DISCOVERY AND VALIDATION OF A HIGH-DENSITY SUB-NEPTUNE FROM THE K2 MISSION

K2 photometry for our target was obtained by the Kepler spacecraft during Campaign 4. This field was observed between 2015 February and April and the data were released on September of the same year. We obtained the decorrelated versions of all the lightcurves in the campaign which were made publicly available for download by Vanderburg & Jonson (2014), using photometry with the optimal aperture, which in the case of our target star corresponded to a ≈3 pixel radius around the target, or an aperture of ≈12'' radius. We performed a transit search using a box least squares (Kovács et al. 2002) algorithm. Once a periodic signal is detected along with the best-fit depth, the transit event is flagged as a pontential planetary candidate if (1) the depth is at least 3σ larger than the average noise level of the lightcurve (denoted by σ) and (2) if there are three or more transit events. Initially, because of the latter requirement, the lightcurve of the target star was not flagged by our transit search pipeline. However, we also performed visual inspection of all the lightcurves, revealing this interesting candidate. In order to double check that this was indeed an astrophysical signal and not a spurious signal arising from the decorrelation method used to obtain the lightcurve, we also inspected the detrended lightcurves released by the Kepler team using the PDC-MAP algorithm (Stumpe et al. 2012), and the same signal was observed at the exact same times as the signals observed in the Vanderburg & Jonson (2014) photometry. We were thus confident that the signal is of astrophysical origin, and proceeded to analyze the light curve.

A median filter with a 41 point (∼20.5 hr) window was used to further filter long-term variations of this target. The resulting median filter was smoothed using a Gaussian filter with a five-point standard-deviation, and this smoothed light curve was used to normalize the light curve. Using this normalized lightcurve, an initial fit using our transit-fitting pipeline (see below) revealed a P = 41.7 day period for this candidate and a lightcurve whose shape resembled that of a planetary transit, with a duration consistent with that of a planetary companion. Using the parameters obtained from this initial fit, we removed outliers from the out-of-transit data, discarding any points deviating more than 3σ from the median flux. The resulting normalized version of this lightcurve is shown in Figure 1. No other significant signals were found in the photometry.

Zoom In Zoom Out Reset image size

A high-resolution spectrum of this target was taken on October 21 with the CORALIE spectrograph mounted on the 1.2 m Euler Telescope in La Silla Observatory in order to obtain rough spectral parameters of the stellar host, and define whether this was a giant or a dwarf star. Data were reduced and analyzed using the procedures decribed in Jordán et al. (2014). The analysis of the CORALIE spectra gave T_eff = 5600 K, log(g) = 4.4 dex, [Fe/H] = 0.0 dex and $v\sin (i)=2.5$ km s⁻¹, which revealed that the star was a dwarf solar-type star. In addition, no secondary peak was seen on the cross-correlation function indicating no detectable spectroscopic binary. Because of this, the target was promoted to our list of planetary candidates despite the lack of high-resolution imaging needed to rule out potential blend events.

High-precision radial velocities (RVs) were obtained from the HARPS spectrograph mounted on the 3.6 m telescope at La Silla between 2015 October and December in order to measure the reflex motion of the star due to the hypothetical planet producing the transit signal. The observations covered our predicted negative and positive quadratures, along with epochs in between, to probe possible long-term trends in the RVs indicative of a possible massive companion. A total of 23 spectra were taken with the simultaneous Thorium–Argon mode; the HARPS pipeline (DRS, version 3.8) was used to reduce these spectra and to obtain the (drift-corrected) radial velocities, which are calculated via cross-correlation with a G2V mask which is appropiate for the stellar type of the host (see Section 3.1). The typical precision was ∼3 m s⁻¹ for each individual RV measurement. For each spectrum, the bisector span, S-index, and the integrated flux of the H_α and He i lines were obtained to monitor the activity of the host star and study its influence on the RVs (Santos et al. 2010; Jenkins et al. 2011). The measured RVs, along with these various calculated activity indicators, are given in Table 1. Although the times are given in UTC, they were converted to TBD (which is the timescale used by Kepler) for our joint analysis, which we describe in Section 3.2.

