A Second Terrestrial Planet Orbiting the Nearby M Dwarf LHS 1140

MEarth-South is a telescope array consisting of eight 40 cm telescopes at the Cerro Tololo International Observatory in Chile. The telescopes are operated on a (nearly) automated basis and take data on every clear night. The observational strategy and the data reduction process are described in greater detail in Irwin et al. (2015) and Dittmann et al. (2017b). We note that the MEarth data are corrected for the effect of differential color extinction by the Earth's atmosphere. The primary driver of differential color extinction in our data is the variability in the amount of precipitable water vapor in the atmosphere over the course of a night. Because the MEarth targets are mid-to-late M dwarfs and typically the reddest object in any observing field, our target stars are more sensitive to changes in water vapor than the field reference stars. We correct for this effect by measuring a "common mode" for all of our target M dwarfs. The common mode is defined as the average differential (i.e., relative flux) light curve of all M dwarfs currently being observed by MEarth-South in time bins of 0.02 days (28.8 minutes). This correlated behavior serves as a good proxy for the local change in precipitable water vapor on these timescales.

MEarth data are reduced in real time during the night. During the course of normal operations, MEarth-South is able to identify potential transits in-progress, and instead of proceeding to the next star in its target list, it can automatically decide to halt the data collection of other targets in order to collect additional high-cadence data around the star showing signs of a potential transit event. Normal operations are then resumed if the event is deemed to be spurious or the flux from the star has returned to its normal level. This MEarth "trigger" mode enables us to potentially confirm a planetary transit in real time and follow the transit all the way through egress. For a more detailed description of the MEarth trigger mode, see Berta et al. (2012).

MEarth-South has been in operation since 2014 January, and we have been observing LHS 1140 since the beginning of this survey. Since 2014, we have taken 28382 observations of LHS 1140 with MEarth-South. The observations contain the discovery data of LHS 1140 b (Dittmann et al. 2017a), including the high-cadence follow-up observations of LHS 1140 b's transits. The majority of these data, however, are standard monitoring observations of LHS 1140. Since 2015 October, the monitoring of LHS 1140 is carried out with two telescopes, with several exposures per visit.

We observed four transits of LHS 1140 b as part of the Spitzer DDT program 13174 (PI: Dittmann). These data were taken with the Infrared Array Camera at 4.5 μm. The goal of this program is to better determine the parameters of LHS 1140 b and to probe the system for signs of a transiting exomoon or transit timing variations. A manuscript analyzing these four transits is currently under preparation. Here, we present one of these transit observations, which fortuitously captured a transit of both LHS 1140 b and LHS 1140 c.

Observations spanned from 2018 April 18 03:21:22 to 2018 April 18 09:27:59 UTC. We placed LHS 1140 in the portion of the detector that is well-characterized for the purpose of obtaining high-precision light curves. We obtained the data in subarray mode and utilized a small 16 × 16 pixel area of the detector for our observation. During the first 78 minutes of the observations, the centroid of LHS 1140 wandered nearly 0.3 pixels (037) before settling into the "sweet-spot" for the rest of the observations. In order to correct for intra-pixel sensitivity variations, we calibrate our data with the pixel-level decorrelation algorithm, which depends on stable pointing to be effective. We exclude this portion of the data and only use data for which the target has settled onto the same portion of the detector. Subsequently, 50% of the image centroids fall within 0.069 pixels of the median centroid location and 95% of the image centroids fall within 0.150 pixels of the median centroid location. We then carry out the data reduction as described in Dittmann et al. (2017b). After obtaining the pixel-level coefficients for our observations, we apply these coefficients to the unbinned and unnormalized data. We sum the values of these weighted pixels in order to obtain the total flux from LHS 1140. We apply a new outlier rejection criteria to these unbinned data. We reject all data points 7.5 median absolute deviations from a 12-minute wide sliding median in order to eliminate single point outliers, resulting in the rejection of eight total data points (0.09% of the total number of data points).

Our final data set consists of 135 sets of 64 individual subarray images, each with an integration time of 2 s (our final datacube does not contain a full set of 64 images, as it was the last datacube of the observing program), for a total of 8545 data points. These data were calibrated with the Spitzer pipeline version S19.2.0 and the timestamps of each data point are calculated at the solar system barycenter in the TDB time system (Eastman et al. 2010).

