The Kepler Giant Planet Search. I. A Decade of Kepler Planet-host Radial Velocities from W. M. Keck Observatory

When fitting systems with transiting planets, (i) the orbital period and phase of the transiting planets are known, and (ii) the assumption of long-term orbital stability implies that additional planets, particularly massive giant planets, do not have orbits so close to the transiting planets that they would destabilize the system. These key advantages of searching for new planets in a system with known transiting planets, as compared to a blind planet search, inspired us to develop a custom algorithm for identifying nontransiting planets in the Kepler systems, which we call the "KGPS" algorithm (Figure 2). The key steps are as follows:

• 1.  
  Search for additional published RVs with RV precision of <5 m s⁻¹. Other instruments included HARPS-N (Cosentino et al. 2012) and SOPHIE (Perruchot et al. 2011).
• 2.  
  Model and optimize a linear combination of orbits for the N transiting planets. We assumed that the transiting planets were in circular orbits. This assumption is motivated by previous studies showing that the stars with multiple transiting planets typically have e < 0.1 (Mills et al. 2019a; He et al. 2020; Yee et al. 2021). Further, introducing eccentricity and argument of periastron as free parameters for planets with low RV semi-amplitudes can absorb noise and/or signals from nontransiting planets. Additional free parameters were the RV zero-point γ and jitter (σ_j , which is added to the measured errors in quadrature) for the HIRES RVs, as well as γ and σ_j terms for any additional instruments. (Note that the only cases where two or more observatories were used had many more than 10 RVs.) For data sets with at least 20 degrees of freedom (dof) and for planets larger than 1.15 R_⊕, we also allowed the N RV semi-amplitudes of the planets (K₁,...,K_N ) to vary, allowing a determination of the masses of the small planets. In cases with fewer than 20 dof or for planets smaller than 1.15 R_⊕, we used the Weiss & Marcy (2014) mass–radius relation to estimate the most likely mass of each transiting planet to determine fixed values of K₁,...,K_N . This assumed mass–radius relationship did not produce any results that were inconsistent with our observations. We used the software package RadVel to develop and optimize all of our models (Fulton et al. 2018). Note that for fixed orbital periods, phases, and a circular orbit, this is a linear problem (in parameters $K\cos \omega$, $K\sin \omega$, γ, and σ_j ), so there is a single global best-fit solution (Wright & Howard 2012).
• 3.  
  After subtracting the best N-planet fit, take a "fast periodogram" of the residual RVs, using the algorithm fasper developed in Press & Rybicki (1989). Assess whether there is a significant peak by estimating the false-alarm probability (FAP) given the peak height and the number of effectively independent frequencies considered: $\mathrm{FAP}\approx 1-{[1-\exp (-{P}_{\max })]}^{2{N}_{\nu }/{f}_{\mathrm{over}}}$, where ${P}_{\max }$ is the maximum power and N_ν /f_over is the total number of frequencies sampled divided by the typical rate of oversampling the average Nyquist frequency (Press & Rybicki 1989). For all targets, we oversampled the Nyquist frequency by f_over = 100.
• 4.  
  If there is a significant peak (FAP < 0.05), assess whether the period of the peak is an appropriate guess for the orbit of an additional planet. This involves checking that the period is (1) within the window function (<4× the RV baseline), (2) longer than 10.0 days (signals near peaks in the window function of the RVs, such as 1 day, are often aliases of long-period structure visible by eye in the RVs), and (3) distinct from the periods of other planets in the model (with a minimum period ratio of 1.15). If any of these conditions are not met, instead propose including a linear RV trend in the model, effectively adding one free parameter (dv/dt) to represent an additional high-mass, long-period body. In cases where at least one condition (1–3) is not met and with at least 15 dof, we also included RV curvature (d² v/dt²).
• 5.  
  If the new proposed period passes the above criteria, fit a new model with all of the previous planets, plus a planet at the new period. We modeled each new planet orbit with two to four free parameters: the RV semi-amplitude K_N+1 and the time nearest conjunction, t_c (even if the planet does not transit). Note that the orbital period was fixed during this step. We used the phase of the Lomb–Scargle periodogram at each period to guess a time of conjunction. In cases with at least 15 dof, we also included two eccentricity parameters, modeled using the basis of $\sqrt{e}\sin \omega$ and $\sqrt{e}\cos \omega$, which effectively assert a uniform prior for e ∈ [0, 1) and ω ∈ [0, 2π).
• 6.  
  Compute and compare the Bayesian information criterion (BIC, with difference ΔBIC) of the N + 1-planet and N-planet models (Kass & Raftery 1995). In our formulation, ΔBIC = BIC(N + 1-planet model) – BIC(N-planet model), with negative values indicating that an N+1-planet model is preferred. While some authors consider ΔBIC of −10 to be sufficient to distinguish the models, we adopt a more conservative ΔBIC threshold of −20. This is particularly important since we effectively tried thousands of different orbital periods before selecting the best one to model via the Lomb–Scargle periodogram. The choice of −20 is arbitrary but based on a qualitative calibration based on trial and error; a substantially lower value leads to several dubious planetary signals, whereas a substantially higher value misses signals that are apparent by eye. The choice of ΔBIC of −20 is also advantageous, since it matches that adopted by Rosenthal et al. (2021) for claiming a planet detection. For trends, we only required ΔBIC of −10, since we only searched for one trend per system (as compared to thousands of orbital periods).
• 7.  
  While the N + 1-planet model is preferred, increment N and repeat steps 3–6.
• 8.  
  When the search has converged on a preferred model, use a Markov Chain Monte Carlo (MCMC) algorithm to explore uncertainties in the free parameters. We used RadVel, which employs the affine-invariant sampler of Goodman & Weare (2010), as implemented for Python in emcee (Foreman-Mackey et al. 2013). Our priors were as follows: (a) the RV semi-amplitudes are positive (K₁,...,K_N > 0), (b) eccentricities are bounded between 0 and 1, and (c) the RV jitter is bounded between 0 and 20 m s⁻¹.

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Table 2. Radial Velocities from Keck-HIRES of 63 Kepler Planet-hosting Stars

Star	BJD –2450000	RV (m s−1)	RV Error (m s−1)	S-value	log ${R}_{\mathrm{HK}}^{{\prime} }$
hip94931	6109.920119	14.01	1.12	0.176	−5.013
hip94931	6110.830622	17.28	1.26	0.178	−5.008
hip94931	6111.896948	13.29	1.15	0.177	−5.011
hip94931	6112.869244	14.57	1.18	0.180	−5.001
hip94931	6113.809189	20.16	1.22	0.175	−5.016
hip94931	6114.831727	19.32	1.22	0.177	−5.010
hip94931	6115.853397	17.51	1.05	0.175	−5.018
hip94931	6133.929013	18.50	1.22	0.176	−5.012
hip94931	6134.967381	19.87	1.25	0.177	−5.012
hip94931	6138.030813	14.89	1.27	0.178	−5.008

Note. Columns are star name, barycentric Julian date, RV, RV error, Mount Wilson S-value, and stellar activity index log ${R}_{\mathrm{HK}}^{{\prime} }$. The first few rows of the table are shown for content.

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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The KGPS algorithm includes steps to account for the number of RVs and the dof to reduce model complexity and speed up model selection compared to purely blind algorithms (e.g., rvsearch; Rosenthal et al. 2021). The KGPS algorithm uses tools from the astronomy-specific Python packages astropy and radvel (Astropy Collaboration et al. 2013, 2018; Fulton et al. 2018; Astropy Collaboration et al. 2022). The output of the KGPS search algorithm for each system is shown in Table 3.

One nuisance parameter in our model is the stellar jitter, σ_j . This parameter is added in quadrature to the intrinsic RV errors in a manner that accounts for stellar-induced RV variability. Often, the stellar jitter is measured empirically for a given star based on the rms of the RV residuals. However, in cases where we have only 10 RVs per star and are looking to detect nontransiting planets, a larger-than-expected rms of the RV residuals could be due to nontransiting planets rather than stellar variability.

To disentangle whether the scatter in the RVs is likely caused by a nontransiting planet or by stellar physics, we make empirical estimates of the expected stellar jitter for each star in our sample. The empirical estimates are based on measured RV scatter for large samples of stars with many observations and known stellar properties. Our stellar jitter estimate incorporates Mount Wilson calcium activity index (S_HK), measured as an excess above an empirically determined activity baseline (ΔS = S_HK − S_BL) as a function of photometric color (B − V)

$$\begin{eqnarray*}&&\begin{array}{l}{\sigma }_{j,{\rm{S}}}=2.3+17.4\,{\rm{\Delta }}S\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (0.4\lt B-V\lt 0.7)\\ {\sigma }_{j,{\rm{S}}}=2.1+4.7\,{\rm{\Delta }}S\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (0.7\lt B-V\lt 1.0)\\ {\sigma }_{j,{\rm{S}}}=1.6-0.003\,{\rm{\Delta }}S\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (1.0\lt B-V\lt 1.3)\\ {\sigma }_{j,{\rm{S}}}=2.1+2.7\,{\rm{\Delta }}S\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (1.3\lt B-V\lt 1.6)\end{array}\end{eqnarray*}$$

(Isaacson & Fischer 2010), the line-of-sight projected stellar rotational velocity $v\sin i$

$$\begin{eqnarray}&&{\sigma }_{j,\mathrm{rot}}=0.876+0.140\,\displaystyle \frac{v\sin i}{\mathrm{km}\,{{\rm{s}}}^{-1}}+0.009\,{\left(\displaystyle \frac{v\sin i}{\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{2}\,{\rm{m}}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 1 }$$

