AD Leonis: Radial Velocity Signal of Stellar Rotation or Spin–Orbit Resonance?

Spectroscopic data of AD Leo were obtained from two sources. We downloaded the publicly available data products of HARPS (Mayor et al. 2003) from the European Southern Observatory archive and processed them with the TERRA algorithms of Anglada-Escudé & Butler (2012). As a result, we obtained a set of 47 radial velocities with a baseline of 3811 days and root-mean-square (rms) estimate of 22.59 ms⁻¹. Given a mean instrument uncertainty of 0.94 ms⁻¹, there are variations in the data that cannot be explained by instrument noise alone. The majority (28) of these were obtained over a short period of 76 days between JDs 2453809 and 2453871, enabling the detection of the signal at a period of 2.23 days (Bonfils et al. 2013; Reiners et al. 2013). The HARPS radial velocities of AD Leo are given in Table 1 and plotted in Figure 1.

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The second set of spectroscopic data was obtained by the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) of the Keck I telescope. This set of 42 radial velocities has a baseline of 3841 days and an rms of 18.93 ms⁻¹. With an average instrument uncertainty of 1.90 ms⁻¹, this data also indicates excess variability that cannot be explained by pure instrument noise. The HIRES velocities of AD Leo are also shown in Figure 1, as published in Butler et al. (2017).

To study the photometric variability of AD Leo, we obtained ASAS (Pojmański 1997, 2002) V-band photometry data⁸ from the ASAS-North (ASAS-N) and ASAS-South (ASAS-S) telescopes. We only selected the "Grade A" data from the set, removed all 5σ outliers, and obtained sets of 316 and 319 photometric measurements with baselines of 2344 and 2299 days, respectively. The apertures with the least amount of variability showed brightnesses of 9263.8 ± 25.2 and 9333.0 ± 28.3 mmag. These values are somewhat different, and we thus only compared the two time series by assuming an offset that was a free parameter of the model. The ASAS data are given in Tables 7 and 8 and plotted in Figure 2, together with the estimated long-period cycle that was clearly present in the data as also observed by Buccino et al. (2014). We note that Kiraga & Stepien (2007) did not discuss AD Leo when publishing photometric rotation periods of nearby M dwarfs.

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We also obtained the raw MOST photometry data⁹ as discussed in Hunt-Walker et al. (2012). The data are presented in Figure 3 for visual inspection and consist of 8592 individual observations over a baseline of roughly 9 days. The MOST observing run is denoted in Figure 2 by a vertical line.

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As reported by Hunt-Walker et al. (2012), the MOST high-cadence photometry showed clear evidence in favor of a photometric rotation period of 2.23 days, as well as several flare events (Figure 3). In an attempt to obtain constraints for the variability of the photometric signal in the MOST data, we split the raw data (N = 8592) in half. We then analyzed the two sets to quantify any changes in the phase, period, and amplitude of the photometric signal over the MOST baseline of 8.95 days.

According to our results, the phase of a sinusoid changed from 1.06 ± 0.08 to 1.85 ± 0.10 rad (when using standard 1σ uncertainty estimates) from the first half of the data to the second. Variability was also detected in the photometric period and amplitude that changed from 2.289 ± 0.019 to 2.145 ±0.011 days and from 0.2609 ± 0.0046 to 0.2242 ± 0.0037 ADU pix⁻¹ s⁻¹, respectively. Given that the median times of the two MOST data halves differ by only 3.38 days, this evolution takes place on a timescale comparable to the star's rotation period. It is thus evident that the spot patterns and active/inactive areas on the star's surface giving rise to the clear photometric rotation signal experience rapid evolution and change considerably over a period of only a few days.

The variability of the photometric signal on short timescales indicates that the signal detected in ASAS-N data is only an average over a baseline of roughly 3000 days. This implies that the photometric amplitude of the signal detected in ASAS-N appears lower than it actually is because it corresponds to an average over different phases in the star's activity cycle. This interpretation is supported by Spiesman & Hawley (1986), who estimated the amplitude to be 12 mmag.

We also obtained and analyzed selected HARPS and HIRES activity indicators: the S-index measuring the emission of Ca ii H & K lines for HARPS and HIRES with respect to the continuum and the line bisector span (BIS) and full width at half maximum (FWHM) for HARPS.

