Revisiting WASP-47 with ESPRESSO and TESS

The Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO; Pepe et al. 2021) is a new high-resolution, visible spectrograph operating at the Very Large Telescope (VLT) at ESO's Paranal Observatory in Chile. ESPRESSO can be fed from any of the 8.2 m Unit Telescopes and can also be fed simultaneously by all four. The three main exoplanet science objectives of ESPRESSO are to find new Earth-mass planets orbiting in the habitable zone of Sun-like stars, to characterize the atmospheres of exoplanets, and to precisely determine the masses of low-mass transiting exoplanets. ESPRESSO has already been used to confirm TESS discoveries, including LP 714-47 b (TOI-442 b; Dreizler et al. 2020), TOI-130 b (Sozzetti et al. 2021), and the planets in the TOI-178 system (Leleu et al. 2021). ESPRESSO radial-velocity measurements have also been used to improve upon the precision of the masses of LHS-1140 b and c (Lillo-Box et al. 2020).

ESPRESSO operates in the wavelength range 380–788 nm, and is designed to achieve a radial-velocity precision of 10 cm s⁻¹ for bright stars (V < 8 mag) in order to be capable of detecting Earth-mass planets around solar-type stars. This unprecedented radial-velocity precision is achieved both by building on and improving the technologies used in the HARPS spectrograph for stability and calibration accuracy but also through the increased light-collecting capacity of the VLT Unit Telescopes compared to the ESO 3.6 m.

WASP-47 was observed by ESPRESSO between the dates of 2019 August 6 and September 30 (Run ID: 0103.C-0422; PI: Bayliss), using an exposure time of 1250 s and with a limit of z < 1.5 for each observation. We reduced all the spectra using the ESPRESSO reduction pipeline (version 2.2.1) through the ESOReflex workflow environment (Freudling et al. 2013). The signal-to-noise achieved from our observations ranged from 40 to 60 at λ = 550 nm. The radial-velocity cross-correlation functions (CCFs) were computed using a G9 mask, and the pipeline automatically extracts diagnostic information on the CCFs, including the FWHM of the CCF. In total, WASP-47 was observed 25 times by ESPRESSO. Two of these observations show anomalous CCF profiles likely caused by cloud coverage and/or moon contamination, and are not used in the analysis. An additional four observations were identified as having been taken during a transit of WASP-47 b. We also remove these from our analysis so that the Rossiter–McLaughlin signal of WASP-47 b (Sanchis-Ojeda et al. 2015) does not affect our radial-velocity model and analysis. This left us with 19 radial-velocity measurements from ESPRESSO for our analysis. We run spectral analysis on our coadded ESPRESSO spectra using the ESPRESSO-DAS pipeline. The values we derive for effective temperature and metallicity are consistent with those derived by Vanderburg et al. (2017).

In addition to the ESPRESSO data, we utilize a number of archival radial-velocity measurements of WASP-47, which were obtained using the HARPS-N (69 data points with mean a precision of 3.2 m s⁻¹; Vanderburg et al. 2017), HIRES (43 data points with mean a precision of 2.0 m s⁻¹; Sinukoff et al. 2017), PFS (26 data points with mean a precision of 3.2 m s⁻¹; Dai et al. 2015), and CORALIE (52 data points with mean a precision of 12.5 m s⁻¹; Neveu-VanMalle et al. 2016) spectrographs. We did not perform any additional reduction of these data sets.

As we discuss in Section 3, a degree of variability in the radial-velocity data indicates relatively strong stellar activity on WASP-47. In order to help understand and account for this activity, we turned to the high-precision time-series photometry for WASP-47 from the Kepler space telescope (Borucki et al.2010).

WASP-47 was observed by the Kepler space telescope during Campaign 03 of the K2 mission (Howell et al. 2014). WASP-47 was observed continuously for 67 days between the dates of 2014 November 17 and 2015 January 23. WASP-47 was observed in the short-cadence mode, and a short-cadence light curve was extracted by Becker et al. (2015) using the method of Vanderburg et al. (2015). We accessed this light curve for the transit analysis performed in this work. ³

WASP-47 was observed by the Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al. 2014) during Sector 42 between the dates of 2021 August 21 and September 15. WASP-47 fell on Camera 1 CCD 4. We utilized the 20 s cadence light curve produced by the SPOC pipeline (Jenkins et al. 2016), which we downloaded from MAST. We use the PDCSAP_FLUX time series in this work, which has had instrumental and blending effects corrected in the light curve (Jenkins et al. 2016).

