The CARMENES Search for Exoplanets around M Dwarfs: A Low-mass Planet in the Temperate Zone of the Nearby K2-18

CARMENES (Calar Alto search for M dwarfs with Exo-earths with Near-infrared and optical Echelle Spectrographs) is a pair of high-resolution echelle spectrographs (Quirrenbach et al. 2014) mounted on the 3.5 m telescope of the Calar Alto Observatory (CAHA) in Spain. The VIS channel covers the wavelength range from 0.52 to 0.96 μm and has a spectral resolution R = 94,600 (Quirrenbach et al. 2016), with a demonstrated precision similar to HARPS and better than Keck/HIRES (Trifonov et al. 2018).

We monitored K2-18 between 2016 December and 2017 June with CARMENES. In total 58 spectra were obtained that were reduced and extracted using the CARACAL pipeline (Caballero et al. 2016; Zechmeister et al. 2018). The pipeline implements the standard method for reducing a spectrum, i.e., each spectrum was corrected for bias, flat-field, and cosmic rays, followed by a flat-relative optimal extraction of the 1D spectra (Zechmeister et al. 2014) and wavelength calibration. In order to get precise RVs, we use the data products from the SERVAL pipeline (Zechmeister et al. 2018), which uses a least-squares fitting algorithm. Following the approach by Anglada-Escudé & Butler (2012), a high signal-to-noise ratio spectrum is constructed by a suitable combination of the observed spectra and used as a template to measure the RVs. The SERVAL-estimated RVs were additionally corrected for small night-to-night systematic zero-point variations, as explained in Trifonov et al. (2018). The origin of the offsets is still unclear, but they are probably due to systematics in the wavelength solution and a slow drift in the calibration source during the night. The time series is shown in the left panel of Figure 6. The optical differential RV measurements and the activity indicators (see Section 3) used in the analysis are reported in Table 6.

We monitored the host star K2-18 for photometric variability with the robotic 1.2 m twin-telescope STELLA on Tenerife (Strassmeier et al. 2004) and its wide-field imager WiFSIP. From 2017 February until 2017 June, we obtained blocks of four exposures in Johnson B and four exposures in Cousins R over 33 nights. The exposure time was 120 s in B and 60 s in R. The data reduction was performed identically to previous host star monitoring campaigns with STELLA (Mallonn et al. 2015; Mallonn & Strassmeier 2016). The bias and flat-field correction was made with the STELLA data reduction pipeline. We performed aperture photometry with the software Source Extractor (Bertin & Arnouts 1996). For differential photometry we divided the flux of the target by the combined flux of an ensemble of comparison stars. The flux of these stars was combined after giving them an optimal weight according to the scatter in their light curves (Broeg et al. 2005). We verified that the selection of comparison stars did not significantly affect the variability signal seen in the differential light curve of K2-18. The nightly observations were averaged, and a few science frames were discarded owing to technical problems. The final light curves contain 29 data points in B and 28 data points in R and are shown in Figure 1.

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Active regions, in the form of dark spots and bright plages, rotate with the stellar surface and produce photometric as well as RV variability. The observed RV signal is often detected at the stellar rotation period (${P}_{\mathrm{rot}}$) and its harmonics (${P}_{\mathrm{rot}}$/2, ${P}_{\mathrm{rot}}$/3, ...) (Boisse et al. 2011). Its amplitude and phase may also vary in time owing to the evolution of the active regions. Therefore, contemporaneous photometry and RV observations are important to determine the stellar rotation period and to differentiate between a planetary and stellar activity signals.

The photometric and spectroscopic observations were performed during the same observational season in 2017. In order to estimate the stellar rotation period, we followed the classical approach by applying the Generalized Lomb–Scargle periodogram (GLS; Zechmeister & Kürster 2009) to the photometric data sets. The GLS analysis showed a peak at ∼40 days in the B band and a peak at ∼39 days in the R band. To assess the false-alarm probability (FAP) of the signals, we applied the bootstrap randomization technique (Bieber et al. 1990; Kuerster et al. 1997). This is done by computing the GLS of a set obtained by randomly shuffling the observed magnitudes with the times of observations. We repeated this 10,000 times, and the FAP is defined as the number of times where the periodogram of the randomized data sets shows a GLS power as high as or higher than that of the original data set. We found that the FAP is <10⁻⁴ in the B band and FAP = 2 × 10⁻⁴ in the R band. The top panel of Figure 2 shows the periodogram of the data taken with the B filter, and the bottom panel shows the periodogram of those taken with the R filter.

