Accurate Empirical Radii and Masses of Planets and Their Host Stars with Gaia Parallaxes

Essentially all exoplanets with measured masses and radii are those found in transiting systems. Unfortunately, as is well known, the masses and radii of transiting planets are generally not measured directly; the planets' masses and radii depend, through direct transit observables, on the assumed masses and radii of their host stars. The observables are the depth of the transit, which (in the absence of limb darkening) is simply ${\delta }_{\mathrm{tr}}={k}^{2}$, where $k\equiv {R}_{p}/{R}_{\star }$ and ${R}_{p}$ and ${R}_{\star }$ are the radius of the planet and star, respectively, and the velocity semi-amplitude ${K}_{\mathrm{RV}}$, which is given by

$$\begin{eqnarray}&&{K}_{\mathrm{RV}}\equiv {\left(\displaystyle \frac{2\pi G}{P}\right)}^{1/3}\displaystyle \frac{{M}_{p}\,\sin \,i}{{({M}_{\star }+{M}_{p})}^{2/3}}{(1-{e}^{2})}^{-1/2},\end{eqnarray} \tag{ 1 }$$

where P, e, and i are period, eccentricity, and inclination of the planet's orbit, respectively, ${M}_{p}$ is the mass of the planet, and ${M}_{\star }$ is the mass of the star. The eccentricity of the orbit can be determined from the precise shape of the Doppler reflex (radial velocity, or RV) motion of the star, and the inclination can be measured from the relative duration of the ingress/egress τ and full-width half-maximum T of the transit (Carter et al. 2008). Thus, in order to estimate ${R}_{p}$ and ${M}_{p}$, one must be able to measure ${R}_{\star }$ and ${M}_{\star }$.

Unfortunately, it is not possible to estimate the mass or radius of the host purely from photometric follow up of the primary transit and RV measurements of the host star, regardless of how precise these measurements are. This is due to a well-known degeneracy, first pointed out in the case of transiting planets by Seager & Mallén-Ornelas (2003). As they note, the only parameter about the star that can be directly measured from observables is the ratio of the semimajor axis of the orbit a to the radius of the star $a/{R}_{\star }$ (Winn 2010),

$$\begin{eqnarray}&&\displaystyle \frac{a}{{R}_{\star }}=\displaystyle \frac{{\delta }_{\mathrm{tr}}^{1/4}}{\pi }\displaystyle \frac{P}{\sqrt{T\tau }}\left(\displaystyle \frac{\sqrt{1-{e}^{2}}}{1+e\sin \omega }\right),\end{eqnarray} \tag{ 2 }$$

where ω is the argument of periastron, which is also an observable from the RV curve. However, this quantity is closely related to the density of the star (Seager & Mallén-Ornelas 2003; Winn 2010),

$$\begin{eqnarray}&&{\rho }_{\star }=\displaystyle \frac{3\pi }{{{GP}}^{2}}{\left(\displaystyle \frac{a}{{R}_{\star }}\right)}^{3}-{k}^{3}{\rho }_{p}\simeq \displaystyle \frac{3\pi }{{{GP}}^{2}}{\left(\displaystyle \frac{a}{{R}_{\star }}\right)}^{3},\end{eqnarray} \tag{ 3 }$$

where ${\rho }_{p}$ is the density of the planet (and is typically $\sim {\rho }_{* }$) and the last equality follows from the fact that typically k³ ≪ 1.

Thus ${\rho }_{* }$ can essentially be inferred from direct observables. Nevertheless, there remains a one-parameter degeneracy that makes it impossible to estimate the mass and radius of the star independently. All transiting-planet systems (with only photometry of the primary transit and RV observations) are subject to this degeneracy. To break the degeneracy, one must bring in additional external constraints, such as a measurement of the surface gravity of the star, $\mathrm{log}g$ (which is a direct observable from high-resolution spectra), astroseismological inferences of the stellar mass and radius (e.g., Huber et al. 2013), or a measurement of the radius of the star (which, as we will show is a direct observable from the bolometric flux and effective temperature of the star, and the distance to the system).

Up until now, these observables of the host stars, while preferred because they are direct, have been either poorly measured, subject to systematic errors, or not constrained at all.

Instead, most authors typically use theoretical and/or empirically calibrated relations between observable properties of the star. Stellar evolution is reasonably well understood, and it is known that a star of a given effective temperature, metallicity, and density cannot have an arbitrary mass and radius. Indeed, to the first order, these three parameters essentially fix the luminosity and age of the star, and thus its radius and mass. Therefore, adopting these constraints, while not direct, typically leads to much more precise estimates of the parameters of the system.

Nevertheless, they are subject to uncertainties in stellar evolution models and second-order parameters (i.e., stellar rotation), and/or inaccurate calibrations of the empirical relations. One might therefore be concerned that these estimates, while precise, are not accurate. One clear demonstration of this is the case of KELT-6b (Collins et al. 2014), where the parameters inferred using the Yonsei-Yale isochrones (Demarque et al. 2004) to break the degeneracy disagreed significantly (by as much as 4σ) from those inferred using the Torres et al. (2010) empirical relations. Likely this was due to the low metallicity [Fe/H] ≈ −0.3 of the KELT-6 host star and the fact that neither the isochrones nor the empirical relations were well-calibrated at such low metallicities.

While slightly erroneous inferences about the properties of individual systems (as in the case of Collins et al. 2014) are troubling, the difficulty with estimating accurate parameters of host stars and thus their transiting planets can be, and indeed has proven to be, quite deleterious in some cases, sometimes leading to markedly incorrect or inconsistent inferences about individual systems or even entire populations of planets. An early example of this is the case of the supermassive (∼12 ${M}_{J}$) planet XO-3 b (Johns-Krull et al. 2008), in which initial estimates of ${R}_{p}$ differed by nearly a factor of two, from ∼1.2 ${R}_{J}$  to ∼2.1 ${R}_{J}$. The latter value would have implied that the planet was highly inflated relative to standard models, which would have been particularly interesting given its relatively large inferred mass. With improved photometry and thus an improved estimate of ${\rho }_{\star }$, Winn et al. (2008) were able to demonstrate that the true planetary radius was likely at the lower end of this range. Indeed, as we show in this work, our revised determinations with the Gaia distance—which places the star at a significantly shorter distance than previously assumed—reveal the planet to be ${M}_{p}$ ≈ 7 ${M}_{J}$ and ${R}_{p}$ ≈ 1.4 ${R}_{J}$. We defer additional case studies of problems arising from current poor constraints on models and empirical relations to the discussion (Section 4).

