K2-97b: A (RE-?)INFLATED PLANET ORBITING A RED GIANT STAR

In the K2 extension to the NASA Kepler mission, multiple fields along the ecliptic are observed almost continuously for approximately 80 days (Howell et al. 2014). EPIC 211351816 (K2-97) was selected for observation as a part of K2 Guest Observer Proposal GO5089 (PI: Huber) and observed in Campaign 5 of K2 during the first half of 2015. As the Kepler telescope now has unstable pointing due to the failure of two of its reaction wheels, it is necessary to correct for the pointing-dependent error in the flux received per pixel. We produced a light curve by simultaneously fitting thruster systematics, low-frequency variability, and planet transits with a Levenberg–Marquardt minimization algorithm, using a modified version of the pipeline from Vanderburg et al. (2016).

We also analyzed the PDC-MAP light curve provided by the K2 Science Office (Smith et al. 2012; Stumpe et al. 2012) as well as the detrended light curves created with the methods of Vanderburg et al. (2016), Petigura (2015), and Aigrain et al. (2016). The use of different light curves resulted in statistically significant differences in the transit depth, illustrating the additional systematic uncertainties introduced by light-curve reductions (see Section 5.1 for more details). However, the results from all light curves analyzed were broadly consistent with the modified Vanderburg et al. (2016) results (see Discussion). Figure 1 shows our adopted light curve for K2-97.

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Natural guide-star AO images of K2-97 were obtained through the broad K' filter (λ_center = 2.124 μm) with the Near-Infrared Camera (NIRC2) at the Keck-2 telescope on Mauna Kea during the nights of UT 2016 March 19 and May 12. The narrow camera (pixel scale 001) was used for both sets of observations. No additional sources were detected within ∼3'' of the star. The contrast ratio of the detection limit is more than 7 magnitudes at 05; brighter objects could be detected to within 015 of the star.

We obtained a high-resolution, high-signal-to-noise spectrum of K2-97 using the High Resolution Echelle Spectrometer (HIRES) on the 10 m Keck-I telescope at the Mauna Kea Observatory on the Big Island of Hawaii. HIRES provides spectral resolution of roughly 100,000 in a wavelength range of 0.3–1.0 μm (Vogt et al. 1994). We also obtained medium-resolution optical and infrared spectra using the Supernova Integrated Field Spectrograph (SNIFS) on the 2.2 m University of Hawaii telescope and SpeX on the 3 m Infrared Telescope Facility (IRTF), providing a spectral resolution of 1000–2000 over a wavelength range from 0.3 to 5.5 μm (Lantz et al. 2004; Rayner et al. 2003).

We joined and flux calibrated the SNIFS and SpeX spectra following the method outlined in Mann et al. (2015). We first downloaded photometry from the Two-Micron All-Sky Survey (2MASS, Skrutskie et al. 2006), AAVSO All-Sky Photometric Survey (APASS, Henden et al. 2012), and The Wide-field Infrared Survey Explorer (Wright et al. 2010). The spectrum and all photometry were converted to physical fluxes using the appropriate zero-points and filter profiles (Cohen et al. 2003; Jarrett et al. 2011; Mann & von Braun 2015). We scaled the optical and NIR spectra to match the photometry and each other in overlapping regions (0.8–0.95 μm), accounting for correlated errors in the flux calibration. Regions of high telluric contamination or missing from our spectrum (e.g., beyond 2.4 μm) were replaced with a best-fit atmospheric model from the BT-SETTL grid (Allard et al. 2011, 2013). The final calibrated and combined spectrum is shown in Figure 2.

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Radial velocity measurements were obtained between 2016 January 27 and May 16 using the HIRES on the Keck-I Telescope at the Mauna Kea Observatory in Hawaii and the Levy spectrometer on the Automated Planet Finder (APF) telescope at Lick Observatory in California. The specific measurements are listed in Table 1. The nine spectra observed were obtained using an iodine cell. Measurements with the Keck telescope achieved a precision of greater than 1 m s⁻¹, whereas the APF measurements have measurement uncertainties of ∼30 m s⁻¹. We collected three measurements with Keck/HIRES and six with APF.

