Four Sub-Saturns with Dissimilar Densities: Windows into Planetary Cores and Envelopes

K2-27 hosts a single transiting sub-Saturn, K2-27b, with P = 6.77 days that was first confirmed in Van Eylen et al. (2016b) using RVs. We obtained 15 spectra of K2-27 with HIRES between 2015 February 5 and 2016 July 17. Van Eylen et al. (2016b) observed this star with HARPS, HARPS-N, and FIES. We included 6 and 19 measurements from HARPS and HARPS-N, respectively in our RV analysis. We did not include the six FIES measurements because their uncertainties are much larger (≈30 ${\rm{m}}\,{{\rm{s}}}^{-1}$), compared to uncertainties from the HIRES, HARPS, and HARPS-N spectra (≈4–5 ${\rm{m}}\,{{\rm{s}}}^{-1}$), and hence add little additional information to constrain the RV fits while increasing model complexity.

We first modeled the combined RVs assuming circular orbits and no additional acceleration term, $\dot{\gamma }$. We then allowed $\dot{\gamma }$ to vary, but found that these models produced a negligible improvement in the BIC (${\rm{\Delta }}\mathrm{BIC}=-1$). We then allowed for eccentricity to float and found a non-zero eccentricity of $e=0.251\pm 0.088.$ Compared to the circular models, the eccentric fits were strongly favored (${\rm{\Delta }}\mathrm{BIC}=-14$). We verified that the eccentricity is detected in both HIRES and HARPS+HARPS-N data sets by fitting these subsets independently. These data sets yield consistent and non-zero eccentricities. We adopted the parameters from the eccentric model as the preferred parameters. The most probable eccentric model is shown in Figure 1. The K2-27 system parameters are summarized in Table 3. We measured a Doppler semi-amplitude of ${K}_{}=11.8\pm 1.8$ ${\rm{m}}\,{{\rm{s}}}^{-1}$, which is consistent with ${K}_{}=10.8\pm 2.7$ ${\rm{m}}\,{{\rm{s}}}^{-1}$ reported by Van Eylen et al. (2016b), but with smaller uncertainties, due to our additional measurements. We measured a mass of ${M}_{{\rm{P}}}=30.9\pm 4.6$ ${M}_{\oplus }$ and a density of $1.87\pm 0.41$ g cm⁻³. As we discuss in Section 5, K2-27b has a relatively high mass for its size, implying a large core mass.

Zoom In Zoom Out Reset image size

Table 3.  System Parameters of K2-27

	Value	Reference
Stellar Parameters
Identifier	EPIC-201546283
${T}_{\mathrm{eff}}$ (K)	$5248\pm 60$	A
$\mathrm{log}g$ (dex)	$4.48\pm 0.05$	A
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	$0.13\pm 0.04$	A
$v\sin i$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	$2.3$	A
${M}_{\star }$ (${M}_{\odot }$)	${0.866}_{-0.023}^{+0.029}$	A
${R}_{\star }$ (${R}_{\odot }$)	$0.885\pm 0.043$	A
age (Gyr)	${10.3}_{-5.2}^{+3.3}$	A
Apparent V (mag)	$12.64\pm 0.02$	A
	Planet b
Transit Model
P (days)	$6.771315\pm 0.000085$	B
T0 (BJD-2454833)	$1979.84484\pm 0.00057$	B
${R}_{{\rm{P}}}$ (${R}_{\oplus }$)	$4.48\pm 0.23$	A
a (au)	$0.06702\pm 0.00071$	A
${S}_{\mathrm{inc}}$ (${S}_{\oplus }$)	${116}_{-12}^{+16}$	A
${T}_{\mathrm{eq}}$ (K)	$902\pm 28$	A
Circular RV Model
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$9.9\pm 2.0$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-2.7\pm 2.0$	A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-0.3\pm 3.8$	A
${\gamma }_{\mathrm{HARPS}-{\rm{N}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$6.4\pm 2.2$	A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)	A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${6.1}_{-1.4}^{+2.0}$	A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${6.9}_{-3.2}^{+5.9}$	A
${\sigma }_{{\rm{jit,HARPS-N}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$4.7\pm 2.4$	A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$26.7\pm 5.3$	A
ρ (g cm−3)	$1.61\pm 0.42$	A
Eccentric RV Model (Adopted)
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$11.8\pm 1.8$	A
e	$0.251\pm 0.088$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-2.2\pm 1.6$	A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$0.2\pm 2.7$	A
${\gamma }_{\mathrm{HARPS}-{\rm{N}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$6.4\pm 2.2$	A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)	A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${4.4}_{-1.2}^{+1.8}$	A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${4.1}_{-2.6}^{+4.8}$	A
${\sigma }_{{\rm{jit,HARPS-N}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${4.8}_{-2.1}^{+2.6}$	A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$30.9\pm 4.6$	A
ρ (g cm−3)	$1.87\pm 0.41$	A

Note. A: this work; B: Crossfield et al. (2016).

Download table as:  ASCIITypeset image

K2-32 hosts three planets, K2-32b, K2-32c, and K2-32d, having orbital periods of P = 8.99 days, 20.66 days, and 31.7 days, respectively, which are near the 3:2:1 period commensurability. The planets were first confirmed in Sinukoff et al. (2016) using multiplicity arguments (Lissauer et al. 2012). We obtained 31 spectra of K2-32 with HIRES between 2015 June 06 and 2016 August 20. Dai et al. (2016) obtained 43 spectra with HARPS and 6 with PFS, which we included in our RV analysis. We first modeled the combined HIRES, HARPS, and PFS RVs assuming circular orbits and no additional acceleration term, $\dot{\gamma }$. When allowing $\dot{\gamma }$ to float, we found that $\dot{\gamma }=2.3\pm 1.8$ ${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$, consistent with zero at the 2σ level. This differs from $\dot{\gamma }={34.0}_{-9.7}^{+9.9}$ ${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$ reported by Dai et al. (2016). All but five of the RVs from Dai et al. (2016) were collected within a 25 day window, and thus do not provide much leverage on $\dot{\gamma }$. The positive $\dot{\gamma }$ measured by Dai et al. (2016) is driven by two RV measurements that fall above our best-fit curve. With our combined data set spanning two observing seasons, we place tighter limits on $\dot{\gamma }$ and on the presence of long-period companions having P ≳ 2 years. Nonetheless, the BIC preferred fixing $\dot{\gamma }$ at 0 ${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$ (${\rm{\Delta }}\mathrm{BIC}=1$).

Next, we considered eccentric models. In the circular models, ${K}_{{\rm{b}}}=1.6\pm 1.0$ ${\rm{m}}\,{{\rm{s}}}^{-1}$ and ${K}_{{\rm{c}}}=2.3\pm 1.1$ ${\rm{m}}\,{{\rm{s}}}^{-1}$. Given that the reflex motion due to K2-32c and K2-32d were only detected at the 2σ level, we cannot place meaningful constraints on their eccentricities. Therefore, we fixed their eccentricities at zero in our fits and allowed the eccentricity of K2-32b to float. We found that the eccentricity of K2-32b was consistent with zero, and placed an upper limit of ${e}_{{\rm{b}}}\lt 0.23$ (95% conf.). We therefore adopted the circular model for the K2-32 system parameters. The physical properties of the K2-32 system along with our circular and eccentric models are listed Table 4. The best-fit circular model is shown in Figure 2.