During both orbits of Sector 42, the Earth crossed the field of view of Camera 1, with the Moon also crossing the field of view during the first orbit. ⁴ Due to the significant increases in scattered light in the background during these events, the PDCSAP_FLUX time series spans only 4.11 days in the first orbit and 9.48 days in the second, giving a total of around half the nominal 27 day coverage for a given sector. The TESS light curve is shown in Figure 2.

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We analyzed the TESS data in order to refine the planetary parameters of the WASP-47 planets. It is not known if WASP-47 c transits the host star (Neveu-VanMalle et al. 2016). However, based on the best ephemeris for WASP-47 c it is not expected to transit during the TESS monitoring; the next conjunction of WASP-47 c is predicted to occur just under 60 days after the end of the TESS observations. We searched the light curve for evidence of any previously unknown transiting planets. We mask out the transits of WASP-47 b, d, and e, and search for additional transit signals using Box Least Squares (Kovács et al. 2002) but we do not find any evidence for a previously unknown transiting planet. Therefore, we consider just WASP-47 b, d, and e in this analysis.

Table 1. Key Stellar Properties for WASP-47

Property	Value	Source
TIC	102264230	TICv8
R.A. (deg)	331.20309414396	Gaia EDR3
decl. (deg)	−12.01907283535	Gaia EDR3
μR.A. (mas)	15.074 ± 0.020	Gaia EDR3
μdecl. (mas)	−41.467 ± 0.020	Gaia EDR3
Parallax (mas)	3.7010 ± 0.0201	Gaia EDR3
V (mag)	11.936 ± 0.046	TICv8
B (mag)	12.736 ± 0.024	TICv8
TESS (mag)	11.29 ± 0.0061	TICv8
Gaia G (mag)	11.7817 ± 0.0028	Gaia EDR3
Gaia bp (mag)	12.1716 ± 0.0029	Gaia EDR3
Gaia rp (mag)	11.2294 ± 0.0038	Gaia EDR3
J (mag)	10.613 ± 0.022	2MASS
H (mag)	10.310 ± 0.022	2MASS
K (mag)	10.192 ± 0.026	2MASS
M* (M⊙)	1.040 ± 0.031	V2017
R* (R⊙)	1.137 ± 0.013	V2017
ρ* (g cm−3)	0.998 ± 0.014	Section 3.1
Teff (K)	5552 ± 75	V2017
[Fe/H]	0.38 ± 0.05	V2017
$\mathrm{log}g$ (cgs)	4.3437 ± 0.0063	V2017

Note. TICv8 - Stassun et al. (2019); Gaia EDR3 - Gaia Collaboration et al. (2021); 2MASS - Skrutskie et al. (2006); V2017 - Vanderburg et al. (2017).

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We model the transit light curves of the three planets using batman (Kreidberg 2015) with the following free parameters: the times of transit center, T_C , the orbital periods of the planets, P, the planet-to-star radius ratios, R_P /R_*, the orbital inclinations, i, and the stellar density, ρ_*, from which we can compute the scaled axes, a/R_*. For T_C and P we use wide uniform priors centered on the values derived by Becker et al. (2015). For R_P /R_* and i we use uniform priors between 0 and 1 and between 0° and 90°, respectively. For ρ_* we use another wide uniform prior between 0.9 and 1.1, based on the prior knowledge of the stellar parameters from Vanderburg et al. (2017). We use a quadratic limb-darkening law with two independent sets of coefficients for the K2 and TESS data. We sampled for these coefficients using the parameterization of Kipping (2013). For the analysis of WASP-47 b and WASP-47 e we adopted circular orbits based on the findings of Vanderburg et al. (2017) that the tidal circularization timescales for these two planets are significantly shorter than the age of the system. We allow for an orbit for WASP-47 d and impose a half-Gaussian prior with a width of 0.014 and centered on 0 for e_d . This constraint on the eccentricity of WASP-47 d comes from the dynamical analyses performed independently by Becker et al. (2015) and Weiss et al. (2017). This eccentricity is taken into account when calculating the value of a/R_* from ρ_* for the orbit of WASP-47 d. We explore the parameters using a Monte Carlo Markov Chain analysis using the emcee Ensemble Sampler (Foreman-Mackey et al. 2013). A total of 48 walkers were run for a burn-in of 3000 steps followed by a further 10,000 steps per chain to sample the posterior distribution. We calculated the autocorrelation lengths, τ, for each parameter and find τ ≪ N/100, where N is the chain length, indicating good convergence for all parameters except ω_d , which has τ = N/26.5. This is unsurprising, as due to the eccentricity of WASP-47 d being low and consistent with 0 at 2σ, from the transit light curves alone we cannot place strong constraints on ω_d and so do not expect excellent convergence. The transit models recovered from this analysis are plotted in Figure 3. The planetary radii we derive are reported in Table 2. From this analysis, we obtain a stellar density of ρ_* = 0.998 ±0.014 g cm⁻³, which we note is consistent with the current best estimates for the stellar mass and radius (Vanderburg et al.2017).

