Zodiacal Exoplanets in Time (ZEIT) VIII: A Two Planet System in Praesepe from K2 Campaign 16

Young planets offer a direct view of the formation and evolution processes that produced the diverse population of mature exoplanet systems known today. The repurposed Kepler mission K2 is providing the first sample of young transiting planets by observing populations of stars in nearby, young clusters or stellar associations. We report the detection and confirmation of two planets transiting K2-264, an M2.5 dwarf in the 650 Myr old Praesepe open cluster. Using our notch-filter search method on the K2 lightcurve, we identify planets with periods of 5.84 d and 19.66 d. This is currently the second known multi-transit system in open clusters younger than 1 Gyr. The inner planet has a radius of 2.27$_{-0.16}^{+0.20}$ R$_\oplus$ and the outer planet has a radius of 2.77$_{-0.18}^{+0.20}$ R$_\oplus$. Both planets are likely mini-Neptunes. These planets are expected to produce radial velocity signals of 3.4 and 2.7 m/s respectively, which is smaller than the expected stellar variability in the optical ($\simeq$30 m/s), making mass measurements unlikely in the optical, but possible with future near-infrared spectrographs. We use an injection-recovery test to place robust limits on additional planets in the system, and find that planets larger than 2 R$_\oplus$ with periods of 1-20 d are unlikely.


INTRODUCTION
Planets and their host stars can change dramatically over their lifetimes. Their structural, orbital, and atmospheric properties are all expected to evolve through interactions with their host star (e.g., Ehrenreich et al. 2015), the protoplanetary disk from which they formed (e.g., Cloutier & Lin 2013), other planets in the system (e.g., Chatterjee et al. 2008), and the greater stellar environment (e.g., Cai et al. 2018). Understanding the underlying drivers and relative importance of these evolutionary mechanisms is critical for revealing the early sculpting of planetary systems and the conditions that give rise to the diversity of mature planetary systems revealed by Kepler and earlier exoplanet surveys (e.g., Lissauer et al. 2014;Inamdar & Schlichting 2016).
The Zodiacal Exoplanets in Time (ZEIT) survey (Mann et al. 2016a) was designed to identify and characterize transiting planets in these young clusters and star forming regions using the precise photometry from K2 (Van Cleve et al. 2016). The greater goal is to better understand how planets form and evolve by comparing the statistical properties of exoplanets of different ages and arXiv:1808.07068v2 [astro-ph.EP] 6 Sep 2018 to older systems found during the original Kepler mission (Borucki et al. 2010; Thompson et al. 2018). Thus far we have identified planets in Hyades , Praesepe (Mann et al. 2017b), and Upper Scorpius (Mann et al. 2016b), many of which were found near-simultaneously by similar surveys focusing on exoplanets in young stellar associations (e.g., Obermeier et al. 2016;David et al. 2016;Pepper et al. 2017;Ciardi et al. 2018;Livingston et al. 2018).
Multi-transiting planetary systems are uniquely useful for studying stellar and planetary properties. In cases where the planets' eccentricities can be independently constrained (e.g., through dynamics; Deck & Agol 2016;Gillon et al. 2017), multi-transiting systems can be used to constrain stellar densities with a precision that rivals eclipsing binaries (e.g., Mann et al. 2017a). Even with no information on the host star properties, differences between the measured transit duration of planets in the same system can be used to measure the relative eccentricities (Kipping et al. 2012). Multi-transit systems where planets undergo transit timing variations offer the best opportunity to measure the masses of small planets (e.g., Deck & Agol 2015;Hadden & Lithwick 2017a). Lastly, these systems provide a measurement of the mutual inclination of planets, a probe of the entropy of a system and hence the role of dynamical disruptions from their (expected) initially flat configuration (e.g., Figueira et al. 2012;Ballard & Johnson 2016).
Multi-transiting systems in clusters offer a unique route to study the dynamical properties of planets with known (young) ages. So far, there is only one known multi-transiting system in an open cluster: K2-136, a three-planet system in the 650 Myr old Hyades cluster Livingston et al. 2018;Ciardi et al. 2018). Here, we present the discovery of two planets transiting the 650 Myr old Praesepe cluster star K2-264 (JS 597;Jones & Stauffer 1991) from its K2 light curve. K2-264 hosts two super-Earth/mini-Neptunesized planets in short (≈6 and ≈20 day) period orbits. We describe our discovery and follow-up observations in Section 2, and we describe our analysis to determine stellar properties in Section 3. In Section 4 we place limits on additional planets in the system, and in Sections 5 and 6 we describe our transit fitting to determine stellar parameters and our false positive probability analysis. Finally, in Section 7, we discuss the implications from discovering a second multi-transiting cluster system.  Stumpe et al. 2012) prior to public release of the data on 30 May 2018. K2-264 was selected as part of four K2 guest observer programs in Campaign 16 1 , three of which selected K2-264 on the basis of membership in the Praesepe cluster.
We applied our detrending and transit search pipeline, which is described in detail in Rizzuto et al. (2017), to the data for K2-264. To summarize the process, we first removed the K2 roll or flat-field systematic, caused by the instability of the K2 pointing and pixel-response variations using the method of Vanderburg & Johnson (2014). This produced a cleaned lightcurve that was mostly free of instrumental systematics but still contains signals from stellar variability and transiting planets. We removed the astrophysical variability with a "notchfilter," which fits a 1-day window of the light curve as a combination of an outlier-resistant second-order polynomial and a trapezoidal notch. The inclusion of the notch allows aggressive detrending outside the notch without over-fitting that may remove or weaken transit-like signals. This window is then moved along each point in the light-curve, detrending the variability signal from the entire dataset. The periodic transits were then identified using the Box Least Squares algorithm (Kovács et al. 2002) on the detrended lightcurve. Figure 1 shows the rotational variability, detrended lightcurve, and detected transit signals.
Once the two transiting planets were detected, we reextracted the data using a simultaneous fit to the K2 roll systematic, low-frequency stellar variability, and transits, including outlier rejection as described in Vanderburg et al. (2016). The final lightcurve, following flattening by removal of the best-fit low-frequency variability and significant outliers, was then used for our MCMC transit fitting described below.

