11institutetext:  1 Aix Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric
Joliot-Curie, 13388 Marseille, France
2 Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51,
Versoix, Switzerland
3 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, UK
4 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4
7AL, UK
5 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
6 Université Nice Sophia Antipolis, Observatoire de la Côte d’Azur,
Département Cassiopée CNRS/UMR 6202, BP 4229, 06304, Nice, France
7 Space sciences, Technologies and Astrophysics Research (STAR) Institute,
Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium
8 Centre for Exoplanet Science, SUPA School of Physics and Astronomy,
University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
9 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, United Kingdom
10 Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland
11 INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania,
Italy
12 INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122
Padova, Italy
13 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,
A-8042 Graz, Austria
14 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The
Netherlands
15 Department of Space, Earth and Environment, Chalmers University of
Technology, Onsala Space Observatory, 439 92 Onsala, Sweden
16 Instituto de Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP,
Rua das Estrelas, 4150-762 Porto, Portugal
17 Instituto de Astrofisica de Canarias, 38200 La Laguna, Tenerife, Spain
18 Departamento de Astrofisica, Universidad de La Laguna, 38206 La Laguna,
Tenerife, Spain
19 Institut de Ciencies de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n,
08193 Bellaterra, Spain
20 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
21 Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary
22 Depto. de Astrofisica, Centro de Astrobiologia (CSIC-INTA), ESAC campus,
28692 Villanueva de la Cañada (Madrid), Spain
23 Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do
Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal
24 Center for Space and Habitability, University of Bern, Gesellschaftsstrasse
6, 3012 Bern, Switzerland
25 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
26 Department of Astronomy, Stockholm University, AlbaNova University Center,
10691 Stockholm, Sweden
27 Center for Space and Habitability, Gesellsschaftstrasse 6, 3012 Bern,
Switzerland
28 Institute of Planetary Research, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany
29 Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005
Paris, France
30 ESTEC, European Space Agency, 2201AZ, Noordwijk, NL
31 Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund,
Sweden
32 Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C,
B-4000 Liège, Belgium
33 Dipartimento di Fisica, Universita degli Studi di Torino, via Pietro Giuria
1, I-10125, Torino, Italy
34 University of Vienna, Department of Astrophysics, Türkenschanzstrasse 17,
1180 Vienna, Austria
35 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry
CV4 7AL, United Kingdom
36 Science and Operations Department - Science Division (SCI-SC), Directorate
of Science, European Space Agency (ESA), European Space Research and
Technology Centre (ESTEC), Keplerlaan 1, 2201-AZ Noordwijk, The Netherlands
37 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121
Budapest, Konkoly Thege Miklós út 15-17, Hungary
38 ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány
1/A, 1117
39 IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77
av. Denfert-Rochereau, 75014 Paris, France
40 Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie
Curie, 98bis blvd. Arago, 75014 Paris, France
41 Department of Astrophysics, University of Vienna, Tuerkenschanzstrasse 17,
1180 Vienna, Austria
42 Institute of Optical Sensor Systems, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany
43 Dipartimento di Fisica e Astronomia ”Galileo Galilei”, Universita degli
Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy
44 ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2, CH-8093
Zurich, Switzerland
45 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK
46 Zentrum für Astronomie und Astrophysik, Technische Universität Berlin,
Hardenbergstr. 36, D-10623 Berlin, Germany
47 Institut für Geologische Wissenschaften, Freie Universität Berlin, 12249
Berlin, Germany
48 ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, 9700
Szombathely, Szent Imre h. u. 112, Hungary
49 MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent Imre h. u. 112,
Hungary
50 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, United Kingdom

# Connecting photometric and spectroscopic granulation signals with CHEOPS and
ESPRESSO

S. Sulis1 M. Lendl2 H. M. Cegla3,4, UKRI Future Leaders Fellow L. F. Rodríguez
Díaz5 L. Bigot6 V. Van Grootel7 A. Bekkelien2 A. Collier Cameron8 P. F. L.
Maxted9 A. E. Simon10 C. Lovis2 G. Scandariato11 G. Bruno11 D. Nardiello12 A.
Bonfanti13 M. Fridlund14,15 C. M. Persson15 S. Salmon2 S. G. Sousa16 T. G.
Wilson8 A. Krenn2,13 S. Hoyer1 A. Santerne1 D. Ehrenreich2 Y. Alibert10 R.
Alonso17,18 G. Anglada19,20 T. Bárczy21 D. Barrado y Navascues22 S. C. C.
Barros16,23 W. Baumjohann13 M. Beck2 T. Beck10 W. Benz10,24 N. Billot2 X.
Bonfils25 L. Borsato12 A. Brandeker26 C. Broeg10,27 J. Cabrera28 S. Charnoz29
C. Corral van Damme30 Sz. Csizmadia28 M. B. Davies31 M. Deleuil1 A. Deline2 L.
Delrez32,7 O. D. S. Demangeon16,23 B.-O. Demory24 A. Erikson28 A. Fortier10,24
L. Fossati13 D. Gandolfi33 M. Gillon32 M. Güdel34 K. Heng24,35 K. G. Isaak36
L. L. Kiss37,38 J. Laskar39 A. Lecavelier des Etangs40 D. Magrin12 M. Munari11
V. Nascimbeni12 G. Olofsson26 R. Ottensamer41 I. Pagano11 E. Pallé17 G.
Peter42 G. Piotto12,43 D. Pollacco35 D. Queloz44,45 R. Ragazzoni12,43 N.
Rando30 H. Rauer28,46,47 I. Ribas19,20 M. Rieder10 N. C. Santos16,23 D.
Ségransan2 A. M. S. Smith28 M. Steinberger13 M. Steller13 Gy. M. Szabó48,49 N.
Thomas10 S. Udry2 N. A. Walton50 D. Wolter28

(accepted to A&A)

###### Abstract

Context. Stellar granulation generates fluctuations in photometric and
spectroscopic data whose properties depend on the stellar type, composition,
and evolutionary state. Characterizing granulation is key for understanding
stellar atmospheres and detecting planets.

Aims. We aim to detect the signatures of stellar granulation, link
spectroscopic and photometric signatures of convection for main-sequence
stars, and test predictions from 3D hydrodynamic models.

Methods. For the first time, we observed two bright stars
($T_{\mathrm{eff}}=5833$ K and $6205$ K) with high-precision observations
taken simultaneously with CHEOPS and ESPRESSO. We analyzed the properties of
the stellar granulation signal in each individual dataset. We compared them to
Kepler observations and 3D hydrodynamic models. While isolating the
granulation-induced changes by attenuating and filtering the p-mode
oscillation signals, we studied the relationship between photometric and
spectroscopic observables.

Results. The signature of stellar granulation is detected and precisely
characterized for the hotter F star in the CHEOPS and ESPRESSO observations.
For the cooler G star, we obtain a clear detection in the CHEOPS dataset only.
The TESS observations are blind to this stellar signal. Based on CHEOPS
observations, we show that the inferred properties of stellar granulation are
in agreement with both Kepler observations and hydrodynamic models. Comparing
their periodograms, we observe a strong link between spectroscopic and
photometric observables. Correlations of this stellar signal in the time
domain (flux versus radial velocities, RV) and with specific spectroscopic
observables (shape of the cross-correlation functions) are however difficult
to isolate due to S/N dependent variations.

Conclusions.  In the context of the upcoming PLATO mission and the extreme
precision RV surveys, a thorough understanding of the properties of the
stellar granulation signal is needed. The CHEOPS and ESPRESSO observations
pave the way for detailed analyses of this stellar process.

###### Key Words.:

¡ Techniques: radial velocities and photometric - Sun: granulation - stars:
atmospheres - Methods: data analysis¿

##  1 Introduction

Stellar convection transports energy from the stellar interior to its surface
in late-type stars. The properties of this complex and multiscale plasma
mixing process are key for understanding stellar structure and evolution, as
the dynamics of the convective cells shape angular momentum transport within
the star, impact the thermal stellar stratification, mix the chemical
elements, and generate the surface acoustics modes (see e.g., Dravins &
Nordlund 1990a; Stein & Nordlund 2001; Alvan et al. 2014; Houdek & Dupret
2015; Salaris & Cassisi 2017; Brun & Browning 2017; Philidet et al. 2020).

This stellar phenomenon is well-studied for the Sun where it is visible in the
form of granules. For other stars, granulation is studied through indirect
techniques. Two of them are photometry – brightness fluctuations – and
spectroscopy – radial velocity (RV) changes.

Through photometric observations, the properties of granulation as a function
of stellar parameters have been revealed: CoRoT observations have shown that
the granulation timescale and amplitudes decrease with the increasing
characteristic frequency of the acoustic modes, the so-called frequency at
maximum power $\nu_{max}$, which depends on the stellar surface gravity and
temperature (Michel et al. 2008). In line with model predictions (see e.g.,
Svensson & Ludwig 2005; Ludwig 2006; Ludwig & Steffen 2013; Trampedach et al.
2013; Tremblay et al. 2013; Beeck et al. 2013; Samadi et al. 2013; Beeck et
al. 2015), numerous studies based on Kepler observations (Mathur et al. 2011;
Bastien et al. 2013; Kallinger et al. 2014; Bastien et al. 2016; Pande et al.
2018; Bugnet et al. 2018; Tayar et al. 2019; Sulis et al. 2020a; Rodríguez
Díaz et al. 2022) have also shown that the granulation properties are
dependent on the stellar fundamental parameters ($T\rm_{eff}$, ${\rm log}g$,
and [Fe/H]). Granule size increases with lower stellar surface gravities
and/or larger effective temperatures, and therefore with a decreasing
$\nu_{max}$. The stellar photometric signal is the net contribution of all the
bright granules and dark intergranular lanes on the stellar surface, which
reduces the disk-integrated fluctuations (compared to the scale of the
granules) and therefore depends on the stellar radius (Trampedach et al. 1998;
Ludwig 2006).

In spectroscopic observations, granulation produces shifts and asymmetries in
the spectral lines (i.e., the line bisectors, see Dravins et al. 1981;
Nordlund 1985; Dravins 1987; Asplund et al. 2000; Asplund et al. 2000; Gray
2009; Nordlund et al. 2009; Dravins et al. 2021). Studying RV time series of a
small sample of main-sequence and subgiant stars, Bastien et al. (2014) found
a correlation between the RV root-mean-square (RMS) evolving on timescales
shorter than $8$ hours (or the “8-hour RV scatter”) and the stellar surface
gravity. This implies a narrow relation between the RV amplitudes driven by
stellar granulation signals and the associated intensities (Bastien et al.
2013; Kallinger et al. 2016). As a result, various empirical relations that
predict the amplitude of the granulation signal in RV from photometric
observations have been derived (see e.g., Bastien et al. 2014; Cegla et al.
2014; Oshagh et al. 2017), but they have shown significant discrepancies in
the predicted RV.

Three-dimensional (3D) hydrodynamical (HD) and magneto-hydrodynamical (MHD)
simulations of stellar convection have been developed since the 1980s
(Nordlund 1982; Nordlund 1985; Nordlund & Dravins 1990; Dravins & Nordlund
1990b). While computationally expensive (leading to long time series difficult
to generate), they have been recently used to generate disk-integrated spectra
(Cegla et al. 2019) and RV time series (Sulis et al. 2020b) of solar
granulation. They have also been used to generate synthetic brightness
fluctuations whose standard deviation and autocorrelation time match those of
Kepler targets from dwarfs to red giants with an overall very good agreement
(Rodríguez Díaz et al. 2022). Moreover, Cegla et al. (2019) predicted
correlations between the photometric and spectroscopic signals of stellar
granulation, but this has yet to be confirmed with high-precision
observations. They also predict that such observed correlations may allow us
to mitigate a significant fraction of the granulation variability in RV
observations (Cegla et al. 2019).

The variability induced by granulation constitutes a significant noise source
hampering the detection of smaller stellar signals, like low-amplitude
acoustic or gravity modes (see e.g., García & Ballot 2019 for solar-like
oscillations, Rodríguez et al. 2016 for M-dwarfs pulsations, and Appourchaux
et al. 2010 for g-modes in the Sun) and planetary signals (Dumusque et al.
2011; Meunier et al. 2015; Meunier & Lagrange 2020). From solar observations,
we learn that granulation can generate variability with RMS around $40$ parts-
per-million (ppm) in photometry (Dravins 1988, Fröhlich et al. 1997; Aigrain
et al. 2004; Sulis et al. 2020a) and around $30$-$46$ cm/s in RVs (Elsworth et
al. 1994; Pallé et al. 1999; Garcia et al. 2005; Appourchaux et al. 2018;
Collier Cameron et al. 2019; Dumusque et al. 2021). This is significant
compared to the signals expected from Earth-like planets around Sun-like
stars: a transit depth of $84$ ppm and a Keplerian RV signal of $9$ cm/s
amplitude (Perryman 2018). Since the correlation timescales of the
granulation-induced noise are similar to the ingress/egress transit duration
of long orbital period exoplanets, this stellar signal affects the planetary
parameters inferred on the individual transits (Sulis et al. 2020a). Thus, the
correlated noise due to stellar granulation can be seen as a source of
information to study stellar physics properties but at the same time as a
nuisance signal for the detection of exoplanets and stellar oscillation modes.
In both cases it needs to be understood, quantified and if possible mitigated.

In this paper, we have three objectives. First, we aim to detect the
signatures of stellar granulation based on the granulation indicators that
have been recently developed, to test their predictions against high-precision
CHEOPS measurements, and to compare the performances of CHEOPS for probing the
photometric signature of stellar granulation to Kepler and TESS data. Second,
we aim to study the link between the spectroscopic and photometric signatures
of convection for main-sequence stars with high-precision ESPRESSO
observations, taken contemporaneously with CHEOPS. Third, we aim to test the
predictions from 3D hydrodynamical models of convection.
For these purposes, we present the analyses of high-precision CHEOPS and
ESPRESSO measurements of two bright stars with different effective
temperatures.

The paper is organized as follows. We describe the data reduction
methodologies in Sec. 2. We refine the stellar parameters of these two targets
in Sec. 3. We analyze the stellar granulation signals for each CHEOPS, TESS
and ESPRESSO observations in Sec. 4. We study the links between granulation
and stellar properties in Sec. 5. We compare our results with the predictions
from 3D hydrodynamical convection models in Sec. 6. We constrain the link
between spectroscopic and photometric signatures in Sec. 7. We investigate the
relationship between the shape of the cross-correlation functions, flux (as
measured by CHEOPS), and RV in Sec. 8. We conclude in Sec. 9.

##  2 Observations and data reduction

We selected HD 67458 ($T_{\mathrm{eff}}=5833$ K) and HD 88595
($T_{\mathrm{eff}}=6205$ K) as good targets based on their stellar parameters,
their low apparent magnitudes, their moderate level of magnetic activity, the
a priori absence of identified planets, and the excellent CHEOPS and ESPRESSO
visibility windows during the time that was allocated to our program. Our
dataset includes $35$ CHEOPS orbits, six dedicated nights at ESPRESSO/VLT, and
five TESS sectors of observations. We note this is the first time that high-
precision spectroscopic data with ESPRESSO have been taken during several full
nights for single stars.