Zoom In Zoom Out Reset image size

Table 4.  K2-32 System Parameters

	Value			Reference
Stellar Parameters
Identifier	EPIC-205071984
${T}_{\mathrm{eff}}$ (K)	$5275\pm 60$			A
$\mathrm{log}g$ (dex)	$4.49\pm 0.05$			A
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	$-0.02\pm 0.04$			A
$v\sin i$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	$0.7$			A
${M}_{\star }$ (${M}_{\odot }$)	$0.856\pm 0.028$			A
${R}_{\star }$ (${R}_{\odot }$)	${0.845}_{-0.035}^{+0.044}$			A
age (Gyr)	$7.9\pm 4.5$			A
Apparent V (mag)	$12.31\pm 0.02$			A
	Planet b	Planet c	Planet d
Transit Model
P (days)	$8.99213\pm 0.00016$	$20.6602\pm 0.0017$	$31.7154\pm 0.0022$	B
T0 (BJD-2454833)	$2076.91832\pm 0.00055$	$2128.4067\pm 0.0032$	$2070.7901\pm 0.0026$	B
${R}_{{\rm{P}}}$ (${R}_{\oplus }$)	$5.13\pm 0.28$	$3.01\pm 0.25$	$3.43\pm 0.35$	A
a (au)	$0.08036\pm 0.00088$	$0.1399\pm 0.0015$	$0.1862\pm 0.0020$	A
${S}_{\mathrm{inc}}$ (${S}_{\oplus }$)	${77.7}_{-8.3}^{+10.8}$	${25.6}_{-2.7}^{+3.6}$	${14.5}_{-1.5}^{+2.0}$	A
${T}_{\mathrm{eq}}$ (K)	$817\pm 25$	$619\pm 19$	$537\pm 16$	A
Circular RV Model (Adopted)
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$5.63\pm 0.91$	$1.6\pm 1.0$	$2.3\pm 1.1$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-1.69\pm 0.85$			A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$1.07\pm 0.84$			A
${\gamma }_{\mathrm{PFS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-6.7\pm 3.2$			A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)			A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${3.77}_{-0.65}^{+0.81}$			A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$4.13\pm 0.73$			A
${\sigma }_{{\rm{jit,PFS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${6.5}_{-2.4}^{+4.3}$			A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$16.5\pm 2.7$	$\lt 12.1$ (95% conf.)	$10.3\pm 4.7$	A
ρ (g cm−3)	$0.67\pm 0.16$	$\lt 2.7$ (95% conf.)	${1.38}_{-0.67}^{+0.92}$	A
Eccentric RV Model
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$5.60\pm 0.93$	$1.7\pm 1.0$	$2.4\pm 1.1$	A
e	<0.23 (95% conf.)	Unconstrained	Unconstrained	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-1.70\pm 0.87$			A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$1.15\pm 0.85$			A
${\gamma }_{\mathrm{PFS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-6.7\pm 3.4$			A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)			A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${3.88}_{-0.65}^{+0.82}$			A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$4.13\pm 0.73$			A
${\sigma }_{{\rm{jit,PFS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${6.6}_{-2.5}^{+4.6}$			A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$16.5\pm 2.8$	$\lt 12.7$ (95% conf.)	$10.9\pm 4.9$	A
ρ (g cm−3)	${0.58}_{-0.13}^{+0.16}$	$\lt 2.5$ (95% conf.)	${1.29}_{-0.61}^{+0.86}$	A

Note. A: this work; B: Crossfield et al. (2016). Because the 2σ confidence interval on ${K}_{{\rm{c}}}$ includes zero, we report upper limits on the mass and density of K2-32.

Download table as:  ASCIITypeset image

For K2-32b, we measured ${M}_{{\rm{P}}}=16.5\pm 2.7$ ${M}_{\oplus }$, which is lower than, but within, the 1σ confidence interval of ${M}_{{\rm{P}}}$ = 21.1 ± 5.9, measured by Dai et al. (2016). We measure a density of ρ = $0.67\pm 0.16$ g cm⁻³, which is low compared to other sub-Saturns of similar size with RV mass measurements. However, when compared to the ensemble of mass measurements from TTVs and RVs, K2-32b is of intermediate density (see Section 5).

Figure 2 shows that the Dai et al. (2016) measurements were not well-sampled during the quadrature times of K2-32c and K2-32d, leading to poor constraints on their Doppler semi-amplitudes with significant covariance between ${K}_{{\rm{c}}}$ and ${K}_{{\rm{d}}}$ due the 3:2 period ratios of the planets. Our combined data set has more uniform sampling in phase with minimal covariance between ${K}_{{\rm{c}}}$ and ${K}_{{\rm{d}}}$.

We achieve marginal detections of K2-32c and K2-32d, which have masses of $6.2\pm 3.9$ ${M}_{\oplus }$ and $10.3\pm 4.7$ ${M}_{\oplus }$, respectively. Because K2-32c is not quite a 2σ detection, we conservatively report an upper limit of ${M}_{{\rm{P}}}\lt 12.1$ ${M}_{\oplus }$ (95% confidence) to be conservative. Continued Doppler monitoring of K2-32 is necessary to place tighter constraints on the masses and orbits of K2-32c and K2-32d. Although such observations are challenging with current instruments given the faint host star (V = 12.3 mag), the dynamical richness of this system makes these observations worthwhile.

K2-39 hosts a single transiting planet, K2-39b, with P = 4.60 days, which was first confirmed by Van Eylen et al. (2016a). When fitting the photometry, Van Eylen et al. (2016a) and Crossfield et al. (2016) arrived at different values of the planet-to-star radius ratio, ${R}_{{\rm{P}}}$/${R}_{\star }$, of 1.93 ± 0.1% and 2.52 ± 0.27%, respectively. The Crossfield et al. (2016) solution favored a grazing impact parameter of $b={1.10}_{-0.09}^{+0.07}$ and is responsible for the larger inferred radius ratio. Motivated by this discrepancy, we re-examined the K2-39 photometry produced by the k2phot pipeline¹⁸ and two other publicly available light curves produced by the k2sff and k2sc pipelines (Vanderburg & Johnson 2014; Aigrain et al. 2015). For each light curve, we masked out the in-transit points and modeled the out-of-transit photometry with a Gaussian Process (Rasmussen & Williams 2005) using a squared-exponential kernel with a correlation length of 2 days. We then performed a standard MCMC exploration of the likelihood surface using the batman (Kreidberg 2015) and emcee Python packages to map out parameter uncertainties and covariances. Figure 3 summarizes the results of these different fits. The k2phot, k2sc, and k2sff light curves yielded ${R}_{{\rm{P}}}$/${R}_{\star }$ of ${1.85}_{-0.05}^{+0.15}$%, ${1.76}_{-0.06}^{+0.15}$%, and ${1.66}_{-0.05}^{+0.12}$%, respectively. We also note the significant correlation between ${R}_{{\rm{P}}}$/${R}_{\star }$ and b at large values of b, which is responsible for the rather asymmetric uncertainties. Because ${R}_{{\rm{P}}}$/${R}_{\star }$ differs by 2σ–3σ among the different reductions, we combined the three sets of MCMC chains and adopt ${R}_{{\rm{P}}}/{R}_{\star }=1.79\pm 0.13$% as an intermediate value of the planet-to-star radius ratio with more conservative errors.

