# The impact of two non-transiting planets and stellar activity on mass
determinations for the super-Earth CoRoT-7b

Ancy Anna John,1,2 Andrew Collier Cameron, 1,2 Thomas G. Wilson 1,2

1SUPA, School of Physics & Astronomy, University of St Andrews, North Haugh,
St Andrews, KY169SS, UK
2Centre for Exoplanet Science, University of St Andrews, North Haugh, St
Andrews, KY169SS, UK
E-mail<EMAIL_ADDRESS>
###### Abstract

CoRoT-7 is an active star, whose orbiting planets and their masses have been
under debate since their initial detection. In the previous studies, CoRoT-7
was found to have two planets, CoRoT-7b and CoRoT-7c with orbital periods 0.85
and 3.69 days, and a potential third planet with a period$\sim$9 days. The
existence of the third planet has been questioned as potentially being an
activity-induced artefact. Mass of the transiting planet CoRoT-7b has been
estimated to have widely different values owing to the activity level of the
parent star, the consequent RV “jitter”, and the methods used to rectify this
ambiguity. Here we present an analysis of the HARPS archival RV (RV) data of
CoRoT-7 using a new wavelength-domain technique, scalpels, to correct for the
stellar activity-induced spectral line-shape changes. Simultaneous modelling
of stellar activity and orbital motions, identified using the ${\ell_{1}}$\-
periodogram, shows that scalpels effectively reduce the contribution of
stellar variability to the RV signal and enhance the detectability of
exoplanets around active stars. Using kima nested-sampling package, we
modelled the system incorporating a Gaussian Process together with scalpels.
The resultant posterior distributions favoured a three-planet system
comprising two non-transiting planets, CoRoT-7c and CoRoT-7d with orbital
periods 3.697±0.005 and 8.966±1.546 days, in addition to the known transiting
planet. The transiting planet CoRoT-7b is found to be a rocky super-Earth with
a mass of Mb=6.06±0.65 $M_{\earth}$. The determined masses of Mc=13.29±0.69
$M_{\earth}$ and Md=17.14±2.55 $M_{\earth}$ suggest the non-transiting planets
CoRoT-7c and CoRoT-7d to be structurally similar to Uranus and Neptune.

###### keywords:

planets and satellites: detection, techniques: radial velocities, stars:
activity, line: profiles, individual: CoRoT-7 – planetary system.

††pubyear: 2021††pagerange: The impact of two non-transiting planets and
stellar activity on mass determinations for the super-Earth CoRoT-7b–22

##  1 Introduction

Orbiting planets exert a gravitational pull on their host star, which results
in a periodic variation of the star’s velocity along the line of sight of
observation. This wobbling effect can be inferred as radial velocity (RV)
information, which is a measure of the associated wavelength shifts. Shortly
after the detection of first exoplanet 51 Pegasi-b, Saar & Donahue (1997)
recognized that stellar activity severely impacts the measurement of RVs. If
the variable stellar surface phenomena influence the RV measurement (by
modifying the line shape or its position), then exoplanet detectability and
characterization are also affected. These include stellar oscillations,
granulation, spots, and faculae/plages, and long-term magnetic activity
cycles(Saar & Donahue, 1997; Dumusque et al., 2011). The most confusing
signals are those caused by the presence of active regions on the stellar
surface, which can show periodicities and amplitudes similar to the ones
generated by real planetary signals. Studies by Figueira et al. (2010) and
Robertson, Roy, & Mahadevan (2015) show that these signals may be harder to
disentangle.

Although the stellar signal significantly impacts the detection and
characterization, RVs have been widely used to detect exoplanets, especially
for low mass planets (Meunier, 2021). These low mass and/or long period
planets induce RV semi-amplitudes similar to or smaller than those produced by
stellar activity (López-Morales et al., 2016; Dai et al., 2017; Haywood et
al., 2018). Therefore, it is highly important to put effort in understanding
and correcting for stellar activity, as it is essential for accurate RV
detection and hence for characterizing interesting new exoplanets. Instrument
stability is no longer the dominant obstacle for the precise and accurate mass
determination of low-mass planets orbiting bright stars, considering the
advanced latest generation of instruments. However, it is often limited by the
intrinsic stellar variability (Saar & Donahue, 1997; Queloz et al., 2001;
Haywood et al., 2014). So, caution is needed while characterizing planets
discovered with the RV method around stars that are active/potentially active,
as this method is highly prone to the uncertainty arising from the host star’s
activity.

Various analysis techniques have been developed to understand and mitigate the
impact of stellar activity effects. The early efforts include decorrelation
against proxy indicators of activity, or simply rejecting a planet candidate
if the same periodicity was found in the FWHM (Full Width Half Maximum), BIS
(Bisector Inverse Slope) or in the chromospheric Ca II H&K emission flux, as
in the RV (e.g., Hatzes et al., 2010). Aigrain, Pont, & Zucker (2012) used
well-sampled light curves to predict the stellar variability expected in RV
measurements. Known as the FF′ method, this approach uses the light curve and
its first derivative to approximate the effects of spots and faculae, and
requires no information about the rotation period or spot modelling. It does,
however, require simultaneous high-precision photometry. Recently, Gaussian
Process regression has been used to model the correlated noise induced by
stellar activity in the RV signal, by fitting the orbital signal
simultaneously (Haywood et al., 2014). Although GP regression with a quasi-
periodic kernel is successful at fitting rotationally modulated activity,
other signals are also present.

As noted by Dumusque et al. (2015) and Cretignier et al. (2021), different
systematics on the spectra such as micro-tellurics, tellurics, colour
variation due to airmass and detector stitching can also affect the RV, on
timescales very different from those modelled with a single GP kernel. YARARA
(Cretignier et al., 2021) and WOBBLE (Bedell et al., 2019) were developed on
this ground as post-processing pipelines to cope with these features in the
spectra. YARARA uses Principal Component Analysis on HARPS spectral time
series to correct for the known non-planetary systematics without using models
for the different effects. WOBBLE is a data-driven method that can be used to
infer the stellar spectra, telluric spectra and RVs simultaneously from the
data (Bedell et al., 2019).

A complementary approach is to model the modes of variation present in the
spectral line profile independently of their temporal behaviour, as explored
by YARARA. While a planet produces a shift of all spectral lines, the stellar
activity affects the shape of the spectral lines. The FourIEr phase SpecTrum
Analysis (FIESTA, a.k.a. $\Phi$ESTA) (e.g., Zhao & Tinney, 2020; Zhao & Ford,
2022) uses the real and imaginary parts of the Fourier transform of the line
profile to disentangle apparent RV shift due to spectral line-profile
variability from true Doppler shift. Concurrently, de Beurs et al. (2021) also
developed machine learning techniques such as linear regression and neural
networks to separate activity signals from true center-of-mass RV shifts using
the average spectral line-shape variations, without any time-domain
information. Another recent investigation (Klein et al., 2022) employs Doppler
imaging to simultaneously model the activity-induced distortions and the
planet-induced shifts in the line profiles. It is challenging to differentiate
between the effects induced by a planet and stellar activity, as the latter
has varied effects on individual spectral lines (Dumusque et al., 2015).

Keeping this in mind, a new algorithm (scalpels\- Self-Correlation Analysis of
Line Profiles for Extracting Low-amplitude Shifts ) has been developed by
Collier Cameron et al. (2021) to decouple the RV information from the time
domain aiming to separate the Doppler shifts caused by orbital motion from the
apparent shifts caused by spectral line-shape variability triggered by stellar
activity. By analysing orthogonal modes of variation in the translationally-
invariant autocorrelation function of the line profile, this method of shape-
shift separation offers precise Doppler detection and characterization of
exoplanets around well-observed, bright main-sequence stars across a wide
range of orbital periods, especially for lower-mass planets.

Collier Cameron et al. (2021) demonstrated the algorithm using observations of
the solar spectrum from HARPS-N. They successfully verified and validated that
the algorithm can accurately detect multiple simulated planets injected into
heliocentric solar observations, spanning a wide range of orbital periods.
Wilson et al. (2022) also used the scalpels to clean up HARPS RVs for the
stronger detection of two sub-Neptunes in the TOI-1064 system.

Here we present the first in-depth investigative analysis to study the
potential of this approach to “clean-up” the RV of a stellar target other than
the Sun and thus to improve the fidelity of RV measurements. We apply scalpels
to the archival HARPS data on the CoRoT-7 system, which has been studied
extensively with time-domain methods, but for which the precise system
architecture remains a subject of debate. In Section 2 we summarize the
history of investigations of this system. In Section 3.1 we describe the
version of scalpels employed here and its joint use with the ${\ell_{1}}$
periodogram method and kima in the following subsections. In subsequent
subsections of Section 4 we examine the evidence for a third planet in the
system and the impact of the number of non-transiting planets on the planetary
mass of transiting planet CoRoT-7b extensively, followed by a discussion and
conclusion in Section 5 and 6.

##  2 Target and Observations

CoRoT-7 is a G9V type main-sequence star of V magnitude = 11.7, slightly
cooler ($T_{\rm eff}$ = 5250 ± 60 K) and younger (1.2–2.3 Gyr) than our Sun
(Hatzes et al., 2010). Here, we revisit the published HARPS RV measurements of
CoRoT-7. Léger et al. (2009) reported the detection of the first ever known
transiting super-Earth, CoRoT-7b with an orbital period of 0.85 days and a
measured radius of 1.68 ± 0.09$R_{\earth}$, which had the smallest
exoplanetary radius at that time. Following this discovery, a 4-month
intensive HARPS RV follow-up campaign was launched in order to measure the
mass of CoRoT-7b. These observations were carried out under ESO Program IDs
082.C-0120, 082.C-0308(A) and 282.C-5036(A) using the HARPS instrument on ESO
3.6 m telescope at La-Silla. CoRoT-7 was observed for 3 nights in 2008 March
and then observed continuously from 2008 November to 2009 February, spanning
over 4 months. In immediate course, Queloz et al. (2009) revealed CoRoT-7c,
the second planet with an orbital period of 3.69 d.

From the large amplitude of photometric modulation, the authors anticipated
the variations in RV to be highly perturbed by the activity of the host star.
Aiming to remove the RV variations caused by stellar activity, Queloz et al.
(2009) applied a harmonic decomposition to the data, preceded by a pre-
whitening procedure. In this ’cleaning’ procedure, the period of the stellar
rotation signal is identified by means of a Fourier analysis, and a sine wave
fitted with this period is subtracted from the data. The process of the
removal of the next strongest signals (which are the sinusoids at integer
multiples of the stellar rotation frequency) from the residuals was repeated
until the noise limit was achieved. The 0.85d signal associated with CoRoT-7b
was found to agree with the CoRoT transit ephemeris, and thus confirmed its
planetary nature. In this significant analysis, the author claims, apart from
the 0.85d and 3.69d signals, all other resultant signals to be linked to the
harmonics of stellar rotation period. In the study which revealed the
discovery of CoRoT-7b, Léger et al. (2009) performed a harmonic decomposition
of the rotational period and up to the first three harmonics to filter out the
activity signal from RV variations caused by the orbiting planet.