Target | Visit | File key | Starting date | $T$ | $\tau$ | $N$ | DC | $\sigma_{TOT}$
---|---|---|---|---|---|---|---|---
| ref. number |  | [UTC] | [hrs] | [s] |  | [$\%$] | [ppm]
| Visit 1 | CH_PR100022_TG001101_V0200 | 2021-01-17 | 8.15 | 34.3 | 789 | $92.5$ | 114
HD 67458 | Visit 2 | CH_PR100022_TG001301_V0200 | 2021-01-18 | 8.61 | 34.3 | 834 | $92.5$ | 110
| Visit 3 | CH_PR100022_TG001201_V0200 | 2021-01-21 | 8.15 | 34.3 | 807 | $94.6$ | 125
| Visit 1 | CH_PR100022_TG001601_V0200 | 2021-02-13 | 8.06 | 37 | 728 | 93.5 | 100
HD 88595 | Visit 2 | CH_PR100022_TG001501_V0200 | 2021-02-14 | 8.15 | 37 | 698 | 88.6 | 104
| Visit 3 | CH_PR100022_TG001401_V0200 | 2021-02-15 | 8.14 | 37 | 718 | 91.2 | 108
| Visit 4 | CH_PR100022_TG001701_V0200 | 2021-02-22 | 8.14 | 37 | 738 | 93.8 | 103
Table 1: CHEOPS observations: file keys referring to the files name in the CHEOPS database, dates (UTC), total observation duration ($T$), integration time ($\tau$), number of observations ($N$), duty cycle (DC), and standard deviation of the light curve ($\sigma_{TOT}$). Target | Night | Starting date | $T$ | Min/Max | $N$ | Min/Max | DC | $\sigma_{TOT}$ | ${\rm log}R^{\prime}_{\rm HK}$
---|---|---|---|---|---|---|---|---|---
| ref. number | [UTC] | [hrs] | airmass |  |  $\sigma_{RV}$ [m/s] | [$\%$] | [m/s] |
| Night 1 | 2021-01-17 | $4.85$ | [1, 2.0] | $155$ | [$0.4,1.1$] | $91.8$ | $2.6$ | -4.97
HD 67458 | Night 2 | 2021-01-18 | $8.48$ | [1, 1.8] | $273$ | [$0.5,2.7$] | $87.5$ | $2.0$ | -5.20
| Night 3 | 2021-01-21 | $4.75$ | [1, 1.7] | $126$ | [$0.4,0.9$] | $72.0$ | $1.1$ | -4.96
HD 88595 | Night 1 | 2021-02-14 | $8.8$ | [1, 2.3] | $308$ | [$0.7,2.3$] | $95.0$ | $2.54$ | -4.96
| Night 2 | 2021-02-22 | $9.0$ | [1, 1.9] | $313$ | [$0.7,3.7$] | $94.0$ | $2.26$ | -4.97
Table 2: ESPRESSO observations: date of observation (UTC), total observation
duration ($T$), Min and Max values of airmass during the data acquisition,
number of observations ($N$), Min and Max RV errorbars ($\sigma_{RV}$), duty
cycle (DC), standard deviation of the detrended RV time series
($\sigma_{TOT}$), and median value of the ${\rm log}R^{\prime}_{\rm HK}$
activity indicator.

###  2.1 CHEOPS

CHEOPS (Benz et al. 2021) observed the two bright stars in the visible
wavelength range ($330-1100$ nm). HD 67458 was observed during three visits of
$T\sim 8.15$ hours each, with a time sampling of $\Delta t=34.3$ seconds. Each
of these measurements resulted from $7$ individual images taken with an
exposure time of $\tau_{exp}=4.9$ seconds and stacked on-board by coadding
them pixel-by-pixel. HD 88595 was observed during four visits of $T\sim 8.15$
hours each, with a time sampling of $\Delta t=37$ seconds. Each of these
measurements resulted from $10$ stacked images taken with an exposure time of
$\tau_{exp}=3.7$ seconds.

The duty cycle of each visits was close to $90\%$, with some gaps present
within the light curves. The origin of these gaps is due to Earth occultations
and the South Atlantic Anomaly (SAA) crossing of the satellite. Each series
has one or two large gaps $<17.5$ min and five to ten short gaps $<2$ min. In
total, the three visits of HD 67458 contain $N=\\{789,807,834\\}$
measurements, while the four visits of HD 88595 contain
$N=\\{718,698,728,738\\}$ measurements.

All CHEOPS observations were processed with the Data Reduction Pipeline (DRP,
version 13.1.0), which is described in Hoyer et al. (2020). We used the light
curves provided by the DRP. We test different apertures to minimize the
contribution from the background contaminants, and finally choose the default
aperture of $25$ pixels radius.

We detrended each CHEOPS visit from the satellite systematics using the
Pycheops software111https://github.com/pmaxted/pycheops (version 1.0.6, Maxted
et al. 2022). We first corrected for contaminating flux from background
sources, with the contamination estimation in Pycheops based on the Gaia DR2
catalog. We then performed a $5\sigma$ clipping to each visit, before
detrending from the x and y centroid variations, roll angle, background, and
smear systematics. The main characteristics of these observations are listed
in Table 1, and the detrended light curves are shown in the top rows of Fig.
1.

Table 3: TESS observations: date of observation (UTC), total observation duration ($T$), integration time ($\tau$), number of observations ($N$), duty cycle (DC), and standard deviation of the light curve ($\sigma_{TOT}$). Target | Sector | Starting date | $T$ | $\tau$ | $N$ | DC | $\sigma_{TOT}$
---|---|---|---|---|---|---|---
| number | [UTC] | [days] | [s] |  | [$\%$] | [ppm]
| 7 | 2019-01-08 | 24.4 | 120 | 16283 | 92.5 | 134
HD 67458 | 8 | 2019-02-02 | 24.6 | 120 | 13369 | 75.5 | 275
| 34 | 2021-01-14 | 25.0 | 120 | 16802 | 93.2 | 92
HD 88595 | 9 | 2019-02-28 | 24.0 | 120 | 15164 | 87.6 | 175
| 35 | 2021-02-09 | 23.8 | 120 | 12866 | 74.9 | 269

Figure 1: Detrended CHEOPS light curves (top row), ESPRESSO radial velocity
time series (middle row), and TESS light curves (bottom row) for HD 67458
(left panels) and HD 88595 (right panels). In each panel, the colors indicate
a different set of observations (CHEOPS visits, ESPRESSO nights, TESS
sectors). Observations are y-shifted for visibility, and the dates of each
time series are normalized to start at $0$ (with a display that goes from the
earliest date on the top, to the latest one on the bottom).

###  2.2 ESPRESSO

ESPRESSO (Pepe et al. 2013) observed the two targets in high-resolution
spectroscopy ($R\sim 140000$) in the wavelength range $\lambda\in[380,780]$
nm, during six full nights222Program IDs: 106.2186.001 to 106.2186.006.. Each
measurement resulted from an exposure time of $\tau_{exp}=60$ seconds. The
ESPRESSO observations have been taken simultaneously or contemporaneously with
the CHEOPS observations (see Fig. 2). They are the first ESPRESSO observations
taken during full nights for such a program.

Among these six nights of observations at the VLT, one333Program ID
106.2186.005 (UTC time: 2021/02/15). encountered significant technical
problems (PLC-ADC communication issues). We remove this RV time series from
this study and are left with three RV time series for the solar-like star HD
67458, and two for the hotter star HD 88595.

All ESPRESSO data were processed with the Data Reduction Software (DRS)
provided with the instrument and publicly available from the ESO pipeline
repository444www.eso.org/sci/software/pipelines/. We refer to Pepe et al.
(2021) for a short description of the main processing steps. Radial velocities
were obtained by cross-correlation with binary F9 and F7 masks for HD 67458
and HD 88595, respectively. We note that we had to process the HD 88595 data
enforcing an F7 spectral type to override the (wrong) F5 spectral type
provided in the OCS.OBJ.SP.TYPE FITS keyword. Given the exquisite short-term
RV precision required to characterize stellar granulation, we paid extra
attention to the instrumental drift measured on fiber B and the blue-to-red
flux balance in the CCF computation. Both effects may introduce instrumental
RV systematics within the night if not properly calibrated out.

From the processed RV time series of the solar-like star HD 67458, we removed
the mean value and we clipped out the $2\sigma$ outliers for all nights. The
standard deviation of the three final RV time series are $2.6$, $2.0$ and
$1.1$ m/s, respectively. The errorbars are between $0.4$ and $2.7$ m/s. We
find that the RV dispersion changed significantly between the different
nights, but also during a given night. In particular, during the first night
we found an RMS of $\sim 3.4$ m/s during the first two hours of acquisition
and $\sim 2.0$ m/s during the rest of the night. As we detail in Sec. 4.1,
this discrepancies are certainly of instrumental origin.

From the processed RV time series of the F star HD 88595, we also removed the
mean value and we clipped out the $5\sigma$ outliers for both nights. The
standard deviations of the two final RV time series are $2.54$ and $2.26$ m/s
respectively. The errorbars are between $0.7$ and $3.7$ m/s.
The characteristics of these observations are listed in Table 2, and the RV
time series are shown in the middle rows of Fig. 1. For both targets, we also
report in Table 2 the median value of the chromospheric activity indicator
${\rm log}R^{\prime}_{\rm HK}$ (Noyes et al. 1984), which is based on the
intensity of the Ca II H&K reemission lines. For both stars, we found ${\rm
log}R^{\prime}_{\rm HK}\sim-4.9$, which is consistent with relatively inactive
stars (Hall 2008). Moreover, as no significant variation of this activity
indicator is observed during the nights, the stellar magnetic activity is
probably not at the origin of the variability observed in the data of HD
67458.

###  2.3 TESS

TESS (Ricker et al. 2015) observed the two stars in the red-optical bandpass
($600-1100$ nm). HD 67458 was observed in sectors {7, 8, 34}, and HD 88595 in
sectors {9, 35}.

In this work we made use of the short-cadence light curves released by the
TESS team. However, we do not use the Presearch Data Conditioning Simple
Aperture Photometry (PDCSAP) flux because sometimes they are affected by some
systematic errors due to over-corrections and/or injection of spurious
signals. The light curves we used were obtained applying Cotrending Basis
Vectors (CBVs) to the Simple Aperture Photometry (SAP) flux as done in
Nardiello et al. (2021). The CBV were obtained by using the SAP light curves
of the stars in the same Camera/CCD in which the targets are located and
following the procedure described in detail in Nardiello et al. (2019, 2020).
For the analysis of the light curves, we rejected the points with DQUALITY>0,
as recommended by the TESS Science Data Products Description
Document555https://archive.stsci.edu/files/live/sites/mast/files/home/missions-
and-data/active-missions/tess/_documents/EXP-TESS-ARC-ICD-TM-0014-Rev-F.pdf.
Additionally, we clipped out the points corresponding to a sky background
value $>3040$ e-/s in sector 35 (since this sector was affected by known
technical issues such as thermal stability and telescope pointing). We finally
clipped out the $3\sigma$ outliers of each sectors for both targets. The main
characteristics of these observations are listed in Table 3, and the
lightcurves are shown in the bottom rows of Fig. 1.

For comparison with the CHEOPS observations in the following, we extracted
subseries of $8$-hours duration from each TESS sector. Then, we removed the
subseries affected by large gaps and kept the subseries for which the duty
cycle was $>90\%$. We obtained $L=172$ subseries for HD 67458, and $L=113$ for
HD 88595.

Figure 2: Contemporaneous observations of HD 67458 (two first rows) and HD
88595 (two last rows) taken with CHEOPS and ESPRESSO (same color code as in
Fig. 1: black, gray, red and blue dots). For each star, there are two
simultaneous sets of CHEOPS/ESPRESSO observations shown with the dashed and
dotted vertical lines.

###  2.4 The Sun as a reference star

As a reference star, we refer throughout the paper to the Sun that has been
observed continuously from space with the Solar and Heliospheric Observatory
(SoHO) since 1996. On board SoHO, the VIRGO instrument measures the solar
spectral irradiance with the three channels sun photometer (SPM) at wavelength
of 402 (blue), 500 (green) and 862 (red) nm (Fröhlich et al. 1995; Fröhlich et
al. 1997; Jiménez et al. 2002). In parallel, the GOLF spectrophotometer
observes the solar disk-integrated position of the Sodium doublet lines at
$\lambda=5895.9$ and $5889.9\leavevmode\nobreak\ \AA$, from which it extracts
the projected radial velocities (Boumier & Dame 1993; Gabriel et al. 1995;
Garcia et al. 2005; Appourchaux et al. 2018). Together, VIRGO and GOLF
observations form a unique set of high-precision solar observations with an
excellent duty cycle of almost $96\%$ over the past $26$ years. In the
following, we use both datasets sampled at one point per minute, taken out of
the initial higher cadence sequences.

For GOLF observations, we used the level-2 GOLF
data666www.ias.u-psud.fr/golf/templates/access.html calibrated as described in
Appourchaux et al. (2018). To be conservative, we only selected the year 1996
from this dataset since, after that, the detector was affected by an
instrumental failure. This year corresponds to a solar cycle minima, leading
then to a negligible impact of the magnetic regions (which is out-of-scope of
the present study). We divided this time series into subseries of $8$-hours
duration to compare them with our ESPRESSO set of observations. From the total
number of available $8$-hours subseries, we selected $75$ subseries with a
condition of regular sampling (i.e., we only took the subseries that avoid
gaps).

For VIRGO observations, we also selected the observations taken in 1996 to be
consistent with the selected GOLF data. We corrected them from the
instrumental degradation over time following the recipe described in Sec. 2 of
Sulis et al. (2020a). We then divided them into $8$-hours subseries to mimic
the duration of CHEOPS observations, and we picked $265$ of these subseries
with a condition of regular sampling. Throughout the paper, we use these VIRGO
and GOLF subseries as references for comparing the properties of solar
granulation with our two main-sequence targets.

In addition, we also compare our set of observations in Sec. 4.1 with ground-
based solar observations taken by the HARPS-N spectrograph, for which the
first three years of observations (2015-2018) have been made recently
available777https://dace.unige.ch/sun/ (Collier Cameron et al. 2019, Dumusque
et al. 2021). We extracted $298$ daily subseries from this released dataset.
The HARPS-N subseries have a median duration $T\sim 6.75$ hours and a sampling
rate $\Delta t\sim 5.4$ min. In Appendix C, we compare these different sets of
solar observations.

Table 4: Properties of the stars HD 67458 and HD 88595.