Zoom In Zoom Out Reset image size

We obtained 42 spectra of K2-39 with HIRES between 2015 August 10 and 2016 August 21. Van Eylen et al. (2016a) obtained 7 spectra with HARPS, 6 with PFS, and 17 with FIES, which we included in our analysis. In contrast to our K2-27 analysis (Section 4.1), we included FIES RVs because of the larger number of measurements (17 as opposed to 6 for K2-27) and because they are the only non-HIRES data set with a sufficient time baseline to resolve a long timescale activity signal, which we discuss below.

The RVs exhibited a large amplitude variability that was not associated with K2-39b that motivated searches for additional non-transiting planets. Figure 4 shows three successive Keplerian searches in the K2-39 RVs using a modified version of the Two-dimensional Keplerian Lomb–Scargle (2DKLS) periodogram (O'Toole et al. 2009; Howard & Fulton 2016). We observed a peak in the periodogram at P = 4.6 days, which corresponds to the period of K2-39b. When we measured the change in χ² (periodogram power) between a one-planet fit and a two-planet fit, then between a two-planet model and a three-planet model (lower two panels in Figure 4) we found several peaks that fall above the 10% empirical false alarm threshold (eFAP; Howard & Fulton 2016). However, inspection of the S-values (Isaacson & Fischer 2010; see Figure 4), which can correlate with stellar activity that can lead to spurious Doppler signals, showed significant long-term variability that is associated with the RV peak seen at ≈330 days and is likely not associated with another planet. We modeled out the activity signal to search for additional non-transiting planets. As a matter of convenience, we modeled the activity signature as a Keplerian and removed it from the time series. A subsequent search of the residual RVs revealed no other significant signals.

Zoom In Zoom Out Reset image size

We modeled the combined HIRES, HARPS, FIES, and PFS RVs using two Keplerians: one for K2-39b and one as a convenient description of the stellar activity. We first assumed circular orbits and no additional acceleration term, $\dot{\gamma }$. We found that models that included $\dot{\gamma }$ were not favored by the BIC and fixed $\dot{\gamma }$ = 0 ${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$ in subsequent fits. Next, we allowed the eccentricity of K2-39b to float, and found that this model was preferred over the circular model (${\rm{\Delta }}\mathrm{BIC}=-7$). We adopted the eccentric model, which is shown in Figure 5. The properties of the K2-39 system are listed in Table 5.

Zoom In Zoom Out Reset image size

Table 5.  K2-39 Planet Parameters

	Value		Reference
Stellar Parameters
Identifiers	EPIC-206247743
	TYC-5811-835-1
${T}_{\mathrm{eff}}$ (K)	$4912\pm 60$		A
$\mathrm{log}g$ (dex)	$3.58\pm 0.05$		A
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	$0.43\pm 0.04$		A
$v\sin i$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	$0.1$		A
${M}_{\star }$ (${M}_{\odot }$)	${1.192}_{-0.070}^{+0.085}$		A
${R}_{\star }$ (${R}_{\odot }$)	$2.93\pm 0.21$		A
Age (Gyr)	${6.7}_{-1.3}^{+1.7}$		A
Apparent V (mag)	$10.83\pm 0.08$		A
Transit Model
P (days)	$4.60497\pm 0.00077$		B
T0 (BJD-2454833)	$2152.4315\pm 0.0058$		B
${R}_{{\rm{P}}}/{R}_{\star }$ (%)	$1.79\pm 0.13$		A
${R}_{{\rm{P}}}$ (${R}_{\oplus }$)	$5.71\pm 0.63$		A
a (au)	$0.0574\pm 0.0012$		A
${S}_{\mathrm{inc}}$ (${S}_{\oplus }$)	$1356\pm 175$		A
${T}_{\mathrm{eq}}$ (K)	$1670\pm 54$		A
Circular RV Model
P (days)	fixed	329 ± 10	B
T0(BJD)	fixed	2456940 ± 16	B
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$12.7\pm 1.3$	17.4 ± 2.8	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$2.1\pm 2.2$		A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-3.4\pm 3.9$		A
${\gamma }_{\mathrm{PFS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$1.3\pm 3.6$		A
${\gamma }_{\mathrm{FIES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$2.7\pm 3.1$		A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)		A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$7.42\pm 0.86$		A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$6.7\pm 1.4$		A
${\sigma }_{{\rm{jit,PFS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$5.8\pm 1.4$		A
${\sigma }_{{\rm{jit,FIES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$7.2\pm 1.6$		A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$37.3\pm 4.3$	⋯	A
ρ(g cm−3)	${1.10}_{-0.31}^{+0.44}$	⋯	A
Eccentric RV Model (Adopted)
P (days)	fixed	327.2 ± 9.8	B
T0 (BJD)	fixed	2456614 ± 25	B
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$13.8\pm 1.4$	18.4 ± 2.9	A
e	${0.152}_{-0.068}^{+0.084}$	$\cdots$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$1.9\pm 2.2$		A
${\gamma }_{\mathrm{HARPS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$24486.6\pm 3.9$		A
${\gamma }_{\mathrm{PFS}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-1.7\pm 3.5$		A
${\gamma }_{\mathrm{FIES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$24574.5\pm 3.0$		A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)		A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$7.57\pm 0.86$		A
${\sigma }_{{\rm{jit,HARPS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$6.3\pm 1.4$		A
${\sigma }_{{\rm{jit,PFS}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$5.0\pm 1.5$		A
${\sigma }_{{\rm{jit,FIES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$6.7\pm 1.6$		A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$39.8\pm 4.4$	⋯	A
ρ (g cm−3)	${1.17}_{-0.32}^{+0.47}$	⋯	A

Note. A: this work; B: Crossfield et al. (2016). We model a large amplitude (≈20 ${\rm{m}}\,{{\rm{s}}}^{-1}$) stellar activity signal by introducing an additional circular Keplerian.

Download table as:  ASCIITypeset image

For K2-39b, we found a mass of ${M}_{{\rm{P}}}=39.8\pm 4.4$ ${M}_{\oplus }$, which agrees with ${M}_{{\rm{P}}}={50.3}_{-9.4}^{+9.7}$ ${M}_{\oplus }$ reported by Van Eylen et al. (2016a) at the 1σ level. The additional RVs improved the precision of the mass measurement by roughly a factor of two. Our derived planet radius of ${R}_{{\rm{P}}}=5.71\pm 0.63$ ${R}_{\oplus }$ is substantially smaller than ${R}_{{\rm{P}}}=8.2\pm 1.1$ ${R}_{\oplus }$ reported by Van Eylen et al. (2016a). This is largely due to the smaller stellar radius (see Section 3). Our adopted value of ${R}_{{\rm{P}}}/{R}_{\star }=1.79\pm 0.13$% is also smaller than the Van Eylen et al. (2016a) value of ${R}_{{\rm{P}}}/{R}_{\star }$ = 1.93 ± 0.1%. Although the difference in ${R}_{{\rm{P}}}/{R}_{\star }$ is not as large in a fractional sense as the difference in ${R}_{\star }$, it also contributes to a smaller derived ${R}_{{\rm{P}}}$. Thus, our derived density of $\rho ={1.17}_{-0.32}^{+0.47}$ g cm⁻³ is significantly larger than $\rho ={0.50}_{-0.17}^{+0.29}$ g cm⁻³, reported by Van Eylen et al. (2016a).