We intensively monitored LHS 1140 with the HARPS in the frame of the M dwarf program (ESO Id: 191.C-0873(A), 198.C-0838(A); PI: Bonfils) and a survey dedicated to LHS 1140 (ESO Id: 0100.C-0884(A); PI: Astudillo-Defru). HARPS is a fiber-fed echelle spectrograph located at the 3.6 m telescope at the La Silla observatory in Chile. Its resolving power is R = 115,000, and it has a wavelength coverage between 380 and 690 nm over 72 orders (Mayor et al. 2003). HARPS achieves a sub-ms⁻¹ long-term precision thanks to its temperature- and pressure-controlled environment and a simultaneous wavelength calibration through a second fiber. We opted to place the calibration fiber on the sky to avoid contamination of the science spectra from a calibration spectrum, notably in the region of the Ca ii H&K lines. This choice does not degrade our RV precision, because for a star with the brightness of LHS 1140, our RV precision is dominated by the photon-noise of the stellar spectrum, as opposed to the calibration of the spectrograph.

We obtained 294 spectra of LHS 1140 between 2015 November 23 and 2018 January 15, spanning 784 days. In general, HARPS observations consist of two spectra per night with an exposure time of 1800 s each. On some nights, we observed LHS 1140 one or three times. We discarded one observation (BJD: 2457686.6227) with an exposure time of only 1.8 s, so our analysis was done over 293 radial velocity measurements.

The HARPS Data Reduction Software (DRS; Lovis & Pepe 2007) automatically reduces the spectra and computes the stellar radial velocity. The latter is done by cross-correlating spectra with a mask designed to match the spectral lines. The DRS uses a mask containing the vast majority of the absorption lines present in the spectrum of an M dwarf; however, a nonnegligible amount of lines are left out. To recover as much of the Doppler signal as possible, we follow the recipe described in Astudillo-Defru et al. (2015, and references therein): the radial velocities derived by the DRS are used to shift all spectra to a common reference frame, then the median is computed to obtain an enhanced stellar spectrum (template). This template is Doppler shifted in a range of velocities to maximize its likelihood with each spectrum, resulting in the set of radial velocities used hereafter.

Articles describing the use of artificial neural networks to classify ground-based data in the exoplanet field have only recently appeared in the literature (known examples include Dittmann et al. 2017a and Armstrong et al. 2018). Nevertheless, given that the first planet in the LHS 1140 system was discovered by Dittmann et al. (2017a) via a machine-learning process, we decided to use a similar approach to search for additional potential transit events in the MEarth data. MEarth generates many "triggers" per night (see Section 3.1), most of which can be explained by variations in the atmosphere (clouds, changes in the column density of precipitable water vapor), rapid stellar variability (e.g., stellar flares), or instrumental systematics. Such false signals can often be identified by correlating the change in flux with a series of atmospheric and instrumental parameters such as the common mode (defined in Section 3.1), the FWHM or pixel position of the target, or simultaneous variations in the fluxes of background stars. A small minority of triggers correspond to genuine transit events, which are not expected to be significantly correlated with any of these parameters. Our machine-learning approach seeks to take advantage of this distinction—that is, we would like to eliminate as many triggers as possible based on systematics, and keep the rest as potential transit candidates.

After testing several neural network designs including feedforward and recurrent neural networks (see Murphy 2012 for more information about neural networks), we settled for a simple feedforward neural network (FNN) consisting of an input layer, a single hidden layer, and an output node. The network design employed in this paper is similar to the one used by Dittmann et al. (2017a) and can be summarized as follows. The input layer was composed of 10 neurons for atmospheric and instrumental parameters including the airmass, angle of the frame relative to the reference frame, rms of the fluxes of background stars, and the FWHM, ellipticity and pixel position of the target, as well as the common mode with its time derivative and offset from a 3-hr median. The hidden layer was made up of 10 neurons as well, each fully connected to every input node. Finally, the FNN had a single output node representing the probability that the given input corresponds to a trigger. The data set, which was randomly split up into a training set (90%) and a validation set (10%), consisted of 1539 real triggers from MEarth between the years of 2014 and 2017. We complemented this set with an equal number of non-triggers, which were selected randomly from the regular, low-cadence observations while ensuring that each star contributed the same amount of triggers and non-triggers (however, the number of triggers contributed by each star varied considerably). After training the network, we reclassified the 1539 real triggers, and selected the events where the FNN generated the lowest probabilities of being an expected trigger (that is, caused by atmospheric or instrumental effects) for further inspection. These events, unlike the majority of triggers, did not exhibit a detectable correlation with the atmospheric and instrumental parameters that we used as inputs.