###  2.1 The second non-transiting planet candidate

Table 1: A timeline of wide range of reported planet masses, especially for
CoRoT-7b. It is evident that the mass of CoRoT-7c is nearly consistent. The
measured orbital periods for all (proposed and confirmed) planets are also
included.

| Planetary masses | Orbital periods
---|---|---
Authors | CoRoT-7 b | CoRoT-7 c | CoRoT-7 d | CoRoT-7 b | CoRoT-7 c | CoRoT-7 d
Leger et al.(2009)a |  < 21$M_{\earth}$ | – | – | 0.85 days | – | –
Queloz et al.(2009)b |  4.8 ± 0.8$M_{\earth}$ |  8.4 ± 0.9$M_{\earth}$ | – | 0.85 days | 3.69 days | –
Hatzes et al. (2010)b |  6.9 ± 1.4$M_{\earth}$ |  12.4 ± 0.42$M_{\earth}$ |  16.7 ± 0.42$M_{\earth}$ | 0.85 days | 3.69 days | 9.02 days
Ferraz-Mello et al. (2011)b |  8.0 ± 1.2$M_{\earth}$ |  13.6 ± 1.4$M_{\earth}$ | – | 0.85 days | 3.69 days | –
Boisse et al. (2011)b |  5.7 ± 2.5$M_{\earth}$ |  13.2 ± 4.1$M_{\earth}$ | – | 0.85 days | 3.69 days | –
Pont, Aigrain, & Zucker (2011)b |  2.3 ± 1.8$M_{\earth}$ | – | – | 0.85 days | – | –
Hatzes et al. (2011)b |  7.4 ± 1.2$M_{\earth}$ | – | – | 0.85 days | 3.69 days | 9.02 days
Haywood et al. (2014)c |  4.7 ± 0.9$M_{\earth}$ |  13.5 ± 1.08$M_{\earth}$ | – | 0.85 days | 3.69 days | 8.58 days (deemed as activity)
Tuomi et al. (2014)d |  4.8 ± 2.3$M_{\earth}$ |  11.8 ± 4.1$M_{\earth}$ |  15.4 ± 6.1 $M_{\earth}$ | 0.85 days | 3.71 days | 8.89 days (unsure about origin)
Barros et al.(2016)e |  5.5 ± 0.8$M_{\earth}$ | – | – | 0.85 days | – | –
Faria et al.(2016)c |  5.5 ± 0.8$M_{\earth}$ |  12.6 ± 0.7$M_{\earth}$ | – | 0.85 days | 3.69 days | 8.58 days (deemed as activity)

Data used : a CoRoT photometric LRa01& SOPHIE RV data, b HARPS RV 2008-9, c
HARPS RV 2012, d HARPS-TERRA RV 2008-9, e CoRoT photometric LRa06

Subsequently, the presence of a potential third planet, CoRoT-7d, with an
orbital period of 9.02d and a mass of 16.7±0.42 $M_{\earth}$ was disclosed by
Hatzes et al. (2010), making CoRoT-7 a compact system. Hatzes et al. (2010)
applied the pre-whitening procedure to the BIS, FWHM and CaII H & K emission
lines (spectral quantities used as measures of the host star’s activity) owing
to their in-dependency to the planetary dynamics. As no significant signal
came up at 0.85 and 3.69 days in any of these indicators, the planetary
candidacies of CoRoT-7b & CoRoT-7c were strengthened.

However, the calculated masses were not in good agreement with the previous
studies. This discrepancy in the mass calculation could possibly be rooted in
the difference in the methods adopted to cancel out the effect of stellar
activity signals from the RV data. Furthermore, no signal was found in the
periodogram of activity indicators at 9.02 days, and thus this RV signal was
attributed to a third-planetary companion in the system, CoRoT-7d. This signal
was previously detected by (Queloz et al., 2009) as well, but had been
attributed to a ‘two frequency beating mode’ arising from an amplitude
modulation of a 61 days signal. It was therefore deemed to be associated with
stellar activity, as this 61 days signal is quite close to twice the stellar
rotation period. Subsequently, Pont, Aigrain, & Zucker (2011) argued that the
data cannot be used to search for additional (non-transiting) planets in the
3–10 d period range and that claims of the detection of such planets
(‘CoRoT-7c’ and ‘CoRoT-7d’) do not withstand scrutiny.

Aiming to settle these arguments, simultaneous photometric and spectroscopic
observations were obtained in 2012 from CoRoT and HARPS (ESO Program ID
088.C-0323) for 26 consecutive clear nights in a row from 2012 January 12 to
February 6, with multiple well-separated measurements on each night adding up
to 77 observations in total (Haywood et al., 2014). These RV data were
reprocessed in the way as the 2008–2009 data (Queloz et al., 2009) using the
HARPS data analysis pipeline. Altogether, the system was observed for RV
studies using HARPS in 2008, 2009 and 2012 with a total of 177 measurements.

The goal of the 2012 observations was to observe the system with CoRoT and
HARPS simultaneously, and to use photometry as a proxy for activity-driven RV
variations. Along with updating the planetary parameters for CoRoT-7b, Haywood
et al. (2014) analysed this system further for the evidence of the additional
planetary companion reported by (Hatzes et al., 2010) at the period of 9.02
days. Although, they found a strong peak in the 6-10 days range in the Lomb-
Scargle periodogram, the marginal likelihoods calculated from their GP
regression analysis favoured the two-planet solution over a three-planet one
including the 9-day signal. They concluded that this signal was more likely
associated with the second harmonic of the stellar rotation at $\sim$7.9 d.
However, they successfully confirmed CoRoT-7c along with CoRoT-7b and improved
their planetary parameters. Faria et al. (2016) carried out a model-comparison
study employing GP regression to model the activity alongside simultaneous
orbit fitting in a nested-sampling scheme using the full CoRoT-7 RV data set.
While they found evidence for a weak signal at 9 days, the odds’ ratio again
favoured the 2-planet solution. Tuomi et al. (2014) presented an analysis of
the HARPS- TERRA (Template-Enhanced RV Re-analysis Application) velocities of
CoRoT-7 that suggested a system of two planetary companions (CoRoT-7b &
CoRoT-7c), possibly three (CoRoT-7d).

###  2.2 The mass of transiting planet

Table 1 lists the reported planet masses for CoRoT-7b spanning a wide range
from 2.3$M_{\earth}$ to 8$M_{\earth}$, which is not ideal. This is plausibly
due to the activity level of the star that contributes a significant amount of
RV “jitter” and how the various methods correct for this ambiguity. It also
depends on how many non-transiting Keplerian signals are included in the
model. The method of removal of activity signal will also affect the RV
amplitude, which in turn can bias the mass determination of companions. All
the aforementioned mass determinations used the same HARPS RV data set, which
clearly highlights the trouble in planets’ mass determination around an active
star like CoRoT-7 (Hatzes et al., 2011).

Collectively, these studies outline the critical role of stellar activity and
non-transiting signals in contaminating the RV observations not only in this
particular system but also in general. Therefore, it is crucial to understand
this role to enhance our ability to detect low-mass planets and thereby
measure their masses precisely. The present study aims to redetermine the
planet masses using the entire set of HARPS RV measurements of CoRoT-7
obtained so far. We explore techniques for determining the masses and orbital
elements of planets discovered around active stars. We expect this study to
offer precise RV measurements corrected for stellar activity and hence to
resolve the existing debates on the number of companions in CoRoT-7 planetary
system and to update their masses with improved precisions.

##  3 Methods

###  3.1 SCALPELS

The analyses of the CoRoT-7 system architecture described above employed time-
domain methods such as pre-whitening and GP regression to model the
contribution of stellar activity to the RV signal. Collier Cameron et al.
(2021) developed the scalpels algorithm as an alternative, wavelength-domain
method for separating Doppler shifts of dynamical origin from apparent
velocity variations arising from variability-induced changes in the stellar
spectrum.

scalpels seeks to decouple the effects of genuine dynamical Doppler shifts
from spurious variations caused by line shape changes arising from stellar
activity and instrumental systematics. scalpels uses the translational
invariance property of the autocorrelation function (ACF; Adler&Konheim 1962)
of the cross-correlation function (CCF) to isolate the effects of shape
changes in the CCF from shifts. In their analysis, the ACF, $A(\delta v)$ has
been described as the expectation value of the vector cross-product of the CCF
with itself at a sequence of lags $\delta v$:

$A(\delta v)={\rm
E}(\mathbf{\mathrm{CCF}}(v)\cdot\mathbf{\mathrm{CCF}}(v+\delta v))$ (1)

Collier Cameron et al. (2021) carried out a Principal-Component Analysis of
the CCF and its ACF, using data from the HARPS-N solar telescope feed. The
temporal variability of the CCF contains both line shape changes and Doppler
shifts. They found that the majority of time variations of the CCF in general
appear similar to those of the shift-invariant profile shape changes probed by
the ACF. The principal modes of variability in the CCF or ACF can be isolated
by calculating the Singular-Value Decomposition (SVD) of the ensemble of CCFs
or ACFs respectively, and can be expressed as:

$\mathbf{C}(v_{i},t_{j})=\left<\mathbf{C}(v_{i})\right>+\mathbf{U}_{C}(t_{j})\cdot{\rm
diag}(\mathbf{S}_{C})\cdot\mathbf{P}_{C}(v_{i}).$ (2) $\mathbf{A}(\delta
v_{i},t_{j})=\left<\mathbf{A}(\delta
v_{i})\right>+\mathbf{U}_{A}(t_{j})\cdot\rm{diag}(\mathbf{S}_{A})\cdot\mathbf{P}_{A}(\delta\mathit{v_{i}}).$
(3)

The perturbations resulting from solar activity were isolated by projecting
the RVs on to the time-domain subspace spanned by the amplitude coefficients
$\mathbf{U}_{A}$ of the ACF’s principal components (Collier Cameron et al.,
2021). On the contrary, dynamical shifts due to the planets are preserved when
projected onto the orthogonal complement of time-domain subspace.

The variation of the shape-driven component of the RV with time can be
obtained from the sum of scaled velocity contributions from all principal
components of ACF,
$\mathbf{v}_{\parallel}=\mathbf{U}_{A}\cdot\hat{\mathbf{\alpha}}$, where
$\hat{\mathbf{\alpha}}$ is the vector of response factors obtained by taking
the inner product $\mathbf{U}_{A}^{T}$ with the mean-subtracted time series of
observed velocities. The transformation from observed to shape-driven
velocities can be considered as a linear projection of the observed velocities
into a subspace spanned by the columns of $\mathbf{U}$:

$\mathbf{v}_{\parallel}=\mathbf{U}_{A}\cdot\mathbf{U}_{A}^{T}\cdot(\mathbf{v}_{\rm
obs}-\left<v\right>_{\rm obs}).$ (4)

These shape-driven perturbations are strongly correlated with the observed
RVs, as seen from the first panel of Figure 13, which could reliably reproduce
the stellar activity dominated long-term and short-term fluctuations (Collier
Cameron et al., 2021). Even though this offers a linear decorrelation, the
time-coefficients of the scalpels basis functions exhibit time lags relative
to the observed RV signals, which needs to be addressed.

$\mathbf{v}_{\perp}$ is defined as the residual velocity which lies outside
the $\mathbf{U}_{A}$ subspace, and can also be considered as a projection into
the orthogonal complement of the ACF subspace, where the planet signals are
sought:

$\mathbf{v}_{\perp}=(\mathbf{v}_{\rm obs}-\left<v\right>_{\rm
obs})-\mathbf{v}_{\parallel}=(\mathbf{I}-\mathbf{U}_{A}\cdot\mathbf{U}_{A}^{T})\cdot(\mathbf{v}_{\rm
obs}-\left<v\right>_{\rm obs})$ (5)

To avoid overfitting, a reduced-rank version of $\mathbf{U}$ is employed.
Collier Cameron et al. (2021) describe the determination of optimal rank using
a leave-one-out cross-validation (LOOCV). This method is found to be efficient
in identifying the number of leading columns of $\mathbf{U}$ that contains
significant profile information.