K2-108, listed as EPIC-211736671 in the Ecliptic Planet Input Catalog (Huber et al. 2016), is a V = 12.3 mag star observed during K2 Campaign 5. We identified K2-108 as a likely planet according to our team's standard methodology, described in detail in Crossfield et al. (2016). In brief, we identified a set of transits having P = 4.73 days and elevated K2-108 to the status of "planet candidate." We fit the light curve according to standard procedures and show the best-fitting model light curve in Figure 6. Follow-up spectroscopic observations revealed that K2-108 is a metal-rich ($[\mathrm{Fe}/{\rm{H}}]=0.33\pm 0.04$ dex), slightly evolved G star having a radius of ${R}_{\star }=1.75\pm 0.14$ ${R}_{\odot }$. The spectroscopically determined stellar parameters along with the results from our light-curve fitting are listed in Table 6.

Zoom In Zoom Out Reset image size

Table 6.  System Parameters of K2-108

	Value	Reference
Stellar Parameters
Identifier	EPIC-211736671
${T}_{\mathrm{eff}}$ (K)	$5474\pm 60$	A
$\mathrm{log}g$ (dex)	$3.99\pm 0.05$	A
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	$0.33\pm 0.04$	A
$v\sin i$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	$\lt 2$	A
${M}_{\star }$ (${M}_{\odot }$)	${1.121}_{-0.053}^{+0.065}$	A
${R}_{\star }$ (${R}_{\odot }$)	$1.75\pm 0.14$	A
age (Gyr)	$7.8\pm 1.5$	A
Apparent V (mag)	$12.33\pm 0.01$	A
	Planet b
Transit Model
P (days)	$4.73401\pm 0.00024$	A
T0 (BJD-2454833)	$2312.0965\pm 0.0019$	A
${R}_{{\rm{P}}}$ (${R}_{\oplus }$)	$5.28\pm 0.54$	A
a (au)	$0.0573\pm 0.0010$	A
${S}_{\mathrm{inc}}$ (${S}_{\oplus }$)	$762\pm 100$	A
${T}_{\mathrm{eq}}$ (K)	$1446\pm 48$	A
Circular RV Model
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$19.4\pm 2.3$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-4.0\pm 1.6$	A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	0 (fixed)	A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${6.0}_{-1.1}^{+1.5}$	A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$55.1\pm 6.8$	A
ρ (g cm−3)	${2.05}_{-0.54}^{+0.75}$	A
Eccentric RV Model (Adopted)
K (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$21.3\pm 1.4$	A
e	$0.180\pm 0.042$	A
${\gamma }_{\mathrm{HIRES}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	$-2.63\pm 0.91$	A
$\dot{\gamma }$ (${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$)	$-11.0\pm 2.3$	A
${\sigma }_{{\rm{jit,HIRES}}}$ (${\rm{m}}\,{{\rm{s}}}^{-1}$)	${2.86}_{-0.78}^{+1.01}$	A
${M}_{{\rm{P}}}$ (${M}_{\oplus }$)	$59.4\pm 4.4$	A
ρ (g cm−3)	${2.22}_{-0.55}^{+0.77}$	A

Note. A: this work.

Download table as:  ASCIITypeset image

We obtained 20 spectra of K2-108 with HIRES between 2015 December 23 and 2016 November 25. We first considered circular models with no acceleration term, $\dot{\gamma }$. We found that eccentric models with non-zero $\dot{\gamma }$ (shown in Figure 7) were favored over circular models (${\rm{\Delta }}\mathrm{BIC}=-25$). The parameters are summarized in Table 6. At $59.4\pm 4.4$ ${M}_{\oplus }$, K2-108b is remarkably massive for a $5.28\pm 0.54$ ${R}_{\oplus }$ planet, implying a large heavy-element component.

Zoom In Zoom Out Reset image size

Although our RV analysis verified the planetary nature of the transiting object, we assessed the possibility of additional stellar companions in the photometric aperture that could appreciably dilute the observed transit, resulting in an incorrectly derived planetary radius. From the light curve fits, the planet-to-star radius ratio is ${R}_{{\rm{P}}}/{R}_{\star }=2.82\pm 0.17 \% .$ For an additional star to significantly alter the observed transit depth requires a flux ratio, ${F}_{2}/{F}_{1}\approx \sigma ({({R}_{{\rm{P}}}/{R}_{\star })}^{2})/{({R}_{{\rm{P}}}/{R}_{\star })}^{2}\approx 0.10$, or ΔV ≈ 2.5 mag (Ciardi et al. 2015).

A search for secondary spectral lines in the HIRES template spectrum (Kolbl et al. 2015) revealed no additional stellar companions having ΔV ≲ 5 mag and ΔRV ≳ 15 $\mathrm{km}\,{{\rm{s}}}^{-1}$, corresponding to a physical separation of ≲1 $\mathrm{au}$.

We obtained high-resolution speckle imaging at 692 nm with the Differential Speckle Survey Instrument (DSSI; Horch et al. 2012), a visitor instrument used at the Gemini-North 8.1 m telescope on 2016 January 13. The observations revealed no additional companions with ΔV < 2.5 mag in the K2 photometric aperture down to separations of 80 mas, or ≈32 au in projected separation at the distance¹⁹ of K2-108. Additional high-resolution imaging with DSSI at 880 nm and Keck/NIRC2 at the K-band also revealed no additional companions. Of the three high-resolution images, the DSSI image at 692 nm (shown in Figure 8) provides the tightest contrast curve in the Kepler bandpass. All imaging data sets are available on the Exoplanet Follow-up Observing Program (ExoFOP) website.²⁰

Zoom In Zoom Out Reset image size

A stellar companion having a ≈ 1–40 $\mathrm{au}$ would have evaded the aforementioned follow-up observations. However, such an object would induce a reflex acceleration on K2-108, which would be easily detectable by our RVs:

$$\begin{eqnarray*}&&\dot{\gamma }\approx \displaystyle \frac{{{GM}}_{\star }}{{a}^{2}}\approx 120\,{\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}\,\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right){\left(\displaystyle \frac{a}{40\mathrm{au}}\right)}^{-2}.\end{eqnarray*}$$

Over the ∼1 year observation baseline, we see only weak evidence for long-term acceleration ($\dot{\gamma }=-11.0\pm 2.3$ ${\rm{m}}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$). To be consistent with the observed $\dot{\gamma }$, a companion at $\approx 40\,\mathrm{au}$ would be $\lesssim 0.1\,{M}_{\odot }$ and much too faint to alter the derived planet radius. At smaller separations, the limits on the mass of a putative companion grow more stringent.