The likelihood ratio periodogram of the HIRES S-index showed some evidence (in excess of 5% but not 1% FAP) for a 150 day periodicity (Figure 5, top left panel). This variability could be connected to photometric variability detected in the ASAS-S photometry. We also detect a broad maximum in the periodogram of HARPS S-indices at a period of 300 days (Figure 5, top right panel). However, we could not observe any evidence for periodicities in the HARPS BIS and FWHM values (Figure 5). This result is consistent with that of Reiners et al. (2013), who discussed hints of evidence for periodicities in the HARPS BIS values, but the corresponding periodic signals only barely exceeded 5% FAP in their analyses.

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We modeled the radial velocities by accounting for a Keplerian signal (f_k), reference velocity γ_l of instrument l, linear trend ($\dot{\gamma }$), linear dependence of the velocities on the activity indices ξ_i,j,l with parameter c_j,l, and moving average component with exponential smoothing accounting for some of the red features (e.g., Baluev 2009; Tuomi et al. 2014) in the radial velocity noise. The statistical model for a measurement m_i,l at epoch t_i,l is thus

$$\begin{eqnarray}{m}_{i,l} & = & {f}_{k}({t}_{i,l})+{\gamma }_{l}+\dot{\gamma }+\displaystyle \sum _{j}{c}_{j,l}{\xi }_{i,j,l}\\ & & +{\phi }_{l}\exp \left\{\displaystyle \frac{{t}_{i-1}-{t}_{i}}{\tau }\right\}{r}_{i-1,l}+{\epsilon }_{i,l},\end{eqnarray} \tag{ 1 }$$

where τ was set equal to 4 days because we expect to see correlations on that timescale but not on longer timescales (Tuomi et al. 2014) and r_i,l represents the residual after subtracting the deterministic part of the model from the data. The Gaussian random variable _i,l represents the white noise in the data: it has a zero mean and a variance of σ_i² + σ_l², where σ_l is a free parameter for all instruments quantifying the excess white noise, or "jitter", in the data.

The combined HARPS and HIRES radial velocities of AD Leo contained a periodic signal that was clearly identified by our DRAM samplings of the parameter space (Figure 6). We show the HARPS and HIRES radial velocities folded on the phase of the signal in Figure 7 for visual inspection. This signal was supported by both data sets; the maximized log-likelihood (natural logarithm) increased from −199.6 to −179.1 for HIRES and from −196.6 to −153.7 for HARPS, exceeding any reasonable statistical significance thresholds.¹¹ Using the BIC to determine the significance of the signal (see Schwarz 1978; Feng et al. 2016), we obtain an estimate for the logarithm of the Bayes factor in favor of a model with one signal of 52.25—considerably in excess of the detection threshold of 5.01 corresponding to a situation where the model is 150 times more probable (e.g., Kass & Raftery 1995). This signal satisfied all the signal detection criteria of Tuomi (2012); i.e., in addition to satisfying the significance criterion, it corresponded to a unique posterior probability maximum constrained from above and below in the period and amplitude spaces.

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As also reported by Bonfils et al. (2013) and Reiners et al. (2013), we observed a negative correlation between HARPS radial velocities and BIS values. The linear parameter c_{j, l} (see the model above) that quantifies this dependence of the HARPS velocities on the observed BIS values was found to have a value of −2.12 [−2.96, −1.30], which is significantly different from zero at a 7.7σ level. Similar strong correlations were not found with HARPS FWHM and S-index. There was a weaker correlation between the HIRES velocities and S-index with a parameter value of 1.62 [0.21, 3.18] ms⁻¹, which indicates a 3.1σ significance.

We have tabulated the parameters of the radial velocity signal, assuming it has a Keplerian shape, in Table 1, together with the "nuisance parameters" in the statistical model. We also tabulated the estimates when not including the correlations between velocities and the activity indicators in the model—i.e., when fixing parameters c_i,l = 0 for all indices and both instruments. We note that the two solutions are not statistically significantly different for the Keplerian parameters. This is demonstrated by the fact that the 99% credibility intervals of the parameters of the signal are not distinct between the two models.¹² This implies that the properties of the signal are independent of whether or not we account for the correlations between the velocities and activity data. However, as discussed above, there is a significant correlation between the HARPS radial velocities and BIS values, and we thus consider the solution that includes the correlations to be more trustworthy (the solution on the left-hand side in Table 1).

