Identifying Young Kepler Planet Host Stars from Keck–HIRES Spectra of Lithium^*

First, we utilized PyRAF's continuum routine to remove the blaze function present in every spectrum. We applied this technique with the following options: a 50-piece cubic spline fit, a low rejection criterion of 2.0σ, a high rejection criterion of 3.0σ, and 50 outlier rejection iterations. The output (normalized) spectrum is the input spectrum divided by the continuum-fit spline function. Outlier rejection allows continuum to ignore any biasing effects from peaks (remaining cosmic rays) and troughs (absorption lines).

Next, we applied a Doppler correction to shift the spectrum into its rest frame. We determined the Doppler correction velocity by cross-correlating the rest-wavelength, National Solar Observatory (NSO) solar spectrum with the object spectrum. The NSO spectrum is an extremely high-resolution and high-S/N solar spectrum collected with the Brault National Solar Observatory Fourier Transform Spectrometer (Wallace et al. 2011). Many of the same absorption lines appear in the solar and HIRES spectra due to the similar T_eff of the Sun and our sample's stars. Therefore, we employed cross-correlation through PyRAF's xcsao routine. This routine succeeded for all stars in our sample. Figure 2 displays the final product of these two routines. The example spectrum exhibits a strong Li feature, unlike the solar spectrum, but both include significant Fe lines. The difference in depth of the Fe lines results from a combination of temperature, metallicity, rotational velocity, and spectral resolution effects. Cooler, higher-metallicity stars such as the Sun display stronger Fe I lines when compared to hotter, lower-metallicity stars such as KOI 274, even at similar $v\sin i$ and spectral resolution.

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In Figure 3, we illustrate the structure of spectra of multiple stars around the 6708 Å Li feature for stars with a range of T_eff. These particular stars were chosen because of their similar A(Li), [Fe/H], and small $v\sin i$. As a function of T_eff, stellar lithium features vary significantly in strength. Note how the Fe lines become slightly stronger as T_eff decreases from top to bottom, while the Li feature becomes much stronger as T_eff decreases. This illustrates the strong relationship between the Li EW and T_eff. In the hotter stars, more Li is ionized, so the Li I feature weakens.

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Next, we measured the Li EW in the normalized and shifted spectra. The National Institute for Standards and Technology (NIST) has the Li I resonance doublet listed with one transition at 6707.76 and the other at 6707.91 Å. We also had to account for the Fe I line that occurs at 6707.44 Å. Because of the wide variety of spectra at different T_eff, [Fe/H], and $v\sin i$, it was difficult to find an automated EW measurement routine that was effective for all spectra in our sample.

After testing multiple automated fitting routines/packages, we concluded that Levenberg–Marquardt FIT (LMFIT; Newville et al. 2014) works well for our purposes; namely, LMFIT can simultaneously fit the Fe line and both Li lines while also providing bounds on each of the fit parameters, unlike other oversimplified methods such as numerical integration or singular Gaussian fits. LMFIT is a nonlinear least-square minimization and curve fitting package for Python, which allows users to specify their own composite functions, bounds on parameters, and more. We used a four-component composite model. This model consisted of a constant = 1 continuum level, one Gaussian for the Fe I line at 6707.44 Å, one Gaussian for the Li I line at 6707.76 Å, and one Gaussian for the Li I line at 6707.91 Å.

In our model, we did not allow the continuum level to vary; we operated under the assumption our continuum normalization requires no adjustment near the Li feature. This assumption is sufficient because the vast majority of CKS stars have $v\sin i$ < 15 km s⁻¹, in addition to all having T_eff > 4500 K. Therefore, we do not expect significant blending of lines due to the stars' small $v\sin i$, nor significant spectral veiling from the molecular/metal absorption lines of M dwarfs near the Li doublet. For each of the Gaussians, we implemented similar bounds on the three fitting parameters. We limited their amplitudes to [−1.0, 0.0] to prevent any positive noise fits. We limited the Gaussian widths (σ) to [0.05, 0.10] to prevent any unphysical, noise-dominated fits. Also, we bounded the set of Gaussian centers to the Fe I line center at 6707.44 Å with physically required separations of 0.32 and 0.47 Å for the Li I 6707.76 and 6707.91 Å lines, respectively, while allowing the group as a whole to shift ±0.06 Å. This gives LMFIT the flexibility to shift to fit noisy line profiles but not by more than a resolution element (∼0.12 Å). Unlike other routines, LMFIT sufficiently fits Li absorption features with varying peaks and widths due to the wide range of stellar properties (T_eff, $\mathrm{log}g$, $v\sin i$, etc.) within the 1305 spectra.

Next, we computed the Li EW. See Figure 4 for an illustration of this method. The trough of the Fe line is not centered with respect to the Fe label. This is because the fit was improved by shifting slightly to the right. We emphasize that the calculated EWs do not include contributions from the Fe I line, as illustrated by the blue-filled area in Figure 4. In weak to moderate Li features like those in Figure 4, the feature has a slight asymmetry, caused by a difference in intensity between the smaller wavelength (greater intensity) and larger wavelength (lesser intensity) lines. This can be seen easily with very high resolution and sufficiently high S/N spectra as discussed in Reddy et al. (2002). In our fit, a slight asymmetry is present in the skewed Gaussian from the sum of the blended Li lines.

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We defined the uncertainty in our EW measurement, σ_EW, as the quadratic sum of the S/N-per-pixel-dependent Equation (7) (σ_UL) in Cayrel (1988), and the average of the difference of measured EWs when modifying the continuum level $\pm \tfrac{1}{{\rm{S}}/{{\rm{N}}}_{\mathrm{res}}}$ where S/N_res is the signal-to-noise per resolution element (Bertran de Lis et al. 2015). Due to the limitations of our abundance-determination software, we report and flag our EW measurements according to the following criteria (all reported uncertainties are σ_EW): if the measured EW > σ_UL + ${\sigma }_{\mathrm{EW}}$, we flag the point as a Li detection and report the measured EW; if σ_UL < EW < σ_UL + ${\sigma }_{\mathrm{EW}}$, we report the measured EW but flag the point as an upper limit; if the measured EW < σ_UL, we report EW = σ_UL and flag the point as an upper limit. See Table 1 for the entire sample's reported EWs.