Parameters | Values | Source
---|---|---
| HD 67458 | HD 88595 |
Target names | HIP 39710 | HIP 50013 | Simbada
| TIC 154234114 | TIC 5506063 |
| Gaia EDR3 5596797039059063680 | Gaia EDR3 5669349344592951936 |
Spectral type | G0V | F7V | Simbad
Right Ascension (ep=J2000) | 08 07 00.522 | 10 12 37.96 | Simbad
Declination (ep=J2000) | -29 24 10.57 | -19 09 10.94 | Simbad
Gaia G-band magnitude | 6.64 | 6.34 | Gaia archiveb
Distance [pc] | 25.68 | 42.01 | This workc (IRFM)
Effective Temperature (K) | $5833\pm 62$ | $6205\pm 35$ | This work (spectroscopy)
Metallicity [Fe/H] | $-0.18\pm 0.04$ | $0.17\pm 0.03$ | This work (spectroscopy)
$\rm[Mg/H]$ | $-0.14\pm 0.06$ | $0.10\pm 0.07$ | This work (spectroscopy)
$\rm[Si/H]$ | $-0.20\pm 0.04$ | $0.12\pm 0.05$ | This work (spectroscopy)
$\rm[Ti/H]$ | $-0.17\pm 0.06$ | $-0.01\pm 0.05$ | This work (spectroscopy)
${\rm log}g$ [cgs] | $4.37\pm 0.10$ | $3.99\pm 0.06$ |  This work (spectroscopy)
| 4.390 $\pm$ 0.026 | 4.148 $\pm$ 0.017 | This work (from $R_{*}$ and $M_{*}$)
Radius ($R_{\odot}$) | $1.021\pm 0.020$ | $1.616\pm 0.017$ | This work (IRFM)
Mass ($M_{\odot}$) | $0.935^{+0.042}_{-0.043}$ | $1.351^{+0.031}_{-0.058}$ | This work (isochrones)
Age [Gyr] | $7.7\pm 2.1$ | $2.8\pm 0.4$ | This work (isochrones)
$\nu_{\rm max}$ [$\mu$Hz] | $2756\pm 169$ | $1534\pm 85$ | Eq. (2), This work (ESPRESSO)
Mean ${\rm log}R^{\prime}_{\rm HK}$ | $-4.99\pm 0.14$ | $-4.96\pm 0.15$ | This work (spectroscopy)
Rotation period [days] | $10.57\pm 0.06$ | $3.1151\pm 0.0003$ | This work (TESS photometry)
$v\sin{i}$ [km/s] | $2.179\pm 0.2$ | $7.24\pm 0.35$ | HARPSd, This work
Stellar inclination [degrees] | $\sim 26$ | $\sim 16$ | This work

Notes. aSIMBAD astronomical database from the Centre de Données astronomiques
de Strasbourg (http://simbad.u-strasbg.fr/simbad/). bArchive of the Gaia
mission of the European Space Agency (https://gea.esac.esa.int/archive/). c
Values are in agreement with the Gaia Early Data Release 3 (catalog
http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=I/352 from Bailer-Jones et
al. 2021). d HARPS data initially published in Soto & Jenkins (2018) and
reanalyzed in this work.

##  3 Derivation of stellar parameters

First concerning the stellar atmospheric parameters, we coadded the individual
exposures taken with ESPRESSO (see Sec. 2.2) after correcting for each
individual radial velocity shift. This was done for each target in order to
create a combined spectra with higher signal-to-noise ratio. We used each
ESPRESSO master spectrum to derive the stellar spectroscopic parameters
($T_{\mathrm{eff}}$, ${\rm\log}g$, micro-turbulence, [Fe/H]) and the
respective uncertainties following the ARES+MOOG methodology as described in
Sousa et al. (2021); Sousa (2014); Santos et al. (2013). The ARES
code888https://github.com/sousasag/ARES (Sousa et al. 2007; Sousa et al. 2015)
was used to measure in a consistent way the equivalent widths of iron lines
included in the line list presented in Sousa et al. (2008). Briefly, ARES+MOOG
performs a minimization process looking for the ionization and excitation
equilibrium to find convergence on the best set of spectroscopic parameters.
For the computation of the iron abundances we make use of a grid of Kurucz
model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (v2019)
(Sneden 1973). In addition we have also used the IDL package Spectroscopy Made
Easy (SME) to do the spectral analysis for these stars (Valenti & Piskunov
1996; Piskunov & Valenti 2017). This code utilizes an input of stellar
parameters to perform radiative transit calculations in order to synthesize
models, that through an iterative minimizing procedure with the observed
spectrum as a template, arrive at a set of final stellar parameters. In this
process one varies one parameter while keeping the other fixed and works with
several different atmospheric models and atomic and molecular line lists from
VALD (Piskunov et al. 1995). In this case we utilized again a grid of Kurucz
model atmospheres (Kurucz 1993).

For HD 67458 both spectral analyses provided completely consistent parameters
and we selected the values given by ARES+MOOG. For HD88595 we rely on the
parameters derived by SME, mostly because of the higher $v\sin{i}$ for this
star which degrade a bit the precise measurements of the equivalent widths in
the ARES+MOOG method. The adopted spectroscopic parameters are listed in Table
4.

We determined the stellar radii of HD 67458 and HD 88595 using a modified IRFM
method in a Markov-Chain Monte Carlo (MCMC) approach (IRFM; Blackwell &
Shallis 1977; Schanche et al. 2020). This was done by computing the bolometric
fluxes for the targets by fitting stellar atmospheric models to broadband
photometry that are converted to effective temperatures and angular diameters
using the physical relationships between these parameters. Utilizing the
target’s parallaxes, we subsequently determined the radii from the angular
diameters. For HD 67458 and HD 88595, we use Gaia, 2MASS, and WISE broadband
photometry (Skrutskie et al. 2006; Wright et al. 2010; Gaia Collaboration et
al. 2021) with the ATLAS catalog of stellar atmospheric models (Castelli &
Kurucz 2003), and the offset-corrected Gaia EDR3 parallax (Lindegren et al.
2021) and find $R_{\star}=1.021\pm 0.020\leavevmode\nobreak\ R_{\odot}$ and
$R_{\star}=1.616\pm 0.017\leavevmode\nobreak\ R_{\odot}$, respectively. These
radii are reported in Table 4.

Adopting $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ as basic input set, we
then derived the isochronal mass $M_{\star}$ and age $t_{\star}$ of each star
from two different stellar evolutionary models. In detail, we used the
isochrone placement algorithm (Bonfanti et al. 2015; Bonfanti et al. 2016),
which interpolates the input parameters within precomputed grids of
PARSEC999PAdova & TRieste Stellar Evolutionary Code:
http://stev.oapd.inaf.it/cgi-bin/cmd (Marigo et al. 2017) isochrones and
tracks, to retrieve a first pair of mass and age values. To improve the
convergence, we also accounted for the stellar $v\sin{i}$, coupling the
isochronal interpolation scheme with gyrochronology as outlined in Bonfanti et
al. (2016). The second pair of mass and age values, instead, was computed
through CLES (Code Liégeois d’Évolution Stellaire; Scuflaire et al. 2008),
which builds the best-fit stellar track according to the input parameters
following the Levenberg-Marquadt minimization scheme as explained in Salmon et
al. (2021). Finally, for each star and for each output parameter, we merged
the two respective distributions derived from PARSEC and CLES, after checking
their mutual consistency through a $\chi^{2}$-based criterion (see Bonfanti et
al. 2021 for further details). We obtained
$M_{\star}=0.935_{-0.043}^{+0.042}\,M_{\odot}$ (resp.
$M_{\star}=1.351_{-0.058}^{+0.031}\,M_{\odot}$) and $t_{\star}=7.7\pm 2.1$ Gyr
(resp. $t_{\star}=2.8\pm 0.4$ Gyr) for HD 67458 (resp. HD 88595). The computed
stellar parameters are listed in Table 4.

The uncertainties associated with each of the stellar parameters are those
obtained by the above procedure, which is the one applied to all CHEOPS
exoplanet host targets (see again Bonfanti et al. 2021, for full details).
This procedure is performed here on two bright main-sequence stars with high
quality spectra, detailed abundances, and numerous and accurate broadband
photometric measurements. Concerning the stellar surface gravity, we want to
recall that its spectroscopic determination is prone to several problems that
directly affect the accuracy of the derived value. Problems such as the
assumption of plane parallel stellar atmosphere models, or the use of the
ionization balance where few optimal lines of ionized iron is present, can
affect strongly the accuracy of the spectroscopic analysis for the ${\rm
log}g$. Fortunately these do not strongly affect the determination of other
atmospheric parameters. For example, it was demonstrated that when using
equivalent-width methods the other atmospheric parameters are mostly
independent from the surface gravity (e.g., Torres et al. 2012). In our
procedure to derive the stellar radius, mass, and age, the spectroscopic
surface gravity is only marginally used as a prior for the radius
determination (where it is allowed to vary within the MCMC), but is not used
anymore in deriving mass and age. Given these considerations, for HD 88595 the
spectroscopic ${\rm log}g$ is reasonably consistent with the one computed from
the derived mass and radius values (see Table 4), which in turn very well
agree with the asteroseismic $\nu_{\rm max}$ measured for this star (see Sec.
4.3). Concerning the solar-like star HD 67458, the spectroscopic ${\rm log}g$
and the one derived from mass and radius are in perfect agreement (see Table
4).

Finally, analyzing the TESS observations based on the generalized Lomb-Scargle
periodograms (Zechmeister & Kürster 2009), we were able to constrain the
rotation period of the F star ($P_{\rm rot}\simeq 3.1151\pm 0.0003$ days) and
the G star ($P_{\rm rot}\simeq 10.57\pm 0.06$ days). We report these rotation
periods in Table 4 and the details in Appendix. A. Using the known $v\sin{i}$
and $R_{\star}$ from Table 4, this gives a stellar inclination of $i\simeq
16^{\circ}$ for the F star, and $i\simeq 26^{\circ}$ for the G star. Both
stars are then seen nearly pole-on.

##  4 Granulation signals in high-precision photometric and spectroscopic
observations

Stellar granulation generates stochastic fluctuations in both photometric and
spectroscopic observations. These fluctuations are correlated over timescales
from some minutes to several hours, depending on the stellar parameters. The
objective of this section is to identify the contribution of the stellar
granulation among the various sources of noise present in the CHEOPS, TESS and
ESPRESSO datasets.

We first look at the behavior of these fluctuations in the time domain, and
how their amplitudes evolve over different timescales (Sec. 4.1). We then
analyze how they behave when using the common observational strategies to
mitigate them (i.e., long exposure time for RV data acquisition or light curve
binning over short timescales), and we discuss these behaviors in the context
of small exoplanets detection (Sec. 4.2). We conclude this section by
analyzing the observations in the frequency domain (periodograms), and in
particular we show how the instrumental noise (dominated by photon noise)
impacts the characterization of the stellar granulation signal (Sec. 4.3). The
main conclusions of this section are summarized in Table. 6.

###  4.1 Amplitude of the granulation signal

Figure 3: Comparison of the stellar signal amplitude at short time scale for
different stars (the Sun, HD 67458, and HD 88595). Left: Comparison of the F8
metric computed on the $8$-hours subseries of VIRGO observations (red, green,
and blue histograms) and CHEOPS observations of HD 67458 and HD 88595 (colored
dashed and dotted vertical lines, respectively). The F8 metric has been
slightly modified from the initial definition of Bastien et al. (2013) to
include fluctuations evolving on timescales from $5$ min to $8$ hours. F8
values computed on each TESS $8$-hour subseries are shown for comparison
(magenta and cyan histograms). Right: Same for spectroscopic solar
observations with GOLF (black histogram) and HARPS-N (red histogram) compared
with ESPRESSO observations of HD 67458 and HD 88595 (colored dashed and dotted
vertical lines, respectively).

We assume that each time series results from the contribution of three
phenomena: instrumental noise, stellar oscillations, and stellar granulation.
We neglect stellar magnetic activity because i) the chromospheric activity
indicator ${\rm log}R^{\prime}_{\rm HK}$ indicates that the two stars are
relatively inactive (${\rm log}R^{\prime}_{\rm HK}\leavevmode\nobreak\ \sim$
-4.9), ii) no correlation between ${\rm log}R^{\prime}_{\rm HK}$ and RV is
observed, and iii) our observations are very close in time so any signature of
magnetic activity (spot or plages) should act as a trend over the nightly
datasets. The instrumental noises are dominated here by the photon noise and
affect differently the CHEOPS, TESS and ESPRESSO observations. The stellar
oscillations (or p-modes) evolve on short timescales for both stars, typically
over $10$ min or less. The stellar granulation, which we seek to identify
here, evolves on longer timescales. We expect the amplitudes of the stochastic
fluctuations generated by the stellar granulation to follow a Gaussian
distribution (see Sec. 2.3 of Sulis et al. 2020a), and to decrease with
increasing stellar surface gravity (Bastien et al. 2013). We therefore expect
the overall amplitude of the granulation signal to be comparable to the solar
values extracted from the visible wavelengths for HD 67458, and to be larger
for HD 88595. In addition, since the granulation signal amplitude decreases at
redder wavelengths (Planck’s law), we expect the amplitude of the granulation
signal to be smaller in the TESS passband than in the CHEOPS one.

We start by extracting the contribution of photon noise, which dominates the
high-frequency part of our time series. To this end, we filter each time
series with a high-passband filter with cut-off frequencies $\nu_{cut}$. We
visually determine these cut-off frequencies $\nu_{cut}$ for each dataset by
examining the flat regions of the different periodograms (see details in Sec.
4.3). We then measure the standard deviation of each filtered time series
($\sigma_{w}$) that corresponds to one visit (CHEOPS), one night (ESPRESSO),
or one sector (TESS). For TESS, we report the median value of the standard
deviations calculated on each $8$-hour subseries contained in a given sector.
For HD 67458, we obtain: $\sigma_{w}=\\{79,81,93\\}$ ppm (CHEOPS),
$\sigma_{w}=\\{97,93,85\\}$ ppm (TESS), and $\sigma_{w}=\\{2.1,1.6,0.95\\}$
m/s (ESPRESSO). For HD 88595, we obtain: $\sigma_{w}=\\{77,75,77,78\\}$ ppm
(CHEOPS), $\sigma_{w}=\\{181,177\\}$ ppm (TESS), and
$\sigma_{w}=\\{1.37,1.49\\}$ m/s (ESPRESSO).
For CHEOPS observations, the observed white Gaussian noise (WGN) amplitudes
are in remarkably good agreement with the predictions from the CHEOPS Exposure
Time Calculator101010 https://cheops.unige.ch/pht2/exposure-time-calculator.
Indeed, including contributions of instrumental (readout, smearing,
quantization and dark current), background (sky and straylight) and photon
noise (which is by far the dominating factor), the ETC predicts white noise
levels at the cadence of the downloaded images of $85$ ppm and $71$ ppm for HD
67458 and HD 88595, respectively. We observe larger white noise levels for
TESS observations (especially for HD 88595) than for CHEOPS. This is also in
agreement with expectations based on the characteristics of the two satellites
(equivalent collective area, photometric performances at a given stellar
magnitude, see e.g., Futyan et al. 2020). For ESPRESSO observations, the
observed white noise amplitudes do vary significantly between the different
nights of HD 67458. This would mean than the global dispersion $\sigma_{TOT}$
(given in Table 2) is driven by short term instrumental noise. In order to
identify whether these variations were of instrumental origin, we looked at
many indicators such as atmospheric conditions, signal-to-noise ratio,
instrumental drifts, or pipeline quality controls. Unfortunately, we could not
clearly identify a source for this high-frequency variability. However, this
excess white noise with no apparent structure will not affect the conclusions
presented in the rest of this study.

We continue with the extraction of the granulation signal. To this end, we use
a slight modification of the $8$-hour flicker (or F8) metric, defined in
Bastien et al. (2013). We start by binning each time series into $5$-minute
intervals (and not $30$-minutes as originally defined in Bastien et al.
(2013), since we would miss the typical timescales of stellar granulation). We
then use a $8$-hours length boxcar filter111111We use the function
convolution.Box1DKernel available from the Python package www.astropy.org. to
remove the long term stellar activity (we note the small impact of this
additional step since the length of our subseries are short). Results are
shown and compared to the Sun in photometry (right) and spectroscopy (left) in
Fig. 3.