Table 1.  Maximum a Posteriori Estimates and 99% Credibility Intervals of the Parameters of a Model with One Keplerian Signal and with or without Correlations with the Activity Indicators

Parameter	Full Model	No Activity Correlations
K [ms−1]	19.11 [15.69, 22.54]	23.18 [19.68, 26.67]
P [days]	2.22579 [2.22556, 2.22593]	2.22579 [2.22566, 2.22592]
e	0.015 [0, 0.147]	0.028 [0, 0.161]
ω [rad]	4.7 [0, 2π]	5.1 [0, 2π]
M0 [rad]	6.2 [0, 2π]	0.3 [0, 2π]
a [au]	0.024 [0.021, 0.026]	0.024 [0.021, 0.027]
$m\sin i$ [M⊕]	19.7 [14.6, 25.3]	23.8 [17.9, 29.8]
$\dot{\gamma }$ [ms−1 yr−1]	−1.50 [−3.00, −0.31]	−1.68 [−2.46, −0.89]
σHARPS [ms−1]	5.52 [4.27, 6.90]	6.58 [5.36, 8.09]
σHIRES [ms−1]	8.40 [7.07, 9.88]	8.74 [7.47, 10.31]
ϕHARPS	−0.10 [−0.94, 0.80]	−0.66 [−1, 0.07]
ϕHIRES	0.61 [0.05, 1]	0.27 [−0.39, 0.88]
cBIS,HARPS	−2.12 [−2.96, −1.29]	⋯
cFWHM,HARPS	0.230 [−0.141, 0.600]	⋯
cS,HARPS [ms−1]	−0.14 [−2.82, 2.55]	⋯
cS, HIRES [ms−1]	1.62 [0.21, 3.18]	⋯

Note. Parameters K, P, e, ω, and M₀ are the Keplerian parameters: radial velocity amplitude, signal period, eccentricity, argument of periapsis, and mean anomaly with respect to epoch t = 2,450,000 JD, respectively. The uncertainty in stellar mass has been accounted for when estimating the minimum mass and semimajor axis corresponding to the signal under planetary interpretation for its origin.

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A true Keplerian Doppler signal of planetary origin cannot depend on the wavelength range of the spectrograph. Although Reiners et al. (2013) tested the wavelength dependence of the signal in the AD Leo velocities by measuring the amplitude of the signal for different subsets of the 72 HARPS orders, they did not account for red noise or correlations between the velocities and activity indicators. We repeated this experiment by calculating the weighted mean velocities for six sets of 12 orders. Apart from the bluest 24 HARPS orders that were found to be heavily contaminated by activity (see also Anglada-Escudé & Butler 2012), we found the parameters of the signal to be independent of the selected wavelength range. This is demonstrated in Figure 8, where we have plotted the signal amplitude, period, and phase (when assuming a circular solution) as a function of wavelength. In Table 4, the radial velocities for individual orders are given. The remarkable stability of the signal represents the hallmark of a Keplerian signal of planetary origin and is difficult to interpret as a signal that is caused by starspots corotating on the stellar surface. Although the bluest orders do not appear to agree with the rest of the orders in this respect, we note that the solution for the first 12 orders is actually a highly eccentric solution that arises from the activity-induced variations (and correspondingly higher rms) at the bluest wavelengths.

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The standard deviation of the radial velocities from the bluest 12 orders was found to be 77.89 ms⁻¹, whereas that of the next bluest 12 orders was 31.07 ms⁻¹. This arises due to a combination of lower signal-to-noise and the bluer orders being more prone to activity-induced variability. As a consequence, it was not possible for us to detect the signal correctly in the bluest 12 orders and probably caused biases in the estimated parameters of the signal for the second bluest 12 orders (Figure 8). However, when using the so-called differential velocities of Feng et al. (2017) as activity proxies, we could see the signal as a clear probability maximum at the same period in the period space for all six wavelength ranges. Although independent detection of the signal was only possible for the four redmost sets of 12 HARPS orders, the differential velocities helped remove activity-induced variability such that the signal could be seen throughout the HARPS wavelength range.