Table 1.  Pipeline Results for Kepler Planet Host Stars

Obs Code	Obs Date	KOI	KepMag	S/N	Teff (K)	$\mathrm{log}g$ (dex)	(Fe/H)	ξ (km s−1)	EWLi (mÅ)	A(Li) (dex)
j122.742	2011 Jun 16	1	11.34	39	5819	4.40	0.01	1.04	85.5 ± 7.9	2.62 ± 0.08
j122.92	2011 Jun 13	2	10.46	39	6449	4.13	0.20	1.77	82.9 ± 7.3	3.11 ± 0.07
j122.81	2011 Jun 13	3	9.17	41	4864	4.50	0.33	0.54	4.2 ± 5.1	<−0.40
j70.1247	2009 Jun 05	6	12.16	119	6348	4.36	0.04	1.58	15.4 ± 2.2	2.16 ± 0.08
j74.509	2009 Jul 31	7	12.21	126	5827	4.09	0.18	1.17	54.0 ± 2.1	2.36 ± 0.06
j70.1251	2009 Jun 05	8	12.45	89	5891	4.54	−0.07	1.05	54.5 ± 3.6	2.42 ± 0.06
j77.875	2009 Oct 05	10	13.56	74	6181	4.24	−0.08	1.46	19.7 ± 3.6	2.15 ± 0.10
j72.483	2009 Jul 03	17	13.30	77	5660	4.15	0.36	1.09	2.3 ± 2.3	<0.75
j72.487	2009 Jul 03	18	13.37	73	6332	4.12	0.02	1.66	52.3 ± 3.8	2.75 ± 0.06
j93.303	2010 Jun 26	20	13.44	103	5927	4.01	0.03	1.30	40.7 ± 2.6	2.31 ± 0.06
j76.1283	2009 Sep 05	22	13.44	79	5891	4.21	0.21	1.19	25.8 ± 3.7	2.04 ± 0.09
j126.89	2011 Jul 09	41	11.20	207	5854	4.07	0.10	1.21	34.3 ± 1.3	2.15 ± 0.06
j90.90	2010 May 01	42	9.36	212	6306	4.28	−0.01	1.57	15.7 ± 1.3	2.14 ± 0.06
j124.500	2011 Jun 23	46	13.77	50	5661	4.07	0.39	1.12	98.2 ± 5.5	2.57 ± 0.07
j120.1133	2011 May 26	49	13.70	46	5779	4.34	−0.06	1.10	3.8 ± 4.6	<1.07
j130.1072	2011 Aug 18	63	11.58	181	5673	4.68	0.25	0.94	90.3 ± 1.6	2.41 ± 0.06
j76.1081	2009 Sep 04	64	13.14	92	5357	3.86	0.09	1.01	3.3 ± 2.9	<0.62
j76.1276	2009 Sep 05	69	9.93	163	5594	4.41	−0.09	0.97	1.2 ± 1.6	<0.38
j97.1478	2010 Aug 24	70	12.50	133	5508	4.47	0.11	0.91	3.3 ± 2.1	<0.71

Note. Observational and stellar data for the CKS stars analyzed in this paper. The stellar parameters T_eff, $\mathrm{log}g$, and [Fe/H] have uncertainties of 60 K, 0.10 dex, and 0.04 dex, respectively; these are the adopted values from Petigura et al. (2017). The KepMag column is from the NASA Exoplanet Archive. The other columns (S/N, ξ, EW_Li, and A(Li)) and their uncertainties, where relevant, were computed by our pipeline. We employ Takeda et al. (2013)'s treatment of ξ. All items in the table with < symbols in the final column indicate stars with EW_Li < σ_UL + ${\sigma }_{\mathrm{EW}}$, which we have flagged as upper limits in our analysis (and downward arrows in our plots). The full table, in machine-readable format, can be found online.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In Figure 5, we plot the measured EW of all CKS stars as a function of T_eff. The upper limits (gray downward arrows) are stars with measured Li EW < σ_UL + ${\sigma }_{\mathrm{EW}}$. From this plot, we can identify young stars: those with large EWs at low T_eff. Any Kepler planet host stars with EWs located far above the "slipper" are particularly young. At higher T_eff, the slipper is less well-defined largely because we do not have as many of these larger stars, and those that we do have are close to the Li "dip" observed in the Hyades, as discussed in Boesgaard et al. (2016).

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We utilized Model Atmosphere in Radiative and Convective Scheme (MARCS; Gustafsson et al. 2008) model atmospheres to convert EW to A(Li). Unlike Kurucz model grids, MARCS grids include the microturbulence parameter (ξ) in addition to T_eff, $\mathrm{log}g$, and [Fe/H]. In particular, we chose MARCS plane-parallel grids because our sample is primarily composed of main sequence dwarfs. We adopted the microturbulent description of Equations (1) and (2) in Takeda et al. (2013). We then interpolated from the discrete MARCS grids to the model atmospheres representing the adopted CKS stellar parameters.

We determined A(Li) using MOOG (Sneden et al. 2012). This code performs a variety of LTE analysis and spectrum synthesis tasks. We used its blends routine, which computes A(Li). Blends fits abundances of species by using a given model atmosphere to match blended-line EWs. We utilized ⁷Li hyperfine splitting transition wavelengths (Sansonetti et al. 1995) and gf values (Yan & Drake 1995) from Table 3, adapted from Andersen et al. (1984), in Smith et al. (1998) as our line list. We did not include the nearby Fe I line within our line list because we only computed the Li feature's EW using LMFIT. We then applied blends to determine A(Li) using this line list, in addition to the Li EW and interpolated model atmospheres from Sections 3.2 and 3.3, respectively.

We calculated the uncertainty, σ_A(Li), using a similar method to Ramírez et al. (2012). First, we varied each of the MOOG input parameters individually (T_eff, $\mathrm{log}g$, [Fe/H], and EW_Li) according to their internal CKS 1σ errors and then recalculated A(Li). This resulted in two Li abundances, one corresponding to the stellar model after a 1σ increase in the varied stellar parameter, A(Li)₊, and the other corresponding to the stellar model after a 1σ decrease in the varied stellar parameter, A(Li)₋. Next, we calculated the largest deviation of the upper and lower bound A(Li) values from the A(Li) corresponding to the adopted parameter. We repeated this process for the rest of the MOOG input parameters, and then added the largest deviations of each input parameter in quadrature to determine σ_A(Li).

As Bertran de Lis et al. (2015) discussed in their Appendix, adding separate errors in quadrature is insufficient because of the nonlinear transformation between stellar parameters/equivalent widths and abundances. Therefore, our reported A(Li) errors are quantitatively incorrect. However, because we use the largest deviations from A(Li) as our adopted individual uncertainties and then add them in quadrature, we posit that we overestimate the true abundance errors on one side due to the asymmetric distribution of abundances. To appropriately determine each σ_A(Li), we would need to perform a Markov chain Monte Carlo error analysis, which would require an extreme amount of computing time (MOOG would need to be run >1000 times per star) for the 1305 stars in our sample. In reality, the errors in the individual A(Li) are not particularly important for the results of this paper. Consequently, we conservatively estimate σ_A(Li) in the symmetric manner described above.

To ensure the accuracy and precision of MOOG's blends routine, we compared our EW-A(Li) results to spectral synthesis A(Li) using the same stellar parameters and the Li-blend EW for a subset of 18 stars. We found the measurements to be consistent within 4%.

Unfortunately, deriving precise ages from measurements of EW_Li and subsequent computation of A(Li) is quite difficult due to the inability of current models (Xiong & Deng 2009) to fit observed abundances. In Figure 7, the distribution of Kepler planet host stars resembles that of the Hyades much better than these Li depletion model isochrones. The orange bars represent the binned median A(Li) for stars with T_eff spanning the width of the bar. The orange curve is a cubic spline interpolation of the orange median A(Li) bars and serves as a statistical representation of the median A(Li) stars at each effective temperature. These stars may represent an empirical isochrone, much like the Hyades do. This curve appears similar in shape to the Hyades curve, while it intersects multiple theoretical model isochrones from Xiong & Deng (2009). Because the theoretical models are unable to match both the Hyades and the distribution of Kepler planet host stars, the ages indicated by each of the black curves prove unreliable. Although numerical ages are difficult to determine, we can use A(Li) versus T_eff plots to distinguish between young (i.e., <650 Myr) and old systems.

Stars deplete their surface Li over time. However, the rate of depletion varies with T_eff. Cooler stars (K type and later) deplete their Li faster because their convective zone depths are larger than hotter (G type and earlier) stars. Therefore, we expect to see more stars with high A(Li) at higher T_eff, as demonstrated in Figure 7. Naturally, we see a higher proportion of upper limits at cool temperatures compared to hot temperatures. We expect younger stars to have higher A(Li) than other stars at their T_eff. With this in mind, we can pick out the youngest stars as those most significantly above the blue Hyades curve and our orange empirical median curve.

