K2-79b and K2-222b: Mass Measurements of Two Small Exoplanets with Periods beyond 10 days that Overlap with Periodic Magnetic Activity Signals

The HARPS radial velocity (RV) survey first revealed the existence of a subpopulation of super-Earth and Neptune-like planets in tight orbits (Mayor & Udry 2008; Lovis et al. 2009). A few years later, NASA's Kepler mission found that these small planets, with sizes between that of Earth and Neptune, are the most abundant type of exoplanets with periods less than ≈100 days (e.g., Borucki et al. 2011; Batalha et al. 2013; Fressin et al. 2013). These types of planets are not found in the Solar System, and their existence was not predicted by contemporary planet-formation models (Ida & Lin 2004a, 2004b; Mordasini et al. 2009a, 2009b). Subsequent studies, using a greater number of small exoplanets from Kepler, revealed relatively few planets with radii between 1.5 R_⊕ and 2 R_⊕, now commonly referred to as the radius valley (Fulton et al. 2017; Zeng et al. 2017a, 2017b; Van Eylen et al. 2018). There also appears to be a scarcity of exoplanets in the sub-Saturnian desert, with radii larger than 4 R_⊕ but smaller than a gas giant (Zeng et al. 2018). Exoplanets in the sub-Saturnian desert appear separated from sub-Neptunes by the radius cliff, a steep drop-off in exoplanet occurrence starting near 3 R_⊕ and attributed to the nonlinear solubility of hydrogen in magma at relatively high pressures (Kite et al. 2019).

The exoplanet community has increased efforts to measure masses of small exoplanets to develop a more complete picture of the processes involved in their formation. Bulk densities measured for exoplanets on both sides of the radius valley suggest that those above the gap have very different compositions from those below (Rogers 2015; Zeng et al. 2018, 2019). Exoplanets below the gap have relatively high densities, consistent with rocky compositions. Exoplanets above the gap have lower densities that are degenerate, consistent with rocky cores surrounded by envelopes of liquid and/or gas.

Several mechanisms have been proposed in the literature to explain the observed small-planet radius valley: photo-evaporation, internal heat-powered mass loss, and water-rich versus water-poor formation. Photo-evaporation refers to the shedding of a planet's envelope from prolonged exposure to its host star (Hansen & Murray 2012; Mordasini et al. 2012b, 2012a; Alibert et al. 2013; Chiang & Laughlin 2013; Chatterjee & Tan 2014; Coleman & Nelson 2014; Lee et al. 2014; Raymond & Cossou 2014; Lee & Chiang 2016). Like photo-evaporation, core-powered mass loss also causes an exoplanet to shed its envelope, but is instead powered by radiation from the hot rocky core as it cools and winds in the hot gaseous envelope (Owen & Wu 2016; Ginzburg et al. 2016). Successful models of photo-evaporation, core-powered mass loss, and atmospheric "boil off" thus far have assumed that all planets above the gap host hydrogen/helium (H or H/He) envelopes, excluding water from their simulations. However, simulations including water and ices have been carried out to form water-rich exoplanets with rocky or rocky/icy cores that accrete significant amounts of water and gaseous envelopes if massive enough (Zeng et al. 2018, 2019).

Comparing small planets of varying surface gravity across a range of stellar irradiation levels can help identify driving factors behind the observed population of small exoplanets (Lopez & Rice 2018; Cloutier & Menou 2020). So far, the majority of small planets with mass measurements occur at relatively short orbital periods (P_orb < 10 days), and therefore are subject to high irradiation levels. Mass and radius measurements of small planets at longer periods are therefore essential for developing a more complete understanding of small exoplanet formation.

Measuring small exoplanet masses at longer orbital periods presents challenges, including hard-to-detect small-amplitude Doppler signals, and potential interference between periodic signals from stellar activity and P_orb. The second problem is even more challenging than initially recognized because P_orb need not overlap perfectly with P_rot or one of its multiples/harmonics for the stellar activity and Doppler signals to interfere (Nava et al. 2020). Mortier et al. (2016) tackled the feat of determining the mass on an exoplanet with P_orb near P_rot, but generally these cases have been set aside to measure masses of exoplanets with less-challenging interference from stellar magnetic active regions.

In this paper we present a method to mitigate these potential interference effects when RV and activity-indicator observations are available for a target over multiple observing seasons. We apply our method to two exoplanets discovered by the K2 mission, K2-79b and K2-222b, whose orbital periods overlap with observed activity signals from their host stars. K2-79b was first detected and validated by Crossfield et al. (2016), and then redetected by Mayo et al. (2018) and Kruse et al. (2019). K2-222b was detected as a planet candidate by Mayo et al. (2018) and Petigura et al. (2018). It was validated by Mayo et al. (2018) and then later redetected by Livingston et al. (2018). Given their radii of 4.09 R_⊕ and 2.35 R_⊕, respectively, measuring masses for these two planets is particularly interesting, since they lie in the sub-Saturnian desert and near the upper edge of the radius valley where their compositions could be water rich or water poor.

In Section 2, we describe the data sets utilized in our analysis, and in Section 3 we detail the characterization of the two host stars. In Section 4 we describe our investigations of potential stellar activity signals in the light curve (LC) and RV data sets of each target. Section 5 describes our Markov Chain Monte Carlo (MCMC) fits to determine the exoplanet parameters. We report our results and discuss the scientific implications for K2-79b and K2-222b in Section 7.

2.1. K2 Photometry

We analyzed 3161 and 3424 photometric K2 observations of K2-79 and K2-222, respectively, collected in long-cadence (29.4 minutes) mode (Figures 1(a) and (b); Tables 1 and 2). K2-79 was observed between 2015 February 10 and 2015 April 20 and K2-222 between 2016 January 6 and 2016 March 23. The K2 spacecraft operated with only two of the four original reaction wheels, causing the spacecraft to drift over time, with intermittent thruster firings to keep desired targets in the telescope's field of view. This led to long-term trends in the LC of a given star, due to the stars drifting over time across pixels with different quantum efficiencies. We corrected for thruster firings first, followed by systematic instrumental effects and other low-frequency signals, according to the technique developed by Vanderburg & Johnson (2014). The top panels in Figures 1(a) and (b) show the LCs of each target after correcting for those effects. After deriving a first-pass systematics-corrected LC, we refined the correction by simultaneously fitting for spacecraft systematics, low-frequency variability, and exoplanet transits following Vanderburg et al. (2016). The bottom panels in the same figures show the resulting flattened LCs with exoplanet transits isolated. We set the error of each point in a given LC data set equal to the standard deviation of the out-of-transit points in the flattened LC. In the case of the brighter star, K2-222, the second half of the observations have increased scatter due to uncorrected K2 systematics. We therefore calculate errors for that LC in two segments separated at BJD–2400000 = 57,438 days, where the start of the increased scatter begins (see Figure 1(b), bottom panel).