Another signpost that a periodic signal in radial velocity is a Keplerian one, caused by a planet orbiting the star, is time invariance. We tested this time invariance by looking more closely at a 60 day period during which HARPS was used to observe AD Leo 28 times: over nine (N = 14) and eight (N = 8) consecutive nights and then six times over a period of 11 nights. These observations are treated as independent data subsets and plotted in Figure 9, together with the estimated Keplerian curve. As can be seen in Figure 9, over this period of 60 days, the periodic variability is remarkably stable and consistent with a time-invariant signal.

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Due to the lack of another period of observations with suitably high observational cadence, we then analyzed in combination the HARPS data not included in the above 60 day period and HIRES data. This combined data set with 19 HARPS and 42 HIRES radial velocities showed evidence for a consistent periodicity with the 60 day HARPS observing run. Assuming zero eccentricity, the signals in the 60 day observing run and the rest of the data have amplitudes of 23.13 [19.28, 26.24] and 18.78 [11.68, 25.87] ms⁻¹, periods of 2.22290 [2.21810, 2.22645] and 2.22576 [2.22507, 2.22601] days, and phases of 4.08 [0, 2π] and 1.83 [0.28, 4.81] rad, respectively. Although the solution given the 60 day HARPS observing run is rather uncertain (the former solution above), this demonstrates that the signal is stable, with a precision of almost two orders of magnitude better than what was observed for the photometric rotation signal in the MOST data in Section 4.1.

To investigate whether starspots corotating on the stellar surface could produce coherent radial velocity signals over a period of several years, such as the baseline of the radial velocity data of AD Leo of 4733 days, we generated artificial radial velocities to study the detectability of evolving signals in the available radial velocity data.

We generated artificial data sets with the same properties as were observed for the HARPS and HIRES data (Table 1) and added one sinusoidal signal in the data with P = 2.23 days and K = 20.0 ms⁻¹. Because the MOST photometry indicated that, within a period of 9 days, the photometric signal evolved considerably in phase, period, and amplitude, we changed the phase of the signal by an angle of ψ t, where ψ is a constant and t is time. Parameter ψ was selected to have values $\psi \in \tfrac{1}{10}[0,1,\,...,\,7]$ rad week^–1—i.e., the signal was made to vary linearly as a function of time from 0 to 0.7 rad week^–1. We denote these data sets as S1, S2, ..., S8, respectively. Although only a toy model, this still enabled us to study the sensitivity of our signal detection to evolving signals. To avoid analyzing the artificial data with the same model as was used to generate it (i.e., committing an "inverse crime"; Kaipio & Somersalo 2005), we used a noise model with third-order moving average terms with second- and third-order components fixed such that ϕ₂ = 0.3 and ϕ₃ = 0.1, respectively, whereas we only applied the first-order moving average model when analyzing the data.

The simulated radial velocity data sets were generated such that the injected signal was varied less rapidly than the photometric one in the MOST data and did not disappear, contrary to what appears to be the case for the photometric signal in the ASAS data (Figure 4). Although we expected a slightly variable signal to cause a clear probability maximum due to the concentration of HARPS data on a period of 60 days (Section 5.3), it was also expected that a more rapidly varying signal would make the detection less probable or impossible. This is indeed what happened, as can be seen in Figure 10, where we have plotted the posterior probability densities as functions of signal periods given artificial data sets with signals whose phases evolve. While the signal was very clear for the simulated data set S1 with a stationary signal (Figure 10, top left panel), it was also clearly detected for sets S2 and S3, for which the evolution in the signal phase was 0.1 and 0.2 rad week^–1, respectively. With a more rapidly evolving phase of 0.3 rad week^–1, the signal and its alias became less and less unique (S4), whereas even more rapidly evolving signals could not be detected as unique solutions despite the fact that the posterior densities showed hints of periodicities close to the period of the injected signal and its daily alias (Figure 10, bottom panels).

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The stationary signal (S1) corresponds to the only simulated case where the signal was detected as clearly as in the actual HARPS and HIRES radial velocity data. For the simulated set S1, the signal was detected with a logarithm of Bayes factor of 58.77—close to the value obtained for the actual data of 52.25. For sets S2 and S3, this value decreased to 21.24 and 6.51; the latter one is only barely above the detection threshold of 5.01. Together with the fact that the signals lose their uniqueness when the signal varies more than 0.2 rad week^–1, this implies that an evolving signal would be unlikely to cause the observed radial velocity variability. This suggests that only signals that do not vary as a function of time can be detected as strong probability maxima in the AD Leo radial velocity data (Figure 6).