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Table 2. Planetary Parameters Derived in this Work in Sections 3.1 and 3.6

Parameter	Symbol	Unit	Prior	Value
Planetary parameters
WASP-47 e
Time of conjunction	TC	BJD (TDB)	${ \mathcal N }\left(2457011.34863,0.00030\right)$	2457011.34862 ± 0.00030
Orbital period	P	days	${{ \mathcal N }}_{{ \mathcal U }}\left(0.789595,0.000005,0.,\infty \right)$	0.7895933 ± 0.0000044
RV semiamplitude	Ke	m s−1	${{ \mathcal N }}_{{ \mathcal U }}\left(5.,10.,0.,\infty \right)$	4.55 ± 0.37
Radius ratio	Re /R*		${ \mathcal U }\left(0.,1.0\right)$	0.01458 ± 0.00013
Planet mass a	Me	M⊕		6.77 ± 0.57
Planet radius a	Re	R⊕		1.808 ± 0.026
Planet density a	ρe	g cm−3		6.29 ± 0.60
WASP-47 b
Time of conjunction	TC	BJD (TDB)	${ \mathcal N }\left(2457007.932103,0.000019\right)$	2457007.932103 ± 0.000019
Orbital period	P	days	${{ \mathcal N }}_{{ \mathcal U }}\left(4.1591492,0.0000006,0.,\infty \right)$	4.1591492 ± 0.0000006
RV semiamplitude	Kb	m s−1	${{ \mathcal N }}_{{ \mathcal U }}\left(150.,50.,0.,\infty \right)$	140.84 ± 0.40
Radius ratio	Rb /R*		${ \mathcal U }\left(0.,1.0\right)$	0.10191 ± 0.00022
Planet mass a	Mb	M⊕		363.6 ± 7.3
Planet radius a	Rb	R⊕		12.64 ± 0.15
Planet density a	ρb	g cm−3		0.989 ± 0.040
WASP-47 d
Time of conjunction	TC	BJD (TDB)	${ \mathcal N }\left(2457006.36955,0.00035\right)$	2457006.36955 ± 0.00035
Orbital period	P	days	${{ \mathcal N }}_{{ \mathcal U }}\left(9.03055,0.0002,0.,\infty \right)$	9.03055 ± 0.00008
RV semiamplitude	Kd	m s−1	${{ \mathcal N }}_{{ \mathcal U }}\left(5.,10.,0.,\infty \right)$	4.26 ± 0.37
Radius ratio	Rd /R*		${ \mathcal U }\left(0.,1.0\right)$	0.02876 ± 0.00017
Planet mass a	Md	M⊕		14.2 ± 1.3
Planet radius a	Rd	R⊕		3.567 ± 0.045
Planet density a	ρd	g cm−3		1.72 ± 0.17
Orbital eccentricity	ed		${{ \mathcal N }}_{{ \mathcal U }}\left(0.,0.014,0.,1.\right)$	$0.010{\,}_{-0.007}^{+0.011}$
Argument of periastron	ωd	deg	${ \mathcal U }\left(-180.,180.\right)$	$16.5{\,}_{-98.6}^{+84.2}$
WASP-47 c
Time of conjunction	TC	BJD (TDB)	${ \mathcal N }\left(2457763.4,20.\right)$	2457763.1 ± 4.3
Orbital period	P	days	${{ \mathcal N }}_{{ \mathcal U }}\left(590,15.,0.,\infty \right)$	588.8 ± 2.0
RV semiamplitude	Kc	m s−1	${{ \mathcal N }}_{{ \mathcal U }}\left(30.,15.,0.,\infty \right)$	31.04 ± 0.40
Planet minimum mass a	${M}_{c}\,\sin \,i$	M⊕		398.9 ± 9.1
Orbital eccentricity	ec		${ \mathcal U }\left(0.,1.\right)$	0.295 ± 0.016
Argument of periastron	ωc	deg	${ \mathcal U }\left(-180.,180.\right)$	112.0 ± 4.3

Note.

^a Parameter derived from fitted parameters.

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Compared to the parameters from just the K2 photometry alone, the inclusion of the TESS photometry into the analysis does not provide additional information on the radii of the planets. This is a result of both the reduced time coverage of the TESS photometry compared to the K2 (13.59 days versus 67 days) and the significantly reduced photometric precision (3040 ppm-per-minute versus 350 ppm-per-minute). In particular, WASP-47 d transited just once during the available TESS photometry (see Figure 2), and so the recovery of the transit in the TESS data is marginal.