The solar photometric values are extracted from the narrow passbands of the
VIRGO red, green and blue SPM channels. To compare CHEOPS and VIRGO
observations, as done in Basri et al. (2010) and Salabert et al. (2016) for
Kepler, we therefore need to consider a combination of the red ($862$ nm) and
green ($500$ nm) channels. The CHEOPS values for HD 67458 are then expected to
fall in the interval defined by the red and green histograms of Fig. 3. The
TESS passband being redder, the TESS values for HD 67458 are expected to fall
closer to the red histogram. This is however not what we observe, with F8
values of $\\{67,55,58\\}$ ppm for the three CHEOPS visits, and
F8$\in[50,106]$ ppm for the whole set of $8$-hours subseries from TESS
observations. We explain these discrepancies by two reasons. First, the level
of white noise is significantly larger in both CHEOPS and TESS observations
than in solar observations. Then, the discrepancy with solar observations is
larger for TESS observations since the granulation amplitude decrease with
increasing wavelengths (see Fig. 4). In TESS data, the high-frequency noise
masks the granulation signal and does not allow to identify it.

Computing now the F8 metric on the four CHEOPS visits of the F star HD 88595,
we obtain F8$=\\{63,67,72,55\\}$ ppm. As expected from a star with a lower
surface gravity, these values are above the solar ones (at all wavelengths).

On the right panel of Fig. 3, we show the histogram of the F8 metric computed
on each $8$-hour solar GOLF (black) and HARPS-N (red) subseries. We observe F8
values in the interval $[0.46,0.75]$ m/s for GOLF and $[0.25,1.2]$ m/s for
HARPS-N. Although these solar values are in agreement, we are not surprised by
the larger RV dispersion of HARPS-N data because GOLF data are obtained using
one line (Sodium Doublet), that is one height in the atmosphere, whereas
HARPS-N uses a series of lines in the visible range, that is an average of
various contributions at different heights.

Computing the F8 metric over the three ESPRESSO observation series for the G
star, we obtain F8$=\\{2.0,1.2,0.8\\}$ m/s; which are larger but still in
agreement with the right tail of the HARPS-N solar data distribution. We
recall that the first night of observations was affected by large variations
during the night (see Sec. 2.1), certainly of instrumental origin, and should
be considered with caution. On the other hand, we measure stable F8 values for
the two nights of the F star with, as expected, values larger than the solar
ones (F8$=\\{2.07,2.25\\}$ m/s).

Finally, we note that, if we drastically filter out the long periods ($>3$
hours instead of $>8$ hours), the F8 values do not change significantly.
Indeed, we observe a decrease of only $6-10$ ppm for the CHEOPS observations
of HD 88595, which is the star with the fastest rotation rate. This sanity
check supports the assumption that the impact of long-term stellar magnetic
activity is negligible in our analyses.

Figure 4: Decrease in the amplitude of the granulation signal as a function of
wavelength. The granulation amplitude is estimated by the RMS of 265 solar
$1$-day subseries (VIRGO), since the level of white noise is negligible in
solar observations. The different solar values are extracted from Fig. 4 of
Sulis et al. (2020a). On the right y-axis, we show the spectral response
function of the CHEOPS (black) and TESS (gray) telescopes.

###  4.2 Mitigation of the granulation signal

To reduce the contribution of the stellar oscillations and granulation for the
detection and characterization of exoplanets, the common strategy is to
average the observations. In radial velocity, this results in the use of a
longer exposure time than the stellar p-modes timescales (see Chaplin et al.
2019 for a recent optimization of this exposure time), followed by a binning
of data points taken over the course of a night to reduce stellar granulation
signals (Hatzes et al. 2011; Dumusque et al. 2011). In photometry, this
results in binning the data points over some minutes, depending on the
characteristics of the studied planetary transit. Below, we first study how
the high precision ESPRESSO observations behave when using this mitigation
strategy, then we turn to the CHEOPS photometric observations. In both cases,
we compare with the expected signal amplitude of an Earth-like planet that
would orbit in the habitable zone (HZ) of the two stars.

In Fig. 5, we show the decreases of the RV amplitudes as a function of the
binning timescale $\tau$. We compare them with the decrease expected for WGNs
of the same variance, that we take as a reference to determine if the RV RMS
at a given $\tau$ is dominated by white noise or by stellar signals. The
decrease of WGNs behaves as $\sigma_{TOT}\sqrt{N_{\rm bin}/N}$, with $N_{\rm
bin}$ the number of binned data points.

For the solar-like star HD 67458 (top panel), we observe a different behavior
for the three nights: one is consistent with the behavior of a white noise,
while the two others are not. At $\tau=20$ min we get an RV RMS of $\sim 0.4$
m/s for the last night (magenta line) and the WGNs, while we get an RV RMS of
$0.7-0.8$ m/s for the other nights (red and green lines). To determine if
these two behaviors are consistent with solar observations, we compute the RV
RMS as a function of $\tau$ for each of the $75$ GOLF solar $8$-hours
subseries and for each corresponding WGN. We see that solar RV RMS
measurements121212We note that the reported RV RMS are also consistent with
the predictions based on solar-like RV simulations from Meunier et al. (2015),
with values around $30$ to $40$ cm/s for a solar-like star at $\tau=1$ hour.
show a large dispersion but do not match the decrease observed for a WGN. The
RV RMS derived on HD 67458 dataset are statistically compatible with solar
data at large $\tau$. Comparing with the RV semi-amplitude of an Earth-mass
planet in the habitable-zone ($K\sim 9$ cm/s for HD 67458, see Appendix. B),
this demonstrates that this strategy is not robust to mitigate enough the
short-timescale stellar signal. This is in total agreement with Meunier et al.
(2015), who found that we need to bin over $\tau>8$-hours to get down to the
level of an Earth-like RV signature. For the F star HD 88595 (bottom panel),
we observe for both nights a behavior inconsistent with a WGN. At $\tau=20$
min, we read RV RMS values of $1.7$ and $1.1$ m/s for the first and second
nights, while we read an RMS of $0.7$ m/s for the WGN. This demonstrates the
failure of this observational strategy for a slightly evolved star as HD
88595, with an RV dispersion remaining above $50$ cm/s even after a binning of
$\tau=150$ min. For comparison the RV semi-amplitude of an Earth-mass planet
in the habitable-zone of this star would be $K\sim 5.6$ cm/s (see Appendix.
B).

While it is clear that the RV RMS at large $\tau$ is not driven by WGN, it
remains to estimate the contribution of the stellar oscillation modes.
Following Chaplin et al. (2019), we estimate the exposure time needed to
mitigate the contribution of these modes down to the $10$ cm/s level for our
two stars131313https://github.com/grd349/ChaplinFilter. We find
$\tau_{osc}\sim 12$ min and $\sim 85$ min for HD 67458 and HD 88595,
respectively (see dotted vertical lines in Fig. 5). Since Chaplin et al.
(2019)’s methodology does not include the contribution of the stellar
granulation signal (stochastic correlated noise) but is designed for the
p-modes mitigation (which behaves at first approximation as pure sines for
oscillations), we conclude that the remaining signal contribution at
timescales $\tau>\tau_{osc}$ is dominated by stellar granulation.

Figure 5:  Effect of temporal binning on the RV RMS of HD 67458 (top) and HD
88595 (bottom). Each color (red, green and magenta) represents one night of
ESPRESSO observations. The behavior of WGNs having the same variance as the
considered dataset is shown by the colored dotted lines. RMS values obtained
on the $75$ solar GOLF subseries are shown in yellow in the top panel, and the
corresponding WGNs in blue (median values are shown by the yellow and blue
solid lines). The RV semi-amplitude of an Earth-like planet orbiting in the HZ
of each star are shown by the dashed horizontal lines. The exposure times
needed to mitigate the p-modes oscillations down to the $\sim 10$ cm/s level
are shown by the dotted vertical lines.

Figure 6:  Effect of temporal binning on the RMS of the photometric data of HD
67458 (three red solid lines for the three CHEOPS visits) and HD 88595 (four
green solid lines for the four CHEOPS visits). The behavior of WGNs of the
same variance than each CHEOPS visit are shown with the dashed gray lines. RMS
values obtained on the $265$ solar VIRGO subseries are shown with the yellow
beam (median value in solid), and the corresponding behavior of WGNs in blue
(median value in solid). Prediction from the CHEOPS ETC are shown with the
black thin lines with dots. The transit depth of an Earth-size planet orbiting
in the HZ of each star are shown by the dashed horizontal lines. Their typical
transit ingress duration are shown by the vertical dotted and dashed lines.

In Fig. 6, we show the decreases of the CHEOPS photometric amplitudes as a
function of the binning timescale $\tau$. Compared to WGNs of same input
variance, these observations show larger amplitude at $\tau<150$ min. This
indicates that the signal remaining at large $\tau$ comes from stellar
variability.

It can be argued that the instrumental noises contain not only white noise
components but also red noises. To evaluate this, we computed from the CHEOPS
ETC the full expected noise taking into account the impact of both red and
white noise components, and we check how it behaves when we integrate
observations on timescales up to $6$ hours. We note that in addition to the
main contributions already listed in Sec. 4.1, these new estimates include
also variable dark current, instabilities in gain, quantum efficiency and
analog electronics, as well as timing errors, and flat field homogeneities.
Comparing these estimates with CHEOPS observations in Fig. 6, we still observe
discrepancies indicating that the excess signal at $\tau<150$ min is driven by
stellar variability.

Finally, since we expect the p-modes photometric contribution to be smaller
than in spectroscopic observations, we can conclude that the significant
difference between our observations and WGNs is driven by stellar granulation.

This stellar signal may bias the inferred parameters of long-period transiting
exoplanets (from which few transits may be observable, see e.g., Sulis et al.
2020a). Indeed, their amplitudes are comparable to the transit depth of Earth-
size planets, which are $80.6$ ppm for HD 67458 and $32.2$ ppm HD 88595 (see
Appendix. B). Moreover, these signals’ amplitudes remain significant
(brightness RMS $\in[32-76]$ ppm) even after a binning of $\tau\sim 7.5-8.4$
min, which are the typical durations of the transit ingresses of Earth-size
planets in the HZ (see Appendix. B). Accessing how this stellar signal is
correlated is therefore important to infer accurate and precise exoplanet
parameters.

###  4.3 Periodograms of the granulation signal

Figure 7: Periodograms of HD 67458 (left) and HD 88595 (right). From top to bottom: Lomb-Scargle periodograms of CHEOPS, ESPRESSO and TESS observations computed on the whole dataset are shown in black. Their respective averaged periodograms, resulting from the average of the periodograms of each CHEOPS visits, ESPRESSO nights, or TESS $8$-hours subseries, are shown in red. Yellow lines represent the best-fitting Harvey models, given in Eq.(1), that fits the Lomb-Scargle periodograms of each datasets. The different components of this model (white noise, oscillation modes, granulation, supergranulation) are represented in dotted yellow. When relevant, the different cut-off frequencies are represented by vertical lines: $\nu_{cut}$ in green (used to infer the level of white noise), $\nu_{max}$ in blue (frequency only relevant for ESPRESSO data of HD 88595), $f_{H}$ in cyan (frequency generally close to $\nu_{cut}$), and $f_{g}$ in magenta.  Table 5: Best-fitting parameters $\\{a_{1},b_{1},c_{1}\\}$ associated to the stellar granulation component in the Harvey functions given in Eq. (1) and found for CHEOPS, ESPRESSO and TESS datasets of HD 67458 and HD 88595. The best-fitting parameters found for the solar VIRGO data without and with white noise added are shown in the two last rows. Symbol $\emptyset$ highlights the nondetection cases (i.e., parameters $\\{a_{1},b_{1},c_{1}\\}$ consistent with zero within their $1\sigma$ uncertainties). Parameter $a_{1}$ is in units of ppm${}^{2}/\mu$Hz or m2/s${}^{2}/\mu$Hz, and parameter $b_{1}$ in units of $\mu$Hz.  Target | Instrument | $a_{1}$ | $b_{1}$ | $c_{1}$
---|---|---|---|---
HD | CHEOPS | $0.64\pm 0.01$ | $2005\pm 44$ | $3.04\pm 0.19$
67458 | ESPRESSO | $938\pm 15$ | $110\pm 12$ | $3.44\pm 0.02$
| TESS | $\emptyset$ | $\emptyset$ | $\emptyset$
HD | CHEOPS | $1.44\pm 0.02$ | $875\pm 18$ | $2\pm 0.01$
88595 | ESPRESSO | $2000\pm 3$ | $270.0\pm 0.3$ | $2.0\pm 0.01$
| TESS | $\emptyset$ | $\emptyset$ | $\emptyset$
Sun | VIRGO | $0.86\pm 0.08$ | $2065.9\pm 147$ | $3.8\pm 0.66$
| VIRGO + WGN | $0.31\pm 0.05$ | $3405\pm 332$ | $3.3\pm 1.1$

The power spectral density of stellar granulation is well-known to act as a
red-like noise, that is a power increase in a given frequency range. For main-
sequence stars, this frequency range correspond to $\nu>10\leavevmode\nobreak\
\mu$Hz.

In Fig. 7, we show the Lomb-Scargle periodograms (Scargle 1982) of each
CHEOPS, ESPRESSO and TESS datasets. To fit these periodograms, classical
models are Harvey-functions (Harvey 1988). These functions are defined as the
sum of Lorentzian functions parameterized by different timescales and
amplitudes to distinguish the stellar activity components that dominate
different frequency regions in the periodogram (generally attributed from high
to low frequencies to: instrumental noise, stellar oscillation modes,
granulation, supergranulation, and active regions). Some debates about the
exact shape of these Lorentzian functions and the number of free parameters to
use exist (see e.g., Mathur et al. 2011; Kallinger et al. 2014). In this work,
we chose to model each periodogram shown in Fig. 7 with a Harvey-function of
the form (Kallinger et al. 2014):

$P_{H}(\nu_{k}^{+})\\!:=\eta^{2}(\nu_{k}^{+})\left(\sum_{i=1}^{2}\frac{a_{i}}{1+\Big{(}\frac{\nu_{k}^{+}}{b_{i}}\Big{)}^{c_{i}}}\\!\\!+\\!\\!P_{osc}\exp\left({\frac{-(\nu_{k}^{+}-\nu_{max})^{2}}{2\sigma_{osc}^{2}}}\right)\right)+\sigma_{H}^{2},$
(1)

where the set of parameters $\\{a_{i},b_{i},c_{i}\\}_{i=1,2}$ collects the
amplitude, characteristic frequency and power of the Harvey functions for the
stellar granulation signal ($i=1$) and the low frequency region ($i=2$).
Parameters $\\{P_{osc},\nu_{max},\sigma_{osc}\\}$ refer to the oscillation
p-modes signals, which are only clearly detected in the ESPRESSO observations
of HD 88595 (see the middle right panel of Fig. 7). Parameter $\sigma_{H}^{2}$
refers to the variance of the high-frequency noise component, assumed to be a
WGN in model (1). Notation $\nu_{k}^{+}$ means that positive Fourier
frequencies are considered, and $\eta:={\rm
sinc}(\frac{\pi}{2}\frac{\nu_{k}^{+}}{\nu_{Ny}})$ is an attenuation factor
based on the Nyquist frequency (defined for a regular sampling with time step
$\Delta t$ as $\nu_{Ny}=1/2\Delta t$). For each star and each periodogram, we
infer the parameters of model (1) using first a nonlinear least-squares
minimization141414We use the LMFIT Python package, see
https://lmfit.github.io/lmfit-py/ (Newville et al. 2014).. Then, we use the
EMCEE package (Foreman-Mackey et al. 2013) to calculate with MCMC the
probability distribution for each parameters, from which we take the median
values and the $1\sigma$ uncertainties. We note that large uniform priors are
used to fit all the parameters of the model except the power index
$\\{c_{i}\\}$ which we consider $\leq 10$ to avoid being too sensitive to
large local variations between two peaks of the periodogram. Estimated values
that are relevant to the granulation signals $\\{a_{1},b_{1},c_{1}\\}$ are
reported in Table 5. The best-fitting models are shown in Fig. 7 (yellow
lines). From these fits, we conclude that the signature of granulation is
evident from both CHEOPS and ESPRESSO observations of HD 88595. For HD 67458,
only the CHEOPS observations show a clear granulation signal. We note an
increase of the periodogram at frequency $\nu<200\leavevmode\nobreak\ \mu$Hz
for both stars that could be the signature of supergranulation in the ESPRESSO
data, but the length of our observations are too short to conclude on the
nature of this signal. On the contrary, TESS is blind to the granulation
signal due to the high level of WGN and the small amplitude of this signal in
this passband (see Fig. 4).

