Characterizing K2 Candidate Planetary Systems Orbiting Low-mass Stars. III. A High Mass and Low Envelope Fraction for the Warm Neptune K2-55b^*

K2-55 (EPIC 205924614) was observed by the NASA K2 mission during Campaign 3, which extended from 2014 November 14 until 2015 February 3. Like the majority of K2 targets, K2-55 was observed in long-cadence mode using 30-minute integrations. The K2 photometry of K2-55 is publicly available on MAST.¹⁸

Although subsequent spectroscopic analyses have revealed that K2-55 is a dwarf star, the target was initially proposed by Dennis Stello on behalf of the KASC Working Group 8, the astroSTEP and APOKASC collaborations, and the GALAH team. Interestingly, K2-55 was not included in guest observer proposals focused on dwarf stars. For more details about the inclusion or exclusion of K2-55 in various K2 guest observer proposals, see the Appendix.

The K2 mission does not provide official lists of planet candidates, but K2-55b was detected by multiple teams using independent pipelines. The candidate was initially reported by Vanderburg et al. (2016) as a 4.4 $\,{R}_{\oplus }$ planet in a 2.8 day orbit around a 4237 K star with a radius of roughly 0.65 ${R}_{\odot }$. Vanderburg et al. (2016) calculated the stellar properties using the $V-K$ color–temperature relation from Boyajian et al. (2013) and flagged the star as a possible giant.

Schmitt et al. (2016) also reported the discovery of K2-55b as PHOI-3 b, a transiting planet with a planetary/stellar radius ratio of ${0.0574}_{-0.0010}^{+0.0032}$ and an orbital period of 2.8 days. Schmitt et al. (2016) did not characterize the host star and therefore did not report a physical planet radius for PHOI-3 b. They did obtain Keck/NIRC2 imaging to search for nearby stellar companions and reported a lack of stellar companions between 025 and 200 from the target with sensitivities of Δm = 4.00 and Δm = 6.07, respectively.

K2-55b was also detected by Barros et al. (2016), who reported transit events with a depth of 0.372% and a total duration of 2.093 hr, and by Crossfield et al. (2016). In addition to rediscovering the planet, Crossfield et al. (2016) used the VESPA framework (Morton 2012, 2015) to validate K2-55b as a bona fide planet with a radius of 3.82 ± 0.32 $\,{R}_{\oplus }$. The Crossfield et al. (2016) false-positive analysis incorporated K-band high-contrast imaging acquired with Keck/NIRC2 and high-resolution spectra obtained with Keck/HIRES that restricted the possibility of stellar blends. Specifically, the AO imagery ruled out the presence of stars Δm_Ks = 8 fainter than K2-55 at a separation of 05 and Δm_Ks = 9 fainter at a separation of 1''. Similarly, a spectroscopic search for secondary stellar lines in the Keck/HIRES spectra (Kolbl et al. 2015) placed a limit of 1% on the brightness of any secondary stars within 04. Overall, Crossfield et al. (2016) calculated a false-positive probability (FPP) of 1.7 × 10⁻⁹, well below their adopted validation threshold of FPP < 1%.

In their analysis, Crossfield et al. (2016) assumed R_⋆ =0.630 ± 0.050 $\,{R}_{\odot }$, M_⋆ = 0.696 ± 0.047 $\,{M}_{\odot }$, and T_eff =4456 ± 148 K. These initial estimates were based on the optical and near-infrared photometry available in the Ecliptic Plane Input Catalog (EPIC, Huber et al. 2016).

Martinez et al. (2017) and Dressing et al. (2017a) later revised the classification of K2-55 by acquiring near-infrared spectra at NTT/SOFI (R ≈ 1000) and IRTF/SpeX (R ≈ 2000), respectively. Dressing et al. (2017a) classified the star as a K7 dwarf with ${R}_{\star }={0.715}_{-0.040}^{+0.043}\,{R}_{\odot }$, M_⋆ = 0.688 ± 0.015 ${M}_{\odot }$, and ${T}_{\mathrm{eff}}={4300}_{-100}^{+107}$ K. Martinez et al. (2017) reported consistent but less precise parameters of R_⋆ = 0.769 ± 0.063 ${R}_{\odot }$, M_⋆ = 0.785 ± 0.059, and T_eff = 4240 ± 259 K. These temperature constraints are consistent with the estimate of 4422 K from Gaia DR2 (Gaia Collaboration et al. 2016, 2018). For the remainder of this paper, we adopt the stellar classification from Dressing et al. (2017a) with the larger mass error of ±0.069 $\,{M}_{\odot }$ reported by Dressing et al. (2017b). Note that this revised stellar radius is 13% larger than the value used in Crossfield et al. (2016), suggesting that the planet is larger than previously reported by Crossfield et al. (2016).

After classifying cool dwarfs hosting K2 candidate planetary systems in Dressing et al. (2017a), we combined our revised stellar classifications with new transit fits of the K2 photometry to produce a catalog of planet properties for K2 cool dwarf systems. As explained in Dressing et al. (2017b), we estimated the planet properties by using the BATMAN Python package (Kreidberg 2015) to generate a transit model based on the formalism presented in Mandel & Agol (2002). We then estimated the errors on planet properties by running a Markov chain Monte Carlo analysis using the emcee Python package (Goodman & Weare 2010; Foreman-Mackey et al. 2013).

During the transit analysis, we varied the orbital period (P), the time of transit (T_C), the planet-to-star radius ratio (R_p/R_⋆), the scaled semimajor axis (a/R_⋆), the inclination (i), the eccentricity (e), the longitude of periastron (ω), and two quadratic limb-darkening parameters (u₁ and u₂). We fit for $\sqrt{e}\cos \omega$ and $\sqrt{e}\sin \omega$ to increase the efficiency of sampling low-eccentricity orbits (e.g., Eastman et al. 2013) and projected the limb-darkening parameters into the q₁ − q₂ coordinate-space proposed by Kipping (2013). We also incorporated our knowledge of the stellar density by including a prior on the scaled semimajor axis (Seager & Mallén-Ornelas 2003; Sozzetti et al. 2007; Torres et al. 2008).

