K2-288Bb: A Small Temperate Planet in a Low-mass Binary System Discovered by Citizen Scientists

The first step in our follow-up process was observing K2-288 with the near-infrared cross-dispersed spectrograph, SpeX (Rayner et al. 2003, 2004) on the 3 m NASA Infrared Telescope Facility. The observations were completed on 2017 July 31 UT (Program 2017A019, PI C. Dressing). The target was observed under favorable conditions, with an average seeing of ∼08. We used SpeX in its short cross-dispersed mode (SXD) with the 03 × 15'' slit, covering 0.7–2.55 μm at a resolution of R ≈ 2000. The target was observed for an integration time of 120 s per frame at two locations along the slit in 3 AB nod pairs, leading to a total integration time of 720 s. The slit position angle was aligned to the parallactic angle in order to minimize differential slit losses. After observing K2-288, we immediately observed a nearby A0 standard, HD 23258, for later telluric correction. Flat and arc lamp exposures were also taken for wavelength calibration. The spectrum was reduced using the SpeXTool package (Vacca et al. 2003; Cushing et al. 2004).

SpeXTool uses the obtained target spectra, A0 standard spectra, and flat and arc lamp exposures to complete the following reductions: flat-fielding, bad pixel removal, wavelength calibration, sky subtraction, and flux calibration. The package yields an extracted and combined spectra. The resulting two spectra have signal-to-noise ratios (S/Ns) of 106 in the J-band (∼1.25 μm), 127 in the H-band (∼1.6 μm), and 107 per resolution in the K-band (∼2.2 μm). The reduced spectra is compared to late-type standards from the IRTF Spectral Library (Rayner et al. 2009) across the JHK-bands in Figure 3. Upon visual inspection, K2-288 is an approximate match to the M2/M3 standard across all three bands. This is consistent with the spectral type estimated using the NIR index based ${{\rm{H}}}_{2}{0}_{K2}$ method of Rojas-Ayala et al. (2012), M2.0 ± 0.6, and the optical index based TiO5 and CaH3 methods of Lépine et al. (2013), M3.0 ± 0.5.

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We used the SpeX spectrum to approximate the fundamental parameters of the star (metallicity, [Fe/H]; effective temperature, T_eff; radius, ${R}_{* };$ mass, ${M}_{* };$ and luminosity, L_*) following the prescription presented in Dressing et al. (2017). Specifically, we estimate the stellar T_eff, R_*, and L_* using the relations of Newton et al. (2015), the metallicity using the relations of Mann et al. (2013a), and M_* by using the Newton et al. (2015) T_eff in the temperature–mass relation of Mann et al. (2013b). Newton et al. (2015) used a sample of late-type stars with measured radii and precise distances to develop a relationship between the equivalent widths of H-band Al and Mg lines and fundamental parameters. Mann et al. (2013a) used a set of wide binaries with solar type primaries and M-dwarf companions to calibrate a relationship between metallicity and the strength of metallicity sensitive spectroscopic indices. Mann et al. (2013b) derived an empirical effective temperature relationship using a sample of low-mass stars with measured radii and distances and temperature sensitive indices in the near-infrared spectra of low-mass stars. Using the same samples, they then derived additional empirical relations for T_eff–R_*, T_eff–M_*, and T_eff–L_*. Our application of these empirical relationships to the SpeX spectrum of K2-288 following the prescription of Dressing et al. (2017) results in T_eff = 3479 ± 85 K, R_* = 0.47 ± 0.03 ${R}_{\odot }$, M_* =0.38 ± 0.08 ${M}_{\odot }$, log($L/{L}_{* }$) = −1.53 ± 0.06, and [Fe/H] =−0.06 ± 0.21. The estimated stellar parameters are consistent with the M2.5 spectral type measured from the spectrum. We note that these values apply to the blended spectrum of a binary system and are not indicative of the final stellar parameters for the components in the system. We discuss the discovery and properties of the binary in Sections 3.2, 3.3, and 5.1.