In this study, we apply this variant of the scalpels method to the CoRoT-7
system together with a GP regression. The host star’s high level of stellar
activity has been a persistent obstacle to previous efforts using other
methods to determine the number of non-transiting planets present, and to
determine their masses. The system’s intrinsic scientific interest complements
its suitability as a test target of the scalpels method.

###  3.2  $l_{1}$-periodogram

The ${\ell_{1}}$\- periodogram (Hara et al., 2017) is primarily designed to
identify the sparsest set of orbital signals that fit the observed RVs. This
sparsity is achieved by simultaneously fitting Keplerian signals on an
optimally sampled grid of frequencies to the data, regularized by minimizing
the $\ell_{1}$ norm of their amplitudes. Hara et al. (2017) present the
algorithm as a tool to search for a representation of the input signal as a
sum of a few sinusoidal components, that is, a representation which is sparse
in the frequency domain.

As the primary step, the data is normalized to the mean of the RV data to get
an estimate of stellar noise. The selection of frequency grids ${\omega}$,
covariance matrix $\mathbf{V}$ and the width of averaging interval ${\eta}$
are done in succession. This algorithm suggests that RV measurements can be
well approximated by the linear combination of a few vectors of
$e^{-i{\omega}t}$ and $e^{i\omega t}$, which spans the columns of the
$\mathbf{A}$ matrix.

The noise is assumed to be drawn from a Gaussian with the covariance matrix
$\mathbf{V}$. If the stellar rotation information $P_{\rm rot}$ is given, the
covariance matrix uses a quasi-periodic kernel. The contribution of activity
to the covariance matrix is then modelled as the product of an exponential
correlation term and a quasi-periodic term.

${V}^{({\rm
act})}_{ij}=\exp\left(-\frac{(t_{i}-t_{j})^{2}}{2\tau^{2}}\right)\left(0.5+{\rm
cos}\left(\pi\frac{(t_{i}-t_{j})}{P_{\rm rot}}\right)\right)$ (6)

The full covariance matrix including the noise terms can be then expressed as:

$\mathbf{V}=\mathbf{\delta}_{\rm k}+{\sigma_{R}^{2}}{V}^{({\rm
act})}_{ij}+\mathbf{I}{\sigma_{W}^{2}}+{\sigma_{C}^{2}}$ (7)

where $\mathbf{\delta_{k}}$ is a diagonal matrix spanned by the RV
uncertainties and $\sigma_{W}$ is the white noise amplitude, which we
considered as 0.1 ms-1. $\sigma_{R}$ stands for the red noise, which is 0.5
ms-1 here multiplied with an identity matrix $\mathbf{I}$. In this simple
covariance model, a block-diagonal calibration-noise covariance component
${\sigma_{C}}$ is also included to account for the night-to-night calibration
errors, which are known to be present in HARPS data at the $\sim$0.5 ms-1
level.

A remarkable advantage of the method is the use of $\ell_{1}$ norm weighting
to find periodicity in unevenly sampled signals. Even without any prior
knowledge of the number of planets, the intrinsic signal information can be
efficiently decoded. The algorithm assigns an amplitude to every frequency on
the grid. It then tensions the goodness of fit to the data (using $\chi^{2}$,
which is the "$\ell_{2}$ norm") against the $\ell_{1}$ norm (median absolute
deviation) of all the amplitudes on the frequency grid. The goal is to
identify the minimal set of sinusoids that give an optimal fit to the data.
The optimal solution is then found iteratively.

Moreover, we also used its capability to fit a set of externally-determined
basis functions simultaneously with the orbit fit. If the columns of the
reduced-rank matrix $\mathbf{U}_{A}$ are used for this purpose, this is
equivalent to the simultaneous sinusoidal orbit fitting method described in
Section 3.3. This inter-operability with the scalpels enables us to perform
shape-shift signal separation simultaneously with identifying the minimal set
of orbit signals present.

###  3.3 Simultaneous sinusoidal planet orbit fitting

We employed the shape-signal separation simultaneously with fitting the
orbits, given prior knowledge of the orbital periods obtained from the
$\ell_{1}$-periodogram analysis. A simultaneous modelling was chosen to be the
best way to figure out the orbital solution, as a simple periodogram has the
limitation of fitting only a single sinusoid per frequency sample, so that the
interaction between various signals may lead to inaccurate amplitude
estimation. To determine the impact of the scalpels signal separation,
(Collier Cameron et al., 2021) used this approach in the solar RV data, with
properly injecting a few planet signals that should be orthogonal to all
elements of $\mathbf{U}_{A}$. This prevented the signals from being partially
absorbed during the scalpels projection.

Estimation of parameters and signal separation can be achieved in a single
linear calculation, once the candidate signal periods have been determined via
periodogram search or through prior knowledge of transits. In the case of
CoRoT-7, we have an estimate of the periods of the three candidate signals
from the blind periodogram search and $\ell_{1}$-periodogram, while the period
and phase of the CoRoT-7b signal were obtained more precisely from transit
observations.

As described by Collier Cameron et al. (2021), the net orbital velocity vector
$\mathbf{v}_{\rm orb}$ for a set of $n$ planet signals can be modelled as the
product of a set of coefficient pairs $\mathbf{\theta}_{\rm
orb}=\\{A_{1},B_{1},\cdots,A_{n},B_{n}\\}$ with an array of time-domain
function pairs
$\mathbf{F}=\\{\cos\omega_{1}t_{j},\sin\omega_{1}t_{j},\cdots,\cos\omega_{n}t_{j},\sin\omega_{n}t_{j}\\}$,
where $\omega_{k}$ is the orbital frequency of the $k^{th}$ planet:

$\mathbf{v}_{\rm orb}=\mathbf{F}\cdot\mathbf{\theta}_{\rm orb}$ (8)

If we consider the planet orbits to be circular, $\mathbf{v}_{\rm orb}$ from
Equation 8 could then provide a complete model of the RV data as the sum of
shift (planet)-driven velocity variations and the shape (stellar
activity)-driven velocity variations (Collier Cameron et al., 2021). Treating
this as a least-square problem enabled us to solve for the unknowns, the
amplitudes, and phases of $\mathbf{F}$. The vector ${\mathbf{\theta}_{\rm
orb}}$ can then be evaluated by minimizing the $\chi^{2}$. Similar
Implementation has been employed by (Wilson et al., 2022) as well. The
detailed algorithm is given in Appendix B.2.

###  3.4 Trans-dimensional Nested Sampling using _kima_

We used the kima package of Faria et al. (2018) to sample from the posterior
distribution of the orbital model parameters. kima employs a Diffusive Nested
sampling algorithm of Brewer, Pártay, & Csányi (2010) for calculating the
evidence or fully marginalized likelihood not only for a model with a fixed
number of Keplerian signals $N_{p}$, but also after marginalizing over
$N_{p}$. The joint posterior distribution was calculated using the prior
${p(\Theta\,|\,M)}$ , likelihood ${p(D\,|\,\Theta,M)}$ and evidence
information ${p(D\,|\,M)}$ from the Bayes theorem:

$\mathbf{p}(\Theta\,|\,{D,M})=\frac{\mathbf{p}(\Theta\,|\,M)\,\,\mathbf{p}(D\,|\,\Theta,M)}{\mathbf{p}(D\,|\,M)}$
(9)

where $M$ is the model with $\Theta$ as the vector of all considered
parameters and $D$ is the RV data series. Undertaking a similar approach to
Haywood et al. (2014) and Faria et al. (2016), we also included a quasi-
periodic Gaussian process(GP) to account for the correlated noise with the
objective of investigating the improvement in the posterior when using the
scalpels shift RVs instead of the raw RVs uncorrected for the shape
variations.

The kima covariance matrix also includes a white-noise term. Unlike the
${\ell_{1}}$\- periodogram, kima does not yet model calibration noise by
imposing correlations between observations made on the same night.
Nonetheless, this white-noise contribution serves a similar purpose and
enables us to optimise the amplitude of the calibration noise used in the
${\ell_{1}}$\- periodogram. The quasi-periodic component of the GP is defined
by the kernel $k(t,t^{\prime})$ (a.k.a. covariance function):

${k(t,t^{\prime})}={{\eta_{1}}^{2}}\rm
exp\left(-\frac{(t-t^{\prime})^{2}}{2{\eta_{2}}^{2}}-\frac{2}{\eta_{4}^{2}}\rm
sin^{2}\left(\frac{\pi(t-t^{\prime})^{2}}{{\eta_{3}}}\right)\right)$ (10)

The hyperparameters of $k(t,t^{\prime})$ are represented by the variables
$\eta_{1}$, $\eta_{2}$, $\eta_{3}$ and $\eta_{4}$. $\eta_{1}$ and $\eta_{2}$
define the amplitude of correlation between RVs at different time separations
and the evolution timescale of active regions. The correlation timescale
$\eta_{3}$ reflects the stellar rotation period, $\eta_{4}$ governs the
variation timescale of GP latent functions in relation to $\eta_{3}$. Smaller
values of $\eta_{4}$ indicate more short-scale structures within a single
stellar rotation period. We used this GP to model the correlated noise
occurring on time-scales of the order of stellar rotation period and its
harmonics.

We also computed the False Inclusion Probability (FIP) and True Inclusion
Probability (TIP) from the joint posterior distribution of orbital parameters
and $N_{p}$. Hara et al. (2021) introduces the FIP as the probability that
there is no planet with period $P$ in a given frequency interval $I$. On the
other hand, the TIP is defined as the counterpart of the FIP, that is, the
probability that there is a minimum of one planet with $P$ $\epsilon$ $I$. The
TIP is calculated by averaging the planet detection over the possible number
of planets, $\frac{N_{p}(i)}{\Sigma\,\,N_{p}}$. Hara et al. (2021) presents
this detection criterion as an efficient way to evaluate the reliability of
significance levels, by effectively accounting for aliases and favouring to
discard the presence of planets under a certain confidence level.

Figure 1: The spectroscopic information measured during the HARPS run spanning
over November-2008 and February-2009 is given in the left figure. The RV,
FWHM, Area, BIS and S-index data have been plotted against the Barycentric
Julian date and shown in each panels from bottom to top. The figure at the
right shows the respective representation of spectroscopic data over the span
of the 2012 HARPS run. The dashed grey lines running horizontally in the top
panels of both figures show the mean values of S-index, indicating the
difference in the level of stellar activity during both observing runs. There
are arrows given in the RV, FWHM and Area panels of the left figure,
pinpointing that the RV peaks occur $\sim$ 2 days before the FWHM and Area
peaks. The respective uncertainties are shown using grey error bars.

###  3.5 Stacked Bayesian General Lomb-Scargle Periodograms

To probe the statistical implication of the planet candidate signals, a method
of stacking the Bayesian General Lomb-Scargle (BGLS) periodograms has been
used. This tool was developed for identifying the periodicities caused by
stellar activity, and to show how it can be used to track the signal-to-noise
ratio (SNR) of the detection over time (Mortier & Collier Cameron, 2017).
Adding in more observations, should increase the power or SNR measure of the
signal, if the signal is real and coherent. On the contrary, if the signal is
incoherent or short-lived, adding more data would cause the SNR or Lomb-
Scargle power to decrease (e.g., Howard et al., 2011; Hatzes, 2013; Mortier &
Collier Cameron, 2017). Hence, signals arising from stellar activity can be
identified by monitoring their variable and incoherent nature. To track the
significance of periodic signals, the BGLS periodogram was calculated for a
smaller data subset and recalculated repeatedly with subsequent addition of
observations.