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To get a better estimate of the stellar rotation period, we fit both bands simultaneously with a sine wave function and forced both light curves to have the same frequency (f_BR) and phase (ϕ_BR), but we allowed the offsets (γ_B and γ_R) and amplitudes (A_B and A_R) to be different for each band. In total we fit for six parameters (f_BR, ϕ_BR, γ_B, γ_R, A_B, and A_R) and performed a Markov Chain Monte Carlo (MCMC) using the emcee ensemble sampler (Foreman-Mackey et al. 2013). We adopted flat uniform priors for all parameters and estimate the rotation frequency to be 0.02524 ±0.00032 day⁻¹ (39.63 ± 0.50 days). This value is in agreement with the one estimated using the K2 photometry, where Cloutier et al. (2017) derived a value of ${38.6}_{-0.4}^{+0.6}$ days using Gaussian processes and Stelzer et al. (2016) derived a value of 40.8 days using an autocorrelation function (private comm.).

We estimated a photometric variability of 8.7 ± 0.5 mmag in B and a smaller variability of 6.9 ± 0.5 mmag in R. This difference is expected when the photometric variability is due to cool spots, since the contrast between the spots and the photosphere decreases at redder wavelengths. Figure 1 shows the photometric variations in the B filter (in blue) and the R filter (in red) and the best-fit model. In Tables 4 and 5 we provide the differential photometry in B and R bands, respectively.

The most common and widely used spectroscopic activity indicators can be divided into two different types: the chromospheric and the photospheric ones. The chromospheric activity indicators measure the excess of flux in the cores of, e.g., Ca ii H and K, calcium infrared triplet (Ca ii IRT), Na i doublet, and Hα lines. The cores of these lines have their origin in the stellar chromosphere, and hence they trace stellar magnetic activity. The photospheric activity indicators measure the degree of asymmetry in the line profile. The presence of spots on the photosphere distorts the spectral lines, and therefore periodic variability of the FWHM and bisector span (BS) of the cross-correlation function (CCF) could indicate the presence of spots. Zechmeister et al. (2018) recently showed that the chromatic index is also an important photospheric indicator (see below).

The SERVAL pipeline provides the line indices of the Ca ii IRT, Hα, and Na i doublet. The three Ca ii IRT lines are centered at 8498.02, 8542.09, and 8662.14 Å; the Hα line is centered at 6562.81 Å; and the Na i D lines are centered at 5889.95 and 5895.92 Å. The pipeline also computes the differential line width (dLW) and the chromatic RV index. The former is a measure of the relative change of the width of the average absorption line, and the latter is a measure of the wavelength dependency on the RV. We refer the reader to Zechmeister et al. (2018) for a detailed description of how the various activity diagnostics are computed.

We performed a period search analysis using GLS to search for a significant periodicity that could be related to stellar activity. Figure 3 (panels 3–6) displays the periodograms of the indicators that show a significant peak. Although we inspected a wide range of frequencies, we only show the frequency range of interest that covers the stellar rotation frequency, the planetary frequency of the transiting planet, and the potential 9-day signal (see Section 4.3). All three Ca ii IRT indices show a clear dominant peak at ∼36 days with FAP = 3 × 10⁻⁴, <10⁻⁴, and =10⁻⁴ for the Ca ii IRT 1, Ca ii IRT 2, and Ca ii IRT 3 lines, respectively, which was determined via bootstrap. The Hα periodogram shows three peaks at 29, 36, and 45 days, with FAP = 3.7 × 10⁻³ at 36 days. The origin of the signal of both indicators is consistent, within the frequency resolution, with the rotational period of the star derived from photometry (Section 3.1). Similar to the photometric data, we fit the Ca ii IRT indices simultaneously with a sine wave function, forcing them to have the same frequency and phase, but allowed the offsets and amplitudes to vary. Figure 4 shows the Ca ii IRT line indices along with the best-fit sinusoidal model. The Na i doublet and dLW periodograms, however, are free from significant peaks even though the Na i lines were expected to be good activity indicators for early M dwarfs (Gomes da Silva et al. 2011; Robertson et al. 2015). We report the data of the activity indicators in Table 6.