NIR Spectra from SPEX
On 2 June 2018 we observed K2-264 with the InfraRed Telescope Facility (IRTF) SpeX mediumresolution spectrograph . We used the 0.3 slit in SXD mode, which yielded a spectral resolution of R 2000 from 0.7 to 2.55 µm. Extraction and calibration of the spectrum, including flat, bias, and wavelength correction, was carried out using the Spextool package , which incorporates the xtellcor package (Vacca et al. 2003) to correct for telluric contamination. The observation was taken in poor conditions at high airmass and had a median signal-tonoise ratio (SNR) per pixel of 15 in the first two orders covering 1.4-2.5 µm, and SNR of 10 in the two orders covering 0.9-1.3 µm. Given the low SNR of the spectrum, we did not attempt to extract stellar properties such as T eff , log g, or metallicity. Figure 2 displays the resulting spectrum of K2-264. We measured a radial velocity from the spectrum by cross-correlating with a similar spectral-type standard. This was done over each order using the tellrv package 2 (Newton et al. 2014). After correcting for Barycentric motion, the cross-correlation yielded a radial velocity for K2-264 of 26±6 km/s, which is within ∼1-σ of the expected expected radial velocity for a Praesepe member.

Literature Photometry and Astrometry
Photometry from multiple all-sky surveys were compiled to build a full SED for K2-264 . Optical g r i z magnitudes were taken from the PanSTARRS point source catalog (Flewelling et al. 2016). Near-IR J,H and Ks photometry was taken from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), the r' magnitude was taken from the Carlsberg Meridian Catalog (CMC15; Muiños & Evans 2014). Mid-IR magnitudes in the W1-4 bands were taken from the Wide-Field Infrared Survey Explorer (WISE; Wright et al. 2010 allax, and G, RP , and BP magnitudes were taken from the Gaia mission second data release (Gaia Collaboration et al. 2018). These data for K2-264 are shown in Table 1.