We also find an oscillation frequency at maximum power $\nu_{\rm max}=1534\pm
85$ $\mu$Hz from ESPRESSO observations of HD 88595, which is consistent with
the prediction from the asteroseismic scaling relation (Kjeldsen & Bedding
1995):

$\nu_{\rm max}=\nu_{\rm
max,\odot}\;\Big{(}\frac{M_{*}}{M_{\odot}}\Big{)}\;\Big{(}\frac{R_{*}}{R_{\odot}}\Big{)}^{-2}\Big{(}\frac{T_{\rm
eff}}{T_{\rm{}_{eff},\odot}}\Big{)}^{-1/2},$ (2)

that gives $\nu_{\rm max}=$ 1542.5$\pm$60 $\mu$Hz, with
${T_{\rm{}_{eff},\odot}}=5777$ K and $\nu_{\rm max,\odot}=3090$ $\mu$Hz (Huber
et al. 2011) and stellar parameters from Table 4.

Table 6: Summary of the data analyses: target name, instrument, reference of
the observations, total dispersion of the time series ($\sigma_{TOT}$),
estimated level of white noise⋆ from the high frequency noise filtering
($\sigma_{w}$, see Sec. 4.3), estimated granulation amplitude from F8 (F8, see
Sec. 4.1), and from predictions from HD simulations (F8HD, see Sec. 6). Units
are in ppm or m/s. The last column indicates if the granulation signal is
detected (yes/no/marginal).

Target | Instrument | Reference | $\sigma_{TOT}$ | $\sigma_{w}$ | F8 | F8HD | Detection ?
---|---|---|---|---|---|---|---
HD 67458 | CHEOPS | visit 1 | 114 | 79 | 67 | 48 | marginal
| CHEOPS | visit 2 | 110 | 81 | 55 | 48 | marginal
| CHEOPS | visit 3 | 125 | 93 | 58 | 48 | marginal
| TESS | sector 7 | 101 - [91, 116] | 97 - [88, 106] | 73 - [54, 106] | $<48$ | no
| TESS | sector 8 | 98 - [93, 213] | 93 - [86, 122] | 72 - [59,80] | $<48$ | no
| TESS | sector 34 | 89 - [81, 101] | 85 - [79, 99] | 61 - [51,76] | $<48$ | no
| ESPRESSO | night 1 | 2.6 | 2.1 | 2.0 | - | no
| ESPRESSO | night 2 | 2.0 | 1.6 | 1.2 | - | no
| ESPRESSO | night 3 | 1.1 | 0.95 | 0.8 | - | no
HD 88595 | CHEOPS | visit 1 | 100 | 77 | 63 | 66 | yes
| CHEOPS | visit 2 | 104 | 75 | 67 | 66 | yes
| CHEOPS | visit 3 | 108 | 77 | 72 | 66 | yes
| CHEOPS | visit 4 | 103 | 78 | 55 | 66 | yes
| TESS | sector 9 | 199 - [166, 258] | 181 - [156, 223] | 87 - [67,128] | $<66$ | no
| TESS | sector 35 | 190 - [163, 293] | 177 - [159, 204] | 98 - [70,149] | $<66$ | no
| ESPRESSO | night 1 | 2.54 | 1.37 | 2.07 | - | yes
| ESPRESSO | night 2 | 2.26 | 1.49 | 2.25 | - | yes

⋆We note that the analyses of TESS data are based on a large number of
$8$-hours subseries, we then indicate the median values, and the min/max
values under intervals.

In each panel of Fig. 7, we have indicated the cut-off frequencies $\nu_{cut}$
that delineate the white noise and granulation regimes (green vertical lines).
In the CHEOPS and ESPRESSO datasets, $\nu_{cut}$ correspond to periods in the
interval $[244,810]$ s. In the TESS dataset, $\nu_{cut}$ corresponds to
periods around $100$ min. We note that, in the dataset of ESPRESSO
observations of HD 88595, the approximation of white noise is only partially
true (the high frequency part of the periodogram is not perfectly flat). These
levels of white noise impact the detection of the granulation signal, with the
periodogram slope in the frequency region of stellar granulation that
decreases with the increase of WGN. Without correction of this WGN, the
inferred Harvey parameters from model (1) may show discrepancies around their
expected values in the Hertzsprung-Russell (HR) diagram. To illustrate this
effect, we compare the periodograms of CHEOPS observations of the solar-like
star HD 67458 and of the solar VIRGO observations with WGN added (because
intrinsically the WGN component is very low in solar data). For this
comparison, we compute the averaged periodogram defined as in Sulis et al.
(2017):

$P_{L}(\nu_{k}):=\frac{1}{L}\sum_{\ell=1}^{L}P^{\ell}(\nu_{k}),$ (3)

with $P^{\ell}$ the periodogram computed on $L=3$ solar $8$-hours subseries
taken randomly in the VIRGO sample. For the case with WGN, we added to each
$L$ solae subseries a WGN with standard deviation $\sigma_{w}$. The value of
$\sigma_{w}$ has been scaled until the high-frequency region of the VIRGO
periodograms matches the one of CHEOPS observations. This corresponds to
$\sigma_{w}\approx 30$ ppm. The averaged periodograms of the solar data (with
and without WGN added) as well as the best-fitting Harvey functions are shown
in Fig. 8. The best-fitting parameters related to the solar granulation signal
component in Eq. (1) are given in Table 5. We see the impact of WGN on the
Harvey parameters and particularly the decrease of the power index value
($c_{1}$ parameter) with the increase of $\sigma_{w}$. When parameter $c_{1}$
is fixed in Eq. (1) (for example $c_{1}=2$ corresponds to a standard Harvey
model), this can bias the comparison of the best-fitting Harvey parameters
derived for stars with different apparent magnitude (i.e., different levels of
white noise). Since Harvey functions are classical empirical functions used to
model the stars’ power spectral density, when we compare the inferred Harvey
parameters for stars of different apparent magnitude, one needs to be careful
about the impact of this white noise on the fitted parameters. The
interpolation of the inferred parameters to their value at a “reference level”
of white noise is necessary to avoid any bias.

Figure 8: Impact of the WGN on the periodograms. Top: Comparison of the
periodogram of HD 67458 (gray) and the averaged periodograms of three solar
$8$-hour subseries (green). Best-fitting models resulting from Harvey-
functions (Eq.(1)) fits on these CHEOPS and VIRGO periodograms are shown in
yellow and green, respectively. Bottom: Same comparison but with the averaged
periodogram of three solar subseries with the WGN added (pink). The associated
best-fitting Harvey function is shown in red. We see how the WGN impacts the
periodogram shape (and so the extraction of the granulation properties) by
comparing the green and red solid lines. In both panels, the different cut-off
frequencies $\nu_{cut}$, $f_{H}$, and $f_{g}$ inferred on CHEOPS observations
are represented by the green, cyan and magenta vertical lines, respectively.

##  5 Relationship between granulation and stellar properties

###  5.1 Computing the flicker index

In this section, we made use of the flicker index (Sulis et al. 2020a). This
index is a granulation indicator that has been defined as the slope of the
averaged periodogram $P_{L}$ in Eq. (3) in the frequency region where the
granulation signal dominates. Related to the parameter $c_{1}$ in Harvey
functions (Eq. (1)), this index has been shown to be correlated to the stellar
fundamental parameters once it is corrected from the influence of the high-
frequency white noise component.

Figure 9: Posterior distribution and correlation between all free parameters
involved in model (4) fitted to the averaged periodogram of HD 88595 CHEOPS
observations. Frequency $f_{g}$ is not optimally constrained since the
duration of the time series is relatively short ($<8$ hours).

To compute the averaged periodograms of TESS observations, we consider all the
available $8$-hour subseries that have a duty cycle $\geq 90\%$. This
corresponds to $L=172$ subseries for HD 67458, and $L=113$ for HD 88595. Since
the duty cycle of the individual CHEOPS and ESPRESSO visits of both stars is
excellent (see Tables. 1 and 2), we use our whole set of observations to
compute $P_{L}$. The resulting averaged periodograms are shown in red in Fig.
7.

We then fit each averaged periodogram with a model defined as a sum of power
laws functions of the form:

$\log
P_{L}(\nu_{k}^{+})=\sum_{i=1}^{3}-\alpha_{i}\log(\nu_{k}^{+})+\beta_{i},$ (4)

with $\alpha_{i}=\\{\alpha_{H},\alpha_{g},\alpha_{L}\\}$ the periodogram
slopes in three particular frequency regimes split by two cut-off frequencies
$\\{f_{g},f_{H}\\}$, and $\beta_{i}=\\{\beta_{H},\beta_{g},\beta_{L}\\}$ the
corresponding amplitudes. Model (4) takes the form of straight lines in the
log-log space. Parameters $\\{\alpha_{H},\beta_{H}\\}$ represent the
periodogram in the high frequency regime (dominated by the photon/instrumental
noises, and also stellar oscillations for ESPRESSO observations of HD 88595),
parameters $\\{\alpha_{g},\beta_{g}\\}$ in the regime dominated by the
granulation signal, and parameters $\\{\alpha_{L},\beta_{L}\\}$ in the regime
dominated by low-frequency stellar signal. We note that photometric and RV
observations are expected to be sensitive to different noise sources in the
low frequency region (e.g., the signal of supergranulation is large in RV
observations but negligible in photometric observations, see Sec. 7.1).

Therefore, model (4) has eight free parameters (indices $\alpha$ and
amplitudes $\beta$ for each three PSD regimes, plus two cut-off frequencies
$\\{f_{g},f_{H}\\}$ that mark these three regimes out). Using the MCMC scheme
described in Sec. 4.2 of Sulis et al. (2020a), we fit model (4) to each
averaged periodogram. An example MCMC posterior is shown in Fig. 9. The best
fitting parameters for the flicker indexes ($\alpha_{g}$), the flicker power
amplitudes ($\beta_{g}$) and the two cut-off frequencies are listed in Table
7.

For reference, the flicker index inferred from the VIRGO averaged periodogram
in the frequency range $\nu\in[550,2083]$ $\mu$Hz is $\alpha_{g}=1.33\pm 0.03$
(for the red SPM channel). We note that consistent indexes are found for all
SPM channels, as shown in Sulis et al. 2020a). The flicker index inferred from
GOLF averaged periodogram in the frequency range $\nu\in[550,1236]$ $\mu$Hz is
$\alpha_{g}=1.34\pm 0.14$, consistent with VIRGO observations. Note the
difference in the cut-off frequency $f_{H}$ between the two instruments.

From Table 7, we read a flicker index for the solar-like star HD 67458 smaller
than the one found on solar VIRGO data. As discussed in Sec. 4.2, this is due
to the large level of white noise in CHEOPS data. For comparing different
stellar observations, we need to correct the flicker index values for this
effect. This will be done in Sec. 5.2. On the other hand, the flicker index
deduced from the ESPRESSO periodogram of HD 67458 is consistent with zero,
leading us to the conclusion that the granulation signal is this dataset is
not clearly identified. On the contrary, indexes inferred from CHEOPS and
ESPRESSO averaged periodograms of HD 88595 are both large (i.e.,
$\alpha_{g}>>0$), and are consistent within their $1\sigma$ errorbars.

Finally, the flicker indexes inferred from TESS averaged periodograms of HD
67458 and HD 88595 are $\alpha_{g}<0.1$, both consistent with a nondetection.
Analyzing the two TESS sectors of HD 88595 allows, however, a marginal
detection of the granulation signal with $\alpha_{g}=0.26\pm 0.36$ for sector
9 (non detection), and $\alpha_{g}=0.71\pm 0.39$ for sector 35 (marginal
detection). We investigate the influence of the high-frequency noise level,
the duration of the subseries, and the temporal sampling on the inferred
flicker index in Appendix. E.

Table 7: Best-fitting parameters of model (4) fitted to CHEOPS and ESPRESSO averaged periodograms of HD 67458 and HD 88595. Frequencies $f_{H}$ and $f_{g}$ are expressed in $\mu$Hz. The last row indicates the flicker index based on solar VIRGO and GOLF averaged periodograms. Target | Parameter | Photometry | Spectroscopy
---|---|---|---
| $\alpha_{g}$ | $0.52^{+0.12}_{-0.13}$ | $0.02^{+0.43}_{-0.28}$
HD 67458 | $\beta_{g}$ | $-4.88^{+0.74}_{-0.42}$ | $-12.60^{+2.82}_{-2.08}$
| $f_{H}$ | $3636^{+1645}_{-577}$ | $1680^{+711}_{-125}$
| $f_{g}$ | $480^{+26}_{-132}$ | $442^{+189}_{-10}$
| $\alpha_{g}$ | $0.61^{+0.49}_{-0.25}$ | $0.77^{+38}_{-0.28}$
HD 88595 | $\beta_{g}$ | $-8.08^{+3.59}_{-1.80}$ | $-18.49^{+2.75}_{-2.15}$
| $f_{H}$ | $1036^{+450}_{-29}$ | $1104^{+7}_{-8}$
| $f_{g}$ | $351^{+2}_{-37}$ | $209^{+111}_{-5}$
Sun | $\alpha_{g}$ | $1.33\pm 0.03$ | $1.34\pm 0.14$

###  5.2 Comparison with Kepler bright stars

To compare the flicker indexes for stars of different apparent magnitudes
(i.e., different white noise levels), we have to interpolate the behavior of
the periodogram’s slope at a constant white noise level. Following Sulis et
al. (2020a), we target the level $\sigma_{w}=30$ ppm (hereafter
$\sigma_{30}$), which was arbitrarily chosen as a good compromise for all
studied stars (since all estimates of $\sigma_{w}$ are above $30$ ppm). For
each photometric dataset (VIRGO’s three SPM channels, CHEOPS observations), we
then applied the following procedure.

  1. (a)

We added a synthetic white Gaussian noise of standard deviation
$\sigma_{w}^{(i)}$ to each $L$ available light curves.

  2. (b)

We compute the averaged periodogram $P_{L}^{(i)}$ with Eq.(3) using these $L$
new light curves.

  3. (c)

We evaluate the flicker index $\widehat{\alpha}_{g}(\sigma_{w}^{(i)})$ by
fitting model (4) to $P_{L}^{(i)}$.

