Spitzer transit follow-up of planet candidates from the K2 mission

TL;DR

This study uses Spitzer transit photometry to validate and refine the properties of eight K2-discovered exoplanet candidates, improving their ephemeris accuracy and identifying promising targets for future characterization.

Contribution

It provides improved planet parameters, validates new candidates, and demonstrates the effectiveness of combined K2 and Spitzer data for exoplanet follow-up.

Findings

Enhanced ephemeris precision by an order of magnitude.

Validated the planetary nature of EPIC 205686202.01.

Identified promising targets for radial velocity and atmospheric studies.

Abstract

We present precision 4.5 $\mu$m Spitzer transit photometry of eight planet candidates discovered by the K2 mission: K2-52 b, K2-53 b, EPIC 205084841.01, K2-289 b, K2-174 b, K2-87 b, K2-90 b, and K2-124 b. The sample includes four sub-Neptunes and two sub-Saturns, with radii between 2.6 and 18 $R_\oplus$, and equilibrium temperatures between 440 and 2000 K. In this paper we identify several targets of potential interest for future characterization studies, demonstrate the utility of transit follow-up observations for planet validation and ephemeris refinement, and present new imaging and spectroscopy data. Our simultaneous analysis of the K2 and Spitzer light curves yields improved estimates of the planet radii, and multi-wavelength information which help validate their planetary nature, including the previously un-validated candidate EPIC 205686202.01 (K2-289 b). Our Spitzer observations yield an order of magnitude increase in ephemeris precision, thus paving the way for efficient future study of these interesting systems by reducing the typical transit timing uncertainty in mid-2021 from several hours to a dozen or so minutes. K2-53 b, K2-289 b, K2-174 b, K2-87 b, and K2-90 b are promising radial velocity (RV) targets given the performance of spectrographs available today or in development, and the M3V star K2-124 hosts a temperate sub-Neptune that is potentially a good target for both RV and atmospheric characterization studies.

Tables

Table 1. Table 1: Spitzer observing log

EPIC	Name	Int. Time	Duration	Start Date
		[sec]	[hr]	[JD]
203776696	K2-52	30	10.6	2457337.6776
204890128	K2-53	6	8.1	2457347.8974
205084841		30	7.8	2457334.7969
205686202	K2-289	12	8.9	2457554.5479
210558622	K2-174	2†	13.5	2457533.4483
210731500	K2-87	30	11.8	2457519.4901
210968143	K2-90	12	10.4	2457531.5406
212154564	K2-124	30	8.3	2457590.9404

Table 2. Table 2: Radial velocities of K2-289 obtained with Subaru/HDS.

BJD	RV (relative)	Error
	[m s-1]	[m s-1]
2457509.063075	-5.95	23.95
2457510.065150	21.30	21.79
2457510.086821	62.27	22.18
2457511.067174	-10.08	21.78
2457511.088864	-16.61	24.39