Archival Imaging
We examined archival imaging observations of K2-264 from several different surveys to search for nearby stars that might contribute the transit signals we see.
In particular, we examined observations from the Palomar Observatory Sky Survey (POSS-I) to identify background stars at the present-day position of K2-264 , and we used observations from the Pan-STARRS survey to search for nearby faint companions.
We first used the POSS images of K2-264 to rule out the presence of bright background stars at the presentday position of the target star. K2-264 was observed by POSS in 1950, when its position was ∼2.7 away from its present day position due to the star's proper motion (µ = 40.1 ± 0.1 mas yr −1 ). K2-264's PSF partially overlaps its present-day position, but it is still possible to rule out some nearby companions. Based on nearby stars observed at the same time, we estimate that if there was a background star at the present-day position of K2-264 brighter than R ∼ 19 mag, we would be able to detect it. Since we see no evidence for such a star, we can rule out the presence of these background companions about three magnitudes fainter than K2-264. We show the POSS image in Figure 3.
We also used observations from the Pan-STARRS survey to search for and rule out faint stars near the position of K2-264. Neither a query of the Pan-STARRS archive point source catalog nor visual inspection of images revealed any nearby stars closer to K2-264 than 30". With Pan-STARRS, we can rule out nearby stars to fairly faint limits (r 20). The Pan-STARRS image is shown in Figure 3.

Companion Constraints from Gaia Data Release 2
While detection limits for additional sources surrounding stars in the Gaia second data release (Gaia Collaboration et al. 2018) are not characterized by the Gaia team, limits can be estimated using populations of known binaries detected in ground-based imaging surveys. Ziegler et al. (2018a) used a sample of 620 binary companions to Kepler Objects of Interest (KOIs) detected with Robo-AO imaging to characterize the detectability of companions 1-4" from a primary in the Gaia second data release. Ziegler et al. (2018a) find that companions with separations <1 arcsecond are not listed as separate sources in the Gaia catalog, and provide contrast limits out to separations of 4". This method can be extended to smaller separations by examining the quality of the Gaia astrometric fit, in particular the significance of the "extra-error" term. In order to do this we supplemented 363 high-confidence binary companions to KOIs identified by Robo-AO (Law et al. 2014;Baranec et al. 2016;Ziegler et al. 2017Ziegler et al. , 2018b with 93 companions detected at ρ < 1 using imaging or aperture mask interferometry with the Near Infrared Camera (NIRC2) on the Keck 2 telescope by Kraus et al. (2016). The higher spatial resolution of Keck, particularly when combined with aperture mask-ing, provided companions down to ρ 10-20 mas. The Robo-AO LP600 filter is very similar to the Gaia G bandpass (Ziegler et al. 2018a), however the companions from Kraus et al. (2016) were detected in K-band. Under the assumption that these companions were very likely to be bound due to the small separations, we interpolated Gaia G band primary-secondary contrasts using the K-band contrast, the primary estimated effective temperature from Kraus et al. (2016), and a 2 Gyr PARSEC 1.2s isochrone (Chen et al. 2014).
We then queried the Gaia second data release (DR2) catalog in a 10 arcsecond cone around each KOI with a detected companion. We assessed detection by Gaia on the basis of three separate conditions: (1) The companion was identified as a unique source in the catalog at the expected position angle and separation and with the expected contrast (2) The companion was not resolved as a distinct source in the Gaia catalog, but the astrometric extra error significance (D) was >10-σ. This was only used for companions with separations <1 arcsecond. (3) The primary star was missing from the Gaia catalog, again this condition was only applied to companions with separations of ρ < 1 . Our interpretation assumes that clear binaries where the astrometric solution was extremely poor were removed from the second data release, which is listed in Gaia Collaboration et al. (2018) as the intended operating procedure employed by the Gaia data reduction team. Finally, if the contrast of the companion and the magnitude of the primary would indicate a Gaia G magnitude of the secondary of >21 mag, we removed that companion from the test sample as it falls below the faint limit for the Gaia survey and may not be robustly detected. Figure 4 displays the separation and contrast of the recovered and non-recovered companions in the Gaia second data release. We find similar magnitude limits in the 1-3 arcsecond range as Ziegler et al. (2018a), with 50% recovery for ∆G = 3 mag at 1 arcsecond and for ∆G = 6 mag at 3". Inside 1 arcsecond, companions with ∆G < 2 mag are reliably detected on the basis of the astrometric fit down to separations of 80 mas.
There were no sources within 35 arcsecond of K2-264 in Gaia DR2, and the astrometric extra error significance for K2-264 is D = 4.98-σ. The Gaia DR2 release notes (Gaia Collaboration et al. 2018) state that for stars with well behaved astrometry, D should be considered as a half-normal with mean zero and spread of unity. Furthermore, Gaia DR2 astrometry may contain instrument and attitude modelling errors that may in- 2 mag at separations of 80-1000 mas.