The Levy Spectrograph is a high-resolution slit-fed optical echelle spectrograph mounted at one of the two Nasmyth foci of the APF designed specifically for the detection and characterization of exoplanets (Burt et al. 2014; Fulton et al. 2015). Each spectrum covers a continuous wavelength range from 3740 to 9700 Å. We observed K2-97 using a 10 wide decker for an approximate spectral resolution of R = 100,000. Starlight passed through a cell of gaseous iodine, which serves as a simultaneous calibration source for the instrumental point-spread function (PSF) and wavelength reference. We measured relative RVs using a Doppler pipeline descended from the iodine technique in Butler et al. (1996). We forward-modeled 848 segments of each spectrum between 5000 and 6200 Å. The model consists of a stellar template spectrum, an ultra-high-resolution Fourier transform spectrum of the iodine absorption of the Levy cell, a spatially variable PSF, a wavelength solution, and RV. Traditionally, a high-signal-to-noise iodine-free observation of the same star is deconvolved with the instrumental PSF and used as the stellar template in the forward modeling process. However, in this case the star is too faint to collect the signal-to-noise needed for reliable deconvolution in a reasonable amount of time on the APF. Instead, we simulated this observation by using the SpecMatch software (Petigura 2015) to construct a synthetic template from the Coelho (2014) models and best-fit stellar parameters.

In order to obtain precise values for the stellar parameters, we collected a moderate signal-to-noise iodine-free observation using the HIRES spectrograph on the Keck I telescope (Vogt et al. 1994). We measured the effective temperature (${T}_{\mathrm{eff}}$), surface gravity ($\mathrm{log}g$), iron abundance ($[\mathrm{Fe}/{\rm{H}}]$), and rotational velocity of the star using the tools available in the SpecMatch software package (Petigura 2015). We first corrected the observed wavelengths to be in the observers rest frame by cross-correlating a solar model with the observed spectrum. Then, we fit for ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, $[\mathrm{Fe}/{\rm{H}}]$, Vsin(i), and the instrumental PSF using the underlying Bayesian differential-evolution Markov Chain Monte Carlo machinery of ExoPy (Fulton et al. 2013). At each step in the MCMC chains, a synthetic spectrum is created by interpolating the Coelho (2014) grid of stellar models for a set of ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ values and solar alpha abundance. We convolved this synthetic spectrum with a rotational plus macroturbulence broadening kernel using the prescriptions of Valenti & Fischer (2005) and Hirano et al. (2011). Finally, we performed another convolution with a Gaussian kernel to account for the instrumental PSF, and compared the synthetic spectrum with the observed spectrum to assess the goodness of fit. The priors are uniform in ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$, but we assign a Gaussian prior to the instrumental PSF that encompasses the typical variability in the PSF width caused by seeing changes and guiding errors. Five echelle orders of the spectrum were fit separately and the resulting posterior distributions were combined before taking the median values for each parameter. Parameter uncertainties were estimated as the scatter in spectroscopic parameters given by SpecMatch relative to the values for 352 stars in the Valenti & Fischer (2005) sample and 76 stars in the Huber et al. (2013b) asteroseismic sample. Systematic trends in SpecMatch values as a function of ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ relative to these benchmark samples were fit for and removed in the final quoted parameter values. Initial fits to the stellar spectrum for ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, $[\mathrm{Fe}/{\rm{H}}]$, and Vsin(i) were made without asteroseismic constraints, and were found to be in good agreement with the asteroseismic quantities. A prior was applied to the value for $\mathrm{log}g$ based on the asteroseismic estimate of 3.26 ± 0.015 (see Section 3.2), which resulted in convergence to the values listed in Table 2.

Stellar oscillations are a powerful tool to determine precise fundamental properties of exoplanet host stars (e.g Christensen-Dalsgaard et al. 2010; Gilliland et al. 2011; Huber et al. 2013b). The top panel of Figure 3 shows the power spectrum calculated from the K2 data after removing the transits from the light curve. We detect a strong power excess with regularly spaced peaks near ∼220 μHz (75 minutes), typical for an oscillating low-luminosity red giant star.