TESS data have provided a useful method for refining the ephemerides for known transiting hot Jupiters by increasing the baseline of transit observations (Shan et al. 2021). We are able to take advantage of this method by increasing the baseline of observations to refine the orbital periods of the WASP-47 planets. For the hot Jupiter WASP-47 b our results yield an orbital period of 4.1591492 ± 0.0000006 days, which is significantly (4.15σ) longer than the current literature value (Becker et al. 2015; see Figure 4). We also reduce the uncertainty on this period by a factor of seven. WASP-47 b displays transit timing variations with an amplitude on the order of 1 minute (Becker et al. 2015); however, these cannot account for the offset in transit times observed, which has a magnitude of 17.4 minutes. For WASP-47 d we find an orbital period of 9.03055 ± 0.00008 days, which is slightly shorter than the Becker et al. (2015) period, but only differs by just over 1σ. We have reduced the uncertainty on this period by a factor of 2.4. For the super-Earth WASP-47 e our measured orbital period of 0.789595 ± 0.000005 days agrees with the Becker et al. (2015) results, but again we significantly improve the precision on this measurement by a factor of 3.6.

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We modeled our ESPRESSO radial-velocity data along with the archival data from HARPS-N, HIRES, and CORALIE using the exoplanet Python package (Foreman-Mackey et al.2020). The exoplanet package allows for robust probabilistic modeling of astronomical time-series data using PyMC3. We used exoplanet to model the multiple radial-velocity data sets using the No U-Turns Sampler Hamiltonian Monte Carlo method.

The free system parameters included in the analysis were the orbital periods of the planets, P_i , the times of conjunction, T_C,i , and the radial-velocity semiamplitudes, K_i , where i represents the planets b, c, d, and e. For P_i and T_C,i we used Gaussian priors taken from the posteriors of the transit analysis. We also fitted for the eccentricities and arguments of periastron of planets c and d, e_c , e_d , ω_c , ω_d . As with the transit analysis, we fix e_b = e_e = 0, and for e_d we imposed a half-Gaussian prior with a width of 0.014 and centered on 0. For e_c we use a uniform prior constraining the eccentricity to be between 0 and 1. For ω_c and ω_d we used uniform priors between −180° and +180°. We also include a white noise jitter term, σ, and a systemic radial velocity, γ, for each instrument. For CORALIE, we use independent jitter and systemic velocity terms for the data taken before and after the upgrade in November 2014.

The archival PFS data (Dai et al. 2015) has been excluded from prior analyses on WASP-47 due to the presence of large scatter in the radial velocities and the high risk of contamination from systematic errors (Vanderburg et al. 2017). From our initial modeling, we also find a large jitter term is required for the PFS data. Motivated by this, we investigated the effect had by excluding the PFS data from our analysis. We find that the derived parameter values, specifically K_i , are unaffected by including the PFS data, and that the maximum log-likelihood value obtained during the sampling increases when the PFS data are not included in the analysis. Therefore, we also exclude the PFS data from our analysis.

From this initial modeling, we found a significant jitter term was required for our ESPRESSO data. The value required was σ_ESP = 3.25 m s⁻¹, compared to the median photon-limited uncertainty of 0.55 m s⁻¹ for the ESPRESSO radial velocities. Motivated by this we searched for evidence of periodicity in the residuals to the initial model. This excess scatter in the ESPRESSO radial velocities suggests the presence of additional noise that we have not accounted for in our model. This additional noise is likely to arise from the stellar activity of WASP-47. We studied the available data for WASP-47 to investigate any evidence for stellar variability.

Vanderburg et al. (2017) used the HARPS-N spectra to derive a maximum stellar rotational velocity of 2 km s⁻¹ based on the line broadening of the stellar absorption lines. This implies a minimum rotation period of ${P}_{\mathrm{Rot};\min }$ = 28.762 days. Independently, Sanchis-Ojeda et al. (2015) derived a value of $v\sin i={1.8}_{-0.16}^{+0.24}$ km s⁻¹ from their Rossiter–McLaughlin analysis of WASP-47 b. From this measurement we estimate a rotation period of WASP-47 of P_Rot = 31.96 ± 4.36 days. These P_Rot estimates are assuming WASP-47 is aligned along our line of sight. This is a reasonable assumption given the Rossiter–McLaughlin measurement of a spin-aligned orbit for WASP-47 b (Sanchis-Ojeda et al. 2015). These P_Rot limits and estimates will be important for properly interpreting our rotation analyses.