STELLAR PARAMETERS
Effective Temperature and Bolometric Flux: We simultaneously solved for the spectral type and bolometric flux (F bol ) by fitting the literature photometry (Section 2.3) using a grid of M-dwarf templates, following the technique outlined in the previous papers in this series (e.g., Mann et al. 2017bMann et al. , 2018. For the templates, we used a set of flux-calibrated templates of members of the Hyades open cluster, which were observed as part of programs to characterize nearby M dwarfs (Gaidos et al. 2014). We first filled missing regions of the tem-plate spectra for which data were not available with BT-SETTL atmosphere models (Allard et al. 2011) of the corresponding temperature, and then reddened each template according to the E(B-V) value for Praesepe from Taylor (2006). For each template, we computed synthetic magnitudes using the filter profiles and zeropoints from Evans et al. (2018) Mann et al. (2015) for other optical bands, and Cohen et al. (2003) for 2MASS. We compared these synthetic magnitudes to the archival values, letting the template choice and overall flux levels shift as free parameters. For each template, we computed F bol by integrating over the full spectrum. Our final adopted spectral type and F bol were those corresponding to the best-fit template weighted by the χ 2 values from the comparison between observed and synthetic (from the templates) photometry). This method yielded a spectral type of M2.5(±0.5) and and F bol of 3.068±0.068×10 −11 erg cm −2 s −1 . The errors in F bol account for variations due to many possible template fits, uncertainties in the cluster reddening, and uncertainties arising from interpolating over gaps in the spectrum. We show the best-fit template and a comparison to the photometry in Figure 5.
To determine R * , M * , and ρ * we used the empirical M K S -R * relation from Mann et al. (2015)  Boltzmann equation, our bolometric flux, temperature of the best-fit template star, and the Gaia parallax. We find that the radius corresponding to the best-fit temperature is 0.475±0.018 R , which agrees with the radius from the M K S -R * very closely.
To calculate the total stellar luminosity, we combined our F bol value with the Gaia parallax, which yielded 0.0330 ± 0.0012L . Joining this with our radius determination and the Stefan-Boltzmann equation produced a T eff of 3580±70 K. This T eff value was consistent with the assigned value for our best-fit template (3560±60 K) derived by comparison to BT-SETTL models (Allard et al. 2011), as described in Mann et al. (2013).
Rotation Period: To determine the rotation period, we took the K2 roll-corrected light curve prior to removing the stellar variability, masked out the transits from the data, and computed a Lomb-Scargle periodogram spanning periods of 1-40 days. We fit a Gaussian to the largest peak in the periodogram to find the period at the peak power, and conservatively estimate the uncertainty as the standard deviation of the Gaussian divided by the periodogram power. We find the rotation period to be 22.8±0.6 days. The rotation period of K2-264 lies directly on the Praesepe rotation-mass sequence. Figure  6 shows the rotation period of K2-264 in relation to the host stars of the seven other known Praesepe members with transiting planets (Mann et al. 2017b) from K2 Campaign 5, and the full Praesepe population (Douglas   0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Mass (