In particular, we identified two triggers belonging to LHS 1140 with very low trigger probabilities: one from 2014 September 15 UTC, which was the basis of the LHS 1140 b discovery by Dittmann et al. (2017a), and another trigger from 2016 August 14 UTC (illustrated in Figure 1), which did not overlap with any of the predicted transits of planet b. The estimated probabilities for the two events to be triggers were 23.2% and 24.5%, respectively, which placed them among the top 5% events with the lowest trigger probabilities. The latter event was then selected as a potential transit candidate.

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We also decided to carry out a phase-folded box model search in the photometry using the Box-fitting Least Squares (BLS) algorithm (Kovács et al. 2002). In the case of LHS 1140, we tested over 1.6 million evenly spaced orbital frequencies corresponding to periods between 0.2 and 10 days. Our frequency spacing ensured that the maximum shift in orbital phase between two consecutive tested frequencies was always less than one half of the spacing between the time bins (we binned the observed fluxes into 10 minute error-weighted intervals). After fitting a box model to each orbital frequency, we imposed 4 selection criteria on the BLS spectrum to eliminate false detections. First, we ignored any periods close to half a day or any integer number of days to avoid detecting variability that can be attributed to the Earth-based observing strategy. Second, we ignored any anti-transits (where the fitted transit depth was negative) as well as transit depths less than 0.1%. We also required that the significance of the fitted model at a given period be at least 90%, estimated by repeatedly refitting the data after adding normally distributed random offsets to the measured fluxes according to their respective uncertainties, and rejecting any outcomes that resulted in anti-transits. Finally, we calculated Δχ², the difference in χ² between the best-fit box model and a constant flux model, for each orbital frequency. In order to eliminate signals that are dominated by observations from a single night, we required that no night contribute more than 70% of the total Δχ²-difference (Burke et al. 2006). We then ordered the remaining orbital frequencies in decreasing order of Δχ².

After masking out the predicted transits of LHS 1140 b, we investigated the 16 highest Δχ²-peaks in the BLS spectrum. One of these orbital periods, P = 3.77797 days, produced a phase-folded model that back-predicted a transit on 2016 August 14, with the ingress and egress times consistent with the MEarth trigger described in Section 4.1. Overall, the in-transit observations spanned 35 different nights, with no single night contributing more than 19% to the total Δχ²-difference. The predicted transit depth from the box model was 0.192%, which corresponds to an Earth-sized planet. The phase-folded model is displayed in Figure 2.

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We show our calibrated Spitzer data in Figure 3. In this plot, we see the transit of LHS 1140b followed by a second transit event of smaller depth and shorter duration. This second transit event is not associated with shifts in the pixel position of the centroid of the target. The duration of this event is approximately 74 minutes with a depth of 3 mmag.

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The discovery of the Spitzer transit event (by J.A.D.) happened independently of identifying the trigger or phase-folded transits of LHS 1140 c in the MEarth data (led by K.M.) and independently of the HARPS radial velocity identification of LHS 1140 c (led by N.A-D.). We discuss the simultaneous fitting of these transits with radial velocity observations in Section 5.1.

Using HARPS radial velocities, we unveiled the signal of LHS 1140 c independently of the photometry. A Generalized Lomb–Scargle (GLS; Zechmeister & Kürster 2009) periodogram analysis of the RVs reveals several interesting signals, shown in Figure 4. The most significant peak lies close to 25 days, which corresponds to the orbital period of LHS 1140 b. After subtracting a Keplerian model for LHS 1140 b from the RVs, a strong signal emerges near the stellar rotation period P_rot = 131 ± 5 days as well as the 3.8 day period, matching the one found by the BLS analysis in Section 4.2. After subtracting a two-planet model, we produced a periodogram displayed in the third panel of Figure 4, exhibiting several low-frequency peaks at around 65 days and longer periods. At least two of these peaks, at 68 and 132 days, can be attributed to the stellar rotation period P_rot and its harmonic P_rot/2. Collections of starspots can often be successfully modeled as a series of no more than the first three or four harmonics of P_rot (Jeffers & Keller 2009; Boisse et al. 2011; Haywood 2015), such as in the cases of Corot-7 (Queloz et al. 2009; Boisse et al. 2011) and HD 189733 (Boisse et al. 2011). Therefore, we proceeded to generate a model for stellar activity that consists of a sum of two sinusoidal signals with periods initialized at P_rot and P_rot/2 (adding in further harmonics was not necessary in this case). Subtracting this model from the residuals significantly reduces any residual power in the low-frequency signals beyond 65 days (displayed in the bottom panel of Figure 4). Thus, we see no evidence that the previously significant GLS peaks between 65 and 130 days were caused by anything other than stellar activity.