(Chontos et al. 2022), and the stellar mass M_⋆ (Luhn et al. 2020)

$$\begin{eqnarray*}&&\begin{array}{l}{\sigma }_{j,{\rm{M}}}=2.5\,{\rm{m}}\,{{\rm{s}}}^{-1}\ ({M}_{\star }\lt 1.2)\\ {\sigma }_{j,{\rm{M}}}=3.5\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (1.2\lt {M}_{\star }\lt 1.3)\\ {\sigma }_{j,{\rm{M}}}=4.0\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (1.3\lt {M}_{\star }\lt 1.5)\\ {\sigma }_{j,{\rm{M}}}=5.0\,{\rm{m}}\,{{\rm{s}}}^{-1}\ (1.5\lt {M}_{\star }\lt 1.7)\\ {\sigma }_{j,{\rm{M}}}=5.5\,{\rm{m}}\,{{\rm{s}}}^{-1}\ ({M}_{\star }\gt 1.7).\end{array}\end{eqnarray*}$$

We computed the expected jitter from each of these relations and chose the mean value as the expected intrinsic stellar jitter for each star in our sample. In addition to the physical jitter, we defined "photometric" jitter by empirically fitting a lower envelope to the observed rms of the RVs as a function of the stellar flux relative to a V = 12.0 star (F/V₁₂) for our sample:

$$\begin{eqnarray}&&{\sigma }_{j,\mathrm{phot}}=\displaystyle \frac{0.81}{F/{V}_{12}}+2.2\,{\rm{m}}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 2 }$$

We combined the expected intrinsic stellar jitter and the photometric jitter for a total expected value of

$$\begin{eqnarray}&&{\mathrm{rms}}_{\mathrm{expected}}=\sqrt{\mathrm{Max}{({\sigma }_{j,{\rm{S}}},{\sigma }_{j,\mathrm{rot}},{\sigma }_{j,{\rm{M}}})}^{2}+{\sigma }_{j,\mathrm{phot}}^{2}}.\end{eqnarray} \tag{ 3 }$$

The expected jitter values and observed rms of the RV residuals are given in Table 4. We plotted the observed rms as a function of expected rms in Figure 3. The rms of the RVs for each system is shown twice: once before any planets are modeled (open triangles), and once after fitting the small transiting planets and any nontransiting companions as prescribed in Figure 2 (filled circles). Systems for which our model improved the rms of the RVs by at least a factor of 5 are highlighted in blue. These are the systems in which we have detected giant planets.

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Table 4. Stellar Properties and RV Data Summary

KOI	Kepler No.	Gaia mag	M⋆	[Fe/H]	NTP	NSS	NGP	NSC	NRV	ΔT	RV rms	σjit	dv/dt	$\lt M\sin i\,$ at 10 au
			(M⊙)							(yr)	(m s−1)	(m s−1)	(S/N)	(MJ, 99.7% conf.)
41	100	11.1	1.12	0.12	3	1	0	0	134	13.4	3.3	4.8	2.7	0.22
69	93	9.9	0.89	−0.20	1	0	0	0	184	13.1	2.6	4.9	112.3	7.41
70	20	12.5	0.93	0.04	5	1	0	0	143	9.1	5.4	2.7	2.0	1.43
72	10	10.9	0.90	−0.17	2	1	0	0	226	10.0	4.0	4.8	0.1	0.29
82	102	11.5	0.80	0.10	5	0	0	0	73	9.3	3.9	3.2	1.6	0.75
84	19	11.9	0.88	−0.12	1	1	0	0	124	7.9	5.0	2.7	3.2	1.39
85	65	11.0	1.24	0.09	3	0	1	0	79	11.1	7.4	5.0	1.5	1.51
94	89	12.2	1.20	0.04	4	0	0	0	72	12.7	8.9	4.8	1.8	1.44
103	⋯	12.6	0.93	−0.07	1	0	0	0	21	12.2	8.0	5.2	0.3	1.23
104	94	12.9	0.82	0.22	1	0	1	0	39	12.0	5.4	3.4	5.4	2.89
108	103	12.2	1.15	0.14	2	0	0	0	28	12.2	6.6	4.7	0.4	0.93
111	104	12.6	0.84	−0.53	3	0	0	0	44	11.3	4.5	4.5	0.1	0.33
116	106	12.8	0.97	−0.10	4	1	0	0	48	9.8	3.5	4.6	1.2	0.69
122	95	12.3	1.12	0.33	1	0	0	0	36	7.9	4.9	4.8	1.1	1.07
123	109	12.3	1.03	−0.06	2	0	0	0	38	10.9	6.3	4.5	2.3	0.67
137	18	13.5	0.97	0.32	3	0	0	0	25	8.8	5.2	2.7	0.2	1.02
142	88	13.1	0.98	0.29	1	0	2	0	58	9.0	8.3	2.7	2.0	1.96
148	48	13.1	0.91	0.26	3	0	2	0	59	12.8	4.6	2.9	6.3	1.97
153	113	13.5	0.77	0.12	2	0	0	0	42	12.1	8.1	1.8	0.8	1.23
157	11	13.7	1.00	0.05	6	0	0	0	31	11.6	7.8	4.7	1.2	1.58
244	25	10.6	1.14	−0.15	2	1	0	0	69	11.2	5.9	4.6	0.1	0.55
245	37	9.5	0.80	−0.43	3	0	0	0	218	12.4	3.2	2.7	1.4	0.26
246	68	9.9	1.06	0.11	2	0	2	0	96	12.2	2.9	4.5	0.3	0.21
260	126	10.5	1.08	−0.32	3	0	0	0	35	5.3	9.8	4.7	1.0	12.03
261	96	10.3	1.01	0.01	1	0	0	0	55	10.9	6.7	5.2	3.0	1.08
262	50	10.4	1.10	−0.16	2	0	0	0	39	9.8	13.9	4.9	0.8	7.43
265	507	11.9	1.17	0.12	1	1	0	0	49	10.8	3.3	3.0	0.1	0.51
273	454	11.4	1.08	0.34	1	0	2	0	116	12.1	5.7	2.8	8.7	2.51
274	128	11.3	1.12	−0.13	2	0	0	0	17	7.9	8.1	3.4	0.3	1.50
275	129	11.6	1.24	0.25	2	0	1	0	35	7.8	4.3	3.2	2.1	4.09
277	36	12.1	0.97	−0.22	2	0	0	0	25	9.2	7.5	2.9	0.1	0.99
281	510	11.9	0.86	−0.50	1	0	0	0	11	10.1	4.1	2.7	0.4	0.66
282	130	11.8	0.90	−0.26	3	0	0	0	10	6.9	5.9	4.3	0.1	1.91
283	131	11.5	1.06	0.18	2	0	0	0	46	6.9	4.8	5.0	1.8	1.23
285	92	11.7	1.25	0.19	3	0	0	0	23	9.9	5.8	3.2	0.1	0.94
292	97	12.8	0.91	−0.22	1	0	0	0	31	8.0	5.8	4.6	1.3	1.24
295	⋯	12.2	0.97	−0.22	2	0	0	0	14	10.7	11.7	4.5	4.7	4.11
299	98	12.9	0.98	0.09	1	0	0	0	42	7.9	7.1	2.7	0.5	1.19
305	99	13.0	0.81	0.13	1	0	0	0	45	7.0	5.3	3.4	0.5	1.18
316	139	12.7	1.08	0.38	3	0	1	0	38	11.8	4.0	2.7	0.4	1.34
321	406	12.5	1.06	0.30	2	0	0	0	58	11.9	4.6	2.7	0.3	0.54
351	90	13.7	1.11	0.14	8	0	1	0	34	11.2	9.9	4.5	0.5	1.80
365	538	11.2	0.87	−0.18	2	0	0	0	28	7.0	3.1	2.8	0.3	0.83
370	145	11.9	1.20	0.01	2	0	0	0	11	6.8	12.9	4.7	1.2	6.33
377	9	13.8	1.03	0.00	3	0	0	0	49	11.7	10.0	5.0	0.4	1.89
623	197	11.8	0.85	−0.59	4	0	0	0	11	7.0	9.0	4.4	0.1	3.05
701	62	13.7	0.68	−0.37	5	0	0	0	17	4.8	5.3	4.6	0.6	2.56
719	220	12.9	0.73	0.08	4	0	0	0	10	9.8	5.4	1.8	0.1	1.36
1241	56	12.4	1.46	0.42	2	0	2	0	47	10.1	5.7	3.5	2.0	6.98
1442	407	12.5	1.08	0.38	1	0	1	0	98	11.0	4.9	2.7	8.4	4.27
1612	⋯	8.7	1.02	−0.27	1	0	0	0	56	11.1	4.6	4.4	1.3	1.42
1692	314	12.5	1.00	0.34	2	0	0	0	14	10.8	13.1	2.9	0.8	3.18
1781	411	12.2	0.83	0.18	3	0	0	0	14	9.7	20.0	3.9	2.2	4.32
1909	334	12.7	1.01	0.03	3	0	0	0	11	6.1	8.8	4.6	0.7	3.27
1925	409	9.4	0.93	0.09	1	1	0	0	97	10.7	3.2	2.7	0.6	0.39
1930	338	12.1	1.06	−0.02	4	0	0	0	11	7.0	8.8	4.9	0.4	3.37
2169	1130	12.4	0.94	0.06	4	0	0	1	34	10.2	6.5	2.8	0.5	19.14
2687	1869	10.1	1.01	−0.01	2	0	0	0	26	9.3	7.3	2.8	2.1	1.90
2720	⋯	10.3	1.00	−0.15	1	0	0	0	22	6.3	4.4	4.6	1.3	1.43
2732	403	12.8	1.21	0.00	4	0	0	0	11	6.0	9.2	4.8	0.3	3.74
3083	⋯	12.8	1.16	0.29	3	0	0	0	11	6.2	6.4	2.7	0.2	2.23
3158	444	8.6	0.73	−0.54	5	0	0	1	207	9.8	2.3	2.8	35.7	5.67
3179	1911	12.1	0.99	−0.04	1	0	0	0	11	7.9	9.9	5.0	0.1	3.17

Note. Columns are KOI, Kepler number (e.g., "10" corresponds to Kepler-10), Gaia magnitude, stellar mass and metallicity, the number of transiting planets, number of RV-detected sub-Saturns, number of RV-detected giant planets, number of RV-detected stellar companions, number of RVs, RV baseline, rms of the RV residuals to the best-fit model, the expected stellar jitter computed in Section 4.2, the measured RV trend, and the 3σ upper limit on $M\sin i$ at 10 au based on the trend. KOI-142, KOI-351, and KOI-1241 each have one giant planet interior to 1 au that counts toward the total number of giant planets in this table. KOI-142 c was detected based on TTVs, whereas KOI-351 h and KOI-1241 c are transiting planets with $M\sin i\gt 0.3{M}_{{\rm{J}}}$ that are interior to 1 au. Additional columns associated with this table are available in the machine-readable version.