Additionally, we can define a sample of stars that are younger than the Hyades by computing:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\rm{A}}(\mathrm{Li}),\mathrm{Hyades}}={\rm{A}}(\mathrm{Li})\,-\,{\rm{A}}{(\mathrm{Li})}_{\mathrm{Hyades}}\end{eqnarray} \tag{ 1 }$$

for each star in the sample, where A(Li) is the computed value from the pipeline and A(Li)_Hyades is the interpolated value of the Hyades at the star's T_eff. Stars that are younger than the Hyades will have Δ_A(Li),Hyades > 0. Table 2 includes all young CKS stars with detected Li (no upper limits are included in the table). The subtraction of the empirical Hyades curve is useful because it removes the offset caused by the T_eff dependence of A(Li).

Table 2.  Stars Younger than the Hyades

KOI	S/N	Teff (K)	$\mathrm{log}g$ (dex)	(Fe/H)	ξ (km s−1)	EWLi (mÅ)	A(Li) (dex)	ΔA(Li),Hyades (dex)
1	39	5819	4.40	0.01	1.04	85.5 ± 7.9	2.62 ± 0.08	0.08
2	39	6449	4.13	0.20	1.77	82.9 ± 7.3	3.11 ± 0.07	0.59
46	50	5661	4.07	0.39	1.12	98.2 ± 5.5	2.57 ± 0.07	0.47
63	181	5673	4.68	0.25	0.94	90.3 ± 1.6	2.41 ± 0.06	0.27
98	71	6500	4.22	0.05	1.78	52.1 ± 4.0	2.87 ± 0.06	0.62
119	47	5681	3.86	0.37	1.19	100.6 ± 6.4	2.60 ± 0.07	0.43
149	42	5708	3.95	0.03	1.18	55.3 ± 7.2	2.27 ± 0.09	0.02
323	53	5529	4.72	0.10	0.84	52.9 ± 5.4	1.98 ± 0.09	0.42
331	42	5506	3.90	0.11	1.08	40.2 ± 6.8	1.91 ± 0.11	0.44
660	53	5320	3.89	−0.07	0.98	25.6 ± 5.4	1.49 ± 0.13	0.61
684	59	5287	3.84	0.10	0.98	26.3 ± 5.0	1.47 ± 0.12	0.67
720	50	5260	4.68	0.04	0.71	14.5 ± 5.5	1.11 ± 0.23	0.39
853	28	4876	4.73	−0.02	0.47	43.8 ± 11.0	1.17 ± 0.20	1.18
1019	156	5018	3.55	0.17	0.92	7.1 ± 1.7	0.51 ± 0.14	0.32
1117	48	6513	4.16	−0.02	1.82	38.3 ± 6.4	2.72 ± 0.10	0.53
1175	43	5640	3.80	0.10	1.19	46.8 ± 7.1	2.12 ± 0.10	0.08
1199	39	4772	4.53	0.11	0.48	85.9 ± 8.7	1.28 ± 0.13	1.40
1208	46	6417	4.16	−0.07	1.73	48.3 ± 6.1	2.77 ± 0.08	0.12
1221	106	5002	3.62	0.33	0.89	9.0 ± 2.4	0.52 ± 0.16	0.36
1230	51	5119	3.35	0.01	1.04	26.8 ± 5.4	1.31 ± 0.13	0.92
1413	34	5253	3.79	−0.03	0.98	18.1 ± 8.3	1.26 ± 0.29	0.56
1438	33	5722	3.94	0.19	1.19	64.2 ± 9.5	2.36 ± 0.10	0.07
1463	136	6532	4.19	−0.04	1.83	38.2 ± 2.0	2.73 ± 0.05	0.66
1800	43	5621	4.69	0.07	0.90	100.1 ± 6.4	2.49 ± 0.08	0.51
1839	51	5517	4.67	0.17	0.85	99.4 ± 5.3	2.28 ± 0.08	0.77
1864	42	5620	3.93	0.12	1.14	52.1 ± 7.7	2.15 ± 0.10	0.18
1985	42	4950	4.64	0.01	0.54	10.3 ± 5.9	0.62 ± 0.40	0.55
2033	38	5051	4.53	−0.12	0.64	12.9 ± 7.5	0.61 ± 0.40	0.35
2035	48	5558	4.67	0.10	0.87	56.2 ± 5.5	2.06 ± 0.09	0.38
2046	45	5579	4.07	0.23	1.07	44.4 ± 6.4	2.04 ± 0.10	0.25
2115	13	5239	4.71	0.13	0.68	95.3 ± 23.1	1.90 ± 0.20	1.24
2175	42	5459	3.91	0.13	1.06	43.8 ± 7.1	1.91 ± 0.11	0.61
2228	41	6656	4.19	−0.10	1.95	47.0 ± 6.7	2.92 ± 0.09	4.38
2261	36	5176	4.70	0.06	0.65	31.4 ± 7.7	1.34 ± 0.17	0.83
2479	43	5372	3.88	0.08	1.02	28.7 ± 6.7	1.61 ± 0.14	0.59
2516	35	5431	3.91	0.30	1.04	19.1 ± 8.1	1.48 ± 0.26	0.27
2541	40	5090	3.70	0.15	0.91	15.9 ± 6.9	0.96 ± 0.27	0.62
2639	24	5583	3.89	−0.02	1.13	99.4 ± 13.2	2.50 ± 0.11	0.69
2640	48	4896	3.02	−0.13	1.01	10.6 ± 5.6	0.63 ± 0.34	0.61
2675	45	5756	4.63	0.13	1.00	82.1 ± 6.2	2.53 ± 0.07	0.13
2678	43	5416	4.70	0.12	0.78	136.4 ± 6.4	2.42 ± 0.09	1.26
2748	44	5499	4.00	−0.06	1.05	33.0 ± 6.2	1.81 ± 0.12	0.37
2769	54	5787	3.96	−0.02	1.22	69.4 ± 5.8	2.47 ± 0.07	0.00
2831	38	5752	3.92	−0.01	1.21	65.8 ± 7.1	2.40 ± 0.09	0.02
2859	46	5260	4.51	−0.07	0.76	12.4 ± 6.1	0.98 ± 0.32	0.26
2885	45	5492	3.93	−0.36	1.07	27.5 ± 6.1	1.70 ± 0.13	0.29
2891	47	6142	4.02	0.14	1.51	99.8 ± 5.8	2.99 ± 0.06	0.01
3012	39	5493	4.16	−0.50	1.00	24.0 ± 7.3	1.63 ± 0.18	0.22
3202	41	5262	3.80	0.03	0.98	23.9 ± 6.7	1.40 ± 0.17	0.68
3239	47	5668	4.66	0.09	0.94	124.5 ± 5.7	2.70 ± 0.07	0.58
3244	58	4970	3.13	−0.03	1.02	25.1 ± 4.8	1.11 ± 0.12	1.01
3371	45	5428	4.56	0.00	0.84	54.1 ± 7.2	1.98 ± 0.10	0.78
3473	37	5157	4.65	0.05	0.66	82.6 ± 7.4	1.85 ± 0.11	1.38
3835	45	5013	4.70	0.04	0.56	118.8 ± 6.1	1.92 ± 0.11	1.73
3871	49	5193	4.62	0.07	0.69	11.0 ± 5.6	0.87 ± 0.33	0.32
3876	40	5720	4.64	0.12	0.98	119.8 ± 6.7	2.73 ± 0.07	0.45
3886	47	4760	2.94	0.20	0.96	11.5 ± 6.3	0.45 ± 0.36	0.58
3891	52	5080	3.75	−0.10	0.89	21.6 ± 5.2	1.08 ± 0.15	0.76
3908	49	5721	3.88	0.03	1.21	58.1 ± 6.6	2.30 ± 0.09	0.02
3936	55	5081	4.66	0.17	0.61	162.5 ± 4.8	2.03 ± 0.12	1.71
3991	41	5606	4.62	−0.02	0.92	96.7 ± 6.6	2.48 ± 0.08	0.56
4004	52	5739	4.68	−0.05	0.97	78.6 ± 5.2	2.48 ± 0.07	0.13
4146	50	5092	4.64	0.24	0.62	22.4 ± 5.9	0.67 ± 0.19	0.32
4156	41	5807	4.05	0.24	1.17	96.8 ± 7.1	2.69 ± 0.07	0.18
4226	39	5844	3.87	0.35	1.28	82.6 ± 7.6	2.62 ± 0.08	0.03
4556	48	5437	3.83	−0.11	1.07	25.5 ± 6.0	1.61 ± 0.14	0.39
4613	46	5443	4.55	−0.13	0.85	23.6 ± 6.7	1.45 ± 0.17	0.20
4647	31	5166	3.81	0.23	0.92	23.2 ± 9.6	1.18 ± 0.25	0.69
4663	52	5545	3.88	0.30	1.11	55.1 ± 6.6	2.12 ± 0.09	0.49
4686	47	5698	3.90	0.33	1.19	75.6 ± 5.9	2.44 ± 0.07	0.22
4745	14	4781	4.56	0.02	0.47	78.8 ± 14.6	1.41 ± 0.16	1.53
4763	36	5695	3.98	0.17	1.17	52.4 ± 7.8	2.23 ± 0.10	0.02
4775	43	5210	3.80	0.47	0.95	36.6 ± 6.3	1.48 ± 0.11	0.90
4811	31	5572	3.90	0.05	1.12	39.9 ± 9.4	1.97 ± 0.15	0.22
4834	50	5030	3.75	0.40	0.86	19.5 ± 5.4	0.86 ± 0.16	0.64
5057	38	5004	3.30	0.00	0.99	26.5 ± 7.1	1.17 ± 0.16	1.01
5107	27	4933	3.04	−0.13	1.03	27.4 ± 10.4	1.11 ± 0.24	1.05
5119	35	4984	3.25	0.08	0.99	13.7 ± 7.3	0.84 ± 0.35	0.71
6676	34	6493	4.18	−0.19	1.79	31.1 ± 8.1	2.59 ± 0.15	0.31
6759	33	5494	3.90	0.36	1.08	66.2 ± 8.2	2.16 ± 0.09	0.75