Table 3. Table 3: Transit parameter estimates

Name	$a$	$b$	$i$	$R_{p,K}$	$R_{p,S}$	$R_{p,C}$	log(${\sigma }_K$)	log(${\sigma }_S$)	$T_{14}$	$T_{23}$	$\eta$	$R_{p,\max }$
	[$R_{\star }$ ]		[degrees]	[$R_{\star }$ ]	[$R_{\star }$ ]	[$R_{\star }$ ]			[days]	[days]		[$R_{\star }$ ]
K2-52 b	${5.2}_{-0.3}^{+0.2}$	${0.32}_{-0.18}^{+0.14}$	${86.46}_{-1.91}^{+2.04}$	${0.072}_{-0.001}^{+0.001}$	${0.078}_{-0.002}^{+0.002}$	${0.075}_{-0.001}^{+0.001}$	-7.01	-5.70	${0.222}_{-0.003}^{+0.003}$	${0.187}_{-0.003}^{+0.002}$	${0.84}_{-0.02}^{+0.01}$	${0.085}_{-0.008}^{+0.013}$
K2-53 b	${26.3}_{-0.4}^{+0.4}$	${0.31}_{-0.10}^{+0.07}$	${89.32}_{-0.16}^{+0.22}$	${0.028}_{-0.001}^{+0.001}$	${0.031}_{-0.002}^{+0.002}$	${0.030}_{-0.001}^{+0.001}$	-8.83	-6.61	${0.145}_{-0.003}^{+0.003}$	${0.136}_{-0.003}^{+0.004}$	${0.94}_{-0.00}^{+0.00}$	${0.033}_{-0.002}^{+0.002}$
205084841.01	${17.8}_{-0.6}^{+0.5}$	${0.62}_{-0.03}^{+0.03}$	${87.99}_{-0.17}^{+0.16}$	${0.127}_{-0.002}^{+0.002}$	${0.117}_{-0.002}^{+0.002}$	${0.122}_{-0.002}^{+0.002}$	-6.36	-5.25	${0.189}_{-0.002}^{+0.003}$	${0.125}_{-0.003}^{+0.003}$	${0.66}_{-0.02}^{+0.02}$	${0.203}_{-0.014}^{+0.016}$
K2-289 b	${22.5}_{-0.3}^{+0.3}$	${0.60}_{-0.02}^{+0.02}$	${88.48}_{-0.07}^{+0.07}$	${0.081}_{-0.001}^{+0.001}$	${0.082}_{-0.002}^{+0.001}$	${0.081}_{-0.001}^{+0.001}$	-7.42	-6.24	${0.168}_{-0.002}^{+0.002}$	${0.130}_{-0.002}^{+0.002}$	${0.77}_{-0.01}^{+0.01}$	${0.128}_{-0.005}^{+0.005}$
K2-174 b	${26.6}_{-0.2}^{+0.2}$	${0.02}_{-0.02}^{+0.03}$	${89.95}_{-0.05}^{+0.04}$	${0.033}_{-0.000}^{+0.000}$	${0.038}_{-0.001}^{+0.001}$	${0.036}_{-0.001}^{+0.001}$	-9.25	-6.80	${0.242}_{-0.002}^{+0.002}$	${0.225}_{-0.002}^{+0.002}$	${0.93}_{-0.00}^{+0.00}$	${0.036}_{-0.001}^{+0.001}$
K2-87 b	${14.4}_{-0.3}^{+0.3}$	${0.57}_{-0.03}^{+0.03}$	${87.74}_{-0.17}^{+0.17}$	${0.045}_{-0.001}^{+0.001}$	${0.055}_{-0.002}^{+0.002}$	${0.050}_{-0.001}^{+0.001}$	-7.84	-5.94	${0.190}_{-0.002}^{+0.002}$	${0.164}_{-0.003}^{+0.003}$	${0.86}_{-0.01}^{+0.01}$	${0.074}_{-0.004}^{+0.005}$
K2-90 b	${33.4}_{-0.3}^{+0.3}$	${0.44}_{-0.05}^{+0.05}$	${89.24}_{-0.09}^{+0.09}$	${0.036}_{-0.001}^{+0.001}$	${0.041}_{-0.002}^{+0.002}$	${0.038}_{-0.001}^{+0.001}$	-8.33	-6.37	${0.123}_{-0.003}^{+0.003}$	${0.112}_{-0.003}^{+0.003}$	${0.91}_{-0.01}^{+0.01}$	${0.048}_{-0.003}^{+0.004}$
K2-124 b	${28.0}_{-1.7}^{+1.5}$	${0.47}_{-0.14}^{+0.10}$	${89.03}_{-0.27}^{+0.32}$	${0.069}_{-0.002}^{+0.002}$	${0.068}_{-0.003}^{+0.003}$	${0.069}_{-0.002}^{+0.002}$	-6.89	-5.73	${0.070}_{-0.002}^{+0.002}$	${0.058}_{-0.003}^{+0.003}$	${0.84}_{-0.03}^{+0.02}$	${0.089}_{-0.013}^{+0.016}$