Table 1.  Radial Velocities Obtained with the HARPS Spectrograph along with Various Activity Indicators

BJD	RV	σRV	BIS	σBIS	SH,K	${\sigma }_{{S}_{H,K}}$	Hα	${\sigma }_{{H}_{\alpha }}$	He i	${\sigma }_{{H}_{\alpha }}$
(UTC)	(m s−1)	(m s−1)	(m s−1)	(m s−1)	(dex)	(dex)	(dex)	(dex)	(dex)	(dex)
2457329.63450	−20333.9	4.4	35.0	6.2	0.1748	0.0063	0.10151	0.00013	0.50230	0.00081
2457329.67362	−20340.1	3.6	28.9	5.0	0.1535	0.0050	0.10337	0.00013	0.50279	0.00081
2457329.72375	−20337.9	3.9	34.2	5.6	0.1864	0.0053	0.10367	0.00013	0.50146	0.00081
2457330.80181	−20343.1	2.6	22.6	3.7	0.1483	0.0035	0.10167	0.00013	0.50787	0.00082
2457331.63418	−20342.5	2.4	23.2	3.4	0.1551	0.0029	0.10171	0.00013	0.51233	0.00083
2457331.68695	−20338.4	2.0	11.0	2.8	0.1573	0.0026	0.10209	0.00013	0.50301	0.00081
2457332.64705	−20335.7	2.6	20.2	3.7	0.1549	0.0038	0.10236	0.00013	0.50221	0.00081
2457332.72713	−20338.4	2.0	12.9	2.9	0.1459	0.0025	0.10178	0.00013	0.50273	0.00081
2457336.65528	−20345.3	3.2	20.8	4.5	0.1701	0.0042	0.10397	0.00013	0.49984	0.00081
2457336.73328	−20339.8	3.6	9.5	5.1	0.1547	0.0047	0.10187	0.00013	0.49884	0.00081
2457339.70924	−20343.1	4.9	14.2	6.9	0.1985	0.0068	0.09939	0.00013	0.49982	0.00081
2457339.72063	−20339.2	4.1	31.1	5.8	0.1738	0.0058	0.10496	0.00013	0.50575	0.00082
2457340.69354	−20340.0	3.7	5.9	5.3	0.1833	0.0056	0.10217	0.00013	0.51486	0.00083
2457340.70475	−20336.6	3.1	23.9	4.3	0.1687	0.0044	0.10083	0.00013	0.50038	0.00081
2457341.74523	−20336.6	3.3	18.7	4.7	0.1658	0.0043	0.10536	0.00013	0.50213	0.00081
2457341.75600	−20335.3	3.2	25.4	4.5	0.1491	0.0043	0.10274	0.00013	0.50658	0.00082
2457348.80101	−20330.3	3.1	21.6	4.4	0.1858	0.0056	0.10339	0.00013	0.50277	0.00081
2457360.62435	−20337.2	2.2	14.8	3.2	0.1581	0.0029	0.10354	0.00013	0.50080	0.00081
2457360.63915	−20336.8	2.1	11.0	3.0	0.1585	0.0027	0.10237	0.00013	0.50273	0.00081
2457361.66418	−20337.3	3.1	30.7	4.4	0.1719	0.0042	0.10607	0.00013	0.50020	0.00081
2457361.67814	−20337.0	2.9	32.8	4.1	0.1533	0.0038	0.10231	0.00013	0.49660	0.00080
2457362.66191	−20331.0	2.2	15.4	3.1	0.1517	0.0031	0.10313	0.00013	0.49866	0.00081
2457362.67602	−20335.6	2.2	13.5	3.1	0.1575	0.0032	0.10479	0.00013	0.50121	0.00081