However, the difficulties with interpreting the properties of planetary populations due to uncertainties about the properties of the host stars became quite prominent and acute with the discoveries of thousands of planets via Kepler (Borucki et al. 2010). Here the difficulties were threefold. First, the Kepler transiting-planet hosts tended to be fairly faint compared to those found via ground based transit surveys, making characterization of the host stars more difficult. Second, the shear number of hosts made systematic assays of their properties via high-resolution spectroscopy extremely resource-intensive; this was obviously exacerbated by the faintness of the hosts. Finally, the wide Kepler bandpass, poor cadence, and/or low signal-to-noise ratio of the majority of the transit signals made estimates of the ingress/egress time, and thus stellar densities, generally imprecise. This has led to herculean efforts to characterize the properties of the host stars, often resulting in quite different conclusions as to the radius distribution of the Kepler target stars and thus their transiting planetary companions (see, e.g., Mann et al. 2012; Pinsonneault et al. 2012; Dressing & Charbonneau 2013; Huber et al. 2014; Gaidos et al. 2016).

Three recent advances now permit the determination of accurate and empirical radii and masses for a large sample of transiting planets. First, there now exist all-sky, broadband photometric measurements for stars spanning a very broad range of wavelengths, from the GALEX far-UV at ∼0.15 μm to the WISE mid-IR at ∼22 μm. These measurements permit construction of spectral energy distributions (SEDs) that effectively sample the majority of the flux for all but the hottest stars. Consequently, the bolometric fluxes (${F}_{\mathrm{bol}}$) and in turn the stars' angular radii (Θ, via the stellar effective temperature, ${T}_{\mathrm{eff}}$) can in principle be determined in a largely empirical manner. Using a set of eclipsing binary stars as benchmarks, Stassun & Torres (2016a) showed that with such data ${F}_{\mathrm{bol}}$ can be measured with a precision that is typically ≲3%. Second, the Gaia mission's first data release (Gaia Collaboration et al. 2016) has delivered trigonometric parallaxes for ∼2 × 10⁶ stars in common with Tycho-2 (Høg et al. 2000), with a precision for the best ∼10% of stars of 240 μas. These parallaxes permit Θ to be converted to ${R}_{\star }$. Third, a large sample of transiting planets orbiting stars that are sufficiently bright to have been included in Tycho-2, and consequently in the Gaia first data release, have been published with quantities that follow from direct observables such as the stellar density, ${\rho }_{\star }$, the ratio of planet-to-star radii, and the orbital radial-velocity semi-amplitude. With ${R}_{\star }$, these quantities yield R_p as well as the stellar mass, which in turn yields the planet mass.

In this paper, we perform this procedure to measure ${F}_{\mathrm{bol}}$ and Θ for 498 planet host stars, which will serve as fundamental stellar parameters for use with upcoming data releases from Gaia. We also report empirical stellar and planet radii and masses as described above for 116 stars that host transiting planets, have the necessary direct observables published in the literature, and have parallaxes newly reported in the Gaia first data release (Gaia Collaboration et al. 2016). We additionally perform this procedure for 242 stars that host non-transiting (radial-velocity only) planets, for which the newly derived planet properties remain modulo factors of sin i.

Importantly, the stellar and planet properties that we determine are independent of stellar models and of empirically calibrated stellar relations; thus the properties that we determine for the stars and their planets do not require the assumption that individual systems behave according to theoretical expectation or within the limits of mean relations. We argue, moreover, that the stellar and planet properties that we determine are empirical and accurate, even if the Gaia parallaxes do not yet yield precisions that rival those typically achieved via model-dependent analyses reported in the literature. However, the Θ measurements that we determine are sufficiently accurate and precise that upcoming, improved parallax measurements from Gaia should enable the stellar and planet properties to be re-determined with accuracies and precisions superior to those currently available in many cases—in an entirely empirical, model-free fashion. As we enter the Transiting Exoplanet Survey Satellite (TESS) era of superb precision in the transit parameters of very bright stars, such empirical and accurate stellar properties will become even more important than they have been for Kepler targets for which precision follow up often proved challenging.

In Section 2, we describe our study sample, the data used, and our methodological approach. The primary results of this study are presented in Section 3, including ${F}_{\mathrm{bol}}$, Θ, ${R}_{\star }$, and ${M}_{\star }$, followed by the planet properties, R_p and M_p. In Section 4, we present additional motivation for the importance of reducing reliance on stellar models, explore the degree to which the approach laid out in this paper is truly empirical, discuss our results in the context of previously published results, and briefly discuss the prospects for improving on the stellar and planet properties reported here with the anticipated advent of improved parallaxes from the Gaia second data release. Finally, Section 5 provides a summary of our results and conclusions.

We began by selecting all planet-hosting stars found in the exoplanets.org database (Han et al. 2014, accessed on 2016 August 31) and added 12 well-characterized transiting planets that were present in the NASA Exoplanet Archive⁵ but missing from exoplanets.org. We then selected systems with host stars that are also present in the Tycho-2 catalog, resulting in 560 unique stars. Of these, 62 stars were removed because they lacked one or more of the minimal set of parameters required for our analysis (see Section 2.2); nearly all of these were Kepler planets for which radial-velocity semi-amplitudes were not reported. The remaining 498 stars form our master study sample for which we perform our SED fitting procedures, resulting in fundamental ${F}_{\mathrm{bol}}$ and Θ measurements, as discussed below. The Gaia DR1 provides parallaxes for 358 of these stars, of which 116 were listed as hosting transiting planets and 242 were listed as hosting radial-velocity planets. We updated the XO-3 stellar radius and distance from the exoplanets.org database to the more recent values reported in Winn et al. (2008; see Section 4).