Note. CKS stars with Li detections younger than the Hyades. The stellar parameters T_eff, $\mathrm{log}g$, and [Fe/H] have uncertainties of 60 K, 0.10 dex, and 0.04 dex, respectively; these are the adopted values from Petigura et al. (2017). The other columns (S/N, ξ, EW_Li, A(Li), and Δ_A(Li),Hyades) and their uncertainties, where relevant, were computed by our pipeline. We employ Takeda et al.'s (2013) treatment of ξ. We do not include upper limits in this table.

A machine-readable version of the table is available.

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Figure 7 indicates that stars with A(Li) > 1.0 and T_eff < 5300 K are the youngest systems due to their unusually high A(Li). For comparison, the Hyades is 650 Myr according to estimates from Perryman et al. (1998) and Brandt & Huang (2015). Therefore, stars above the blue Hyades curve should be younger than 650 Myr. In essence, we can determine the relative ages of systems in Figure 7 by subtracting the blue Hyades curve from the Kepler planet host star data points. These A(Li)–T_eff plots also allow the qualitative comparison of average system properties above and below the Hyades.

For a comparison to our CKS sample, we present Figure 8, which contains data from Ramírez et al. (2012). We plot the same structures as those in Figure 7. In Ramírez et al. (2012), the Li desert (outlined as a gray trapezoid in Figures 7 and 8) was emphasized as an area without stars. The authors argued that the Li desert is a physical phenomenon caused by short-lived processes on the stellar surface that deplete Li for 1.1–1.3 M_⊙ stars. These processes are not well understood, but the observational evidence is hard to ignore. However, our sample produces two stars within this desert. If the Li desert is indeed a physical gap, we conclude that errors in both A(Li) and T_eff can account for this discrepancy. Moreover, Figure 7 (our sample) has a large number of upper limits when compared to Figure 8 (Ramírez et al. 2012). This is due to our low median S/N ≈ 45 compared to the median S/N ⪆ 100 from Ramírez et al. (2012). Figure 7 has more cool stars than Figure 8, which has more hot stars. This is an important distinction between the two samples. Faint stars in the CKS sample were chosen because of higher planet multiplicity and/or the presence of interesting planets. Because of this, we have a larger fraction of low T_eff stars compared to Ramírez et al. (2012). Curiously, only one star appears above the Hyades at T_eff ≈ 5900–6300 K in Figure 7, while Figure 8 has ≳10 stars in the same area. Based on the simple assumption of a uniform distribution of stellar ages, Figure 7 should have a few stars above the Hyades at these T_eff. This observation puzzles us.

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We are also puzzled by the large number of stars within the Li dip in Figure 8. According to the Li depletion mechanisms discussed in Xiong & Deng (2009), the Li dip is a result of gravitational settling of Li into progressively hotter radiative zones in the stellar interior, where it is burned in (p, α) reactions. Why so many stars from Ramírez et al. (2012) reside in the dip perplexes us. We do not see the same within the CKS sample, but we do not have many stars in that range of T_eff.

Table 3 displays the full list of the CKS planets with significant Li detections that are younger than the Hyades. However, we do not incorporate stars with upper limits because their A(Li) are unreliable, nor do we include false positive planet detections in this table.