Download table as:  ASCIITypeset image

Archival imaging was obtained from the STScI Digitized Sky Survey⁸ at the EPIC coordinates of our target. Data are from the Palomar Observatory Sky Survey (POSS). In Figure 2 we show the best images among the available archival images in terms of the measured FWHM. We show images taken at two epochs and with two filters: one obtained in 1995 using the RG610 filter (red⁹ , 590–715 nm), taken by the POSSII-F and one using the RG9 filter (near-infrarred¹⁰ , 700–970 nm) obtained in 1996 by the POSSII-N. For reference, we show the aperture used to obtain our K2 photometry (black circle, 12'') along with circles with 5'' (white solid line) and 2'' (white dashed line) radii which are centered on the centroid of our target star, which was obtained by fitting a 2D Gaussian to the intensity profile.

Zoom In Zoom Out Reset image size

New imaging was obtained using the Las Cumbres Observatory Global Telescope Network (LCOGT). Four images were taken using the SBIG camera with the Bessel R filter on UT 2015/12/27 from the Cerro Tololo Interamerican Observatory (CTIO). Our target star reached close-to saturation counts (∼47,000 counts) in order to have enough photons to observe the close-by stars present in the POSS images. Figure 3 shows the resulting image obtained by median-combining our four images, along with the same circles as those drawn in Figure 2.

Zoom In Zoom Out Reset image size

Given that the largest potential source of false-positive detections in our case comes from blended eclipsing binary systems mimicking a planetary transit event, we note that, since the depth of the observed transit is ∼0.05%, if a blended eclipsing binary system was responsible of the observed depth, then assuming a total eclipse of the primary (which is the worst case scenario; all other scenarios should be easier to detect), the eclipsed star would have to be ∼8.23 mag fainter than our target star in the Kepler bandpass. We can confidently rule out such a bright star down to a distance of 9'' of the target star with the POSSII and LCOGT images. For reference, the closest star to the left of the target star (indicated with a red circle) in Figures 2 and 3 is ∼8.2 mag fainter than the target star in the R band. As can be seen on the images, a star that bright would be evident in the archival POSS images and/or on our new LCOGT images at distances larger than 9''.

AO imaging was obtained using the MagAO+Clio2 instrument mounted at the Magellan Clay telescope in Las Campanas Observatory on December 6 using the K_s filter with the full Clio2 1024 × 512 pixel frames of the narrow camera (f/37.7). The natural guide star system was used and, because our target is relatively bright, it was used as the guide star. A total of 32 images with exposure times of 30 s each were taken in five different positions of the camera (nodding), all of them at different rotator offset angles. Due to a motor failure of the instrument, the nodding and rotation patterns were not able to cover the full 16'' × 8'' field of view around the star. However, it gave us enough data to rule out stars within a 2'' radius. We follow methods similar to those described in Morzinski et al. (2015) to reduce our images, which we briefly describe here; a Python implementation of such methods is available at Github.¹¹ First, the images are corrected by dark current but not flat fielded, because the flats show an uneven flux level as a result of optical distortions and not of intrinsic pixel sensitivities (see Section A.3 in Morzinski et al. 2015, for a detailed explanation of this effect). A bad pixel mask provided by Morzinski et al. (2015) is used to mask bad pixels. After these corrections are applied to each image, we obtain a median image using our 32 frames in order to get an estimate of the background flux, which we then subtract from each of the individual frames. In order to further correct for differences in the sky backgrounds of each image, we apply a 2D median filter with a 200 pixel (≈3'') window which takes care of large-scale fluctuations of each image. The background-subtracted images are then merged by first rotating them to the true north (using the astrometric calibration described in Morzinski et al. 2015) and combined using the centroid of our target star (obtained by fitting a 2D Gaussian to the profile) as a common reference point between the images. Our resulting AO image, obtained by combining our 32 images, is shown in Figure 4. A 2D Gaussian fit to the target star gives a FWHM of 02, which we set as our resolution limit.