For each of the transiting-planet hosts, we adopted the literature values from the exoplanets.org database for each of the following quantities: the orbital period, $P$, and its uncertainty; the transit depth, ${\delta }_{\mathrm{tr}}$, and its uncertainty; the ratio of the orbital semimajor axis to the stellar radius, $a/{R}_{\star }$, and its uncertainty; the orbital inclination angle to the line of sight, i, and its uncertainty; and the orbital radial-velocity semi-amplitude, ${K}_{\mathrm{RV}}$, and its uncertainty. In this paper we analyze only the "b" planet in each system; however, the stellar properties that we newly determine here (e.g., ${F}_{\mathrm{bol}}$, Θ, ${R}_{\star }$, ${M}_{\star }$) can be readily applied to other planets in the case of currently known or future discovered additional planets.

We also adopted the parallax measurements newly available from Gaia (Gaia Collaboration et al. 2016). We adopt the Gaia parallax, π, and its uncertainty as provided by the Gaia first data release⁶ (see Table 4). We note that, at the time of this writing, the Gaia π values potentially have systematic uncertainties that are not yet fully characterized but that could reach ∼300 μas.⁷ Preliminary assessments suggest a global offset of −0.25 mas (where the negative sign indicates that the Gaia parallaxes are underestimated) for π ≳ 1 mas (Stassun & Torres 2016b), corroborating the Gaia claim, based on comparison to directly measured distances to well-studied eclipsing binaries by Stassun & Torres (2016a). Gould & Kollmeier (2016) similarly claim a systematic uncertainty of 0.12 mas. Casertano et al. (2016) used a large sample of Cepheids to show that there is likely little to no systematic error in the Gaia parallaxes for π ≲ 1 mas, but found evidence for an offset at larger π consistent with Stassun & Torres (2016b). Thus the available evidence suggests that any systematic error in the Gaia parallaxes is likely to be small. Thus, for the purposes of this work, we use and propagate the reported random uncertainties on π only, emphasizing that (a) the fundamental ${F}_{\mathrm{bol}}$ and Θ measurements that we report are independent of π and (b) additional (or different) choices of π uncertainties may be applied to our ${F}_{\mathrm{bol}}$ and Θ measurements following the methodology, equations, and error propagation coefficients supplied below.

The assembled set of literature parameters for the 498 stars in our master sample—including the 116 transiting-planet hosts and the 242 non-transiting-planet hosts in our study for which Gaia DR1 parallaxes are available—are tabulated in Tables 5 and 6, respectively.

For all planets in the study sample, we first collect high-quality spectroscopic values of host star ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [Fe/H], as described in Section 2.3, and broadband photometric data, as described in Section 2.4. Then, for each planetary system, we fit a spectroscopic parameter-based stellar atmosphere model to a broadband photometry-based SED, as described in Section 2.5. We directly sum the unreddened SED model over all wavelengths to obtain ${F}_{\mathrm{bol}}$ and use the Stefan–Boltzmann law to calculate the host-star radius, ${R}_{\star }$, from ${F}_{\mathrm{bol}}$, ${T}_{\mathrm{eff}}$, and the Gaia parallax, as described in Section 2.6.

For the transiting planets, we calculate the planet radius ${R}_{p}$ from the transit depth, ${\delta }_{\mathrm{tr}}$ ≡ (${R}_{p}$/${R}_{\star }$)², and the empirically calculated stellar radius, ${R}_{\star }$. The stellar density, ${\rho }_{\star }$, is calculated from the transit model parameter $a/{R}_{\star }$ and the orbital period, P, (see Section 2.6) and then the stellar mass, ${M}_{\star }$, is calculated from ${\rho }_{\star }$ and ${R}_{\star }$. Planet mass, ${M}_{p}$, is then empirically calculated from ${M}_{\star }$, the radial-velocity semi-amplitude, ${K}_{\mathrm{RV}}$, and the orbital period, P, and eccentricity, e (see Section 2.7). Finally, we calculate planet surface gravity, $\mathrm{log}{g}_{p}$, from ${K}_{\mathrm{RV}}$, the transit parameters $a/{R}_{\star }$ and ${\delta }_{\mathrm{tr}}$, and P, e, and orbital inclination, i, and we calculate the insolation at the planet, $\langle F\rangle$, from ${F}_{\mathrm{bol}}$, the distance to the star, and the semimajor axis of the orbit, a (see Section 2.7).

For the non-transiting planets, we calculate ${M}_{\star }$ from our empirical ${R}_{\star }$ and the spectroscopic $\mathrm{log}g$. We also calculate the minimum planet mass, ${M}_{p}$ $\sin \,i$, in the same way that we calculate ${M}_{p}$ for the transiting planets, but with the spectroscopic $\mathrm{log}g$-based stellar mass rather than the generally higher accuracy transit-based ${\rho }_{\star }$. We calculate the insolation at the planet in the same way as for the transiting planets.

We determine uncertainties for all calculated parameters by propagating the measured parameter uncertainties through the relevant equations. We also include the effects of typical parameter correlations for the transit, orbital, and RV parameters, as part of error propagation (see Section 2.7).

For most of the stars in our study sample, values of ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [Fe/H] are available in the exoplanet archives as obtained from the original literature. We adopted these values if no other independent, spectroscopic values were available. However, where possible, we opted to instead use ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [Fe/H] from the recently updated PASTEL catalog (Soubiran et al. 2016), which compiles stellar properties determined from high-quality⁸ spectroscopic analyses in the literature. We did this in order to ensure that the stellar properties used were as independent as possible from the transit-based parameters, so as to avoid hidden correlations in the derived parameters (see also Section 2.7).

Where multiple values were available in the PASTEL catalog, we adopted the mean and uncertainty on the mean. Where no values were available in the PASTEL catalog, we adopted the values reported in the archive. For a small number of stars with values in neither PASTEL nor in the archive, we adopted the values reported by Santos et al. (2013). Finally, for a few stars our initial SED fits clearly indicated a poor choice of ${T}_{\mathrm{eff}}$, so we adopted an alternate ${T}_{\mathrm{eff}}$ from the literature, as indicated in Tables 5 and 6.