Because a linearly evolving phase is an unrealistic description of the potentially very complex patterns of starspot evolution, we also tested a simple stochastic variability model. When assuming that the phase of the signal was a random variable drawn from a Gaussian probability distribution centered at the current phase and a standard deviation σ_ψ ranging from 0 to 0.7 rad, as was the case for the linearly evolving ψ above, we updated the phase randomly after 2–4 rotation periods and again searched for signals in the generated artificial data sets. For such stochastic variability, the signal could not be detected in the data for σ_ψ > 0.3 rad. Although detection was possible for a phase changing more slowly than that, our test again suggested that variable signals generally cannot be detected unless the rate of variability is low.

It thus appears that, unlike the photometric signal (Figure 3) that varies in phase, amplitude, and period, the radial velocity signal of AD Leo appears to be caused by a stationary process. At least, it can be said that the signal in AD Leo radial velocities is consistent with a stationary signal.

The ASAS survey has been used to identify several nearby M dwarfs with short (<10 days) photometric rotation periods (Kiraga & Stepien 2007). We identified 10 targets in a sample of 360 nearby M dwarfs (Tuomi et al. 2018) for which the ASAS-S photometry shows evidence for periodicities shorter than 10 days (Table 2). We have tabulated the corresponding significant (exceeding 0.1% FAP) photometric periodicities with P_rot < 10 days in Table 2 and interpret them as photometric rotation periods of the stars with period P_rot. We have also plotted the likelihood ratio periodograms of these targets in Figure 12 to visually demonstrate the significance of the detected photometric rotation periods. All these targets have also been observed spectroscopically with HARPS, HIRES, and/or the Planet Finding Spectrograph (PFS), and we thus examined the radial velocity data sets in order to search for counterparts of the photometric periodicities.

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We also investigated whether the available radial velocities of the targets in Table 2 showed evidence for signals that could be interpreted as counterparts of the photometric rotation periods. Apart from GJ 494 and GJ 1264 that only had two and one radial velocity measurements available, respectively, and GJ 841A, whose HARPS spectra were contaminated and resulted in radial velocities varying at a 10 km s⁻¹ level, we searched for such counterparts of photometric rotation periods with posterior samplings. The resulting estimated posterior densities as functions of signal periods are shown in Figure 13. We also show the posterior given AD Leo (GJ 388) data for the same period space between 1 and 12 days.

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Table 4.  Example of HARPS Radial Velocity of AD Leo for All 72 Individual Orders v_i, i = 1, ..., 72