Table 3.  Planets Younger than the Hyades

KOI	Teff (K)	EWLi (mÅ)	A(Li) (dex)	ΔA(Li),Hyades (dex)	Planet Number	Rp (R⊕)	Period (days)	Fp (F⊕)
1	5819	85.5 ± 7.9	2.62 ± 0.08	0.08	1	${14.3}_{-1.4}^{+1.4}$	${2.5}_{-0.0}^{+0.0}$	${890.7}_{-184.9}^{+184.9}$
2	6449	82.9 ± 7.3	3.11 ± 0.07	0.59	1	${13.4}_{-2.0}^{+2.0}$	${2.2}_{-0.0}^{+0.0}$	${3029.6}_{-931.2}^{+931.2}$
46	5661	98.2 ± 5.5	2.57 ± 0.07	0.47	1	${5.7}_{-0.7}^{+0.7}$	${3.5}_{-0.0}^{+0.0}$	${1030.6}_{-279.4}^{+279.4}$
⋯	⋯	⋯	⋯	⋯	2	${1.2}_{-0.2}^{+0.2}$	${6.0}_{-0.0}^{+0.0}$	${497.1}_{-132.2}^{+132.2}$
63	5673	90.3 ± 1.6	2.41 ± 0.06	0.27	1	${6.1}_{-0.5}^{+0.5}$	${9.4}_{-0.0}^{+0.0}$	${109.0}_{-18.3}^{+18.3}$
98	6500	52.1 ± 4.0	2.87 ± 0.06	0.62	1	${8.5}_{-1.0}^{+1.0}$	${6.8}_{-0.0}^{+0.0}$	${572.2}_{-142.3}^{+142.3}$
119	5681	100.6 ± 6.4	2.60 ± 0.07	0.43	2	${7.2}_{-0.9}^{+0.9}$	${190.3}_{-0.0}^{+0.0}$	${7.1}_{-1.9}^{+1.9}$
⋯	⋯	⋯	⋯	⋯	1	${7.9}_{-1.0}^{+1.0}$	${49.2}_{-0.0}^{+0.0}$	${42.5}_{-11.5}^{+11.5}$
149	5708	55.3 ± 7.2	2.27 ± 0.09	0.02	1	${5.5}_{-0.8}^{+0.8}$	${14.6}_{-0.0}^{+0.0}$	${204.7}_{-64.9}^{+64.9}$
323	5529	52.9 ± 5.4	1.98 ± 0.09	0.42	1	${2.0}_{-0.2}^{+0.2}$	${5.8}_{-0.0}^{+0.0}$	${167.5}_{-28.2}^{+28.2}$
331	5506	40.2 ± 6.8	1.91 ± 0.11	0.44	1	${4.2}_{-0.7}^{+0.7}$	${18.7}_{-0.0}^{+0.0}$	${141.5}_{-40.4}^{+40.4}$
660	5320	25.6 ± 5.4	1.49 ± 0.13	0.61	1	${3.9}_{-0.7}^{+0.7}$	${6.1}_{-0.0}^{+0.0}$	${664.0}_{-228.0}^{+228.0}$
684	5287	26.3 ± 5.0	1.47 ± 0.12	0.67	1	${8.7}_{-3.7}^{+3.7}$	${4.0}_{-0.0}^{+0.0}$	${1201.6}_{-389.0}^{+389.0}$
720	5260	14.5 ± 5.5	1.11 ± 0.23	0.39	1	${3.0}_{-0.2}^{+0.2}$	${5.7}_{-0.0}^{+0.0}$	${130.6}_{-22.2}^{+22.2}$
⋯	⋯	⋯	⋯	⋯	2	${2.8}_{-0.2}^{+0.2}$	${10.0}_{-0.0}^{+0.0}$	${61.2}_{-10.5}^{+10.5}$
⋯	⋯	⋯	⋯	⋯	4	${1.6}_{-0.1}^{+0.1}$	${2.8}_{-0.0}^{+0.0}$	${337.1}_{-57.2}^{+57.2}$
⋯	⋯	⋯	⋯	⋯	3	${2.7}_{-0.2}^{+0.2}$	${18.4}_{-0.0}^{+0.0}$	${27.3}_{-4.6}^{+4.6}$
853	4876	43.8 ± 11.0	1.17 ± 0.20	1.18	1	${2.7}_{-0.3}^{+0.3}$	${8.2}_{-0.0}^{+0.0}$	${51.8}_{-8.9}^{+8.9}$
⋯	⋯	⋯	⋯	⋯	2	${2.5}_{-0.4}^{+0.4}$	${14.5}_{-0.0}^{+0.0}$	${24.2}_{-4.1}^{+4.1}$
1117	6513	38.3 ± 6.4	2.72 ± 0.10	0.53	1	${1.9}_{-0.3}^{+0.3}$	${11.1}_{-0.0}^{+0.0}$	${328.7}_{-88.0}^{+88.0}$
1175	5640	46.8 ± 7.1	2.12 ± 0.10	0.08	1	${2.6}_{-0.4}^{+0.4}$	${31.6}_{-0.0}^{+0.0}$	${89.8}_{-25.7}^{+25.7}$
⋯	⋯	⋯	⋯	⋯	2	${2.3}_{-0.5}^{+0.5}$	${17.2}_{-0.0}^{+0.0}$	${204.7}_{-58.4}^{+58.4}$
1199	4772	85.9 ± 8.7	1.28 ± 0.13	1.40	1	${2.5}_{-0.2}^{+0.2}$	${53.5}_{-0.0}^{+0.0}$	${4.0}_{-0.7}^{+0.7}$
1208	6417	48.3 ± 6.1	2.77 ± 0.08	0.12	1	${8.4}_{-1.2}^{+1.2}$	${700.0}_{-0.0}^{+0.0}$	${1.2}_{-0.3}^{+0.3}$
1221	5002	9.0 ± 2.4	0.52 ± 0.16	0.36	1	${4.7}_{-0.7}^{+0.7}$	${30.2}_{-0.0}^{+0.0}$	${133.6}_{-39.3}^{+39.3}$
⋯	⋯	⋯	⋯	⋯	2	${3.8}_{-0.6}^{+0.6}$	${51.1}_{-0.0}^{+0.0}$	${66.2}_{-19.4}^{+19.4}$
1230a	5119	26.8 ± 5.4	1.31 ± 0.13	0.92	1	${38.7}_{-6.6}^{+6.6}$	${165.7}_{-0.0}^{+0.0}$	${30.4}_{-10.6}^{+10.6}$
1413	5253	18.1 ± 8.3	1.26 ± 0.29	0.56	1	${3.6}_{-0.7}^{+0.7}$	${12.6}_{-0.0}^{+0.0}$	${311.4}_{-113.2}^{+113.2}$
⋯	⋯	⋯	⋯	⋯	2	${3.0}_{-0.5}^{+0.5}$	${21.5}_{-0.0}^{+0.0}$	${153.6}_{-55.8}^{+55.8}$
1438	5722	64.2 ± 9.5	2.36 ± 0.10	0.07	1	${2.5}_{-0.4}^{+0.4}$	${6.9}_{-0.0}^{+0.0}$	${532.5}_{-137.9}^{+137.9}$
1463	6532	38.2 ± 2.0	2.73 ± 0.05	0.66	1	${23.0}_{-2.8}^{+2.8}$	${1064.3}_{-0.0}^{+0.0}$	${0.7}_{-0.2}^{+0.2}$
1800	5621	100.1 ± 6.4	2.49 ± 0.08	0.51	1	${6.3}_{-0.5}^{+0.5}$	${7.8}_{-0.0}^{+0.0}$	${126.0}_{-21.2}^{+21.2}$
1839	5517	99.4 ± 5.3	2.28 ± 0.08	0.77	1	${2.3}_{-0.2}^{+0.2}$	${9.6}_{-0.0}^{+0.0}$	${88.5}_{-14.9}^{+14.9}$
⋯	⋯	⋯	⋯	⋯	2	${2.3}_{-0.2}^{+0.2}$	${80.4}_{-0.0}^{+0.0}$	${5.2}_{-0.9}^{+0.9}$
1864	5620	52.1 ± 7.7	2.15 ± 0.10	0.18	1	${2.3}_{-0.3}^{+0.3}$	${3.2}_{-0.0}^{+0.0}$	${1482.2}_{-431.3}^{+431.3}$
1985	4950	10.3 ± 5.9	0.62 ± 0.40	0.55	1	${2.5}_{-0.3}^{+0.3}$	${5.8}_{-0.0}^{+0.0}$	${93.0}_{-15.8}^{+15.8}$
2033	5051	12.9 ± 7.5	0.61 ± 0.40	0.35	1	${1.5}_{-0.1}^{+0.1}$	${16.5}_{-0.0}^{+0.0}$	${25.3}_{-4.3}^{+4.3}$
2035	5558	56.2 ± 5.5	2.06 ± 0.09	0.38	1	${2.2}_{-0.2}^{+0.2}$	${1.9}_{-0.0}^{+0.0}$	${766.9}_{-131.4}^{+131.4}$
2046	5579	44.4 ± 6.4	2.04 ± 0.10	0.25	1	${2.5}_{-0.4}^{+0.4}$	${23.9}_{-0.0}^{+0.0}$	${70.7}_{-22.3}^{+22.3}$
2115	5239	95.3 ± 23.1	1.90 ± 0.20	1.24	1	${3.1}_{-0.3}^{+0.3}$	${15.7}_{-0.0}^{+0.0}$	${33.1}_{-5.7}^{+5.7}$
2175	5459	43.8 ± 7.1	1.91 ± 0.11	0.61	1	${3.0}_{-0.4}^{+0.4}$	${26.8}_{-0.0}^{+0.0}$	${84.7}_{-25.3}^{+25.3}$
⋯	⋯	⋯	⋯	⋯	2	${3.5}_{-0.5}^{+0.5}$	${72.4}_{-0.0}^{+0.0}$	${22.7}_{-6.7}^{+6.7}$
2261	5176	31.4 ± 7.7	1.34 ± 0.17	0.83	1	${1.2}_{-0.1}^{+0.1}$	${4.0}_{-0.0}^{+0.0}$	${193.2}_{-33.0}^{+33.0}$