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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More than half of the points in Figure 3 are above the "expected" curve. One possibility is that our method for estimating the expected RV scatter by averaging the three jitter estimates from the literature is overly optimistic. For example, in cases where one factor contributes strongly to the RV scatter (such as stellar activity), the rms of the residuals would be better described as the maximum than the mean of the three values. However, when we adopt the maximum of the three jitter estimates, the majority of predictions overestimate the total rms because taking the maximum estimated jitter is a conservative approach. Another possibility is that there might still be planets of various masses that we have not yet detected in our sample that contribute significantly to the rms of the residual RVs.

A careful study of RV variability across a large sample of stars that incorporates multiple jitter predictors (stellar activity, mass, rotation, and photon limits) and identifies as many low-mass planets as possible would enable a more precise estimate of the jitter of each star, but such an effort is beyond the scope of this paper.

The KGPS algorithm yielded several systems for which the best-fit model included one or more nontransiting, eccentric companions. The KGPS algorithm has limitations in its ability to identify the most likely orbital periods of long-period companions, particularly in cases where the companion is eccentric and has a longer period than the RV baseline. For every system in which one or more nontransiting companions were detected, we examined whether a superior fit could be found by eye to the best-fit model from the KGPS algorithm. We then used the best fit (as determined either by eye or by the KGPS algorithm) to seed a new MCMC analysis in which the orbital periods and eccentricities of the nontransiting companion(s) were allowed to vary, as well as any transiting companions suspected of having significant RV amplitudes. The MCMC chains had 48 walkers each and ran until the chains were "well mixed," based on having a maximum Gelman−Rubin statistic of ≤1.03, a minimum chain length to autocorrelation length of 75, a maximum relative change in autocorrelation time of ≤.01, and at least 1000 independent draws. In cases where the periodogram yielded peaks of similar heights but at different orbital periods, we tested models with the companion at each of the possible periods.

In most cases, the MCMC algorithm yielded nearly identical results to the KGPS algorithm. A few notable exceptions are KOI-148, KOI-246, and KOI-316, in which an additional long-period planet was detected by eye. In KOI-148 and KOI-246 these additional companions are based on low-amplitude, moderately eccentric signals with periods comparable to the RV baseline, whereas in KOI-316 a high-amplitude signal is missed by the KGPS algorithm because the signal has very uneven time sampling, leading to a low-significance peak in the periodogram.

Other substantial updates were in KOI-94 and KOI-2169. In KOI-94, the RV data yield a significant improvement in the orbital period of one of the transiting planets (KOI-94 d), a warm Jupiter-sized planet with an RV semi-amplitude of ∼20 m s⁻¹. The system is known to have planet–planet interactions (based on observed TTVs; Masuda et al. 2013), and the baseline from our RV survey (12 yr) is substantially longer than the baseline of Kepler photometry (4 yr), and so it is not surprising that we are detecting a modest deviation from the best-fit linear ephemeris for this planet. In KOI-2169, the orbit of the stellar-mass companion is very eccentric and has a much longer period than the RV baseline, which resulted in a poor fit in the KGPS algorithm. We used the publicly available algorithm rvsearch to explore a fine grid in the period−eccentricity solution space for this companion and identify a better fit (Rosenthal et al. 2021).

The system architectures are presented in Figure 4. Each row corresponds to a planetary system, and the systems are ranked by the semimajor axis of the innermost known planet (typically, a transiting planet). Each body (planetary or stellar) orbiting the primary star is indicated with a circle, and the circle size is scaled to the square root of the body's mass or $M\sin i$, if known. In cases where the RVs were insufficient to meaningfully constrain the planet mass, a mass–radius relationship was assumed (Weiss & Marcy 2014). Each circle is colored by the object's eccentricity, if known (the transiting planets are assumed to have zero eccentricity).

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A few notable patterns emerge in Figure 4. The two most massive objects in our survey, Kepler-444 BC (represented by a single circle with the combined mass of Kepler-444 B and Kepler-444 C) and Kepler-1130 B, are also the most eccentric objects in our survey, and they accompany systems of high-multiplicity, small planets: the Kepler-444 planets are Mars sized, and the Kepler-1130 planets are Earth sized. In these systems, the periastron passage distances of the stellar-mass companion(s) with respect to the primary are 20 and 4 au, respectively, well inside the typical physical extent of a forming protoplanetary disk of ∼100 au (Weiss et al. 2022). It is likely that these stellar-mass companions truncated the protoplanetary disks in which the small planets formed. Such disk truncation would prevent the flow of dust and rocky material at the truncation radius, reducing the rocky material available in the innermost ∼1 au of the disk from which the planets formed. Discovering more systems with architectures analogous to Kepler-444 and Kepler-1130 is critical for building a sample from which to study the effect of eccentric S-type binary stars on the formation and architectures of planetary systems.

In contrast, the Jovian-mass planets we detected are not associated with particularly small or particularly close-in planets. Figure 5 shows the semimajor axes and masses of the bodies of Figure 4, including the transiting planets and their RV-detected companions. Systems that contain a Jovian-mass or larger body are shown in color, with the color corresponding to the mass of the most massive companion detected in the system. Two of these massive companions are transiting, KOI-94 d (Kepler-89 d) and KOI-351 h (Kepler-90 h), whereas the other companions were all detected via RVs (and in the case of Kepler-444, also high spatial resolution imaging) and either do not transit or have not yet been detected in transits. The KGPS survey spans seven orders of magnitude in companion mass and four orders of magnitude in orbital separation.

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Figure 6 shows how the KGPS I companions compare to field star companions from the California Legacy Survey (CLS; Rosenthal et al. 2021) in their eccentricity and semimajor axis distributions, for companions with $M\sin i\gt 0.3{M}_{{\rm{J}}}$. These surveys have different selection functions, and so any comparison between them should be examined with caution. An important distinction at short orbital periods is that the KGPS planets at a < 0.8 au are generally detected via transits, ³² whereas the CLS planets are all detected via RVs. There might also be differences at long periods because the CLS planets were detected in a 30 yr baseline, whereas the KGPS planets were detected in a 15 yr baseline, although the longest-period object is actually in the KGPS sample. We compare these samples for $M\sin i\gt 0.3{M}_{{\rm{J}}}$, for which the RVs of both surveys were sufficient to determine precise eccentricities, based on the results of Table 6. KGPS companions (blue) have lower eccentricities than the CLS companions (gray). The difference in the distributions of KGPS and CLS eccentricities is statistically significant based on the Anderson–Darling tests (p = 0.006) and marginally significant based on the Kolmogorov–Smirnov test (p = 0.048). One possible reason for an astrophysical difference in their eccentricity distributions is that the KGPS companions all have at least one surviving planet with a < 1 au in addition to any planets with a > 1 au, whereas the CLS planets are not required to have inner companions, and so they can have a higher envelope on eccentricity as a function of semimajor axis. Future work exploring the relationship between eccentricity, planet multiplicity, and orbital stability will clarify whether the discrepancy in eccentricity between KGPS and CLS is (1) statistically significant and (2) dynamical in origin.

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It is desirable to have a constraint on the $M\sin i$ upper limit for companions in each system based on the RVs. For each system, we used the best-fit model output from KGPS but allowed dv/dt to vary to test the hypothesis that the RVs have some long-term trend consistent with the presence of a long-period companion on a circular orbit (P ≳ 4× the RV baseline, typically ≳10 au). The absence of a significant trend or significant peaks in the Lomb–Scargle periodogram via KGPS generally rules out the presence of any companions with $M\sin i\gtrsim 1{M}_{{\rm{J}}}$ within 10 au with 3σ confidence, although the exact upper limits depend on the RV baseline, the number of RVs, and rms. The 3σ upper limits on $M\sin i$ for all the Kepler systems are shown in Figure 7. All of the cases where the upper limit on $M\sin i$ is larger than 10M_J at 10 au correspond to systems with detected long-period planets. There is typically sufficient freedom in the fit to the orbital parameters of the long-period nontransiting planets, especially massive ones, that a large value of dv/dt cannot be ruled out. These same dv/dt upper limits typically correspond to $M\sin i\lt 1{M}_{{\rm{J}}}$ at 5 au, which is interior to the nominal lowest period of 4× the RV baseline for most systems, but in combination with a nondetection in the Lomb–Scargle periodogram, the lack of a significant trend rules out the majority of possible Jovian companions interior to 5 au. Detailed injection recovery to determine each star's sensitivity to planets as a function of mass and orbital period will be addressed in Paper II of this series.

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Kepler-444 (KOI-3158, HIP 94931) is a bright (V = 8.9), nearby (d = 35.7 pc) K dwarf (component A) hosting five Mars-sized transiting planets within 0.1 au. The planetary system, including asteroseismology of the host star, was first characterized in detail by Campante et al. (2015). The system has a wide-separation M dwarf binary companion (components B and C) characterized by Dupuy et al. (2016), which has a nearly edge-on orbit and high eccentricity with periastron passage of just a few astronomical units. Mills & Fabrycky (2017) conducted a detailed analysis of the TTVs of the planets and measured mass upper limits for two of the planets comparable to that of Mars.