Here, we put the four sub-Saturns presented in this paper in the context of other planets in their size class. Starting with the database of exoplanet properties hosted at the NASA Exoplanet Archive (NEA; Akeson et al. 2013), we constructed a list of sub-Saturns from the literature that have densities measured to 50% or better. We supplemented the list with additional measurements that have yet to be ingested into the NEA and removed a few planets with unreliable measurements. Including the measurements from this work, we found 19 sub-Saturns that passed our quality cuts; they are listed in Table 7.

Table 7.  Sub-Saturns with Well-measured Densities

Name	NP	${R}_{{\rm{P}}}$	${M}_{{\rm{P}}}$	ρ	e	${T}_{\mathrm{eq}}$	${f}_{\mathrm{env}}$	$[\mathrm{Fe}/{\rm{H}}]$
		${R}_{\oplus }$	${M}_{\oplus }$	g cm−3		K	%	dex
Kepler-4 b	1	${4.00}_{-0.21}^{+0.21}$	${24.5}_{-3.8}^{+3.8}$	${2.09}_{-0.44}^{+0.53}$	⋯	1597	${6.7}_{-1.4}^{+1.3}$	+0.17
GJ 436b	1	${4.17}_{-0.17}^{+0.17}$	${22.1}_{-2.3}^{+2.3}$	${1.67}_{-0.26}^{+0.30}$	${0.1383}_{-0.0002}^{+0.0002}$	659	${12.6}_{-1.9}^{+1.9}$	⋯
Kepler-11 e	6	${4.19}_{-0.09}^{+0.07}$	${8.0}_{-2.1}^{+1.5}$	${0.60}_{-0.14}^{+0.15}$	${0.0120}_{-0.0060}^{+0.0060}$	630	${14.9}_{-0.7}^{+0.7}$	−0.04
Kepler-413 b	1	${4.35}_{-0.10}^{+0.10}$	${51.0}_{-21.0}^{+22.0}$	${2.40}_{-1.00}^{+1.00}$	${0.1185}_{-0.0017}^{+0.0018}$	348	${10.8}_{-2.0}^{+3.8}$	⋯
K2-27b	1	${4.48}_{-0.23}^{+0.23}$	${30.9}_{-4.6}^{+4.6}$	${1.88}_{-0.38}^{+0.46}$	${0.2510}_{-0.0880}^{+0.0880}$	910	${13.9}_{-2.5}^{+2.4}$	+0.13
Kepler-223 e	4	${4.60}_{-0.41}^{+0.27}$	${4.8}_{-1.2}^{+1.4}$	${0.27}_{-0.09}^{+0.11}$	${0.0510}_{-0.0190}^{+0.0190}$	944	${16.5}_{-2.6}^{+2.5}$	+0.06
HAT-P-11 b	1	${4.73}_{-0.16}^{+0.16}$	${25.7}_{-2.9}^{+2.9}$	${1.34}_{-0.20}^{+0.22}$	${0.1980}_{-0.0460}^{+0.0460}$	861	${17.1}_{-1.5}^{+1.5}$	+0.31
K2-32b	3	${5.13}_{-0.28}^{+0.28}$	${16.5}_{-2.7}^{+2.7}$	${0.67}_{-0.15}^{+0.18}$	⋯	815	${22.5}_{-3.1}^{+3.0}$	−0.02
Kepler-25 c	3	${5.20}_{-0.09}^{+0.09}$	${24.6}_{-5.7}^{+5.7}$	${0.96}_{-0.23}^{+0.24}$	⋯	1018	${21.4}_{-1.3}^{+1.3}$	−0.04
Kepler-223 d	4	${5.24}_{-0.45}^{+0.26}$	${8.0}_{-1.3}^{+1.5}$	${0.30}_{-0.07}^{+0.10}$	${0.0370}_{-0.0170}^{+0.0180}$	1040	${22.3}_{-3.2}^{+3.4}$	+0.06
K2-108b	1	${5.28}_{-0.54}^{+0.54}$	${59.4}_{-4.4}^{+4.4}$	${2.21}_{-0.59}^{+0.90}$	${0.1800}_{-0.0420}^{+0.0420}$	1440	${16.0}_{-5.4}^{+4.8}$	+0.33
Kepler-18 c	3	${5.49}_{-0.26}^{+0.26}$	${17.3}_{-1.9}^{+1.9}$	${0.57}_{-0.10}^{+0.12}$	⋯	979	${25.7}_{-2.8}^{+2.6}$	+0.19
K2-24 b	2	${5.68}_{-0.56}^{+0.56}$	${21.0}_{-5.4}^{+5.4}$	${0.62}_{-0.22}^{+0.31}$	⋯	766	${28.4}_{-6.2}^{+7.4}$	+0.42
K2-39b	1	${5.71}_{-0.63}^{+0.63}$	${39.7}_{-4.6}^{+4.6}$	${1.16}_{-0.34}^{+0.54}$	${0.1500}_{-0.0760}^{+0.0760}$	1689	${18.1}_{-4.8}^{+5.2}$	+0.43
Kepler-101 b	2	${5.77}_{-0.79}^{+0.85}$	${51.1}_{-4.7}^{+5.1}$	${1.46}_{-0.50}^{+0.90}$	${0.0860}_{-0.0590}^{+0.0800}$	1547	${19.5}_{-6.9}^{+7.9}$	+0.33
Kepler-87 c	2	${6.14}_{-0.29}^{+0.29}$	${6.4}_{-0.8}^{+0.8}$	${0.15}_{-0.03}^{+0.03}$	${0.0390}_{-0.0120}^{+0.0120}$	440	${35.6}_{-3.7}^{+3.4}$	−0.17
HATS-7 b	1	${6.31}_{-0.38}^{+0.52}$	${38.1}_{-3.8}^{+3.8}$	${0.83}_{-0.17}^{+0.23}$	⋯	1070	${33.3}_{-5.8}^{+6.1}$	+0.25
HAT-P-26 b	1	${6.33}_{-0.36}^{+0.81}$	${18.8}_{-2.2}^{+2.2}$	${0.40}_{-0.10}^{+0.15}$	${0.1240}_{-0.0600}^{+0.0600}$	981	${35.8}_{-7.5}^{+6.9}$	−0.04
CoRoT-8 b	1	${6.39}_{-0.22}^{+0.22}$	${69.9}_{-9.5}^{+9.5}$	${1.47}_{-0.25}^{+0.27}$	⋯	844	${32.2}_{-3.1}^{+3.5}$	+0.30
Kepler-56 b	3	${6.51}_{-0.28}^{+0.29}$	${22.1}_{-3.6}^{+3.9}$	${0.44}_{-0.09}^{+0.10}$	⋯	1479	${26.3}_{-2.4}^{+2.4}$	+0.37
Kepler-18 d	3	${6.98}_{-0.33}^{+0.33}$	${16.4}_{-1.4}^{+1.4}$	${0.26}_{-0.04}^{+0.05}$	⋯	784	${44.5}_{-3.6}^{+3.2}$	+0.19
Kepler-79d	4	${7.16}_{-0.16}^{+0.13}$	${6.0}_{-1.6}^{+2.1}$	${0.09}_{-0.03}^{+0.03}$	${0.0250}_{-0.0230}^{+0.0590}$	626	${42.6}_{-3.3}^{+2.5}$	−0.02
K2-24 c	2	${7.82}_{-0.72}^{+0.72}$	${27.0}_{-6.9}^{+6.9}$	${0.31}_{-0.10}^{+0.14}$	⋯	605	${57.3}_{-9.9}^{+9.1}$	+0.42