We observed K2-288 on 2017 August 18 UT with the HIRES spectrometer (Vogt et al. 1994) on the Keck I telescope. The star was observed following the standard California Planet Survey (CPS, Marcy et al. 2008; Howard et al. 2010) procedures with the C2 decker, $0\buildrel{\prime\prime}\over{.} 87\times 14\buildrel{\prime\prime}\over{.} 0$ slit,and no iodine cell. This set-up provides wavelength coverage from 3600 to 8000 Å at a resolution of R ≈ 60000. We integrated for 374 s, achieving 10000 counts on the HIRES exposure meter for an S/N of ∼25 per pixel at 5500 Å. The target was observed under favorable conditions, with seeing ∼1''. During these observations, we noted that the intensity distribution of the source in the HIRES guider images was elongated approximately along the SE–NW axis. We observed K2-288 again on 2017 August 19 UT using the same instrument settings and integration time, but in slightly better seeing. A secondary component was partially resolved in the guider images at ∼1'' to the SE. This observation prompted adaptive optics (AO) imaging using Keck NIRC2 to fully resolve the binary (see Section 3.3). Following the identification of the secondary, we observed K2-288 again with Keck HIRES on 2017 September 6 UT with an integration time of 500 s in <1'' seeing, achieving an S/N ∼20 per pixel at 5500 Å. During these observations, we oriented the slit to be perpendicular to the binary axis (PA = 330°) and shifted the slit position to center it on the secondary (K2-288B). All HIRES spectra were reduced using standard routines developed for the CPS (Howard et al. 2010).

Visual inspection of the reduced blended and secondary spectra revealed morphologies and features consistent with low-activity M dwarfs. All spectra exhibited Hα absorption, with no discernible emission in the line cores or wings. Weak emission cores were visible in the Ca ii H&K lines; however, we did not measure their strengths due to the very low S/N (≤3) at short wavelengths. Such weak emission is often observed even in low-activity M dwarfs. We derived stellar parameters from the spectra using the SpecMatch-Emp code (Yee et al. 2017).²⁷ SpecMatch-Emp is a software tool that uses a diverse spectral library of ∼400 well-characterized stars to estimate the stellar parameters of an input spectrum. The library is made up of HIRES spectra taken at high S/N (>100 per pixel). SpecMatch-Emp finds the optimum linear combination of library spectra that best matches the unknown target spectrum and interpolates the stellar T_eff, R_*, and [Fe/H]. SpecMatch-Emp performs particularly well on stars with T_eff < 4700 K, so it is well suited to K2-288, a pair of M dwarfs. SpecMatch-Emp achieves an accuracy of 70 K in T_eff, 10% in R_*, and 0.12 dex in [Fe/H] (Yee et al. 2017). The library parameters are derived from model-independent techniques (i.e., interferometry or spectrophotometry) and therefore do not suffer from model-dependent offsets associated with low-mass stars (Newton et al. 2015; Dressing et al. 2017). Our SpecMatch-Emp analysis of the blended spectra resulted in mean parameters of T_eff = 3593 ± 70 K, ${R}_{* }=0.44\pm 0.10{R}_{\odot }$, and [Fe/H] = −0.29 ± 0.09. Consistent with an M2.0 ± 0.5 spectral type following the color–temperature conversions of Pecaut & Mamajek (2013).²⁸ The SpecMatch-Emp analysis of the secondary spectrum resulted in T_eff = 3456 ± 70 K, ${R}_{* }\,=0.41\pm 0.10{R}_{\odot }$, and [Fe/H] = −0.21 ± 0.09. The spectroscopic temperature of the secondary is approximately 150K cooler than the blended spectrum. This is consistent with an M3.0 ± 0.5 spectral type (Pecaut & Mamajek 2013). The HIRES stellar parameters for the blended spectrum are also consistent with the SpeX parameters within uncertainties. Since the HIRES spectra are blended or only partially resolved, we only use the metallicities in subsequent analyses. As expected for stars in a bound system, the metallicities from the different spectra are consistent. The measured metallicities are provided in Table 1.

The standard CPS analyses of the HIRES spectra also provide barycentric corrected radial velocities (RV). Our two epochs of blended HIRES spectra provide a mean RV of 73.0 ± 0.3 km s⁻¹. The partially resolved secondary spectrum yields RV = 70.2 ± 0.3 km s⁻¹. These RVs are broadly consistent but differ at the 9σ level, potentially due to orbital motion. To search for additional stellar companions at very small separations, we performed the secondary line search algorithm presented by Kolbl et al. (2015) on the HIRES spectra. This analysis did not reveal any significant signals attributable to additional unseen companions in the system at ΔRV ≥10 km s⁻¹ and ΔV ≲ 5 mag. We report the weighted mean HIRES RV in Table 1.