et al. 2017).
Membership in the Praesepe cluster: The kinematics, position, and photometry of K2-264 all place it as a high confidence member of the Praesepe cluster. Combining our RV measurement for K2-264 with the Gaia data release 2 position, proper motions, and parallax measurements allow calculation of the three dimensional space velocity to be (U, V, W ) = (37.3 ± 4.6, −18.0 ± 2.3, −14.7 ± 3.4) km/s. This agrees with the 3D space velocity of Praesepe derived from the locus of the known members updated with Gaia DR2 astrometry of (42,-20,-10) km/s with intra-cluster dispersion of 1-2 km/s (Kraus & Hillenbrand 2007). Figure 7 shows the proper motion offset from the Praesepe veloc-  (2007) with membership probability greater than 95%. The intra-cluster dispersion appears to be 1-2 mas/yr, or equivalently 1-2 km/s. The proper motion of K2-264 is highly consistent with the projected sky motion of the Praesepe cluster.
ity projected onto the plane of the sky for K2-264 and the Praesepe members of Kraus & Hillenbrand (2007) with membership probability greater than 95%. Here we take the members from Kraus & Hillenbrand (2007), but plot the de-projected proper motions from Gaia DR2. K2-264 falls within the range of the velocity dispersion of the members. The Gaia DR2 positions and parallax (π = 5.36±0.06 mas; D = 186.6 +2.1 −4.1 pc) place K2-264 on the periphery of the central core of the Praesepe cluster. Figure 8 shows the spatial position slices of known Praesepe members and the position of K2-264 in relation to the cluster. We calculate a kinematic and spatial membership probability of ∼97% for K2-264 using the Bayesian membership selection method of Rizzuto et al. (2011Rizzuto et al. ( , 2015. In Figure 6 we also show the Gaia (BP-RP, G) colormagnitude diagram of Praesepe members from Kraus & Hillenbrand (2007). The single and binary star sequences are clearly visible, and K2-264 falls directly on the single star sequence of Praesepe members. In combination with the kinematic and rotational match to the cluster population, this makes membership in Praesepe highly likely. In addition, the narrow single stars sequence rules-out an unresolved companion to K2-264 contributing more that 10-20% of the total observed flux. This is consistent with the lack of companions with ∆G 2 mag determined from the Gaia astrometry in Section 2.5.
Metallicity: Given the strong membership of K2-264 in the Praesepe cluster, we can assign the bulk cluster metallicity of the Praesepe population to it. A value of [Fe/H] = 0.12 (Boesgaard et al. 2013) is used when required for other calculations and model fitting.

LIMITS ON ADDITIONAL PLANETS
We tested the sensitivity of the combination of our transit search and detrending pipeline and the K2 data for K2-264 using the method described in Rizzuto et al. (2017). We injected a series of synthetic planet signals with random parameters into the raw K2 photometry using the BATMAN model of Kreidberg (2015). We used orbital periods of 1-30 days and planet radii of 0.5-10 R ⊕ , and allow orbital phase and impact parameter to have values within the interval (0,1). We fixed the eccentricity to zero in these simulations, as it does not significantly alter detectability of a transit, but would significantly increase the required number of trials to obtain a dense enough mapping of parameter space. We randomly injected 5000 trial planet signals for this test. More information regarding this process can be found in Rizzuto et al. (2017).
For each injected planet, we applied the corrections for the K2 pointing and stellar variability, and searched for planets using the BLS algorithm, retaining signals with power spectrum peaks of >7-σ. If a planet was detected within 1% of both the injected period and injected orbital phase, we flagged it as recovered. Figure 9 displays the results of the injection-recovery testing. We found that the combination of the K2 data and our search methodology is sensitive to 1.7 R ⊕ planets at orbital periods of 1-10 days, 2.0 R ⊕ planets at orbital periods of 10-20 days, and 3.4 R ⊕ planets out to periods of 25 days at the 90% recovery level.

TRANSIT FITTING
To determine transit parameters for K2-264, we fit the cleaned and detrended K2 lightcurve with a Markov Chain Monte Carlo (MCMC) as described in Mann et al. (2016aMann et al. ( , 2018 and Johnson et al. (2017). In summary, our MCMC fitting is based on the combination of the BATMAN transit model code (Kreidberg 2015) with the Affine-invariant MCMC code emcee (Foreman-Mackey et al. 2013). The BATMAN transit models were computed including oversampling and binning to the ∼30 minute K2 long-cadence exposures. We implemented a quadratic limb-darkening law, and used the triangular sampling method of Kipping (2013). The free parameters in our model that are different for each planet are the planet-star radius ratio (R P /R * ), orbital period (P), epoch of first transit mid-point (T 0 ), impact parameter (b), and two parameters used in place of eccentricity and argument of periastron ( √ e sin ω, √ e cos ω). These parameters were all fit simultaneously with a common stellar density (ρ * ) and the limb darkening parameters (q 1 and q 2 ).
We applied a Gaussian prior on stellar density ρ * , de-  Completeness map for additional planets in the K2-264 system, produced from injection-recovery testing of our search pipeline. Each point represents an injected planet signal, with blue points indicating recovery and red points non-recovery. White stars are the two detected planets b and c. Our pipeline and the K2 data for K2-264 are sensitive to planets as small as ∼1-2 R⊕ at orbital periods of 1-20 days.
termined from our SED fitting described in Section 3. We also applied a Gaussian prior of the limb darkening parameters determined from the Limb Darkening Tool Kit (LDTK; Parviainen & Aigrain 2015) using the Husser et al. (2013) models, the Kepler filter response function, and the stellar parameters from Section 3. The priors computed were 0.42±0.10 and 0.38±0.05 for u 1 and u 2 respectively. The Gaussian prior was applied after conversion to the triangular sampling parameterization for quadratic limb darkening of Kipping (2013). All other parameters were explored with uniform priors with physical boundaries (e.g., 0 < b < 1). We ran the MCMC chain for 200,000 steps, with 50,000 steps of burn-in. The transit fit parameters and other derived quantities are reported in Table 2. For each value, we report the median, with errors derived from the 16 th and 84 th percentile values from our fit posteriors. The best fit transit models are shown in Figure 10. We also show posterior distributions for a subset of parameters (R p /R * , e, b, ρ * ) in Figure 11. Both planets have most likely eccentricities close to zero, which is expected for multiple systems of shortperiod planets, though both the eccentricity and impact parameters for the planets are not well-constrained by the K2 data. Both planets are also similar in size with radii of 2.27 +0.20 −0.16 R ⊕ and 2.77 +0.20 −0.18 R ⊕ .