⋯	⋯	⋯	⋯	⋯	2	${0.9}_{-0.2}^{+0.2}$	${6.6}_{-0.0}^{+0.0}$	${97.9}_{-16.7}^{+16.7}$
2479	5372	28.7 ± 6.7	1.61 ± 0.14	0.59	1	${2.2}_{-0.4}^{+0.4}$	${25.5}_{-0.0}^{+0.0}$	${93.5}_{-33.1}^{+33.1}$
2516	5431	19.1 ± 8.1	1.48 ± 0.26	0.27	1	${1.3}_{-0.2}^{+0.2}$	${2.8}_{-0.0}^{+0.0}$	${1611.6}_{-440.9}^{+440.9}$
2541	5090	15.9 ± 6.9	0.96 ± 0.27	0.62	1	${2.4}_{-0.5}^{+0.5}$	${7.4}_{-0.0}^{+0.0}$	${748.1}_{-245.6}^{+245.6}$
⋯	⋯	⋯	⋯	⋯	2	${2.6}_{-0.4}^{+0.4}$	${20.5}_{-0.0}^{+0.0}$	${192.6}_{-63.3}^{+63.3}$
2639	5583	99.4 ± 13.2	2.50 ± 0.11	0.69	1	${3.6}_{-0.7}^{+0.7}$	${25.1}_{-0.0}^{+0.0}$	${115.2}_{-40.2}^{+40.2}$
⋯	⋯	⋯	⋯	⋯	2	${9.7}_{-61.1}^{+61.1}$	${2.1}_{-0.0}^{+0.0}$	${3092.3}_{-1075.3}^{+1075.3}$
2675	5756	82.1 ± 6.2	2.53 ± 0.07	0.13	1	${2.2}_{-0.2}^{+0.2}$	${5.4}_{-0.0}^{+0.0}$	${243.0}_{-40.8}^{+40.8}$
⋯	⋯	⋯	⋯	⋯	2	${1.0}_{-0.1}^{+0.1}$	${1.1}_{-0.0}^{+0.0}$	${2017.0}_{-341.1}^{+341.1}$
2678	5416	136.4 ± 6.4	2.42 ± 0.09	1.26	1	${1.8}_{-0.2}^{+0.2}$	${3.8}_{-0.0}^{+0.0}$	${260.9}_{-44.5}^{+44.5}$
2748	5499	33.0 ± 6.2	1.81 ± 0.12	0.37	1	${2.5}_{-0.4}^{+0.4}$	${23.2}_{-0.0}^{+0.0}$	${95.0}_{-27.1}^{+27.1}$
⋯	⋯	⋯	⋯	⋯	2	${2.0}_{-0.3}^{+0.3}$	${5.8}_{-0.0}^{+0.0}$	${604.6}_{-175.2}^{+175.2}$
2859	5260	12.4 ± 6.1	0.98 ± 0.32	0.26	1	${0.7}_{-0.1}^{+0.1}$	${3.4}_{-0.0}^{+0.0}$	${272.5}_{-46.3}^{+46.3}$
⋯	⋯	⋯	⋯	⋯	3	${0.7}_{-0.1}^{+0.1}$	${4.3}_{-0.0}^{+0.0}$	${202.9}_{-34.3}^{+34.3}$
⋯	⋯	⋯	⋯	⋯	4	${0.7}_{-0.1}^{+0.1}$	${2.9}_{-0.0}^{+0.0}$	${342.1}_{-58.6}^{+58.6}$
⋯	⋯	⋯	⋯	⋯	2	${0.6}_{-0.1}^{+0.1}$	${2.0}_{-0.0}^{+0.0}$	${557.7}_{-95.1}^{+95.1}$
3371	5428	54.1 ± 7.2	1.98 ± 0.10	0.78	1	${1.3}_{-0.1}^{+0.1}$	${58.1}_{-0.0}^{+0.0}$	${7.4}_{-1.3}^{+1.3}$
⋯	⋯	⋯	⋯	⋯	2	${1.1}_{-0.1}^{+0.1}$	${12.3}_{-0.0}^{+0.0}$	${59.2}_{-10.1}^{+10.1}$
3473b	5157	82.6 ± 7.4	1.85 ± 0.11	1.38	1	${7.0}_{-2.8}^{+2.8}$	${27.6}_{-0.0}^{+0.0}$	${14.4}_{-2.5}^{+2.5}$
3835	5013	118.8 ± 6.1	1.92 ± 0.11	1.73	1	${2.6}_{-0.2}^{+0.2}$	${47.1}_{-0.0}^{+0.0}$	${6.0}_{-1.0}^{+1.0}$
3876	5720	119.8 ± 4.3	2.73 ± 0.06	0.45	1	${1.9}_{-0.2}^{+0.2}$	${19.6}_{-0.0}^{+0.0}$	${42.4}_{-7.2}^{+7.2}$
3886a	4760	11.5 ± 6.3	0.45 ± 0.36	0.58	1	${28.6}_{-8.5}^{+8.5}$	${5.6}_{-0.0}^{+0.0}$	${7866.2}_{-4770.1}^{+4770.1}$
3891b	5080	21.6 ± 5.2	1.08 ± 0.15	0.76	1	${7.1}_{-2.2}^{+2.2}$	${47.1}_{-0.0}^{+0.0}$	${58.2}_{-20.0}^{+20.0}$
3908	5721	58.1 ± 6.6	2.30 ± 0.09	0.02	1	${2.9}_{-0.4}^{+0.4}$	${59.4}_{-0.0}^{+0.0}$	${37.4}_{-10.8}^{+10.8}$
3936	5081	162.5 ± 4.8	2.03 ± 0.12	1.71	2	${1.6}_{-0.2}^{+0.2}$	${13.0}_{-0.0}^{+0.0}$	${36.8}_{-6.2}^{+6.2}$
3991	5606	96.7 ± 6.6	2.48 ± 0.08	0.56	1	${1.4}_{-0.2}^{+0.2}$	${1.6}_{-0.0}^{+0.0}$	${1053.8}_{-179.7}^{+179.7}$
4004	5739	78.6 ± 5.2	2.48 ± 0.07	0.13	1	${1.1}_{-0.1}^{+0.1}$	${4.9}_{-0.0}^{+0.0}$	${256.4}_{-43.6}^{+43.6}$
4146	5092	22.4 ± 5.9	0.67 ± 0.19	0.32	1	${0.8}_{-0.1}^{+0.1}$	${3.5}_{-0.0}^{+0.0}$	${219.8}_{-37.4}^{+37.4}$
4156	5807	96.8 ± 7.1	2.69 ± 0.07	0.18	1	${1.4}_{-0.2}^{+0.2}$	${4.9}_{-0.0}^{+0.0}$	${786.5}_{-209.0}^{+209.0}$
4226	5844	82.6 ± 7.6	2.62 ± 0.08	0.03	1	${2.4}_{-0.3}^{+0.3}$	${49.6}_{-0.0}^{+0.0}$	${51.0}_{-11.8}^{+11.8}$
4613	5443	23.6 ± 6.7	1.45 ± 0.17	0.20	1	${0.7}_{-0.1}^{+0.1}$	${2.0}_{-0.0}^{+0.0}$	${674.4}_{-115.6}^{+115.6}$
4647	5166	23.2 ± 9.6	1.18 ± 0.25	0.69	2	${2.1}_{-0.3}^{+0.3}$	${12.0}_{-0.0}^{+0.0}$	${290.1}_{-87.2}^{+87.2}$
⋯	⋯	⋯	⋯	⋯	1	${2.5}_{-0.4}^{+0.4}$	${37.9}_{-0.0}^{+0.0}$	${62.1}_{-18.7}^{+18.7}$
4663	5545	55.1 ± 6.6	2.12 ± 0.09	0.49	1	${1.0}_{-0.1}^{+0.1}$	${5.7}_{-0.0}^{+0.0}$	${679.8}_{-166.5}^{+166.5}$
4686	5698	75.6 ± 5.9	2.44 ± 0.07	0.22	1	${1.2}_{-0.2}^{+0.2}$	${11.8}_{-0.0}^{+0.0}$	${262.5}_{-60.0}^{+60.0}$
4745	4781	78.8 ± 14.6	1.41 ± 0.16	1.53	1	${2.2}_{-0.2}^{+0.2}$	${177.7}_{-0.0}^{+0.0}$	${0.8}_{-0.1}^{+0.1}$
4763	5695	52.4 ± 7.8	2.23 ± 0.10	0.02	1	${2.0}_{-0.3}^{+0.3}$	${56.4}_{-0.0}^{+0.0}$	${30.2}_{-8.7}^{+8.7}$
4775	5210	36.6 ± 6.3	1.48 ± 0.11	0.90	1	${1.9}_{-0.3}^{+0.3}$	${16.4}_{-0.0}^{+0.0}$	${177.1}_{-56.9}^{+56.9}$
4811	5572	39.9 ± 9.4	1.97 ± 0.15	0.22	1	${2.5}_{-0.4}^{+0.4}$	${21.7}_{-0.0}^{+0.0}$	${121.5}_{-37.9}^{+37.9}$
4834	5030	19.5 ± 5.4	0.86 ± 0.16	0.64	1	${1.4}_{-0.2}^{+0.2}$	${3.7}_{-0.0}^{+0.0}$	${1420.4}_{-427.7}^{+427.7}$
5057	5004	26.5 ± 7.1	1.17 ± 0.16	1.01	1	${11.0}_{-2.4}^{+2.4}$	${493.7}_{-0.0}^{+0.0}$	${7.4}_{-3.2}^{+3.2}$
5107	4933	27.4 ± 10.4	1.11 ± 0.24	1.05	1	${18.1}_{-3.2}^{+3.2}$	${169.3}_{-0.0}^{+0.0}$	${71.1}_{-23.8}^{+23.8}$
5119	4984	13.7 ± 7.3	0.84 ± 0.35	0.71	1	${12.7}_{-3.0}^{+3.0}$	${143.2}_{-0.0}^{+0.0}$	${35.1}_{-13.2}^{+13.2}$