Table 4. Table 8: Limb darkening priors

	K2		Spitzer
Name	$q_1$	$q_2$	$q_1$	$q_2$
K2-52	$𝒩(0.339,0.024)$	$𝒩(0.237,0.030)$	$𝒩(0.026,0.005)$	$𝒩(0.174,0.022)$
K2-53	$𝒩(0.490,0.012)$	$𝒩(0.372,0.016)$	$𝒩(0.047,0.002)$	$𝒩(0.198,0.019)$
205084841	$𝒩(0.392,0.034)$	$𝒩(0.271,0.028)$	$𝒩(0.035,0.005)$	$𝒩(0.165,0.023)$
K2-289	$𝒩(0.481,0.011)$	$𝒩(0.339,0.012)$	$𝒩(0.043,0.002)$	$𝒩(0.203,0.023)$
K2-174	$𝒩(0.561,0.007)$	$𝒩(0.463,0.005)$	$𝒩(0.055,0.001)$	$𝒩(0.213,0.019)$
K2-87	$𝒩(0.453,0.008)$	$𝒩(0.327,0.007)$	$𝒩(0.043,0.001)$	$𝒩(0.153,0.015)$
K2-90	$𝒩(0.553,0.003)$	$𝒩(0.448,0.003)$	$𝒩(0.058,0.001)$	$𝒩(0.205,0.021)$
K2-124	$𝒩(0.530,0.023)$	$𝒩(0.266,0.029)$	$𝒩(0.038,0.001)$	$𝒩(0.082,0.036)$

Table 5. Table 9: The effect of the choice of limb darkening law on estimates of R p subscript 𝑅 𝑝 R_{p} / R ⋆ subscript 𝑅 ⋆ R_{\star} , for a range of host star effective temperatures. The subscripts K 𝐾 K and S 𝑆 S denote the Kepler and Spitzer (4.5 μ ​ m 𝜇 m \mu\mathrm{m} ) bandpasses, respectively.

Name	$R_{p,K}$	$R_{p,S}$
	[$R_{\star }$ ]	[$R_{\star }$ ]
linear
205084841.01	${0.12171}_{-0.00187}^{+0.00231}$	${0.11602}_{-0.00225}^{+0.00237}$
K2-174 b	${0.03605}_{-0.00179}^{+0.00244}$	${0.03892}_{-0.00143}^{+0.00129}$
K2-124 b	${0.06791}_{-0.00255}^{+0.00398}$	${0.07005}_{-0.00318}^{+0.00373}$
quadratic
205084841.01	${0.12134}_{-0.00179}^{+0.00208}$	${0.11616}_{-0.00248}^{+0.00223}$
K2-174 b	${0.03593}_{-0.00160}^{+0.00301}$	${0.03864}_{-0.00133}^{+0.00129}$
K2-124 b	${0.06761}_{-0.00226}^{+0.00333}$	${0.07011}_{-0.00342}^{+0.00317}$
square-root
205084841.01	${0.12187}_{-0.00202}^{+0.00215}$	${0.11586}_{-0.00232}^{+0.00246}$
K2-174 b	${0.03615}_{-0.00183}^{+0.00284}$	${0.03868}_{-0.00130}^{+0.00131}$
K2-124 b	${0.06770}_{-0.00243}^{+0.00367}$	${0.06994}_{-0.00329}^{+0.00326}$
logarithmic
205084841.01	${0.12134}_{-0.00172}^{+0.00197}$	${0.11584}_{-0.00229}^{+0.00246}$
K2-174 b	${0.03597}_{-0.00180}^{+0.00260}$	${0.03881}_{-0.00137}^{+0.00124}$
K2-124 b	${0.06774}_{-0.00219}^{+0.00374}$	${0.07043}_{-0.00340}^{+0.00324}$

Equations

$$\beta =\frac{{\sigma }_M}{{\sigma }_0}\sqrt{\frac{N(M-1)}{M}},$$

$$\Delta S^t=\frac{{\sum }_{i=1}^9{c}_i{P}_i^t}{{\sum }_{i=1}^9{P}_i^t}+M_{tr}(\mathbf{\theta },t)+\epsilon (\sigma ),$$

$$q_1$$

$$q_2$$

$$\eta =\frac{T_{23}}{T_{14}}$$

$$R_{p,max}/R_{\star }=\frac{1-\eta }{1+\eta }$$

$${\rho }_{\star ,LC}=\frac{3\pi (a/R_{\star }{)}^3}{G{P}^2}$$

Full text

planets and satellites: detection – planets and satellites: fundamental parameters – techniques: photometric