3.3.1. Photometric Rotation Analysis

We used the K2 data of WASP-47 to search for photometric signs of stellar activity. Stellar spots are cooler and less bright than the majority of the stellar surface. As the star rotates, these spots rotate into and out of view of the telescope, resulting in a sinusoidal-like variation in the photometric data at the rotation period of the star, or some harmonic of this period. During the production of the K2 short-cadence light curve, long-term photometric variations were fitted and removed using a spline (Becker et al. 2015). It is these exact variations that we want to study here. Therefore, we use a different long-cadence K2 light curve that had been reduced using the "self-flat-fielding" method (Vanderburg & Johnson 2014). This reduction technique significantly improves upon the precision of the raw K2 photometry and in most cases gets to a precision within a factor of two of the photometry delivered by Kepler during the nominal mission (Vanderburg & Johnson 2014). More importantly, this method preserves long-term astrophysical variations in the photometry.

We used the orbital parameters from Becker et al. (2015) to remove any data points taken during a transit of any of the three WASP-47 transiting planets. We also excluded the data points during two sharp ramps. These two ramps are related to the settling of the roll of the spacecraft at the start of the campaign and after the change in the roll direction around 50 days into the campaign. We fit quadratic polynomials to the two remaining continuous data chunks in order to remove long-term systematic trends in the data believed to be spacecraft systematics rather than astrophysical in nature. The detrended K2 photometry is shown in Figure 5.

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We ran a Lomb–Scargle analysis on the remaining out-of-transit photometry. The resultant periodogram is shown in the bottom panel of Figure 5. The K2 photometry displays some sign of periodicity on periods which are similar to those present in the ESPRESSO data. However, due to the 67 day length of the K2 photometry and the detrending methods applied, this period search becomes less sensitive for longer-period signals, especially for signals around half the monitoring length and longer. The significant peak in the K2 periodogram is at a period of 16.26 ± 1.94 days. This period is shorter than the ${P}_{\mathrm{Rot};\min }$ limit set by Vanderburg et al. (2017). Periodic signals are expected in photometric time series at the second harmonic of the rotation period (Clarke 2003), particularly in cases where there are multiple active regions of the surface of the star where the largest peak in the periodogram can be at half the true rotation period (e.g., McQuillan et al. 2013). Therefore, this 16.26 ± 1.94 day signal likely corresponds to half the true rotation period of WASP-47, giving a rotation period of P_Rot = 32.5 ± 3.9 days, which is consistent with the prediction from the Sanchis-Ojeda et al. (2015) $v\sin i$ measurement.

We also turn to theoretical predictions for stellar rotation periods to confirm that this value is a physically reasonable rotation period for WASP-47. We use the model from Barnes (2007), which estimates P_Rot given the stellar magnitudes in the B and V passbands and the age of the star. For WASP-47, we have magnitudes V = 11.936 mag and B = 12.736 mag, and by assuming a solar age of 4.5 Gyr we find a predicted rotation period P_{Rot ;pred} = 34.96 days.

We note that Hellier et al. (2012) searched the WASP photometry of WASP-47 for signs of stellar rotation and did not find any significant rotational modulation, placing an upper limit of 0.7 mmag for the amplitude of any such modulation. From the K2 photometry in Figure 5, the peak-to-peak amplitude of the modulation is approximately 0.1 mmag, therefore the nondetection of rotational modulation in the WASP photometry is not inconsistent with the K2 detection. Similarly, due to the short time coverage and low precision, the TESS data are unable to help constrain the stellar rotation.

3.3.2. Spectroscopic Rotation Analysis

We search for periodic signals in our spectroscopic ESPRESSO data that might provide evidence that the excess scatter seen in the ESPRESSO radial velocities arises as a result of stellar activity. We ran a generalized Lomb–Scargle analysis on the residuals to the model from the fitting in Section 3.2, which revealed periodicities at periods of roughly 35 and 18 days, as shown in Figure 6. We also searched for periodic signals in the ESPRESSO CCF FWHM measurements, as the FWHM of the CCF has been shown to act as a stellar activity indicator (Boisse et al. 2011; Oshagh et al. 2017). Stellar spots suppress the flux contributions from different sections of the stellar surface to the stellar lines and thus and modify the CCF profile. Spots on the limb of the star suppress the wings of the line, resulting in the CCF profile showing a smaller FWHM. Conversely, spots on the center of the stellar disk affect the center of the stellar lines, resulting in a wider CCF FWHM (Boisse et al. 2011). Periodic signals with the same periods as those found in the radial-velocity residuals were found in the CCF FWHM measurements (see Figure 6). Boisse et al. (2011) also find that the stellar activity causes the CCF contrast, the fractional height of the CCF peak, to anticorrelate with the FWHM. We compare the contrast and FWHM for the ESPRESSO CCFs and find a strong anticorrelation (see Figure 7). This anticorrelation is in line with the predictions and strengthens the confidence that the variations in the CCF FWHM, and by extension the ESPRESSO radial-velocity residuals, are a result of stellar activity.