Zoom In Zoom Out Reset image size

The limiting contrasts in our AO observations in the K_s band were estimated as follows. First, a 2D Gaussian fit to the target star was made and used to remove it from the image. Although a 2D Gaussian does not perform a perfect fit at the center, the fit is good enough for the wings of the point-spread function (PSF), which is our aim. Then, at each radial distance n × FWHM away from the target star, where n = 1, 2, ..., 15 is an integer, a fake source was injected at 15 different angles. Sources with magnitude differences from 11 to 0 were injected in 0.1 steps, and a detection was defined if three or more pixels were 5σ above the median flux level at that position. The results of our injection and recovery experiments are plotted in Figure 5.

Zoom In Zoom Out Reset image size

Only one source was detected at ∼2'' from the target. The shape and position of this object is inconsistent with a speckle but is very faint: we measure a magnitude difference of ΔK_s = 9.8 with the target, which is thus just above our contrast level at that position (see Figure 4, the source is indicated with a gray circle in the upper right). A careful assessment of the PSF shape, however, made it inconsistent with the object having the same PSF shape as our star. Comparing its PSF with known "ghosts" on the image, on the other hand, revealed that this source is not of astrophysical but of instrumental origin.

To search for companions at larger separations, lucky imaging was obtained with AstraLux Sur mounted on the New Technology Telescope at La Silla Observatory (Hippler et al. 2009) on 2015 December 24 using the i' band. Figure 4 shows our final image obtained by combining the best 10% images with a drizzle algorithm. Because the PSF shape obtained for our lucky imaging is complex and we already ruled out companions inside a 2'' radius with Magellan+Clio2, and given that our objective with lucky imaging was to rule out companions at larger angular distances, we did not perform PSF substraction algorithms to obtain the 5σ contrasts at those distances. Instead, we used simple aperture photometry to estimate the 5σ contrasts outside the 2'' radius by performing a procedure similar to that described in Wöllert et al. (2015). In summary, we estimated the noise level in a 5 × 5 box at each radial distance at 15 different angles for distances larger than 2'' from the estimated centroid of the image (where the contribution of the target star's PSF to the background level is low), and calculated the magnitude contrast by obtaining the flux of the target star using a 5 pixel radius around it and a 5 pixel radius about the desired distance from the star, where 5σ counts are summed to each pixel at that distance before performing the aperture photometry. Then, the magnitude contrast at a given distance is obtained as the average value obtained at the different angles. The resulting 5σ contrasts are presented in Figure 5. We study the constraints that our archival, new, AO and lucky imaging put on the false-positive probabilities and transit dilutions on the next section.