t [JD–2450000]	v1 (ms−1)	${\sigma }_{{v}_{1}}$ (ms−1)	v2 (ms−1)	${\sigma }_{{v}_{2}}$ (ms−1)	v3 (ms−1)	${\sigma }_{{v}_{3}}$ (ms−1)
2986.8579	502.80	300.23	−1153.97	276.03	190.56	289.89
3511.5468	−38.00	141.87	−94.04	133.81	521.06	126.51
3520.5205	158.29	191.61	471.33	187.46	278.08	169.40
3543.4808	138.40	240.76	17.90	227.65	−75.73	216.12
3544.4518	−377.68	171.55	−354.19	162.54	−156.08	148.84
3550.4596	1176.80	321.81	−744.10	286.01	369.98	296.29
3728.8648	−8.87	162.99	195.14	155.38	−16.01	144.67
3758.7540	81.83	129.82	50.39	127.00	−21.92	111.59
3760.7547	−18.18	120.27	−472.43	116.04	168.59	107.86
3761.7800	184.64	107.38	19.01	103.78	−113.55	90.69
3783.7255	−158.69	113.05	103.10	112.32	150.24	96.51
3785.7264	−1.90	111.54	−155.30	115.46	257.99	99.78
3809.6599	49.90	79.20	−143.19	82.32	24.83	69.91
3810.6767	314.57	108.86	296.13	96.35	110.75	84.68
3811.6750	123.42	98.04	200.36	93.21	−131.53	81.88
3812.6637	−157.04	166.80	538.03	159.24	−261.14	145.15
3813.6584	231.57	89.82	−45.62	88.21	−17.41	78.00
3814.6540	−91.03	152.77	−217.45	148.51	−72.68	132.42
3815.5702	−14.76	121.71	232.38	119.64	−153.38	107.72
3815.6211	−217.01	103.38	−41.35	100.24	−227.38	88.13
3815.7354	−400.83	109.28	5.05	105.98	−46.12	90.78
3816.5433	35.14	127.58	194.62	123.77	−150.30	111.89
3816.6552	−186.00	107.40	−201.14	109.34	−39.90	92.84
3816.7211	127.14	116.73	65.59	118.16	−218.59	100.93
3817.5500	154.66	94.25	−95.49	95.87	−76.58	81.27
3817.6749	84.06	103.83	83.03	99.67	100.74	88.41
3829.6146	−247.02	97.13	76.68	95.42	118.33	81.55
3830.5397	164.12	119.88	−63.25	112.66	−100.52	101.58
3831.6694	224.15	89.65	−165.35	91.34	177.34	76.56
3832.6497	124.92	92.06	−128.90	95.22	−233.24	82.47
3833.6269	−130.12	79.78	25.88	76.59	65.00	66.56
3834.6103	−208.36	85.12	−248.17	79.18	99.07	68.95
3835.6551	−73.31	98.09	−53.44	95.03	−223.53	84.78
3836.6165	−166.97	87.31	118.87	87.04	−320.86	75.65
3861.5934	629.82	127.62	106.85	126.00	−329.65	110.16
3863.5687	−70.57	103.04	−450.80	96.21	76.85	85.86
3864.5340	25.72	106.61	156.79	108.93	147.68	92.01
3867.5417	−415.38	118.93	107.27	115.52	−124.68	101.36
3868.5177	108.62	115.43	−265.25	112.28	−256.27	98.14
3871.5622	−94.14	105.93	−71.85	104.76	118.22	92.04
6656.8493	291.80	204.95	−289.60	194.67	142.49	175.07
6656.8602	247.43	200.15	−233.94	189.16	542.26	168.08
6657.8522	−208.53	212.18	534.53	194.33	278.32	173.81
6658.8644	−313.55	204.48	−212.75	197.12	−231.53	169.65
6658.8755	−42.68	197.72	117.97	189.36	−112.47	162.47
6659.8581	−465.41	291.66	1210.13	257.40	−2.09	247.89
6797.5118	126.76	105.52	271.71	102.69	216.04	90.55

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Table 6.  Example of HARPS Radial Velocity of GJ 674 for All 72 Individual Orders v_i, i = 1, ..., 72