The RVs of Kepler-444 (Figure 8) exhibit a linear trend that is broadly consistent with the orbit described in Dupuy et al. (2016). An updated study that uses Gaia data and dynamical analysis confirms the eccentricity of the BC pair with respect to the primary found previously (0.8; Stalport et al. 2022). However, a separate analysis by Zhang et al. (2023) that includes direct adaptive optics of the BC pair, Gaia data, and new HIRES RVs of the systemic BC component finds an eccentricity of BC orbit of 0.55, which changes the periastron passage of the BC component from 5 to 8 au. The slight increase in periastron passage suggests a less severely truncated, more massive disk in which the planets formed than what was originally proposed in Dupuy et al. (2016), resolving the apparent conflict between the estimated primordial solid mass in the disk and the total planet mass.

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The periodogram of our best-fit residual to the trend includes a marginally significant peak at 72 days. The KGPS automated pipeline preferred including a planet at this period (based on ΔBIC = −27), but the Keplerian associated with this signal had an unphysically high eccentricity (of 0.6, our imposed upper limit) that would result in orbit crossing. We do not believe that this signal is associated with a planet; it is probably related to some complexity in the long-term RV signal of the Kepler-444 ABC system, and it might even be related to the orbit of the BC pair. Further analysis is needed to test the significance of the 72 day signal and identify its origin.

Kepler-100 (KOI-41) is a V = 11.2 Sunlike star at d = 307 pc. Transiting planets Kepler-100 b, c, and d, which have orbital periods of 6.89, 12.82, and 35.33 days and radii of 1.3, 2.2, and 1.6 Earth radii, respectively, were confirmed with RVs in Marcy et al. (2014). The mass measurements at the time were of order 2σ significance and/or mass upper limits. A detailed photodynamical analysis, which reproduced the transit depth variations of the planets as well as their TTVs, resulted in more precise masses: M_b = 5.1 ± 1.7 M_⊕, M_c = 14.6 ± 2.8 M_⊕, and M_d = 1.1 ± 0.5 M_⊕ (Judkovsky et al. 2022). The architecture of this system cannot be explained by photoevaporation alone because a low-mass, gas-enveloped planet exists interior to a highly irradiated rocky planet, and so other physical mechanisms (including atmospheric mass loss) were essential for sculpting the planet compositions (Owen & Campos Estrada 2020).

This star was observed for multiple programs at Keck Observatory. It was originally selected as a bright Kepler planet-hosting star for follow-up in Marcy et al. (2014). After several years of low-cadence observations, this target was selected for moderate-cadence analysis to improve the mass measurements of the transiting planets. Our analysis of the full RV time series spanning 2009–2022 (Figure 9) yields M_b = 5.5 ± 1.3 M_⊕, M_c = 3.8 ± 1.7 M_⊕, and M_d = 1.2 ± 1.4 M_⊕, in agreement with the TTV solution. In addition, we discover a nontransiting planet, Kepler-100 e, at P_e = 60.88 ± 0.04 days and with minimum mass M_e sini_e = 24.8 ± 3.5 M_⊕. Fixing the eccentricity of Kepler-100 e at zero results in M_b = 4.8 ± 1.3 M_⊕, M_c = 4.3 ± 1.7 M_⊕, M_d = 2.4 ± 1.8 M_⊕, and M_e = 25 ± 3 M_⊕, yielding an overall upper limit on the mass of planet d: M_d < 7.8 M_⊕ (3σ conf.). A joint analysis of the TTV and RVs would yield even better masses and orbital constraints of the four known planets. In our 12 yr RV baseline, we find no significant trend in the RV residuals (rms = 3.3 m s⁻¹), consistent with $M\sin i\lt 0.06{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 0.23{M}_{{\rm{J}}}$ at 10 au (3σ conf.), after assuming a four-planet model (including nontransiting planet e). A run of high-cadence RVs collected in 2022 as part of a student project are not presented here.

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Kepler-93 (KOI-69) has one transiting planet that is notable for having one of the most precise radius measurements of a super-Earth yet reported: R_b = 1.48 ± 0.02 R_⊕, which has a radius uncertainty of 120 km (Ballard et al. 2014). The precision of the planet radius measurement is based on asteroseismology of the host star (for a precise stellar radius) and transits observed by the NASA Spitzer space telescope (for a precise transit depth). RVs from Keck/HIRES yielded a planetary mass of M_b = 3.8 ± 1.5 M_⊕ and also a long-term trend (Marcy et al. 2014). Additional RVs from TNG/HARPS-N improved the mass characterization of the planet, M_b = 4.0 ± 0.7 M_⊕, a value consistent with an Earthlike rocky composition for the planet (Dressing et al. 2015).

With 64 new RVs from Keck/HIRES collected since Marcy et al. (2014), we have extended the RV baseline of Kepler-93 to 12 yr (Figure 10). The RV trend is dv/dt = 0.0373 ± 0.0003 m s⁻¹ day⁻¹. For a companion at 15 au (which corresponds to 4 times the RV baseline), this trend yields a minimum mass of $M\sin i\gt 15{M}_{{\rm{J}}}$, meaning that the companion is stellar or a brown dwarf, rather than planetary. The additional RVs also improve the mass precision for the transiting planet to a 6σ detection: M_b = 3.6 ± 0.6 M_⊕.

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Kepler-20 (KOI-70) has five transiting planets that are notable for their unusual size ordering, with alternating super-Earth-sized and Mars-sized planets. Buchhave et al. (2016) characterized this system using RVs from HARPS-N and HIRES, finding a nontransiting planet with a period of 34.9 days in between the 19.6 and 77.6 day transiting planets. We have collected one new HIRES RV since 2016 that is consistent with the solution from Buchhave et al. (2016). While the single new RV data point does not substantially improve the mass measurement of the transiting planets, it does further constrain the nondetection of an RV trend, yielding an upper limit of $M\sin i\lt 0.35{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 1.4{M}_{{\rm{J}}}$ at 10 au (3σ conf., Figure 11).

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Kepler-10 (KOI-72) contains the first rocky planet discovered by the Kepler mission with P_b < 1 day and a sub-Neptune-sized planet at P_c = 45.45 days (Batalha et al. 2011). RVs were collected on both Keck-HIRES and TNG-HARPS-N, and while both sets of data are consistent with a rocky composition for planet b, the data were discrepant regarding the mass and composition of planet c, with HIRES data favoring a low mass M_c = 7 M_⊕ and HARPS-N data favoring a high mass M_c = 17 M_⊕ (Dumusque et al. 2014; Weiss et al. 2016). The discrepancy might be caused by additional nontransiting planets in the system sampled at different phases by the two instruments, an idea that is supported by the detection of TTVs of planet c (Weiss et al. 2016). Modeling the RVs with Gaussian processes appears to somewhat resolve the discrepancy, although the host star is an old thick-disk star that is not expected to be magnetically active (Rajpaul et al. 2017).

We have collected 14 new HIRES RVs of Kepler-10 since Weiss et al. (2016), producing an RV baseline of 12 yr (Figure 12). The residuals to the best two-planet fit have rms = 4.6 m s⁻¹ (HIRES) and 3.9 m s⁻¹ (HARPS-N) and exhibit no RV trend, yielding an upper limit of $M\sin i\lt 0.11{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 0.46{M}_{{\rm{J}}}$ at 10 au (3σ conf.) for additional companions. Note that there is a third low-mass planet candidate (m_p ∼ 1 M_⊕) in the system, as indicated in Weiss et al. (2016), but the period is ambiguous. Because the third planet candidate is low-mass and has P < 200 days, our choice to exclude it from our model here does not substantially affect the upper limits on Jovian planets at long periods. However, we do detect significant structure in the periodogram of the RV residuals, with a prominent peak at P = 25 days (FAP = 0.04, ΔBIC = −18.9). This period was previously modeled as the stellar rotation period in a fit that used a Gaussian process method to remove stellar noise (Rajpaul et al. 2017), although a rotation period of 25 days is uncharacteristically fast for a 10 Gyr old thick-disk star. We consider this signal as a third planet candidate in the system, to be confirmed with more RVs. A careful analysis that jointly fits the HIRES and HARPS-N RVs as well as the TTVs might yield an improved characterization and/or confirmation of the planet candidate at 25 days.

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KOI-82 (Kepler-102) is a system with five transiting planets, three of which are smaller than Earth. The system was confirmed in Marcy et al. (2014) with HIRES RVs establishing mass upper limits for the planets. The star is active (with a visible stellar rotation signal in the Kepler photometry, and Mount Wilson S_HK = 4.48), which contributes to substantial RV scatter. A joint analysis of HIRES and HARPS-N RVS that included a Gaussian process trained on the rotational modulation in the Kepler light curve found M_d = 2.5 ± 1.4 M_⊕, M_e = 4.7 ± 1.7 M_⊕ (Brinkman et al. 2022). Here we employ a simple circular fit to planets d and e based on the HIRES data alone, with masses for the other (smaller) planets based on an empirical mass–radius relationship (Weiss & Marcy 2014). The HIRES RVs extend from 2010 April to 2021 June, comprising a baseline of 11 yr. The residuals to our best fit have rms = 3.9 m s⁻¹ and no RV trend, yielding a 3σ upper limit of $M\sin i\lt 0.2{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 0.8{M}_{{\rm{J}}}$ at 10 au for additional companions (Figure 13). Note that the Lomb–Scargle periodogram of the RVs has a marginal peak at 28 days, which is consistent with the stellar rotation identified in the Kepler light curve (Brinkman et al. 2022).