To obtain the properties of the host star, we made use of both photometric and spectroscopic observables of our target. For the former, we retrieved B, V, g, r and i photometric magnitudes from the AAVSO Photometric All-sky Survey (APASS, Henden & Munari 2014) and J, H and K photometric magnitudes from 2MASS for our analysis. For the spectroscopic observables, we used the Zonal Atmospherical Stellar Parameter Estimator (ZASPE, R. Brahm et al. 2016, in preparation) algorithm using our HARPS spectra as input. ZASPE estimates the atmospheric stellar parameters and $v\sin i$ from our high-resolution echelle spectra via a least-squares method against a grid of synthetic spectra in the most sensitive zones of the spectra to changes in the atmospheric parameters. ZASPE obtains reliable errors in the parameters, as well as the correlations between them, by assuming that the principal source of error is the systematic mismatch between the data and the optimal synthetic spectra, which arises from the imperfect modeling of the stellar atmosphere or from poorly determined parameters of the atomic transitions. We used a synthetic grid provided by R. Brahm et al. (2016, in preparation) and the spectral region considered for the analysis was from 5000 to 6000 Å, which includes a large number of atomic transitions and the pressure-sensitive Mg Ib lines. The resulting atmospheric parameters obtained through this procedure were T_eff = 5766 ± 99 K, log(g) = 4.5 ± 0.08, [Fe/H] = −0.15 ± 0.05 and $v\sin (i)=3.3\pm 0.31$ km s⁻¹. With these spectroscopic parameters at hand and the photometric properties, we made use of the Dartmouth Stellar Evolution Database (Dotter et al. 2008) to obtain the radius, mass, age and distance to the host star using isochrone fitting with the isochrones package (Morton 2015). We took into account the uncertainties in the photometric and spectroscopic observables to estimate the stellar properties, using the emcee (Foreman-Mackey et al. 2013) implementation of the affine invariant Markov Chain Monte Carlo (MCMC) ensemble sampler proposed in Goodman & Weare (2010) to explore the posterior parameter space. We obtained a radius of ${R}_{* }={0.928}_{-0.040}^{+0.055}{R}_{\odot }$, mass ${M}_{* }={0.961}_{-0.029}^{+0.032}{M}_{\odot }$, age of ${3.3}_{-1.5}^{+1.9}$ Gyr and a distance to the host star of ${152.1}_{-7.4}^{+9.7}$ pc. The distance to the star was also estimated using the spectroscopic twin method described in Jofré et al. (2015), which is independent of any stellar models. The values obtained were 158.3 ± 5.4 pc when using 2MASS J band photometry and 160.0 ± 5.7 pc if H band photometry was used instead, where the stars HIP 1954, HIP 36512, HIP 49728 and HIP 58950 were used as reference for the parallax. Those values are in very good agreement with the value obtained from isochrone fitting. The stellar parameters of the host star are sumarized in Table 2.

Table 2.  Stellar Parameters of K2-56

Parameter	Value	Source
Identifying Information
EPIC ID	210848071	EPIC
2MASS ID	03343623+2035574	2MASS
R.A. (J2000, h:m:s)	03h34m3623	EPIC
Decl. (J2000, d:m:s)	20°35'5723	EPIC
R.A. p.m. (mas yr−1)	36.7 ± 0.7	UCAC4
Decl. p.m. (mas yr−1)	−51.8 ± 1.3	UCAC4
Spectroscopic properties
Teff (K)	5766 ± 99	ZASPE
Spectral Type	G	ZASPE
[Fe/H] (dex)	−0.15 ± 0.05	ZASPE
$\mathrm{log}{g}_{* }$ (cgs)	4.5 ± 0.08	ZASPE
$v\sin (i)$ (km s−1)	3.3 ± 0.31	ZASPE
Photometric properties
Kp (mag)	11.04	EPIC
B (mag)	11.728 ± 0.044	APASS
V (mag)	11.038 ± 0.047	APASS
g' (mag)	11.352 ± 0.039	APASS
r' (mag)	11.872 ± 0.050	APASS
i' (mag)	10.918 ± 0.540	APASS
J (mag)	9.770 ± 0.022	2MASS
H (mag)	9.432 ± 0.022	2MASS
Ks (mag)	9.368 ± 0.018	2MASS
Derived properties
M* (M⊙)	${0.961}_{-0.029}^{+0.032}$	isochrones+ZASPE
R* (R⊙)	${0.928}_{-0.040}^{+0.055}$	isochrones+ZASPE
ρ* (g cm−3)	${1.70}_{-0.26}^{+0.20}$	isochrones+ZASPE
L* (L⊙)	${0.88}_{-0.12}^{+0.15}$	isochrones+ZASPE
Distance (pc)	${152.1}_{-7.4}^{+9.7}$	isochrones+ZASPE
Age (Gyr)	${3.34}_{-1.49}^{+1.95}$	isochrones+ZASPE

Note. Logarithms given in base 10.