In order to systematize and simplify our procedures, we opted to assemble for each host star the available broadband photometry from the following large, all-sky catalogs (listed here in approximate order by wavelength coverage) via the VizieR ⁹ query service.

• 1.  
  GALEX All-sky Imaging Survey (AIS): FUV and NUV at ≈0.15 μm and ≈0.22 μm, respectively.
• 2.  
  Catalog of Homogeneous Means in the U BV System for bright stars from Mermilliod (2006): Johnson U BV bands (≈0.35–0.55 μm).
• 3.  
  Tycho-2: Tycho B (B_T) and Tycho V (V_T) bands (≈0.42 μm and ≈0.54 μm, respectively).
• 4.  
  Strömgren Photometric Catalog by Paunzen (2015): Strömgren uvby bands (≈0.34–0.55 μm).
• 5.  
  AAVSO Photometric All-Sky Survey (APASS) DR6 (obtained from the UCAC-4 catalog): Johnson BV and SDSS gri bands (≈0.45–0.75 μm).
• 6.  
  Two-Micron All-Sky Survey (2MASS): JHK_S bands (≈1.2–2.2 μm).
• 7.  
  All-WISE: WISE1–4 bands (≈3.5–22 μm).

We found BV, JHK_S, and WISE1–3 photometry—spanning a wavelength range ≈0.4–10 μm—for nearly all of the stars in our study sample. Most of the stars also have WISE4 photometry and many of the stars also have Strömgren and/or GALEX photometry, thus extending the wavelength coverage to ≈0.15–22 μm. We adopted the reported measurement uncertainties unless they were less than 0.01 mag, in which case we assumed an uncertainty of 0.01 mag. In addition, to account for an artifact in the Kurucz atmospheres at 10 μm (see, e.g., Stassun & Torres 2016a, and note that the contribution to ${F}_{\mathrm{bol}}$ for λ ≥ 10 μm is ≲10⁻⁴), we artificially inflated the WISE3 uncertainty to 0.1 mag unless the reported uncertainty was already larger than 0.1 mag. Stassun & Torres (2016a) found no evidence for systematic effects in any of the other passbands, although individual outlier cases do occur; see Section 2.5. The assembled SEDs are presented in the Appendix.

To measure the ${F}_{\mathrm{bol}}$ of each of the 498 stars in our master sample, we followed the SED fitting procedures described in Stassun & Torres (2016a). Briefly, the observed SEDs were fitted with standard stellar atmosphere models and ${F}_{\mathrm{bol}}$ was measured by summation of the model after correction for extinction. For the stars in our sample with ${T}_{\mathrm{eff}}$ > 4000 K, we adopted the atmospheres of Kurucz (2013), whereas for the stars with ${T}_{\mathrm{eff}}$ < 4000 K we adopted the NextGen atmospheres of Hauschildt et al. (1999). As discussed in Stassun & Torres (2016a), because the photometric data span such a large portion of the SED, and because there is only one free parameter (A_V) in the SED fit, the determination of ${F}_{\mathrm{bol}}$ via this procedure is virtually model independent, the model atmosphere serving essentially as an "intelligent interpolation" between the photometric measurements.

As summarized in Tables 5 and 6, for each star we have ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [Fe/H]. We interpolated in the model grid to obtain the appropriate model atmosphere for each star in units of emergent flux. To redden the SED model, we adopted the interstellar extinction law of Cardelli et al. (1989) with the usual ratio of total-to-selective extinction, R_V = 3.1.¹⁰ We then fitted the atmosphere model to the flux measurements to minimize χ² by varying just two fit parameters: extinction (A_V) and overall normalization (effectively the ratio of the stellar radius to the distance, ${R}_{\star }^{2}/{d}^{2}$). (The uncertainty in the adopted stellar ${T}_{\mathrm{eff}}$ is handled in a later step via the propagation of errors through the stellar angular radius, Θ; see Section 2.6.) We estimated A_V from the three-dimensional Galactic dust model of Amôres & Lépine (2005, Model A) for each star's line of sight and distance, but we allowed the fit to vary A_V from this initial guess by as much as 20%, limited by the maximum line-of-sight extinction from the dust maps of Schlegel et al. (1998). The best-fit model SED with extinction is shown for each star in the Appendix, and the best-fit A_V and reduced χ² values (χ_ν²) are given in Table 7.

Inspection of the SEDs shows generally excellent fits (see below for discussion of χ²), with only a few exhibiting clear outlier behavior. These include HD 208527, α Ari, XO-4, 6 Lyn, HD 190360, HAT-P-15, HD 132563, and 14 And. Such outliers could occur for a number of possible reasons, including close neighbors that are unresolved in the available photometric catalogs and/or excesses arising from circumstellar material; in principle, multi-component SED fits could improve these in the future. Here we simply note these few cases and remind the reader that these are readily flagged by the ${\chi }_{\nu }^{2}$ of the fit provided in Table 7. These are also excluded from our analyses below.

The primary quantity of interest for each star is ${F}_{\mathrm{bol}}$, which we obtained via direct summation of the fitted SED, without extinction, over all wavelengths. The formal uncertainty in ${F}_{\mathrm{bol}}$ was determined according to the standard criterion of Δχ² = 2.30 for the case of two fitted parameters (e.g., Press et al. 1992), where we first renormalized the χ² of the fits such that ${\chi }_{\nu }^{2}=1$ for the best-fit model. Because ${\chi }_{\nu }^{2}$ is in almost all cases greater than 1 (see Table 7), this χ² renormalization is equivalent to inflating the photometric measurement errors by a constant factor and results in a more conservative final uncertainty in ${F}_{\mathrm{bol}}$ according to the Δχ² criterion.