Note. List of sub-Saturns having densities measured to 50% or better, assembled from the NASA Exoplanet Archive, this work, and other sources. N_P—total number of detected planets in the system, e—orbital eccentricity; we do not report eccentricity if only upper limits are available. ${T}_{\mathrm{eq}}$ refers to the blackbody temperature (i.e., assuming zero albedo). ${f}_{\mathrm{env}}$—"envelope fraction," i.e., fraction of planet's mass in H/He in the two-component modeling of planet mass and radius, described in Section 5.2. Notes on individual systems: Kepler-4 b—Borucki et al. (2010b), GJ 436b—Butler et al. (2004), Kepler-11 e—Lissauer et al. (2013), Kepler-413 b—Kostov et al. (2014), K2-27b—This work, Kepler-223 e—Mills et al. (2016), HAT-P-11 b—Bakos et al. (2010), K2-32b—This work, Kepler-25 c—Marcy et al. (2014), Kepler-223 d—Mills et al. (2016), K2-108b—This work, Kepler-18 c—Cochran et al. (2011), K2-24 b—Petigura et al. (2016), K2-39b—This work, Kepler-101 b—Bonomo et al. (2014), Kepler-87 c—Ofir et al. (2014), HATS-7 b—Bakos et al. (2015), HAT-P-26 b—Hartman et al. (2011), CoRoT-8 b—Bordé et al. (2010), adopted 3 ± 1 Gyr; Kepler-56 b—Huber et al. (2013), Kepler-18 d—Cochran et al. (2011), Kepler-79d—Jontof-Hutter et al. (2014), K2-24 c—Petigura et al. (2016).

Download table as:  ASCIITypeset image

In Figure 9, we show the measured masses and sizes of these planets, highlighting the measurements from this work. Remarkably, for sub-Saturns, there is little correlation between a planet's mass and size. These planets have a nearly uniform distribution of mass from ${M}_{{\rm{P}}}\approx 6\mbox{--}60$ ${M}_{\oplus }$. We note that the hottest planets tend to have higher masses, while cool planets have both high and low masses. Figure 10 shows planet mass and radius, cast in terms of planet size and planet density. The decreasing densities toward larger sizes can be understood simply in terms of larger radii, given there is no strong trend of larger planet masses with larger planet size.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

One advantage of sub-Saturns is that in this size range, both heavy elements and low-density gaseous envelopes contribute significantly to the total planet mass. A consequence is, to first order, sub-Saturns can be approximated as two-component planets consisting of a rocky heavy-element core, surrounded by an envelope of H/He (Lopez & Fortney 2014; Petigura et al. 2016). Here we use the results of Lopez & Fortney (2014), who simulated the internal structure and thermal evolution of planets with solar-composition H/He envelopes atop Earth-composition cores. These simulations were run over a wide swath of parameter space for planets with different masses, ${M}_{{\rm{P}}}$, different envelope fractions, ${f}_{\mathrm{env}}={M}_{\mathrm{env}}/{M}_{{\rm{P}}}$, and on orbits receiving different levels of incident stellar irradiation, ${S}_{\mathrm{inc}}$. The model planets were allowed to evolve over time and Lopez & Fortney (2014) noted the planet radii at specified intervals. The result of these simulations is a four-dimensional grid of planet radius, ${R}_{{\rm{P}}}$, sampled at various combinations of (${M}_{{\rm{P}}}$, ${f}_{\mathrm{env}}$, ${S}_{\mathrm{inc}}$, age).

We used this grid to solve for the values of the planet ${f}_{\mathrm{env}}$ that are consistent with the observed (${M}_{{\rm{P}}}$, ${R}_{{\rm{P}}}$, ${S}_{\mathrm{inc}}$, and age). We drew ${M}_{{\rm{P}}}$, ${R}_{{\rm{P}}}$, and ${S}_{\mathrm{inc}}$ measured posterior distributions and interpolated the model grid in order to derive ${f}_{\mathrm{env}}$. We assumed a uniform age of 5 Gyr. With the exception of a few stars analyzed with asteroseismology, most of the system ages are derived from isochrone fitting and are thus uncertain at the ≈2–3 Gyr level. Fortunately, the derived ${f}_{\mathrm{env}}$ fraction is not sensitive to the adopted system age. Adopting a uniform age of 2 Gyr resulted in a typical change in the derived ${f}_{\mathrm{env}}$ of ≈2%. The resulting values of ${f}_{\mathrm{env}}$ are listed in Table 7.

We have made several approximations when computing ${f}_{\mathrm{env}}$. We have ignored the possibility of water or other volatile ices contributing significantly to the heavy-element cores of these planets. Lopez & Fortney (2014), however, showed that including ices in the core does not significantly alter the radius–composition relationship for planets in this size range. Because the H/He envelope represents most of the planet volume, our inferred ${f}_{\mathrm{env}}$ does not depend sensitively on the detailed composition of the planet core. We have also assumed that all the heavy elements are concentrated in the planets' cores rather than being distributed throughout the envelope.

If photo-evaporation plays a dominant role in sculpting the gaseous envelopes of sub-Saturns, one might expect the envelope fraction, ${f}_{\mathrm{env}}$, to correlate with the energy it receives from its host star. Figure 11 shows the total planet mass (${M}_{{\rm{P}}}$) and envelope fraction (${f}_{\mathrm{env}}$) as a function of the blackbody equilibrium temperature. As a matter of convenience, we include core mass (${M}_{\mathrm{core}}$) and envelope mass (${M}_{\mathrm{env}}$), which can be trivially computed from ${M}_{{\rm{P}}}$ and ${f}_{\mathrm{env}}$. We do not observe a strong one-to-one correlation envelope fraction and equilibrium temperature. We do, however, observe that cool planets span a large range of ${f}_{\mathrm{env}}\approx 10 \% \mbox{--}50 \%$ while hot planets span a more narrow range of ${f}_{\mathrm{env}}\approx 10 \% \mbox{--}20 \% .$ Perhaps photo-evaporation excludes planets from occupying certain domains in the ${f}_{\mathrm{env}}$–${T}_{\mathrm{eq}}$ plane.

Zoom In Zoom Out Reset image size

To further explore the possible role of photo-evaporation, we considered two quantities that are more directly related to a planet's susceptibility to photo-evaporation: XUV heating and planet binding energy. XUV heating is the total lifetime-integrated XUV flux incident at a planet's orbit multiplied by the planet's current cross section. Figure 12, which is updated from Lopez & Fortney (2014), shows the planet XUV heating (in ergs) versus planet binding energy (in erg). This sort of diagram has been used as supporting evidence for the role of photo-evaporation in sculpting the envelopes of highly irradiated sub-Neptunes (Lecavelier Des Etangs 2007; Lopez et al. 2012; Owen & Wu 2013). The dashed line shows the envelope survival threshold predicted by photo-evaporation and thermal evolution models in Lopez et al. (2012), and the absence of planets with gaseous envelopes above this line suggests that planets near this threshold have experienced photo-evaporation. Although a few sub-Saturns lie close to this threshold, most lie well below it, indicating that they are sufficiently massive or are on wide enough orbits to be immune to significant photo-evaporation.