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In addition to the indicators provided by SERVAL, we computed the CCF for each spectrum by cross-correlating the spectrum with a weighted binary mask that was built by co-adding all the observed spectra of the star itself. We selected around 4000 deep, narrow, and unblended lines, which were weighted according to their contrast and inverse FWHM. We computed one CCF for each spectral order, and the final CCF was computed by combining all the individual CCFs according to signal-to-noise ratio. A Gaussian function was fitted to the combined CCF. From this, the FWHM and BS were derived. A period analysis of the FWHM and BS does not show significant periods. The lines in a typical M dwarf spectrum are blended and, thus, may mask changes in the FWHM and BS, which could be the reason why these indicators do not show a variability. Another reason is probably the low projected rotational velocity of the star (v sin i). Reiners et al. (2018b) imposed an upper limit on v sin i at 2 km s⁻¹. However, from the stellar radius and rotation period (Table 2), we estimate a true equatorial velocity v of only 0.53 km s⁻¹. The spot–BS relationship from Saar & Donahue (1997) predicts, for v sin i = 0.53 km s⁻¹, a bisector variability of 0.01 ${\rm{m}}\,{{\rm{s}}}^{-1}$, which is too small to measure.

The star shows photometric variability with a stellar rotation period of 39.63 ± 0.50 days. The semi-amplitude is 0.87% in the B band and 0.69% in the R band. K2-18 also shows chromospheric variability in the Ca ii IRT and Hα lines with a period consistent with the rotation period derived from photometry within the frequency resolution. Figure 5 shows the variations of the Ca ii IRT second index and the best-fit model (solid black curve) and the photometric variability of K2-18 in the B band (dashed blue curve). There is an anticorrelation between the photometric and the chromospheric variability. The chromosphere shows variations that are 180° out of phase with the photosphere. Similar trends are seen with the first and third Ca ii IRT indices and the Hα line. This demonstrates that for high Ca ii emission values, a minimum in the photometric light curve is observed. This is expected if active chromospheric regions are present on top of a photospheric spot. This is not the first time that an anticorrelation between the chromosphere and photosphere of M dwarfs is observed. Bonfils et al. (2007) reported an anticorrelation for GJ 674, which is also an early M2.5 dwarf. It would be worth checking whether the anticorrelation will hold for late M dwarfs.

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We conclude that K2-18 is a moderately active star and there is an anticorrelation between the photospheric and chromospheric variations, which is consistent with the previous results of Radick et al. (1998) for younger, more active stars. Finally, although Hα is a good activity indicator (Kürster et al. 2003; Hatzes et al. 2015; Robertson et al. 2015; Jeffers et al. 2018), the Ca ii IRT lines show a significantly stronger peak compared to Hα. Ca ii IRT lines are thus good chromospheric activity proxies (see discussion by Martin et al. 2017) and provide a promising approach to detect stellar activity signals in M dwarfs, where the signal-to-noise ratio is too low to measure Ca ii H and K lines, especially for mid- and late M dwarfs. This is also in agreement with the findings of Robertson et al. (2016).

Benneke et al. (2017) analyzed the K2 and Spitzer light curves and derived an orbital period of $P={32.939614}_{-0.000084}^{+0.000101}$ days. To ensure that we have detected the planet signal with high significance, we performed a periodogram analysis for the RVs obtained with CARMENES-VIS. The RV measurements show a peak at 34.97 days with an FAP < 10⁻⁴ (Figure 3, panel 1). This peak is approximately the mean of the planetary orbital frequency and the stellar rotation frequency (0.02524 day⁻¹), as measured in Section 3.1. The peak in the periodogram is therefore not centered at the orbital period of the planet, but is shifted halfway between the stellar rotation frequency and the planetary orbital frequency. This shows that the RVs are contaminated by stellar activity, which is conceivable since the star is moderatively active (Section 3).