Mortier & Collier Cameron (2017) recommend that one should not solely rely on
this approach to establish the planetary nature of a signal. This can be used
as an additional test to reinforce a finding. We are aiming is to devise a
workflow by combining the methods mentioned in Section 3.1, 3.2, 3.3, 3.4 and
3.5.

##  4 Data analysis and results

###  4.1 Time series analysis

In addition to measuring the RV, the HARPS Data Reduction Software returns
several shape diagnostics for the CCF and flux indices for known activity-
sensitive spectral lines. The RV data and supplementary information including
the S-index, BIS, Area and FWHM from 2008-9 and 2012 campaigns are shown in
left and right panels of Figure 1. The quantity ’Area’ is the product of the
FWHM of the CCF and its contrast, i.e., the fractional depth of the CCF
measured relative to the continuum level. The y-axis values for each parameter
in both observing seasons are set to have the same limits for easy comparison.

The supporting measurements of the ’activity indicators’ such as the FWHM and
BIS of CCF and the Ca II H and K values from HARPS allow us to examine the
intrinsic stellar variability. Queloz et al. (2009) suggests that one can
check two simple characteristics to look for changes in the shape of CCF: the
width parameter FWHM and its BIS, computed following Queloz et al. (2001). The
chromospheric activity is typically quantified by the S-index, as noted by
Wilson (1978). The S-index varies linearly with the chromospheric emission
flux in the cores of the Ca II H and K lines in the near ultraviolet (Salabert
et al., 2016). From both panels of the Figure 1, the mean value for S-index
can be calculated as 0.3569 in the 2008-9 season and 0.2759 in 2012 and is
denoted by the grey horizontal dashed line. The lower mean value in 2012
indicates that the star was less active during 2012.

Similarly, the RV measurements possessed an RMS scatter of 10 ms-1 in the
2008-9 run and a lesser scatter value of about 6.8 ms-1 in 2012, showing that
the amplitudes of RV variability were lower in 2012 than 2008-9. The mean
level of RV amplitudes were comparable (31.193 and 31.182) during the two
campaigns. However, both sets exhibit obvious multi-periodic variability
structures. The FWHM variations exhibit a fairly smooth periodic pattern, with
a period of $\sim$23 days in both seasons, as seen in Queloz et al. (2009).

A close inspection of Figure 1 shows a temporal offset between FWHM, Area and
the RV. The RV maxima occur $\sim$2 days prior to those of FWHM and Area. In
the Sun, Collier Cameron et al. (2021) observed similar time-lags between the
RV, FWHM and BIS. Such temporal shifts between RV and several activity proxies
were reported in previous studies as well (e.g., Santos et al., 2014; Queloz
et al., 2009). While Queloz et al. (2009) suggest these time-structured
variations between these two parameters to be associated with star-spot
related variability in CoRoT-7, Collier Cameron et al. (2021) note that the
same phenomenon is present even when the RV signal is dominated by facular
suppression of the convective blueshift. Haywood et al. (2014) detail the
suppression of convective blueshift by active regions surrounding star-spots
to have a much greater impact on RV than the flux blocked by star-spots,
particularly in slowly rotating stars like Sun and CoRoT-7.

The BGLS periodograms for the above-mentioned activity indicators over the
entire span of observation are given in Figure 18.

###  4.2 Blind Periodogram search

We applied scalpels to the barycentric RVs of CoRoT-7, and examined the
results using a Generalized Lomb-Scargle (GLS) periodogram (Zechmeister &
Kürster, 2009) for different subsets of the data (2008-9, 2012 and combined),
given the varied level of activity in both seasons as discussed in Section
4.1. The first two curves in Figure 2, shown in blue and orange in all panels,
show the observed barycentric RVs from which their own mean is subtracted
($\mathbf{v}_{\rm obs}-\left<v\right>_{\rm obs}$), and the shape-driven
($\mathbf{v}_{\parallel}$ ) RV perturbations derived using the scalpels
projection respectively. The green curve shows the cleaned shift-driven
velocities, obtained when the shape-driven variations are subtracted from the
observed RVs. Histograms showing the reduction of RMS scatter after separating
the shape-driven signals are presented for all three analyses in Section B.1.

Figure 2: The top panel corresponds to the data from 2008-9 run, while the
bottom panel shows 2012 data. In both panels, the blue curve at the top is the
barycentric RV subtracted from its own mean. The middle orange curve
represents the shape-driven component obtained from the scalpels projection,
while the green curve manifests the ’cleaned’ RV, which is the shift-driven
component. Please note that an offset of $\pm 60$ m s-1 is introduced for
clarity. The relatively small uncertainties are shown as semi-transparent
error bars to avoid overcrowding.

Figure 3: The left, middle and right figures represents 2008-9, 2012 and
combined data sets respectively. In each figure, the top periodogram (blue) is
for the measured velocities derived from the CCF. The middle trace (orange) is
for the shape-driven velocities ($\mathbf{v}_{\parallel}$ ) obtained from the
scalpels projection. The third periodogram (green) is the shift-driven
velocities ($\mathbf{v}_{\perp}$ ) remaining after the subtraction of shape-
driven velocities from the observations. Light blue dotted vertical lines
denotes the orbital periods of all confirmed and potential planetary
companions

Along with the confirmed planet signals with 0.85 day and 3.69 day orbital
periods, a third signal with a period of 8.96 days and an amplitude of $\sim$4
ms-1 also appears in all shift-driven periodograms, with no significant
counter-part in the shape-driven periodogram as seen from Figure 3. This
signal, noted previously by Hatzes et al. (2011) as being attributable to a
third planet CoRoT-7d, was suppressed in the GP regression analysis of Haywood
et al. (2014), attributing it to be more likely an activity signal. However,
we found no significant signal at 8.96 days period in the activity-induced
shape-driven periodogram, despite the dataset chosen. This argues against an
activity-related origin.

A curious feature of these periodograms is that in the 2012 and combined data,
the spurious 1-day periodicity seen in in the raw observations is almost
entirely accounted for in the shape signal. In the 2012 season, the target was
observed 2 or 3 times per night over 26 consecutive clear nights. The target
was therefore observed at a range of airmasses an a clear correlation was
observed between the airmass and one of the SCALPELS basis vectors
($\mathbf{U}_{2}$). We consider it possible that chromatic extinction could
produce a subtle change in CCF shape with airmass, and hence with local
sidereal time, despite the care with which the HARPS DRS fits the blaze
function and normalizes the continuum prior to cross-correlation. A detailed
investigation is, however, beyond the scope of this study.

A strong peak in the combined observed periodogram shows up at around 23 days
with an amplitude of 6.42 ms-1, which is the stellar rotation period. Although
a shift signal at this period is present, the appearance of a shape-driven
signal at the same period casts doubt on a planetary interpretation of this
signal. While the presence of a shift signal without a counterpart in the
shape signal points strongly to a planetary origin, we cannot completely
ignore the possibility of some form of activity in CoRoT-7 producing a shape
change in the profile that closely mimics a shift. Therefore, we have to
investigate further the nature of the 23 days signal.

The complex frequency structure of the periodogram for the combined 2008-9 and
2012 data (Figure 3) arises from both cycle-count uncertainty and aliasing. In
principle, all of this structure should be attributable to a finite number of
orbital signals combined with a quasi-periodic activity signal and noise.
Given the difficulty in interpreting the system using standard GLS
periodograms, specially when the window function is complex and there are
multiple, non-harmonic signals present, we recommend performing further
analyses to understand similar complex systems to overcome these inherent
problems.

Figure 4:  $\mathbf{Top}$: The ${\ell_{1}}$\- periodogram of the raw RVs of
CoRoT-7 after unpenalizing the transiting 0.85 days signal of CoRoT-7b, with
the strongest spikes marked with red dots at the top. Traces of peaks for the
planetary signals of CoRoT-7c (3.7 days) and CoRoT-7d (8.9 days) along with
the stellar rotation signal ($\sim$23 days) are present after the dominant
long-term activity signals. $\mathbf{Bottom}$: The ${\ell_{1}}$\- periodogram
for the Shift RVs after correcting for the shape variations, showing the
strongest peaks at orbital period of planets (8.96 and 3.71 days) and the
stellar rotation period with the activity trends suppressed significantly. The
corresponding False Alarm Probabilities (FAP) are also listed above each
panel.

###  4.3 $l_{1}$-periodogram

Figure 4 shows the ${\ell_{1}}$\- periodograms, after un-penalizing the
0.85-day signal of the transiting planet. In the ${\ell_{1}}$\- periodogram
terms, this means that we are always including the 0.85 days planet in the
fit. The top panel represents the ${\ell_{1}}$\- periodogram for the raw RVs,
dominated by the long-term activity signals.

An $\ell_{1}$ periodogram analysis for the Shift RVs is shown in the bottom
panel of Figure 4. After decorrelating the stellar activity-induced shape
variations using the scalpels $\mathbf{U}$ vectors, the two non-transiting
planet candidates dominate the periodogram. Now the 4 signals with the greater
significance and lowest false alarm probabilities are found at periods 3.70
days, 8.96 days and a closely spaced pair (23.69 and 22.94 days) around the
stellar rotation period. The long-term signals are weakened considerably. All
other strong peaks that showed up in the periodogram are identifiable as one-
day aliases ($|\frac{1}{P}\pm 1|^{-1}$) of these four signals.

The presence of two closely-spaced signals near 23d in the both RV sets
suggest that, in the $\ell_{1}$ periodogram’s sparse representation of the
data, the change in the amplitude of this signal between 2008-9 and 2012 is
modelled as a 1990 days beat pattern between the two closely-spaced sinusoids.

Figure 5:  The periodograms of RVs derived from the barycentric CCFs when the
signal separation and orbit fitting of the sinusoidal signals of periods
derived from the analysis associated with Figure 4 have been performed
simultaneously. $\mathbf{Left}$: The top three traces are as defined in the
caption of Figure 3.The fourth trace (red) shows the periodogram of the fitted
model of all the signals, including the 3 candidate planet signals (dark blue
vertical bars) and 3 pairs of closely spaced beat periods associated with the
rotation period and its harmonics (magenta vertical lines) found by the
${\ell_{1}}$\- periodogram. The orbital model signal (red) is now much closer
in appearance to the shift signal (green), leaving little residuals in the
bottom trace (purple), obtained after subtracting both the shape-driven and
orbital RV models from the onserved RVs. Each trace is offset by $\sim 10$ m
s-1 for better illustration. $\mathbf{Right}$:The resultant histogram of
residual RVs shows significantly reduced RMS scatter.

Figure 6:  $\mathbf{Bottom}$: The time-domain RV plots obtained from the
simultaneous modelling for 2008-9 and 2012 data series are given in the left
and right figures, colour-coded corresponding to each periodograms shown in
the previous panels. Semi-transparent error bars are also included in the
respective curves.