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Kepler-19 (KOI-84) was first confirmed and characterized in Ballard et al. (2011). The system has one transiting planet (b) with P_b = 9.3 days and R_b = 2.2 R_⊕. Significant TTVs of Kepler-19 b indicated the presence of a nontransiting perturber (Ballard et al. 2011). Precise RVs were collected with HARPS-N (101) in order to conduct a joint TTV-RV analysis (Malavolta et al. 2017), which resulted in a mass measurement of M_b = 8.4 ± 1.6 M_⊕ and the detection of two nontransiting planets: Kepler-19 c (P_c = 28.7 days, M_c = 13.1 ± 2.7 M_⊕) and Kepler-19 d (P_d = 63 days, M_d = 20.3 ± 3.4 M_⊕). The mass and radius of Kepler-19 b are consistent with a rocky interior overlaid with a substantial volatile envelope, and the masses of the nontransiting planets are also consistent with volatile-enveloped planets (Weiss & Marcy 2014; Rogers 2015).

We collected 24 HIRES RVs between 2009 and 2017, resulting in a baseline of 8 yr (Figure 14). We jointly analyze the HIRES and HARS-N RVs. We recover the planet at 28.54 days (Kepler-19 c, FAP = 0.02, ΔBIC = − 24), although we do not detect the planet at 63 days (Kepler-19 d). In the periodogram of the residuals, the highest peak is near 38 days and has FAP = 0.15. We obtain masses of M_b = 7.5 ± 1.9 M_⊕ and M_c = 17.3 ± 3.0 M_⊕. A joint analysis of all the RVs and TTVs is needed to reconfirm the planet at 63 days. The residual rms of the RVs is 4.7 m s⁻¹ with no significant trend, yielding an upper limit of $M\sin i\lt 0.3{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 1.8{M}_{{\rm{J}}}$ at 10 au (3σ conf.).

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KOI-85 (Kepler-65) has three transiting planets originally validated in Chaplin et al. (2013), all with P < 10 days and R_p < 4 R_⊕. A nontransiting planet was identified in 7 yr of Keck-HIRES RVs (Mills et al. 2019b). The compact configuration of the three inner planets generates TTVs, and a photodynamical analysis of the Kepler photometry and HIRES RVs yielded precise masses for the four known planets: ${M}_{b}={2.4}_{-1.6}^{+2.4}\,{M}_{\oplus }$, M_c = 5.4 ± 1.7 M_⊕, M_d = 4.14 ± 0.80, and ${M}_{e}={260}_{-50}^{+200}\,{M}_{\oplus }$ (planet e is nontransiting and decoupled from the inner planets, and so its inclination is not well constrained). We have collected 76 HIRES RVs between 2011 May and 2022 June (including two new RVs since Mills et al. 2019b), providing a baseline of 11 yr (Figure 15). After fitting a four-planet model (including planet e), the residual RVs have an rms = 7.0 m s⁻¹ and no apparent trend, yielding a 3σ upper limit of $M\sin i\lt 0.30{M}_{{\rm{J}}}$ at 5 au or $M\sin i\lt 1.2{M}_{{\rm{J}}}$ at 10 au.

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KOI-94 (Kepler-89) is a V = 12.2 late F-type star (T_eff = 6000 K) with R_⋆ = 1.20 M_⊙. It has four transiting planets, including a warm Jupiter (KOI-94 d, P_d = 22.34 days, R_d = 11.3 ± 1.1 R_⊕) surrounded by three sub-Neptunes (P_b = 3.73 days, P_c = 10.4 days, P_d = 54.3 days). Hirano et al. (2012) characterized the photometry of the system and detected a planet–planet occultation, thereby inferring that the line-of-sight projected mutual inclination between KOI-94.01 and KOI-94.03 is −115 ± 055. The system was confirmed with RVs in Weiss et al. (2013), in which the mass of KOI-94 d was measured (M_d = 106 ± 11 M_⊕) and upper limits were set for the smaller transiting planets. A contemporaneous analysis of TTVs in this system yielded a substantially different mass for the warm Jupiter (Masuda et al. 2013), although an erroneously low stellar mass was used in the TTV analysis (1.0 vs. 1.27 M_⊙), contributing to the discrepancy.

Several transits of the Kepler-89 planets (two of planet d, four of planet c, one of planet e, and none of planet b) were identified in TESS photometry, yielding an updated solution to the best fit to the TTVs (Jontof-Hutter et al. 2022). A four-planet model fit the observed TTVs poorly (reduced χ² = 2.48), and Jontof-Hutter et al. (2022) proposed two possible five-planet models that better reproduced the TTVs, although still with reduced χ² > 1.8. In model A the fifth planet candidate was at 118.0 days and had 7.0 M_⊕, whereas in model B the fifth planet was at 39.99 days and had 0.71 M_⊕.

KOI-94 was selected for follow-up as part of a program to survey stars with at least three transiting planets from 2015 onward. We have continued monitoring KOI-94 with HIRES and have collected 44 new RVs since 2013 and 97 RVs total on 73 unique nights (Figure 16). Note that RVs from 2012 August 10, which were part of an RM observation (Albrecht et al. 2013), have been removed from our analysis. Because the planets have significant TTVs, we tested a model in which we allowed the period of the warm Jupiter (which is strongly detected in the RVs) to vary, but with a prior informed from the TTV analysis of Jontof-Hutter et al. (2022): P_d = 22.3425 ±0.0004. For this model, the best-fit mass from the RVs is M_d = 72 ± 9 M_⊕, which significantly reduces the tension between the RV solution and the TTV solution. We tentatively detected a moderate eccentricity of the warm Jupiter of e_d = 0.2 ± 0.1, although (1) this eccentricity might be from mixing of near-resonant signals and (2) the introduction of a fifth planet to the model might reduce the eccentricity. We also tested a model with zero eccentricity for the warm Jupiter (which might be more realistic for a planet in a near-resonant chain) and found M_d = 77 ± 8 M_⊕. We prefer the zero-eccentricity model (Figure 16), although both models had very similar results for the masses of the transiting planets. The residual RVs have rms = 8.9 m s⁻¹, no significant peak in the Lomb–Scargle periodogram (right panel of Figure 16), and no apparent trend, yielding an upper limit of $M\sin i\lt 0.90{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 3.6{M}_{{\rm{J}}}$ at 10 au) for additional companions. These upper limits do not exclude the putative TTV planet candidates identified in Jontof-Hutter et al. (2022). The star's moderate rotation ($v\sin i$ = 7.5 km s⁻¹) produces relatively large RV errors, which will make an RV detection of the putative fifth planet challenging. Additional RVs collected in 2022 as part of a student project are not presented here and are consistent with the nondetection of long-period giant companions.

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Kepler-1710 (KOI-103) is a V = 12.6 Sunlike star. It has one transiting planet with P_b = 14.91 days and R_b = 3.20 R_⊕, which was recently reconfirmed with ExoMiner, a machine-learning validation tool (Valizadegan et al. 2022). We have collected 21 HIRES RVs with over a decade of baseline (2009–2021; Figure 17). After fitting for the transiting planet, the residual RVs have an rms of 8.3 m s⁻¹ and no trend, yielding a 3σ upper limit of $M\sin i\lt 0.3{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.1{M}_{{\rm{J}}}$ at 10 au) for any additional companions. The RV scatter in this system is unexplained in terms of the expected amplitude of the transiting planet and the stellar activity (log R ${}_{\mathrm{HK}}^{{\prime} }=-4.77$). There are no stellar companions within 2'' of the primary identified with either imaging or spectroscopy that could be causing increased RV scatter. Perhaps there are unidentified nontransiting planets that are confounding a mass measurement of the transiting planet.

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Kepler-94 (KOI-104) is a V = 12.9 late K-type or early M-type star with discrepant various temperatures reported in the literature (T_eff = 4200–4750 K; Muirhead et al. 2012; Yee et al. 2017; Fulton & Petigura 2018; Berger et al. 2018; Brewer & Fischer 2018). The system has one short-period transiting sub-Neptune (P_b = 2.51 days, R_b = 3.5 R_⊕). The transiting planet was confirmed with Keck-HIRES RVs in Marcy et al. (2014) (M_b = 10.8 ± 1.4 M_⊕), which also revealed a long-period companion with P_c = 820 ± 3 days and $M\sin {i}_{c}\,=9.7{M}_{{\rm{J}}}\pm 0.6{M}_{{\rm{J}}}$. We have collected 10 new RVs since Marcy et al. (2014), which yield refined properties for the long-period object: P_c = 816.3 ± 0.7 days, $M\sin {i}_{c}=8.9{M}_{{\rm{J}}}\pm 0.2{M}_{{\rm{J}}}$, and e_c = 0.35 ± 0.0086 (Figure 18). Because $M\sin i$ corresponds to a minimum mass, it is still unclear whether the nontransiting object is below or above the deuterium-burning limit of 13M_J. The HIRES RVs extend from 2010 to 2022, composing a 12 yr baseline. The residuals to our best two-planet fit (including the nontransiting companion) have rms = 5.4 m s⁻¹ and no apparent RV trend, consistent with a 3σ upper limit of $M\sin i\lt 0.30{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.2{M}_{{\rm{J}}}$ at 10 au) on any additional companions.

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Brewer & Fischer (2018) reported unusual abundances for this star. The star is apparently rich in nitrogen ([N/H] = 0.46, [C/H] = 0.19, [O/H] = 0.10) and very rich in sodium ([Na/H] = 0.70). The star is also rich in iron and nickel ([Fe/H]=0.38, [Ni/H] = 0.41), as well as magnesium and aluminum ([Mg/H] = 0.30, [Al/H] = 0.51). The uncertainties are reported as less than 0.1 dex for all abundances, although the star is a late K/early M type, for which determining abundances can be challenging. It is noteworthy that this star has apparently high metallicities for metals and also has a super-Jovian companion (which is rare for late K-type and early M-type stars). Perhaps the high metal abundances permitted the formation of the planet through core accretion, as suggested based on the correlation between giant planet occurrence and host star metallicity (Fischer & Valenti 2005). A more detailed examination of the host star properties and abundances and their possible relationship to the planet properties is warranted but is beyond the scope of this paper.