After the binary companion was identified, we observed K2-288 with high-resolution AO imaging at the Keck Observatory. This was completed in order to ensure our transit signal was due to the presence of an exoplanet and not the stellar companion. The observations were made on Keck II with the NIRC2 instrument behind the natural guide star AO system. These observations were completed on 2017 August 20 UT in the standard three-point dither pattern used with NIRC2. This observing mode was chosen to avoid the typically noisier lower left quadrant of the detector. We observed K2-288 in the narrow-band Br–γ, the H-continuum, and J-continuum filters. Using a step-size of $3^{\prime\prime}$, the dither pattern was repeated three times, with each dither offset from the previous by 05. We used integration times of 6.6, 4.0, and 2.0 s, for the the narrow-band Br–γ, the H-continuum, and J-continuum respectively, with the co-add per frame for a total of 59.4, 36.0, and 26.1 s. The narrow-angle mode of the camera allowed for a full field of view of $10^{\prime\prime}$ and a pixel scale of approximately $0\buildrel{\prime\prime}\over{.} 009942$ per pixel. The Keck AO observations clearly detected a nearly equal brightness secondary $0\buildrel{\prime\prime}\over{.} 8$ to the southeast of the primary target. We also observed K2-288 on 2017 December 29 UT in the broader J and K_p filters through poor and variable seeing (∼1''–2''). The binary was clearly resolved, but the images were of much lower quality than the 2017 August 20 observations and are not used in any subsequent analyses.

The resulting NIRC2 AO data have a resolution of $0\buildrel{\prime\prime}\over{.} 049$ (FWHM) in the Br–γ filter, $0\buildrel{\prime\prime}\over{.} 040$ (FWHM) in the H-cont, and $0\buildrel{\prime\prime}\over{.} 039$ (FWHM) in the J-cont filter. Fake sources were injected into the final combined images with separations from the primary in multiples of the central source's FWHM in order to derive the sensitivities of the data (Furlan et al. 2017). The 5σ limits on the sensitivity curves are shown in Figure 4. The separation of the secondary was measured from the Br–γ image and determined to be ${\rm{\Delta }}\alpha =0\buildrel{\prime\prime}\over{.} 259\pm 0\buildrel{\prime\prime}\over{.} 001$ and ${\rm{\Delta }}\delta \,=-0\buildrel{\prime\prime}\over{.} 746\pm 0\buildrel{\prime\prime}\over{.} 001$, corresponding to a position angle of PA ≈1598 east of north. The blending caused by the presence of the secondary was taken into account in the resulting analysis, to obtain the correct transit depth and planetary characteristics (Ciardi et al. 2015).

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The blended 2MASS JHK-magnitudes of the system are: J = 10.545 ± 0.020 mag, H = 9.946 ± 0.023 mag, and ${K}_{s}=9.724\pm 0.018$ mag. The primary and secondary have measured magnitude differences of ${\rm{\Delta }}J=0.997\pm 0.009$ mag, ΔH = 0.990 ± 0.005 mag, and ${\rm{\Delta }}{K}_{s}=0.988\pm 0.004$ mag. Br–γ has a central wavelength that is sufficiently close to Ks to enable the deblending of the 2MASS magnitudes into the two components. The primary star has deblended real apparent magnitudes of J₁ = 10.910 ± 0.027 mag, H₁ = 10.313 ±0.021 mag, and Ks₁ = 10.092 ± 0.023 mag, corresponding to (J − H)₁ = 0.597 ± 0.033 mag and ${\left(H-{K}_{s}\right)}_{1}=0.221\,\pm 0.031$ mag. The secondary star has deblended real apparent magnitudes of J₂ = 11.907 ± 0.0269 mag, H₂ = 11.303 ±0.021 mag, and Ks₂ = 11.079 ± 0.023 mag, corresponding to (J − H)₂ = 0.604 ± 0.033 mag and ${\left(H-{K}_{s}\right)}_{2}=0.223\,\pm 0.031$ mag. We derived the approximate deblended Kepler magnitudes of the two components using the $({Kepmag}\,-{Ks})\ \mathrm{versus}\ (J-{Ks})$ color relationships described in Howell et al. (2012). The resulting deblended Kepler magnitudes are Kep₁ = 13.46 ± 0.09 mag for the primary and Kep₂ =14.49 ± 0.10 mag for the secondary, with a resulting Kepler magnitude difference of ΔKep = 1.03 ± 0.12 mag. These deblended magnitudes were used when fitting the light curves and deriving true transit depth.