In order to reduce the likelihood of systematic biases in our planet properties, we fit the K2 photometry returned by three different data reduction pipelines. First, we analyzed the photometry returned by the K2SFF pipeline (Vanderburg & Johnson 2014; Vanderburg et al. 2016) and found a planet/star radius ratio of ${R}_{p}/{R}_{\star }={0.056}_{-0.001}^{+0.002}$. Next, we re-fit the transit parameters using photometry reduced with the K2SC pipeline (Aigrain et al. 2016) and the k2phot pipeline (Petigura et al. 2015). In both cases, we found consistent planet/star radius ratios of ${R}_{p}/{R}_{\star }={0.056}_{-0.001}^{+0.002}$ and ${R}_{p}/{R}_{\star }={0.055}_{-0.001}^{+0.002}$, respectively.

All of these values are in agreement with the previous estimate of R_p/R_⋆ = 0.0552 ± 0.0013 (Crossfield et al. 2016), which was based on fits to the k2phot photometry. Vanderburg et al. (2016) and Schmitt et al. (2016) found larger (but also consistent) values of ${R}_{p}/{R}_{\star }=0.05814$ and ${R}_{p}/{R}_{\star }\,={0.0574}_{-0.0019}^{+0.0032}$, respectively.

Combining the stellar radius of ${R}_{\star }={0.715}_{-0.040}^{+0.043}\,{R}_{\odot }$ (Dressing et al. 2017a) with the planet-to-star radius ratio of ${R}_{p}/{R}_{\star }={0.056}_{-0.001}^{+0.002}$ yields a planet radius of ${4.38}_{-0.25}^{+0.29}\,{R}_{\oplus }$ (Dressing et al. 2017b). Our estimate is consistent with the radius of 4.63 ± 0.40 $\,{R}_{\oplus }$ estimated by Martinez et al. (2017), but significantly larger than the value of 3.82 ± 0.32 $\,{R}_{\oplus }$ found by Crossfield et al. (2016). We attribute the planet radius discrepancy to differences in the assumed stellar radius; the revised estimates determined by Martinez et al. (2017) and Dressing et al. (2017a, 2017b) were larger than the value assumed by Crossfield et al. (2016).

In order to refine the transit ephemeris estimated from the K2 data, we observed an additional transit of K2-55b using the Infrared Array Camera (IRAC) on the Spitzer Space Telescope (GO 11026, PI Werner). We began monitoring K2-55 at BJD = 2457430.636 (2016 February 12) and collected data points every 12 s until BJD = 2457430.891 for a total observation period of 6.1 hr. Based on our previous analysis of the K2 photometry, we expected that K2-55b would begin transiting 2 hr into the requested observation window and finish egress 1.9 hr later. Our planned observation therefore included 4.2 hr of out-of-transit flux baseline to aid our analysis of the transit event.

Prior to beginning our science observations, we obtained 30 min of "pre-observation" data to allow the telescope temperature to stabilize after slewing from the preceding target (Grillmair et al. 2012). We conducted these pre-observations in peak-up mode using the Pointing Calibration and Reference Sensor to improve the positioning of K2-55 during our science observations. For both sets of observations, we elected to conduct observations in Channel 2 (4.5 μm) rather than Channel 1 (3.6 μm) due to the lower amplitude of intra-pixel sensitivity variations visible in Channel 2 data (Ingalls et al. 2012).

Between 2016 August 12 and 2016 December 25, we obtained 12 observations of K2-55 using the HIRES (Vogt et al. 1994) on the 10 m Keck I Telescope on the summit of Maunakea. HIRES is a slit-fed spectrograph and a demonstrated single-measurement precision of approximately 1.5 m s⁻¹ for observations with a signal-to-noise ratio (S/N) of 200 and 1 m s⁻¹ for S/N of 500 (Fischer et al. 2016). Although the spectrometer has a wavelength range of 364–797 nm, we restricted our radial velocity (RV) analysis to the 510–620 nm region covered by the iodine reference cell, which was mounted in front of the spectrometer entrance slit for all of our RV observations. Following standard California Planet Search (CPS) procedures (Howard et al. 2010), we obtained our RV observations using the "C2" decker (087 ×14'' slit) for a spectral resolution of 55,000. We terminated the exposures after 45 minutes giving an S/N pixel⁻¹ = 60–90 near 550 nm, depending on sky conditions.

On 2016 September 22, we also obtained a higher resolution "template" observation with the iodine cell removed to aid in the process of separating the stellar and iodine spectra. Our template observation was taken using the "B3" decker (057 × 14'' slit) to reach a higher resolution of roughly 70,000. As in previous CPS publications, we determined RVs by forward-modeling the iodine-free template spectra, a high-quality iodine transmission spectrum, and the instrumental response (Marcy & Butler 1992; Valenti et al. 1995; Butler et al. 1996; Howard et al. 2009). We present the measured RVs and uncertainties in Table 1.

We considered a variety of fixed and variable apertures when extracting the photometric light curves from the Spitzer observations. Our investigation was motivated by previous Spitzer analyses demonstrating that a wise choice of extraction aperture can minimize the scatter and red-noise component of the resulting residuals (Knutson et al. 2012; Lewis et al. 2013; Todorov et al. 2013; Kammer et al. 2015; Benneke et al. 2017). As in earlier studies, our full photometry extraction procedure included determining and removing the sky background, estimating the position of the star on the detector array using flux-weighted centroiding, and then summing the total flux within particular circular apertures.

In the fixed case, we tested 36 aperture radii spanning the range between 1.5 pixels and 5 pixels at 0.1 pixel spacing. When exploring time variable apertures, we began by determining the scaling of the noise pixel parameter $\beta \,={({\displaystyle \sum }_{n}{I}_{n})}^{2}/({\sum }_{n}\,{I}_{n}^{2})$, where I_n is the intensity measured in pixel n (Mighell 2005). We then rescaled the noise pixel aperture radius as $r=a\sqrt{\beta }+c$, where we considered scaling factors 0.6 ≤ a ≤ 1.2 and shifts −0.8 ≤ c ≤ 0.4.