  4. (d)

We performed steps (a) to (c) for $\sigma_{w}^{(i)}=0$ (initial conditions) to
$\sigma_{w}^{(i)}=1000$ ppm.

This gives us the empirical curve $\widehat{\alpha}_{g}(\sigma_{w}^{(i)})$,
which behaves as a decreasing exponential function of the form (see Eq. (5) in
Sulis et al. 2020a):

$\widehat{\alpha_{g}}(\sigma_{w}^{(i)})=a_{e}\leavevmode\nobreak\
e^{-b_{e}\leavevmode\nobreak\ \sigma_{w}^{(i)}}+c_{e},$ (5)

with $\\{a_{e},b_{e},c_{e}\\}\in\mathbb{R}$, that are found correlated with
the stellar parameters151515We note that we do not find coefficients
$\\{a_{e},b_{e},c_{e}\\}$ consistent with the predictions from Eq.(6) of Sulis
et al. (2020a). This may be due to the small number ($<5$) of targets with
mass $<1$ $M_{\odot}$ and/or $>1.5M_{\odot}$ that were used to derive the
Eq.(6) of Sulis et al. (2020a). This may also be due to uncertainties on the
stellar parameters of the Kepler sample or to the few number of CHEOPS visits
that leads to an approximate interpolation of the curve
$\widehat{\alpha}_{g}(\sigma_{i})$ involved in Eq.(5).. This behavior is
clearly shown in Fig. 10 for the three VIRGO SPM channels. For the red
channel, we read $\widehat{\alpha}_{g}(\sigma_{30})=0.92\pm 0.14$ (see yellow
star). The flicker index $\alpha_{g}=0.52^{+0.12}_{-0.13}$ inferred on the
CHEOPS data of HD 67458 (see Table 7) is shown at $\sigma_{w}\approx 87$ ppm
(value computed using the three visits, see yellow square symbol). Based on
the fit of model (5) to the empirical curve
$\widehat{\alpha}_{g}(\sigma_{w}^{(i)})$ of HD 67458, we found
$\widehat{\alpha_{g}}(\sigma_{30})=0.89^{+0.12}_{-0.13}$ (see yellow star
symbol). The flicker index of HD 67458 is therefore in complete agreement with
the solar flicker index at comparable white noise levels. For HD 88595, we
found $\alpha_{g}=0.61^{+0.49}_{-0.25}$ at $\sigma_{w}=77$ ppm (see Table 7)
and $\widehat{\alpha_{g}}(\sigma_{30})=2.08^{+0.49}_{-0.25}$ based on the
interpolation.

We compare these values with the brightest stars (magnitude $<10$) in the
Kepler sample studied in Sulis et al. (2020a). This sample includes $87$ stars
(among which the $17$ solar-like stars, with inferred flicker indexes added to
Fig. 10), from which the flicker indexes have been inferred from averaged
periodograms computed based on hundreds of $1$-day subseries and then
interpolated at the white noise level $\sigma_{30}$ using Eq.(5). Parameters
$\widehat{\alpha}_{g}(\sigma_{30})$ and the flicker cut-off frequency $f_{g}$
are shown as a function of the stellar surface gravity161616The ${\rm log}g$
for the Kepler targets was obtained by different techniques, from the highest
priority to the lowest ones, depending on availability: asteroseismology,
high-resolution spectroscopy, low-resolution spectroscopy, flicker,
photometric observations, and finally the KIC (see
https://exoplanetarchive.ipac.caltech.edu/docs/KSCI-19097-004.pdf).  in Fig.
11. The two bright stars observed by CHEOPS (pentagon and triangle symbols for
HD 67458 and HD 88595, respectively) are in complete agreement with Kepler
(gray dots) and solar (star symbol) predictions. The inferred parameters are
listed in Appendix. F (see Table 11, which is also available online).

We note that the frequency $f_{g}$ behaves similarly to the convective
characteristic timescale ($1/b_{1}$) that is fitted empirically by the usual
Harvey functions in Eq. (1) assuming an exponential decay with time. They are
both strongly correlated with the stellar maximum oscillation frequency
$\nu_{max}$. The frequency $f_{g}$ can be interpreted as the upper tail of the
distribution of the granule cells’ lifetime (Seleznyov et al. 2011). From Fig.
11, we read that the maximum correlation timescale for our sample stars ranges
between $\sim 22$ min ($750\leavevmode\nobreak\ \mu$Hz) and $\sim 2.7$ hours
($100\leavevmode\nobreak\ \mu$Hz).

Figure 10: Estimated values of the flicker index ($\alpha_{g}$) as a function
of the white noise level ($\sigma_{w}$) added in the VIRGO time series (red,
blue, and green SPM channels, see Fig. 19 of Sulis et al. 2020a). The red star
corresponds to the flicker index of the red channel measured at
$\sigma_{w}=\sigma_{30}=30$ ppm. Black dots show the flicker indexes inferred
from the averaged periodograms of Kepler Sun-like stars. The yellow square
shows the flicker index inferred from the averaged periodogram of HD 67458
(CHEOPS), and the yellow star shows its interpolation at $\sigma_{30}$.

Figure 11: Flicker parameters as a function of the stellar surface gravity.
The flicker indexes (left) have been interpolated to a fixed level of white
noise ($\sigma_{w}=30$ ppm) for this comparison. The flicker frequencies are
shown in the right panel. In both panels, the gray dots indicate values
obtained for the sample of Kepler bright targets (magnitude $<10$) involved in
Sulis et al. (2020a). The pentagon symbols indicate values obtained for HD
67458 and the triangle symbols for HD 88595 based on CHEOPS high-precision
observations. The star symbols indicate the value obtained for the Sun (VIRGO
observations). The square symbols indicate the values obtained from HD
simulations of $17$ stars (see Sec. 6). The color code indicates the stellar
metallicity [Fe/H] (not shown for Kepler targets).

##  6 Predictions from 3D hydrodynamic models of convection

###  6.1 Description and predictions for HD 67458 and HD 88595

The minimum level of stellar activity of a late type star is due to surface
convective motions that produces stochastic variations of the light curves.
The amplitude and time scale of these fluctuations depend on the spectral type
and generally increase toward red giant type or earlier type (F star) due to
more vigorous convection in these stars. In this work we aim at reproducing
this stellar noise with the use of state-of-the-art 3D hydrodynamical
simulations. These simulations do not account for magnetic fields and
therefore no plage or spot but they showed a remarkable agreement with
bolometric light variations of KEPLER targets (Rodríguez Díaz et al. 2022).
Previous studies have also used these kind of models to study the granulation
signal (see e.g., Ludwig 2006; Ludwig & Steffen 2013; Trampedach et al. 2013;
Tremblay et al. 2013; Samadi et al. 2013), and compared their results with
observations. However, those studies could not reproduce the observational
trends with high accuracy or without introducing the Mach number, a quantity
that is very difficult to compute for observational data.

Therefore, we generated long time series of box-in-a-star type 3D
hydrodynamical simulations across the HR diagram using the STAGGER-code
(Nordlund & Galsgaard 1995; Magic et al. 2013). This code solves the equations
for the conservation of mass, momentum, and energy, as well as the radiative
transfer equation assuming local thermodynamic equilibrium (LTE). For more
information about the code we refer to Rodríguez Díaz et al. (2022) and
references therein. The box-in-a-star type means that the 3D models are
centered around the stellar photospheres. That is, they cover the photosphere,
the superadiabatic region, and the quasi-adiabatic deeper convective layers,
where a flat entropy profile is ensured at the bottom boundary. These layers
are distributed in a specific 3D Cartesian geometry.

The 3D models are defined by three stellar parameters: the effective
temperature $T\rm_{eff}$, the surface gravity ${\rm\log}g$, and the
metallicity [Fe/H]. $T\rm_{eff}$ is defined by the entropy value at the bottom
boundary of the models, while [Fe/H] is defined by the abundance of chemical
elements present in the models.

Each model contains typically $10$ granules, whose sizes are a few tens of
pressure scale heights. This means that the sizes of the granular cells are
bigger for models representing early-type stars or evolved stars.

Realistic radiative transfer is performed using long characteristic along
several rays at different inclinations across the simulation domain in order
to account for heating and cooling in the energy equation (e.g., Stein &
Nordlund 2003). These radiative intensities are integrated in wavelengths and
inclinations to give bolometric fluxes which can be compared to observations.
To compare with observations, this quantity has to be rescaled by the number
of granules visible on the disk, as Trampedach et al. (1998) and Ludwig (2006)
proposed. With this scaling, Rodríguez Díaz et al. (2022) was able to
determine the standard deviation of the stellar disk-integrated intensity from
the small box models. We refer the reader to Rodríguez Díaz et al. (2022) for
a more detailed description of the method.

Rodríguez Díaz et al. (2022) found that at solar metallicity the standard
deviation of the flux scales like $\sigma_{HD}\sim\nu_{\rm max}^{-0.567}$, and
the characteristic timescale follows $\tau_{ACF}\sim\nu_{\rm max}^{-0.997}$
(see Table 4. in their paper). We note that the characteristic timescale is
defined as the autocorrelation time of the entire time series. It is a
different timescale than the one related to the flicker frequency $f_{g}$
(measured on the observed periodogram). For consistency with how we evaluate
the granulation amplitude in Sec. 4.1, we have also determined a relation for
F8 based on these 3D HD models (see App. D for details). We found
F8${}_{HD}\propto\nu_{\rm max}^{-0.530}$ .

Using such scaling and the parameters given in Table 4, the two values
predicted by the 3D simulations are F8${}_{HD}\approx 48\,{\rm ppm}$ (HD
67458) and F8${}_{HD}\approx 66\,{\rm ppm}$ (HD 88595). The predicted
characteristic timescales are $\tau_{ACF}\approx 203.2\,{\rm s}$ (HD 67458)
and $\tau_{ACF}\approx 369.3\,{\rm s}$ (HD 88595). For HD 67458, the F8 values
inferred from CHEOPS dataset are larger than the predictions from HD
simulations, indicating that white noises are dominating the dataset. For HD
88595, the inferred and predicted F8 are in very good agreement (see Sec. 4.1
and Table 6).

###  6.2 Flicker indexes and relationship with the stellar properties

From the total sample of $27$ stars simulated with the 3D models in Rodríguez
Díaz et al. (2022), we select the $17$ main-sequence stars. The selected stars
have $T\rm_{eff}\in[4727,6485]$ K, ${\rm log}g\in[3.25,4.44]$ cgs, and
[Fe/H]$\in[-2,0.5]$.

The length of all the synthetic time series correspond to at least $1000$
convective turnover times, with these turnover times that are adapted to the
stellar properties of the synthetic target stars. For all of these target
stars, we compute the classical periodogram and scale the two cut-off
frequencies $f_{g}$ and $f_{c}\geq f_{H}$ with the stellar parameters, with
$f_{c}$ the corner frequency that marks the p-modes frequency domination
regime out (see Sulis et al. 2020a). We finally evaluate the flicker indexes
as in Sec. 5.1 based on these synthetic periodograms of stellar granulation.
Results are shown with the square symbols in Fig. 11. The color code indicates
the stellar metallicity (not given for Kepler stars). We note that the HD
simulated time series do not contain white noise, but only granulation signals
and are therefore used as reference.

The flicker index indicators $\alpha_{g}$ derived from the HD convection
models are in good agreement with the bright Kepler and CHEOPS targets. In
line with Rodríguez Díaz et al. (2022), this demonstrates the success of these
3D models in reproducing realistic photometric time series of stellar
granulation. We however note a larger dispersion of the simulations compared
to observations for stars with ${\rm log}g=4.0$. This dispersion seems to be
strongly correlated with the stellar metallicity: low metallicity shows
smaller flicker index. While the results are different at other ${\rm log}g$
values, this needs to be investigated further based on a larger synthetic
stellar population.

The flicker frequency $f_{g}$ derived from the HD convection models are also
in good agreement with the values inferred from Kepler and CHEOPS
observations. We note however a slight shift compared to Kepler values at
${\rm log}g=4.0$.

##  7 Link between the spectroscopic and photometric signatures

Only a few variations of the physical granulation properties were observed
during the solar magnetic cycles (Garcia et al. 2005), with $2\%$ variation in
the density and mean granules’ area observed during the solar cycle (Ballot et
al. 2021). This cannot be directly verified on other stars since their
surfaces cannot be resolved on the scale of the granulation cells. However,
this can also be confirmed indirectly by studying variations in their power
spectra (Seleznyov et al. 2011; Muller et al. 2018; Sulis et al. 2020a). A
comparison of the current periodogram of HD 88595 with future observations
taken in a few years would allow us to confirm if the granulation signal is
indeed stationary with the stellar magnetic cycle. Without having any dataset
– to our knowledge – to verify this statement, we assume in this section that
photometric observations of the granulation signal can allow us to predict its
signature (amplitude and timescales) in spectroscopic observations (e.g., in
line with techniques that have been developed for magnetic activity, such the
FF’ technique described in Aigrain et al. (2012)). This is important since
high-precision photometric surveys allow the accumulation of ten to hundred of
continuous $1$-day observation of stars from space missions, while RV ground-
based surveys typically have poor sampling to characterize this noise source
for exoplanet detection (typically: one to three data points per night spread
over long term campaigns).

###  7.1 Comparison of the periodograms

In the left panel of Fig. 12, we compare the (arbitrarily normalized171717We
note that this normalization factor has no impact on the inferred flicker
index values, which are based on the periodograms’ slope and not their
amplitudes.) averaged periodograms of solar VIRGO (red) and GOLF (black)
observations. We observe comparable periodogram’ slopes on data taken in
photometry and spectrophotometry with flicker indexes $\alpha_{g}=1.33\pm
0.03$ and $\alpha_{g}=1.34\pm 0.14$, respectively (see Sec. 5.1). In both
dataset, the level of white noise is low and does not affect the
characterization of the stellar granulation signal. We note that, in the
frequency region where the supergranulation starts to dominate
($\nu<200\leavevmode\nobreak\ \mu$Hz), the periodogram of brightness becomes
flat while it still increases in the periodogram of RV.

In the two other panels, we show the averaged periodograms of CHEOPS and
ESPRESSO observations of HD 67 458 and HD 88595. The flicker index deduced
from the ESPRESSO periodogram of HD 67458 is consistent with zero (see Table
7), but its value remain consistent within their $1\sigma$ errorbars with the
flicker index inferred from CHEOPS observations. While we assert a marginal
detection of the granulation signal for this star, this could explain the
visual good match of the two periodograms; the white noise being too large to
infer a precise flicker index value. More subseries ($>>3$) would be needed to
improve the precision on this parameter. On the other hand, the flicker
indexes of HD 88595 inferred on CHEOPS ($\alpha_{g}=0.61^{+0.49}_{-0.25}$) and
ESPRESSO data ($\alpha_{g}=0.77^{+38}_{-0.28}$; see Table 7) are in complete
agreement within their $1\sigma$ errorbars.

This indicates that the properties (i.e., correlation timescales where this
signal dominates, amplitudes distribution, and type of correlation with the
flicker index) of the stellar granulation signal in RV data could be predicted
from high-precision photometric observations (CHEOPS and PLATO) of bright
stars.