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We also used the Ca ii H and K lines in the ESPRESSO spectra to determine values of the activity index $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$. Running a further Lomb–Scargle analysis on this activity indicator reveals a periodic signal in the range 30–40 days with a peak around 33 days (Figure 6).

We expect the spectroscopic signals from stellar activity to manifest at periods equal to P_Rot and at the P_Rot/2 and P_Rot/3 harmonics (Boisse et al. 2011). From the ESPRESSO radial-velocity periodogram (top panel of Figure 6), we see a signal close to P = 35 days, with harmonics close to P/2 and P/3. These peaks are also seen in the CCF FWHM periodogram, although an extra peak is seen close to 27 days that is likely due to the Moon. The $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ activity index from the ESPRESSO spectra has a peak in the periodogram at approximately 33 days.

The periodicities detected in the ESPRESSO radial-velocity residuals and activity indicators are consistent with the 32.5 ± 3.9 day rotation period derived from the K2 photometry for WASP-47. Based on the continuous coverage and very high precision of the K2 data, we take the K2 P_Rot value as the most probable rotation period.

We utilize a Gaussian process (GP) kernel with a periodicity close to P_Rot = 32.5 days in order to accurately model the radial-velocity variability due to spot rotation. We use the celerite2 rotation term kernel constructed from two simple harmonic oscillator (SHO) terms at P_Rot and P_Rot/2, implemented using the exoplanet (Foreman-Mackey et al. 2020) and celerite2 (Foreman-Mackey et al. 2017; Foreman-Mackey 2018) Python packages. A single SHO term is given by

$$\begin{eqnarray}\begin{array}{rcl}\kappa (\tau ) & = & {\sigma }_{{GP}}^{2}\exp \left(-\displaystyle \frac{\pi \tau }{{{QP}}_{\mathrm{Rot}}}\right)\left[\cos \left(2\pi \eta \displaystyle \frac{\tau }{{P}_{\mathrm{Rot}}}\right)\right.\\ & & +\left.\displaystyle \frac{1}{2\eta \,Q}\sin \left(2\pi \eta \displaystyle \frac{\tau }{{P}_{\mathrm{Rot}}}\right)\right],\end{array}\end{eqnarray} \tag{ 1 }$$

where $\eta =| 1-{(4{Q}^{2})}^{-1}{| }^{1/2}$. The rotation term kernel takes as its hyperparameters the standard deviation of the process, σ_GP , which is related to the amplitude of the variability signal; the stellar rotation period, P_Rot; the signal quality, Q; the difference in signal quality between the P_Rot and P_Rot/2 modes, ΔQ, such that ${Q}_{{P}_{\mathrm{Rot}}}={Q}_{{P}_{\mathrm{Rot}}/2}+{\rm{\Delta }}Q$; and the mix factor between the two modes, f, such that ${\sigma }_{{P}_{\mathrm{Rot}}}^{2}=f{\sigma }_{{P}_{\mathrm{Rot}}/2}^{2}$. Such a kernel has been designed to be a good model for variability due to stellar rotation and has been used successfully in previous exoplanet high-precision radial-velocity analyses (e.g., Osborn et al.2021).

During the analysis, we used the same kernel to fit the radial-velocity measurements and the CCF FWHM activity indicators of the ESPRESSO observations. By modeling the variations in CCF FWHM simultaneously with the radial velocities, we get a better estimate of the impact of the stellar variability on the radial-velocity measurements. This allows us to limit any impact on the measured planetary radial-velocity signals due to overfitting of the GP, and so extract better-quality mass measurements. A similar method was used by Osborn et al. (2021) to account for the stellar variability of TOI-755.

For this method, during the sampling we use the same P_Rot, Q, ΔQ, and f hyperparameters for all data sets and a different σ_GP and mean for each activity or radial-velocity time series. We use a wide Gaussian prior centered on 35 days for P_Rot taken from our Lomb–Scargle analysis. The Q hyperparameter is related to the damping timescale of the SHO modes, which in a physical sense is related to the decay timescale of the stellar spots. As such, we also implement a prior enforcing Q > π, which in turn ensures the spot decay timescale is longer than P_Rot. This requirement comes from our prior knowledge of the lifetimes of active stellar regions (Donahue et al. 1997). Avoiding low values of Q also helps prevent the GP overfitting the data (Kosiarek & Crossfield 2020).