To assess the FAP of the planetary signal and, hence, to confirm the detection of the planet, we applied the bootstrap randomization technique. Unlike the previous analysis where we computed the GLS for the randomly shuffled data set (see Section 3.1), this time we fitted an adapted model to the randomized data points. The model employed the known ephemeris of the planet from Benneke et al. (2017), assumed zero eccentricity, and had only the RV semi-amplitude K_b and the RV zero-point (offset) as free parameters. We performed this 100,000 times and found that the FAP to infer a K_b amplitude as large as (or larger than) the one estimated from the original data is <10⁻⁵ and the FAP to get a χ² as small as (or smaller than) the one from the original fit and finding at the same time that K_b is positive is also <10⁻⁵. This ensures that given the known ephemeris of the planet, we are confident that there is a signal at the known ephemeris, which can be a combination of the planet and activity signals. In Section 4.2 we address several tests that we performed to check whether the RVs and, therefore, the planetary amplitude are affected by activity.

Signals that are sampled at discrete times can produce fake signals in the periodogram that are due instead to observational patterns. In order to check for periodicities due to sampling, we applied the GLS on the window function (WF), which is a periodogram analysis of the observation times. The GLS shows a peak at 32.2 days (Figure 3, panel 2) which is very close to the orbital period of the planet. The reason for that peak is because we aimed to observe the star on a daily basis. However, some nights were lost as a result of bad weather, and more importantly, during dark nights, roughly for a couple of lunar cycles, another instrument was mounted on the telescope, and no observations were carried out with CARMENES. This pattern could have caused the peak in the WF that is close to the lunar synodic cycle.

The presence of a peak in the WF would lead to the detection of the wrong frequency when there is a signal in the data. Dawson & Fabrycky (2010) showed that the reported periods of 55 Cnc e and HD 156668b from their respective discovery papers were actually wrong and affected by daily aliases. In the case of K2-18 b, first we have evidence that the star is moderately active (Section 3), and as a result, we anticipate the presence of a signal in the RVs close to the stellar rotation frequency. Second, the planet transits (Montet et al. 2015; Benneke et al. 2017), and therefore we expect another signal in the data close to the orbital period of the planet. However, the proximity of the stellar rotation frequency to the planetary orbital one makes separating them challenging, since the frequencies are not resolved given the time span of the data set.

Given the presence of the peak in the WF and assuming the presence of one signal in the data (either the planetary signal or the stellar rotational period), is it possible to retrieve the signal at the right frequency? To answer this question, we generated a single synthetic sinusoidal signal sampled at times identical to the real RVs. The uncertainty of every point corresponded to the uncertainties derived from the RVs. We generated two different sets, each with an amplitude of 3 ${\rm{m}}\,{{\rm{s}}}^{-1}$, one set using the stellar rotation frequency and a second set using the planetary frequency. Finally, for the synthetic data generated using the rotational frequency, instead of fixing the phase, we covered a grid of phases $[-\pi ,-0.9\pi ,\,...,\,\pi ]$. For the planetary signal we assumed that the phase is well constrained. We then did a periodogram analysis for each set and could recover a peak at the true frequency. This test shows that even though the WF shows a peak, we can still retrieve the signal at the right frequency (planet frequency or the stellar rotation frequency) given the data sampling. Hence, the data set is not affected by aliases.

In short, the planet's orbital period is 32.94 days (Benneke et al. 2017), and the stellar rotation period is ∼40 days. Not only are the RVs affected by activity, but the WF also shows a peak close to 32.2 days, caused by observational patterns in the way the data were sampled. Previous studies (Robertson & Mahadevan 2014; Vanderburg et al. 2016) showed the difficulty in detecting RV planets in orbits close to the stellar rotation period. Hatzes (2013) and Rajpaul et al. (2016) demonstrated that the WF can give rise to fake signals in the periodogram that mimicked the presence of a planet around α Cen B, which was reported by Dumusque et al. (2012). In the case of K2-18 b, the planet transits, and hence its existence is undeniable. However, a closer look at the WF is needed to check whether the RV signal of the planet is present in the data. This case demonstrates the difficulty in detecting nontransiting low-mass planets not only at orbits close to the stellar rotation period but also when observational patterns are present in the data.