Kepler-103 (KOI-108) is a Sunlike star with two transiting planets: a sub-Neptune (R_b = 3.37 R_⊕) with P_b = 16.0 days, and a sub-Saturn (R_c = 5.14 R_⊕) with P_c = 180 days. The planets were confirmed with RVs from Keck-HIRES, with reported best-fit masses of 9.7 ± 8.6 and 36 ± 25 for planets b and c, respectively (Marcy et al. 2014). Although the planet masses were not detected with high confidence, these values correspond to upper limits on the planet masses that demonstrate that they are planetary (rather than stellar). We have collected six HIRES RVs since 2013, which have not substantially improved the planet mass estimates (Figure 19). The rms of the RV residuals is 6.8 m s⁻¹. The 12 yr RV baseline, which extends from 2009 to 2022, has no trend, placing a 3σ upper limit of $M\sin i\lt 0.24{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.0{M}_{{\rm{J}}}$ at 10 au). TTVs have been identified by Gajdoš et al. (2019), but the planet masses were not measured based on the TTVs.

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Kepler-106 (KOI-116) is a Sunlike star with four small transiting planets, with masses and/or mass upper limits published in Marcy et al. (2014). A periodogram of the RV residuals (rms = 6.0 m s⁻¹) reveals peaks near P = 90, 180, and 365 days (Figure 20). These peaks could be driven by the seasonal window function, combined with the seasonal downward trend of the RVs in 2012, 2013, and 2014. The star is not particularly active and does not have S-value variations that correspond to these seasonal RV slopes. The RV slopes are likely caused by some combination of a (real) giant planet with an undetermined period and seasonal aliasing. Another possibility is that planets at 90, 180, and/or 365 days would continue the near-resonant chain of the inner four transiting planets. If the planet candidate at 90 days is real (FAP = 0.003, ΔBIC = − 43), it produces an RV semi-amplitude of 7.5 m s⁻¹ and has $M\sin i=50\,{M}_{\oplus }$. Continued monitoring at a variety of cadences, including RVs early and late in the Kepler season, will help resolve what additional planet(s) exist(s). TTVs have been identified by Gajdoš et al. (2019), but the planet masses were not measured. Although the mass, period, and veracity of planets with P < 1 yr are uncertain, the RVs have no significant long-term trend, yielding a 3σ upper limit of $M\sin i\lt 0.25{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.0{M}_{{\rm{J}}}$ at 10 au). This mass upper limit does not change with the addition of a fifth planet at 90 days.

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Kepler-95 (KOI-122) has one transiting planet at P_b = 11.5 days and R_b = 3.4 R_⊕. The planet was confirmed with Keck-HIRES RVs in Marcy et al. (2014), which yielded a mass of M_b = 13.0 ± 2.9 M_⊕. The star is Sunlike, but with high metallicity ([Fe/H] = 0.3). We have collected three new RVs since 2013, which have not substantially improved the mass of the transiting planet (Figure 21). The RV baseline extends from 2009 to 2017 with no trend, yielding a 3σ upper limit of $M\sin i\lt 0.30{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.2{M}_{{\rm{J}}}$ at 10 au) for additional planets in the system.

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Kepler-109 (KOI-123) is a Sunlike star with two transiting planets at P_b = 6.48 and P_c = 21.2 days, both of which are smaller than Neptune (R_b = 2.37 R_⊕, R_c = 2.52 R_⊕). The planets were confirmed with Keck-HIRES RVs in Marcy et al. (2014), which established mass upper limits for the planets. Since 2013, we have collected 24 new RVs, resulting in 38 total RVs collected between 2009 and 2021 (an 11 yr baseline). Our best two-planet fit yields mass upper limits of M_b = 5.0 ± 4.4 M_⊕ and M_c = 5.5 ± 3.6 M_⊕, consistent with both planets being low-density sub-Neptunes with extended gas envelopes (Figure 22). The RV residuals have rms = 6.4 m s⁻¹ and no apparent trend, yielding a 3σ upper limit of $M\sin i\lt 0.16{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 0.65{M}_{{\rm{J}}}$ at 10 au) for additional planets in the system.

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Kepler-18 (KOI-137) is a Sunlike star with three transiting planets interior to 15 days (P_b = 3.50 days, P_c = 7.64 days, P_d = 14.86 days) that were first announced and confirmed in Cochran et al. (2011). The innermost planet, Kepler-18 b, is notably smaller than the other two planets (R_b = 2.0 R_⊕, R_c = 5.49 R_⊕, R_d = 6.98 R_⊕). The planets are near (but not in) a 4:2:1 Laplace resonant chain, and planets c and d have significant TTVs as the result of their dynamical interactions. Cochran et al. (2011) also collected RVs with Keck-HIRES to determine a Doppler mass for the planets, in comparison to (and in combination with) the TTV-determined masses. Their best-fit masses using TTVs and RVs were M_b = 6.9 ± 3.4 M_⊕, M_c = 17.3 ± 1.9 M_⊕, and M_d = 16.4 ± 1.4 M_⊕.

The solutions in Cochran et al. (2011) were based on Kepler photometry from quarters 0 to 6, whereas a more recent analysis by Hadden & Lithwick (2017) was based on the full Kepler time series (quarters 0–17). One challenge of TTVs for near-resonant systems is that the determination of mass and eccentricity can be degenerate (Lithwick et al. 2012). In Kepler-18, the mass of planet c is sensitive to whether a high-mass (and low-eccentricity) prior was adopted, with the solution for the default prior yielding M_c = 12.9 ± 6.0 M_⊕ and the high-mas prior yielding M_c = 21.6 ± 3.6 M_⊕, whereas the mass of planet d did not sensitively depend on the prior (M_d = 16.2 ± 1.4 M_⊕; Hadden & Lithwick 2017). Planet b has flat TTVs, and the reanalysis in Hadden & Lithwick (2017) did not update the mass of this planet.

Since 2011, we have collected 11 new RVs of Kepler-18. Our analysis of the RVs alone (without incorporating TTVs) yields M_b = 12.9 ± 4.1 M_⊕, M_c = 19.5 ± 5.1 M_⊕, and M_d = 24.0 ± 6.5 M_⊕, consistent with gas-enveloped compositions for all three planets (Figure 23). There is no apparent RV trend over the 9 yr baseline (2009–2018), yielding a 3σ upper limit of $M\sin i\lt 0.40{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.6{M}_{{\rm{J}}}$ at 10 au). This system would benefit from a joint analysis of the new RVs and TTVs to improve the masses and eccentricities of the planets.

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KOI-142 (Kepler-88) is a system with one transiting planet (P_b = 10.95 days, R_b = 3.43, R_⊕) that has TTVs with semi-amplitude of half a day (0.05 of the orbital period). The system was first noted in Steffen et al. (2013) and was colloquially dubbed the "King of TTVs" on account of the large TTVs of planet b. Nesvorný et al. (2013) conducted an in-depth dynamical analysis of the system, finding a unique solution for the mass and period of a giant planet perturbing the transiting planet from just outside the 2:1 mean motion resonance. The giant planet was confirmed with RVs collected at SOPHIE (Barros et al. 2014). This system is unique in that it is the only Kepler system we followed up for which TTVs clearly indicated an outer perturber. However, the period of the perturber was well-known (22 days). The TTVs gave no prior knowledge about whether this system would harbor a giant planet beyond 1 au.

A total of 8 yr of Keck-HIRES RVs were also collected and analyzed in a joint RV and photodynamical model in Weiss et al. (2020), which confirmed and refined the mass of the giant planet in resonance and also revealed a distant Jovian-mass companion at 2.6 au. Although the host star is Sunlike, it has high metallicity ([Fe/H] = 0.3). We have collected two additional Keck-HIRES RVs since 2020, which are consistent with the solution in Weiss et al. (2020; see Figure 24). Because our baseline is comparable to the longest detected orbit in the system, any putative trend in the residuals to our best three-planet fit is not well constrained, and so our upper limit on additional companions is only $M\sin i\lt 0.67{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 2.7{M}_{{\rm{J}}}$ at 10 au). Continued low-cadence monitoring will improve the ephemeris determination of the distant Jovian planet and might yield additional long-period planets.

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KOI-148 (Kepler-48) has three transiting planets with dynamical interactions that produce TTVs, which confirmed their planetary nature (Steffen et al. 2012). Marcy et al. (2014) presented 28 RVs of the system collected with HIRES, yielding masses for the transiting planets and revealing the existence of one nontransiting giant planet (Marcy et al. 2014). There are four stellar companions within 6'' of the primary summarized in Marcy et al. (2014). For the RV measurements, care was taken to align the field to avoid secondary light into the slit. The primary star is Sunlike.

We have obtained 28 new RVs with HIRES since Marcy et al. (2014), resulting in 56 RVs total, which we used to improve the characterization of the masses and orbits of the planets (Figure 25). We detected a new nontransiting planet, Kepler-48 f, based on our long-term monitoring, with high confidence (FAP = 2 × 10^–8, ΔBIC = −530). This planet's orbital period (P_f = 5400 days) is distinct from the peak periodicity of the Mount Wilson S-values (∼2500 days). In order of increasing orbital distance, the planets have periods P_b = 4.78 days, P_c = 9.67 days, P_d = 42.9 days, P_e = 1001 days, and P_f = 5205 days and minimum masses of 6.1 M_⊕, 11.6 M_⊕, 8 M_⊕, 2.1M_J, and 1.3M_J. We assume that the three transiting planets are circular, and we find upper limits on the eccentricities of the nontransiting planets of e_e < 0.09 and e_f < 0.36 (3σ confidence).

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Because our baseline is comparable to the longest detected orbit in the system, any putative trend in the residuals to our best five-planet fit is not well constrained, and so our upper limit on additional companions is only $M\sin i\lt 0.50{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 2.0{M}_{{\rm{J}}}$ at 10 au). Future RVs will improve the characterization of the period, eccentricity, and minimum mass of the long-period companion and will also place better constraints on the presence of additional companions.