We also investigated whether binning the data before fitting would improve performance. For each choice of aperture, we generated eight binned versions of K2-55 photometry using between 2 and 9 points per bin for effective integration times of 24–108 s. We analyzed these data sets along with the unbinned data.

Finally, we experimented with trimming the data. As noted by Chen et al. (2018), proper trimming of pre- and post-transit data can improve the quality of fits to Spitzer data exhibiting curved systematics. We considered 21 possible trim durations ranging between 0 hr and 1 hr at either end of the light curve and allowed the ending trim duration and the starting trim duration to assume different values.

We selected the ideal binning, aperture, and pre- and post-transit trimming by fitting the Spitzer light curve using the full range of parameter choices and inspecting each fit. After extracting and trimming each light curve, we fit the systematics and transit signal as described in Section 4.2, rebinned the residuals in progressively larger bins, checked the scaling of the noise with increasing bin size, and assessed how well each fit reproduced the expected square-root noise scaling. We performed this initial parameter exploration using an eccentric model for the orbit of K2-55b (e = 0.125, ω = 196°), but the estimated transit properties are nearly identical for eccentric and circular orbits (see Section 4.3 and Table 2).

Table 2.  Transit and Systematic Parameters from the Photometric Analysis

		Model
Parameter	Units	Spitzer Circular	Spitzer+K2 Circular	Spitzer+K2 Fixed e	Spitzer+K2 Variable e
T0a	days	$-6.13\times {10}^{-5}{}_{-0.0012}^{+0.0013}$	7.96 × 10−5 ± 0.00019	$-1.83\times {10}^{-5}{}_{-0.00024}^{+0.00022}$	$2.27\times {10}^{-13}{}_{-0.00024}^{+0.00021}$
P	days	2.849274 (fixed)	${2.84927265}_{-6.42\times {10}^{-6}}^{+6.87\times {10}^{-6}}$	${2.84927261}_{-6.38\times {10}^{-6}}^{+6.94\times {10}^{-6}}$	${2.84927252}_{-6.60\times {10}^{-6}}^{+7.01\times {10}^{-6}}$
Rp/R⋆,K2	⋯	⋯	${0.0559}_{-0.0012}^{+0.0030}$	${0.559}_{-0.0011}^{+0.0029}$	${0.0561}_{-0.0013}^{+0.0031}$
${R}_{p}/{R}_{\star ,S}$	⋯	${0.0562}_{-0.0025}^{+0.0030}$	${0.0557}_{-0.0023}^{+0.0022}$	${0.0557}_{-0.0023}^{+0.0022}$	${0.0557}_{-0.0022}^{+0.0023}$
$a/{R}_{\star }$	⋯	${9.53}_{-3.06}^{+1.54}$	${10.55}_{-1.38}^{+0.64}$	${10.86}_{-1.37}^{+0.64}$	${10.50}_{-1.37}^{+1.14}$
i	deg	${86.82}_{-4.07}^{+2.26}$	${88.05}_{-1.75}^{+1.36}$	${88.17}_{-1.62}^{+1.27}$	${87.98}_{-1.70}^{+1.33}$
σK2	⋯	⋯	${0.000167}_{-3.7\times {10}^{-6}}^{+3.9\times {10}^{-6}}$	${0.000167}_{-3.7\times {10}^{-6}}^{+3.9\times {10}^{-6}}$	${0.000167}_{-3.8\times {10}^{-6}}^{+3.9\times {10}^{-6}}$
σS	⋯	${0.0024}_{-7.3\times {10}^{-5}}^{+7.7\times {10}^{-5}}$	${0.0024}_{-7.3\times {10}^{-5}}^{+7.8\times {10}^{-5}}$	${0.0024}_{-7.2\times {10}^{-5}}^{+7.6\times {10}^{-5}}$	${0.0024}_{-7.1\times {10}^{-5}}^{+7.5\times {10}^{-5}}$
$\sqrt{e}\sin \omega$	⋯	⋯	⋯	−0.10 (fixed)	$-{0.08}_{-0.18}^{+0.21}$
$\sqrt{e}\cos \omega$	⋯	⋯	⋯	−0.34 (fixed)	$-{0.29}_{-0.09}^{+0.12}$
e	⋯	⋯	⋯	0.125 (fixed)	${0.127}_{-0.055}^{+0.057}$
ω	rad	⋯	⋯	−2.86 (fixed)	$-{2.88}_{-0.64}^{+0.61}$

Note.

^aFor ease of comparison, we display the time of transit center minus BKJD = 2150.42286667.

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For the remainder of the paper, we investigate the light curve produced using the best combination of fit parameters: a fixed aperture radius of 2.7 pixels, a binning of 3 points per bin (36 seconds per bin), a starting trim of 0.3 hr, and no trimming at the end of the light curve. As shown in Figure 1, this light curve has the lowest red-noise component and the lowest scatter of all of the light curves we considered.

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We analyzed our Spitzer data using the pixel-level decorrelation (PLD) technique first introduced by Deming et al. (2015) and later modified by Benneke et al. (2017). Specifically, we modeled the observed flux D(t_i) at each timestamp t_i as the multiplicative combination of a sensitivity function S(t_i) and a transit model f(t_i). We then maximized the likelihood

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \sum _{i=1}^{N}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^{2}}}\exp \left(-\displaystyle \frac{{(D({t}_{i})-S({t}_{i})\cdot f({t}_{i}))}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 1 }$$

where σ is a photometric scatter parameter fit simultaneously with S(t_i) and f(t_i). We allowed σ to vary between 0.00001 and 0.3. For the instrument model S(t_i), we assumed that the sensitivity can be described by the linear combination of the raw counts D_k (t_i) of each pixel k within a 5 × 5 pixel region centered on the star and a linear ramp with slope m:

$$\begin{eqnarray}&&S({t}_{i})=\displaystyle \frac{{\displaystyle \sum }_{k=1}^{25}{w}_{k}{D}_{k}({t}_{i})}{{\displaystyle \sum }_{k=1}^{25}{D}_{k}({t}_{i})}+m\cdot {t}_{i},\end{eqnarray} \tag{ 2 }$$

where the w_k are the time-independent PLD weights given to each pixel.