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The power excess can be characterized by the frequency of maximum power (${\nu }_{\max }$) and the average separation of modes with the same spherical degree and consecutive radial order (${\rm{\Delta }}\nu$). To measure ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, we analyzed the K2SC light curve of this system (Aigrain et al. 2016) using the method of Huber et al. (2009), which corrects the background granulation noise by fitting a two-component Harvey model (Harvey 1985) in the frequency domain. The frequency of maximum power was then measured from the peak of the heavily smoothed, background-corrected power spectrum, and ${\rm{\Delta }}\nu$ was measured using an autocorrelation of the power spectrum. We calculated uncertainties using 1000 Monte Carlo simulations as described in Huber et al. (2011), yielding ν_max = 223.7 ± 5.4 μHz and Δν = 16.83 ± 0.17 μHz.

The bottom panel of Figure 3 shows an échelle diagram, which stacks radial orders on top of each other, showing the asymptotic spacing of oscillation modes with the same spherical degree l. The échelle diagram of K2-97 shows the characteristic signature of nearly vertically aligned quadrupole (l = 2) and radial (l = 0) modes, while the dipole modes (l = 1) show a more complex distribution due to the coupling of pressure modes with gravity mode in the core (known as mixed modes, e.g., Dziembowski et al. 2001; Bedding et al. 2010; Montalbán et al. 2010). The position of the l = 0 ridge agrees with the expected value for a low-luminosity RGB star (Huber et al. 2010; Corsaro et al. 2012).

To estimate stellar properties from ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, we use the scaling relations of Brown et al. (1991) and Kjeldsen & Bedding (1995):

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\approx {f}_{{\rm{\Delta }}\nu }{\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{0.5},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\approx \displaystyle \frac{g}{{{\rm{g}}}_{\odot }}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{-0.5}.\end{eqnarray} \tag{ 2 }$$

Equations (1) and (2) can be rearranged to solve for mass and radius:

$$\begin{eqnarray}&&\displaystyle \frac{M}{\ {{\rm{M}}}_{\odot }}\approx {\left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{f}_{{\rm{\Delta }}\nu }{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{1.5},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{R}{{R}_{\odot }}\approx \left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{f}_{{\rm{\Delta }}\nu }{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{0.5}.\end{eqnarray} \tag{ 4 }$$

Our adopted solar reference values are ν_max,⊙ = 3090 μHz and Δν_⊙ = 135.1 μHz (Huber et al. 2011), as well as T_eff,⊙ = 5777 K.

Equations (1)–(4) are not exact, particularly for stars that are significantly more evolved than the Sun. Empirical tests using interferometry and open clusters and individual frequency modeling have illustrated that the relations typically hold to ∼5% in radius and ∼10% in mass. Comparisons to model frequencies have also demonstrated that the ${\rm{\Delta }}\nu$ scaling relation shows systematic deviations of up to a few percent as a function of ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$ (White et al. 2011). We accounted for this through the correction factor f_Δν in Equations (1)–(4), which we determined by iterating the spectroscopic ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$ as well as the asteroseismic mass and $\mathrm{log}g$ using the model grid by Sharma et al. (2016). The converged correction factor was f_Δν = 0.994, and our final adopted values for the stellar radius, mass, $\mathrm{log}g$ and density are listed in Table 2.

To estimate a stellar age, which cannot be derived from scaling relations alone, we used evolutionary tracks from Bressan et al. (2012). Matching the asteroseismic radius to an isochrone with the best-fit asteroseismic mass and $[\mathrm{Fe}/{\rm{H}}]$ = +0.42 dex from spectroscopy (see Table 3) yielded ∼7.8 ± 2 Gyr. An independent analysis using the BAyesian STellar Algorithm, which is based a grid of BaSTI models and has been applied to model several dozen Kepler exoplanet host stars (Silva Aguirre et al. 2015), yielded strongly consistent results. The stellar age can be constrained more precisely by modeling individual asteroseismic frequencies, but such modeling is beyond the scope of this paper.