Figure 12: Comparison of the averaged periodograms of photometric (red) and
spectroscopic (black) observations. Left: Solar VIRGO (red SPM channel) and
GOLF observations. Middle: CHEOPS and ESPRESSO observations of HD 67458.
Right: Same for HD 88595. The periodograms have been arbitrarily normalized to
compare their slope in the frequency regime that is dominated by stellar
granulation. When relevant, the different cut-off frequencies $\nu_{cut}$,
$\nu_{max}$, $f_{H}$, and $f_{g}$ represented by the green, blue, cyan and
magenta vertical lines, respectively.

###  7.2 Prediction from empirical laws

Some relations between the amplitudes of the stellar granulation signal in
spectroscopy (RV RMS) and photometry (F8 metric) have been derived in the
literature. We aim to test these predictions with the CHEOPS and ESPRESSO
datasets of HD 67458 and HD 88595.

A first relation has been derived in Bastien et al. (2014), based on a small
sample of $12$ stars observed first by Kepler and later by RV ground-based
surveys at the Keck and Lick observatories. The stars in this sample have been
characterized as “chromospherically quiet stars” since they showed low-
amplitude photometric variability ($\leq 3$ ppt) in their Kepler light curves.
However, the RV precision was limited to $\geq 4$ m/s leading to a difficult
characterization of the stellar granulation signal. Fitting a linear law
between F8 and the corresponding RV RMS for each stars, the relation:

${\rm RV\leavevmode\nobreak\ RMS}=(31.99\pm 3.95)\times F8+(3.46\pm 1.19)$ (6)

has been derived with RV RMS in m/s and F8 in ppt (see also Sec.3.5 of Tayar
et al. 2019).

A second relation has been derived in Cegla et al. (2014), but based on
indirect RV measurements. This study used a statistically more robust stellar
sample ($944$ stars) observed by Kepler in photometry and with the GALEX
survey (Martin et al. 2005) in far ultraviolet (FUV). From the FUV
observations, the authors converted the data to chromospheric activity proxies
(${\rm log}R^{\prime}_{\rm HK}$) using the conversion extracted from Findeisen
et al. (2011), and then converted again to an RV RMS using the relation given
in Saar et al. (2003)181818We note that they found similar RV RMS based on
another work of Wright (2005), but higher RV RMS values when using predictions
from Santos et al. (2000).. This work led to the linear relations:

$\begin{split}{\rm RV\leavevmode\nobreak\ RMS}&=18.04\times
F8+0.98\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if $T\rm_{eff}<6000$
K},\\\ {\rm RV\leavevmode\nobreak\ RMS}&=84.23\times
F8-3.35\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if $T\rm_{eff}\geq
6000$ K}.\end{split}$ (7)

These relations were derived from targets deemed to be on the so called
flicker floor in Cegla et al. (2014) and therefore convection dominated; stars
with large amplitude photometric variations did not show a clear correlation
between F8 and RV RMS.

Finally, Oshagh et al. (2017) extended the work of Bastien et al. (2014) with
$9$ new targets observed with K2 and HARPS. However, their HARPS RV
measurements contained only few data points ($<12$) and the RV RMS remained
large ($>10$ m/s) meaning the RV signal was dominated by other noise sources
than granulation (e.g., active regions). We then do not consider this study in
the present work.

Table 8: Radial velocity RMS for HD 67458, HD 88595, and the Sun as predicted from Eqs.(6) and (7) based on the F8 metric computed on our CHEOPS and VIRGO observations. The last column shows the RV RMS measured on ESPRESSO observations (see Table 2). Radial velocity RMS for the Sun are based on the $75$ GOLF subseries. All RV units are in m/s.  Target | RV RMS predicted from | Measured
---|---|---
| Eq.(6) | Eq.(7) | RV RMS
HD 67458 | $4.31\pm 1.19$ | $1.46$ | $1.1,{2.6}$
HD 88595 | $5.59\pm 1.22$ | $2.27$ | $2.26,2.54$
Sun | $4.10\pm 1.19$ | $1.34$ | $[0.87,1.63]$

We note that in Eqs.(6) and (7) the F8 metric has been derived based on the
initial definition, that is using data binned into $30$-min intervals
($\tau$). However, this $\tau$ is not optimal for main-sequence stars since
the granulation timescales are shorter than $30$ minutes. We therefore
investigated different binning intervals. To reduce the contribution of the
high-frequency noises, we found that a data binning of $\tau=15$ minutes and
$\tau=5$ minutes (as in Sec. 4.1) was sufficient to extract the signal of
granulation in the CHEOPS data of HD 67458 and HD 88595, respectively. In
particular, since ESPRESSO data of HD 67458 are dominated by photon noise we
have to use a more drastic data binning than in Sec. 4.1 to reduce its
contribution in the measured RV data and makes an approximation comparison
with the literature possible. We then derive the RV RMS following the
relations given in Eqs.(6) and (7). Results are shown in Table 8. Clearly,
comparing these predictions with the RV RMS found on our ESPRESSO time series
(see last columns) are in favor of Eq.(7) derived by Cegla et al. (2014).

We note however the high sensitivity of this relation to the length of the
temporal binning $\tau$ used to compute Eq.(7). Indeed, computing the F8
metric with $\tau=\\{5,15,30\\}$ min in Eq.(7) leads to a predicted RV RMS of
$\\{2.0,1.61,1.4\\}$ m/s for HD 67458, $\\{1.46,1.34,1.26\\}$ m/s for the Sun,
and $\\{2.27,1.05,-0.10\\}$ m/s for HD 88595 (the last value representing a
limit of Eq.(7) for F8 values $<39.77$ ppm). The choice of $\tau$ for
computing the F8 metric does not significantly affect the predictions for the
Sun (where the level of high-frequency noise in VIRGO data is negligible). For
HD 67458, using a too short temporal binning for computing the F8 metric of
the solar-like star HD 67458 makes the time series completely dominated by the
high-frequency noise. Using a temporal binning of $30$-minutes, as in Cegla et
al. (2014), leads to predicted RV RMS of similar order of magnitude. For the F
star HD 88595, the relation in Eq.(7) is not adapted to $\tau>15$ minutes.
Computing the F8 metric based on a temporal binning that is adapted to the
granulation timescales, as proposed in Bugnet et al. (2018), may allow the
photometric versus RV relation derived in Cegla et al. (2014) to be refined.

In addition, we compare the measured RV RMS of our two targets and the Sun
with the stellar surface gravity and metallicity in Fig. 13. The dots
represent the predicted values from Eq. (7), both are in agreement with
ESPRESSO observations (see the interval delimited by the dashed horizontal
lines), as shown in Table 8. In this figure, we also represent the targets
involved in the study of Tayar et al. (2019), who computed the RV RMS with Eq.
(7) using an F8 metric coming from the stellar parameters. This leads us to
the conclusion that, although the predictions of Cegla et al. (2014) are rough
estimates (i.e., no confidence interval is given while the granulation signal
behaves – a priori – as Gaussian colored noise with amplitudes that should be
defined with mean and standard deviation parameters, see Sulis et al. 2020a),
they lead to values that are quite consistent with CHEOPS and ESPRESSO
observations.

Figure 13: Radial velocity RMS as a function of the stellar surface gravity.
Individual small dots represent the values for the $129\leavevmode\nobreak\
055$ stars of the APOGEE sample studied in Tayar et al. (2019). For this
stellar sample, the RV RMS was computed based on Eq.(7) (Cegla et al. 2014).
The two big dots represent the RV RMS predicted from Eq.(7) for HD 67458 and
HD 88595. The star symbol represents the RV RMS predicted from Eq.(7) for the
Sun. The color code for all targets represents the stellar metallicity [Fe/H].
The blue and red horizontal dashed lines show the RV RMS values inferred from
ESPRESSO observations of HD 88595 and HD 67458, respectively.

##  8 Analysis of CCF shape

Table 9: Correlations with RV as assessed by Spearman’s Rank ($\rho_{S}$) for
the binned datasets. Also listed in brackets is the average S/N from echelle
order 10 per night; strong correlations with low S/N may be spurious (HD
67458), see for example Fig. 14. “N.O” means there was no overlap between
CHEOPS and ESPRESSO observation for this night (see Fig. 2)

.

| HD 67458 | HD 88595
---|---|---
Indicator | Night 1 (88) | Night 2 (68) | Night 3 (87) | Night 1 (263) | Night 2 (285)
FWHM | $-0.75$ | -0.92 | $-0.77$ | $-0.31$ | 0.71
Contrast | $-0.68$ | -0.86 | -0.83 | $-0.26$ | $-0.14$
EW | $-0.67$ | $-0.89$ | $-0.82$ | $-0.26$ | 0.26
BIS | 0.35 | 0.05 | 0.56 | -0.77 | 0.31
Curvature | -0.11 | 0.1 | 0.82 | -0.77 | -0.77
Ab | -0.28 | 0.01 | -0.15 | 0.6 | 0.94
Flux | N.O | $-0.56$ | 0.73 | 0.0 | 0.03

We know from solar observations that granulation gives rise to a “C”-shaped
bisector in most stellar absorption lines; the uprising/blue-shifted granules
contribute more overall to the stellar lines due to their larger brightness
and surface area, while the dark downfalling and red-shifted intergranular
lanes serve to depress the redward wing. As the granules evolve over time, and
the ratio of granule to intergranular lane changes, we expect to see a
corresponding change in line shape, asymmetry and overall brightness. Such
changes in line shape may then manifest as RV shifts. Both 3D MHD and HD
simulations predict correlations between granulation-induced line shape and
both RV and brightness (Cegla et al. 2019; Dravins et al. 2021). Consequently,
in this section we explore variations in the shape of the cross-correlation
functions (CCFs) and RVs, as measured from the ESPRESSO observations, and
brightness as measured from CHEOPS. We examine CCFs rather than individual
stellar lines due to the increase in S/N; we note that due to the weighting in
the CCF template mask, the ‘C’-shaped bisector of the individual lines is not
preserved in the CCF, but the overall shape should respond to the
corresponding temporal variations experienced by the individual lines.

Before we can explore the impact of granulation, we must exclude the impact of
the p-mode oscillations; this is particularly important in RV-space where they
may dominate over the granulation-induced shifts. We take two approaches to
disentangling the granulation and p-mode effects.

###  8.1 Data binning

First, we bin the individual CCFs following Chaplin et al. (2019), to
optimally average out the p-modes. Using the OscFilter script provided by
Chaplin et al. (2019), we predicted in Sec. 4.2 that an exposure time of
$\sim$12 min should result in a remaining RMS (due to the p-modes) of $10$
cm/s for HD 67458. However, to reach a similar (p-mode) RMS for the hotter and
slightly evolved HD 88595 would require binning over $\sim$85 min. We note
that we have not propagated the uncertainties in the stellar parameters
through OscFilter nor accounted for the variability in mode amplitudes. From
Table 1 of Chaplin et al. (2019), we expect these uncertainties to be of order
a few minutes for HD 67458, but they may be as large as $50$ minutes or more
for HD 88595. For HD 67458, we account for this uncertainty by averaging over
$20$ minute time bins; unfortunately, with $85$ minutes time bins for HD 88595
we are left with only $6$ points per night so we do not attempt to account for
this uncertainty here.

For each time bin, we sum all the CCFs that fall within the bin and then
continuum normalize; uncertainties on the CCF flux are propagated accordingly.
We then fit a Gaussian to the binned CCFs to determine the RV, full width at
half maximum (FWHM), and contrast. To investigate any potential trends with
S/N, we add the individual S/N, provided by the DRS for each order, in
quadrature. We follow Cegla et al. (2019) and measure a variety of shape
diagnostics for the binned CCFs, including the FWHM, contrast, equivalent
width (EW), bisector inverse span (BIS), bisector curvature, and bisector
amplitude (Ab). Both the BIS and bisector curvature split the CCF bisector
into discrete regions. Here we use the standard definitions for BIS and
curvature (BIS: top region is bounded by 10-40% of the line depth, bottom
region by 55-90% of the line depth; curvature: top region is 20-30%, middle is
40-55%, 75-95% of the line depth); we explore optimizing these definitions in
Appendix G191919We are able to improve the correlation between BIS or
curvature and RV by tweaking the top, bottom, and middle definitions, which
may indicate we are identifying regions more sensitive to granulation.
However, these regions change across nights and further exploration is needed
to determine the root cause; see Appendix G for more details.. For nights with
overlapping simultaneous CHEOPS and ESPRESSO observations (see Fig. 2), we
average the CHEOPS light curves over the same time bins as the ESPRESSO data.

In each instance, we explore the relationship between the shape indicator or
brightness and RV on a night by night basis for each target. If we are able to
detect the impact of granulation on the CCF shape, we expect to see
correlations between the shape indicators, flux and RV that are consistent
across the various observing nights and independent of S/N. Irregardless of
the observations here, the CCF contrast, and to a lesser extent FWHM, is often
correlated with S/N, especially at low S/N due to difficulties in accurately
subtracting the background (bias level, dark current, diffuse inter-order
background, sky background, etc.). Hence, any correlation between shape
indicator and RV that is S/N-dependent should be taken with caution.

We use the Spearman’s Rank correlation coefficient202020We use the spearmanr
function from the scipy Python library. ($\rho_{S}$) to assess, in a
nonparametric way, the strength of any potential correlations between the
shape indicator and the measured RV shift. The results are shown in Table 9,
and a subset of these are shown in Figs. 14 and 15 for HD 67458 and HD 88595,
respectively.

For HD 67458, the FWHM, contrast, and EW show clear correlations with RV that
are consistent across all three nights (see Fig. 14); however, these
correlations are dependent on the S/N, as shown by the data color-coded, with
the S/N from the 10th echelle order (there does not appear to be a dependence
on

Figure 14: Relationship between various shape indicators and brightness
(y-axis) vs RV for HD 67458 shown as individual data points for CCFs binned in
20 minute time intervals, where errorbar color, and lines connecting data in
time, indicates the night of the observation ($n1,n2,n3$). Each point is
color-coded by the average S/N of the 10th echelle order. The Spearman’s
correlation coefficients is indicated by $\rho_{s}$ for each night.

Figure 15: Same as Fig. 14, but for HD 88595. For this star, the CCFs are
binned over $85$ minutes.

airmass). The remaining shape indicators do not show any consistent or strong
correlations with RV. The flux appears to correlate with RV, but changes from
anticorrelated in the second night to positively correlated in the third
night. The peak-to-peak RV variation of $\sim$5 m/s is also potentially larger
than we might expect based on solar observations of granulation. For all these
reasons, the correlations we do see are likely due to difficulties in
accurately subtracting the sky background, rather than driven by true
convection changes. Future observations under more ideal observing conditions
may be key to confirm the nature of the observed correlations; alternatively,
a more refined treatment of the p-modes may also help.

For HD 88595, with the exception of the bisector curvature, we do not see any
strong correlations that are consistent across nights (see Fig. 15). Even with
the bisector curvature there are still some inconsistencies; although the
strength (and sign) of the correlation is similar, there are offsets in the
actual values of the curvature and RV. We find that the bisectors between the
two nights show a different shape, indicating the overall shape of the CCF is
different between nights (see Fig. 21). We also note that for the flux-RV
exploration in the first night, there is a lack of CHEOPS observations at the
start of the night that means there is no overlap in the first time bin and
limited overlap in the second bin (the latter corresponding to the data point
in bottom right subplot of Fig. 15 with the highest flux).