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We fit a joint model of the Spitzer observations, MEarth-South observations, and the radial velocity measurements from HARPS to simultaneously constrain the masses, radii, and orbital parameters of both planets, as well as the radial velocity variability due to the stellar activity of LHS 1140. Our Spitzer data are shown in Figure 3 and described in Section 3.2. We have observed one transit of LHS 1140 c with Spitzer, and we present one transit of LHS 1140 b with Spitzer here as well. The remaining transits of LHS 1140 b from the Spitzer program will be the subject of a future publication. We do not currently have a full transit observation of LHS 1140 c with MEarth-South, but we fit the transit observations obtained on 35 individual transit nights (see Section 4.2), including the night of the trigger.

We fit our photometric observations with the batman code (Kreidberg 2015). batman is an optimized python implementation of the analytical model for transit light curves from Mandel & Agol (2002) and its erratum. We model the radial velocity variations of LHS 1140 from the orbits of the two planets (consistent with the transit ephemeris) and we adopt a simple sinusoidal model to describe the radial velocity variations of LHS 1140 due to the activity on the star.

We initiate the physical parameters for LHS 1140 b with the values found in Dittmann et al. (2017a). We adopt limb-darkening coefficients from Claret et al. (2012) for a 3300 K star with log(g) = 5.0 in a Cousins I filter. The Cousins I filter has a similar effective wavelength to the MEarth bandpass. For the Spitzer transits, we adopt limb -darkening coefficients from Claret et al. (2013), which are calculated for the Spitzer bandpass. We initiate the period of LHS 1140 c at the best-fit period in the phase-folded detection and a T₀ estimated from the midpoint of the Spitzer transit. The radial velocity amplitude for LHS 1140 c is initialized to an amplitude of 3 m s⁻¹, which is the approximate amplitude a rocky composition planet would exhibit at this orbital period. In order to efficiently explore parameter space, we use the emcee code (Foreman-Mackey et al. 2013). emcee is a python implementation of the Affine-Invariant Markov Chain Monte Carlo (MCMC) sampler (Goodman & Weare 2010). Each model is initiated with 100 walkers in a Gaussian ball located at this initial solution. We run each chain for 100,000 steps and discard the first 10% of steps so that the solution may "burn-in" independent of the choice of initialization. We show the results of this analysis in Table 2.

Table 2.  Model Parameters from the Joint Photometry and Radial Velocity fit in Section 5.1