Kepler-113 (KOI-153) is a V = 13.7 K-type star (T_eff = 4791 K; Fulton & Petigura 2018). The system has two transiting sub-Neptunes (R_b = 1.82 ± 0.6 R_⊕, R_c = 2.18 ± 0.06 R_⊕) with periods of P_b = 4.75 days and P_c = 8.93 days. The planets were confirmed with Keck-HIRES RVs in Marcy et al. (2014), which yielded a mass upper limit for the outer planet and a 2.7σ measurement for the inner planet of 11.7 ± 4.2 M_⊕. With 18 new HIRES RVs since 2013 (42 RVs total), we have refined the planet masses, finding M_b = 7.0 ± 3.3 M_⊕ and M_c = 2.5 ± 2.5 M_⊕ (Figure 26). Planet b, which is near the distinction between planets with rocky surfaces and planets with volatile envelopes (Weiss & Marcy 2014; Rogers 2015; Fulton et al. 2017), is consistent with having a predominantly rocky (by volume) composition, with a possible thin gas envelope, whereas planet c is consistent with having a larger gas envelope more typical of sub-Neptunes. The residual RVs (rms = 8.2 m s⁻¹) have structure at 55 and >500 days, although the false-positive probability (0.16) is too high to consider these signals as planet candidates. The possibility of an additional planet in the system was also noted in Marcy et al. (2014), but at periods of 1.065, 16, 0.984, and 0.515 days, none of which have become stronger signals with more data. Additional RVs will help determine whether one or both of the new candidate periods are associated with bona fide planets. There are no strong peaks in the periodogram of the RV residuals. There is no apparent trend in the RV residuals time series to our two-planet fit, yielding a 3σ upper limit of $M\sin i\lt 0.31{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.3{M}_{{\rm{J}}}$ at 10 au).

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Kepler-11 (KOI-157) is a faint Kepler star (V = 14) with six transiting planets that has been extensively studied as an exemplary multiplanet system with significant TTVs (Lissauer et al. 2011). The planets range in size from 1.8 to 4.2 R_⊕ and have ultralow densities, based on their TTVs (Lissauer et al. 2013), and the host star is a solar twin (Bedell et al. 2017). A subset of the RVs presented here were used in a joint RV and TTV analysis, the results of which were that (i) the RVs alone ruled out rocky compositions for the planets, (ii) the combined RV-TTV analysis was consistent with low densities for the planets, and (iii) the RVs reduced the mass upper limit of planet g, which is dynamically decoupled from the inner five planets and is therefore not well characterized by their TTVs (Weiss 2016). Between 2010 and 2022, we collected 31 RVs (Figure 27). The RVs provide mass upper limits for the transiting planets, as described in Weiss (2016). The residual RVs to our best six-planet fit have rms = 7.3 m s⁻¹ and no apparent trend, yielding an upper limit of $M\sin i\lt 0.6{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.6{M}_{{\rm{J}}}$ at 10 au, 3σ conf.)

Kepler-25 (KOI-244) is a Sunlike star with two transiting planets near the 2:1 mean motion resonance. Dynamical interactions between the transiting planets produce TTVs, which confirm the planetary nature of the system (Steffen et al. 2012). Marcy et al. (2014) collected 62 RVs of this system with Keck-HIRES. The RVs also revealed a nontransiting planet at P_d = 91 or 122 days. Hadden & Lithwick (2014) performed N-body forward modeling to fit the TTVs of the transiting planets and found upper limits on the planet masses. A photodynamical analysis that jointly fit the Kepler photometry and the HIRES RVs (including five new RVs) revealed a reduction in the planet mass–eccentricity degeneracy when using photometry alone (Mills et al. 2019b). The results are consistent with a low-eccentricity resonant state but do not indicate that the planets are definitively librating within the resonance. Two complete transits were observed with Keck-HIRES Doppler spectroscopy to measure the RM effect, finding the orbital plane of the planets and the axis of stellar rotation to be well aligned (Albrecht et al. 2013). The detection of the planets transiting in NASA TESS photometry was leveraged to update their orbital ephemerides, improving the precision of the linear ephemerides by an order of magnitude (Battley et al. 2021). Jontof-Hutter et al. (2022) combine legacy Kepler photometry with new TESS photometry to determine that the TTV solution does not change with the inclusion of TESS data for this system.

We have collected two new RVs since 2019, bringing the total number of non-RM RVs to 69 (Figure 28). The new RVs and updated ephemerides have not substantially changed the characterization of any of the known planets. However, the nondetection of an RV trend in the residuals to a three-planet model (including planet d at either 91 or 122 days) over the baseline of 2011–2022 yields a 3σ upper limit on additional long-period companions of $M\sin i\lt 0.70{M}_{{\rm{J}}}$ at 10 au.

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Kepler-37 (KOI-245) is a V = 9.7 G-type (T_eff = 5300 K) star. The system has three transiting sub-Neptunes (P_b = 13.4 days, P_c = 21.3 days, P_d = 39.8 days), one of which is smaller than Mercury (R_b = 0.30 R_⊕), that were identified and confirmed in Barclay et al. (2013). The star has solar-like oscillations, which enabled a precise determination of the physical stellar properties through the combination of asteroseismology and spectroscopy. Marcy et al. (2014) used Keck-HIRES RVs to place upper limits on the masses of transiting planets b, c, and d. A fourth transiting candidate at P = 51 days was identified in the photometry but was suspected to be due to noise or an instrumental false positive (Barclay et al. 2013). Nonetheless, the fourth candidate has been considered in various TTV analyses (Hadden & Lithwick 2014; Holczer et al. 2016). Most recently, Rajpaul et al. (2021) further examined Kepler photometry data, concluding that Kepler-37 e, with a 51 day period, should be stripped of its confirmed/validated status, citing the extremely weak photometric evidence. Assuming a circular orbit near 51 days, we find marginal evidence for an RV signal at P_e = 50.25 ± 0.15 days, with K_e = 1.6 ± 0.3 m s⁻¹ and M_e = 8.4 ± 1.7 M_⊕. We consider models both with and without a transiting planet candidate at this period.

Since the publication of Marcy et al. (2014), we have acquired 48 new RVs on Keck-HIRES, resulting in 81 RVs from HIRES and 218 RVs total (including literature RVs from HARPS-N; Figure 29). We applied models with either three or four small planets (to account for the candidate at 51 days) to the combined HIRES and HARPS-N RVs. In both cases, there is significant structure in the residual RVs, with peaks near 460 and 2000 days. When either the 460 day peak or 2000 day peak is fit, the other peak disappears, and so one of these peaks is an alias.

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We initially adopted the signal at 2000 days to test as a new planet candidate in the system. The amplitude of this signal is low (K = 3.3 m s⁻¹), and the fit is quite eccentric (e = 0.5). The effective temperature of this star (6265 K) places it very near the Kraft break, and the slightly evolved stellar surface gravity (log g = 4.31) means that this star is unlikely to be heavily influenced by chromospheric activity. However, there appears to be modest correlation between the Mount Wilson S-values and RVs, with the S-value periodogram peaking at approximately 2000 days (Figure 29). Further RV, stellar activity, and photometric monitoring, as well as a joint fit to the RVs and TTVs, is needed to ascertain whether the long-period signal is planetary in nature. For the purposes of this work, we do not consider the signal a detected planet since its provenance is questionable. Whether or not the long-period signal is modeled with a Keplerian and removed, the RV residuals yield upper limits of $M\sin i\lt 0.22{M}_{{\rm{J}}}$ at 10 au with 3σ conf.

Kepler-68 (KOI-246; Figure 30) is a system with a sub-Neptune-sized and an Earth-sized transiting planet. The transiting planets were validated via photometric techniques, including odd–even tests and color diagnostics, and dynamically confirmed using RVs in Gilliland et al. (2013). The stellar mass and radius were determined via asteroseismology. The RVs also revealed a nontransiting planet, Kepler-68 d, with a best-fit period of 581 days. Additional RVs were obtained and published in Marcy et al. (2014), resulting in an update of the period of the nontransiting planet to 625 days and modestly refined transiting planet masses. Another update in Mills et al. (2019b) again revised the period of the nontransiting planet to 634 days and improved the masses of the transiting planets via dynamical analysis.

With several additional seasons of RVs, we confirm the properties of planet d: P_d = 633 days, $M\sin {i}_{d}\,=0.765{M}_{{\rm{J}}}\,\pm 0.045{M}_{{\rm{J}}}$, e_d = 0.19 ± 0.05 (Figure 30). We also announce a nontransiting planet, Kepler-68 e. The peak of our Lomb–Scargle periodogram is at 362 days, which is likely a yearly alias of the (presumed) long period of planet e. The Mount Wilson S-values also have a peak at 362 days, which is likely due to the window function. We obtained a new RV template observation and reran our barycentric correction analysis for this target, but the peak at 362 days persisted even after our new reduction and RV determinations. The best-fit model includes planets at both 362 and 633 days with moderate eccentricities that would result in orbit crossing, and so this configuration is unlikely. Our interpretation is that the peak at 362 days is the alias of a planet, Kepler-68 e, at a period of P_e ≈ 4400 days with $M\sin {i}_{e}\approx 0.4{M}_{{\rm{J}}}$ and e_e ≈ 0.4. Adopting these properties in our four-planet model removes the 362 day peak from the Lomb–Scargle periodogram of the residuals. The mass upper limit of an additional (third) giant planet is not well constrained because of the uncertainty in the orbit of planet e, yielding a 3σ upper limit of $M\sin i\lt 0.11{M}_{{\rm{J}}}$ at 5 au (0.43M_J at 10 au). If we ignore the second giant planet and instead consider the residuals to a three-planet model, the upper limits on an additional companion increase by 50%.