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2.2. HARPS-N Spectroscopy

We collected 79 spectra of K2-79 over four seasons between 2015 November 4 and 2019 December 29 (Figure 2; Table 5), and 63 spectra of K2-222 over three seasons between 2016 August 14 and 2019 December 23 (Figure 3; Table 3). All spectra were collected with HARPS-N, the high-precision spectrograph mounted on the Telescopio Nationale de Galileo at the Observatorio del Roque de los Muchachos in La Palma, Spain (Cosentino et al. 2012). The spectrograph covers wavelengths in the range 383–690 nm, with average resolving power R = 115,000. Observations for K2-79 were taken with 30 minutes exposure times, yielding a mean signal-to-noise ratio (S/N) of 32.67 in order 60 (627.077–634.166 nm) of the spectrum. For K2-222, observation exposure times ranged between 15–30 minutes, yielding a mean S/N of 74.11 in order 60. The observations were collected as part of the HARPS-N collaboration's Guaranteed Time Observations (GTO) program.

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We reduced the spectra and calculated associated RVs and activity indicators using the HARPS-N data-reduction software (DRS; Lovis & Pepe 2007). Activity indicators calculated include the contrast, FWHM, cross-correlation function area (CCF area = contrast × FWHM), bisector inverse span (BIS), and S-index (Figures 2 and 3).

We derived stellar parameters for each target using the Stellar Parameter Classification Tool (SPC; Buchhave et al. 2012, 2014) to analyze the K2-79 and K2-222 HARPS-N spectra and estimate stellar effective temperature (T_eff), surface gravity ($\mathrm{log}g$), metallicity ([Fe/H]), and projected rotational velocity ($v\,\sin \,i$) for each target. Spectra with normalized peak heights of less than 0.9 in the CCF were discarded due to low S/Ns. We aligned all individual spectra to a common RV frame and calculated the maximum stellar rotation period associated with the lower limit of $v\,\sin \,i$ using the equation ${P}_{\mathrm{rot},\max }=2\pi {R}_{* }/v\,\sin \,i$. Final values from SPC are summarized in Table 6.

Table 6. Stellar Parameters for K2-79 and K2-222

Parameter	K2-79	K2-222	Source
EPIC ID	210402237	220709978
R.A. [deg]	55.255896	16.462277	Gaia Collaboration et al. (2021)
Decl. [deg]	13.519365	11.753428	"
Parallax [mas]	3.841 ± 0.015	10.00896 ± 0.06080	"
G	11.799 ± 0.0003	9.3659 ± 0.0003	"
B	12.89 ± 0.02	10.04 ± 0.03	Zacharias et al. (2012)
V	12.07 ± 0.06	9.54 ± 0.03	"
J	10.36 ± 0.02	8.42 ± 0.02	Cutri et al. (2003)
H	10.00 ± 0.02	8.17 ± 0.03	"
K	9.91 ± 0.02	8.11 ± 0.02	"
W1mag	9.83 ± 0.02	8.05 ± 0.02	Cutri et al. (2013)
W2mag	9.86 ± 0.02	8.10 ± 0.02	"
W3mag	9.80 ± 0.06	8.07 ± 0.02	"
Reddening (G-R)	${0.23}_{-0.02}^{+0.03}$	0.0 ${}_{-0.0}^{+0.01}$	Green et al. (2019)
Reddening (B-V)	${0.20}_{-0.02}^{+0.03}$	0.0 ${}_{-0.0}^{+0.01}$	Green et al. (2019)
Extinction, A(V)	${0.81}_{-0.09}^{+0.06}$	${0.03}_{-0.02}^{+0.05}$	Isochrones
Distance [pc]	260 ± 1	99.6 ± 0.2	Isochrones
Mass [M⊙]	1.06 ± 0.05	0.94 ± 0.05	Isochrones*
	0.992 ± 0.103	1.054 ± 0.104	Huber et al. (2016)
Radius [R⊙]	1.269 ± 0.051	1.072 ± 0.043	Isochrones*
	1.1 ± 0.4	1.1 ± 0.5	Huber et al. (2016)
Density [ρ⊙]	0.52 ± 0.02	${0.76}_{-0.04}^{+0.05}$	Isochrones*
	0.75 ± 0.30	0.79 ± 0.30	Huber et al. (2016)
Luminosity [L⊙]	${1.76}_{-0.07}^{+0.06}$	1.34 ± 0.04	Isochrones
Age [Gyr]	6.5 ± 1.3	${7.1}_{-1.7}^{+1.5}$	Isochrones
Eff. temp., Teff [K]	5897 ± 118	5942 ± 119	Combined—A+M & SPC*
	5901 ± 118	6002 ± 120	Isochrones
	5943 ± 119	6000 ± 120	A+M
	5851 ± 117	5883 ± 118	SPC
	5926 ± 86	5818 ± 136	Huber et al. (2016)
Metallicity [Fe/H]	0.035 ± 0.06	−0.315 ± 0.06	Combined—A+M & SPC*
	${0.05}_{-0.04}^{+0.05}$	−0.25 ${}_{-0.11}^{+0.04}$	Isochrones
	0.04 ± 0.04	−0.26 ± 0.04	A+M
	0.03 ± 0.08	−0.37 ± 0.08	SPC
Surface grav, $\mathrm{log}g$ [cgs]	4.26 ± 0.01	4.35 ± 0.02	Isochrones*
	4.21 ± 0.11	4.24 ± 0.12	A+M
	4.29 ± 0.1	4.19 ± 0.1	SPC
Microturb., ξturb [ km s−1]	1.09 ± 0.04	1.14 ± 0.04	A+M
vsini [ km s−1]	2.7 ± 0.5	3.1 ± 0.5	SPC
${P}_{\mathrm{rot},\max }$ [days]	23.8 ± 4.4	17.5 ± 2.8	SPC (calculated from $\mathrm{vsin}$ i)

Note. For values with multiple estimates, we indicate the adopted value with asterisks. Abbreviations for analyses applied to the HARPS-N spectra are as follows: A+M = ARES+MOOG (Sousa 2014), SPC = Stellar Parameter Classification tool (Buchhave et al. 2012, 2014). The adopted isochrone values were obtained by combining results from four different isochrone fits detailed in Section 3. Tables 9 and 10 list stellar parameters for K2-79 and K2-222 obtained with fits using the EXOFAST v2 software (Eastman et al. 2019).

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We also applied the ARES+MOOG equivalent-width method to the K2-79 and K2-222 HARPS-N spectra to calculate T_eff, $\mathrm{log}g$, [Fe/H], and microturbulence (χ_turb) (Sousa 2014) for each target. Spectra associated with 3σ RV outliers were discarded. As with SPC, all used spectra were shifted to a common RV frame before combining them for analysis. For accuracy, the value of $\mathrm{log}g$ was corrected following Equation (3) in Mortier et al. (2014), and its error bars were increased according to Sousa et al. (2011). Final values measured with ARES+MOOG and the combined results with SPC are summarized in Table 6.