Notes. Planets in systems with ${{\rm{\Delta }}}_{{\rm{A}}(\mathrm{Li}),\mathrm{Hyades}}\gt 0.0$, where Δ_A(Li),Hyades is the A(Li) of each point subtracted by the Hyades curve at each T_eff (systems younger than the Hyades). Upper limits in A(Li) are excluded. We list stellar data, produced from our pipeline, in the first five columns, while we list the individual planet information from the CKS in the last four columns. Errors in T_eff are 60 K. The last four columns, in order, include the KOI planet number (which is appended to the KOI number), planet radius in Earth radii, planet orbital period in days, and planet insolation flux in Earth flux. All planets in this table are either confirmed planets (P_planet ≳ 0.99) or planet candidates (P_planet ≳ 0.90)—no false positives are included. The errors in the planet orbital periods from the Archive are orders of magnitude smaller than the listed precision. Any ellipsis (⋯) marks indicate an additional planet which belongs to the system listed above.

^aThese planet sizes are too large to be physical. After further investigation, we found that KOI 1230's companion has been dispositioned as a certified false positive since we last accessed the NASA Exoplanet Archive. KOI 3886's companion has also been dispositioned as a certified false positive. ^bThese systems have been modified with manually calculated R_p based on the transit depth and the stellar radius due to a non-physical planet radius in the CKS catalog.

A machine-readable version of the table is available.

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We begin our age investigation by analyzing trends in stellar properties with A(Li). We again utilize the A(Li) versus T_eff plot much like Figures 7 and 8, but color the points according to the stellar property of interest. First, we investigate [Fe/H]. We find no significant trends in metallicity with age after comparing stars with A(Li) above and below the Hyades, although a clump of low-metallicity points occurs both above and immediately to the right of the Li desert. We arrive at similar conclusions for $\mathrm{log}g$ although there do appear to be some "young" subgiants/giants around T_eff = 5000 K and A(Li) = 1.0. This will become important for clean sample selection later. In addition, the stars with the highest A(Li) at their respective T_eff have high $\mathrm{log}g$.

Figure 9 is a proxy for an HR diagram with $\mathrm{log}g$ versus T_eff and points colored by their A(Li). The youngest systems are the brightest (green and yellow) points on the main sequence at their respective T_eff and $\mathrm{log}g$. Most stars in the sample are main sequence dwarfs, although the sample also includes the horizontal branch of subgiants and then giants at the top of the "tail" at the lowest T_eff and $\mathrm{log}g$. In addition, this plot reveals A(Li)'s temperature dependence, visible in the smooth transition of colors from hotter to cooler effective temperatures. We note that a few stars on the lower envelope of the main sequence in the figure (at the highest $\mathrm{log}g$) typically have larger A(Li) compared to stars at the same T_eff and slightly lower $\mathrm{log}g$. Current stellar evolution models predict that, as stars evolve during their main sequence lifetimes, they gradually increase in luminosity and inflate in size. Therefore, the stars with the highest $\mathrm{log}g$ values at their respective T_eff are likely some of the youngest stars in our sample.

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Just as we investigated stellar parameters versus age in the previous section, we can apply the same analysis to exoplanet parameters. We use the catalog of exoplanet parameters provided by Johnson et al. (2017) and the NASA Exoplanet Archive³ to obtain exoplanet parameters for our systems. We note that many of the NASA Exoplanet Archive planet parameters derive from the detailed stellar analysis of Huber et al. (2014). First, we investigate whether planet multiplicity (number of planets discovered per star) varies with age. In Figure 10, single-planet systems are colored red, while multi-planet systems range from blue (two planets) through bright green (seven planets). We compare points above and below the Hyades, and we find no evidence for multiplicity's dependence on age. Similarly, we investigated whether there are any trends in planet disposition, average planet period, and average planet insolation flux using more A(Li) versus T_eff plots, but no patterns were apparent. Therefore, we conclude that these exoplanetary properties show no dependence on age.

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Finally, we consider the average planet radius in each system. A quick look at Figure 11 does not reveal any clear trends between planet radius and location above/below the Hyades. However, many of the identified young systems in Section 4.1 (those at T_eff < 5500 K and above the Hyades curve) are green- and yellow-colored. Therefore, they have large average planet radii. We find this evidence interesting, as planets are expected to deflate as they age and their star remains on the main sequence (Lopez et al. 2012). We continue with a more thorough investigation.