We generated the transit model f(t_i) by using the BATMAN python package (Kreidberg 2015) to solve the equations of Mandel & Agol (2002). Unlike the K2 photometry, our Spitzer time series contains only a single transit event. We therefore fixed the orbital period to that found by Dressing et al. (2017b) and fit for the transit midpoint T₀, planet/star radius ratio R_p/R_⋆, scaled semimajor axis ratio a/R_⋆, and orbital inclination i. For our adopted model, we assumed that K2-55b had a circular orbit based on our analysis of the RV data (see Section 5), but we note that this choice does not significantly alter the transit profile. We estimated quadratic limb-darkening coefficients in the Spitzer bandpass by interpolating the values tabulated by Claret & Bloemen (2011). Accordingly, we set the coefficients to u₁ = 0.0824 and u₂ = 0.1531. We restricted the orbital inclination to 70° < i < 90° and required that the transit midpoint fall within the Spitzer data set.

In addition to verifying the orbital ephemeris predicted from the K2 data, our Spitzer data also provide an opportunity confirm the depth of the transit event. In Figure 2, we compare the planet/star radius ratios estimated from our independent fits to the K2 and Spitzer data. Although we find tighter radius ratio constraints from the K2 data (${R}_{p}/{R}_{\star }={0.056}_{-0.001}^{+0.002}$) than from the Spitzer data (${0.0562}_{-0.0025}^{+0.0030}$), our results are nearly identical. Table 2 contains all of the model parameters from the Spitzer-only fit.

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After fitting the Spitzer photometry separately, we conducted a joint fit of the Spitzer and K2 photometry to further contrain the planet parameters. For our joint fit, we used fixed quadratic limb-darkening parameters set by consulting the limb-darkening tables in Claret & Bloemen (2011). Specifically, we adopted u₁ = 0.7306 and u₂ = 0.0338 for the Kepler bandpass and u₁ = 0.0824 and u₂ = 0.1531 for the Spitzer bandpass. These values are the parameters estimated by Claret & Bloemen (2011) for a 4250 K star with log g = 4.5 and [Fe/H] = 0.3.

The free parameters in our joint fit were the orbital period P, the transit midpoint T₀, the planet/star radius ratio in both the Spitzer and K2 bandpasses (R_p/R_⋆,Spitzer, R_p/R_⋆,K2), the scaled semimajor axis ratio a/R_⋆, the orbital inclination i, and two photometric scatter terms (σ_Spitzer, σ_K2). As for the Spitzer-only fit, we assumed a circular orbit for K2-55b based on our analysis of the RV data. For comparison, we repeated the analysis using an eccentric orbit (e = 0.125, ω = 196°) and found little variation in the resulting parameters. We also ran a third analysis in which we used the results of our RV analysis to impose Gaussian priors on e and ω and allowed the parameters to vary. All three fits yield consistent planet properties and R_p/R_⋆,Spitzer = 0.056 ± 0.002 in all cases.

We adopt the circular fit as our chosen model and display the results in Figure 3. We also summarize the results in Table 2. The residuals to the full fit follow Gaussian distributions with a median value of −1.1 × 10⁻⁵ and a standard deviation of 0.00017 for the K2 data and 0.0001 and 0.0024, respectively, for the Spitzer data. The primary benefit to analyzing the Spitzer data along with the K2 data is that the errors on the transit midpoint and period decreased by factors of 1.9 and 4.0 compared to analyzing the K2 data alone. Accordingly, the uncertainty on the transit midpoint for an observation in late 2020 has decreased from 30 minutes to 7 minutes, significantly reducing the amount of telescope time needed to ensure that the full transit is observed.

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We tested the influence of our choice of limb-darkening parameters by repeating the variable eccentricity analysis using two different sets of limb-darkening parameters. In particular, we considered one set of alternative parameters corresponding to a 4000 K star with log g = 4.0 and [Fe/H] = 0.2 (u_1,Kepler =0.7858, u_2,Kepler = −0.0163, u_1,Spitzer = 0.0827, u_2,Spitzer =0.1443) and a second set corresponding to a 4500 K star with log g = 5.0 and [Fe/H] = 0.5 (u_1,Kepler = 0.6895, ${u}_{2,{\text{}}{Kepler}}\,=0.067$, ${u}_{1,{Spitzer}}=0.0791$, ${u}_{2,{Spitzer}}=0.1594$). Regardless of our specific choice of limb-darkening parameters, we found consistent results for the planet properties.

Once we had determined the best-fit system parameters, we checked for transit timing variations (TTVs) by inspecting each transit event individually. Specifically, we found the transit midpoints that minimized the difference between the observed data points and the best-fit transit model. We then rescaled the errors so that the reduced χ² was equal to unity and slid the transit model along until the χ² increased by 1. As shown in Figure 4, the transit midpoints we measured for the 24 transits visible in the K2 data are consistent with a linear ephemeris. Although there is a hint of curvature, fitting the transit times with a quadratic ephemeris does not improve the fit enough to justify the introduction of additional free parameters (ΔBIC = 10). Accordingly, we expected that our prediction of the Spitzer transit midpoint would be accurate to within a few hours even in the worst-case scenario. Indeed, our Spitzer-only fit yielded a transit midpoint of BJD = 2457430.75882 within one minute (<1σ) of our predicted value of BJD = 2457430.75902.

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Combining our photometrically derived planet radius estimate of ${4.41}_{-0.28}^{+0.32}\,{R}_{\oplus }$ with our RV mass constraint of ${43.13}_{-5.80}^{+5.98}\,{M}_{\oplus }$, we find that K2-55b has a bulk density of ${2.8}_{-0.6}^{+0.8}$ g cm⁻³. Although K2-55 is only 14% larger than Neptune ($3.87\,{R}_{\oplus }$) and 11% larger than Uranus ($3.98\,{R}_{\oplus }$), it is significantly more massive than either ice giant: K2-55b (${43.13}_{-5.80}^{+5.98}\,{M}_{\oplus }$) is 2.5 times as massive as Neptune ($17.15\,{M}_{\oplus }$), three times as massive as Uranus ($14.54\,{M}_{\oplus }$), and nearly half the mass of Saturn ($95.16\,{M}_{\oplus }$). As a result, the bulk density of K2-55b (${2.8}_{-0.6}^{+0.8}$ g cm⁻³) is 120% and 71% higher than the densities of Uranus (1.271 g cm⁻³) and Neptune (1.638 g cm⁻³), respectively. The interior structure of K2-55b is therefore quite distinct from that of the ice giants in our solar system. Despite the similar sizes of all three planets, K2-55b must have a lower fraction of volatiles or ices than either Uranus or Neptune.