To assess the statistical justification of using a GP to account for stellar activity in the radial-velocity data sets, we perform a simple model-comparison analysis. We calculate and compare the Bayesian information criterion (BIC; Schwarz 1978; Neath & Cavanaugh 2012) for both sets of models, one with the GP and one without. This statistic assesses whether or not the model fit to the data is sufficiently improved to justify the increased complexity of the new model. When comparing models, the model which produces the lower BIC value is preferred.

We calculate independent BIC values for the ESPRESSO, HARPS-N, and HIRES data sets. We calculate the change in the BIC, Δ BIC, between the models with and without the GP included. We find values for the HARPS-N and HIRES data sets of Δ BIC_HARPS−N = 4.88 and ${\rm{\Delta }}\,{\mathrm{BIC}}_{\mathrm{HIRES}}=8.76$. This indicates that the inclusion of GPs for these two data sets is not justified. For the ESPRESSO radial velocities, we calculate a value of Δ BIC_ESPRESSO = −16.86. This is strong statistical evidence for the inclusion of a GP to model the stellar activity in the ESPRESSO radial velocities. It is not unexpected that the use of a GP is justified for ESPRESSO and not HARPS-N and HIRES. The larger telescope diameter of the VLT results in the photon-limited uncertainties of the ESPRESSO radial velocities (0.55 m s⁻¹) being a factor of six smaller than those for HARPS-N (3.2 m s⁻¹) and a factor four smaller than HIRES (2.0 m s⁻¹). This increased precision of the data allows for the robust detection of the stellar activity signal. We do not use GPs for the CORALIE data as the uncertainty is 12.5 m s⁻¹, which is too large to detect the stellar activity signal seen in the ESPRESSO data.

We modeled the ESPRESSO, HARPS-N, HIRES, and CORALIE data using our final model built from Keplerian orbits for the four planets along with a GP to account for the stellar variability signals present in the ESPRESSO data set. The GP was simultaneously fit to the ESPRESSO radial velocities and CCF FWHM measurements, in order to limit the effect of overfitting the GP to the radial velocities. We perform the sampling using the method detailed in Section 3.2. We ran 40 chains for 10,000 steps each, following a burn-in of 4000 steps for each chain. We calculate the Gelman–Rubin statistic (; Gelman & Rubin 1992) for all the chains, and find that all chains have ≲ 1.001, indicating good convergence.

From this analysis we find a radial-velocity semiamplitude for WASP-47 e of K_e = 4.55 ± 0.37 m s⁻¹. This value is consistent with the semiamplitude derived from the initial modeling in Section 3.2 of 4.73 ± 0.39 m s⁻¹, but represents an improvement in the precision of the measurement. This corresponds to an improvement in the mass measurement precision from M_P = 7.03 ± 0.59 M_⊕ for the initial model to M_P = 6.77 ± 0.57 M_⊕ for the final model. We also note that the value of P_b derived from this analysis is consistent with the value derived from the transit analysis in Section 3.1. The stellar rotation period of WASP-47 derived from this analysis is P_Rot = $39.4{\,}_{-4.5}^{+2.2}$ days, which again is consistent with the rotation period derived from the K2 photometry.

The ESPRESSO data are plotted in Figure 8 and the radial-velocity phase folds for all planets and instruments are plotted in Figure 9. The parameters derived are given in Tables 2 and 3.

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Table 3. Additional Parameters from the Sampling in Section 3.6