††facilities: Kepler, Spitzer, Gaia, Keck (NIRC2, HIRES), Subaru (HDS)††software: numpy (Oliphant, 2006), scipy (Jones et al., 2001–present), matplotlib (Hunter, 2007), lmfit (Newville et al., 2014), emcee (Foreman-Mackey et al., 2013), celerite (Foreman-Mackey et al., 2017), batman (Kreidberg, 2015), isochrones (Morton, 2015b), vespa, (Morton, 2015a), k2phot (Petigura et al., 2015), SpecMatch-syn (Petigura, 2015), SpecMatch-emp (Yee et al., 2017b), RadVel (Fulton et al., 2018), IRAF (Tody, 1986, 1993)

1 Introduction

The NASA K2 mission (Howell et al., 2014) has extended the legacy of Kepler by discovering transiting exoplanets and candidate planets at a rate of hundreds per year. In contrast to Kepler, K2 surveyed a wider sky area at the cost of shorter time baseline per field, which has enabled the discovery of planets orbiting brighter stars. In addition to monitoring a greater number of bright stars than Kepler, K2 monitored more low mass stars than Kepler, partly as a result of its community-driven target selection. The result is that planets detected by K2 are generally more amenable to follow-up. To date, K2 has significantly enhanced the number of known small planets orbiting brighter stars than those surveyed by Kepler (Foreman-Mackey et al. 2015; Montet et al. 2015; Vanderburg et al. 2016; Crossfield et al. 2016, hereafter Cr16; Mayo et al. 2018; Livingston et al. 2018a, b), as well as discovering planets in cluster environments (Obermeier et al., 2016; Pepper et al., 2017; David et al., 2016a; Mann et al., 2016a, 2017; Gaidos et al., 2017; Mann et al., 2018; Livingston et al., 2018c; Ciardi et al., 2018; Rizzuto et al., 2018; Livingston et al., 2018d), including a 5–10 Myr planet in the Upper Scorpius star forming region (David et al., 2016b; Mann et al., 2016b). K2 planets will be available for follow-up studies with the James Webb Space Telescope (JWST; Gardner et al., 2006) contemporaneously with the planets expected to be found by the Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015).

Our focus with Spitzer (Werner et al., 2004) is on small planets orbiting K and M dwarfs discovered by K2, which could potentially be good targets for future radial velocity (RV) or atmospheric characterization studies. We have been conducting Spitzer transit observations of planet candidates and using these data to refine estimates of their orbital and physical parameters. As K2 observes each field for approximately 80 days each, our observations play a critical role in refining the ephemerides due to the long time baseline they provide (typically 6–12 months longer). These results are part of an ongoing program, data from which has been used to ensure the feasibility of future study of K2 planets by Beichman et al. (2016), Benneke et al. (2017), Chen et al. (2018), Dressing et al. (2018), and Hardegree-Ullman, K. et al. (in preparation).

In this paper we validate the planet K2-289 b, identify several targets of potential interest for future characterization studies, and demonstrate the utility of transit follow-up observations for ephemeris refinement. When done with a smaller beam, i.e. with Spitzer or CHEOPS (Fortier et al., 2014), such follow-up will prove especially useful in the validation of planet candidates identified by TESS, which will frequently encounter stellar blends due to the $\sim$21″  pixel scale of its detectors. In Section 2 we describe our observations, including K2 and Spitzer photometry, high resolution imaging and spectroscopy, and literature data. In Section 3 we describe the analysis methods used to measure host star and planet properties from these data, as well as our planet validation approach. In Section 4 we present the results of our analyses and discuss the potential for future characterization studies, concluding with a summary in Section 5.

2 Observations

2.1 K2 photometry

The basis for this work is the initial identification of planet candidates in K2 light curves. This process is described in Cr16 and Petigura et al. (2018), but we briefly summarize it here. We use k2phot 111https://github.com/petigura/k2phot to correct the instrumental systematics induced by the roll of the Kepler spacecraft. The resulting corrected light curves are publicly available on the community portal ExoFOP222https://exofop.ipac.caltech.edu. We use TERRA 333https://github.com/petigura/terra to search these light curves for transit signals, and the resulting candidates are then vetted by eye to eliminate instrumental or astrophysical false positives. During this process we also assess the utility of conducting follow-up transit observations with Spitzer. The planets we analyze here were all deemed interesting targets for Spitzer because they were relatively small, temperate, and/or orbit late-type host stars. All of the planets and candidates in this work were previously published by Cr16, with the exception of K2-124 b, which was observed in K2 Campaign 5 and subsequently discovered by our team (Dressing et al., 2017; Petigura et al., 2018; Livingston et al., 2018b).