Figure 1 (left panel) shows the dependence of the fractional ${F}_{\mathrm{bol}}$ uncertainty as functions of the goodness of the SED fit and of the uncertainty on ${T}_{\mathrm{eff}}$. For stars with ${T}_{\mathrm{eff}}$ uncertainties of ≲1%, the ${F}_{\mathrm{bol}}$ uncertainty is dominated by the SED goodness-of-fit. With the exception of a few outliers, we achieve an uncertainty on ${F}_{\mathrm{bol}}$ of at most 6% for ${\chi }_{\nu }^{2}\leqslant 5$. Adopting this as a goodness-of-fit threshold thus yields a sample of 114 transiting-planet host stars with Gaia DR1 parallaxes for which we can in principle achieve an uncertainty in ${R}_{\star }$ of ≈3% (see Equation (4)).

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Figure 1 (right) shows the distribution of fractional ${F}_{\mathrm{bol}}$ uncertainties for the transiting-planet sample (blue) and for all stars (black). For the transiting-planet hosts with ${\chi }_{\nu }^{2}\leqslant 5$, the median uncertainty on ${F}_{\mathrm{bol}}$ is 2% and is at most 5.7%. For the full sample, which includes the radial-velocity sample that is brighter on average than the transit sample, the median uncertainty is 1.7%, with 95% of the sample having an uncertainty of less than 5%.

Finally, we derived stellar radii ${R}_{\star }$ by combining our ${F}_{\mathrm{bol}}$ values derived from SED fitting with our ${T}_{\mathrm{eff}}$ values derived from photometry and spectra. These quantities are related by

$$\begin{eqnarray}&&{\rm{\Theta }}={({F}_{\mathrm{bol}}/{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^{4})}^{1/2},\end{eqnarray} \tag{ 4 }$$

where Θ is the stellar angular radius (${\rm{\Theta }}\equiv {R}_{\star }/d$), d is the distance to the star, and ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant. Errors propagating into ${R}_{\star }$ come from random and systematic errors on ${T}_{\mathrm{eff}}$, and uncertainties on ${F}_{\mathrm{bol}}$, A_V, and d. We note that the calculation of d from π can become non-trivial when ${\sigma }_{\pi }/\pi \gtrsim 20 \%$ (see, e.g., Bailer-Jones 2015). The actual ${\sigma }_{\pi }/\pi$ from Gaia DR1 for our study sample has a median of 2.3%, and is less than 20% for 98% of the sample. We therefore proceed in this paper to compute d via the straightforward 1/π conversion.

For transiting planets, we determine the stellar density, ${\rho }_{\star }$, from the transit model parameter $a/{R}_{\star }$ and the orbital period, P, through the relation

$$\begin{eqnarray}&&{\rho }_{\star }=\displaystyle \frac{3\pi }{{{GP}}^{2}}{(a/{R}_{\star })}^{3}\end{eqnarray} \tag{ 5 }$$

for ${\rho }_{p}\sim {\rho }_{\star }$ and ${k}^{3}\equiv {({R}_{p}/{R}_{\star })}^{3}\ll 1$.

Combining ${\rho }_{\star }$ with our determination of ${R}_{\star }$ provides a direct measure of the stellar mass, ${M}_{\star }$. For the non-transiting planets, we calculate ${M}_{\star }$ from our ${R}_{\star }$ and the spectroscopic $\mathrm{log}g$.

As a check that our procedures result in accurate Θ and ${F}_{\mathrm{bol}}$, we applied our procedures to the interferometrically observed planet-hosting stars HD 189733 and HD 209458 reported by Boyajian et al. (2015). Our SED fits are shown in Figure 2 and the Θ and ${F}_{\mathrm{bol}}$ values directly measured by those authors versus those derived in this work are compared in Table 1, where the agreement is found to be excellent and within the uncertainties.

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Table 1.  Comparison of Stellar Angular Diameters (2 × Θ) and Bolometric Fluxes (${F}_{\mathrm{bol}}$) for Stars with Interferometrically Measured Angular Diameters from Boyajian et al. (2015)

	Boyajian et al. (2015)	This Work
HD 189733 (${T}_{\mathrm{eff}}$ = 4875 ± 43 K)
2 × Θ (mas)	0.3848 ± 0.0055	0.391 ± 0.008
${F}_{\mathrm{bol}}$ (10−8 erg s−1 cm−2)	2.785 ± 0.058	2.87 ± 0.06
HD 209458 (${T}_{\mathrm{eff}}$ = 6092 ± 103 K)
2 × Θ (mas)	0.2254 ± 0.0072	0.225 ± 0.008
${F}_{\mathrm{bol}}$ (10−8 erg s−1 cm−2)	2.331 ± 0.051	2.33 ± 0.05

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From our empirically calculated ${R}_{\star }$, we then directly obtain ${R}_{p}$ via ${R}_{p}/{R}_{\star }=\sqrt{{\delta }_{\mathrm{tr}}}$ for the transiting planets. From our empirically calculated ${M}_{\star }$, we can directly calculate ${M}_{p}$ (${M}_{p}$ $\sin \,i$ for the RV planets) for all samples in the study via

$$\begin{eqnarray}&&{M}_{p}=\displaystyle \frac{{K}_{\mathrm{RV}}\sqrt{1-{e}^{2}}}{\sin \,i}{\left(\displaystyle \frac{P}{2\pi G}\right)}^{1/3}{M}_{\star }^{2/3},\end{eqnarray} \tag{ 6 }$$

in the limit ${M}_{p}\ll {M}_{\star }$.

For the transiting planets, we also directly calculate the planet surface gravity, $\mathrm{log}{g}_{p}$, from the RV, orbital, and transit parameters via

$$\begin{eqnarray}&&\mathrm{log}{g}_{p}=\displaystyle \frac{2\pi {K}_{\mathrm{RV}}\sqrt{1-{e}^{2}}{(a/{R}_{\star })}^{2}}{P\,{\delta }_{\mathrm{tr}}\,\sin \,i}.\end{eqnarray} \tag{ 7 }$$

Note that this is a direct observable and does not depend on the properties of the host star (Winn 2010).

Finally, we directly calculate the insolation at the planet, $\langle F\rangle$, for all samples in the study from the relation,

$$\begin{eqnarray}&&\langle F\rangle ={F}_{\mathrm{bol}}{\left(\displaystyle \frac{d}{a}\right)}^{2},\end{eqnarray} \tag{ 8 }$$

where d is the distance from Earth to the planetary system determined from the Gaia parallax and a is the semimajor axis of the planetary orbit. For the transiting planets, a is determined from $a/{R}_{\star }$ and our empirically determined ${R}_{\star }$. For the non-transiting planets, we use the values of a from the literature.