Finally, with the parallax for each star from the Gaia Early Data Release 3 (Gaia Collaboration et al. 2021) and stellar magnitudes in various filters (B, V, J, H, K, W1, W2, W3; details in Table 6), we used the Isochrones (Morton 2015) software to run four isochrone analyses for each target: Dartmouth models (Dotter et al. 2008) and MIST models (Choi et al. 2016) each paired with both the SPC- and ARES+MOOG-obtained T_eff and metallicity values. The Gaia parallax and magnitudes in various filters are listed in Table 6. We used 400 live points for the Multinest algorithm and combined posteriors from all four isochrone runs, as first presented in Malavolta et al. (2018), to obtain final values for T_eff, $\mathrm{log}g$, [Fe/H], extinction (A(V)), stellar luminosity, radius, mass, density, distance, and age. For both targets, parameter values between our four isochrone runs were in agreement with each other within 1.5σ. The final isochrone values listed in Table 6 were obtained by combining results from the four isochrone fits. When applicable, we inflate errors on values obtained with any method to 2% in temperature and radius, 4% in luminosity, 5% in mass, and 20% in age, according to Tayar et al.'s (2020) approximations for the systematic uncertainty floor associated with current measured parallaxes and bolometric fluxes. The final derived isochrone results had 1σ agreement with relevant parameters measured in our SPC and ARES+MOOG analyses, as well as those previously measured by Huber et al. (2016).

We performed analyses of the K2-79 and K2-222 LCs and the HARPS-N RVs and activity indicators to search for prominent stellar activity signals. The next two subsections detail these analyses.

4.1. Light Curve Stellar Activity Analysis

To probe prominent stellar activity signals, we calculated the Generalized Lomb–Scargle (GLS) periodograms (Zechmeister & Kürster 2009) on the LC of each target with only K2 thruster firings removed (top panels in Figure 1). We also performed auto-correlation function (ACF) analysis on each LC. We utilized the Astropy.timeseries Lomb–Scargle package to calculate all GLS periodograms using the default auto method, which selects the best periodogram algorithm based on the input data. We calculate all periodograms with a sampling of N = 50,000 for periods between 0.1 days and one-half the baseline time span of the LC, or corresponding frequencies (Astropy Collaboration et al. 2013, 2018). We calculate a 1% false alarm probability (FAP) level using the Astropy.timeseries Lomb–Scargle package's default false_alarm_level tool, which calculates the upper limit to the alias-free probability using the approach outlined in Baluev (2008). The resulting periodograms of the transit-masked K2-79 and K2-222 LCs are shown in the top panels of Figure 4.

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To calculate each window function, we produce a signal with the same time stamps as the observations, replacing the LC fluxes with ones. We then calculate a GLS periodogram on the signal with the same period and frequency limits used to calculate the LC periodogram, this time applying the scipy periodogram algorithm, which treats individual points without error. The resulting window functions of the K2-79 and K2-222 LCs are shown in the bottom panels of Figure 4.

In the cases of both targets, the top-panel periodograms are dominated by long-term instrumental systematics that can be seen by eye in the LCs in the top panels of Figure 1. Therefore, for each target, we perform periodogram analyses on two more LCs: one with the best-fit third-degree polynomial subtracted (second panels in Figures 1 and 4) and the other reduced with Everest (Luger et al. 2016), an open-source pipeline for removing instrumental noise signals from K2 LCs (third panels in Figures 1 and 4). We truncated the Everest LCs to match the baseline of the observations used in our analysis and removed extreme outliers. After these cuts, the Everest-reduced LCs still include some points excluded in our reduction, but the window functions for the two sets are identical.

Two statistically significant (FAP < 1%) signals found in both targets' periodograms are likely due to remaining high-frequency instrumental systematics in the LCs. In the K2-79 and K2-222 polynomial- and Everest-reduced LCs, respectively, the first signal occurs at 12.9 days, 12.7 days, 12.9 days, and 12.6 days, and the second at 10.3 days, 9.33 days, 10.7 days and 10.6 days. We attribute the scatter in values to the different reduction methods being applied, but the set of peaks can be spotted by eye in both targets' periodograms.

In the case of K2-79, after removing the best-fit third-order polynomial, additional statistically significant peaks occur in the LC periodogram at 22.9 days (f = 0.0437 day⁻¹), 16.4 days (f = 0.0610 day⁻¹), 6.7 days (f = 0.1500 day⁻¹), and 6.0 days (f = 0.1667 day⁻¹) (Figure 4(a), second panel). After applying the Everest reduction, additional statistically significant peaks occur at 25.9 days (f = 0.0386 day⁻¹), 16.7 days (f = 0.0599 day⁻¹), and 7.0 days (f = 0.1427 day⁻¹) (Figure 4(a), third panel). The strongest peaks at 22.9 days and 25.9 days in the polynomial- and Everest-reduced LCs, respectively, are consistent with the estimated ${P}_{\mathrm{rot},\max }$ (23.8 days ± 4.4 days) within 0.5σ, and may be related to stellar rotation. In the case of K2-222, after removing the best-fit third-order polynomial, additional statistically significant peaks occur at 28.2 days (f = 0.0355 day⁻¹) and 7.6 days (f = 0.1316 day⁻¹) (Figure 4(b), second panel). After applying the Everest reduction, additional statistically significant peaks occur at 25.7 days (f = 0.0389 day⁻¹), 18.8 days (f = 0.0532 day⁻¹), 15.2 days (f = 0.0658 day⁻¹), 8.4 days (f = .1190 day⁻¹), and 6.8 days (f = 0.1471 day⁻¹) (Figure 4(b), third panel). The strongest peaks at 28.2 days and 25.7 days in the polynomial- and Everest-reduced LCs, respectively, are barely within 4 and 3σ of the estimated ${P}_{\mathrm{rot},\max }$ (17.5 days ± 4.4 days).

We calculated the ACF by applying discrete shifts to the polynomial- and Everest-reduced LCs and cross-correlating the shifted LCs with the original. The ACF shows repeating peaks separated by P_quasi, a timescale often related to P_rot, with the correlation power of each peak dropping off at a rate related to the evolution timescale of magnetic active regions, τ. We utilized the scipy nonlinear least-squares function to fit an underdamped simple harmonic oscillator curve to each ACF, described by

$$\begin{eqnarray}&&y={e}^{-t/\tau }\left[A\cos \left(\displaystyle \frac{2\pi t}{{P}_{\mathrm{quasi}}}\right)+{y}_{0}\right],\end{eqnarray} \tag{ 1 }$$

where A is the ACF amplitude, and y₀ is the oscillator offset from zero. The fits provide estimates of P_quasi and τ for each star (McQuillan et al. 2014; Giles et al. 2017). For each fit we also tried a model accounting for the common scenario of two magnetic active regions evenly spaced around the star. We did this by including a second cosine term, $B\cos \left(\tfrac{2\pi t}{{P}_{\mathrm{quasi}}}\right)$, inside the brackets of Equation (1). However, all of the best fits returned an amplitude consistent with zero for B. We also place an upper bound of 100 days on τ, as it will suffice to identify a stable (τ ≪ P_quasi) signal with only 35–40 day ACF baselines.