2.2 Spitzer photometry

Spitzer presents several advantages over ground-based transit follow-up observations: its position in space enables precise photometry unaffected by the Earth’s atmosphere; its Earth-trailing orbit frees it from the scheduling constraints imposed by the day/night cycle on Earth; the diminished effects of limb darkening in the infrared enable precise estimation of transit model parameters; the 4.5 $\mu\mathrm{m}$ Infrared Array Camera (IRAC, Fazio et al., 1998) bandpass (in conjunction with the Kepler bandpass) provides a relatively broad wavelength baseline which facilitates planet validation. In addition, our high-cadence Spitzer observations provide better sampling of the transit than the 30 minute cadence of K2.

We conducted our transit observations using the IRAC 4.5 $\mu\mathrm{m}$ channel as part of Spitzer cycle 11 GO program 11026 (P.I. Werner). We chose integration times between 2–30 seconds to keep the detector in the linear regime and minimize downlink bandwidth. Target acquisition places the stars on the “sweet spot” of the detector, which has been well-characterized for the purpose of precise time-series photometry (Ingalls et al., 2012), and falls within the region of the detector accessible in sub-array mode (used for observing bright stars). Following the guidelines for high precision Spitzer photometry (Grillmair et al., 2012), we performed $\sim 30$ minutes of integrations on an empty field before each transit observation, which can help mitigate systematics induced by thermal settling of the spacecraft. See table Table 1 for details of the observations.

2.3 High resolution imaging

High resolution imaging is important for detecting stellar companions and constraining the probability of chance alignments with background sources within the K2 and Spitzer photometric apertures, and thus plays a critical role in assessing the false positive probability (FPP) of a planet candidate. In this work, we utilize imaging previously published by Cr16 along with additional AO imaging from Keck/NIRC2 (Wizinowich et al., 2014). Cr16 did not obtain imaging of EPIC 205084841, so we used NIRC2 in natural guide star mode to observe the star in $K$ band on UT July 9, 2017. Using the image reduction and analysis methods described in Cr16, we find the star to be single and rule out companions above the contrast curve shown in Figure 1 at the 5$\sigma$ confidence level. We also utilize a $J$ band image of K2-289, which was obtained with NIRC2 on UT April 1, 2015 and made available on ExoFOP, but was not published in Cr16. As in the $K$ band image reported by Cr16, the companion is clearly detected in $J$ band, thus providing useful color information (see Section 4.1).

2.4 Spectroscopy

Our team has primarily used Keck/HIRES (Vogt et al., 1994) to obtain high resolution spectra of candidate host stars, enabling the measurement of more robust planet properties as well as detecting (or ruling out) double-lined spectroscopic binaries (see Cr16 and Petigura et al. (2018) for more details). We use SpecMatch-syn to measure precise stellar parameters from the spectra of stars hotter than $\sim$4200 K, which matches spectra to an interpolated grid of models from Coelho et al. (2005). For cooler stars, we use SpecMatch-emp, which matches spectra to a spectral library of 404 standard stars (Yee et al., 2017a).

We also conducted spectroscopic observations of K2-289 using the High Dispersion Spectrograph (HDS) mounted on the Subaru 8.2 m telescope between UT 2016 April 29 and May 2. We employed the standard I2a setup, which covers $4940-7590\mathrm{\AA}$, and the image slicer #2, achieving a spectral resolution of $R\sim 80,000$. For the RV measurements, we used the iodine (I2) cell on three consecutive nights. We also obtained the stellar spectrum without the I2 cell for the template of RV measurements. The HDS data were reduced using standard IRAF routines, by which we extracted one-dimensional, wavelength-calibrated spectra of K2-289. We then put those spectra through a RV analysis pipeline (Sato et al., 2002, 2012), which does a forward modeling of each observed spectrum to measure the RV relative to the template. Table 2 lists the extracted relative RVs and their internal errors.