Table 3.  Posterior Probabilities from Light-curve and Radial Velocity MCMC Modeling

Parameter	Median	84.1%	15.9%	Prior
ρ(g cm−3)	0.020	+0.001	−0.001	${ \mathcal N }$(0.02; 0.001)
T0 (BKJD)	2309.072	+0.007	−0.007	${ \mathcal U }$(1.3; 2.5)
Porb (days)	8.4062	+0.0015	−0.0015	${ \mathcal U }$(8.3; 8.5)
b	0.933	+0.006	−0.007	${ \mathcal U }$(0.0, 1.0 + ${R}_{p}/{R}_{* }$)
${R}_{p}/{R}_{* }$	0.0311	+0.0013	−0.0015	${ \mathcal U }$(0.0, 0.5)
K (m s−1)	103	+8	−8
T${}_{0,\mathrm{RV}}$ (BKJD)	2583.808	+0.007	−0.007	${ \mathcal U }$(0.0, Porb)
lnf	−3.8	+2.8	−3.9	${ \mathcal U }$(−10, 10)
hGP (ppm)	157	+5	−5	${ \mathcal U }$(exp(−12, 0))
λGP (days)	0.057	+0.005	−0.004	${ \mathcal U }$(exp(−10, 10))
σGP (ppm)	189	+4	−4	${ \mathcal U }$(exp(−20, 0))

Note. ${ \mathcal N }$ indicates a normal distribution with mean and standard deviation given respectively. ${ \mathcal U }$ indicates a uniform distribution between the two given boundaries. Ephemerides were fit relative to the first measurement in the sample and then later converted to Barycentric Kepler Julian Date (BKJD). Transit limb darkening parameters γ₁ and γ₂ were fixed to 0.6505 and 0.1041, respectively.

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A model-independent estimate of the distance was found using the bolometric flux of 3.579 ± 0.086 × 10⁻¹³ W m⁻² (uncorrected for extinction) computed from the flux-calibrated spectrum (Section 2.3), the temperature from the high-resolution spectroscopic analysis (Section 3.1), a reddening value of E(B − V) = 0.039 based on the maps of Schlafly & Finkbeiner (2011) and the extinction law of Fitzpatrick (1999). The estimated distance is 763 ± 42 pc, placing the star 350 pc above the galactic plane (b = 27°). The location well above the plane is consistent with the locations of other RGB stars (Casagrande et al. 2016) and justifies our use of the $\infty$ value for reddening.

The transit of K2-97b was first identified by applying the box least-squares algorithm of Kovács et al. (2002) to all targets in our K2 Campaign 5 program. The transits are sufficiently deep to be spotted by eye (see Figure 1) and the combined signal-to-noise is greater than 20, well above commonly adopted thresholds for significant transit events. The transit event was also identified in the planet candidate paper of Pope et al. (2016).

Evolved stars show correlated stellar noise on timescales of hours to weeks due to stellar granulation (Mathur et al. 2012), leading to significant biases in transit parameter estimation (Carter & Winn 2009; Barclay et al. 2015). To account for this, we used Gaussian process estimation, which has been successfully applied to remove correlated noise in transmission spectroscopy, Kepler light curves, and radial velocity data in the past (Gibson et al. 2012; Dawson et al. 2014; Haywood et al. 2014; Barclay et al. 2015; Grunblatt et al. 2015). This is accomplished by describing the covariance of the time-series data as an N × N matrix Σ where

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{ij}}={\sigma }_{i}^{2}{\delta }_{{ij}}+k({t}_{i},{t}_{j})\end{eqnarray} \tag{ 5 }$$

where σ_i is the observational uncertainty, δ_ij is the Kronecker delta, and $k({t}_{i},{t}_{j})$ is the so-called covariance kernel function that quantifies the correlations between data points. The simplest and most commonly used kernel function, the squared-exponential or radial basis function kernel, can be expressed as

$$\begin{eqnarray}&&k({t}_{i},{t}_{j})={h}^{2}\exp \left[-{\left(\displaystyle \frac{{t}_{i}-{t}_{j}}{\lambda }\right)}^{2}\right]\end{eqnarray} \tag{ 6 }$$

where the covariance amplitude h is measured in flux units and the length scale λ is measured in days (Rayner et al. 2004). Previous transit studies have used the squared-exponential kernel to remove correlated noise without removing the transit signal (Barclay et al. 2015).