The results of the ACF analysis are shown in Figure 5(a) for the K2-79 LC and in Figure 5(b) for K2-222. In the case of K2-79, the fit to the ACF of the polynomial-reduced LC reveals the strongest periodicity at P_quasi = 26.6 ± 0.4 days, with an evolution timescale of τ = 72 ± 6 days. The ACF of the Everest-reduced LC reveals the strongest periodicity at P_quasi = 26.1 ± 0.2 days, with an evolution timescale of τ = 100 ± 10 days. In the case of K2-222, the fit to the ACF of the polynomial-reduced LC reveals the strongest periodicity at P_quasi = 28.2 ± 0.2 days, with an evolution timescale of τ = 100 ± 10 days. The ACF of the Everest-reduced LC returns an amplitude consistent with zero, dominated by the leftover long-period trend that can be seen by eye in the LC (Figure 1(b), third panel). The P_quasi values returned by successful fits for both targets are similar to each other and the strongest signals in all LC periodograms (Figure 4). The τ values are also all quite long relative to P_quasi. It is possible but unlikely that both stars coincidentally have similar rotation periods with very slow evolution of their magnetic active regions. Therefore, we cannot rule out the potential that the ACF signal is the result of remaining systematics for either target. We will rely on our analysis of the HARPS-N RVs in the next section to further search for periodic stellar activity signals.

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4.2. RV Stellar Activity Analysis

We first estimated values of P_rot for both targets using the Noyes et al. (1984) correlation between P_rot and $R{{\prime} }_{\mathrm{HK}}$, and applying a reddening correction of E(B-V) = 0.20 in the case of K2-79 (Green et al. 2019). This method yields a value of P_rot,Noyes ≈ 21.3 days in the case of K2-79 (〈 S-index 〉 = 0.159) and P_rot,Noyes ≈ 8.0 days in the case of K2-222 (〈 S-index 〉 = 0.160). We note that lower-metallicity stars have higher $R{{\prime} }_{{HK}}$ values and therefore shorter estimated values of P_rot, and vice versa, when using the Noyes et al. (1984) correlation. Hence, for K2-79 P_rot,Noyes may be slightly overestimated ([Fe/H] = –0.05), and that of K2-222 may be underestimated ([Fe/H] = –0.25). We therefore consult other methods in further attempts to estimate P_rot from the RVs.

To investigate the overall strength of stellar activity signals in the RVs, we calculated Spearman correlation coefficients (r_s ) between the RVs and each of the measured activity indicators. In order to identify changes from season to season, we calculated three sets of correlation coefficients for each target: one for the full RV data set, one with just the first season (S1) RVs, and one with just the second season (S2) RVs. Other observing seasons contained too few points for a meaningful calculation. Figures 6 and 7 show correlation plots and associated r_s values between the RVs and each of the activity indicators for K2-79 and K2-222, respectively. The p-value, a measure of how probable it is that the observed correlation would be produced by a null data set, is also shown. To compare correlation strengths, we establish the following terminology corresponding to the absolute value of r_s :

• 1.  
  <0.20 = no correlation;
• 2.  
  0.20–0.39 = weak correlation;
• 3.  
  0.40–0.59 = moderate correlation;
• 4.  
  ≥ 0.60 = strong correlation.

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In the case of K2-79, the full RV set has weak correlations with the contrast, CCF area, and BIS. The S1 RVs have no correlation with any activity indicators. The S2 RVs have weak correlations with the contrast and CCF area and a moderate anticorrelation with the BIS, where we see the strongest peak at 11.0 days. In the case of K2-222, the full RV set has a moderate correlation with the contrast, where we see the strongest peak at 39.2 days, and a weak correlation with the CCF area. The S1 RVs have weak correlations with the contrast and S-index. The S2 RVs have moderate correlations with the contrast and CCF area, where we see the strongest peak at 41.8 days in both. In the cases of both targets, the S2 RVs show greater overall correlation with activity indicators than the S1 RVs, which we also see in the periodogram analysis.

We calculate GLS periodograms for the RVs and activity indicators of K2-79 and K2-222 using three sets of data for each: the full data set, the first season (S1) data set, and the second season (S2) data set. The strongest periods in stellar activity signals can change significantly from season to season due the sizes, locations, and distribution of stellar features evolving over time. Table 5 provides more detailed information about the full and seasonal data sets. We also calculate the window functions using the method described in Section 4.1. The resultant sets of periodograms for each target are shown in Figures 8 and 9. In the periodograms we only show up to f = 0.20 day⁻¹ because no statistically significant peaks occur beyond that frequency in the case of K2-79, and, in the case of K2-222, statistically significant peaks beyond that frequency are aliases of the 1 day peak in the window function.

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We address statistically significant peaks in the periodograms, as well as those that occur at the known orbital periods of the transiting exoplanets. Although none of the peaks at the known orbital periods meet the traditional criteria for statistical significance (FAP < 1%), the probability of the peaks being false decreases beyond traditional FAP levels with the confirmed existence of exoplanets at those periods.

For both K2-79 and K2-222, magnetic activity signals generally show a stronger potential presence in S2 than in S1. This is apparent both in the relative strengths of the S1 and S2 seasonal-activity-indicator periodograms and the S1 and S2 Spearman correlation coefficients, especially considering the much lower number of data points in S2. For each target, the strongest power in the S2 activity-indicator periodograms is greater than or comparable to the strongest in S1, in spite of far fewer points in the S2 data set (Figures 8 and 9). In the case of K2-79, none of the S1 activity indicators correlate with the RVs, while three of the S2 activity indicators show at least weak correlation (Figure 6). In the case of K2-222, two activity indicators correlate with the RVs in both S1 and S2. There is a correlation between the S1 S-index values and RVs (r_s = 0.22) where no correlation exists in S2, but it is much weaker than the correlations between contrast and RVs for both S1 (r_s = 0.28) and S2 (r_s = 0.57), where the S2 Spearman coefficient is significantly higher (Figure 7).

For both targets, at P_b specifically, seasonal periodograms also show more potential for activity interference in S2 than in S1. Even though the K2-79 and K2-222 seasonal RV periodograms reveal stronger signals near P_b in S1, the seasonal-activity-indicator periodograms show stronger signals at P_b in S2. In the case of K2-79, the S1 RV peak near P_b has a FAP ≈10%, while the S2 RV peak has no statistical significance (FAP <99%). Yet, near P_b, the S1 activity indicators contain no peaks of significance, while the S2 BIS contains a peak with FAP <10%, and the S2 contrast contains a peak with FAP ≈50%. In the case of K2-222, the S1 RV peak near P_b is far stronger (FAP < 10%) than the S2 peak (FAP > 50%), which is expected given the relative sizes of the seasonal data sets. However, near P_b , the S1 activity indicators again contain no peaks of significance, while the S2 contrast contains a peak with FAP ≈50%.

In the case of K2-79 (Figure 8), no statistically significant peaks occur in the full or either of the seasonal RV periodograms. The two strongest peaks occur at the known orbital period of the planet (P_b = 10.99 days; f = 0.09099 day⁻¹) and at f = 0.0882 day⁻¹, an alias of P_b with the strongest peak in the full window function (f_w = 0.0028 day⁻¹). The signal at P_b could potentially be combined with an activity signal, based on the corresponding peaks with FAP ≈ 10% and ≈ 50% in the S2 BIS and contrast periodograms, respectively, and its location less than one day from the first harmonics of the estimated ${P}_{\mathrm{rot},\max }$ and P_rot,Noyes.