Both stars have infrared colors that are consistent with approximately M3V spectral type (Figure 5). However, this is driven by the uncertainties on the component photometry. With an approximate primary spectral type of M2, and ΔJHK ≈1 mag, the secondary is likely about one and a half sub-types later than the primary (Pecaut & Mamajek 2013). It is unlikely that the star is a heavily reddened background star. Based on an extinction law of R = 3.1, an early-K type star would have to be attenuated by more than 1 mag of extinction for it to appear as a mid M-dwarf. The line-of-sight extinction through the Galaxy is only A_V ≈ 0.7 mag at this location (Schlafly & Finkbeiner 2011), making a highly reddened background star unlikely compared to the presence of a secondary companion. Additionally, archival ground-based imaging does not reveal a stationary or slow moving point source near the current location of K2-288, indicating that the imaged secondary at 08 is likely bound (See Section 3.4). Gaia DR2 also provides consistent astrometry for two stars near the location of K2-288 (See Section 4.1).

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K2-288 is relatively bright and has been observed in many seeing limited surveys at multiple wavelengths. Currently available archival imaging of the system spans nearly 65 yr. Over the long time baseline of the available imaging, the large total proper motion of K2-288 (μ = 195.9 mas yr⁻¹) has carried it ≈125. This allows for additional checks for very close background sources at the current location of the system and additional constraints on whether the resolved binary is bound or a projected background source. Figure 6 displays an image of K2-288 from K2 at its current location (left) compared to two epochs of Palomar Observatory Sky Survey (POSS) images (center and right). The green polygon represents the optimal photometric aperture used to extract the K2 light curve. In the POSS images from 1951, there is no source at K2-288's current location down to the POSS I R limit of 20.0 mag (Abell 1966). This indicates that there are no slow moving background stars that are beyond the limits of our AO imaging. Additionally, the lack of a bright source at the current location in archival observations reinforces that the resolved secondary is bound and co-moving with the primary.

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The archival data does reveal a faint point source ∼14'' to the NE of K2-288's current location. This star, 2MASS J03414730+1816135, is relatively slow moving (μ = 32.6 mas yr⁻¹) background source at a distance of 390 pc (Gaia Collaboration et al. 2018). It is ∼5 mag fainter than K2-288 in the Kepler band and falls just outside of the optimal aperture used to produce the K2 light curve. Due to its proximity to the optimal aperture, this background star warrants further examination as the potential source of the transit signal. We used additional light curves generated during the k2phot reduction where the flux was extracted using different size apertures to investigate possible contributions from this faint star. In Figure 7 we show the phase folded transit signal from the light curve extracted with the optimal aperture compared to the same signal extracted using soft-edged circular apertures with radii of 1.5, 3, and 8 pixels. Due to the proper motion of the target and the use of 2MASS coordinates to place the circular apertures, the centers are offset from K2-288's current position by ∼3'' . None the less, the 3 and 8 pixel radius apertures yield phase folded transits with the same approximate depth as the optimal aperture while including more light from the nearby faint background star. The 1.5 pixel circular aperture suffers from a substantial increase in noise because it does not include the brightest pixels of K2-288. These analyses indicate that the faint slow moving background star is likely not the source of the observed transits and the candidate orbits one of the components of resolved binary. This is further reinforced by our detection of a partial transit in Spitzer observations that use an aperture that is much smaller and free of contamination from the faint nearby star (see Section 4.2).

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Astrometry (Lindegren et al. 2018) and photometry (Evans et al. 2018; Riello et al. 2018) of K2-288 obtained by Gaia over the first 22 months of mission operations were made available in the second data release from the mission (DR2 Gaia Collaboration et al. 2018). The DR2 catalog lists two sources within 37 of the 2MASS coordinates of K2-288 (Gaia DR2 44838019758175488 and 44838019756570112). The separation, position angle, and ΔG of these sources are consistent with the results of our Keck AO imaging and the estimated magnitude difference in the Kepler band. Thus, both components of K2-288 were resolved by Gaia. However, the proximity of the sources led to relatively poor fits in the five-parameter astrometric solution for each star. Here we refer to the goodness of fit statistic of the astrometric solution in the along scan direction, astrometric_gof_al in the Gaia DR2 catalog. Good solutions typically have astrometric_gof_al <3, where K2-288 A and B have values of 24.1 and 31.8, respectively. This also leads to significant excess noise in the fit for each star (Gaia DR2 parameter astrometric_excess_noise), 0.41 mas for the primary and 0.77 mas for the secondary. The utility of Gaia DR2 data in identifying binaries has been demonstrated via comparison to a large sample of AO resolved multiple systems from the Kepler planet candidate host sample (Ziegler et al. 2018b). Similarly significant excess noise in Gaia astrometric parameter fits has also been observed in this sample (Rizzuto et al. 2018).