At this stage, it is not clear if these differences in behavior between nights
are driven by changes in the instrumental PSF or stellar changes happening on
longer timescales, such as supergranulation. The two nights are separated in
time by $\sim$8 days and although we would expect the supergranular lifetime
to be longer than on the Sun (which is just under 2 days) this seems likely
too long for the supergranulation to remain correlated; that said, there is
not a strong enough understanding of stellar supergranulation to completely
rule this out. If supergranulation were the dominant effect here (rather than
the short-term granulation), then there might be tentative evidence for
correlations across the nights (that often differ in sign from the nightly
behavior). Unfortunately, the long duration required to mitigate the p-modes
significantly reduces our sampling and prevents us from making strong
conclusions here.

###  8.2 Data filtering

Figure 16: HD 67458 correlation strengths for the relationship between
(ESPRESSO) CCF shape indicator (FWHM, contrast, equivalent width [EW], BIS,
bisector curvature, bisector amplitude [Ab]) or (CHEOPS) flux and RV, as
determined by a Spearman’s rank correlation test, for various Butterworth
filter cutoff frequencies for each night (n1, n2, and n3) as indicated by
marker style. CHEOPS data (last column) was interpolated onto the ESPRESSO
timestamps prior to filtering; no simultaneous observations were performed
during n1.

Motivated by the large timespans required to bin out the p-modes for HD 88595
(and likelihood for a large uncertainty on the optimal binning), we also
attempt to filter out the p-mode impact in the RVs and shape indicators. For
this, we apply a Butterworth filter provided by the Python library scipy.
Here, we use the individual RVs, the FWHM and contrast provided by the
ESPRESSO DRS. We fit each of the remaining shape indicators for each
individual CCF. For nights where there is an overlap between CHEOPS and
ESPRESSO observations, we (linearly) interpolate the CHEOPS lightcurves onto
the ESPRESSO timestamps. Then, on a night by night basis, we apply the
Butterworth filter to the RV and shape indicator time-series with a cutoff
frequency corresponding to 5 min and increasing in 5 min intervals up to 80
and 255 min for HD 67458 and HD 88595, respectively (with the latter being
approximately three times the estimated time to bin the p-modes down to $10$
cm/s RMS for this target). For each cutoff frequency, we asses the correlation
strength between the filter RV and indicator time-series following the
Spearman’s Rank correlation coefficient. We repeat this analysis on the
unfiltered (unbinned) data.

We examine the behavior of each indicator for all observed nights for each
target, as well as the behavior of all indicators across a given night. If we
are filtering out the p-modes with increasing cutoff frequency, then a given
indicator should increase in correlation strength in a consistent manner
across the nights. However, we do run the risk of over-filtering and
flattening the signal, and therefore artificially creating a correlation
(e.g., if both the RV and shape indicator or flux are constant over time then
they will have a perfect correlation). If all indicators grow in correlation
strength in a similar way with cutoff frequency, but behave differently on
different nights, this may be a sign we are over-filtering the data.

In general we find, for both targets and all nights, the correlation strength
between RV and most shape indicators or flux tends to grow as a function of
cutoff frequency, as shown in Figs. 16 and 17. However, the behavior of the
correlation strength as a function of cutoff frequency is largely inconsistent
across nights for most indicators, for both targets. This is consistent with
the binned analysis in Section 8.1, where the sign of the correlations which
did not trend with S/N changed across nights. The inconsistencies across
nights here casts doubt on the detection of granulation-induced shape changes
and the increase in correlation strength with cutoff frequency may indeed be
due to simply flattening both time-series. The exception to this is the
relationships with contrast and bisector curvature for HD 88595, where the
correlation with RV increases in strength for cutoff frequencies corresponding
to longer timescales in a similar manner across both nights. This also adds
further evidence that the strong correlation between bisector curvature and RV
in the binned data in Section 8.1 (Fig. 15) might be indeed be due to
granulation.

The relationship between flux and RV (as a function of cutoff frequency) is
also somewhat consistent across nights; we note that the p-values associated
with the Spearman rank correlations strengths indicate that a window size of
$100-150$ minutes or more may be required to reject the null hypothesis, that
means we may only be able to trust the correlations with a large filtering
window. This behavior might explain why we were unable to detect an RV-flux
correlation in the $80$-minute binned data in Section 8.1. Interestingly, the
difference in strength of the p-values associated with the Spearman rank
correlation may further indicate the difference in data quality between nights
(e.g., Fig. 5 shows how the RV RMS of the first nights is consistently larger
than for the second night).

Nonetheless, a further refinement of the treatment of the p-modes is required
to confidently confirm the nature of this behavior; this is beyond the scope
of this paper and will be the subject of a future study. Additionally, future
observations would also be useful to confirm the origin of the CCF shape
differences between nights and the relationship between granulation-induced RV
shifts and flux.

Figure 17: Same as Fig. 16, but for HD 88595.

##  9 Conclusions

The first goal of this paper was to detect the stellar granulation signal of
two bright stars for the first time with high-precision photometric CHEOPS
observations taken simultaneously or contemporaneously with high-resolution
spectroscopic ESPRESSO observations. We detected the stellar granulation
signal for the F star HD 88595 in both datasets, but only in the CHEOPS
dataset for the solar-like star HD 67458 (Sec. 4). In particular, we observed
significant variabilities between the three nights of observations of HD 67458
with ESPRESSO. Although we could not clearly identify the origin of these
variabilities, we showed that they are induced by an excess of white noise in
the observations masking the signatures of the stellar granulation (Sec. 4.1).
We demonstrated the sensitivity of high-precision photometric CHEOPS
observations to this stellar signal, which was not detected on TESS
observations (large photon noise and small amplitude of the granulation signal
in the TESS passband).

We also showed that the temporal binning of RV time series (i.e., the
technique that is classically used to mitigate the amplitude of this stellar
signal) is limited as this signal remains correlated over timescales
$\tau>10-15$ min (Sec. 4.2). In particular, to reach the RV amplitude level of
Earth-mass planets in the HZ of our two stars, we estimated that observations
would need to be binned over $\tau\geq 180$ minutes for the G star (in
agreement with the predictions of Meunier et al. 2015) and over $\tau\gg 250$
minutes for the F star.

Computing the flicker index indicator, defined as the slope of the
periodograms, we demonstrated that both this indicator and the flicker
frequency are correlated with the stellar parameters (Sec. 5), which is in
agreement with previous studies (Sulis et al. 2020a) based on Kepler
observations.

The second goal of this study was to test the predictions from 3D hydrodynamic
models of convection (Sec. 6). We observed photometric amplitudes with CHEOPS
that are consistent with the expectation from 3D models: $48$ ppm for the
solar-like star and $66$ ppm for the F star. The flicker index and flicker
frequency derived on synthetic granulation time series generated by these 3D
model predictions are also in agreement with both Kepler and CHEOPS
observations.

The last goal of this study was to link the spectroscopic and photometric
signatures of convection for main-sequence stars with high-precision
observations (Sec. 7). We observed a very good match between the periodograms
of CHEOPS and ESPRESSO observations for the F star in the frequency regime
dominated by stellar granulation. This was quantitatively confirmed by similar
flicker indexes inferred on both datasets and this indicates that the
spectroscopic signal of stellar granulation can be predicted by photometric
observations. Unfortunately, the RV precision of the ESPRESSO observations of
the solar-like star HD 67458 was not good enough to precisely characterize
this stellar signal (the errorbars were between $0.4$ and $2.7$ m/s against a
signal with amplitude $<1$ m/s). Additional RV observations with precision
$<1$ m/s are then still needed to confirm the strong link that we already
observed in the Sun with VIRGO and GOLF observations.

Comparing the RV RMS predictions from photometric measurements lead us to
invalidate the empirical relations derived in Bastien et al. (2014), while
confirming the relations derived in Cegla et al. (2014). We however
demonstrated the high sensitivity of the data binning – used to extract the
photometric granulation signal – on these RV RMS predictions. We argued that
the amplitude of this stellar signal needs to be adapted on the stellar
dependent granulation timescales and high-frequency noise level that affects
its characterization. A perspective of this study will be to use a more robust
technique to do so, as the one developed in Bugnet et al. (2018). Indeed, in
the context of the upcoming ESA PLATO mission and the extreme precision radial
velocity (EPRV) surveys, the interplay between the photometric and
spectroscopic observables will be key for mitigating this stellar noise source
that limits the detection and characterization of both stellar oscillations
and exoplanet signatures. In line with Sulis et al. (2020a), we expect a high
sensitivity to this stellar phenomenon with the future high-precision
photometric PLATO observations (at least in the $24$-camera mode).

Finally, we also explored whether it was possible to detect the impact of
granulation-induced changes on the CCF shape, corresponding induced RV shifts
and brightness changes (Sec. 8). This analysis was complicated by the presence
of p-modes excited by the granulation and occurring on similar timescales. The
RV shifts induced by the p-modes dominate over the granulation and efforts to
mitigate and/or disentangle the p-modes likely also squash the granulation
signal. Moreover, the long timescales required to bin or filter out the
p-modes for the hotter F star (HD 88595) mean that even though the granulation
signal is also expected to be larger, it is still equally tricky to
disentangle. Nonetheless, there are potential hints that the CCF bisector
curvature may provide information on the granulation signal, at least for the
F star. If the p-modes are sufficiently filtered (i.e., with a cutoff
frequency corresponding to $\sim$100-150 minutes or more), there also appears
to be a correlation observed between flux and RV for the F star that is
consistent across nights. Unfortunately, the G star (HD 67458) observations
seem to limited by signal-to-noise as the apparent correlations between shape
and RV are signal-to-noise dependent. Future observations are needed to
further determine the impact of granulation on the overall brightness, CCF
shape and net RV shifts. Additionally, a more refined treatment of the p-modes
may also be key to further unveiling the granulation behavior; this is
currently under analysis and will be the subject of a forthcoming paper.

###### Acknowledgements.

The authors thank the anonymous referee for her/his helpful comments that
improved the quality of the manuscript. Overall, the paper has been
significantly improved in terms of both science and writing. CHEOPS is an ESA
mission in partnership with Switzerland with important contributions to the
payload and the ground segment from Austria, Belgium, France, Germany,
Hungary, Italy, Portugal, Spain, Sweden, and the United Kingdom. The CHEOPS
Consortium would like to gratefully acknowledge the support received by all
the agencies, offices, universities, and industries involved. Their
flexibility and willingness to explore new approaches were essential to the
success of this mission. The authors acknowledge the ESPRESSO project team for
its effort and dedication in building the ESPRESSO instrument. This paper used
data taken during the program IDs: 106.2186.001, 106.2186.002, 106.2186.003,
106.2186.004, 106.2186.005, 106.2186.006. This paper includes data collected
by the TESS and Kepler missions, which are publicly available from the
Mikulski Archive for Space Telescopes (MAST). Funding for these missions is
provided by NASA’s Science Mission directorate. The VIRGO and GOLF instruments
onboard SoHO are a cooperative effort of scientists, engineers, and
technicians, to whom we are indebted. SoHO is a project of international
collaboration between ESA and NASA. We acknowledge X. Dumusque, Astronomy
Department Geneva Univ., Switzerland, for the public HARPS-N data that have
been made available.
This research has made use of the VizieR catalog access tool, CDS, Strasbourg,
France (DOI : 10.26093/cds/vizier). The original description of the VizieR
service was published in 2000, A&AS 143, 23
We acknowledges M. Oshagh for sharing the dataset from Oshagh et al. (2017).
SS acknowledges support from CNES, the Programme National de Planétologie
(PNP), and the Programme National de Physique Stellaire (PNPS) of CNRS-INSU.
M.L. and S.S acknowledge support of the Austrian Research Promotion Agency
(FFG), under project number 859724 (GRAPPA). M.L. acknowledges support of the
Swiss National Science Foundation under grant number PCEFP2_194576. The
contributions of M.L. and A.K. have been carried out within the framework of
the NCCR PlanetS supported by the Swiss National Science Foundation. H.M.C.
acknowledges support from the UKRI under a Future Leader Fellowship (grant
number MR/S035214/1). Funding for the Stellar Astrophysics Centre is provided
by The Danish National Research Foundation (Grant agreement no.: DNRF106). L.
F. R. D. acknowledges support from the Independent Research Fund Denmark
(Research grant 7027-00096B). The numerical results presented in this work
were obtained at the Centre for Scientific Computing, Aarhus
http://phys.au.dk/forskning. Part of the calculations have also been performed
using the OCA/SIGAMM mesocentre. This research was undertaken with the
assistance of resources provided at the NCI National Facility systems at the
Australian National University through the National Computational Merit
Allocation Scheme supported by the Australian Government. V.V.G. is an
F.R.S-FNRS Research Associate. ACC acknowledges support from STFC consolidated
grant numbers ST/R000824/1 and ST/V000861/1, and UKSA grant number
ST/R003203/1. PM acknowledges support from STFC research grant number
ST/M001040/1. LBo, GBr, VNa, IPa, GPi, RRa, GSc, VSi, and TZi acknowledge
support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. MF and CMP gratefully
acknowledge the support of the Swedish National Space Agency (DNR 65/19,
174/18). S.G.S. acknowledge support from FCT through FCT contract nr.
CEECIND/00826/2018 and POPH/FSE (EC). ACC and TGW acknowledge support from
STFC consolidated grant numbers ST/R000824/1 and ST/V000861/1, and UKSA grant
number ST/R003203/1. SH gratefully acknowledges CNES funding through the grant
837319. This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (project Four Aces. YA and MJH acknowledge the support of the Swiss
National Fund under grant 200020_172746. We acknowledge support from the
Spanish Ministry of Science and Innovation and the European Regional
Development Fund through grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R,
PGC2018-098153-B-C33, PGC2018-098153-B-C31, ESP2017-87676-C5-1-R,
MDM-2017-0737 Unidad de Excelencia Maria de Maeztu-Centro de Astrobiología
(INTA-CSIC), as well as the support of the Generalitat de Catalunya/CERCA
programme. The MOC activities have been supported by the ESA contract No.
4000124370. S.C.C.B. acknowledges support from FCT through FCT contracts nr.
IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF and JL acknowledge their role as
ESA-appointed CHEOPS science team members. ABr was supported by the SNSA. This
project was supported by the CNES. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (project Four Aces, grant agreement No.
724427). It has also been carried out in the frame of the National Centre for
Competence in Research “PlanetS” supported by the Swiss National Science
Foundation (SNSF). A.De. acknowledges the financial support of the SNSF. The
Belgian participation to CHEOPS has been supported by the Belgian Federal
Science Policy Office (BELSPO) in the framework of the PRODEX Program, and by
the University of Liège through an ARC grant for Concerted Research Actions
financed by the Wallonia-Brussels Federation. L.D. is an F.R.S.-FNRS
Postdoctoral Researcher. This work was supported by FCT - Fundação para a
Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020
- Programa Operacional Competitividade e Internacionalizacão by these grants:
UID/FIS/04434/2019, UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-AST/32113/2017
& POCI-01-0145-FEDER- 032113, PTDC/FIS-AST/28953/2017 &
POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT. B.-O.D.
acknowledges support from the Swiss National Science Foundation
(PP00P2-190080). DG gratefully acknowledges financial support from the CRT
foundation under Grant No. 2018.2323 “Gaseousor rocky? Unveiling the nature of
small worlds”. M.G. is an F.R.S.-FNRS Senior Research Associate. KGI is the
ESA CHEOPS Project Scientist and is responsible for the ESA CHEOPS Guest