In order to better compare K2-55b to other exoplanets, we queried the Confirmed Planets table from the NASA Exoplanet Archive²⁰ (Akeson et al. 2013) and selected all planets orbiting single stars²¹ with densities measured to better than 50% as of 2018 March 28. In Figure 6, we place K2-55b and the other well-constrained planets on the mass–radius diagram. K2-55b resides near several other planets with masses $\gt 30\,{M}_{\oplus }$ and radii $\lt 6\,{R}_{\oplus }$: K2-27 b (Van Eylen et al. 2016; Petigura et al. 2017), K2-39 b (Van Eylen et al. 2016; Petigura et al. 2017), K2-98 b (Barragán et al. 2016), K2- 108 b (Petigura et al. 2017), Kepler-101 b (Bonomo et al. 2014), and WASP-156 b (Demangeon et al. 2018). All of these planets orbit stars that are hotter and more massive than K2-55. The coolest host stars are K2-39 (${T}_{\mathrm{eff}}=4912$ K), an evolved star with a radius of $2.93\,{R}_{\odot }$, and WASP-156 (${T}_{\mathrm{eff}}=4910$ K), a metal-rich K3 star with [Fe/H] = 0.24 ± 0.12. K2-55 stands out as the smallest, lowest mass host star harboring a massive transiting planet (${M}_{p}\gt 30\,{M}_{\oplus }$).

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The density of K2-55b (${2.8}_{-0.6}^{+0.8}$ g cm⁻³) is intermediate between the values expected for terrestrial planets and gas giants, suggesting that K2-55b has a heterogeneous composition containing both heavy elements and low-density volatiles. Accordingly, we model K2-55b as a two-layer planet consisting of a rocky core capped by a low-density H/He envelope. We note that that K2-55b might also contain ices (Rogers et al. 2011), but variations in the core water abundance of Neptune-sized planets have a negligible influence on the radius-composition relation compared to changes in the H/He envelope fraction. (Lopez & Fortney 2014). Furthermore, the degeneracies between icy interiors and rocky interiors are impossible to break with mass and radius measurements alone (Adams et al. 2008; Figueira et al. 2009).

For our two-layer model, we use the internal structure and thermal evolution models developed by Lopez & Fortney (2014), who generated an ensemble of model planets spanning a variety of planet masses (M_p), envelope fractions (${M}_{\mathrm{env}}/{M}_{p}$), and planet insolation flux (F_p). Lopez & Fortney (2014) then evolved the planets forward in time and tracked the evolution of the planet radii. The resulting grid of planet masses, radii, envelope fractions, insolation fluxes, and ages has been used to infer the compositions of a multitude of planets (e.g., Wolfgang & Lopez 2015). The studies most germane to our analysis of K2-55b are those of Petigura et al. (2016, 2017), who employed the models of Lopez & Fortney (2014) to analyze a set of sub-Saturns. As defined by Petigura et al. (2016, 2017), "sub-Saturns" are planets with radii of $4\mbox{--}8\,{R}_{\oplus }$. At ${4.41}_{-0.28}^{+0.32}\,{R}_{\oplus }$, K2-55b could therefore be described as a "small sub-Saturn."

The Petigura et al. (2017) planet sample included 19 sub-Saturns with densities measured to precisions of 50% or better. Although tightly restricted in radius to $4.0\,{R}_{\oplus }\lt {R}_{p}\lt 7.8\,{R}_{\oplus }$, the Petigura et al. (2017) sub-Saturn sample spans a broad mass range of $4.8\mbox{--}69.9\,{M}_{\oplus }$ and a correspondingly large density range of 0.09–2.40 g cm⁻³. The observed masses and radii of the planets in their sample could be explained by envelope fractions of 7%–60% H/He by mass.

Interpolating the same Lopez & Fortney (2014) models to investigate the composition of K2-55b, we find that our estimated mass of ${43.13}_{-5.80}^{+5.98}$ and radius of ${4.41}_{-0.28}^{+0.32}\,{R}_{\oplus }$ are consistent with an envelope fraction of 12 ± 3%. This inferred envelope fraction is on the low end of the range observed by Petigura et al. (2017), underscoring the point that K2-55 has an exceptionally low gas fraction for its mass. K2-55b is denser than any of the planets in the Petigura et al. (2017) sample and more massive than all but 4 of the 19 planets they considered.

Considering all planets with $20\,{M}_{\oplus }\lt {M}_{p}\lt 100\,{M}_{\oplus }$ and radii of $3\,{R}_{\oplus }\lt {R}_{p}\lt 17\,{R}_{\oplus }$ (i.e., all of the planets in the right panel of Figure 6), we find that the median host star has an effective temperature of 5449 K and a mass of $0.99\,{M}_{\odot }$. The full range spans 3416–6270 K and $0.47\mbox{--}1.44\,{M}_{\odot }$. As shown in Figure 7, the only host star less massive than K2-55 is GJ 436, further emphasizing that K2-55b may be a curiously massive planet given the mass of its host star. Figure 7 also reveals that K2-55b is denser than all of the planets in the right panel of Figure 6. The combination of our high bulk density estimate for K2-55b (${2.8}_{-0.6}^{+0.8}$ g cm⁻³) and the high metallicity of K2-55 might suggest that K2-55b formed from a protoplanetary disk with an unusually deep reservoir of solid material.