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Before we proceed, we must consider sample biases and eliminate any systems which may introduce biases in age and planet size. Therefore, we utilize similar cuts to produce a clean sample as in Fulton et al. (2017). First, we eliminate all false positives identified in the CKS (Petigura et al. 2017) and stars with spectral S/N < 30. Next, we remove all planets with impact parameter b >0.7 and those orbiting subgiant and giant stars according to Equation (1) in Fulton et al. (2017). Additionally, we remove all planets with low-transit S/Ns (S/N < 10). These cuts ensure that the planetary radii are reliable and that the host stars are main sequence dwarfs (where A(Li) is a reliable indicator of age). We do not remove planets with long orbital periods and large KepMags because these cuts further reduce our sample size, while the likelihood of systematic biases in age and planetary radii of these systems is small.

To make any obscured trends more apparent in Figure 11, we split the sample into two parts: systems with R_avg > 1. 75R_⊕ and those with R_avg ≤ 1. 75R_⊕. We choose 1.75R_⊕ as our separating radius because it corresponds to the trough of the empirical gap revealed in Fulton et al. (2017). Figure 12 displays our "clean" sample in another A(Li)–T_eff comparison plot with the points colored according to their average planet radius.

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We observe from Figure 12 that the stars younger than the Hyades typically have planet companions on the large side of the planet radius gap discovered by Fulton et al. (2017). The prevalence of green points above the Hyades supports the conclusion that younger planets are larger than older planets. However, because this plot assigns an average planet radius to each star, we lose information about individual planetary radii. Therefore, we must perform additional statistical analysis to test our hypothesis that, on average, larger planets orbit younger stars.

To differentiate between young and old systems, we use the Hyades as the dividing line. The Hyades are ∼650 Myr old (Perryman et al. 1998; Brandt & Huang 2015; Boesgaard et al. 2016); systems that fall above (below) it in A(Li) versus T_eff space (see Figure 12) are younger (older) than the Hyades. This statement holds true for the majority of systems, but those close to the Hyades curve are more likely to be on the wrong side of it. Errors in measurement and potential abundance effects (i.e., differences in initial A(Li)) affect our young/old system designation in detail. We do not expect the measurement errors to systematically bias our results. In addition, while abundance effects may introduce a systematic bias (i.e., Kepler stars have systematically higher initial A(Li) compared to the Hyades), ensembles of stars have been shown to give consistent initial A(Li) although the individual scatter may be large (Soderblom 2010). Much like the measurement errors are unlikely to introduce systematic bias, scatter in the initial A(Li) for our Kepler stars should not introduce systematic effects in our reported A(Li). We note that the Hyades curve does not extend far enough at low T_eff for us to compare systems with stellar T_eff ≲ 5100 K. Therefore, we extrapolate the Hyades curve to low T_eff to solve this issue. We choose the observed median curve from the CKS sample as our adopted extrapolation (blue dashed line at low T_eff in A(Li)–T_eff plots) because it appears to follow the A(Li)–T_eff relationship of the Hyades at low T_eff.

Before we begin any detailed statistical tests, we must again eliminate any systems which may introduce bias. We removed all planets/systems discussed at the beginning of Section 4.4, in addition to systems with T_eff > 5500 K because our sample of young systems at higher T_eff is incomplete. However, we do re-include stars with spectral S/N = 10–30 because some of these systems include significant Li detections. We also note that ignoring these stars significantly reduces the size of our available sample. After making these cuts, 257 systems that host 408 exoplanet candidates remained.

We separated the two groups into old (Δ_A(Li),Hyades ≤ 0) and young (Δ_A(Li),Hyades > 0) systems. We placed these points into R_p bins and plotted the resulting normalized histograms (see top plot of Figure 13). There are 285 old planets (purple histogram) and 123 young planets (green histogram). We also performed a two-sided/two-sample Kolmogorov–Smirnov (K-S) test to determine if the two distributions are from different parent populations. We plot the cumulative fraction of planets in R_p and the K-S test result in the bottom panel of Figure 13. With a p-value of 0.004, we reject the hypothesis that the two samples were drawn from the same parent distribution at a statistical significance of ∼3σ. Therefore, we conclude that old and young systems represent distinct populations in R_p space. The median value for the young planets is R_p = 2.13 ± 0.01 R_⊕, while the median value for the old planets is R_p = 1.61 ± 0.01 R_⊕.

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To ensure we are not being fooled by potential confounding factors (such as transit S/N limitations of small planets around larger stars), we plot the stellar radius distributions (Petigura et al. 2017) of stars younger (green) and older (purple) than the Hyades in Figure 14 as we do planets in Figure 13. From Figure 14, we observe that the stars younger than the Hyades are systematically smaller than their older counterparts. This limits the possibilities for artificial inflation of younger planets. We note that the stars are primarily in the range 0.6–1.0 R_⊙. From transit S/N considerations and these stellar radius distributions, it is likely that we can detect smaller planets around the younger stars if they are present. Ultimately, we conclude that our young stars are smaller and unlikely to explain the difference we see in the old and young planet radii distributions.

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As has been demonstrated by Babu & Feigelson (2006), the one sample K-S test and other empirical distribution functions can be unreliable when using them to determine the parameters of best-fit models. Fortunately, we are using a two-sample K-S test to compare two populations within our data, not using it to determine parameters of those populations. Despite the utility of the two-sample K-S statistic, according to Engmann & Cousineau (2011), this statistic is inferior to the two-sample Anderson–Darling (A-D) statistic. The A-D test is more proficient in detecting differences in shift, scale, and symmetry between samples from two different distributions. Thus, we perform a two-sample A-D test on the same sample used for the two-sample K-S test. Our resulting normalized A-D statistic is 5.98, corresponding to a p-value of 0.002 (>3σ). Unsurprisingly, we report a more significant A-D p-value than K-S p-value.

However, these results are not robust: if we remove systems with S/N < 30 from our sample, our K-S p-value rises to 0.48 and our A-D p-value rises to 0.37, which both correspond to a statistical significance of <1σ. This is not sufficiently significant to definitively conclude that exoplanet radii decrease as exoplanets age. Removing these moderate-S/N (10 <S/N < 30) systems reduces the number of old planets from 285 to 212 and the number of young planets from 123 to 43. We argue that including moderate-S/N systems is important for the K-S and A-D tests because without them we are left with a very small sample of younger planets, insufficient for statistical distribution comparisons with the older planets.

We also perform additional statistical analyses, including Pearson and Spearman rank-order correlation coefficient tests of the relation between R_p and Δ_A(Li),Hyades. We use these tests to analyze 20 exoplanet candidates that remain after removing all of the following stars and planets: subgiants and giants, T_eff > 5500 K, upper limit A(Li), stars with Δ_A(Li),Hyades < 0 (old stars), stars with spectral S/N < 10, false positive planets, planets with b > 0.7, and planets with transit S/N < 10. We remove all upper limits in A(Li) because the ages of these systems are inherently uncertain. In addition, we eliminate old stars and planets because these systems/planets contaminate the theoretically predicted relationship between planet size and age in younger systems (Lopez et al. 2012). The Spearman rank-order test is more reliable than the Pearson test because it assumes nothing about the underlying relationship between planet radius and age, unlike the Pearson test which assumes a linear relationship.

We plot the results of the Spearman test in Figure 15, and we report that it returns a positive correlation between planet radius and A(Li) relative to the Hyades. The figure has a strong positive correlation coefficient (r = 0.6465), and a small p-value (p = 0.0021). We performed a similar analysis using the Pearson correlation coefficient test for comparison. Unsurprisingly, the Spearman correlation coefficient is stronger than the Pearson coefficient. Unlike the K-S and A-D test results, these correlation coefficients are robust and not sensitive to changes in parameter ranges. Although the p-value indicates a high significance of correlation between planet radius and the relative age of these systems, our small sample size makes it difficult to trust these p-values at their reported significance. Nevertheless, the combination of the low p-value and the strong positive correlation coefficient between planet radius and age supports the current models of young exoplanet evolution.

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