The astrometric statistics of K2-288 may be improved in later Gaia data releases as more data is obtained for each star. The excess errors are manifested as discrepancies between the astrometric measurements of the components. For example, the parallax of the secondary differs from that of the primary by 0.9 mas, a >4σ difference (when considering the secondary parallax uncertainty). This discrepancy is likely too large to be attributed to binary orbital motion over the time baseline of the Gaia observations. Despite this discrepancy, the distances to the components of the binary are comparable, 70.0 ± 0.4 pc and 65.7 ± 0.9 pc for the primary and secondary, respectively.²⁹ Given supporting evidence that these stars form a moderate separation, bound system—consistent RVs (see Section 3.2), consistent proper motions (see Section 3.4)—we adopt the weighted mean and error of the primary and secondary distances as the distance to the system, 69.3 ±0.4 pc, and include it in Table 1. At this adopted distance, we find the projected separation of the secondary is 54.8 ± 0.4 au. We also use this distance to infer the individual stellar parameters of the components in Section 5.1. Gaia DR2 also provides a radial velocity for the primary, RV = 72.15 ± 1.72 km s⁻¹. This is consistent with the HIRES measured system RV and provides further evidence that there are not additional unseen stellar companions in the system.

EPIC 210693462 was observed by Spitzer from UT 2017 December 11 15:35:58 to 2017 December 11 20:31:28. The observations were conducted with the Infra-Red Array Camera (IRAC; Fazio et al. 2004) at 4.5 μm with an exposure time of 2 s. Because of the small separation of the binary components (∼08) and the pixel scale of IRAC (12), the binary was unresolved in the Spitzer images. Photometry of the blended binary point-spread function (PSF) was obtained using circular apertures and the background was estimated and subtracted following a procedure similar to Beichman et al. (2016). The aperture was then chosen by selecting the light curve with minimal white and red noise statistics, as computed by the standard deviation and the red noise factor β (Pont et al. 2006; Winn et al. 2008, Livingston et al. 2019). Following this procedure, an aperture radius of 2.3 pixels was found to yield the lowest noise, which is consistent with the optimal apertures found in previous studies (e.g., Beichman et al. 2016; Knutson et al. 2012). We then binned the light curve and pixel data to 60 s, as this has been shown to yield an improved systematics correction without affecting the information content of the light curve (e.g., Benneke et al. 2017).

Due to the close separation of the components of K2-288, it is crucial to estimate their individual properties to further evaluate the characteristics of the planet candidate. The spectra obtained using HIRES and SpeX are blended and the stellar parameters estimated in Sections 3.1 and 3.2 are not indicative of the properties of each star, except the metallicities, which should be, and are measured to be, consistent. However, our resolved NIR photometry from Keck AO imaging and the Gaia distance to the system provide a basis for reliably estimating the individual component properties.

We base our approach on that of Dressing et al. (2018), which hinges on using stellar absolute K_s magnitudes (M_K), photometric colors, and calibrated relations to estimate the masses, radii, and effective temperatures of low-mass stars. Dressing et al. (2018) showed that this approach provides fundamental parameter estimates consistent with those calculated using spectroscopic index and equivalent width based relations with comparable levels of precision. We used the adopted system distance of 69.3 ± 0.4 pc and the resolved K_s-band magnitudes of the components to calculate their M_k's. We find ${M}_{{K}_{p}}\,=5.888\pm 0.036$ mag for the primary and ${M}_{{K}_{s}}=6.875\pm 0.036$ mag for the secondary. Throughout the discussion, we use the subscript p to denote the primary and s to denote the secondary.