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In general, Neptune-sized planets are more common than Jupiter-sized planets, but much rarer than smaller planets (e.g., Youdin 2011; Howard et al. 2012; Dressing & Charbonneau 2013; Fressin et al. 2013; Petigura et al. 2013; Fulton et al. 2017). Using the full Kepler data set and sub-dividing the stellar sample by spectral type, Mulders et al. (2015) estimated that planets with radii of $4\mbox{--}5.7\,{R}_{\oplus }$ and periods of 2.0–3.4 days occur at a rate of 0.00022 ± 0.00018 planets per F star, 0.0011 ± 0.0004 planets per G star, 0.0016 ± 0.0008 planets per K dwarf, and <0.0069 planets per M dwarf. The detection of K2-55b is therefore less remarkable for the low mass of the host star than for the intermediate size of the planet: close-in Neptunes seldom occur, regardless of host star spectral type.

The dependence of the hot Neptune occurrence rate on stellar metallicity is more complicated. The increased prevalence of gas giants orbiting metal-rich stars is well established (e.g., Gonzalez 1997; Santos et al. 2004; Fischer & Valenti 2005), but the role of metallicity on the occurrence rates of smaller planets is less well understood. Examining the Kepler planet sample, Buchhave et al. (2014) found that planets larger than $3.9\,{R}_{\oplus }$ orbit stars that are significantly more metal rich than the hosts of smaller planets. Buchhave et al. (2014) also noted that the host stars of 1.7–3.9 $\,{R}_{\oplus }$ planets are more metal rich than the host stars of smaller planets, but Schlaufman (2015) countered that the data are better described by a continuous gradient of increasing metallicity with increasing planet radius from 1 $\,{R}_{\oplus }$ to 4 $\,{R}_{\oplus }$ rather than a sharp metallicity jump at 1.7 $\,{R}_{\oplus }$.

In a related study, Wang & Fischer (2015) observed that planet occurrence is positively correlated with stellar metallicity independent of planet size. In particular, they found that metal-rich stars ([Fe/H] > 0.05) were ${9.30}_{-3.04}^{+5.62}$ times more likely than metal-poor stars ([Fe/H] $\lt -0.05$) to harbor planets with radii of 3.9–22 $\,{R}_{\oplus }$. The metallicity bias appears less pronounced for smaller planets (${2.03}_{-0.26}^{+0.29}$ for $1.7\,{R}_{\oplus }\lt {R}_{p}\lt 3.9\,{R}_{\oplus }$ and ${1.72}_{-0.17}^{+0.19}$ for ${R}_{p}\lt 1.7\,{R}_{\oplus }$), but the metallicity preference might be underestimated due to the observational bias against detecting transiting planets orbiting metal-rich stars due to the shallower transit depths caused by their larger radii.

Considering the possible interplay between planet occurrence, stellar metallicity, and orbital period, Mulders et al. (2016) found that short-period planets ($P\lt 10$ days) are biased toward metal-rich host stars ([Fe/H] ≃ 0.15 ± 0.05 dex) while longer period planets orbit stars with solar-like metallicities. While this trend toward higher stellar metallicities at shorter planet orbital periods is quite pronounced for the smallest planets (<1.7 $\,{R}_{\oplus }$), the trend disappears for larger planets: host stars of $3.9\mbox{--}14\,{R}_{\oplus }$ planets typically have super-solar metallicities of 0.14 ± 0.04 dex regardless of planet orbital period. Accordingly, the realization that the host star of Neptune-sized K2-55b is metal rich ([Fe/H] = 0.376 ± 0.095) would be unsurprising even if the planet had an orbital period significantly longer than the observed value of 2.8 days.

Concentrating on sub-Saturns, Petigura et al. (2017) tested several different theories to explain the large dispersion in planet mass, density, and envelope fraction. Petigura et al. (2017) noted that the envelope fractions of the hottest planets in their sample (${T}_{\mathrm{eq}}\gt 1250$ K) were restricted to a smaller range of $10 \% \lt {M}_{\mathrm{env}}/{M}_{p}\lt 30 \%$, while the cooler planets spanned the full estimated range from 10% to 60%. The lack of hot planets with larger envelope fractions might indicate that photoevaporation prevents close-in sub-Saturns from retaining large quantities of volatiles. However, photoevaporation could not be the only explanation for the observed diversity of sub-Saturn compositions because Petigura et al. (2017) did not observe a strong correlation between present-day planet equilibrium temperature and envelope fraction. In agreement with Petigura et al. (2017), the right panels of Figure 7 do not display a strong relationship between planet density and insolation flux. The most highly irradiated planet (KELT-11b, Pepper et al. 2017) has a bulk density of 0.93 g cm⁻³, but less strongly irradiated planets like K2-55b (${F}_{p}\lt 200\,{F}_{\oplus }$) span a wide range of densities from 0.09 to 2.2 g cm⁻³.

Similarly, Petigura et al. (2017) failed to detect a correlation between host star metallicity and envelope fraction, demonstrating that disk metallicity changes alone cannot explain the observed densities of sub-Saturns. The lack of a correlation between stellar metallicity and envelope fraction was slightly surprising because Thorngren et al. (2016) had previously noted an anticorrelation between planet metal abundance (approximated as ${Z}_{p}={M}_{\mathrm{core}}/{M}_{p}$) and planet mass for planets with masses of $30\mbox{--}3000\,{M}_{\oplus }$. The Petigura et al. (2017) planet sample included more lower mass planets than the original Thorngren et al. (2016) sample, which allowed Petigura et al. (2017) to learn that the previously detected anticorrelation does not appear to extend to planets with masses below $30\,{M}_{\oplus }$. Petigura et al. (2017) suggested that perhaps the extinction of the trend at lower masses is a manifestation of different formation pathways for gas giants and lower mass planets. K2-55b is a more massive sub-Saturn and falls nicely on the relation found by Thorngren et al. (2016) between planet metal enrichment relative to stellar metallicity (${Z}_{\mathrm{planet}}/{Z}_{* }$) and planet mass. Specifically, the Thorngren et al. (2016) relation predicts a planet metal enrichment ratio of ${Z}_{\mathrm{planet}}/{Z}_{* }=24$ for a $44\,{M}_{\oplus }$ planet, and the ratio for K2-55b is ${Z}_{\mathrm{planet}}/{Z}_{* }=26$.