Parameter	Unit	Prior	Value
Instrumental Parameters
γESP	m s−1	${ \mathcal U }\left(-27350.,-26850.\right)$	−27165.95 ± 1.27
γHARPS	m s−1	${ \mathcal U }\left(-27350.,-26850.\right)$	−27041.03 ± 0.55
${\gamma }_{\mathrm{HIRES}}$	m s−1	${ \mathcal U }\left(-250.,250.\right)$	6.50 ± 0.75
γCOR; A a	m s−1	${ \mathcal U }\left(-27500.,-26500.\right)$	−27081.33 ± 1.89
γCOR; B a	m s−1	${ \mathcal U }\left(-27500.,-26500.\right)$	−27067.57 ± 5.96
γESP FWHM	m s−1	${ \mathcal U }\left(7600.,7700.\right)$	7635.6 ± 1.0
${\mathrm{log}}_{10}{\sigma }_{\mathrm{ESP}}$	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-5.,1.\right)$	0.145 ± 0.214
${\mathrm{log}}_{10}{\sigma }_{\mathrm{HARPS}}$	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-5.,1.\right)$	−2.33 ± 1.82
${\mathrm{log}}_{10}{\sigma }_{\mathrm{HIRES}}$	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-5.,1.\right)$	0.488 ± 0.069
${\mathrm{log}}_{10}{\sigma }_{\mathrm{COR};\ {\rm{A}}}$ a	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-7.,2.\right)$	−2.38 ± 3.03
${\mathrm{log}}_{10}{\sigma }_{\mathrm{COR};\ {\rm{B}}}$ a	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-7.,2.\right)$	−2.95 ± 2.75
${\mathrm{log}}_{10}{\sigma }_{\mathrm{ESP}\ \mathrm{FWHM}}$	$\mathrm{log}\,$ m s−1	${ \mathcal U }\left(-7.,1.\right)$	−2.79 ± 0.13
GP Hyperparameters
PRot	days	${{ \mathcal N }}_{{ \mathcal U }}\left(35,10,28,\infty \right)$	$39.4{\,}_{-4.5}^{+2.2}$
$\mathrm{ln}\,Q$		${ \mathcal U }\left(1.15,15\right)$	$2.6{\,}_{-1.1}^{+5.2}$
$\mathrm{ln}\,{\rm{\Delta }}\,Q$		${ \mathcal U }\left(-7,10\right)$	$-0.40{\,}_{-5.04}^{+6.18}$
f		${ \mathcal U }\left(0,1\right)$	$0.52{\,}_{-0.31}^{+0.32}$
σGP, ESP		${ \mathcal U }\left(0,20\right)$	$4.58{\,}_{-2.44}^{+3.83}$
σGP, ESP FWHM		${ \mathcal U }\left(0,100\right)$	$7.04{\,}_{-2.16}^{+4.18}$

Note.

^a COR;A and COR;B refer to the sets of CORALIE data taken before and after the spectrograph was updated in 2014 November.

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With the inclusion of our ESPRESSO data we have obtained a mass of WASP-47 e of 6.77 ± 0.57 M_⊕. This value is consistent at the 1σ level with the mass determined by Vanderburg et al. (2017), but we have improved on the precision they quote on the mass by 15%. With the improved precision of the ESPRESSO data we have uncovered a clear stellar activity signal, which is consistent with the stellar variability seen in the K2 photometry. This allows us to model the stellar activity using a physically motivated GP informed by our knowledge of the stellar rotation.

Our results also allow us to improve the constraints on the bulk density of WASP-47 e. We derive a density ρ_e = 6.29 ± 0.60 g cm⁻³, making WASP-47 e the super-Earth with the second most precisely constrained density to date, behind only 55-Cancri e(Bourrier et al. 2018). Our ESPRESSO data also improves the constraint on M_d . We find a mass 14.2 ± 1.3 M_⊕, which is both consistent with the mass measured by Vanderburg et al. (2017) and a 13% improvement on the precision of the mass measurement. For the hot Jupiter WASP-47 b we improve the precision of the radial-velocity semiamplitude, yet the precision on the mass remains unchanged from the Vanderburg et al. (2017) measurements. From this, we conclude that the primary limiting factor for improving the constraints on M_b arise from the constraints on the stellar mass, which is currently measured to 3% precision, compared to the 0.3% precision to which we have measured K_b . We do not achieve significant improvement on the measurement of the minimum mass of WASP-47 c. This is unsurprising, as our ESPRESSO radial-velocity measurements only cover a small fraction of the orbital period of of WASP-47 c (P_c = 588.4 days).

WASP-47 b and WASP-47 d have already been shown to exhibit significant transit timing variations (TTVs; e.g., Becker et al. 2015; Weiss et al. 2017). Due to the low signal-to-noise of the TESS transits of these planets, we are unable to strongly constrain the transit center times for the individual TESS transits of WASP-47 d or WASP-47 e. However, we can measure the transit times for the hot Jupiter WASP-47 b.

We refit each transit of WASP-47 b in the TESS data in order to measure the individual T_C values, which are reported in Table 4. During this analysis, we fix the transit shape to the model derived in Section 3.6. The same model is used to subtract the transits of WASP-47 d and WASP-47 e from the light curve before fitting. We also perform the same analysis for the WASP-47 b transits in the short-cadence K2 data. The transit times measured are plotted in Figure 10 and provided in Table 4. The uncertainty on an individual T_C from the TESS data is on the order of 1 minute, which is roughly twice the TTV amplitude seen in the K2 data. The TESS transit times display a scatter of a similar magnitude. As such, the T_C measurements are consistent with the TTV predictions from the K2 results (see Figure 10). However, the detection is marginal, and we cannot know the phase of the signal due to the significant time gap between K2 and TESS data.

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