To estimate the masses of the stars, we used the M_K–mass relation presented in Benedict et al. (2016). We estimated stellar mass uncertainties by assuming the errors on our absolute magnitudes and the coefficients in Benedict et al. polynomial relation follow Gaussian distributions and calculated the mass 10⁴ times using Monte Carlo (MC) methods. The median and standard deviation of the resulting distribution were adopted as the mass and associated statistical uncertainty. We then added this uncertainty in quadrature to the intrinsic scatter in the Benedict et al. (2016) relation (0.02 ${M}_{\odot }$). This procedure resulted in mass estimates of M_p = 0.52 ± 0.02 ${M}_{\odot }$ and M_s = 0.33 ± 0.02 ${M}_{\odot }$ .

Our radii estimates use the M_K–radius–[Fe/H] relation from Mann et al. (2015, 2016). In these calculations we used the HIRES measured metallicities attributed to the primary and secondary provided in Table 1. Our approach to radius uncertainty estimates is similar to that used in the mass calculation. We use MC methods assuming Gaussian distributed errors on M_K and [Fe/H] then add the resulting radius uncertainties in quadrature to the scatter in the Mann et al. (2015, 2016) polynomial fit (0.027 ${R}_{\odot }$). This results in radii estimates of R_p = 0.45 ± 0.03 ${R}_{\odot }$ and R_s = 0.32 ± 0.03 ${R}_{\odot }$.

Our effective temperature estimates use the V–J–T_eff–[Fe/H] relation from Mann et al. (2015, 2016). Here we also used the HIRES measured metallicities from Table 1. The calculation also requires an estimate of the V − J color of the stars, which we interpolate from the Pecaut & Mamajek (2013) main-sequence color–temperature table.³⁰ The V − J color and uncertainty is estimated using MC methods during the interpolation. We estimate (V − J)_p = 3.304 ± 0.022 mag for the primary and (V − J)_s = 3.962 ± 0.024 mag for the secondary. We then used the Mann et al. (2015, 2016) relation to estimate the stellar temperatures following the same approach to uncertainty estimation previously described for the mass and radius estimates. We estimate the primary and secondary effective temperatures to be ${T}_{\mathrm{eff},{\rm{p}}}=3584\pm 205$ K and ${T}_{\mathrm{eff},{\rm{s}}}\,=3341\pm 276$ K, respectively. These effective temperatures are consistent with spectral types of M2 ± 1 and M3 ± 1 using the relations of Pecaut & Mamajek (2013). Additionally, the temperature estimated using the resolved primary photometry is consistent with the temperatures estimated from the blended SpeX and HIRES spectra (see Sections 3.1 and 3.2). This result is consistent with the ∼1 mag difference between the components inferred from Keck AO imaging, which reveals that the primary contributes substantially more flux than the secondary and dominates the blended spectra. We also use our calculated M_K mags and the Pecaut & Mamajek (2013) extended table to interpolate luminosities for K2-288 A and B. These values, along with all of the other stellar parameters, are included in Table 1. The parameters estimated in this section are used in subsequent transit modeling analyses.

5.2.1. K2 and Spitzer Transit Modeling

To model the K2 transit, we adopted a Gaussian likelihood function and the analytic transit model of Mandel & Agol (2002) as implemented in the Python package batman (Kreidberg 2015). We used the Python package emcee (Foreman-Mackey et al. 2013) for Markov Chain Monte Carlo (MCMC) exploration of the posterior probability surface. The free parameters of the transit model are: the planet-to-star radius ratio R_p/R_⋆, the scaled semimajor axis a/R_⋆, mid transit time T₀, period P, impact parameter b, and the quadratic limb-darkening coefficients q₁ and q₂ under the transformation from u-space of Kipping (2013). The transit signal was originally identified in the k2phot light curve. However, for this analysis, we fit the transit model to the EVEREST 2.0 (Luger et al. 2018) light curve due to the lower level of residual systematics; the EVEREST 2.0 light curve and best-fit transit model are shown in Figure 9. We model the Spitzer systematics using the pixel-level decorrelation (PLD) method proposed by Deming et al. (2015), which uses a linear combination of the normalized pixel light curves to model the systematic noise caused by motion of the PSF on the detector (see Figure 8). To allow error propagation we simultaneously model the transit and systematics using the parameterization

$$\begin{eqnarray}&&{\rm{\Delta }}{S}^{t}=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{9}{c}_{i}{P}_{i}^{t}}{{\displaystyle \sum }_{i=1}^{9}{P}_{i}^{t}}+{M}_{{tr}}({\boldsymbol{\theta }},t)+\varepsilon (\sigma ),\end{eqnarray} \tag{ 1 }$$