In the case of K2-222 (Figure 9), the strongest peak in the full RV periodogram occurs at f = 0.007 day⁻¹ (P = 146.6 days), notably not at the period of the known exoplanet. An alias of this signal with the strongest peak in the full window function (f = 0.0026 day⁻¹) also occurs at f = 0.0042 day⁻¹. The peak at 146.6 days does not appear to be related to the window function or activity-indicator periodograms; peaks at nearby frequencies do occur in the activity indicators, but not at the exact same frequency, and the signal is much more significant in the RVs. The 146.6 day signal remains the most significant peak (FAP < 1%) when we remove the last three solitary observations and recalculate the periodogram. We therefore consider this signal as a potential second object (P_c?).

The periodogram of the full K2-222 RV set also shows peaks at the period of the known transiting exoplanet (P_b = 15.39 days; f = 0.06498 day⁻¹) and at f = 0.0624 day^–1, an alias of the exoplanet signal with the strongest peak in the full window function (f_w = 0.0026 day⁻¹). The signal at P_b could potentially be combined with an activity signal, based on the corresponding peak with FAP ≈ 10% in the contrast periodogram. We analyze an additional statistically significant peak at f = 0.1229 day⁻¹ (P_act = 8.137 days) that occurs in both the full and S1 periodograms. We believe the signal is most likely a result of stellar activity because of a nearby peak with FAP ≈ 10% in the FWHM periodograms. The signal at P_act,1/2 with FAP ≈ 10% may also be related to activity, as it occurs at approximately one-half of P_act and less than 1 day away from a peak with FAP < 50% in the contrast periodogram. The four prominent peaks surrounding the peak at P_act,1/2 are aliases with the largest peak in the S2 window function.

As discussed above, for both K2-79 and K2-222, the S2 RVs suffer more potential interference from stellar activity at P_b than the S1 RVs. Since each target's S1 and S2 observations were not taken in the same season as the other's, we can rule out any seasonal instrumental effect being the cause of more interference in S2. For K2-79 and K2-222, the small number of observations in S2 will already make for a less precise mass determination, but constructive or destructive interference with stellar activity in the S2 RVs could also affect the accuracy of the exoplanet mass measured from the full RV set. Therefore, in addition to full RV fits, we perform fits to the S1 and S2 RV subsets to confirm agreement between exoplanet masses estimated using the full result and at least the S1 result.

Attempts to fit the K2-79 and K2-222 LCs and RVs separately revealed that for both targets neither of the data sets is sufficient to constrain the orbital eccentricity of the transiting exoplanet. We therefore perform a simultaneous fit to the RVs and flattened LCs (top panels of Figures 2 and 3, bottom panels of Figures 1(a) and (b)) using the publicly available Differential Evaluation MCMC software, EXOFAST v2 (Eastman et al. 2019). We run our EXOFAST v2 fits with maxsteps = 50,000 and nthin = 30, running for a maximum of 50,000 recorded steps while recording the 30th step of each walker. The global model used in EXOFAST v2 includes spectral-energy distribution and integrated Isochrones and Stellar Tracks (MIST) stellar evolutionary models informed by the magnitudes in Table 6 to constrain stellar parameters. In the case of both targets, we utilize isochrones from Choi et al. (2016) to derive a physical star (mass, radius, luminosity, age) based on T_eff, [Fe/H], and $\mathrm{log}g$. Table 8 shows the median and 1σ stellar values returned for K2-79 and K2-222.

For the transit fits, we set the long-cadence flag to account for the 29.4 minutes cadence of the K2 data, and we use the default quadratic limb-darkening law, based on tables reported in Claret & Bloemen (2011). The transit model includes parameters corresponding to each transiting exoplanet's orbital period (P_orb), time of central transit (T_c), orbital inclination (i), orbital eccentricity (e), orbital argument of periastron (ω), and radius relative to its star (R_P /R_*). The RV model is a Keplerian consisting of the following parameters: RV semiamplitude (K), e, P_orb, and ω.

The transit fits place strong constraints on P_orb and T_c. Fit alongside with stellar density, they also drastically reduce the degenerate e–ω parameter space. Pairing the transit and RV fits therefore allows us to break the degeneracy between these two parameters and achieve better estimates of e and ω than were possible with the RVs or transits alone.

For each target, we perform three fits assuming a single planet: one using all available RVs, another with just the S1 RVs, and a third with just the S2 RVs. In the case of K2-222, we perform a second set of these three fits for a two-planet scenario to investigate the nature of the strongest RV periodogram signal at 146.6 days. We fix the eccentricity of the potential second exoplanet to zero, taking into account the exoplanet's relatively long period and gaps in its phase coverage.

Table 7 summarizes the probability distributions of priors applied to the EXOFAST v2 fits for both targets. Priors on stellar parameters were based on the combined result from the SPC tool and ARES+MOOG analyses of the HARPS-N spectra (see Section 3 and Table 6). In the case of the seasonal two-planet fits to K2-222 data, we fix the parameters associated with the potential second exoplanet to the median values returned by the full two-planet fit. This allows us to confirm agreement between the full and seasonal RV sets of the parameters associated with K2-222b, despite the fact that the seasonal baselines are too short to reliably fit the 146.6 day signal of a potential second companion object.

Table 7. Prior Probability Distributions Applied to Parameters Modeled in Our EXOFAST v2 Transit Fits

Target	K2-79	K2-222
Teff [K]	Gaussian(5897, 58.5)	Gaussian(5942, 57.5)
Surface gravity, $\mathrm{log}g$	Gaussian(4.25, 0.15)	Gaussian(4.22, 0.16)
Metallicity [Fe/H]	Gaussian (0.035, 0.06)	Gaussian (–0.315, 0.06)

Note. Stellar priors are based on combined results from the SPC tool and ARES+MOOG analyses of the HARPS-N spectra. The two numbers in each prior are the mean and standard deviation of the Gaussian distribution, respectively.

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Table 8 shows the MCMC posterior median values and 1σ uncertainties of the stellar and exoplanet transit parameters for each target fitted with EXOFAST v2. We also list several other parameters describing the individual transits (depth, duration, impact parameter) and full LCs (baseline flux, variance of transit model residuals). The estimated equilibrium temperature (T_eq) returned by EXOFAST v2 follows Equation (1) of Hansen & Barman (2007), ${T}_{\mathrm{eq}}={T}_{\mathrm{eff}}\sqrt{{R}_{* }/(2{\text{}}a)}$, where a is the orbital semimajor axis and no albedo and perfect energy redistribution are assumed. The stellar values returned by the EXOFAST v2 fits agree with those adopted from our stellar analysis within 1σ, with the exception of K2-222's stellar radius, which sees a slightly higher discrepancy but agrees well within 1.5σ.