3 Analysis

3.1 K2 light curves

As part of our team’s large-scale transit search of the K2 data (Section 2.1), we correct systematics induced by the coupling of spacecraft motion and intra-pixel gain variations with k2phot, which uses a Gaussian Process model (GP; Rasmussen & Williams, 2005). The resulting light curves are essentially free of the large amplitude position-dependent flux variations characteristic of raw K2 photometry, but stellar variability and potentially residual systematic trends must be accounted for to measure precise transit properties. A common approach is to first model and remove out-of-transit variability using various methods, such as a median filter, polynomial, or spline model. These approaches are simple and fast, and usually do not significantly impact the results for stars with low levels of variability and residual systematics. In order to minimize the potential for biased parameter estimates while employing a uniform framework for a range of light curve behaviors, we use the celerite GP framework with a Matern-3/2 kernel and take an approach similar to the “type-II” maximum likelihood estimation (MLE) described in Gibson et al. (2012).

The GP is first trained on the out-of-transit light curve using scipy.optimize, and then the full light curve is analyzed with the same GP in conjunction with the transit model (Section 3.4). We first use the “L-BFGS-B” method (Byrd et al., 1995; Zhu et al., 1997) to find the MLE value of the kernel hyperparameters given the out-of-transit light curve, using the GP likelihood and gradient in celerite. During this stage, we perform iterative outlier rejection to minimize the possibility of biased kernel hyperparameters. Next we use the Nelder-Mead simplex algorithm (Nelder & Mead, 1965) to fit the joint GP and transit model to the full light curve, initialized with the MLE kernel hyperparameters and an initial set of transit parameters from previous analyses. Fitting the GP hyperparameters simultaneously with the transit model helps ensure that we find an optimal noise model that is valid during and out of transit. We then restrict the data to 1-day windows centered on each transit and sample the posterior of the joint GP-transit model, by running emcee for 500 steps initialized with the optimum found in the previous step. This brief sampling stage is especially important if the kernel hyperparameters are initially stuck in a local optimum, and can be used to ensure that the posteriors are unimodal and sharply peaked around these optimal values.

Finally, we fix the hyperparameters to their optima and proceed to run the sampler for 10,000 steps (see also Section 3.4). This approach retains the benefits of a flexible noise model while minimizing computational complexity. Figure 2 and Figure 3 show examples of these fits, with datapoints shown in gray if they were excluded during the iterative outlier rejection performed during the initial GP training stage.

3.2 Spitzer light curve extraction

We extract the Spitzer light curves following the approach taken by Knutson et al. (2012) and Beichman et al. (2016). In brief, we compute aperture photometry using circular apertures centered on the host star, for a range of radii between 2.0 and 5.0 pixels, corresponding to 2.4–6.0″ due to Spitzer’s 1.2″ pixel scale. We used a step size of 0.1 pixels from 2.0 to 3.0, and a step size of 0.5 from 3.0 and 5.0. An important component of precision photometry with Spitzer is the selection of an optimal aperture, due to the fact that significant levels of “red” (correlated) noise present to a varying extent in each time series. Smaller radii tend to have less photon noise due to the decreased sky background in the aperture, while slightly larger radii can sometimes mitigate the inter- and intra-pixel gain variations which are responsible for the correlated noise. Ideally, an optimal radius minimizes both of these effects, although in practice it is common to attempt only to minimize correlated noise or the photon noise (e.g. Knutson et al., 2012; Lewis et al., 2013; Lanotte et al., 2014).