Parameter	Value	Explanation
Pb (days)	24.736959 ± 0.000080	Orbital period of LHS 1140 b
T0,b (BJD)	2456915.71154 ± 0.000040	Transit time of LHS 1140 b
$\tfrac{{r}_{{\rm{b}}}}{{r}_{* }}$	0.07390 ± 0.00008	Ratio of LHS 1140 b and LHS 1140's radii
$\tfrac{{a}_{{\rm{b}}}}{{r}_{* }}$	95.34 ± 1.06	Ratio of semimajor axis to stellar radius for LHS 1140 b
ib (degrees)	(${89.89}_{-0.03}^{+0.05}$	Inclination angle of LHS 1140 b
eb	0 (fixed; <0.094 at 99% confidence)	Eccentricity of LHS 1140 ba
Kb (m s−1)	4.71 ± 0.09	RV amplitude of LHS 1140 ba
Pc (days)	${3.777931}_{-0.000002}^{+0.000004}$	Orbital period of LHS 1140 c
${T}_{0,{\rm{c}}}$ (BJD)	2458226.843169 ± 0.000026	Transit time of LHS 1140 c
$\tfrac{{r}_{{\rm{c}}}}{{r}_{* }}$	0.05486 ± 0.00013	Ratio of LHS 1140 c and LHS 1140's radii
$\tfrac{{a}_{{\rm{c}}}}{{r}_{* }}$	26.57 ± 0.05	Ratio of semimajor axis to stellar radius for LHS 1140 c
ic (degrees)	${89.92}_{-0.09}^{+0.06}$	Inclination angle of LHS 1140 c
ec	0 (fixed; <0.236 at 99% confidence)	Eccentricity of LHS 1140 ca
Kc (m s−1)	2.42 ± 0.10	RV amplitude of LHS 1140 ca
γ (km s−1)	−13.23851 ± 0.00008	RV zero-point of LHS 1140a
K* (m s−1)	3.0 ± 0.1	RV amplitude of stellar activitya
ϕ* (BJD)	2457766.66 ± 0.68	RV phase of stara
P* (days)	132.26 ± 0.37	RV period of stara
a4.5	0.0104 ± 0.0007	Spitzer limb-darkening coefficient
b4.5	${0.2175}_{-0.0042}^{+0.0012}$	Spitzer limb-darkening coefficient
aMEarth	${0.195}_{-0.007}^{+0.022}$	MEarth limb-darkening coefficient
bMEarth	${0.356}_{-0.11}^{+0.013}$	MEarth limb-darkening coefficient

Note.

^aThese values were merely used as intermediate results, and were subsequently supplanted by the values from the Gaussian Process modeling, listed in Table 1.

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We show the results of these fits in Figures 5 and 6. We are able to recover the phased transits of LHS 1140 c in the MEarth-South photometry and simultaneously fit the photometry and radial velocities for both planets. Our planetary parameters for LHS 1140 b are consistent with, but more precise than, those presented in Dittmann et al. (2017a). We find that LHS 1140 c has a transit depth of 3.0 mmag. We present further discussion of the results of these fits and the physical parameters of the LHS 1140 system in Section 6.

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Up to this point, we have modeled the radial velocities using a simplistic model: we have only considered the effects of both of the planets in the system and have modeled the stellar signal as a single sinusoid at the stellar rotation period (measured through our photometry). This analysis assumes that each data point is independent and that there is no correlated noise on any timescale. We now relax these assumptions to more accurately capture the correlated noise and errors present in our radial velocity measurements.

As demonstrated by the GLS periodogram in Figure 4, the radial velocities exhibit significant correlation near the stellar rotation period of 131 ± 5 days as well as its harmonics. The exact properties of these quasi-periodic variations in M dwarfs are yet to be resolved, although they are commonly believed to be linked to features on the stellar surface such as dark spots and granulation. Overall, a constant sinusoidal model (as presumed in Section 5.1) might be an inaccurate representation of activity-induced RV modulations. In pursuit of a more accurate accounting of the latter, we employed Gaussian process (GP) regression (Rasmussen & Williams 2006), which has recently been applied to several exoplanetary systems including Corot-7 (Haywood et al. 2014; Faria et al. 2016), Kepler-21 (López-Morales et al. 2016), Kepler-1655 (Haywood et al. 2018), K-2 18 (Cloutier et al. 2017), Alpha Centauri B (Rajpaul et al. 2015), and LHS 1140 (Dittmann et al. 2017a). Our goal in using GP regression is to make as few assumptions about the exact form of the signal due to stellar activity as possible. We did not include the photometric data in our GP framework in order to make the analysis computationally manageable. Our GP regression was implemented by using a quasi-periodic kernel of the form:

$$\begin{eqnarray}&&k(t,t^{\prime} )={\eta }_{1}^{2}\cdot \exp \left[-\displaystyle \frac{{\left(t-t^{\prime} \right)}^{2}}{{\eta }_{2}^{2}}-\displaystyle \frac{{\sin }^{2}\left(\tfrac{\pi \left(t-t^{\prime} \right)}{{\eta }_{3}}\right)}{{\eta }_{4}^{2}}\right]\end{eqnarray} \tag{ 1 }$$