The presence of low-amplitude peaks in Figure 4 may also indicate the presence
of correlated, non-Gaussian noise, which we expect to be the case here. The
${\ell_{1}}$\- periodogram is clearly successful at shortlisting the most
plausible candidate planet signals in our data and suppressing the parasitic
signals. To validate the physical origin of these signals, we computed
additional $\ell_{1}$ periodograms for the shape-driven RVs and the activity-
indicators, as shown in Figure 16. None of these show traces of peaks at or
around 0.85, 3.69 and 8.96 days, adding to the case for their planetary
origins, especially for the third signal. Having identified the most plausible
signals, we now turn to fitting them simultaneously with scalpels.

###  4.4 Simultaneous sinusoidal planet orbit fitting

In Section 4.2, we projected the observed RVs, including planet signals, into
the space spanned by the scalpels basis vectors. Although, we found that there
is hardly any correlation between the shape and the shift RVs (See 13), if any
of the planet signals are correlated with any of the basis vectors, they may
get partially absorbed in the shape signal. This is problematic, as we cannot
guarantee orthogonality between any irregularly sampled orbital superposition
with the components of the scalpels $\mathbf{U}$ vectors. The orbital
perturbations of any planet and the scalpels projection process must therefore
be modelled self-consistently for the signal separation to recover their semi-
amplitudes as reliably as possible. For this reason, Collier Cameron et al.
(2021) solved simultaneously for shape-driven variations and a set of circular
Keplerian orbits with known periods. Here we use a similar approach, using the
set of orbital frequencies obtained with the ${\ell_{1}}$\- periodogram as
inputs. The approach of obtaining a simultaneous solution involved determining
the shape-driven variations from the difference between observed RVs and the
orbit model velocities (Collier Cameron et al., 2021).

The assumption of circular orbits for candidate signals of the three planets
can be justified by considering the tidal circularization timescales.
According to Dobbs-Dixon, Lin, & Mardling (2004) we could assume negligible
eccentricities for the signals at 0.85 and 3.69 days, having periods less than
6 days and hence accounted for by dissipation of tidal disturbances within
their envelopes that are induced by their host stars. For the third signal at
a period of 8.96 days, we calculated the circularization timescale as
described in Dobbs-Dixon, Lin, & Mardling (2004), and found a value of
$\sim$610 Gyr for an assumed planetary tidal quality factor
$Q_{p}^{\prime}=10^{6}$. We also relaxed the prior on eccentricity for the
analysis in Section 4.5 to allow the data to find the feasible value. The
obtained value for eccentricity falls around 0.057±0.062 (see Figure 20),
again supporting the assumption of circular orbits.

The left panel of Figure 5 shows the RV periodograms when the signal
separation and orbit modelling of all 4 strongest sinusoidal signals having
non-zero FAPs from the ${\ell_{1}}$\- periodogram search along with the
transiting planet signal and two pairs of periods at $P_{\rm rot}/2$ and
$P_{\rm rot}/3$ have been performed simultaneously. The orbital periods of
candidate planets are marked by vertical blue lines at 0.85 days, 3.71 days,
8.96 days and beat periods for stellar rotation periods and harmonics by
magenta lines at 22.94, 23.75, 10.42, 10.74, 7.30 and 7.35 days.

The top three traces are as defined in the Section 4.2, with the only
difference being the shape-driven periodogram here is obtained by subtracting
the orbit model from the raw RVs. The fourth trace (red) shows the periodogram
of the fitted orbit-model of the candidate signals. The bottom periodogram in
purple shows the residuals after subtraction of both the shape-driven and
orbital RV models. It is worth noting the level of matching between both
periodograms (green & red), even with the finest details, in this case.

Table 2: The periods and semi-amplitudes of the strongest signals in the periodograms of raw (${k=0}$) and shape-corrected apparent shift-velocities ($k=k_{\rm max}$) from simultaneous modelling of CCF shape changes and planetary motion, made with prior knowledge of the periods for candidate planets (from the ${\ell_{1}}$\- periodogram) and pairs of beat period at the stellar rotation period and its harmonics are listed below.  P(days) | K(ms-1) |  $\sigma_{K}$(ms-1) | K(ms-1) |  $\sigma_{K}$(ms-1)
---|---|---|---|---
| ${k=0}$ | ${k=0}$ | $k=k_{\rm max}$ | $k=k_{\rm max}$
0.8535 | 4.185 | 0.263 | 3.552 | 0.306
3.6963 | 5.484 | 0.254 | 6.012 | 0.312
8.9674 | 6.877 | 0.324 | 5.482 | 0.380
10.4269 | 0.472 | 0.569 | 1.156 | 0.606
10.7409 | 4.215 | 0.535 | 2.560 | 0.597
7.3039 | 2.173 | 0.536 | 2.391 | 0.620
7.3583 | 3.004 | 0.570 | 3.276 | 0.655
22.9425 | 6.915 | 0.288 | 4.891 | 0.353
23.6918 | 4.077 | 0.273 | 3.311 | 0.348

A closer look at the orange and green periodograms reveal that the balance of
the signal separation has been altered from those in Figure 3, by modelling
the orbital motion explicitly at known periods. The algorithm efficiently
reduces the correlation between the modelled orbit signals and modes of
velocity variation driven by line shape changes alone.

To determine the orbital velocity amplitudes that would be recovered without
applying profile-shape corrections to the velocities, we reduced the number of
principal components $k_{\rm max}$ to zero. We found that, when the sinusoids
are fitted to the observed RVs and left uncorrected for line-profile shape
variations, the semi-amplitudes of the recovered signals depart from the semi-
amplitudes recovered during the simultaneous modelling (see columns 2 and 4 of
Table LABEL:tab:recovery).

We also compared the recovered semi-amplitudes of the four strongest signals
when fitted with and without the 23.69 days beat signal found in the
${\ell_{1}}$\- periodogram search, in the appendix ( See Table 6). The last
two columns of both tables list the semi-amplitudes and associated
uncertainties for the shift-driven RVs acquired in both cases. While the
differences in the RV amplitudes for planets b, c and d between the 4-signal
model and the 5-signal model are small when compared with their uncertainties,
the 5-signal model gives a more faithful approximation to the behaviour of the
23-d activity signal between seasons.

The 23-day peak in the shift signal is therefore best modelled as the beat
pattern between the two closely spaced periods, requiring 2 frequencies to be
fitted for the stellar rotation period. In the 4-signal case, a single
sinusoid is fitted at this frequency, so the 23-day peak shows strong cycle-
count interference. We attribute this effect to the 1990 days beat period
between the 22.7 days and 22.9 days signals being greater than the entire
observation span of $\sim$1195 days. When both 23-day periods found by the
${\ell_{1}}$\- periodogram are fitted, the form of the 23-day peak in the
model signal (red) is much closer in appearance to the shift signal (green)
(See Figures 5 and 15). However, there is still some power in the residuals at
$P_{\rm rot}/2$ and $P_{\rm rot}/3$ (marked with pink vertical lines in Figure
15), when only fitting 5 signals. This indicates that a purely sinusoidal
model for the residual activity signal, even with two closely-spaced
frequencies, might be insufficient.

Therefore, it’s better to model the shift-like component of the activity-
driven RV signal in such a way as to minimise the residuals and make the
residuals as flat as possible. This motivated us to include pairs of sinusoids
for the $P_{\rm rot}/2$ and $P_{\rm rot}/3$ as shown in Figure 5. The RMS
scatter is now significantly reduced to 2.37 ms-1. The resultant orbital
periods, semi-amplitudes, and uncertainties of the sinusoidal signals included
in the orbit model are listed in Table LABEL:tab:recovery.

Hence, we conclude that the beat model is a better description of the long-
term evolution of the 23-day signal, making it unlikely that a fourth planet
signal is superposed on the activity signal at this period. The property of
the 23-d signal, being a quasi-periodic shift-like change in the CCF shape
produced by the activity, complicates the RV analysis.

In brief, all three candidate planet signals reported in the previous sections
were recovered successfully with improved precision in their $K$ amplitudes.
Particularly for the 8.97d signal, no significant counterpart was detected in
the shape RVs obtained from the simultaneous modelling irrespective of the
number of planet sinusoids included, strengthening the evidence for the
existence of CoRoT-7d, the third planet in the system. Our inability to fit
the $\sim$23-day signal satisfactorily with a single sinusoid indicates that
it is likely a stellar rotation signal with variable amplitude and phase.

###  4.5 Nested Sampling using GP +scalpels

Using kima (Faria et al., 2018), we sampled the posterior distribution of
orbital parameters along with a complete Keplerian solver and a Gaussian
Process. The posterior distribution constrained by the RV data was then used
to probe the credible number of planets given the data, the orbital parameters
and the planetary masses. The priors on the GP model parameters hyperpriors
were carefully tailored for CoRoT-7, as given in Table.3.

To account for the stellar activity, we decorrelated the scalpels U-vectors
(when $k=k_{\rm max}$) that are found to be significant contributors to the
shape signal, which are $\mathbf{U_{1}}$, $\mathbf{U_{2}}$ & $\mathbf{U_{3}}$.
Knowing the parameters of one planet provides a small amount of information
about the parameters of another planet (Faria et al., 2016). Keeping this in
mind, we treated the transiting planet CoRoT-7b as a known planet, adopting
the updated transit parameters in Barros et al. (2014). By putting tight
priors on the orbital period P and mid-transit time $T_{0}$ we ensure that the
phase of this transiting planet signal is consistent with the photometry.

For the known planet, we put a lower limit on $K$ derived from the mass it
would have if it were made of compressed water, with a radius of 1.528
$R_{\oplus}$ (Barros et al., 2014) that is around 1.56 $M_{\oplus}$ (according
to the Growth model interpretation of mass distribution proposed by Zeng,
Sasselov, & Jacobsen (2016)), giving an RV amplitude of $K_{min}$ $\sim$ 1.08
${\rm ms^{-1}}$. The upper limit was set to be about a physically-motivated
number of $K_{max}$ $\sim$ 10 m $s^{-1}$ corresponding to the 10 $M_{\oplus}$
expected for a pure iron planet of that size.

Taking into account the existing knowledge of the rotation period from the
literature (e.g., Queloz et al., 2009; Haywood et al., 2014) and the present
work, we restricted the GP hyperparameter $\eta_{3}$ (representing the stellar
rotation period) to a sensible range between 19 and 30 days. We adopted a
prior period distribution for non-transiting planets with periods between 0.5
days and the duration of the 2008-9 data, i.e., 100 days.

As an initial experiment, the number of planets $N_{p}$ was set to be a free
parameter in the MCMC, with a uniform prior between 0 and 5. Considering
$N_{p}$ as an unknown parameter, kima uses a birth-death MCMC proposed by
Brewer (2014), which allows estimating the improvements in evidence values
when the sampler switches between solutions with different $N_{p}$ values.