A typical transit dataset contains significant time both in and out of transit, so we compute relevant noise metrics as a function of radius for different subsets, most of which are fully out-of-transit or between 2nd and 3rd contact (i.e. do not contain ingress or egress). Thus, for a given radius, the ensemble of these values are largely unaffected by the transit signal, and thus reflect only photon noise and systematic noise. To quantify the level of red noise, we compute $\beta$, the factor by which the standard deviation of the observed binned residuals deviate from the theoretical value (Pont et al., 2006; Winn et al., 2008):

[TABLE]

where $\sigma_{M}$ is the standard deviation of the binned residuals (in M bins), $\sigma_{0}$ is the standard deviation of the un-binned residuals, and N is the number of data points per bin. To ensure a robust estimate and focus on the timescales of red noise which could significantly impact transit parameter estimates, we compute the median $\beta$ value for bin widths between 5 and 40 minutes. We divide each flux time series into 10 equal-sized segments and compute both the standard deviation (i.e. overall noise level) and the $\beta$ value (i.e. red noise level) for each segment. The optimal aperture is then the one that minimizes each metric. We then compute the median of the optimal aperture radii for each metric over all segments. Finally, the aperture radius adopted for subsequent analysis is the mean of these two “optimal” radii; the selected aperture thus only “approximately” minimizes both metrics in cases where these two radii are not equal. We chose 10 segments as a tradeoff between having more robust statistics and having enough light curve in each segment to compute red noise on a range of different timescales. We find that the minimum red noise aperture is frequently consistent with the minimum standard deviation aperture to within a few tenths of a pixel, and the optimal radius is typically 2.2–2.4 pixels, which is consistent with the optimal apertures found in previous analyses of Spitzer transit data (e.g. Knutson et al., 2012), in which the residuals computed from the best-fit transit and systematics model are analyzed instead of the raw light curve.

In principle, aperture selection could be handled via Bayesian model selection (i.e. computing the Bayesian evidence), although any improvements might not be significant enough to justify the computational cost. Additionally, it may be fruitful to simultaneously estimate the white and red noise levels using a GP; a sufficient choice of kernel could more fully disentangle these two noise signals, as compared to the standard deviation and $\beta$ factor. We leave an investigation of these possibilities for a future work.

3.3 Spitzer systematics model

We model the systematics inherent to the Spitzer light curves using the pixel-level decorrelation (PLD) method (Deming et al., 2015). In comparisons between various methods used to correct Spitzer systematics (Ingalls et al., 2016), PLD was among the top performers, displaying both high precision and repeatability. PLD uses a linear combination of (normalized) pixel light curves to model the effect of PSF motion on the detector coupled with intra-pixel gain variations, thus it does not require the calculation of centroids. The parametrization of the full model for the transit light curve, including PLD, is:

[TABLE]

where $M_{tr}$ is the transit model, the $c_{i}$ are the PLD coefficients, and $\varepsilon(\sigma)$ are zero-mean Gaussian errors with width $\sigma$. To form a valid set of basis vectors for the instrumental systematics component of the light curve, the astrophysical signal in each individual pixel light curve is removed by normalization (the sum in the denominator of Equation 2). We show an illustrative example of these normalized pixel light curves in Figure 4. In testing, we found that using a $3\times 3$ pixel grid sufficiently captures the information content corresponding to the motion of the PSF on the detector (which is typically $\lesssim$ a few tenths of a pixel). However, Spitzer target acquisition occasionally misses the “sweet spot” pixel, which may yield datasets with more pronounced systematics that could benefit from using a larger pixel grid (such as $5\times 5$). In Figure 5 we plot the full model fit for two datasets (upper panels), as well as the data corrected by subtracting the best-fit PLD noise model (lower panels). In Figure 6 we plot the corrected data and transit models for the remaining Spitzer datasets.

3.4 Transit fitting

To model the transits in both the K2 and Spitzer light curves, we use the analytic model of Mandel & Agol (2002), assuming a circular orbit and a quadratic limb darkening law, as implemented in batman. The free parameters $\boldsymbol{\theta}$ of the transit model are the planet-to-star radius ratio $R_{p}/R_{\star}$, the scaled semi-major axis $a/R_{\star}$, mid transit time $T_{0}$, orbital period $P$, impact parameter $b\equiv a\cos(i)/R_{\star}$, and the modified quadratic limb darkening coefficients $q_{1}$ and $q_{2}$, which efficiently sample the space of physically allowed limb darkening coefficients using the triangular sampling method of Kipping (2013). These transformed coefficients are computed directly from the quadratic limb darkening coefficients $u_{1}$ and $u_{2}$ using equations 17 and 18 of Kipping (2013), reproduced here for convenience:

[TABLE]