We performed joint modeling of the photometric light curves obtained with STELLA and the RV measurements. Similar to Section 3.1, we modeled the photometric data of both bands with a sine wave function and fit for f_BR, ϕ_BR, γ_B, γ_R, A_B, and A_R. We adopted uniform priors for the phase and offsets of the stellar photometric variability. For f_BR, A_B, and A_R we adopted Gaussian priors centered at 0.02524 day⁻¹, 8.7 mmag, and 6.9 mmag, respectively, and with a standard deviation of 0.00032 day⁻¹ and 0.5 mmag for both amplitudes (see Section 3.1). We fit the RV measurements with a Keplerian model assuming a circular orbit (e = 0) and using the combined K2 and Spitzer ephemeris, i.e., we fixed the mid-transit time T₀ and P_b to the values derived photometrically by Benneke et al. (2017) since these parameters are tightly constrained. We accounted for stellar activity in the RV data by assuming that it has a sinusoidal function whose frequency is constrained from the photometric light curves. We let the phase of the stellar activity ϕ_act free and thus fit for the phase, amplitude K_act, and frequency f_BR of the stellar activity. We adopted noninformative priors for the offset, ϕ_act, K_act, and K_b. The joint analysis was then performed using emcee (Foreman-Mackey et al. 2013), and in total we fit for 10 parameters: 6 parameters for the photometric data (mentioned above) and 5 parameters for the RV data (γ, K_b, ϕ_act, K_act, and f_BR); the stellar rotation frequency is the same in both data sets.

The best-fit model gave a planetary semi-amplitude of ${K}_{b}={3.60}_{-0.51}^{+0.53}$ ${\rm{m}}\,{{\rm{s}}}^{-1}$ and a stellar activity semi-amplitude of K_act = 2.72 ± 0.50 ${\rm{m}}\,{{\rm{s}}}^{-1}$, corresponding to a planetary mass of ${M}_{b}={9.07}_{-1.49}^{+1.58}\,{M}_{\oplus }$, using M_* = 0.359 ± 0.047 M_⊙. Figure 13 shows the joint and marginalized posterior constraints on the model parameters. Using the transit depth, R_b/R_*, and stellar radius, R_*, as reported in Benneke et al. (2017) and provided in Table 2, we derive a planetary radius R_b = 2.37 ± 0.22 R_⊕;¹⁵ this corresponds to a planetary density of ${\rho }_{b}={4.18}_{-1.17}^{+1.71}$ g cm⁻³. The v sin i and spot filling factor estimated from photometry yield an RV semi-amplitude of 2.7 ${\rm{m}}\,{{\rm{s}}}^{-1}$ for spots using the relationship by Hatzes (2002), which is in excellent agreement with the one estimated using the RV data. The planetary semi-amplitude value is consistent with the one derived using HARPS RVs by Cloutier et al. (2017) at the 1σ level. The best-fit model and the phased RVs are shown in Figure 6. We report the stellar and planetary parameters used in this study and the median values of all the parameters, along with the 16th and 84th percentiles of the marginalized posterior distributions, in Table 2.

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To further test whether the activity signal is due to cool spots, we compared the phase shift between the photometric light curve and the RV signal due to activity. Figure 7 shows the phase-folded photometric light curve in the B band in blue and the RV signal in black. When the spot is at the center of the stellar surface (minimum in the photometric light curve), the contribution of the spot to the RV signal is close to zero. As the spot moves along the stellar surface to the receding redshifted limb (zero in the photometric light curve), the star appears to be blueshifted (minimum in the RV curve). Therefore, the phase shift is ∼90°. This is expected if the variations are due to cool spots, which is also consistent with the multiwavelength photometry analysis (Section 3.1). This is only considering the flux effect of dark spots. In general, the RV variations in active regions are induced by two different physical processes: first, the asymmetry in the stellar line profiles created by starspots, and second, the suppression of the convective blueshift effect due to the presence of strong magnetic fields that inhibit convection inside active regions. The convective blueshift effect could explain why the RV curve appears shifted a bit vertically at the minimum phase of the photometric light curve.