Table 8. K2-79 and K2-222 Stellar and Transit-related Planet Parameters based on Median Values and 68% Confidence Interval Returned by EXOFAST v2 Simultaneous RV and LC Fits

Symbol	Parameter and Units	K2-79	K2-222
Stellar Parameters
M*	Mass (M⊙)	${1.066}_{-0.070}^{+0.057}$	${0.989}_{-0.065}^{+0.070}$
R*	Radius (R⊙)	${1.265}_{-0.027}^{+0.041}$	1.115 ± 0.029
L*	Luminosity (L⊙)	1.76 ± 0.10	1.452 ± 0.077
Age	Age (Gyr)	${6.6}_{-2.6}^{+2.9}$	${6.3}_{-3.0}^{+3.4}$
Teff	Effective temperature (K)	${5902}_{-59}^{+86}$	${6000}_{-69}^{+67}$
$\mathrm{log}g$	Surface gravity (cgs)	${4.257}_{-0.043}^{+0.038}$	4.339 ± 0.042
[Fe/H]	Metallicity	${0.053}_{-0.036}^{+0.048}$	−0.143 ${}_{-0.14}^{+0.071}$
Exoplanet Parameters:		b	b
P	Period (days)	${10.99470}_{-0.00047}^{+0.00031}$	15.38857 ± 0.00088
TC	Time of transit (BJD—2.4E6)	${57103.22750}_{-0.00084}^{+0.00076}$	57399.0595 ± 0.0016
i	Inclination (degrees)	88.44 ± 0.44	${89.12}_{-0.41}^{+0.55}$
RP /R*	Radius of planet in stellar radii	${0.02948}_{-0.00037}^{+0.00094}$	${0.01928}_{-0.00035}^{+0.00040}$
Transit Parameters:
F0	Baseline flux	0.9999949 ± 0.0000095	1.0000015 ± 0.0000060
δ	Transit depth (fraction)	${0.000869}_{-0.000022}^{+0.000057}$	${0.000372}_{-0.000013}^{+0.000016}$
T14	Total transit duration (days)	${0.1894}_{-0.0029}^{+0.0026}$	${0.1836}_{-0.0025}^{+0.0028}$
b	Transit impact parameter	${0.46}_{-0.16}^{+0.15}$	${0.31}_{-0.20}^{+0.19}$
σ2	Added variance	2.28 × 10−9 ± 9.6 × 10−10	1.85 × 10−9 ± 3.7 × 10−10

Note. Added variance describes remaining scatter of the data after removal of the transit model.

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Tables 9 and 10 show the posterior median values and 1σ uncertainties of all other parameters from the EXOFAST v2 fits to K2-79 and K2-222, respectively. In the cases of both targets, the obtained K values for K2-79b and K2-222b from the full RVs and seasonal fits agree within 1σ. In the case of K2-222, the one-planet and two-planet fits also return values consistent within 1σ, but the median value for the semiamplitude of the transiting planet, K_b , is notably smaller in the two-planet model.

Table 9. K2-79 Radial Velocity Parameters based on Median Values and 1σ Confidence Interval Returned by EXOFAST v2 Simultaneous RV and LC Fits

Symbol	Parameter and Units	All	S1	S2
Exoplanet Parameters:
Kb	RV semiamplitude (m s−1)	${3.28}_{-0.78}^{+1.0}$	${3.2}_{-1.3}^{+1.1}$	${3.0}_{-1.4}^{+1.6}$
eb	Eccentricity	${0.082}_{-0.056}^{+0.088}$	${0.133}_{-0.095}^{+0.14}$	${0.119}_{-0.079}^{+0.12}$
ωb	Argument of periastron (degrees)	$-{140}_{-130}^{+150}$	${5}_{-50}^{+73}$	160 ±110
HARPS-N Parameters:
γrel	Relative RV offset (m s−1)	$-{0.06}_{-0.73}^{+0.67}$	$-{2.10}_{-0.99}^{+1.2}$	1.1 ± 1.2
σJ	RV jitter (m s−1)	3.55 ± 0.67	3.7 ± 1.1	${2.4}_{-2.4}^{+1.4}$

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Table 10. K2-222 Radial Velocity Parameters based on Median Values and 68% Confidence Interval Returned by EXOFAST v2 Simultaneous RV and LC Fits

Symbol	Parameter and Units	All/1p	All/2p	S1/1p	S1/2p	S2/1p	S2/2p
Exoplanet Parameters:
Kb	RV semiamplitude (m s−1)	${2.18}_{-0.50}^{+0.48}$	${1.65}_{-0.51}^{+0.50}$	${2.26}_{-0.62}^{+0.69}$	${1.76}_{-0.45}^{+0.52}$	${2.52}_{-0.92}^{+1.1}$	${1.95}_{-0.76}^{+0.73}$
eb	Eccentricity	${0.188}_{-0.097}^{+0.11}$	${0.23}_{-0.12}^{+0.11}$	${0.30}_{-0.19}^{+0.13}$	${0.21}_{-0.12}^{+0.13}$	${0.17}_{-0.11}^{+0.14}$	${0.144}_{-0.087}^{+0.12}$
ωb	Argument of periastron (degrees)	${151}_{-44}^{+30}$	${164}_{-38}^{+29}$	${163}_{-64}^{+52}$	${157}_{-52}^{+27}$	${162}_{-81}^{+52}$	${145}_{-87}^{+59}$
Kc	RV semiamplitude (m s−1)	⋯	${2.52}_{-0.51}^{+0.52}$	⋯	≡ 2.52	⋯	≡ 2.52
Pc	Period (days)	⋯	147.5 ± 3.3	⋯	≡ 147.5	⋯	≡ 147.5
ec	Eccentricity	⋯	≡ 0	⋯	≡ 0	⋯	≡ 0
ωc	Argument of periastron (degrees)	⋯	≡ 0	⋯	≡ 0	−	≡ 0
HARPS-N Parameters:
γrel	Relative RV offset (m s−1)	${0.04}_{-0.37}^{+0.35}$	−0.10 ± 0.31	${0.32}_{-0.48}^{+0.43}$	$-{0.01}_{-0.35}^{+0.40}$	$-{0.60}_{-0.72}^{+0.73}$	$-{0.21}_{-0.51}^{+0.52}$
σJ	RV jitter (m s−1)	${1.98}_{-0.31}^{+0.36}$	${1.26}_{-0.40}^{+0.38}$	${2.13}_{-0.48}^{+0.46}$	${1.46}_{-0.46}^{+0.59}$	${2.26}_{-0.76}^{+0.85}$	${0.89}_{-0.89}^{+0.96}$
Model Comparison:
NRV	Number of RV observations	63	63	41	41	19	19
AIC	Akaike information criterion	−1454.3836	−1468.3867	−1538.5536	−1567.6584	−1665.0455	−1667.2941
ΔAIC	AIC difference from adopted	0	−14.0031	−84.1700	−113.2748	−210.6619	−212.9105
BIC	Bayesian information criterion	−1377.2251	−1375.1535	−1464.4513	−1481.2057	−1594.4461	−1584.9282
ΔBIC	BIC difference from adopted	0	+2.0716	−87.2662	−103.9806	−217.2209	−207.7031

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We find evidence in support of a second companion candidate by comparing the robustness of the one-planet and two-planet models using their Bayesian information criterion (BIC) and Akaike information criterion (AIC) values. The BIC value is lower for the one-planet model full fit, but the AIC value is lower for the two-planet model by a much larger margin. The AIC and BIC of the two-planet model are also penalized more due to the higher number of parameters fit than in the one-planet model. Although the AIC and BIC values provide tentative support in favor of a two-planet model, we adopt the one-planet solution for now, considering the significant phase gaps in the coverage of the 147.5 day period returned by our fit.