where ${\rm{\Delta }}{S}^{t}$ is the measured change in signal at time t, M_tr is the transit model (with parameters ${\boldsymbol{\theta }}$), the c_i are the PLD coefficients, P_i^t is the ith pixel value at time t, and ε(σ) are zero-mean Gaussian errors with width σ; we fit for the logarithm of these parameters, denoted as log(σ) in Table 2. We imposed Gaussian priors on the limb-darkening coefficients for both the Kepler and IRAC2 bandpasses, with mean and standard deviation determined by propagating the uncertainties in host star properties (T_eff, log g, and [Fe/H]) via MC sampling an interpolated grid of the theoretical limb-darkening coefficients tabulated by Claret et al. (2012). We performed an initial fit using nonlinear least squares via the Python package lmfit (Newville et al. 2014), and then initialized 100 "walkers" in a Gaussian ball around the best-fit solution. We then ran an MCMC for 5000 steps and visually inspected the chains and posteriors to ensure they were smooth and unimodal, and discarded the first 3000 steps as "burn-in." To ensure that we had collected enough effectively independent samples, we computed the autocorrelation time³¹ of each parameter.

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Table 2.  Transit Parameters

Parameter	Unit	Value
T0-2454833	BJDTDB	${2230.402366}_{-0.001107}^{+0.001162}$
P	days	${31.393463}_{-0.000069}^{+0.000067}$
a	${R}_{\star }$	${110.2}_{-15.5}^{+8.9}$
b	⋯	${0.37}_{-0.25}^{+0.23}$
i	deg.	${89.81}_{-0.17}^{+0.13}$
${R}_{p,K}$	${R}_{\star }$	${0.0356}_{-0.0010}^{+0.0012}$
${R}_{p,S}$	${R}_{\star }$	${0.0487}_{-0.0031}^{+0.0030}$
${u}_{1,K}$	⋯	${0.314}_{-0.057}^{+0.055}$
${u}_{2,K}$	⋯	${0.397}_{-0.033}^{+0.034}$
${u}_{1,S}$	⋯	${0.009}_{-0.007}^{+0.011}$
${u}_{2,S}$	⋯	${0.192}_{-0.014}^{+0.014}$
log(${\sigma }_{K}$)	⋯	$-{8.88}_{-0.12}^{+0.13}$
log(${\sigma }_{S}$)	⋯	$-{6.88}_{-0.04}^{+0.04}$
${\rho }_{\star ,\mathrm{circ}}$	g cm−3	${25.70}_{-9.39}^{+6.77}$
T14	days	${0.088}_{-0.004}^{+0.003}$
T23	days	${0.079}_{-0.003}^{+0.003}$
shape	⋯	${0.91}_{-0.03}^{+0.01}$
${R}_{p,\max }$	${R}_{\star }$	${0.049}_{-0.006}^{+0.019}$

Note. The subscripts K and S refer to the Kepler and Spitzer 4.5 μm bandpasses, respectively. The parameter "shape" is the ratio of T₂₃ to T₁₄, where values close to unity indicate a "box-shaped" transit caused by a small occulting body. The parameter ${R}_{p,\max }$ corresponds to the maximum planetary radius (in units of the stellar radius) allowed by the transit geometry. log(${\sigma }_{K}$) and log(${\sigma }_{S}$) represent the width of the zero-mean Gaussian errors.

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We plot the Spitzer data and resulting transit fit in Figure 10. A significant partial transit, including ingress, is detected at the end of the observing sequence. The time of this transit is shifted from the transit time predicted using K2 data by ∼3σ. This is consistent with previous Spitzer transit observations obtained months to years after the K2 observations (Beichman et al. 2016; Benneke et al. 2017). The Spitzer observations of K2-288 were obtained ∼2.5 yr after K2 C4 and the relative imprecision of the K2 transit ephemeris (a result of detecting only three transits) results in a significant linear drift over this time baseline (see Beichman et al. 2016). We report the median and 68% credible interval of each parameter's marginalized posterior distribution in Table 2.