Zoom In Zoom Out Reset image size

One may interpret present-day stellar metallicity as a proxy for the metal enrichment of the protoplanetary disk because the disk and star formed from the same molecular cloud. However, the mean metallicity of the protoplanetary disk may be different from the local disk metallicity at the location of planet formation. With this caveat in mind, we nonetheless looked for correlations between host star metallicity and the observed properties of sub-Saturns.

In contrast to equilibrium temperatures, we observe a stronger set of correlations between the planetary properties of sub-Saturns and host star metallicity, $[\mathrm{Fe}/{\rm{H}}]$. Figure 13 shows ${M}_{{\rm{P}}}$, ${f}_{\mathrm{env}}$, ${M}_{\mathrm{core}}$, and ${M}_{\mathrm{env}}$ against $[\mathrm{Fe}/{\rm{H}}]$. Sub-Saturns orbiting metal-rich stars tend to be more massive. Interestingly, both the planetary core mass, ${M}_{\mathrm{core}}$, and envelope mass, ${M}_{\mathrm{env}}$, appear to increase with stellar metallicity. This is understandable, however, given that planets with more massive cores should generally be able to accrete larger gaseous envelopes before their disks dissipate (Lee & Chiang 2015). This suggests that the metal-rich hosts had more solids available in their disk, allowing those sub-Saturns to form more massive cores, and that these larger cores were then able accrete more massive envelopes.

Zoom In Zoom Out Reset image size

It is worthwhile to compare the observed ${M}_{{\rm{P}}}$–$[\mathrm{Fe}/{\rm{H}}]$ trend to the stellar metallicity distribution of Kepler planet hosts. Buchhave et al. (2012) observed that planets smaller than $\approx 4\,{R}_{\oplus }$ are found around stars of wide-ranging metallicities (−0.5 ≲ $[\mathrm{Fe}/{\rm{H}}]$ ≲ +0.5 dex), while larger planets are typically found around more metal-rich stars (−0.2 ≲ $[\mathrm{Fe}/{\rm{H}}]$ ≲ +0.5 dex). This is consistent with our sample, constructed from planets found by Kepler, K2, and other surveys. Stellar metallicity of >−0.2 dex seems to be an important criterion for forming sub-Saturns. However, the additional metals in the disk seem to result in more massive final planets.

The path through which increased metallicity produces more massive sub-Saturns could proceed in one of two ways: (1) disks with more solid material form substantially more massive planet cores that grow smoothly into more massive planets or (2) disks with more solid material form slightly more massive cores that perturb neighboring planets causing collisions and mergers. We have a slight preference for the latter explanation, given the eccentricity distribution of sub-Saturns, explored in Section 5.5. It is also plausible that metal-enriched disks could form planet cores more quickly, allowing for a long period of gas accretion. However, we see no evidence of this given the absence of a strong correlation between ${f}_{\mathrm{env}}$ and host star metallicity.

In addition to the trends with stellar metallicity shown in Figure 13, it is also interesting to examine whether there are trends in the heavy-element abundances of sub-Saturns after controlling for the dependence on stellar metallicity. Recently, Thorngren et al. (2016) examined planet metal mass fraction, Z_P, for 47 planets having ${M}_{{\rm{P}}}\approx 30\mbox{--}3000$ ${M}_{\oplus }$ relative to the heavy-element fraction of their parent stars, ${Z}_{\star }={Z}_{\odot }{10}^{[\mathrm{Fe}/{\rm{H}}]}$. Z_P was computed via ${Z}_{{\rm{P}}}={M}_{\mathrm{core}}/{M}_{\mathrm{env}}$ using thermal evolution models similar to those used here. Thorngren et al. (2016) observed an anti-correlation between ${M}_{{\rm{P}}}$ and ${Z}_{{\rm{P}}}/{Z}_{\star }$. They argued that such an anti-correlation can be understood in terms of traditional core-accretion formation theory if one assumes that planets below the isolation mass are able to accrete all of the solids in their isolation zone, typically ≈3.5 Hill radii (Lissauer 1993), but not all of their gas. Given these assumptions, Equation (9) of Thorngren et al. (2016) predicts the planetary metal enrichment:

$$\begin{eqnarray}&&\displaystyle \frac{{Z}_{{\rm{P}}}}{{Z}_{\star }}=3{f}_{H}{f}_{e}\displaystyle \frac{H}{a}{Q}^{-1}{\left(\displaystyle \frac{{M}_{{\rm{P}}}}{{M}_{\star }}\right)}^{-2/3}.\end{eqnarray} \tag{ 1 }$$

Here, f_H ∼ 3.5 is the approximate number of Hill radii from which a planet can effectively accrete solids, f_e ∼ 1 is an enrichment factor to allow for metal enhancement due to radial drift by solids, H is the disk scale height, a is the semimajor axis, and Q is the Toomre disk instability parameter (Toomre 1964). Thorngren et al. (2016) found that Equation (1) can reproduce the observed trend between ${M}_{{\rm{P}}}$ and Z_P/Z_⋆ if f_H = 3.5, f_e = 1, and Q = 5.

Figure 14 compares our sample of sub-Saturns to the predictions of Equation (1). Sub-Saturns are a valuable laboratory for testing the physics of envelope accretion because they are larger than the sub-Neptunes, which typically have ${f}_{\mathrm{env}}\lesssim 10 \%$, and the gas giants, which are nearly entirely envelope (${f}_{\mathrm{env}}\sim 100 \%$). Following Thorngren et al. (2016), we approximate the planetary metal abundance by setting ${Z}_{{\rm{P}}}={M}_{\mathrm{core}}/{M}_{{\rm{P}}}$. For the most massive sub-Saturns (having ${M}_{{\rm{P}}}/{M}_{\star }\gtrsim {10}^{-4}$), where our sample overlaps with the Thorngren et al. (2016) sample, we find good agreement with the predictions of Equation (1), as shown by the dotted line. Below ${M}_{{\rm{P}}}/{M}_{\star }\sim {10}^{-4}$, however, we find a significant increase in dispersion below this relation, with many planets being significantly less enriched in metals than expected from Equation (1).

Zoom In Zoom Out Reset image size

One interpretation is that this increase in scatter simply reflects the natural transition between giant planets, which essentially accrete all of the heavy elements in their feeding zones, and low mass planets, which do not. Equation (1) assumes that the solids in the disk have fully decoupled from the gas, and that a planet can successfully accrete all of the solids near its isolation zone. At lower core masses, gravitational focusing becomes more important, gas damping of planetesimals eccentricities becomes more efficient, and collision and growth timescales become longer. All of these factors mean less massive cores may not accrete all solids within ≈3.5 Hill radii, which may instead be dispersed or incorporated into other planets in compact, multiplanet systems. In summary, we examined whether the correlation between planet metal enrichment and planet masses observed by Thorngren et al. (2016) for planets having ${M}_{{\rm{P}}}$ = 30–3000 ${M}_{\oplus }$ is present among the sub-Saturns. We do not observe a strong correlation, indicating a possible transition in the formation pathways of planets with ${M}_{{\rm{P}}}\lesssim 30$ ${M}_{\oplus }$.