FALSE POSITIVE PROBABILITY
While most planet candidates detected by Kepler and K2 are likely to be bona fide exoplanets, some transitlike signals may be caused by other astrophysical scenarios. We quantified the likelihood of one of these scenarios causing either of the two transit signals we see towards K2-264 using the open-source vespa software package (Morton 2015). Vespa calculates the false positive probability (FPP) of transiting signals using the procedure described by Morton (2012) and Morton et al. (2016). In particular, vespa performs a Bayesian model comparison between several different scenarios which might cause transit-like signals (transiting planets, an eclipsing binary on the target star, an eclipsing binary on a physically bound companion star, or an eclipsing binary on an unassociated background star), and using the transit light curve, stellar parameters, photometric measurements, and observational constraints, determines the likelihood of each model.
In the case of K2-264, we ran vespa using the transit light curve, broadband photometric measurements from the 2MASS survey 5 , and constraints on the presence of nearby stars from the 2MASS J -band image of K2-264 6 . Based on these inputs, vespa calculated an FPP of 4×10 −3 for K2-264 b and 9×10 −4 for K2-264 c. These FPPs do not take into account the fact that candidates in multi-candidate systems are considerably less likely to be false positives than candidates in single-candidate systems (Latham et al. 2011;Lissauer et al. 2012). We take this into account by applying a "multiplicity boost" to the calculated FPPs for K2-264 b and c. Following Lissauer et al. (2012), we divide the calculated FPP for K2-264 b and c by a factor of 25 as K2-264 is a twocandidate system. This agrees with the value derived by Sinukoff et al. (2016) for K2 data. After applying the multiplicity boost, we find FPPs for K2-264 b and c of about 10 −4 and 4 × 10 −5 , respectively. Based on these very low FPPs, we consider both candidates in the K2-264 system to be validated planets.