The high planet mass of K2-55b and the super-solar metallicity of K2-55 are also consistent with the finding by Petigura et al. (2017) that stars with higher metallicities tend to host more massive sub-Saturns. The positive correlation between stellar metallicity and sub-Saturn mass may suggest that more massive planetary cores formed in more metal-rich protoplanetary disks (Petigura et al. 2017). As shown in the bottom panels of Figure 7, the densest sub-Saturns tend to orbit the most metal-rich host stars. This trend is particularly pronounced in the bottom right panel, which displays a clear separation between the denser sub-Saturns and the low-density larger planets.

Intriguingly, Petigura et al. (2017) also noted that more massive sub-Saturns tend to have moderately eccentric orbits and orbit stars without other detected planets, while less massive sub-Saturns tend to follow more circular orbits and reside in systems with multiple transiting planets. As a ${43.13}_{-5.80}^{+5.98}\,{M}_{\oplus }$ planet in a system with no other detected planets, K2-55b might therefore be expected to have an eccentric orbit. Additional observations are required to tighten the constraints on the orbital eccentricity of K2-55b and better distinguish between eccentric and circular models.

Under the core-accretion model of planet formation, planetesimals collide to form protoplanetary cores, which then acquire gaseous envelopes (Perri & Cameron 1974; Mizuno et al. 1978; Mizuno 1980; Stevenson 1982; Bodenheimer & Pollack 1986; Pollack et al. 1996). If the planet core is able to become sufficiently massive before the gaseous disk dissipates (at roughly a few Myr, Williams & Cieza 2011), then the growing planet can enter a phase of runaway accretion in which the envelope grows rapidly. The onset of the "core-accretion instability" occurs when the mass of the planetary core exceeds the "critical core mass," M_crit. While numerous studies have estimated M_crit as roughly $10\,{M}_{\oplus }$ (Ikoma et al. 2000, and references therein), Rafikov (2006, 2011) demonstrated that variations in the assumed disk properties and planetesimal accretion rate can alter M_crit by orders of magnitude, resulting in a wide range of $0.1\mbox{--}100\,{M}_{\oplus }$. In general, M_crit decreases with increasing distance from the star due to the cooler disk temperatures found in the outer disk. M_crit also decreases with increasing mean molecular weight, but this effect can be outbalanced by the stronger trend of increasing M_crit with increasing dust opacity (Hori & Ikoma 2011; Nettelmann et al. 2011; Piso & Youdin 2014).

Although the mass of K2-55b is below the upper end of the $0.1\,{M}_{\oplus }\lt {M}_{\mathrm{crit}}\lt 100\,{M}_{\oplus }$ range found by Rafikov (2006, 2011), the absence of a large volatile envelope for a ${43.13}_{-5.80}^{+5.98}\,{M}_{\oplus }$ planet is noteworthy and at odds with general expectations from core-accretion models. Naïvely assuming that K2-55b formed in situ at 0.0347 au in a minimum mass solar nebula (MMSN, Hayashi 1981) with a ${{\rm{\Sigma }}}_{p}=33{{FZ}}_{\mathrm{rel}}{r}^{-3/2}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ solid surface density profile, a total mass ratio F = 1, and a metal richness ${Z}_{\mathrm{rel}}=0.33$ (Chiang & Youdin 2010), there would have been only $0.01\,{M}_{\oplus }$ of solids available for building K2-55b. Adopting a more massive minimum mass extrasolar nebula (MMEN) solid surface density profile (Chiang & Laughlin 2013; Gaidos 2017) would yield roughly $0.16\,{M}_{\oplus }$ of solids. Although significantly higher than the estimate based on the MMSN, the solid mass locally available in the MMEN model is lower than 0.4% of the present-day mass of K2-55b, indicating that either K2-55b itself or the planetary building blocks that would become K2-55b (e.g., Hansen & Murray 2012; Chatterjee & Tan 2014) must have migrated inward from farther out in the disk.

Acknowledging the puzzling existence of a massive close-in planet with only a modest H/He envelope, we propose four possible formation scenarios for K2-55b:

• 1.  
  Classic type I migration into the inner disk cavity
• 2.  
  Collisions of multiple planets
• 3.  
  Post-formation atmospheric loss
• 4.  
  Formation via less efficient core accretion.

Under the first scenario, uneven torques from the disk on K2-55b would have caused the planet to drift inward toward the host star (Ward 1997; Tanaka et al. 2002). The type I migration²² would have been halted after K2-55b entered the inner cavity between the disk and the star. K2-55b would have therefore escaped runaway accretion because it was trapped at the 2:1 resonance with the disk inner edge (e.g., Kuchner & Lecar 2002) rather than embedded within the disk. Although feasible, this argument is unsatisfying due to the fine-tuning required to have K2-55b cross the disk edge after reaching a high overall mass but before accumulating a substantial envelope.

In the second scenario, K2-55b might have been formed via collisions of smaller planets. For instance, Boley et al. (2016) found that collisions of smaller planets in systems of tightly packed inner planets (STIPs) can produce gas-poor giant planets if the progenitor planets collide after the gas disk has dissipated. Another possible explanation is that the protoplanetary disk orbiting K2-55 might have been slightly misaligned with respect to the host star (e.g., Bate et al. 2010), which could have been orbited by several less massive planets. Once the gas in the disk had dissipated, the continued contraction of the star along the Hayashi track could have driven a resonance through the system (Spalding & Batygin 2016). The resonance would have perturbed the orbits of the smaller planets, causing them to collide with each other and form a more massive planet.

The primary challenge facing the second explanation is that collisional velocities close to the star at the present-day orbital location of K2-55b are high enough that collisions are more likely to result in fragmentation than growth (Leinhardt & Stewart 2012, but see Wallace et al. 2017). Unless the smaller planets collided farther out in the disk where collisional velocities were lower and the newly formed K2-55b subsequently migrated inward to 0.0347 au via planetesimal scattering, this scenario is unlikely to explain the formation of K2-55b. Alternatively, the presence of a gaseous envelope before the collision might have made the collision less destructive (e.g., Liu et al. 2015). The logical observational test for this scenario is to measure the spin–orbit alignment of the system via the Rossiter–McLaughlin effect (McLaughlin 1924; Rossiter 1924), but the host star is too faint to permit such a precise measurement with current facilities.