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Kepler-126 (KOI-260) is a V = 10.5 star near the Kraft break (T_eff = 6200 K). The system has three transiting planets at P_b = 10.5, P_c = 21.9, and P_d = 100 days that were statistically validated (Lissauer et al. 2014; Rowe et al. 2014). All three planets have significant TTVs that have been used to set upper limits on their eccentricities (Van Eylen & Albrecht 2015). Planet d appears to be dynamically decoupled from the inner two planets, and so its TTVs are likely caused by at least one planet that has not been detected yet.

We present 35 RVs from Keck-HIRES, which were collected between 2014 July 23 and 2019 November 11 (Figure 31). The time series began in 2014 for a KOI follow-up program that examined planets in multiplanet systems with period ratios near mean motion resonance. The residual to our best three-planet fit has rms = 9.9 m s⁻¹, which is typical for a star with T_eff = 6150 K and $v\sin i$ = 8.7 km s⁻¹ (Isaacson & Fischer 2010). The nondetection of an RV trend sets an upper limit of $M\sin i\lt 0.8{M}_{{\rm{J}}}$ for a companion at 5 au ($M\sin i\lt 3.2{M}_{{\rm{J}}}$ at 10 au).

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Kepler-96 (KOI-261) is a V = 10.3 Sunlike star with one transiting planet (P_b = 16.2 days, R_b = 2.6 R_⊕). The planet was confirmed with 44 Keck-HIRES RVs, which yielded a mass of 8.4 ± 3.4 M_⊕, consistent with a volatile-enveloped planet (Marcy et al. 2014).

We have collected 11 new RVs since 2014 (Figure 32). A single RV outlier of 20 m s⁻¹ in 2017 might be consistent with the orbit of an eccentric nontransiting planet, although the RVs are also consistent with no giant planet if the 2017 data point is ignored. If the planet is real, it has a period of ∼4000 days and a mass of ∼1M_J. We consider this a planet candidate, although more RVs are needed to confirm its existence and characterize its orbit and mass.

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Kepler-50 (KOI-262) is a V = 10.7 F-type star (T_eff = 6200 K) with two transiting planets (R_b = 1.7 R_⊕, R_c = 2.2 R_⊕) confirmed in Chaplin et al. (2013). The planets' compact architecture (P_b = 7.81 days, P_c = 9.37 days) produces significant TTVs, which yield mass upper limits of M_b < 8.9 M_⊕ and M_c < 8.2 M_⊕ (Steffen et al. 2012). The star has solar-like oscillations that enabled an asteroseismic analysis, including mode splitting that revealed that the stellar rotation axis is closely aligned with the angular momentum vector of the planet orbits (Chaplin et al. 2013).

We observed this target for a KOI follow-up program that characterized planets in multiplanet systems with period ratios near mean motion resonance. We collected 39 RVs between 2012 and 2022 (10 yr baseline; Figure 33). The residual RVs to our best two-planet fit have a high rms (14 m s⁻¹). The star's temperature is near the Kraft break (Kraft 1967), and the star has $v\sin i=10\,$ km s⁻¹, both of which could contribute to the high rms of the residual RVs, although we only expected a jitter of 9 m s⁻¹ for this target. There is no significant trend to the RV residuals, yielding a 3σ upper limit of $M\sin i\lt 0.70{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 2.8{M}_{{\rm{J}}}$ at 10 au).

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Kepler-507 (KOI-265) is an F-type star (T_eff = 6000 K) with one transiting planet (P_b = 3.56 days, R_b = 1.3 R_⊕) that was statistically validated in Morton et al. (2016). The stellar and planetary properties were updated based on the Gaia parallax (Berger et al. 2018). We have collected 49 Keck-HIRES RVs between 2011 and 2022, which yield a mass of M_b = 5.5 ± 1.7 M_⊕ (Figure 34). This star was not part of the Marcy et al. (2014) results owing to stellar properties at the time falling outside of the optimal range for precise RVs.

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The residual RVs have rms = 4.6 m s⁻¹, and there is no significant RV trend, resulting in a 3σ upper limit of $M\sin i\lt 0.3{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1{M}_{{\rm{J}}}$ at 10 au). However, there is a marginally significant peak in the fast periodogram at 94 days (FAP = 0.04) and several harmonics and aliases of that peak. The inclusion of a Keplerian fit at 94 days reduces the rms of the RVs from 4.6 to 3.3 m s⁻¹ and produces ΔBIC = −30. The RV amplitude of the 94 day signal is 4.5 m s⁻¹, corresponding to a planet candidate mass of $M\sin i=35\,{M}_{\oplus }$ (which is well within the upper limits established from the RV trend). More RVs are needed to identify the correct orbital period of this candidate second planet and determine its mass.

KOI-273 (Kepler-454) has one transiting planet at P = 10.57 days. Gettel et al. (2016) used precise RVs from HIRES and HARPS-N to characterize the transiting planet (b), a nontransiting planet at P = 524 days (c), and a linear trend representative of a third companion. The Kepler photometry revealed solar-like oscillations of the host star, enabling a precise characterization of the stellar mass and radius. Our new RVs from Keck-HIRES are lower than expected from the linear trend, indicating the detection of curvature due to the orbit of the distant companion. Our joint fit to the HIRES and HARPS-N RVs yields planet masses of M_b = 6 ± 2 M_⊕, ${m}_{c}\sin i\,=4.6{M}_{{\rm{J}}}\pm 0.2{M}_{{\rm{J}}}$, and ${m}_{d}\sin i=2.6{M}_{{\rm{J}}}\pm 0.3{M}_{{\rm{J}}}$ (Figure 35). Additionally, our new RVs clarify the orbits of the nontransiting planets and detect a marginally significant eccentricity for planet d: P_c = 524 ± 1 days, P_d = 4800 ± 1700 days, and e_d = 0.26 ± 0.04. Our solution for Kepler-454 d is broadly consistent with the RV trend announced in Bonomo et al. (2023). Kepler-454 is another example of the value of long RV time baselines when detecting long-period massive planets. Because our baseline is comparable to the longest detected orbit in the system, any putative trend in the residuals to our best three-planet fit is not well constrained, and so our upper limit on additional companions is only $M\sin i\lt 1.7{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 7.0{M}_{{\rm{J}}}$ at 10 au). Future RVs will improve the characterization of the period, eccentricity, and minimum mass of the long-period companion and will also place better constraints on the presence of additional companions.

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Kepler-128 (KOI-274) is an F-type star with moderate rotation (T_eff = 6000 K, $v\sin i=6\,\mathrm{km}\,{{\rm{s}}}^{-1}$; Fulton & Petigura 2018). The star has two transiting planets with radii R_b = 1.4 R_⊕ and R_c = 1.3 R_⊕. The compact architecture of the transiting planets (P_b = 15.1 days, P_c = 22.8 days) produces significant TTVs (Xie 2014). An N-body dynamical characterization of the TTVs yielded planet masses of M_b = 0.77 ± 0.40 M_⊕ and M_c = 0.90 ± 0.45 M_⊕ (Hadden & Lithwick 2014). The low densities determined from the TTVs (ρ_b = 1.5 ± 0.8 g cm⁻³, ρ_c = 2.1 ± 1.1 g cm⁻³) are distinct from the density of rocky planets at this size (∼7.5 g cm⁻³), suggesting that the planets, though small, might contain low-density volatiles as a major constituent of their bulk compositions.

We have collected 17 RVs using Keck-HIRES between 2012 and 2022 (Figure 36). This number of RVs did not meet our criterion for fitting planet masses (at least 20 dof). Assuming that the planets have typical masses for their sizes (Weiss & Marcy 2014), the residual RVs have rms = 8.1 m s⁻¹, which is typical for a moderately rotating F-type star. There are no strong peaks in the periodogram of the RV residuals and no apparent trends, yielding a 3σ upper limit of $M\sin i\lt 0.37{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 1.5{M}_{{\rm{J}}}$ at 10 au).

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KOI-275 (Kepler-129) has two transiting sub-Neptune-sized planets. Zhang et al. (2021) used precise RVs from HIRES to detect and characterize a nontransiting companion at an orbital distance of a few astronomical units, with a minimum mass near the boundary between planetary- and brown-dwarf-mass objects. Solar-like oscillations of the host star revealed that the star has nonzero obliquity with respect to the transiting planets. The obliquity is consistent with a scenario in which the massive companion torques the inner planets. The inner planets have a coupling timescale much shorter than their precession time, allowing them to remain mutually transiting as their line of nodes precesses around the host star. We present nine new RVs here and their analysis, which result in an updated orbit and mass determination for the nontransiting companion (assuming no RV trend): $M\sin {i}_{d}=6.3{M}_{{\rm{J}}}\pm 0.2{M}_{{\rm{J}}}$, P_d = 1935 days, a_d = 3.26 au, and e_d = 0.07. Our new RVs greatly improve our coverage of the orbit of the companion, allowing us to decisively classify it as a giant planet ($M\sin {i}_{d}\lt 13{M}_{{\rm{J}}}$) rather than a brown dwarf (Figure 37). However, the allowance of an RV trend significantly complicates our interpretation of the RVs, since the earliest RV we collected strongly influences our interpretation of the planet orbit (and any possible trend). If an RV trend is allowed in addition to our best three-planet fit, the constraints on the mass of the detected companion become $M\sin {i}_{d}\,=6.8{M}_{{\rm{J}}}\pm 2.2{M}_{{\rm{J}}}$, which is an order of magnitude worse than if we assume no trend but still yields a planetary mass for the companion (as shown in Figure 37). Allowing an RV trend, our 3σ upper limit is $M\sin i\lt 13{M}_{{\rm{J}}}$ at 5 au ($M\sin i\lt 53{M}_{{\rm{J}}}$ at 10 au). Future RV monitoring will clarify the orbital properties of the super-Jovian companion and any additional planetary- or brown-dwarf-mass companions in the system.