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Even though the star shows periodic photometric variability, there is evidence that the chromosphere does not show strict periodic sine-like variability (see Section 4.3 and Figure 4, where some points deviate from the best-fit curve, especially Ca ii IRT 1 and Ca ii IRT 3). Therefore, modeling the RV signal of stellar activity by a periodic sinusoidal function might not be the best approach. However, we next argue that the derived planetary semi-amplitude is not dependent on our choice of the model used to account for stellar activity. We performed several tests to check this dependency. First, following Baluev (2009), we accounted for stellar activity by adding a constant white-noise term often referred to as the RV jitter term, σ_jitter. The jitter term is treated as an additional source of Gaussian noise with variance ${\sigma }_{\mathrm{jitter}}^{2}$ and is added in quadrature to the estimated RV uncertainties (Ford 2006). We derived a planetary semi-amplitude ${K}_{b}={3.38}_{-0.76}^{+0.75}$ ${\rm{m}}\,{{\rm{s}}}^{-1}$ and an RV jitter ${\sigma }_{\mathrm{jitter}}={3.02}_{-0.53}^{+0.57}$ ${\rm{m}}\,{{\rm{s}}}^{-1}$. The planetary semi-amplitude derived using this model is in agreement with the one derived previously, within the 1σ error bars.

Second, to check whether the RVs are affected by stellar activity, we looked for correlations between the raw RVs and the various activity indicators mentioned in Section 3.2. The top panels of Figure 12 in the Appendix show the measured RVs plotted against the activity indicators and color-coded according to the stellar rotational phase. We did not find a linear correlation between any of these quantities and the measured RVs. However, there is a slight indication that the color-coded data points follow a circular path, especially for Ca ii IRT 2, but not with high significance. We further repeated the same analysis after the removal of the planetary signal and still did not find any significant correlations with the activity indicators. The results are shown in the bottom panel of Figure 12. Despite detecting a signal close to the stellar rotational period in both the RVs and the Ca ii IRT lines, no evident linear or circular correlation is seen, indicating that the relation is quite complex.

Third, we ignored activity and fit the RVs with a single Keplerian signal and fixed T₀ and P_b to the known photometric values. We estimated a planetary semi-amplitude K_b = 3.35 ± 0.47 ${\rm{m}}\,{{\rm{s}}}^{-1}$, which is also in agreement with the previous results. We further divided the data set into two, each containing 29 data points, and repeated the same analysis for the first and second halves of the data. We found similar planetary semi-amplitudes in both cases, and the values are given in Table 1.

Table 1.  Planetary Semi-amplitudes K_b Derived for the Full, First Half, and Second Half of the Data Set Using the Full-λ RVs, the Blue RVs, and the Red RVs

Kb (${\rm{m}}\,{{\rm{s}}}^{-1}$)	Full Set	First Set	Second Set
Full-λ RVs	3.35 ± 0.47	3.23 ± 0.66	3.10 ± 0.68
Blue RVs	3.46 ± 0.55	3.71 ± 0.79	${2.71}_{-0.77}^{+0.80}$
Red RVs	3.29 ± 0.46	2.77 ± 0.65	3.44 ± 0.64

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As a final test,¹⁶ we looked at the RV measurements in the red and blue orders of CARMENES-VIS. If the RVs are dominated by activity due to active regions on the stellar surface, then the planetary semi-amplitude in the blue part of the spectrum should be more affected by activity, whereas the red part should be less affected. As a result, if the star is active, a single Keplerian fit to the data should yield different planetary semi-amplitudes for different orders. The RV measurements for K2-18 are available at 42 orders. We calculated an RV weighted mean average for the first and second half of the orders, which we will refer to as the blue RVs and as the red RVs, respectively, and are reported in Table 6. The blue orders cover the wavelength range from 561 to 689 nm, whereas the red orders cover the range from 697 to 905 nm. We also ignored activity and fit separately the blue and red RVs with a Keplerian model with T₀ and P_b fixed. We did this analysis for the full CARMENES-VIS data set, the first half, and the second half. So, in total we repeated this analysis six times, all of which yielded similar planetary semi-amplitudes within the error bars. The values are reported in Table 1, where we denote the original full wavelength coverage RVs as full-λ RVs. We conclude that the RVs are not dominated by stellar activity and that the estimation of the planetary semi-amplitude is robust and does not depend on the choice of model used to account for stellar activity.