A third possibility is that K2-55b formed as a "regular" sub-Saturn with a typical envelope fraction but then lost most of its envelope to a single late giant impact (e.g., Inamdar & Schlichting 2015, 2016; Liu et al. 2015; Schlichting et al. 2015). More massive planets are less vulnerable to envelope loss via either photoevaporation or impacts (Lopez & Fortney 2013; Inamdar & Schlichting 2015), suggesting that a late giant impact could have had a more catastrophic effect for K2-55b than for a Saturn-mass planet.

Our fourth formation scenario for K2-55b is that the planet formed via "conventional" core accretion, but that our incomplete understanding of core accretion causes us to overestimate the efficiency of planet formation. We note that the relatively small envelopes of Uranus and Neptune mandate that the gas disk dissipated just after the planets reached their final masses (e.g., Pollack et al. 1996; Dodson-Robinson & Bodenheimer 2010) and that producing super-Earths rather than mini-Neptunes requires delaying planet formation until most of the gas is depleted (Lee et al. 2014; Lee & Chiang 2016). Alternatively, super-Earths might form in a gas-rich disk but with dust-rich atmospheres that delay cooling and prevent them from acquiring enough gas to trigger runaway accretion (Lee et al. 2014; Lee & Chiang 2015).

Instead of requiring that the gas in the K2-55 protoplanetary disk dissipated just as K2-55b was beginning to accrete an envelope, an alternative formation scenario is that K2-55b grew via pebble accretion (Lambrechts & Johansen 2012). As the pebbles accreted, they would have heated the growing planet and consequently turned to dust due to the high temperature of the atmosphere. The dusty atmosphere would have inhibited cooling and prevented K2-55b from accreting an envelope (Lega & Lambrechts 2016).

Although the pebble heating explanation is appealing, Lee & Chiang (2015) note that pebble accretion can block runaway accretion only for planets with low-mass cores (${M}_{\mathrm{core}}\lt 5\,{M}_{\oplus }$); a youthful version of K2-55b would be too massive to escape runaway gas accretion. Nevertheless, the modern high density of K2-55b might be attributed to gas-stealing late giant impacts (Inamdar & Schlichting 2015, 2016). If K2-55b actually has an eccentric orbit, tidal heating may have also warmed the planet and helped block runaway accretion (Ginzburg & Sari 2017). While the specific formation pathway for K2-55b is uncertain, the sheer variety of possible explanations demonstrates that further theoretical and observational work is required to better understand core accretion and planet formation in general. Studying additional planets in the same size range as K2-55b will help determine which scenario (or combination of scenarios) best explains the formation of dense Neptune-sized planets.

Although K2-55b alone cannot solve all of the mysteries of planet formation, determining the composition of the envelope may help constrain where and how K2-55b formed. At the most basic level, determining the mean molecular weight of the atmosphere would reveal whether our simplistic two-layer model of a rocky core surrounded by a H/He envelope is sufficient or whether K2-55b is better explained by a lower-density core containing a large admixture of ices and a higher-density water-rich envelope. More sophisticated measurements of the relative abundances of particular molecules would enable tests of the various formation scenarios outlined in Section 6.5 and perhaps spur the genesis of new formation scenarios. For instance, measuring a superstellar C/O ratio would provide further evidence that K2-55b formed beyond the snow line and subsequently migrated inward (Öberg et al. 2011). On the other hand, measuring a substellar C/O ratio could indicate that K2-55b formed inside the ice line (Mordasini et al. 2016).

Transmission spectra would also reveal whether the atmosphere of K2-55b is clear or shrouded by clouds or hazes. Morley et al. (2015) predicted a transition at equilibrium temperatures near 1000 K between predominantly hazy atmospheres for cooler planets and predominantly clear atmospheres for hotter planets. Crossfield & Kreidberg (2017) note that observations of warm Neptunes ($2\,{R}_{\oplus }\lt {R}_{p}\lt 6\,{R}_{\oplus }$, $500\,{\rm{K}}\,\lt {T}_{\mathrm{eff}}\lt 1000\,{\rm{K}}$) are consistent with this theory, but that the observations cannot yet differentiate between high mean molecular weight atmospheres and high-altitude clouds or hazes for the majority of planets with apparently featureless spectra. Furthermore, the Crossfield & Kreidberg (2017) sample contains only six warm Neptunes. K2-55b has an equilibrium temperature of roughly 900 K and would be an interesting addition to this small sample.

In order to test whether such observations might be feasible, we used the publicly available ExoTransmit package (Kempton et al. 2017) to generate model atmospheres for K2-55b. We considered a wide variety of atmospheric compositions with a range of C/O ratios. In all cases, the high surface gravity of K2-55b ($22\,{\rm{m}}\,{{\rm{s}}}^{-2}=2{g}_{\mathrm{Neptune}}$) muted the dynamic range of atmospheric features, rendering detailed atmospheric characterization challenging.

Overall, the full range of transit depths is expected to span approximately 150 ppm if the atmosphere has roughly solar composition. Increasing the C/O ratio of a solar metallicity model atmosphere from C/O=0.2 to C/O=1.2 would increase the transit depth by 50–100 ppm in the most informative regions (2–2.5 μm and 3–4 μm) and produce negligible effects elsewhere in the spectrum. Distinguishing between a water-dominated atmosphere and a carbon dioxide-dominated atmosphere would require detecting differences of roughly 20 ppm. Accordingly, the first-order investigation of whether the atmosphere of K2-55b has a low or high mean molecular weight would be relatively straightforward (assuming that the investigation is not foiled by clouds), but determining detailed molecular abundances would require a more significant investment of telescope time.

The atmosphere of K2-55b could also be probed during secondary eclipse. Assuming an albedo of 0.15 and an equilibrium temperature of 900 K, the estimated secondary eclipse depth is 140 ppm. This modest signal would be challenging to detect with Spitzer (S/N = 0.8), but would be detectable with MIRI on board the James Webb Space Telescope (S/N = 7–8). For reference, GJ 436b has a secondary eclipse depth of 155 ± 22 ppm at 3.6 μm (Morley et al. 2017), but GJ 436 (V = 10.613, Ks = 6.073) is significantly brighter than K2-55 (V = 13.55, Ks = 10.471).