where η₁ represents the amplitude of the covariance function, η₂ is the evolution timescale of activity-inducing stellar surface features, η₃ is the recurrence timescale (the expected period of correlation), and η₄ determines the amount of high-frequency structure in the GP. The Gaussian process framework is extremely versatile, and therefore it is often desirable to constrain the values of the hyperparameters with tight priors based on physical intuition about the system. We decided to adopt our priors from Dittmann et al. (2017a) for consistency purposes. As a result, the recurrence timescale η₃ was constrained by a Gaussian prior centered at the stellar rotation period (131 ± 5 days). The active regions of the star are likely to remain stable over a few rotation periods, so we adopt a Gaussian prior at three times the rotation period (393 ± 30 days) for the evolution timescale η₂. Similarly to Dittmann et al. (2017a), choosing a different value of the same order of magnitude does not significantly change our results. The structure parameter η₄ is constrained by a Gaussian prior centered at 0.5 ± 0.05, which allows for two or three activity-induced peaks in the RV curve per rotation (López-Morales et al. 2016; Haywood et al. 2018). RV models with more than three peaks per rotation would likely be unphysical due to the limb-darkening and foreshortening effects that effectively erase any structure in the photometric and RV signatures (Jeffers & Keller 2009).

We represented each planet i by a Keplerian model with an amplitude K_i, an orbital period P_i, and a time of transit t_T,i. The amplitudes K_i (as well as η₁) were constrained only by a modified Jeffreys prior. In order to accommodate the results of the joint fit from Section 5.1, we implemented tight Gaussian priors on P_i and t_T,i centered at the joint fit results while setting the widths equal to the respective uncertainties. Initially, our models also included eccentricities e_i and arguments of periapsis ω_i. However, we found the improvement in likelihood statistically negligible: the Bayesian information criterion (BIC) increased from 1173 to 1218 when substituting a two-planet eccentric model for a circular one. Thus, we used the eccentric model only to place an upper limit on the orbital eccentricities: we optimized the rest of the parameters assuming e = 0. The parameters were optimized via an affine-invariant MCMC routine (Goodman & Weare 2010). We present a list of all modeled parameters with their appropriate prior distributions in Table 3.

Table 3.  Modeled Parameters and Their Prior Distributions

Parameter	Prior Distribution
P (days)	Gaussian (24.736959, 0.000080), (3.777931, 0.000003)
K (m s−1)	Modified Jeffreys (σRV)
e	Uniform [0, 1)
ω (deg)	Uniform [0, 360)
tT (BJD)	Gaussian (2456915.71154, 0.00004), (2458226.843169, 0.000026)
η1 (m s−1)	Modified Jeffreys (σRV)
η2 (days)	Gaussian (393, 30)
η3 (days)	Gaussian (131, 5)
η4	Gaussian (0.5, 0.05)
RV0	Uniform $(-\infty ,+\infty )$

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Our best fit yielded an amplitude of K_b = 4.85 ± 0.55 m s⁻¹ and a period of P_b = 24.73712 ± 0.00032 days for planet b. Similarly for planet c, we obtain K_c = 2.35 ± 0.49 m s⁻¹and P_c = 3.777950 ± 0.000007 days. Using these results, we can estimate the planetary masses as m_b = 6.98 ± 0.89 M_⊕ and m_c = 1.81 ± 0.39 M_⊕ (including the propagated uncertainties from the stellar mass). We also obtained a GP amplitude of η₁ = 2.68 ± 0.81 m s⁻¹, GP timescales of η₂ = 400 ± 38 days and η₃ = 134.8 ± 3.2 days, and a structure parameter value of η₄ = 0.51 ± 0.06. The latter three are all consistent with our initial priors. Using the eccentric models, we can place upper limits of e_b < 0.060 and e_c < 0.313 on the orbital eccentricities with 90% confidence. However, we note that the posterior probability distribution for the eccentricity of planet c is significantly lopsided, with the 80% confidence limit at e_c < 0.14. We decided to retain the orbital periods and times of transit from the combined fit in Section 5.1 since these are constrained much more tightly by photometry. The uncertainties on all of the parameters were obtained by calculating the 14th and 86th percentiles of the final posterior distributions, and the best-fit values were taken from the medians of the PDFs.

Our best circular RV model is illustrated in Figure 7, and our final values for all modeled and derived parameters are given in Table 1. We note that the 1σ uncertainties for K_b and K_c from our updated RV solution are now consistent with the expected uncertainties based on the Fisher information. Previously, Cloutier et al. (2018) found that the measurement precision of K_b in the 1-planet model of Dittmann et al. (2017a) was about $\sqrt{2}$ times larger than its theoretical value based on the number of RV data points.

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