Table 2.  Stellar and Planetary Parameters for the System K2-18

Parameter	Value
Stellar Parameters
Prot (days)	39.63 ± 0.50
M* (M⊙)a	0.359 ± 0.047
R* (R⊙)a	0.411 ± 0.038
T* (K)a	3457 ± 39
[Fe/H] (dex)a	0.12 ± 0.16
Transit Parameters
Rb/R* (%)a	${5.295}_{-0.059}^{+0.061}$
T0 (BJD)a	${2457264.39144}_{-0.00066}^{+0.00059}$
Pb (days)a	${32.939614}_{-0.000084}^{+0.000101}$
Rb (R⊕)b	2.37 ± 0.22
i (deg)a	${89.5785}_{-0.0088}^{+0.0079}$

		Models
	Planet only	Planet + sine	Planet + jitter
Radial Velocity Parameters
Kb (m s–1)	3.35 ± 0.47	${3.60}_{-0.51}^{+0.53}$	${3.38}_{-0.76}^{+0.75}$
Kact (m s–1)	⋯	2.72 ± 0.50	⋯
σjitter (m s–1)	⋯	⋯	${3.02}_{-0.53}^{+0.57}$
e	0 (fixed)	0 (fixed)	0 (fixed)
Planet Parameters
a (au)a	${0.1429}_{-0.0065}^{+0.0060}$	${0.1429}_{-0.0065}^{+0.006}$	${0.1429}_{-0.0065}^{+0.006}$
Mb (M⊕)	${8.43}_{-1.35}^{+1.44}$	${9.06}_{-1.49}^{+1.58}$	${8.49}_{-1.97}^{+2.08}$
Teq (K)	283 ± 15	283 ± 15	283 ± 15
ρb (g cm−3)	${3.89}_{-1.08}^{+1.58}$	${4.18}_{-1.17}^{+1.71}$	${3.90}_{-1.24}^{+1.77}$

Notes.

^aParameters based on Benneke et al. (2017). ^bRecalculated the value using R_b/R_* and R_* as derived by Benneke et al. (2017).

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We also computed the results of Table 1 using a Keplerian model plus a sinusoidal model to account for activity, where we fit for the stellar rotation frequency. We find that the planetary semi-amplitude is consistent within 1σ when computed for the full data, first half, and second half for the full spectral coverage, the red orders, and the blue orders with one exception, the planetary amplitude computed for the second half in the red order. However, the value is in agreement at the 2σ level. Even though we expect the activity semi-amplitudes to be different in different orders, the semi-amplitudes derived are consistent either at the 1σ or at the 2σ level. This could be explained by the low-amplitude signals in both order ranges, which are on the order of 2.7 ± 0.73 ${\rm{m}}\,{{\rm{s}}}^{-1}$, i.e., a higher precision would be required to differentiate between the activity semi-amplitudes in different orders.

Cloutier et al. (2017) used 75 HARPS RV measurements spanning approximately three seasons of observations to estimate the mass of K2-18 b and to search for additional planetary signals. They reported a nontransiting planet, K2-18 c, with a period of 8.962 ± 0.008 days and a semi-amplitude of 4.63 ± 0.72 ${\rm{m}}\,{{\rm{s}}}^{-1}$. The signal of K2-18 c is stronger than that of K2-18 b (see Figure 2 in Cloutier et al. 2017).

We searched for the signal of the second planet in the CARMENES-VIS data set. As mentioned in Section 4, the periodogram only shows one strong peak at 34.97 days, the combined signal of the ∼33-day-period planet and the stellar rotation period. The second strongest peak is around 9 days, with an FAP > 5% and significantly weaker than in the HARPS data. We then subtracted the signal of the 33-day-period planet and stellar activity from the RVs and performed again a period analysis. We still did not find a strong signal at the period of the supposed second (inner) planet (Figure 3, panel 7).

In order to examine whether the absence of the 9-day signal in the CARMENES-VIS data set is due to bad sampling, we generated a synthetic RV data set assuming that there are two planets in the system and using the real observing times of CARMENES. We set the values of the orbital period, semi-amplitude, and time of inferior conjunction of both planets as derived by Cloutier et al. (2017): P_b = 32.93963 days, P_c = 8.962 days, K_b = 3.18 ${\rm{m}}\,{{\rm{s}}}^{-1}$, K_c = 4.63 ${\rm{m}}\,{{\rm{s}}}^{-1}$, T_0,b = 2,457,264.39157 BJD, and T_0,c = 2,457,264.55 BJD. We further assumed that the uncertainty is the sum of the observational error and a random noise (drawn from a normal distribution centered at 0 and a standard deviation of 0.25 ${\rm{m}}\,{{\rm{s}}}^{-1}$) to attribute to the stellar jitter determined by Cloutier et al. (2017). We then did a periodogram analysis and could recover an extremely strong peak at 8.98 days with an FAP < 0.1%. This shows that our analysis is not affected by poor time sampling.