Download table as:  ASCIITypeset image

We performed a joint analysis of the photometry and the radial velocities using the EXOplanet traNsits and rAdIal veLocity fittER, exonailer, which is made publicly available at Github.¹² For the transit modeling, exonailer makes use of the batman code (Kreidberg 2015), which allows the user to use different limb-darkening laws in an easy and efficient way. If chosen to be free parameters, the sampling of the limb-darkening coefficients is performed in an informative way using the triangular sampling technique described in Kipping (2013). For the quadratic and square-root laws, we use the transformations described in Kipping (2013) in order to sample the physically plausible values of the limb-darkening coefficients. For the logarithmic law we use the transformations described in Espinoza & Jordán (2016), which presents the sampling of the limb darkening parameters for the more usual form of the logarithmic law to allow for easier comparison with theoretical tables (if the geometry of the system is properly taken into account, see Espinoza & Jordán 2015). The code also allows the user to fit the lightcurve assuming either a pure white-noise model or an underlying flicker (1/f) noise plus white-noise model using the wavelet-based technique described in Carter & Winn (2009). For the RV modeling, exonailer assumes Gaussian uncertainties and adds a jitter term in quadrature to them. The joint analysis is then performed using the emcee MCMC ensemble sampler (Foreman-Mackey et al. 2013).

For the joint modeling of the data set presented here, we tried both eccentric and circular fits. For the radial velocities, uninformative priors were set on the semi-amplitude, K, and the RV zero point, μ. The former was centered on zero, while the latter was centered on the observed mean of the RV data set. Note that our priors allow us to explore negative radial velocity amplitudes, which is intentional as we want to explore the possibility of the RVs being consistent with a flat line (i.e., K = 0). Initially a jitter term was added but was fully consistent with zero, so we fixed it to zero in our analysis. As for the non-circular solutions, flat priors were set on e and on ω instead of fitting for the Laplace parameters $e\cos (\omega )$ and $e\sin (\omega )$ because these imply implicit priors on the parameters that we want to avoid (Anglada-Escudé et al. 2013). For the lightcurve modeling, we used the selective resampling technique described in Kipping (2010) in order to account for the 30 min cadence of the K2 photometry, which has as a consequence the smearing of the transit shape. In order to minimize the biases in the retrieved transit parameters we fit for the limb darkening coefficients in our analysis (see Espinoza & Jordán 2015). To decide which limb-darkening law to use, we apply the method described in Espinoza & Jordán (2016) which, through simulations and given the lightcurves properties, aids in selecting the best limb-darkening law in terms of both precision and bias using a mean-squared error (MSE) approach. In this case, the law that provides the minimum MSE is the quadratic law, and we use this law to parametrize the limb-darkening effect. In addition, the K2 photometry is not good enough to constrain the ingress and egress times because only two transits were observed in long-cadence mode, which provides poor phase coverage; this implies that the errors on a/R_* are rather large. Because of this, we took advantage of the stellar parameters obtained with our HARPS spectra, and derived a value for this parameter from them (see Sozzetti et al. 2007) of $a/{R}_{* }={54.83}_{-3.16}^{+2.19}$. This value was used as a prior in our joint analysis in the form of a Gaussian prior. We used the largest of the errorbars as the standard deviation of the distribution, which is centered on the quoted median value of the parameter.¹³ We tried both fitting a flicker-noise model and a white-noise model, but the flicker noise model parameters were consistent with no 1/f noise component, so the fit was finally obtained assuming white noise. 500 walkers were used to evolve the MCMC, and each one explored the parameter space in 2000 links, 1500 of which were used as burn-in samples. This gave a total of 500 links sampled from the posterior per walker, giving a total of 250,000 samples from the posterior distribution. These samples were tested to converge both visually and using the Geweke (1992) convergence test.