Table 3: Prior distribution for model parameters. $\cal{LU}$\- Log Uniform ; $\cal{MLU}$\- Modified Log Uniform; $\cal{K}$\- Kumaraswamy ; $\cal{U}$\- Uniform ; $\cal{G}$\- Gaussian  Notations | Orbital parameters | Priors
---|---|---
$P$ | Orbital Period |  $\cal{LU}$(0.5,100)
$K$ | Semi-amplitude |  $\cal{MLU}$(1,25)
$e$ | Eccentricity |  $\cal{K}$(0.867,3.03)
$\phi$ | Orbital phase |  $\cal{U}$(0, 2$\pi$)
$\omega$ | Longitude of line of sight |  $\cal{U}$(0, 2$\pi$)
| GP parameters |
$\eta_{1}$ | Amplitude of correlations |  $\cal{LU}$(1.6,2.7)
$\eta_{2}$ | Decay timescale |  $\cal{LU}$(15,60)
$\eta_{3}$ | Correlation period |  $\cal{U}$(19,30)
$\eta_{4}$ | Periodic scale |  $\cal{U}$(-1,0)
$s$ | Extra white noise |  $\cal{LU}$(0.5,max $v$)
| For known planet |
$P$ | Orbital Period |  $\cal{G}$(0.85359, 5.7e-7)
$K$ | Semi-amplitude |  $\cal{MLU}$(1.08,10)
$e$ | Eccentricity |  $\cal{U}$(0, .1)
$t_{0}$ | Time of mid-transit |  $\cal{G}$(54398.0776, 0.0007)
$\omega$ | Longitude of line of sight |  $\cal{U}$(-$\pi$,$\pi$)
$\beta$ | Activity indicators |  $\cal{G}$(0,10)

The resultant joint posterior distributions for the semi-amplitudes of the
signals are shown in the top panel of Figure 7. We found that two non-
transiting planets are favoured by the data, considering the ephemeris of the
transiting planet CoRoT-7b as known in the model. Amidst the posterior spread
in the parameter space, these are seen as heavily populated regions at $P_{c}$
= 3.69 days and $P_{d}$ = 8.96 days.

Reviewing this posterior distribution triggered another experiment by fixing
the number of planets as $N_{p}$=2 with the rest of the priors as the same as
in Table.3. The resultant posterior distribution of RV semi-amplitudes is
represented in the bottom panel of Figure 7, with well-constrained peaks
around $P_{c}$ = 3.69 days and $P_{d}$ = 8.96 days, within the orbital period
uncertainties reported in the literature. Supporting the presence of CoRoT-7c
and CoRoT-7d, the same constrained detail can be seen in the posterior
distribution of eccentricities also (See Figure 20). From the 10000 effective
samples in the joint posterior distribution, the evidence log$\cal{Z}$2 for
this model was calculated as log(${p(D\,|\,M)}$) = -475.46.

Table 4: The mass and mean-density (column 4 & 5) calculated for the transiting planet CoRoT-7b from the posterior distributions of orbital period and RV semi-amplitudes (column 2 & 3) considering different models are listed. Important note: CoRoT-7b is considered as a known planet in all these models with transit parameters from Barros et al. (2014) (See Table 3), who updated the planet radius as 1.528 ± 0.065 $R_{\oplus}$. $N_{p}$ is the fixed number of planets in each model. The first three rows give results from models with GP + scalpels and different number of planets, where the last three rows are the results for different models with GP alone and different $N_{p}$.  Model |  $P_{b}$[days] |  $K_{b}$[$ms^{-1}$] |  $M_{b}$ [$M_{\oplus}$] |  $\rho_{b}$[g cm-1] | log $\cal{Z}$ |  $\cal{Z}$n-$\cal{Z}$n+1
---|---|---|---|---|---|---
GP +scalpels |  |  |  |  |  |
$N_{p}$=1 | 0.853592 ± 6.01e-7 | 3.943 ± 0.559 | 5.595 ± 0.794 | 8.641 ± 0.269 | -479.20 | –
$N_{p}$=2 | 0.853592 ± 5.87e-7 | 4.291 ± 0.460 | 6.056 ± 0.653 | 9.355 ± 0.235 | -475.46 | 42.31
$N_{p}$=3 | 0.853592 ± 5.78e-7 | 4.278 ± 0.453 | 6.048 ± 0.643 | 9.322 ± 0.234 | -474.58 | 2.40
GP only |  |  |  |  |  |
$N_{p}=1$ | 0.853592 ± 5.48e-7 | 4.007 ± 0.550 | 5.679 ± 0.781 | 8.786 ± 0.264 | -478.58 | –
$N_{p}=2$ | 0.853592 ± 5.53e-7 | 4.293 ± 0.468 | 6.087 ± 0.664 | 9.394 ± 0.236 | -474.87 | 40.89
$N_{p}=3$ | 0.853592 ± 5.72e-7 | 4.331 ± 0.476 | 6.146 ± 0.676 | 9.459 ± 0.236 | -474.27 | 1.82

If we fix $N_{p}$=1, we recover the orbital parameters for CoRoT-7c only,
similar to those reported by Faria et al. (2016). The evidence is degraded to
log$\cal{Z}$1 = -479.20 for this model. The difference in log evidence between
the model with 2 non-transiting planets and the model with 1 non-transiting
planet is 3.8. This is in the substantial support range of the Jeffreys scale
for Bayesian evidence comparison (Jeffreys 1961). These results therefore
indicate a 2-planet model to be 42.31 times ($\cal{Z}$1-$\cal{Z}$2) more
probable than a 1-planet model. We emphasize that, when saying a 2-planet
model, we are referring to CoRoT-7c and CoRoT-7d, as the model considers the
orbital parameters of transiting planet CoRoT-7b as known.

Figure 7: The posterior distribution for semi amplitudes ($\rm ms^{-1}$) in
log frequency space ($\rm cd^{-1}$) is shown. The transiting 0.85 days planet
is considered as ’known’. The inset plots zoom into the clusterings
corresponding to the $\sim$9 days planet and $\sim$3.7 days planet (right),
showing the interference patterns occurred due to the widely separated
observing seasons. $\mathbf{Top:}$ Setting number of planets, $N_{p}$ as a
free parameter with uniform priors between 0 and 5. $\mathbf{Bottom:}$ $N_{p}$
is fixed at 2.

Figure 8: Top: The FIP periodogram of CoRoT-7 HARPS data, computed using the
joint posterior distributions obtained from KIMA, modelled withscalpels, GP
and $P_{b}$ as known. Bottom: TIP periodogram. Both periodograms strongly
favour 2 planets at orbital periods 3.69 days and 8.96 days, additional to the
known transiting 0.85 day signal.

Figure 9: $\mathbf{x-axis}$: Number of observations, $\mathbf{y-axis}$:
signal-to-noise ratio. The SNR plotted against the number of observations for
the optimal periods of individual candidate signals at periods 3.70 days, 8.97
days and 23.47 days.  Figure 10: The Stacked Bayesian General Lomb-Scargle
periodogram for CoRoT-7 with number of observations in y-axis plotted against
period in days displayed in logarithmic scale. The colour gradient designates
the logarithm of probability, where redder is more promising.

After ‘confidently detecting’ CoRoT-7b and CoRoT-7c, Faria et al. (2016)
pointed out that it is more likely that there are four planets according to
the posterior distribution. Their posterior showed smaller peaks around
$\sim$9 days and $\sim$2 days. However, adding the scalpels basis vectors to
the model for activity decorrelation enabled us to confidently recover the
orbital signal of CoRoT-7d at 8.96 days. We were unable to recover any signal
around 2 days, as it is removed as being some artefact arising from activity,
aliasing or stellar rotation harmonics. In an attempt to validate the
improvement in the model with the inclusion of the scalpels shape-signal
decorrelation vectors along with a GP, we compared the run with the scalpels
against the one without. The results are given in Table 4. We found that de-
trending the RVs for line shape variations using the scalpels basis vectors
yields a model that is 4.14 times more probable than a model that does not
include these activity variations.

####  4.5.1 FIP and TIP calculation

The zoomed-in view of the over density regions in Figure 7 revealed
interference patterns associated with CoRoT-7c and CoRoT-7d at 3.69 days and
8.96 days. It is interesting to consider the two seasons of observations in
2008-9 and 2012 as a ’double-slit experiment’, in which the path lengths
between the light waves from two slits give rise to an interference pattern
due to the phase shift. We see an analogous scenario here, with sharp peaks
separated by a frequency $\sim$ $\frac{1}{1200}$ $cd^{-1}$, accounting for the
time separation (1200 days) between two observing campaigns. This clearly
indicates that both signals are detected unambiguously in both seasons.

We performed the FIP analysis in the frequency space, setting the bin size
equal to the Nyquist frequency resolution of the 2008-9 season rather than the
full duration of the data. Since the probability of each planet is spread over
several interference fringes, we should marginalize over the uncertainty in
the cycle count to establish the existence or otherwise of the planet. A
frequency bin size of 0.005 cycles per day captures all probabilities while
fully resolving the structure of the 2008-9 periodogram.

Hara et al. (2021) presented FIP and TIP testing as good practice for the
analysis of RV data, given that assumptions on the priors and the noise model
(log$\cal{L}$) might be incorrect. Following this idea, we searched for
various planets simultaneously using a frequency window sliding along the
entire frequency space spanned by the posterior distribution of the 2-planet
model (with CoRoT-7b being known). Consequently, two strong peaks were found,
at 3.697 days and 8.965 days, both with significantly minimal FIP and maximal
TIP, reinforcing their planetary nature (Figure 8). To avoid being misled by
the shared power among adjacent interference peaks linked with relevant
signals, a basic histogram was used to find these most probable periods, See
Figure 21.

Being a trans-dimensional sampling algorithm, kima enabled us to directly
sample the joint posterior of the number of planets ($N_{p}$) and other
orbital elements. Several ambiguous detections are prompted at spurious
periods due to the cross-talk between different aliases. FIP and/or TIP
attenuated the invalid detections associated with these periods. Interesting
is the enfeebled TIP and very large FIP ($\sim$1) at the stellar rotation
period of $\sim$23 days and its harmonics, providing more evidence against the
planetary nature of a body at this period. From Figure 5 alone, one could
argue that since the stellar rotation period ($\sim$23 days) is seen in the
shift signal, any other period that also survives when the shape signals are
subtracted could also be activity-driven. However, a better validation for the
dynamical origin of the candidate signals is achieved here through
incorporating a GP to account for any remaining quasi-periodic signals. It is
promising to see that both the non-transiting planet signals with orbital
periods 3.69 and 8.96 days survives the GP and produces a 2-slit interference
pattern as shown in Figure 7.

###  4.6 Stacked Bayesian General Lomb-Scargle periodogram

To investigate the stability of the periodic signals found in the previous
analyses, we used a stacked normalized BGLS periodogram (Mortier & Collier
Cameron, 2017) of the shift-driven RV data. Following the method of Mortier &
Collier Cameron (2017), we derived the stacked periodograms of CoRoT-7 from
the cleaned shift-driven RVs obtained after the scalpels shape-signal
separation. The BGLS periodogram was initially obtained from the first 4 data
points and then refined the progressively by adding more one by one. This
analysis was done for different subsets of data, obtained from the two
observing runs (2008-9 & 2012) separately (See Section E.1) and also for the
entire data set as shown in Figure 10. Figure 10 is colour coded based on the
log probability of the periods in the BGLS periodogram. One should expect to
see the SNR of a planetary signal increasing asymptotically in proportion to
the square root of the number of data points added.

Figure 11: The most plausible composition of the transiting planet CoRoT-7b
backed by the derived mass and radius for minimum core pressure is shown as
obtained from the manipulateplanet tool developed by Zeng, Sasselov, &
Jacobsen (2016). The thick black curve in the ternary diagrams shows the
degeneracy of tri-layer model due to the trade-off between the three
components, but not due to the mass or radius measurement uncertainties.

Fringes began to appear in the stacked BGLS periodogram when analysed the both
data sets together, as seen from Figure 10. These are caused by the
uncertainty in the cycle count elapsed across the three-year gap separating
the 2008-9 and 2012 RV campaigns (Mortier & Collier Cameron, 2017). However,
the combined data set works well for tracking the candidate signals.