We also examined whether the 9-day signal could be caused by stellar activity, since the period is near the fourth harmonic of the stellar rotation period (39.63 days—Section 3; Cloutier et al. 2017). We divided the full CARMENES-VIS data set into two, each consisting of 29 data points, and did a periodogram analysis for each set of the RVs, Ca ii IRT, and Hα lines. Figure 8 shows the periodograms for both data sets. The top left and top right panels show the periodograms of the activity indicators and RVs, respectively, for the first half of the CARMENES-VIS data set. Similarly, the bottom panels show the periodograms for the second half of the data set. The dashed line in the periodograms of the activity indicators shows the stellar rotation period, P_rot, while the dashed line in the RV periodograms indicates the period of the inner planet, P_c, as estimated by Cloutier et al. (2017). Note that for the activity indicators only the periodogram region near the rotation period is shown, whereas for the RVs only the region around the 9-day signal is displayed. The different levels of FAPs are indicated in the plot. The first half of the RV data set does not show a power at the orbital period of the supposed inner planet. That is also true when the Ca ii IRT and Hα lines do not show a consistent peak. The second Ca ii IRT index is the only indicator that shows a somewhat stronger peak with an FAP of ∼1%. The other indicators do not show a prominent peak, and notably Hα shows no power at the stellar rotation period. The signal of the 9-day period appears only in the second half of the RV data set, which occurs at the same time when all the Ca ii IRT and Hα lines show a prominent peak at the stellar rotation period with an FAP < 0.1%, demonstrating that the level of activity increased in this set. This indicates that the signal of the 9-day planet is absent when the star is less active and is present only when the level of activity increases significantly. We thus conclude that the presence of the 9-day signal correlates with the Ca ii IRT and Hα lines.

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This is further illustrated in Figure 9, which shows the periodograms of the full-λ RVs and the blue and red RVs of CARMENES-VIS, which are calculated as explained in Section 4.2. The periodogram for the blue RVs, red RVs, and full-λ RVs is shown in blue, red, and black, respectively. The legend indicates the period with the highest power for the different sets of RVs. The blue, red, and full-λ RVs show a single peak in the first half of the data set (top panel) close to 36 days. In the second half, interestingly the periodogram of the blue RVs shows the highest GLS power close to 9 days, while the red and full-λ RVs show the highest power close to the orbital period of the 33-day-period planet. This further demonstrates that when the level of stellar activity increased, the blue RVs show a period at the fourth harmonic of the stellar rotation period, while the red RVs do not. This is in line with the notion that RV variations due to photometric starspots are wavelength dependent and more prominent in the blue part of the spectrum, while the variations get smaller at redder wavelengths (Reiners et al. 2010). On the other hand, the RV variation of a planetary signal is wavelength independent and should be constant at all wavelengths. This shows the importance of multiwavelength RV measurements to differentiate planetary from stellar activity signals.

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Notably, in the second half of the data set, when the star is relatively more active, the red and full-λ RVs show peaks much closer to the orbital period of the planet and are not shifted in value toward that of the stellar rotation. It seems that the contribution of activity to the RVs appears near the fourth harmonic of the stellar rotation period, and this set shows a clean planetary signal.

We conclude that, although we found evidence of the second planet signal announced recently by Cloutier et al. (2017), the peak is not significant in the CARMENES-VIS data set with an FAP > 5%. The signal is also time and color variable and correlates with stellar activity. Given the sampling and the time baseline of our observations, we conclude that we do not have enough evidence to confirm the presence of the second inner planet, and there is a strong indication that the signal is intrinsic to the star. This also could explain why no transits were observed by K2 (Cloutier et al. 2017), although this can also be explained by misaligned orbits.