Figures 10 and 11 show the phase-folded LCs of K2-79b and K2-222b with the best-fit transit models overplotted. Figures 12 and 13 show the phase-folded RV curves of each transiting exoplanet with the best-fit one-planet Keplerian model overplotted. Figure 14 shows the phase-folded RV curves of K2-222b and the candidate K2-222 c with their best-fit two-planet Keplerian model overplotted.

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Finally, in addition to the simultaneous RV and LC fits with EXOFAST v2, we also fitted the RV data with the RadVel software (Fulton et al. 2018) to see if we could improve the RV solutions using Gaussian processes (GP) regression to model the stellar activity (Haywood et al. 2014). We detail these fits in the next section.

The following RV fits including GPs were not included in the calculation of any final mass results. However, in this section we detail our analysis and results for completeness.

In addition to the simultaneous RV and LC fits, we also use the RadVel software (Fulton et al. 2018) to perform MCMC fits on just the HARPS-N RVs, but this time including a quasiperiodic GP regression to model a stellar activity signal in addition to the exoplanet Keplerian (Haywood et al. 2014). We once again perform fits on the full and seasonal RV sets, and, in the case of K2-222, with both one-planet and two-planet models.

The RadVel model fits for the same Keplerian and instrumental parameters fit by EXOFAST v2. In the two-planet models we once again fix the eccentricity of the potential second exoplanet and, in the seasonal fits, fix all of its associated parameters to the values returned by the full fit. We place priors on P_b , T_C,b , and e_b for both targets based on posteriors from our EXOFAST v2 fits. We set Jeffreys priors on K_b values with an upper limit of 10 m s⁻¹ to explore parameter space around the ≈ 6 m s⁻¹ standard deviation of both RV sets. We place a Gaussian prior on σ_HARPS−N, set equal to the distribution of RV errors (〈RVerr〉, σ_RVerr). Table 11 summarizes the priors applied to the exoplanet parameters in our RadVel fits.

The GP-generated quasiperiodic stellar activity model populates a covariance matrix with the function

$$\begin{eqnarray}&&k(t,t^{\prime} )={A}^{2}\exp \left[-\displaystyle \frac{{\left(t-t^{\prime} \right)}^{2}}{{\tau }^{2}}-\displaystyle \frac{{{\sin }}^{2}\left(\tfrac{\pi (t-t^{\prime} )}{{P}_{\mathrm{quasi}}}\right)}{2{\eta }^{2}}\right],\end{eqnarray} \tag{ 2 }$$

where k(t, t${\rm{^{\prime} }}$ ) is the correlation weight between observations taken at times t and ${{t}}^{{\rm{{\prime} }}}$. A is the mean amplitude of the activity signal, τ is a hyper-parameter related to the evolution timescale of activity features, P_quasi is related to the stellar rotation period, and η describes the level of high-frequency variation expected within a single stellar rotation and is related to the average distribution of activity features on the surface of the star.

We apply Jeffreys priors on the GP stellar amplitudes with the same limits and motivation applied to the K_b priors, and also place a Jeffreys prior on τ investigating a wide range of values between 2 and 100 days. As mentioned above, η is physically related to the average distribution of magnetic active regions on the stellar surface. Models by Jeffers & Keller (2009) demonstrated that even highly complex activity distributions will average to just two to three large active regions in a given rotation. We therefore assign a prior distribution of η = 0.25 ± 0.025, which allows for two to three local minima or maxima per rotation (Haywood et al. 2014; Grunblatt et al. 2015; López-Morales et al. 2016).

For each target, we ran fits with various priors on P_quasi, first with a Jeffreys prior of 2–40 days for each. Based on the posterior distributions returned by those fits, we ran additional fits with Jeffreys priors of 10–40 days and 2–10 days for K2-79 and K2-222, respectively. We also used several Gaussian priors centered around various P_rot estimates with a 20% standard deviation. For K2-79 we ran fits with Gaussian priors on P_quasi centered around 26.35 days, 23.8 days, and 11.0 days based on P_rot,Noyes and the LC ACFs, the $v\,\sin \,i$-calculated ${P}_{\mathrm{rot},\max }$, and the BIS periodogram, respectively. For K2-222 we ran fits with Gaussian priors on P_quasi centered around 28.2 days, 17.5 days, and 8.0 days, based on the polynomial-reduced LC ACF, the $v\,\sin \,i$-calculated ${P}_{\mathrm{rot},\max }$, and P_rot,Noyes plus the RV and FWHM periodogram peaks, respectively. Final exoplanet parameters returned by successful fits agreed within 1σ no matter which priors were used on the GP hyper-parameters. All fits including a Gaussian prior on P_quasi just returned the input prior in their posterior, so we report values from the fits including 10–40 day and 2–10 day Jeffreys priors for K2-79 and K2-222, respectively. Table 11 summarizes the priors placed on the GP hyper-parameters in our reported RadVel fits.

The majority of the RadVel fits failed to converge on a solution for the stellar activity signal. RVs for both K2-79 and K2-222 were not optimally scheduled to characterize stellar activity signals, with never more than a single observation taken per night and several multiple-day gaps in each of the analyzed seasons. It is therefore unsurprising that our models have a difficult time characterizing the stellar activity signal. We define successful fits as those that converge for values of A and P_quasi, at least. Even in successful fits, posterior distributions show τ pushing toward the smallest possible value, consistent with a highly unstable P_quasi signal. This was the case for all fits including a GP-generated model with a quasiperiodic kernel. We include a corner plot for each target in the Appendix for those wishing to view the posteriors from the reported Radvel fits in further detail (Figures 16 and 17). We also tested the performance of GP fits with a simpler squared-exponential kernel, but found they show the same behavior in the τ posteriors as the fits with quasiperiodic kernels and are even less likely to converge on a value for stellar amplitude, A.

