mass of up to 20%, and errors in the radius as large as 100% in some cases. Even though such errors are now usually avoided, many authors still retain the temperatures and metallicities obtained from unconstrained spectroscopic analyses, i.e., without fixing log g to the more accurate values based on the light curve modeling. We demonstrate that this practice can lead to residual biases in M_⋆ of up to 20%, and systematic errors in R_⋆ up to 10% for the hotter stars, which will propagate through to the planetary properties. Such errors can be larger than other observational uncertainties, and may explain part of the radius anomaly of some of the inflated Jovian planets. In order to avoid this, we advocate performing a second iteration on the spectroscopic analysis (particularly when using SPC or SME) that fixes log g to the value inferred from the photometrically determined mean stellar density.

We are grateful to Jeff Valenti for helpful discussions, and to P. Berlind, M. Calkins, G. Esquerdo, D. W. Latham, and R. P. Stefanik for their help in obtaining the TRES spectra used here. We also thank the anonymous referee for helpful comments. G.T. acknowledges partial support for this work from NASA's Origins of Solar Systems program, through grant NNX09AF59G, and from NSF grant AST-10-07992. The research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System Abstract Service.