To analyze the light curves, initial parameter guesses are selected for the kernel function, and then a likelihood of the residuals defined by the kernel function parameters is calculated, where the residuals are equivalent to the light curve with a Mandel–Agol transit model subtracted from it (Mandel & Agol 2002). The logarithm of the posterior likelihood of our model is given as

$$\begin{eqnarray}&&\mathrm{log}[{ \mathcal L }({\boldsymbol{r}})]=-\displaystyle \frac{1}{2}{{\boldsymbol{r}}}^{{\rm{T}}}{{\boldsymbol{\Sigma }}}^{-1}{\boldsymbol{r}}-\displaystyle \frac{1}{2}\mathrm{log}| {\boldsymbol{\Sigma }}| -\displaystyle \frac{n}{2}\mathrm{log}(2\pi ),\end{eqnarray} \tag{ 7 }$$

where r is the vector of residuals of the data after removal of the mean function (in our case, r is the light-curve signal minus the transit model), and n the number of data points.

The GP kernel function and transit model parameters are then fit as free parameters via Markov chain Monte Carlo (MCMC) exploration of parameter space using the Python software package emcee (Foreman-Mackey et al. 2013). The emcee package contains an Affine-invariant MCMC Ensemble sampler, which determines the maximum likelihood parameters through an iterative exploration of parameter space. We draw the planet radius from this MCMC exploration of parameter space, with a 1σ error corresponding to 68% confidence intervals in the MCMC distributions of all free parameters. Along with the planet-to-star radius ratio, the impact parameter, period, and ephemeris of transit were fit simultaneously with the Gaussian process kernel parameters and a photometric jitter term. Limb darkening parameters were fixed to the Claret & Bloemen (2011) stellar atmosphere model grid values closest to the measured temperature, surface gravity, and metallicity of the host star. Initial parameter values and priors were determined via a least-squares transit fit using ktransit (Barclay 2015). The results and priors for this simultaneous parameter fitting are listed in Table 3 and parameter distributions are given in Figure 5.

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To ensure our results were replicable, we performed a second MCMC analysis of the system using additional model parameters using a method very similar to that applied to Kepler-91 by Barclay et al. (2015). Mean stellar density, photometric zero-point, two limb darkening parameters, radial velocity zero point, two Gaussian process hyperparameters, time of mid-transit, orbital period, impact parameter, the scaled planet radius, two eccentricity vectors ($e\sin \omega$ and $e\cos \omega$), radial velocity semi-amplitude, secondary eclipse depth, amplitude of ellipsoidal variations, amplitude of reflected light from the planet, and two uncertainty parameters added in quadrature with the reported uncertainties on radial velocity and photometric data were included in this secondary model. The priors on these parameters were uniform except for a Gaussian prior based on the asteroseismic value of the mean stellar density, priors that kept the two limb darkening parameters physical (Burke 2008) plus Gaussian priors with means taken from Claret & Bloemen (2011) and a standard deviation of 0.4, a prior of $1/e$ on the eccentricity to avoid biasing this value high (Eastman et al. 2012) and an additional prior that took the form of a Beta function with parameters determined by Van Eylen & Albrecht (2015). Additionally, we sampled the logarithm of the Gaussian process hyperparameters, RV semi-amplitude, secondary eclipse depth, ellipsoidal variations, reflected light, and two uncertainty parameters. We ran the MCMC algorithm using 600 walkers and 20,000 steps yielding 12 million samples. We found posteriors on the scaled planet radius of ${0.0296}_{-0.0024}^{+0.0035}$ and an impact parameter of ${0.921}_{-0.032}^{+0.023}$, strongly consistent with our earlier study. A secondary eclipse, ellipsoidal variations, and any reflected light from the planet were not detected. We found an eccentricity of a few percent, marginally inconsistent with zero.

We modeled the APF and Keck radial velocity measurements of the planet with a Keplerian orbital model. Assuming K2-97b would produce the dominant signal in the radial velocity measurements, we assume a circular orbit for the planet and fit the data with a sinusoid with a period set to the orbital period obtained from the transit fitting. Using a Markov chain Monte Carlo method, best-fit values were determined for the phase and amplitude of the radial velocity variations. We applied a velocity shift of 23 m s⁻¹ to the APF measurements relative to the Keck measurements, and additionally fit for a non-zero offset to the resultant sinusoid to account for the different RV zero points of the two instruments. The mass of the planet was then estimated from the Doppler amplitude. The best-fit RV model and relative measurement values are shown in Figure 7. As sub-giant and giant stars are known to have an additional 4–6 m s⁻¹ of velocity scatter due to stellar jitter (Johnson et al. 2007b), we adopted a value of 5 m s⁻¹ and add it to our measurement errors in quadrature.