We determine uncertainties for all calculated parameters by propagating the measured parameter uncertainties through the relevant equations. For the purposes of this paper, we assume first-order linear perturbations for the error propagation and we also include the effects of typical parameter correlations for the transit, orbital, and RV parameters as part of error propagation (see below). In principle, our use of simple linear error propagation could underestimate the true errors for systems where the observational uncertainties are large and thus higher order terms would be needed. However, because this paper utilizes the Gaia DR1 parallaxes, which currently are the limiting factor for most of the systems in our study sample (see Section 3.3), this simplification of approach should be sufficient (see also below). The eventual arrival of the Gaia DR2 parallaxes may very well require more sophisticated error analysis procedures, such as through the use of full MCMC chains.

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It is beyond the scope of this work to calculate parameter correlations for all planets in our study, so we use KELT-15 (Rodriguez et al. 2016) as an exemplar planetary system to estimate parameter correlations and to verify our full error propagation methodology. To be sure, the KELT-15 correlations do not exactly represent the parameter correlations for all planets in our study. Nonetheless, we expect that they are reasonable representations, and since the Gaia DR1 parallaxes currently dominate the errors, this approximation should not compromise the accuracy of our results. To verify this expectation, we compared the results of several systems with and without the KELT-15 covariance terms included in the error propagation and found that indeed the results were identical to within ±1 in the least significant digit of most reported parameter values. Rodriguez et al. (2016) used a custom version of EXOFAST (Eastman et al. 2013) that allows multiple RV and transit data sets to be simultaneously fitted (see Siverd et al. 2012 for more details) to perform a KELT-15 global system fit to spectroscopic parameters, RV data, and photometric follow-up light curves. EXOFAST uses MCMC to robustly determine system parameter uncertainties. We used the resulting MCMC chains to calculate transit parameter correlations for KELT-15b. The resulting parameter correlation values are listed in Table 2 and are adopted for all planets for the purposes of error propagation.

Table 2.  Parameter Correlation Values Used for Error Propagation as Determined from the KELT-15b Global System Fit

Param. 1	Param. 2	Correlation
$a/{R}_{\star }$	${\delta }_{\mathrm{tr}}$	−0.152
$a/{R}_{\star }$	${K}_{\mathrm{RV}}$	−0.134
$a/{R}_{\star }$	$P$	−0.217
$a/{R}_{\star }$	$i$	0.517
$a/{R}_{\star }$	$e$	−0.645
${\delta }_{\mathrm{tr}}$	${K}_{\mathrm{RV}}$	−0.015
${\delta }_{\mathrm{tr}}$	$P$	0.090
${\delta }_{\mathrm{tr}}$	$i$	−0.406
${\delta }_{\mathrm{tr}}$	$e$	−0.007
${K}_{\mathrm{RV}}$	$P$	−0.049
${K}_{\mathrm{RV}}$	$i$	−0.051
${K}_{\mathrm{RV}}$	$e$	0.086
$P$	$i$	−0.043
$P$	$e$	0.366
$i$	$e$	−0.187

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We compare our calculated KELT-15b parameter values and uncertainties to those of Rodriguez et al. (2016) in Table 3. We find host-star and planet radii and masses that are ∼1σ higher than the literature values. Since ${\rho }_{\star }$ and $\mathrm{log}{g}_{p}$ are calculated solely from the transit light-curve and RV parameters (i.e., they do not involve ${R}_{\star }$), those values are nearly identical to the literature values. Our $\langle F\rangle$ is consistent with the literature value. Our propagated stellar and planet radii uncertainties are ∼3× the literature values. The stellar mass uncertainty is significantly higher and the planet mass uncertainty is ∼2× higher than the literature values. To the extent that KELT-15b is representative of the systems in our study sample, this suggests that in fact our simplified linear error propagation approach discussed above results in conservative error estimates.

Table 3.  KELT-15b Empirical Parameter Values and Uncertainties (This Work) Compared to Rodriguez et al. (2016)

Parameter	Units	This Work	Literature
${R}_{\star }$	Stellar Radius (${R}_{\odot }$)	1.783 ± 0.205	1.481 ± 0.066
${M}_{\star }$	Stellar Mass (${M}_{\odot }$)	2.065 ± 0.748	1.181 ± 0.051
${\rho }_{\star }$	Stellar Density (cgs)	0.513 ± 0.056	0.514 ± 0.055
${R}_{p}$	Planet Radius (${R}_{J}$)	1.737 ± 0.203	1.443 ± 0.084
${M}_{p}$	Planet Mass (${M}_{J}$)	1.311 ± 0.434	0.910 ± 0.220
$\mathrm{log}{g}_{p}$	Planet Surface Gravity (cgs)	3.032 ± 0.106	3.020 ± 0.115
$\langle F\rangle$	Incident Flux (109 erg s−1 cm−2)	1.642 ± 0.544	1.652 ± 0.145

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Two fundamental products of this work are the newly determined ${F}_{\mathrm{bol}}$ for each of the 498 planet host stars in our master sample, and from them the newly determined Θ for each of the stars. These are presented in Table 7.

Because the precision on ${F}_{\mathrm{bol}}$ is typically 2% and the precision on ${T}_{\mathrm{eff}}$ is typically 1%–2% (Figure 1), the achieved median precision on Θ is 3% for the transiting hosts and 1.8% for all stars (Figure 3, left panel). In absolute units, the median precision is 0.8 μas for the transiting hosts and 1.6 μas for all stars (Figure 3, right panel). The transiting-planet hosts are at greater distances, on average, than the radial-velocity planet hosts, and thus have smaller absolute Θ uncertainties despite having larger relative Θ uncertainties.

Importantly, these two newly determined properties—${F}_{\mathrm{bol}}$ and Θ—do not depend on d. Thus, these results serve as a fundamental, accurate, and purely empirical data set to permit the re-determination of stellar and planet radii and masses as measurements of d improve.