Following Cr16, we use Gaussian limb darkening priors in our light curve analysis, which we derive from the coefficients for the Kepler and IRAC bandpasses tabulated by Claret et al. (2012). We use a Monte Carlo approach in which we sample the stellar parameters ($T_{\mathrm{eff}}$, $\log g$, $[$Fe/H$]$) of each star and interpolate the tabulated coefficients at the sampled stellar values. Because we sample in $q$-space, we convert the tabulated values of $u_{1}$/$u_{2}$ to $q_{1}$/$q_{2}$ to enable using a Gaussian prior in $q$-space. We then use the mean and standard deviation of the sampled coefficients to define the Gaussian priors (see Appendix A for more details). For the final set of parameter estimates listed in Section 3.5 we also impose a prior on the mean stellar density determined from our isochrones analysis (see Section 3.7), which yields more precise parameter estimates by leveraging more information about the host star. However, we also perform a parallel set of identical analyses without this prior, which provides the opportunity to compare the density from our stellar characterization to the independent measurement of the mean stellar density from the light curve $\rho_{\star,LC}$ (see Equation 7). We discuss this in detail in Section 4.1.

For Markov Chain Monte Carlo (MCMC) parameter estimation, we use emcee, a Python implementation of the affine-invariant ensemble sampler (Goodman & Weare, 2010). We performed an initial optimization with lmfit and positioned 128 “walkers” in a Gaussian ball centered on the optimum. We then ran the sampler for 10,000 steps allowing it to evolve according to the MCMC. Finally, we checked for convergence by visual inspection of the trace, discarding the first 5000 samples as “burn-in,” and computed the autocorrelation time of each parameter using the Python package acor444https://github.com/dfm/acor to ensure we had collected a sufficient number of independent samples after burn-in.

3.5 Simultaneous K2 and Spitzer analysis

In this work we simultaneously model the K2 and Spitzer light curves of each target. This is motivated by several factors. First, the posterior distributions of some transit model parameters are often distinctly non-Gaussian (e.g. $a/R_{\star}$), so simply imposing Gaussian priors derived from previous analysis of the K2 light curves could bias the resulting fits to the Spitzer data. Secondly, we leverage the K2 data to model the Spitzer data, because we simultaneously fit the systematics and transit models. Some transit model parameters ($a/R_{\star}$ and $b$) are shared between Spitzer and K2, while others are distinct. We fit $R_{p}/R_{\star}$ separately for the Kepler and Spitzer bandpasses to enable the detection of false positive scenarios.

Our procedure is as follows. We first fit the K2 data alone, and form Gaussian priors on $T_{0}$ and $P$ based on the mean and standard deviation of the posterior distributions from MCMC. These priors are then used in a fit to the Spitzer transit data alone, yielding an initial set of parameter estimates derived from the Spitzer data. We then fit the K2 and Spitzer data simultaneously, without priors on $T_{0}$ and $P$. This simultaneous fit is responsible for the improvement in the ephemeris estimates, for reasons discussed above, as well as more robust transit shape parameter estimates. We list the transit parameters in Section 3.5, and the ephemeris estimates in Section 3.6.

An additional benefit to this approach is that it allows the high cadence and diminished limb darkening of our Spitzer light curves to yield improved constraints on transit parameters which are sensitive to ingress and egress (e.g. the impact parameter $b$). For example, even though our Spitzer observation (just barely) missed the ingress of K2-289 b, (see Figure 5), the joint analysis of the K2 and Spitzer data yields better constraints on the transit geometry than either the Spitzer-only or K2-only analyses. Figure 8 shows several key posterior distributions for this system from K2-only, Spitzer-only, and our joint analysis of both datasets, illustrating the improved constraints in transit geometry. The left-most panels in Figure 8 show the stellar density from our isochrones analysis as gray bands for comparison to the mean stellar density derived from Equation 7 and the transit fit posteriors (without a density prior). In this case the density estimated from the K2 data alone can be seen to be in mild disagreement with the isochrones density, but the Spitzer data yield an improved density estimate in good agreement. The improvement in parameter estimates afforded by this simultaneous modeling approach can be thought of as a type of “Bayesian shrinkage,” in which the high cadence of Spitzer and high photometric precision of K2 work together to extract higher measurement precision from the data.