The fits to K2-27, K2-39, and K2-108 RVs favored non-zero orbital eccentricities of $0.251\pm 0.088$, ${0.152}_{-0.068}^{+0.084}$, and $0.180\pm 0.042$, respectively. Given that these planets are on short orbital periods of 4.6–6.8 days, respectively, it is worthwhile to consider the extent to which tides are expected to damp away eccentricity. The timescale for eccentricity damping (Goldreich & Soter 1966) is given by

$$\begin{eqnarray}&&{\tau }_{e}=\displaystyle \frac{4}{63}\left(\displaystyle \frac{{Q}^{\prime }}{n}\right)\left(\displaystyle \frac{{M}_{{\rm{P}}}}{{M}_{\star }}\right){\left(\displaystyle \frac{a}{{R}_{{\rm{P}}}}\right)}^{5}.\end{eqnarray} \tag{ 2 }$$

Here, $n=\sqrt{{{GM}}_{\star }/{a}^{3}}$ is the mean motion, and ${Q}^{\prime }$, the modified tidal quality factor, is given by ${Q}^{\prime }=3Q/2{k}_{2}$, where Q is the specific dissipation function and k₂ is the Love number (see Goldreich & Soter 1966; Murray & Dermott 1999; Mardling & Lin 2004). ${Q}^{\prime }$ is quite uncertain even for planets in the solar system. As a point of reference Lainey et al. (2017) give ${Q}^{\prime }\,\approx$ 6000–18,000 for Saturn, based on Cassini ranging data. Tittemore & Wisdom (1989, 1990) give Q ≈ 11,000–39,000 for Uranus based on studies of the Uranian satellites, which translates to ${Q}^{\prime }\approx$ 165,000–585,000, adopting k₂ = 0.104 from Gavrilov & Zharkov (1977). ${Q}^{\prime }$ is even more uncertain for sub-Saturns, which have no solar system analogues. Here, we adopt ${Q}^{\prime }={10}^{5}$ for the sub-Saturns, with the understanding that this estimate is only good to an order of magnitude. Re-writing Equation (2),

$$\begin{eqnarray*}{\tau }_{e} & \sim & 1.8\,\mathrm{Gyr}\left(\tfrac{{Q}^{\prime }}{{10}^{5}}\right)\left(\tfrac{P}{10\,\mathrm{days}}\right)\left(\tfrac{{M}_{{\rm{P}}}}{10\,{M}_{\oplus }}\right){\left(\tfrac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1}\\ & & \times {\left(\tfrac{a}{0.05\mathrm{au}}\right)}^{5}{\left(\tfrac{{R}_{{\rm{P}}}}{4{R}_{\oplus }}\right)}^{-5}.\end{eqnarray*}$$

For K2-27b, we find τ_e ∼ 10 Gyr, comparable to the age of the system, suggesting that if this eccentricity was caused by planet–planet scattering early in the star's lifetime, the eccentricity could persist to the present day. K2-39b and K2-108b are slightly larger than K2-27b and also orbit closer to their host stars. Because the circularization timescale is a strong function of a/${R}_{{\rm{P}}}$, they have substantially shorter τ_e of ∼0.6 Gyr and ∼2 Gyr, respectively. These circularization timescales are formally shorter than the age of their host stars and present some challenges for understanding any present day eccentricities. This tension could be resolved if ${Q}^{\prime }$ is ∼10⁶ as opposed to the assumed value of ∼10⁵. Eccentric orbits could also be maintained by additional yet undetected planets. Deming et al. (2007) proposed such an explanation for GJ 436b, another short-period eccentric sub-Saturn (see Table 7). Assuming ${Q}^{\prime }\sim {10}^{5}$, τ_e for GJ 436b is only ∼0.1 Gyr. Although no additional planets have been detected in the GJ 436 system, Batygin et al. (2009) showed this explanation to be plausible in the context of secular theory.

In Table 7, we also included the measured eccentricities, when available. We broke the sample into low (e < 0.1), moderate (e > 0.1), and poorly constrained eccentricities. In order for a planet to be included in the low/moderate eccentricity bins, its entire $1\sigma$ eccentricity confidence interval must be below/above 0.1. Planets with eccentricity upper limits or constraints straddling 0.1 fall in the poorly constrained category. These different eccentricity categories are color-coded in Figures 11, 13, and 15.

We observed that the highest-mass planets were often the only detected planet in the system. Figure 15 shows ${M}_{{\rm{P}}}$, ${f}_{\mathrm{env}}$, ${M}_{\mathrm{core}}$, and ${M}_{\mathrm{env}}$ as a function of the total number of detected planets in the system. There is a steady decline in the mass of sub-Saturns as overall multiplicity increases. The planets with moderate eccentricities are confined to apparently single systems. These high-mass singles could originally have had neighbors, but were in a dynamically unstable architecture.

Zoom In Zoom Out Reset image size

Such instabilities would eventually lead to close encounters and planet–planet scattering. Because these planets are so deep in the potential wells of their host stars, these scattering events would likely lead to mergers as opposed to ejections from the system. The maximum velocity a planet can impart to its neighbor is the escape velocity,

$$\begin{eqnarray*}&&{v}_{\mathrm{esc}}=11.2\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\left(\displaystyle \frac{{M}_{{\rm{P}}}}{{M}_{\oplus }}\right)}^{1/2}{\left(\displaystyle \frac{{R}_{{\rm{P}}}}{{R}_{\oplus }}\right)}^{-1/2},\end{eqnarray*}$$

and if ${v}_{\mathrm{esc}}$ is smaller than the orbital velocity,

$$\begin{eqnarray*}&&{v}_{\mathrm{orb}}=30\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{a}{1\mathrm{au}}\right)}^{-1/2},\end{eqnarray*}$$

single scattering events cannot lead to ejections. For the sub-Saturns²¹ in Table 7, ${v}_{\mathrm{esc}}/{v}_{\mathrm{orb}}\approx 0.1\mbox{--}0.3$. Any previous dynamical instabilities would likely lead to planet mergers, increasing their total mass. The present-day eccentricities of the more massive sub-Saturns may be a relic of previous scattering and merging events.

It is worth considering the biases associated with the different techniques by which planet mass and eccentricity are measured and whether they could be responsible for the observed mass–multiplicity–eccentricity trends. TTV measurements require multiplanet systems and thus do not contribute to any points in the N_P = 1 bin in Figure 15. Constraining e < 0.1 is challenging with RVs given that one is trying to measure slight deviations from sinusoidal RV curves. Therefore, limitations of the RV and TTV techniques might explain why there are no single planets with secure eccentricity measurements of <0.1.

However, these observational biases cannot explain why planets in multiplanet systems are low mass and preferentially circular. Previous studies (e.g., Weiss & Marcy 2014) have noted that planets with TTV mass measurements are typically less massive than planets with RV mass measurements. TTVs, however, are not blind to high-mass planets; such planets would produce larger TTVs. The lack of high-mass, high-eccentricity planets in multiplanet systems is likely the result of dynamical instabilities. Planets in such systems would likely perturb one another and merge, resulting in high-mass planets in low-multiplicity systems.