We can use the newly available parallaxes from the Gaia first data release to estimate ${R}_{\star }$ and ${M}_{\star }$ from our newly measured ${F}_{\mathrm{bol}}$ and Θ. The linear radii and masses newly determined here for the planet-hosting stars in our sample are presented in Table 7. In Figure 4, we show the achieved precision on the stellar radii and masses as a function of the Gaia distance and the ${T}_{\mathrm{eff}}$ precision. The stellar radii and masses for the transiting-planet host stars are determined with a median precision of 8% and 30%, respectively. Importantly, as shown in the figure, for most of the transiting-planet host stars, the limiting factor on the radius precision is the precision on the current Gaia distance.

Figure 5 presents the H–R diagram of the planet host stars, where all of the quantities represented are now measured entirely empirically. The separation of metal-poor stars below the metal-rich stars on the main sequence is readily apparent.

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The resulting planet radii and masses, R_p and M_p, are presented in Table 7. We also calculate directly the planet surface gravity, $\mathrm{log}{g}_{p}$, and the insolation received by the planet, $\langle F\rangle$. These quantities are displayed together in Figure 6 for the transiting planets. The effect of insolation on the planet radii and surface gravities is evident, in the sense that more highly insolated planets of high mass have significantly larger radii and lower surface gravities at fixed mass. This effect is not clearly evident among lower mass planets below ${M}_{p}\lesssim 0.1\,{M}_{J}$. This may simply be due to the small sample size, but it has been suggested that the lowest mass planets at high insolation may have had their atmospheres photoevaporated (Owen & Wu 2013).

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The precisions on ${R}_{p}$ and ${M}_{p}$ are shown in Figure 7. The median uncertainty on ${R}_{p}$ (we determine ${R}_{p}$ for transiting planets only) is 9%. The median uncertainty on ${M}_{p}$ is 22% for the transiting planets. The median uncertainty on ${M}_{p}$ $\sin \,i$ is 10% for the radial-velocity planets. Three planets with low signal-to-noise ${K}_{\mathrm{RV}}$ have fractional ${M}_{p}$ uncertainties above 0.5 and are not shown on the plot.

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4.1.1. The Importance of Reducing Reliance on, and Testing, Stellar Models and Empirical Relations

4.1.2. Are the Stellar and Planet Properties Truly Empirical?

4.2.1. Previous Compilations of Planet Host-star Radii

4.2.2. Planet and Host-star Radii and Masses: This Study Versus Previously Published Values

While we anticipate that the ${R}_{\star }$, ${M}_{\star }$, ${R}_{p}$, and ${M}_{p}$ values derived from our ${F}_{\mathrm{bol}}$ and Θ determinations will be greatly improved with the Gaia second data release (see below), we can already assess the impact of Gaia parallaxes from the first data release on the inferred stellar and planet properties. We present a direct comparison of our newly determined ${R}_{\star }$ versus those reported in exoplanets.org in Figure 8.

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Our revised ${R}_{\star }$ values range from ∼50% smaller to ∼100% larger than the previous literature values, and these differences are clearly the result principally of the improved Gaia distances, the one exception being XO-3 for which the updated ${R}_{\star }$ is small despite a change in d of nearly 25%. We have traced the XO-3 anomaly to the complicated history of this system in the literature; see below. For the transiting-planet host stars, because the fractional changes in d are generally relatively small, there is good overall agreement with previously reported ${R}_{\star }$; the median difference is 3% with no strong systematic dependence on ${T}_{\mathrm{eff}}$, and for 90% of the sample the change in ${R}_{\star }$ is at most 15%. On the other hand, for the radial-velocity planet host stars, there are significant differences, and in particular among the coolest stars there is a clear tendency for large changes in ${R}_{\star }$. We have compared our adopted ${T}_{\mathrm{eff}}$ with those previously used in the literature and confirm that this is not the cause of these large ${R}_{\star }$ differences (the difference in ${T}_{\mathrm{eff}}$ being 0.0% ± 0.3%). Comparison between the new stellar distances from Gaia and those previously reported in the literature suggests that the errors in the previous distances, particularly among the cooler very nearby stars, may be the culprit; here the differences in d can be as large as ∼100%, most notably for the red giants in the sample (Figure 8, right panel).

Similarly, we compare our newly determined ${M}_{\star }$ to those previously reported in Figure 9, where the ${M}_{\star }$ for the transiting-planet host stars are determined directly from transit-based ${\rho }_{\star }$ and ${R}_{\star }$, whereas for the radial-velocity planet hosts it is determined from the spectroscopic $\mathrm{log}g$ and ${R}_{\star }$. Once again, for the transit-planet host stars, there is good overall agreement. However, again for the radial-velocity host stars, there are significant differences, and these show systematics with ${T}_{\mathrm{eff}}$. We suspect that these differences arise from the fact that, for the transiting systems, we have a very direct measure of ${M}_{\star }$ from ${\rho }_{\star }$ and ${R}_{\star }$, whereas for the radial-velocity planet hosts we are subject to the well-known challenges with spectroscopic $\mathrm{log}g$  (Torres et al. 2012). Recall that we have opted to use spectroscopically determined $\mathrm{log}g$ that are measured independent of the planet discovery papers.

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Mortier et al. (2014) proposed empirical corrections for spectroscopic-based $\mathrm{log}g$ on the basis of transits and asteroseismology. We can use our transit-derived $\mathrm{log}g$ for the transiting-planet host stars to test this (Figure 10). Indeed, we find a tendency for stars with ${T}_{\mathrm{eff}}$ ≳ 6000 K to have spectroscopic $\mathrm{log}g$ that are overestimated by ∼0.1 dex, and for the cooler stars to have spectroscopic $\mathrm{log}g$ that are underestimated by a similar amount. This trend (${\rm{\Delta }}\mathrm{log}g=1.6\times {10}^{-4}{T}_{\mathrm{eff}}-0.95$, spectroscopic minus transit) is somewhat less steep than that proposed by Mortier et al. (2014). Nonetheless, the effect is in the correct sense to at least partially explain the systematics in ${M}_{\star }$ for the radial-velocity planet hosts (Figure 9, right panel).