Table 12 shows final parameter values returned by successful RadVel GP fits. The ≈24 day P_quasi signal returned for K2-79 is consistent with that returned by the LC ACFs and is close in value to the largest peaks in the LC periodogram, P_rot,Noyes, and the ${P}_{\mathrm{rot},\max }$ calculated from $v\,\sin \,i$. A second smaller peak that shows up in the posterior at 12–13 days and the 11 day peak in the BIS periodogram both sit near the first harmonic of these signals, as well. We therefore estimate that K2-79 has a P_rot in the range 21–26 days. The ≈4 day P_quasi signal returned for K2-222 is consistent with the first harmonics of the estimated P_rot,Noyes and the ≈8 day peaks in the RV and FWHM periodograms. While we believe the most likely rotation period of this star is ≈8 days, estimates of P_rot for K2-222 have varied greatly from method to method, and we are less confident in the estimate than that for K2-79. The reported jitter values are low relative to the average RV error for both targets, giving us concern that the GP models could be overfitting noise that is unrelated to stellar activity. However, the exoplanet semiamplitudes returned by the successful GP fits all have 1σ agreement with those in their corresponding EXOFAST v2 fits. We therefore list the RadVel results as further evidence in support of the measured K2-79b and K2-222b semiamplitudes, but adopt final results for the exoplanet characteristics from our EXOFAST v2 fits to the full RV data sets.

Table 12.  Radvel Markov Chain Monte Carlo Posterior Probability Distributions for Exoplanet Parameters and GP Hyper-parameters Fit to the K2-79 and K2-222 RVs

	K2-79	K2-222	K2-222
	All	1p/All	1p/S1
Exoplanet
Pb [days]	10.99475 ± 0.00047	15.3885 ± 0.00088	15.3885 ± 0.0018
TC,b [BJD—2.4E5]	57103.2276 ± 0.0017	57399.0594 ± 0.0032	57399.0593 ± 0.0033
Kb [m s−1]	3.4 ± 1.1	${1.94}_{-0.53}^{+0.51}$	1.81 ± 0.68
eb	${0.078}_{-0.054}^{+0.059}$	${0.159}_{-0.1}^{+0.093}$	0.21 ± 0.14
Arg. of periastron, ωb [deg]	−2.7 ${}_{-2.5}^{+1.6}$	${2.77}_{-0.99}^{+1.2}$	2.8 ± 1.4
Instrument
γHARPS−N	−-0.22 ± 0.93	${0.17}_{-0.40}^{+0.41}$	0.35 ± 0.5
σHARPS−N	${1.7}_{-1.6}^{+1.3}$	${1.21}_{-0.56}^{+0.52}$	1.39 ± 0.54
Stellar Activity
A [m s−1]	${3.1}_{-1.4}^{+1.2}$	${1.55}_{-0.56}^{+0.49}$	${1.5}_{-0.62}^{+0.60}$
τ [days]	${7.6}_{-4.3}^{+11.0}$	${13.7}_{-8.7}^{+11.0}$	${8.7}_{-5.5}^{+13.0}$
Pquasi [days]	${24.2}_{9.5}^{+9.8}$	${4.0}_{-1.2}^{+3.4}$	${4.4}_{-1.8}^{+3.7}$
η	0.252 ± 0.025	0.254 ± 0.025	0.251 ± 0.025

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Figure 15 shows the locations of K2-79b and K2-222b in mass–radius space along with the population of confirmed exoplanets with T_eq between 300 and 3000 K, 1σ mass errors less than 50%, 1σ radius errors less than 20%, and orbiting stars with radii between 0.6 R_⊙ and 2.0 R_⊙. Table 13 summarizes the fitted and derived parameters for the known planets, including both the one-planet (adopted) and two-planet cases for K2-222. With radii of 4.09 and 2.35 R_⊕, K2-79b and K2-222b exist at two important locations: the sub-Saturnian desert and at the upper edge of the radius valley, respectively.

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Table 13. Final Derived Exoplanet Characteristics

Parameter	K2-79b	K2-222b (1-plan)	K2-222b (2-plan)
Porb [days]	${10.99470}_{-0.00047}^{+0.00031}$	15.38857 ± 0.00088	15.38857 ± 0.00088
Mplan [M⊕]	11.8 ± 3.6	8.0 ± 1.8	6.0 ± 1.9
Rplan [R⊕]	${4.09}_{-0.12}^{+0.17}$	${2.35}_{-0.07}^{+0.08}$	${2.35}_{-0.07}^{+0.08}$
ρplan [ρ⊕]	0.17 ± 0.06	0.62 ± 0.15	0.46 ± 0.15
Surface grav. [g⊕]	0.71 ± 0.22	1.45 ± 0.34	1.09 ± 0.35
Irradiation [Flux⊕]	180 ± 8	95 ± 5	95 ± 5
Teq [K]	${1021}_{-20}^{+21}$	${878}_{-15}^{+17}$	${878}_{-16}^{+17}$
ζ	2.21 ± 0.34	1.40 ± 0.16	1.50 ± 0.24

Note. ζ is physically related to the chemical composition of an exoplanet core (see Section 7 and Figure 15).

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Zeng et al. (2021, accepted) provides evidence for two new dimensionless parameters that can be used to describe exoplanet compositions. The first, $\zeta \equiv ({R}_{p}/{R}_{\oplus }){\left({M}_{p}/{M}_{\oplus }\right)}^{-1/4}$, can distinguish between three small exoplanet populations: rocky planets composed of varying ratios of silicates and metals (ζ ≈ 1), water worlds dominated by significant amounts of water and ice (ζ ≈ 1.4), and Neptune-/Uranus-like planets with ice-dominated cores hosting small gaseous envelopes (ζ ≈ 2.2) (see histogram in Figure 15). ζ is physically related to the chemical composition of the exoplanet core as illustrated by the ternary plot in Figure 15. The second parameter, $z\equiv {\int }_{\in \mathrm{envelope}}\tfrac{{dP}}{\rho }/\left(\tfrac{G\cdot {M}_{\oplus }}{{R}_{\oplus }}\right)$, characterizes both the amount and the temperature of the added envelope. We will refer to ζ and z in the following two subsections when discussing potential compositions of K2-79b and K2-222b.

7.1. K2-79b

7.2. K2-222b

7.3. Utilizing Seasonal RV Analyses

The HARPS-N project was funded by the Prodex Program of the Swiss Space Office (SSO), the Harvard University Origin of Life Initiative (HUOLI), the Scottish Universities Physics Alliance (SUPA), the University of Geneva, the Smithsonian Astrophysical Observatory (SAO), the Italian National Astrophysical Institute (INAF), University of St. Andrews, Queen's University Belfast, and University of Edinburgh. This work has been supported by the National Aeronautics and Space Administration under grant No. NNX17AB59G, issued through the Exoplanets Research Program. Parts of this work have been supported by the Brinson Foundation. A.Mo. acknowledges support from the senior Kavli Institute Fellowships. C.A.W. acknowledges support from Science and Technology Facilities Council (STFC) grant No. ST/P000312/1. S.H.S. acknowledges support by NASA Heliophysics LWS grant No. NNX16AB79G. L.Z. acknowledges support by the Sandia Z. Fundamental Science Program by the Department of Energy National Nuclear Security Administration under Awards DE-NA0003904 (to S.B.J.) (principal investigator) with Harvard University.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the GaiaData Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

We also thank Sarah Blunt for graciously answering questions regarding use of the RadVel software.

Figures 16 and 17 contain posterior distributions from successful RadVel Markov chain Monte Carlo fits to the K2-79 and K2-222 RVs, which included stellar activity signals modeled with quasiperiodic GP regression.

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