DISCUSSION
We have reported the discovery and characterization of a two planet transiting system in the Praesepe open cluster. There are now several detected transiting planets in young open clusters and associations observed by K2, though K2-264 is one of only two multiple-planet systems, the other being K2-136, a three transitingplanet system in the Hyades open cluster .
K2-264 b and c are both likely mini-Neptunes, and both sit near the upper envelope of the field masspanet radius distribution, as is seen for other planets in intermediate-age clusters. These two planets continue the trend of young open clusters M dwarfs hosting planets of larger radii than have been observed for planets transiting older field population dwarfs from the original Kepler sample (Mann et al. 2017b;Dressing & Charbonneau 2015). Figure 12 shows the planet radii and host star masses of the M-dwarf hosted young planets identified in the ZEIT survey (Mann et al. 2016a(Mann et al. ,b, 2017b, including K2-264 b/c, compared to older transiting systems. The possible inflation in radii at ∼650 Myr may be a sign of ongoing atmosphere loss (e.g., Lopez et al. 2012). With further completeness testing on the entire sample of Hyades and Praesepe stars observed by K2 a measure of the rate and significance of the potential radius difference could be measured.
Systems with multiple transiting planets offer the potential for many science cases not possible with single transit systems. In particular, eccentricity and stellar density can strongly constraint each other (Van Eylen & Albrecht 2015;Mann et al. 2017a). Planet masses for multiple systems can also be measured from transit timing variations (TTV's) (Hadden & Lithwick 2017b). Though we did not explicitly test for TTV's, the detection of TTV's in the K2 dataset is unlikely; similar size planets show variations of <15 min, which is smaller than the long-cadence timing of 30 mins. In particular, TTV's on planet b due to planet c are expected to be very small (<1 min) given that the orbital periods are very far from a resonance (Agol et al. 2005). One scenario where TTV detection could be possible involves the presence of a third planet in or near e.g., a 2:1 resonance with the inner planet b. Such a planet would have to be approximately earth-mass to have avoided detection in the K2 lightcurve. The TTV amplitude  (Mann et al. 2016a(Mann et al. , 2017b and those presented in this paper from C16, compared to older M-dwarf hosted planets from the original Kepler samples (Dressing & Charbonneau 2015). The 650 Myr Praesepe and Hyades planet population have larger radii than those hosted by older M dwarfs. The single 10 Myr old planet in Upper Scorpius (K2-33 b; Mann et al. 2016b) is also significantly larger than the older planets.
from such a planet on the ephemeris of K2-264 b, assuming zero eccentricities, is 5-15 min depending on the proximity to resonance (Agol et al. 2005).
The currently available long-cadence data from K2 is particularly unsuited to the science cases described above. However, K2-264 is highly amenable to follow-up photometry. Both planets are large enough that groundbased facilities could resolve their transits ( 3 mmag), though the faintness of the host star (r 16mag) may be prohibitive for small apertures at high cadence. Shorter cadence data resolving ingress and egress shapes can place stronger constraints on eccentricity, and offer suggestions as to the types of formation mechanisms responsible for forming these two short-period planets. Space-based follow-up with the Hubble Space Telescope or Spitzer is possible for both planets. In Spitzer channel 1 ( 3.5 µm; Hora et al. 2008) K2-264 is 12 mag (Wright et al. 2010) and in a 2 min exposure a SNR of 500 pmm is possible. This is sufficient to resolve the transit shape from even a single transit.
Follow-up spectroscopy to measure the masses of K2-264 b,c may not be possible. K2-264 shows stellar variability with a period 22.8 days and photometric amplitude of 3%. If the star is seen equator-on, this amplitude of variability is expected to produce RV variability of 30 m/s in a similar band as K2. Using the massradius relation for planets from Weiss & Marcy (2014) and the radii inferred from our transit fitting, we find that K2-264 b,c have likely masses of 5.8 M ⊕ and 7 M ⊕ respectively. Assuming circular orbits and the stellar properties derived above, these masses correspond to radial velocity semi-amplitudes of 3.4 m/s and 2.7 m/s respectively. The amplitude of these signals is significantly smaller than the expected stellar rotations signal. Moving to the near-infrared, where the stellar variability is expected to have significantly smaller amplitude, could alleviate this problem in combination with our prior knowledge of the rotation period of the star.

ACKNOWLEDGMENTS
ACR was supported as a 51 Pegasi b Fellow though the Heising-Simons Foundation. AWM was supported through NASA Hubble Fellowship grant 51364 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. AV's work was performed under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. S.T.D. acknowledges support provided by the NSF through grant AST-1701468. This paper includes data collected by the K2 mission. Funding for the K2 mission is provided by the NASA Science Mission directorate. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper 7 . This work has made use of data from the European Space Agency (ESA) mission Gaia 8 , processed by the Gaia Data Processing and Analysis Consortium (DPAC) 9 . Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. This research has made use of NASA's Astrophysics Data System Bibliographic Services.
Facilities: Kepler, IRTF , Texas Advanced Computing Center    Table 2 continued a Inclination, ω, a/R * and transit duration were not fit as part of our MCMC, but were derived from other fit parameters (see Section 5).
b The most likely eccentricities for both systems is ∼0, and so we report only the 1-σ upper limit.
c R P Teq were calculated using T eff from Section (3). Equilibrium temperature Teq was calculated assuming an albedo of 0.3.