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Based on the updated stellar ${R}_{\star }$ and ${M}_{\star }$, we can compare our newly determined ${R}_{p}$ and ${M}_{p}$ to those previously published. These comparisons are shown in Figures 11 and 12, respectively. The outlier planet radius labeled in Figure 11 is Kepler-454b. Gettel et al. (2016) reported its radius as 2.37 ± 0.13 R_⊕, but the value in exoplanets.org is 2.37 ± 0.13 ${R}_{J}$. The tip of the light gray arrow on the plot shows the proper position of Kepler-454b for the correct value from Gettel et al. (2016). The planet with a mass significantly below the line in Figure 12 is XO-3 b. The distance to XO-3 b is reported as $d={282}_{-23}^{+27}$ pc in exoplanets.org. In the XO-3 b discovery paper, Johns-Krull et al. (2008) reported a mass of ${M}_{p}=13.25\pm 0.64$ ${M}_{J}$ and a radius of ${R}_{p}=1.95\pm 0.16$ ${R}_{J}$ based primarily on a host-star stellar radius derived from the spectroscopic $\mathrm{log}g$. In a follow-up paper, Winn et al. (2008) found a slightly lower mass ${M}_{p}=11.79\pm 0.59$ ${M}_{J}$ and a significantly lower radius ${R}_{p}=1.217\pm 0.073$ ${R}_{J}$ and distance d = 174 ± 18 pc based on the stellar density from the transit light curves.¹² We find a significantly lower empirical value of planet mass ${M}_{p}=7.290\pm 1.188$ ${M}_{J}$ and a slightly higher planet radius ${R}_{p}=1.414\pm 0.121$ ${R}_{J}$ and distance d = 201.7 ± 14.2 pc compared to Winn et al. (2008).

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4.2.3. Occurrence of "Retired A Stars"

Finally, one of the outstanding questions in the literature is that of the so-called "retired A stars." In brief, it has been suggested that planets orbiting evolved subgiants and red giants may serve as probes of planet formation and evolution in massive stars (e.g., Johnson et al. 2007), which is particularly useful given the paucity of known planets orbiting unevolved (i.e., main-sequence) stars more massive than the Sun. The utility of the subgiant and red giant planets in this context is of course predicated on the assumption that the stars are indeed the evolved ("retired") counterparts of stars that were formerly A-type stars on the main sequence. However, this has been called into question (e.g., Lloyd 2011, 2013), considering the challenge of disambiguating stars by mass in the subgiant and red giant regions of the H–R diagram, where ${T}_{\mathrm{eff}}$ and ${L}_{\mathrm{bol}}$ become largely degenerate with ${M}_{\star }$, particularly when uncertainties and systematic errors in the determination of [Fe/H] and $\mathrm{log}g$ are considered. These degeneracies are compounded by uncertainties in the accuracy of stellar evolution models in these parts of the H–R diagram. As a result, the arguments and counter-arguments presented by, e.g., Lloyd (2011, 2013), Johnson & Wright (2013), and Schlaufman & Winn (2013) have largely relied on indirect lines of evidence, such as the proper application of priors, sample biases, Galactic models, and stellar spins. One exception is Johnson et al. (2014), who used asteroseismology to directly measure the mass of one "retired A star" to be ∼2 M_⊙.

With our new, empirically determined stellar properties—including in particular ${M}_{\star }$—we are in a position to assess the occurrence of stars with ${M}_{\star }$ ≳ 1.2 ${M}_{\odot }$ in the subgiant and red giant branches of the H–R diagram, at least among the systems in our study sample. Figure 13 shows the 134 stars in our study with the currently most accurately determined ${M}_{\star }$, for which ${\sigma }_{{M}_{\star }}\lt 15 \%$. Thirty of these are in the region of the H–R diagram often dubbed "retired A stars." The mean and median stellar mass of these 30 stars is ${M}_{\star }$ =1.58 ${M}_{\odot }$ and 1.45 ${M}_{\odot }$, respectively, and the inter-quartile range is 1.22–1.66 ${M}_{\odot }$. Only ∼20% of these stars have ${M}_{\star }$ < 1.2 ${M}_{\odot }$. Thus, we conclude that in our sample there is strong evidence for a preponderance of bona fide "retired A stars," though there do appear to be some stars that are less massive.

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The anticipated second data release from Gaia should enable the fundamental stellar ${F}_{\mathrm{bol}}$ and Θ reported here to achieve their full potential. In particular, whereas the accuracy that we currently achieve on ${R}_{p}$ and ${M}_{p}$ is limited for the transiting planets by the current Gaia parallax error floor of 240 μas (see Figure 4), the parallax precision for the Gaia second data release should be 20 μas for the bright stars in our study sample, and consequently the final ${R}_{p}$ and ${M}_{p}$ will in most cases be limited only by the accuracy of Θ, which in this work we already determine to be 1.8%. Thus, we may expect that the application of our Θ to the Gaia second data release parallaxes should yield ${R}_{p}$ and ${M}_{p}$ for the transiting planets in our sample to ≈3% and ≈5%, respectively, assuming that the uncertainties in the observed ${\delta }_{\mathrm{tr}}$ and ${\rho }_{\star }$ are negligible. At present, the median uncertainties on ${\delta }_{\mathrm{tr}}$ and ${\rho }_{\star }$ for the transit-planet host stars are ≈2% and ≈14%; thus the accuracy in ${R}_{p}$ should indeed be close to ≈3%, whereas for ${M}_{p}$ it would be closer to ≈10%. Still, improvements in the ${\rho }_{\star }$ from high-precision transit observations, such as expected from the bright TESS targets, will allow the methodology laid out here to achieve ≈5% accuracy in ${M}_{p}$ with the Gaia DR2 parallaxes.

Finally, we note that once we have many more precise and accurate empirical measurements of these stars from Gaia DR2, it will then be possible to refine and extend both stellar evolutionary models and empirical relations. These can then be more confidently applied to systems where direct empirical measurements are not possible, in order to derive precise and presumably more accurate parameters for vastly larger numbers of star/planet systems.