t [JD–2450000]	v1 (ms−1)	${\sigma }_{{v}_{1}}$ (ms−1)	v2 (ms−1)	${\sigma }_{{v}_{2}}$ (ms−1)	v3 (ms−1)	${\sigma }_{{v}_{3}}$ (ms−1)
3158.7520	−588.71	263.81	227.27	269.05	1749.74	276.26
3205.5995	12.31	101.82	−47.32	116.04	15.93	90.61
3237.5635	39.12	214.44	−1319.67	226.17	−159.88	200.46
3520.7932	−134.69	121.59	−183.98	138.57	−28.22	105.64
3580.5685	−2.02	89.88	0.17	101.79	−39.91	82.02
3813.8477	−57.20	58.22	−117.54	69.24	35.22	50.26
3816.8691	−166.60	65.66	168.84	77.43	−20.63	58.47
3861.8085	111.50	66.70	−93.26	75.73	25.76	57.69
3862.7863	−41.33	81.56	194.86	92.34	50.25	72.07
3863.8105	16.53	72.25	34.94	82.14	93.66	64.76
3864.7669	190.33	73.11	20.47	83.28	−112.01	65.66
3865.8114	−97.21	63.40	−91.95	74.01	31.54	54.87
3866.7558	70.73	55.05	106.57	61.28	15.81	47.47
3867.8481	3.60	82.11	−120.05	87.68	−36.68	66.88
3868.8393	−32.43	68.11	−25.00	75.97	−88.54	60.51
3869.8038	22.40	71.35	135.15	79.57	−183.62	60.78
3870.7388	−225.05	102.91	−48.89	112.87	93.43	90.79
3871.8606	84.78	75.72	272.50	84.60	0.74	64.68
3882.7445	−142.07	55.19	−6.12	63.29	−1.25	47.93
3886.7489	54.69	51.51	−34.08	58.77	−38.51	44.38
3887.7915	41.26	54.68	−69.51	62.46	43.17	47.53
3917.7229	−67.53	102.54	−307.72	113.21	−164.23	88.00
3919.7271	266.24	139.73	138.67	153.85	118.44	121.60
3921.6271	55.45	55.68	−133.81	65.64	−50.26	51.47
3944.5860	−116.93	111.85	−169.81	119.60	−102.59	102.00
3947.5906	−419.66	215.97	−574.18	224.53	−506.57	208.66
3950.6474	−22.58	87.82	264.96	98.07	−43.89	77.73
3976.5244	36.70	65.73	−172.51	71.14	8.10	55.64
3980.5691	−66.00	120.14	−104.77	130.93	−49.01	104.40
3981.5927	−24.58	86.43	−118.72	96.98	−119.14	76.33
3982.6162	−150.55	85.05	−208.65	97.15	50.91	76.92
3983.5287	−394.07	119.96	387.17	132.16	219.03	103.19
4167.8783	47.99	71.05	89.18	83.42	29.21	64.07
4169.8879	266.39	66.32	73.14	73.76	−145.35	56.81
4171.8966	13.84	64.57	−41.04	77.54	22.22	57.44
4173.8674	−139.35	71.71	−51.11	83.35	−22.61	63.72
4340.6256	−86.51	64.88	156.99	72.66	31.53	56.34
4342.5927	−168.33	84.39	76.99	92.49	−22.83	77.75
4347.5335	−43.39	87.77	−237.71	100.36	6.53	79.37
4349.6053	−288.70	105.74	−242.57	114.94	−88.61	92.15
4388.4992	410.95	131.62	−165.98	142.65	−372.56	113.86
4589.8983	52.11	70.05	−91.71	82.61	193.71	63.27
4666.7202	159.42	107.67	117.36	118.26	−19.54	96.35
4732.4793	172.13	103.85	−16.30	116.38	−135.62	89.29
5784.6229	106.28	140.50	89.25	151.05	−55.52	118.74
5810.5101	−43.88	111.11	−64.36	118.18	268.38	95.76
5815.5504	−416.05	127.38	−204.83	140.64	24.47	107.59
5817.5325	−371.97	129.16	−368.98	128.88	−296.97	108.19
6385.7728	−26.32	123.76	26.87	144.05	109.07	117.45
6386.7932	−116.45	165.94	−427.02	173.78	247.43	151.03
6387.8161	382.30	139.11	−481.63	149.42	207.95	115.49
6388.7910	31.58	142.91	−389.56	155.37	206.52	125.34
6389.7575	121.32	162.90	−384.28	171.63	−52.14	142.02
6390.7806	−53.28	184.35	−725.17	188.55	101.97	164.66
6393.8390	−569.83	175.81	−1284.30	181.88	−152.91	156.36
6394.8345	−35.35	145.96	−726.20	161.48	137.19	126.25
6395.7581	−118.25	141.99	−1190.94	151.95	−266.51	124.99
6396.8055	0.45	151.23	−983.89	160.24	−98.54	130.24
6397.8548	761.62	154.42	−1555.73	160.96	−347.23	132.46
6398.7295	683.26	154.21	−328.82	167.12	−239.14	137.56
6399.7167	145.08	133.42	−315.28	147.14	101.25	112.60
6400.7247	669.04	166.52	−1103.21	179.43	−567.21	146.99
6401.7452	−120.21	150.96	−737.78	162.96	−631.43	129.01
6402.7721	326.55	179.47	−1694.13	186.14	−375.28	159.66

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It can be seen in Figure 13 that only AD Leo (GJ 388; left column, second panel) shows evidence for a unique radial velocity signal in the sense that there are no local maxima exceeding the 0.1% equiprobability threshold of the global maximum. However, there seems to be a reasonable correspondence in the sense that all targets have global posterior maxima at or near the detected photometric periodicities, even though the maxima in the radial velocities are far from unique (Figure 13). It is thus likely that the radial velocities contain periodic variabilities corresponding to the photometric rotation periods. But AD Leo is the exception in this sense, as it is the only one with a unique short-period radial velocity signal.