Except the 23 days signal, all other candidate signals stand out with clearly
growing probability features at periods 0.85, 3.69 and 8.97 days, supporting
the coherent nature of these signals. It is worth noting that the 23.47 days
signal grows unsteadily, with the strength maximizing around 70 to 100
observations and decreasing later on, see also Figure 9. We looked
particularly into the 8.97 days signal to check its behaviour after adding
more data points from the 2012 campaign and found that the SNR constantly
increases with more observations, as expected for a real and coherent signal.

####  4.6.1 Tracking the significance of detection

We tracked the significance of each strong signal in the Figure 10 by plotting
the SNR against the number of observations (Figure 9). The optimal frequency
for each candidate signal was obtained from the dominant peaks in the top line
of the stacked BGLS periodogram.

Each point in Figure 9 represents the maximum SNR for the set of all RVs up to
that point in time. We particularly looked into the 3.69 and 8.97 days signals
and found that they exhibit a monotonic rise towards asymptotic square-root
behaviour, confirming that they are strictly periodic and were present
throughout the observations, hence substantiating the existence of CoRoT-7c
and CoRoT-7d. As a result of the steady growth, their detections reach
$\sim$25$\sigma$ and $\sim$20$\sigma$ respectively.

On the other hand, the detection significance of the 23.47 days signal first
grows rapidly but then levels out and even drops at some point when more
observations are added. This is a clear sign of an incoherent signal. We have
an advantage of having longer-baseline which make this analysis relevant,
which would not be the case for shorter baselines and longer orbital periods.
This evidence again strengthens the activity origin of the 23-day signal, due
to its unstable nature throughout the period of observation.

##  5 Discussion

###  5.1  Mass & Mean density of CoRoT-7b

As discussed in Section 2.1, the mass of the transiting planet CoRoT-7b has
been a subject of debate for more than a decade. Table 1 shows the wide range
of masses as reported in the literature. This can be attributed to the
relatively high level of activity in the host star. Different posterior
distributions for semi-amplitudes, considering different models, are listed in
Table.3. We consider the model with GP + SCALPELS + $N_{p=2}$ \+ CoRoT-7bknown
as the optimal one. Several results motivated us to reach this conclusion. The
major motivation was the significant improvement of evidence ($\cal{Z}$) when
fixing the number of non-transiting planets at 2. This model yielded a RV
semi-amplitude of 4.291 ± 0.46 ms-1 for CoRoT-7b.

The planetary mass was calculated using this semi-amplitude, assuming a
circular orbit. The orbital inclination $i$ and stellar mass $M_{*}$ were
obtained from Barros et al. (2014) as 80.98 ± 0.51 degree and 0.915 ±
0.017$M_{\odot}$ respectively. We determined the mass of CoRoT-7b to be 6.056
± 0.653$M_{\oplus}$, making it a super-Earth. This mass measure is within the
uncertainty limit of several previous studies (e.g., Faria et al., 2016;
Hatzes et al., 2010). Incorporating with the planet radius of 1.528 ± 0.065
$R_{\oplus}$ from (Barros et al., 2014), the planet bulk density was
calculated as 9.355 ± 0.235 g cm-1.

We used the manipulateplanet tool developed by Zeng, Sasselov, & Jacobsen
(2016) to get an estimate on the planetary composition. The derived value of
mass and a minimum core pressure (3020 GPa) supports a planet with an iron
core (Fe : 58.7% by mass), Silicate mantle (MgSiO3 : 40.9%) and a thin layer
of water on the surface (H2O : 0.9%), favouring the rocky composition
discussed in the literature.

Figure 12: Mass–radius relation for planets with masses <10 $M_{\odot}$
represented by grey circles. The plot also includes Earth and Venus, for
reference. The lines show models of different compositions, with solid lines
indicating single composition planets (either H2O, MgSiO3, i.e., rock, or Fe).
The dashed and dotted lines indicate Mg-silicate planets with different
amounts of H2O and Fe. The green circles show the widely different mass
estimates for the transiting planet CoRoT-7b from the literature, where the
refined mass estimate from this study is marked in blue, with significant
improvement in uncertainty. We want to point out the reason for a few points
falling in a horizontal line with our estimate being the similar value of
planetary radius considered in the corresponding analyses.

For the given value of the mass and radius of the planet, we varied the core
pressure in the model to investigate the degeneracy in the composition of the
planet up to a maximum supported core pressure of 3346 GPa. This maximum core
pressure proposes a dense iron core (Fe : 96.3 %) immersed in deep ocean
mantle (H2O : 12.6 %) with a compromised silicate layer(MgSiO3 : 3.2 %). Also,
the lower value of planet mass (M-1$\sigma$) suggests a composition that can
be up to 78.6 % Fe, 6.2% MgSiO3 and 15.1% H2O. However, none of the above
compositions seem to be feasible, considering the high equilibrium temperature
(1756 K) of CoRoT-7b estimated assuming a zero albedo (Barros et al., 2014).
Therefore, the most plausible composition occurs to be the one discussed
earlier in this section, shown in Figure 11 with little or no surface layer of
water. It should be borne in mind that the manipulateplanet tool assumes a
condensed volatile layer typical of a cold planet. More recent studies, (e.g.
Zeng et al., 2021) show that a substantial water layer would establish a deep
steam atmosphere in a planet as strongly-irradiated as CoRoT-7b, increasing
the radius well beyond the observed value.

We also explored the masses of known planet CoRoT-7b for different models, as
tabulated in Table 4. As seen from the table, the mass estimation improves and
settles for the transiting + 2-planet model incorporated with GP regression
and the scalpels decorrelation. This approach takes into account the
contributions of all additional planets whose existence may not be established
conclusively, but whose RV variations influence the mass determination of the
known planet(s).

###  5.2 Masses of non-transiting planets

As listed in Table 5, we also computed the masses of non-transiting planets
from the RV semi-amplitudes obtained from the analysis described in Section
4.5. The inner non-transiting planet CoRoT-7c, with an orbital period of 3.69
days, was found to have a mass of 13.289 ± 0.689 $M_{\oplus}$. This planet
could be structurally similar to Uranus ($\sim$ 14.5 $M_{\oplus}$) in our
Solar System. The RV semi-amplitude corresponding to the second non-transiting
planet CoRoT-7d at 8.97 days suggests a planetary mass of 17.14 ± 2.55
$M_{\oplus}$. The mass of this long-period planet is comparable to that of
Neptune ($\sim$ 17.1 $M_{\oplus}$). However, the bulk density and composition
of these non-transiting planets are unknown, due to the lack of radius
information from the photometry.

Table 5: Planetary parameters for CoRoT-7b, CoRoT-7c and CoRoT-7d obtained from the best model. Planet | P[days] | K[ms-1] | M [$M_{\oplus}$]
---|---|---|---
CoRoT-7b | 0.8535±5.87e-7 | 4.291±0.46 | 6.056±0.653
CoRoT-7c | 3.697 ± 0.005 | 5.757 ± 0.298 | 13.289 ± 0.689
CoRoT-7d | 8.966 ± 1.546 | 5.525 ± 0.792 | 17.142 ± 2.552

###  5.3  Areas for further improvement

The quasi-periodic signal with P$\sim$23 days showed up in the shift-driven
periodogram is very close to the stellar rotation period that stands out as a
clear peak in CoRoT photometry, activity-indicator periodograms and the
scalpels basis-vector periodograms (Figure 18 and 17, marked with the
rightmost dotted lines). This points out that the $\sim$23 days shift signal
is more likely to be a stellar rotation artefact than a planetary reflex-
motion signal. However, the presence of this signal in both shift periodograms
(Figure 3 and Figure 5) and BGLS periodograms of activity indicators
strengthens the conclusion reached in Section 4.2: the property of this signal
that produces a shape change in the profile that also closely mimics a shift.
A note of caution is due here, since this confusing signal survived the
scalpels signal separation by imitating the behaviour of a real planet signal.
We anticipate this to be the effect of time-lags between the temporal
coefficients of $\mathbf{U}$ vectors and the RVs, which makes a linear
decorrelation difficult and incomplete.

##  6 Conclusion

In this paper, we present the results of applying the scalpels to extract
precise RV estimates from the high-resolution spectroscopic planet surveys
without using time-domain information. scalpels construct an orthogonal basis
containing the coefficients of the first few principal components of shape-
induced CCF variations, then project the raw RVs onto this basis to obtain a
time series of shape-induced RV variations. These are then subtracted from the
original RVs to leave shift-only RV variations, where the planet signals are
sought.

By separating the activity-induced line-shape changes from the HARPS archival
RV information of an active-star CoRoT-7, scalpels isolated the planet-induced
shift-driven signals. The consequent shift-driven RVs suggest that CoRoT-7 is
best modelled as a 3-planet system, with quasi-periodic harmonics of the
stellar rotation period also present. An ultra-short period (0.85 days) planet
CoRoT-7b; which is transiting, a second planet CoRoT-7c with orbital period of
3.69 days together with a third companion CoRoT-7d at a long-period orbit of
8.97 days, make CoRoT-7 an ultra-compact planetary system. Both the long-
period planets (CoRoT-7c & CoRoT-7d) are non-transiting.

We used the $\ell_{1}$ periodogram, a compressed-sensing method developed by
Hara et al. (2017), to identify the sparsest set of orbital periods that could
simultaneously fit the scalpels shift signal. The resulting sparse solution
eliminated unwanted alias signals and revealed three dominant candidate
signals. This analysis favoured the results from the scalpels projection, that
CoRoT-7 is better modelled as a 3-planet system than a 2-planet system.
Because the $\sim$23-d signal varies slowly in phase and amplitude, we noted
the usefulness of a split-period representation for reproducing the detailed
frequency structure of the periodogram and for determining the orbital
velocity amplitudes of the three dominant planet signals.

A simultaneous modelling of stellar variability and planetary motion was
performed to study the impact of the scalpels signal separation on the CoRoT-7
system, considering planet signals of known periods. A good match was observed
for a 5-signal model comprising three planet signals + two closely-spaced
periods near 23 d representing the quasi-periodic stellar rotation signal, as
suggested by the $\ell_{1}$ periodogram. This showed the effectiveness of the
scalpels in reducing the contribution of stellar variability to the RV signal
and enabling the detection of exoplanets’ signals in data from active stars.

The posterior probability distributions for the orbital parameters were
determined along with a complete Keplerian solver and a Gaussian Process using
kima (Faria et al., 2018). This GP model, incorporating the scalpels basis
vectors for line-shape decorrelation, is found to be 4 times more probable
than a model without scalpels. The improvement in the evidence again suggested
a system with two non-transiting planets and the known transiting planet.

In addition to improving the reliability of planet detection, our method also
allows more rigorous determination of the masses of known, transiting planets
whose bulk densities reveal their interior composition. This approach takes
into account the contributions of all additional (non-transiting) planets
whose existence may not be established conclusively, but whose RV variations
influence the mass determination of the known planet(s). Combined with
modelling of any surviving shift-like signal from stellar activity, this
approach offers a significant improvement in the precise characterization of
exoplanet systems, especially for the low-mass planets whose RV amplitudes are
close to the detection threshold.

## Acknowledgements

AAJ acknowledges the support from World leading St Andrews Doctoral
Scholarship. ACC and TGW acknowledge support from STFC consolidated grant
numbers ST/R000824/1 and ST/V000861/1, and UKSA grant number ST/R003203/1.
AAJ, ACC and TGW thank Nathan C. Hara for the productive discussions on
l1-periodogram. We are also grateful to Joao P. Faria whose algorithm (kima)
played a significant role in validating our results.