Figures 6 and 7 show close-ups to the phased photometry and radial velocities, respectively, along with the best-fit models for both circular (red, solid line) and non-circular (red, dashed line) fits obtained from our joint analysis of the data set. The lightcurve fits for both models are very similar, but in the RVs the differences are evident. In particular, the eccentric fit gives rise to a slightly smaller semi-amplitude than (yet, consistent with) the one obtained with the circular fit. For the eccentric fit, we obtain $e={0.096}_{-0.066}^{+0.089}$, $\omega ={53}_{-23}^{+17}$ deg and a semi-amplitude of $K={2.9}_{-1.0}^{+1.1}$ m s⁻¹. For the circular orbit, we find a semi-amplitude of $K={3.1}_{-1.1}^{+1.1}$ m s⁻¹. Since the differences on the lightcurves are very small, we analyze the likelihood function of the RV data to compare the models and decide which is preferred by the data. We obtain that both models are indistinguishable, with both the AIC (ΔAIC = 2) and BIC (ΔBIC = 2) values being ∼2. We thus choose the simpler model of those two, which is the circular model, and report the final parameters using this as our final model.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The resulting parameters of our fit are tabulated in Table 3. It is interesting to note that the RV semi-amplitude is inconsistent with zero by almost 3σ. Moreover, we are confident that those variations do not arise from activity as all the correlation coefficients we calculate between our RVs and the different activity indexes given in Table 1 give correlation coefficients which are consistent with 0 at ≈1σ, and all variations of the activity indices at the period and time of transit-center found for our target are consistent with flat lines. Interestingly, the radial-velocity semi-amplitude is large for a planetary radius of only ${R}_{{\rm{p}}}={2.23}_{-0.11}^{+0.14}{R}_{\oplus };$ the $K={3.1}_{-1.1}^{+1.1}$ m s⁻¹ semi-amplitude implies a mass of ${M}_{{\rm{p}}}={16.3}_{-6.1}^{+6.0}{M}_{\oplus }$, which at face value could be consistent with a rocky composition, a rare property for a Neptune-sized exoplanet such as K2-56b. We caution, however, that this interpretation has to be taken with care, as we have poor phase coverage on the "up" quadrature. We put these values in the context of discovered exoplanets of similar size in Section 4.

Table 3.  Orbital and Planetary Parameters for K2-56

Parameter	Prior	Posterior Value
Lightcurve parameters
P (days)	${ \mathcal N }(41.68,0.1)$	${41.6855}_{-0.0031}^{+0.0030}$
${T}_{0}\mbox{--}2450000$ (BJDTDB)	${ \mathcal N }(7151.90,0.1)$	${7151.9021}_{-0.0047}^{+0.0042}$
a/R⋆	${ \mathcal N }(54.83,3.16)$	${55.8}_{-3.3}^{+3.3}$
Rp/R⋆	${ \mathcal U }(0,0.1)$	${0.02204}_{-0.00057}^{+0.00058}$
i (deg)	${ \mathcal U }(80,90)$	89.55${}_{-0.14}^{+0.17}$
q1	${ \mathcal U }(0,1)$	${0.38}_{-0.16}^{+0.29}$
q2	${ \mathcal U }(0,1)$	${0.52}_{-0.30}^{+0.32}$
${\sigma }_{w}$ (ppm)	${ \mathcal J }(50,80)$	${55.00}_{-0.72}^{+0.73}$
RV parameters
K (m s−1)	${ \mathcal N }(0,100)$	${3.1}_{-1.1}^{+1.1}$
μ (km s−1)	${ \mathcal N }(-20.337,0.1)$	$-{20.33638}_{-0.00073}^{+0.00073}$
e	⋯	0 (fixed)
Derived parameters
Mp (M⊕)	⋯	${16.3}_{-6.1}^{+6.0}$
Rp (R⊕)	⋯	${2.23}_{-0.11}^{+0.14}$
ρp (g cm−3)	⋯	${7.89}_{-3.1}^{+3.4}$
$\mathrm{log}{g}_{{\rm{p}}}$ (cgs)	⋯	${3.50}_{-0.21}^{+0.14}$
a (au)	⋯	${0.241}_{-0.017}^{+0.019}$
Vesc (km s−1)	⋯	${30.2}_{-6.2}^{+5.3}$
Teq (K)
Bond albedo of 0.0	⋯	${546}_{-18}^{+19}$
Bond albedo of 0.75	⋯	${386}_{-12}^{+13}$

Note. Logarithms given in base 10. ${ \mathcal N }(\mu ,\sigma )$ stands for a normal prior with mean μ and standard-deviation σ, ${ \mathcal U }(a,b)$ stands for a uniform prior with limits a and b and ${ \mathcal J }(a,b)$ stands for a Jeffrey's prior with the same limits.