We interpret this as an indication that such rapidly rotating M dwarfs do not readily produce clearly distinguishable radial velocity signals at or near the rotation period. This is the case even when the photometric rotation period is readily detectable with ASAS photometry. When searching for signals in the data of the comparison stars (Figure 13), we obtain broadly similar results as those for our artificial data sets with evolving signals (Figure 10); i.e., there are no unique and significant probability maxima. AD Leo clearly represents an exception in both these respects.

It was not possible to study the dependence of the comparison target radial velocity signals on spectral wavelength, as was done for AD Leo (Figure 8). We calculated the radial velocities for the six wavelength intervals, each corresponding to 12 HARPS orders, for GJ 803 that had the most HARPS observations available (N = 20 after removing outliers). Significant periodicities could not be detected in any of these six sets, and we thus could not study the wavelength dependence of signals caused by purely stellar rotation. However, the probability maximum corresponded to different periods in each of the six wavelength intervals. These periods ranged from 2.8 to 6.3 days, but it remains uncertain whether any of them actually correspond to the photometric rotation period of the star of 4.86 days.

Finally, we analyzed the HARPS radial velocity data (see Tables 5 and 6) of GJ 674, a quiescent slowly rotating M dwarf that has been reported to be a host to a candidate planet with a minimum mass of 11.09 M_⊕ with an orbital period of 4.6938 ± 0.007 days (Bonfils et al. 2007). Moreover, as discussed by Bonfils et al. (2007) and Suárez Mascareño et al. (2015), the HARPS radial velocities of GJ 674 also show evidence for the star's rotation period. This period was estimated to be 34.8467 ± 0.0324 days by Bonfils et al. (2007), but other authors provide slightly different estimates of 32.9 ± 0.1 days (Suárez Mascareño et al. 2015) based on spectroscopic activity indices and 33.29 days (Kiraga & Stepien 2007) based on ASAS photometry. Because this target provides an ideal test case for studying the differences between planet- and rotation-induced signals, we examined the wavelength dependence of the two signals based on radial velocity data calculated for different wavelength ranges, as we did for AD Leo in Section 5.3.

As is expected for a planetary signal, the signal of GJ 674b was consistently detected in all but the bluest 12 HARPS orders. For the five reddest sets of 12 orders, we obtained a consistent periodicity of 4.6950 ± 0.0002 days and an amplitude of 8.7 ± 0.3 ms⁻¹ that also agrees very well with the solution reported by Bonfils et al. (2007).

However, the rotation-induced radial velocity signal of GJ 674 was found to be dependent on wavelength. The rotation signal could not be detected in the two bluest sets of 12 orders, 1–12 and 13–24, at all, although the latter showed hints of a periodicity of 36.67 days corresponding to the highest probability maximum in the period space. In radial velocities calculated for orders 25–36, 37–48, 49–60, and 61–72, we detected periodicities of 36.66 ± 0.19, 3.6708 ± 0.0004, 36.18 ± 0.05, and 33.33 ± 0.03, respectively, although the last set of orders had two roughly equally high probability maxima with an alternative solution at a period of 36.66 ± 0.03 days (Figure 14). According to these results, the rotation-induced signal is not found consistently at the same period but varies as a function of wavelength. Moreover, a signal corresponding to the star's rotation period could not be identified in the radial velocities calculated for orders 37–48, but another signal near 4 days was present instead. This signal was also present in orders 49–60 as a secondary solution (Figure 14, bottom left panel). We did not observe significant differences in the amplitudes of the signals as a function of wavelength because the amplitude of the rotation-induced signal had an amplitude below 3 ms⁻¹ with uncertainties of roughly 1 ms⁻¹, making the available precision insufficient for determining possible differences in amplitude as a function of wavelength. These results demonstrate that stellar rotation does not readily produce wavelength-invariant radial velocity signals and that the signal observed in AD Leo radial velocities is thus likely caused by a planet orbiting the star.

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We further highlight the wavelength dependence and independence of the Keplerian signal in GJ 674 data and the rotation-induced signal near 37 days in Figure 15. We note that the estimated second period is not shown for orders 37–48 in the bottom panel of Figure 15 because there are no significant signals at or near the stellar rotation period and the corresponding second signal for these orders has a period of 3.67 days.

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