The Kepler pixels span 4'' on the sky, and thus background eclipsing binaries (EBs) can often cause false positive transit signals (Batalha et al. 2010; Jenkins et al. 2010; Everett et al. 2015). In addition, the K2 light curve was constructed using an aperture that is 7 pixels or 28'' across, exacerbating the possibility of a false positive. As the maximum transit depth of an EB is 50%, such a system would have to be at least as bright as Kepler magnitude (K_P)  ≈  19 to mimic a transit. To identify potential culprits, we searched the photometry database of the Sloan Digital Sky Survey (Data Release 9) for sources within 30'' of K2-97. We identified only a single source (SDSS J083104.13+105112.9) of interest. It has an estimated K_P = 19.05, yet is well outside the photometric aperture and the small fraction of light scattered into the aperture by the Kepler point response function ensures it could not have produced the transit signal. No sources were detected in our Keck II-NIRC 2 AO imaging down to K' = 15.5–18 (02–2''), corresponding to ${K}_{P}\gt 19$ for M dwarf stars that are the most likely components of faint background EBs.

To calculate a false positive probability for the background EB scenario, we followed the method of Gaidos et al. (2016). This discrete (Monte Carlo) Bayesian calculation uses a synthetic population generated by the TRILEGAL galactic stellar population model as priors (v. 1.6; Vanhollebeke et al. 2009) for 10 square degrees at the location of K2-97 on the sky. Likelihoods are calculated by imposing constraints on stellar density from the transit duration and orbital period, and on brightness from the non-detections in the SDSS and NIRC2 images, requiring that the diluted eclipse depth is at least equal to the transit depth. We found that the false positive probability for this scenario is effectively zero because no star from the simulated background population can simultaneously satisfy the stellar magnitude and density constraints. Background stars are either too faint to produce the transit or are ruled out by our high-resolution imaging, and the long transit duration implies a stellar density that is too low for dwarf stars.¹⁷ Low stellar density precludes a companion dwarf EB as the source of the signal; evolved companions are ruled out by our AO imaging to within 02 and stellar counterparts within ∼1 au are ruled out by the absence of a drift in our radial velocity data.

We have described the discovery and characterization of a Jupiter-mass planet on an 8.4-day orbit around a red giant branch star. This object joins a sample of only five other known transiting planets hosted by highly evolved stars (Huber et al. 2013a; Lillo-Box et al. 2014; Barclay et al. 2015; Ciceri et al. 2015; Quinn et al. 2015; Van Eylen et al. 2016). The high metallicity of the host star is also characteristic of the close-in gas giant planet population, suggesting that this system may be simply a successor to such "hot Jupiter" systems.

As the stellar radius of K2-97 has been determined to 3% precision through asteroseismology, the dominant uncertainty in planet radius for this system comes from the transit depth. We compared the star-to-planet radius ratio (${R}_{p}/{R}_{* }$) for this system using light curves produced by the PDC-MAP pipeline (Smith et al. 2012; Stumpe et al. 2012), the K2 "self flat field" (K2SFF) pipeline (Vanderburg et al. 2016), as well as a modified version of the Vanderburg et al. (2016) pipeline, which simultaneously fit thruster systematics, low-frequency variability, and planet transits with a Levenberg-Marquardt minimization algorithm, the K2SC pipeline (Aigrain et al. 2016), and the TERRA pipeline (Petigura et al. 2013). We find that measured transit depths vary by over 30% between the different systematic detrending pipelines we tested. We plot the spread in recovered star-to-planet radius ratios in Figure 6.

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To investigate the differences in ${R}_{p}/{R}_{* }$ recovered from light curves produced from different pipelines, we injected transits modeled from those in the K2-97 system into light curves (with systematics) of 50 stars classified as low-luminosity red giants from our K2 Campaign 5 target list. These light curves were then detrended using both the standard K2SFF method of Vanderburg et al. (2016) as well as the modified method, which detrended instrumental noise and fit the planet transit simultaneously (hereby referred to as K2SFF+). The transit depths in both sets of processed light curves were then fit using a box least-squares search (Kovács et al. 2002) and a Mandel–Agol transit model (Mandel & Agol 2002; Barclay 2015). This transit injection/recovery test revealed that the transit depth was retained with some scatter when both the transit and systematics were fit simultaneously, but when the systematics were fit and removed with the nominal Vanderburg et al. (2016) method, transit depths were reduced by 13% and the planet's radius was underestimated by 8% on average.