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5.2.2. Simultaneous K2 and Spitzer Analysis

We simultaneously fit the K2 and Spitzer light curves to ensure a robust recovery of the transit signal in the Spitzer data, as well as to enable the higher cadence of the Spitzer data to yield improved parameter estimates from the K2 data (Livingston et al. 2019). This is achieved by sharing strictly geometric transit parameters (a/R_⋆ and b), which are bandpass-independent, while using using separate parameters for limb-darkening, systematics, etc., which are bandpass-dependent. R_p/R_⋆ is fit for both the Kepler and Spitzer 4.5 μm bandpasses separately to allow for a dependence of transit depth on wavelength. Any such chromaticity is of particular interest in this case because it contains information about the levels of dilution present at each band, which in turn can determine which component of the binary is the true host of the transit signal. The K2 light curve contains only three transits, and thus only sparsely samples ingress and egress due to the 30-minute observing cadence. The impact parameter b is thus poorly constrained in the fit to the K2 data alone, but the addition of the higher cadence Spitzer transit yields an improved constraint, which in turn yields a more precise measurement of R_p/R_⋆ in the Kepler bandpass. A grazing transit geometry is strongly ruled out, and a significantly larger value of R_p/R_⋆ is detected in the Spitzer bandpass (∼3.7σ), indicating that the component of the binary that hosts the planet candidate is subject to lower levels of dilution at longer wavelengths (see Figure 11).

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We derive the planet parameters for our system assuming two configurations: the planet orbits the primary M2V and the planet orbits the secondary M3V. We complete this analysis using parameters from both our K2 and Spitzer transit fits. Results are presented in Table 3. We use Equations (4) and (6) from Furlan et al. (2017) to account for dilution in the transit for the primary and secondary scenarios, respectively. When estimating the planet radius (R_p) for our K2 derived parameters, we use the ΔKep mag estimated in Section 3.3. When estimating the planet radius from the Spitzer fit, we estimated the ${\rm{\Delta }}\mathrm{IRAC}2$ band magnitude using our resolved component properties and the compiled ${K}_{s}-W2$ colors from Pecaut & Mamajek (2013). We estimate ΔIRAC2 = 0.89 ± 0.03 mag. In each case, our planet radius and equilibrium temperature estimates assume Gaussian distributed uncertainties for the parameters from Tables 1 and 2.

Table 3.  Planet Parameters

Parameter	Unit	Value
Primary
${R}_{p,K}$	R⊕	2.06 ± 0.16
${R}_{p,S}$	R⊕	2.86 ± 0.27
a	au	0.231 ± 0.03
Teq	K	242.85 ± 19.8
Secondary
${R}_{p,K}$	R⊕	1.70 ± 0.36
${R}_{p,S}$	R⊕	2.23 ± 0.47
a	au	0.164 ± 0.03
Teq	K	226.36 ± 22.3

Note. The subscripts K and S refer to the Kepler and Spitzer 4.5 μm bandpasses, respectively.

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We find that the K2 and Spitzer planet radii estimated assuming the candidate orbits the secondary are more consistent than when assuming it orbits the primary. This result, along with the consistency between the stellar density estimated from the transit fit (ρ_* = ${25.70}_{-9.39}^{+6.77}$ g cm⁻³); Table 2) and the estimated density of the M3V companion based on resolved measurements (ρ_* = 14.2 ± 5.0  g cm⁻³ ; Table 1), and the significantly deeper transit in the the Spitzer IRAC2 band, provide evidence that the candidate orbits the secondary component in the system.

Assuming the candidate transits the secondary, we applied the vespa statistical planet validation tool to the system (Montet et al. 2015; Morton 2015). We use the stellar parameters of the secondary provided in Table 1 as input. We also include the contrast curves from our resolved NIR imaging as additional input. vespa returns a false positive probability (FPP) of 7.7 × 10⁻⁹ when using the folded K2 transit. This FPP indicates that the transiting signal is not a bound or background eclipsing binary and we consider the transiting body a validated planet.

We conclude, using evidence provided in this section, that the observed transit is caused by a planet on a 31.39-day orbital period and most likely occurs around the secondary M3V star. We calculate the weighted mean of the K2 and Spitzer transit derived planet radii to arrive at R_p = 1.9 ± 0.3 R_⊕. Given both stellar and planet parameters provided in Tables 1 and 2 respectively, we estimate the equilibrium temperature of the planet to be roughly 226 K. We adopt the following nomenclature for this system: K2-288A is the primary M2V star, K2-288Bb is the secondary M3V, and K2-288Bb is the planet.