Download table as:  ASCIITypeset image

To validate the planet scenario which we have implied in the previous sub-section, we make use of the formalism described in Morton (2012) as implemented on the publicly available vespa ¹⁴ package. In short, vespa considers all the false-positive scenarios that might give rise to the observed periodic dips in the light curve and, using photometric and spectroscopic information of the target star, calculates the false-positive probability (FPP) which is the complement of the probability of there being a planet given the observed signal. Because our archival and modern imaging presented on Section 2.4 rule out any companion at distances larger than 9'' radius, we consider this radius in our search for possible false-positive scenarios using vespa, which considers the area around the target star in which one might suspect false-positives could arise. The algorithm calculates the desired probability as

$$\begin{eqnarray*}&&{\rm{FPP}}=\displaystyle \frac{1}{1+{f}_{{\rm{p}}}P},\end{eqnarray*}$$

where f_p is the occurrence rate of the observed planet (at the specific observed radius) and P = L_TP/L_FP, where TP indicates the transiting-planet scenario and FP the false-positive scenario, and each term is defined as ${L}_{i}={\pi }_{i}{{ \mathcal L }}_{i}$, where π_i is the prior probability and ${{ \mathcal L }}_{i}$ is the likelihood of the ith scenario. For our target, considering all the information gathered and the fact that no secondary eclipse larger than ≈165 ppm (i.e., 3σ) is detected, we obtain a value of P = 4288.79. As for the occurrence rate of planets like the one observed, we consider the rates found by Petigura et al. (2013) for planets between 2 and 2.83 R_⊕ with periods between 5 and 50 days orbiting solar-type stars, which is 7.8%, i.e., f_p = 0.078. This gives us a false-alarm probability of FPP = 3 × 10⁻³. Given that this probability is smaller than the usual 1% threshold (e.g., Montet et al. 2015), we consider our planet validated. We note that this FPP is an upper limit on the real FPP given our AO and lucky-imaging observations. Both observations rule out an important part of the parameter space for blending scenarios between 02 and 5'' from the star, which are the main source of false-positives for our observations.

As will be discussed in the next section, both the planet radius and mass puts K2-56 in a very interesting part of the mass–radius diagram. Therefore, it is important to discuss the constraints that our spectroscopic and new, AO and lucky imaging observations pose on possible background stars that might dilute the transit depth and thus cause us to underestimate the transit radius.

Given that the factor by which the planetary radius is changed by a collection of stars inside the aperture used to obtain the photometry of the target star is given by $\sqrt{1/{F}_{ \% }}$, where F_% is the fraction of the total flux in the aperture added by the star being transited, we estimate that only stars with magnitude differences ≲2 are able to change the transit radius by magnitudes similar to the quoted uncertainties in Table 3. We note that such magnitude differences in the Kepler bandpass are ruled out from 02 to the aperture radius used to obtain the photometry for our target star: our AO and lucky imaging observations rule out companions of such magnitudes from 02 to 5'' (see Figure 5). On the other hand, stars with magnitude differences of that order should be evident on our retrieved archival and new images presented in Section 2.4, at least at distances of 5'' from our target star, and up to and beyond the 12'' aperture used to obtain the K2 photometry. Given that the remaining unexplored area on the sky is very small (only 02 around our target star), and that a star of such magnitude should produce an evident peak on the cross-correlation function on our high-resolution spectra, which is not seen, we confidently consider that our derived transit radius is unaffected by dilutions of background field stars.