We report results from the K2SFF+ light curve as it was demonstrated to preserve transit depth through our transit injection/recovery tests, and its measured transit depth is strongly consistent with transit depths measured from two independently detrended light curves. We add an additional 5% error in planet radius to account for the uncertainty in transit fitting seen in the injection/recovery tests. Current and future studies with injection/recovery tests similar to those performed for Kepler (Petigura et al. 2013; Christiansen et al. 2015) will help resolve this discrepancy between accuracy and precision in measuring transit depths with K2.

We can test planet inflation mechanisms by examining the response of planets to increasing irradiation as the host star leaves the main sequence. In particular, planets with orbital periods of <30 days will experience levels of irradiation comparable to typical hot Jupiters for more than 100 Myr. Following the nomenclature of Lopez & Fortney (2016), if the inflation mechanism requires direct heating and thus falls into Class I, the planet's radius should enter a re-inflated state around a post-main-sequence star. However, if the inflation mechanism falls into Class II, requiring delayed cooling, there should be no effect on planet radius as a star enters the red giant phase, and re-inflation will not occur. K2-97b provides a valuable test for the re-inflation hypothesis, as it is inflated now but orbits at a distance such that it may not have received irradiation above the inflation threshold for its entire existence.

To estimate the change in stellar irradiation over time, we use the Parsec evolutionary tracks (Bressan et al. 2012) with the host star mass and metallicity derived in Section 3.2. Figure 8 shows an HR diagram and incident flux evolution for models with masses of 1.0, 1.15, and 1.3 M_⊙ from the pre-main-sequence to the tip of the red giant branch. We used metallicities of 0.6, 0.42, and 0.34 dex for the 1.0, 1.15, and 1.3 M_⊙ models, respectively, which results in overestimated limits given that metal-poor stars are hotter than metal-rich stars for a fixed mass. We also denote an inflation threshold of 2 × 10⁸ erg s⁻¹ cm⁻² (∼150 F_⊕) following Demory & Seager (2011) and Miller & Fortney (2011), who note that this corresponds to an equilibrium temperature of 990 K assuming a Bond albedo of 0.1, comparable to the temperature at which Ohmic heating may become important (Batygin et al. 2011). None of the 38 transiting giant planets with insolations below this threshold known to date appear to be inflated (Thorngren et al. 2015).

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Figure 8 demonstrates that the incident flux of this planet may have been above the 150 F_⊕ threshold for inflation throughout its main-sequence life. However, it is also possible that the planet experienced a flux below this threshold, depending on the exact mass and metallicity of the star. To estimate the main-sequence incident flux level quantitatively, we performed Monte Carlo simulations by interpolating the evolutionary tracks to randomly sampled values of stellar mass and metallicity as measured for K2-97 and calculated the average incident flux on the main sequence. The resulting distribution yielded an average main-sequence flux of ${170}_{-60}^{+140}$ F_⊕. We also estimated the incident flux evolution using a different set of evolutionary tracks from the MIST database (Choi et al. 2016), which yielded consistent results. Our analysis demonstrates that K2-97b received a main-sequence incident flux, which is close to the inflation threshold, but lower than the typical incident flux for planets with a comparable radius. This suggests that additional inflation occurred after the star evolved off the main sequence.

We illustrate the current constraints on the mass and radius of K2-97b in Figure 9 relative to other known, well-characterized giant planets. The dotted line denotes the empirical threshold for planet inflation put forth by Miller & Fortney (2011). Colors correspond to the incident fluxes on these planets, except in the case of K2-97b, where we have also indicated the incident flux the planet would have received on the main sequence to illustrate how uncharacteristic of the inflated planet population it would have been at that time.

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Furthermore, the energetics of K2-97b indicate that if it was inflated to its current radius while its host star was on the main sequence, the planet would be an outlier within the inflated planet population, with internal heating over an order of magnitude higher than would be expected. We illustrate this in Figure 10, where we plot the intrinsic cooling luminosity predicted by the models of Lopez & Fortney (2016) against incident flux for the known inflated planet population. The radius anomaly, or difference in measured and predicted planet size, is indicated by color. The filled square corresponds to K2-97b today, showing clear agreement with the rest of the inflated planet population energetically. However, the open square with dashed error bars corresponds to the incident flux on the planet when its host star was on the main sequence. The only planet energetically comparable to this scenario is WASP-67b, a planet with less than half the mass around a young star (Hellier et al. 2012). As lower mass planets are easier to inflate, and young planets may still be inflated from their initial formation, it would be very surprising to find a Jupiter-mass, middle-aged planet with similar energetic qualities. This, along with the empirical evidence for the energetic boundary of inflation of 2 × 10⁸ erg s⁻¹ established by Miller & Fortney (2011), suggest that K2-97b was not inflated when its host star was on the main sequence.

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Assuming that the inflation of the planet was due to the deposition of flux into the planet interior, we can use the model of Lopez & Fortney (2016) to estimate the heating efficiency needed to reproduce the current radius of K2-97b. Figure 11 shows the radius evolution of K2-97b as a function of age, given a range of heating efficiencies, a planetary structure of a H/He envelope with a 20 M_⊕ core of heavier elements, and a 1.15 Msun, [Fe/H] = +0.42 dex model for the star. The scenario with no additional interior heating is shown by the dotted line. The planet is consistent with heating efficiencies of ∼0.3%, and inconsistent with a class II scenario with no additional heating at late times. This suggests K2-97b may be the first re-inflated planet discovered.

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Further studies of giant planets around evolved stars will be necessary to confirm this hypothesis. Gas planets at a slightly larger orbital period (∼10–30 days) around a similar star would experience fluxes well below the empirical inflation threshold during the main sequence and would thus provide a clearer picture of the inflation mechanism. Although planets inflated by mechanisms more heavily dependent on factors other than incident flux, such as metallicity, have not been observed around main-sequence stars, these factors could potentially delay contraction at orbital distances beyond the nominal inflation boundary, and thus we cannot completely rule out the possibility that such effects may also be responsible for the inflation of this planet (Chabrier & Baraffe 2007).

The expansion of a star in the red giant phase can extend to au scales, eventually engulfing any short-period planets. We calculate that K2-97b will be engulfed when its host star reaches a radius of ∼18 R_⊙. This provides a conservative upper limit for the remaining lifetime of the planet of ∼200 Myr.

The scarcity of short-period planets orbiting giant stars has been suggested to be a result of tidally driven orbital decay (Schlaufman & Winn 2013). We can estimate the timescale of orbital decay due to tides following the prescription of Schlaufman & Winn (2013):

$$\begin{eqnarray}t & = & 10\,\mathrm{Gyr}\,\displaystyle \frac{{Q}_{* }/{k}_{* }}{{10}^{6}}{\left(\displaystyle \frac{{M}_{* }}{{{\rm{M}}}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{p}}{{{\rm{M}}}_{\mathrm{Jup}}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{{R}_{* }}{{{\rm{R}}}_{\odot }}\right)}^{-5}{\left(\displaystyle \frac{a}{0.06\mathrm{au}}\right)}^{-13/2}\end{eqnarray} \tag{ 8 }$$

Here, Q_* is the tidal quality factor of the star, and k_* its tidal Love number. These values are highly uncertain, but making the usual assumption of ${Q}_{* }/{k}_{* }$ = 10⁶ (Schlaufman & Winn 2013) the decay time is ≈60 Myr. If, however, ${Q}_{* }/{k}_{* }$ = 10², as Schlaufman & Winn (2013) suggest may be the case for sub-giant stars, then t ≈ 6000 years. This indicates that such a low value for ${Q}_{* }/{k}_{* }$ is implausible. Consequently, the discovery of EPIC 211351816.01 along with other planets around evolved such as K2-39b (Van Eylen et al. 2016) and Kepler-91b (Barclay et al. 2015) suggests that observation bias may contribute to the relative paucity of planets detected on short-period orbits around giant stars.