## Data availability

This paper uses of data from the Data & Analysis Center for Exoplanets (DACE)
database dedicated to the visualization, exchange and analysis of extrasolar
planets’ data. The HARPS data products used for the analyses are publicly
available in the DACE platform (https://dace.unige.ch). The PYTHON codes and
notebooks used to generate the results and diagrams in this paper will be made
available through the University of St Andrews Research Portal.

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## Appendix A

###  A.1 SCALPELS

The scalpels identified shape-driven perturbations are strongly correlated
with the observed RVs, as seen from the first panel of Figure 13, which could
reliably reproduce the stellar activity dominated long-term and short-term
fluctuations.

Figure 13: The correlation plots for the observed velocities ($v_{obs}$),
shape-driven component ($v_{\parallel}$) and the shift component ($v_{\perp}$)

## Appendix B

###  B.1 Blind search periodograms

We applied scalpels to the barycentric RVs of CoRoT-7, and examined the
results using a Generalized Lomb-Scargle periodogram (Zechmeister & Kürster,
2009) for different subsets of the data (2008-9, 2012 and combined).
Histograms showing the reduction of RMS scatter after separating the shape-
driven signals are presented for all three analyses in Figure 14.

Figure 14: Histograms showing the decrease in RMS scatter in the shift-driven
velocities (shown in green) during the blind-period search for 2008-9, 2012
and combined data respectively.

###  B.2 Simultaneous sinusoidal fit : Steps

After computing the Singular Value Decomposition(SVD) of the ACF rows;

  1. 1.

Compute
$\mathbf{F}=\\{\cos\omega_{1}t_{j},\sin\omega_{1}t_{j},\cdots,\cos\omega_{n}t_{j},\sin\omega_{n}t_{j}\\}$.

  2. 2.

Concatenate $\mathbf{A}_{(m\times(2n+l))}=\left[\mathbf{F}_{m\times 2n}\ \ \
\mathbf{U}_{A,(m\times l)}\right]$ using reduced rank $l$.

  3. 3.

Compute reduced-rank covariance matrix using:

$\frac{1}{m}\mathbf{R}^{T}_{k_{\rm max}}\cdot\mathbf{R}_{k_{\rm
max}}=\frac{1}{m}(\mathbf{P}_{C,k_{\rm max}}\cdot{\rm
diag}(\mathbf{S}_{C,k_{\rm max}}^{2})\cdot\mathbf{P}_{C,k_{\rm max}}^{T}).$
(11)

  4. 4.

Compute row variances $\sigma^{2}_{j}$ of $\mathbf{R}-\mathbf{R}_{k_{\rm
max}}$

  5. 5.

Compute model of full covariance matrix as:

$\Sigma_{\xi,\eta}(t_{j})\simeq\sigma^{2}_{j}\times{\rm
max}\left(1-\frac{|v_{\xi}-v_{\eta}|}{\delta
v},0\right)+\frac{1}{m}\mathbf{R}^{T}_{k_{\rm max}}\cdot\mathbf{R}_{k_{\rm
max}}.$ (12)

  6. 6.

Compute $\mathbf{C^{\prime}}$ and and ${\rm Var}(\mathbf{v}(t_{j}))$ using:

${\rm
Var}(\mathbf{v}(t_{j}))=\frac{1}{\mathbf{C^{\prime}}(v,t_{j})^{T}\cdot\mathbf{\Sigma}^{-1}(t_{j})\cdot\mathbf{C^{\prime}}(v,t_{j})}.$
(13)

  7. 7.

Construct $\Sigma={\rm Diag}({\rm Var}(\mathbf{v}(t_{j})))$.

  8. 8.

Solve

$(\mathbf{A}^{T}\cdot\mathbf{\Sigma}^{-1}\cdot\mathbf{A})\cdot\mathbf{\theta}=\mathbf{A}^{T}\cdot\mathbf{\Sigma}^{-1}\cdot\mathbf{\mathbf{v}_{\rm
obs}}$ (14)

to obtain $\mathbf{\theta}$ and ${\rm Var}(\mathbf{\theta})=1/{\rm
Diag}(\mathbf{A}^{T}\cdot\mathbf{\Sigma}^{-1}\cdot\mathbf{A})$.

  9. 9.

Partition $\mathbf{\theta}$ into coefficients of columns of $\mathbf{F}$ and
$\hat{\mathbf{\alpha}}$.

  10. 10.

Compute $\mathbf{v}_{\parallel}=\hat{\mathbf{\alpha}}\cdot{\mathbf{U}_{A}}$ in
reduced-rank basis.

  11. 11.

Compute $\mathbf{v}_{\perp}=\mathbf{v}_{\rm obs}-\mathbf{v}_{\parallel}$.

Return: RV amplitudes and variances, $\mathbf{v}_{\parallel}$,
$\mathbf{v}_{\perp}$.

Figure 15: $\mathbf{Left}$: All the signals found by the ${\ell_{1}}$\- periodogram are fitted including the 3 planet candidate signals (blue vertical lines) and 1 pairs of closely spaced beat period associated with the rotation period (magenta vertical lines). The orbital model signal (red) matches with the shift signal (green), except for the harmonics (light pink vertical bars) of the stellar rotation period. $\mathbf{Right}$:The resultant histogram of residuals with significantly reduced RMS scatter is shown. Table 6: $\mathbf{Top}$: The periods and semi-amplitudes of the strongest signals in the periodograms from simultaneous modelling of CCF shape changes and planetary motion, made with prior knowledge of the periods (from l1) of 3 planet candidates an a single period representing the stellar rotation are listed. $\mathbf{Bottom}$ A pair of beat period corresponding to the stellar rotation is included instead of the single period, considering the varying phase and amplitude. P(days) | K(ms-1) |  $\sigma_{K}$(ms-1)
---|---|---
0.8535 | 2.915 | 0.280
3.6963 | 5.112 | 0.302
8.9674 | 3.223 | 0.292
22.9425 | 5.437 | 0.321
0.8535 | 3.099 | 0.284
3.6963 | 5.343 | 0.303
8.9674 | 3.320 | 0.293
22.9425 | 5.142 | 0.326
23.6968 | 2.923 | 0.337

## Appendix C

###  C.1 $l_{1}$-periodograms for shape RVs & activity indicators

The physical origin of the planet candidate signals, were validated by
computing additional $\ell_{1}$ periodograms for the shape-driven RVs and the
activity-indicators, as shown in Figure 16.

Figure 16:  $\mathbf{Top}$: The ${\ell_{1}}$\- periodogram for the shape-
driven RVs and traditional activity indicators like FWHM, Area and the BIS are
shown from top to the bottom panels. None of these show significant signals at
0.85 days, 3.69 days and 8.96 days, adding to the case for a dynamical origin.
The corresponding False Alarm Probabilities (FAP) are also listed above each
panel.

## Appendix D

###  D.1 BGLS periodogram for activity indicators

We studied the shape changes of CCFs comprehensively to investigate intrinsic
stellar variability. The spectral parameters such as FWHM, BIS, and Area are
used for this diagnostic, as these offer different measures of shape changes
in the CCF profile. The Ca ii $H$& $K$ S-index is also used as a proxy for
active regions on the stellar surface. While the FWHM carries information
about change in the width of the line profile, the BIS records the profile
asymmetry. The Area is obtained by taking the product of FWHM and the central
line depth. The time series information of these all these indicators for
different observing runs are detailed in Section 4.1 along with Figure 1.

Here, we analyse the Bayesian Generalized Lomb-Scargle periodograms of the
above-mentioned activity indicators (Figure 18). The periods corresponding to
the stellar rotation period and its harmonics are represented by red dotted
vertical lines, while the candidate signal periods are marked in green. The
primary intention is to examine whether any of these periodograms has peaks at
or near the candidate signals of interest. We also examined the BGLS
periodograms of the first 4 leading principal components that contributed
significantly in modelling the stellar activity (of Figure 17), to understand
how efficiently the Principal Component Analysis of the ACF correlate with the
known aforesaid activity proxies.

We find that the second principal component ($\mathbf{U}_{1}$) exhibits a
strong resemblance to the variability of CCF area. Another correlation is
found between the third principal component($\mathbf{U}_{2}$) and the FWHM,
indicating that the changes in the width of the profile are also reliably
considered while modelling the shape-driven component.

Except the first principal component and the S-index, all other principal
components and activity indicators trace down the stellar rotation period
efficiently and hence show significant power at $\sim$23d. Unsurprisingly, no
traces of power are found in any of the periodograms at 0.85 days, 3.70 days
and 8.97 days , thus ruling out activity as a plausible origin for these
signals. We also noticed that, while $\mathbf{U}_{3}$ traces some stellar
rotation harmonics very efficiently($P_{\rm rot}$,$\frac{P_{\rm
rot}}{2}$,$\frac{P_{\rm rot}}{3}$,$\frac{P_{\rm rot}}{6}$ and $\frac{P_{\rm
rot}}{7}$), the Bisector inverse Slope(BIS) has peaks at almost every
harmonics ($P_{\rm rot}$,$\frac{P_{\rm rot}}{2}$,$\frac{P_{\rm
rot}}{3}$,$\frac{P_{\rm rot}}{4}$,$\frac{P_{\rm rot}}{5}$, $\frac{P_{\rm
rot}}{6}$ and $\frac{P_{\rm rot}}{7}$). In the BIS periodogram, the second
harmonic ($\frac{P_{\rm rot}}{3}$ = 7.8d) has a significantly greater power
than the original stellar rotation period ($P_{\rm rot}$ $\sim$ 23d).

Figure 17: Bayesian Generalized Lomb-Scargle periodograms of the 4 leading
principal components of the residual CCF of RV time series. Except the first,
all other principal components exhibit power around the rotation period and
the first and second harmonics, denoted by red-dotted vertical bars at
$\sim$23 d, $\sim$11.5 days and $\sim$7.6 d. Similar vertical bars in green
marks the periods of candidate planet signals. Figure 18:  Bayesian
Generalized Lomb-Scargle periodograms of activity proxies such as S-index,
Area, Full Width Half Maximum (FWHM) and Bisector span (BIS). The vertical
lines have the same meaning as in the panel above.

## Appendix E

###  E.1 Stacked BGLS periodogram- significance check

The top panel shows the Stacked BGLS periodogram for the 2008-9 observing
season (Queloz et al., 2009) with 106 observations, where the periods of four
candidate signals are marked using arrows. As anticipated, the region around
period $\sim$1 day appears to be crowded due to the presence of aliases of
several signals, which makes it difficult to spot the comparatively weaker
signal at 0.8549 days corresponding to the transiting planet. However, the
zoomed-in plot clearly shows the stable nature of this signal. The 1-day alias
of 0.854 days signal is clearly seen at 5.92 d, appearing to share power with
the real signal.

Another strong signal is present at 3.689 days which becomes apparent after
about 30 observations, growing steadily with more observations (Figure 19&9).
The 8.97 days signal appears slightly different, with the probability passing
through a short-lived maximum at $\sim$ 60th observation and then decaying
briefly before resuming a monotonic increase thereafter, as expected for a
real planetary signal. We expect to get a clear picture of this signal by
analysing the entire data set. There is a broad strong signal around 23.47
days with the SNR showing strong fluctuations over the span of observations,
which is not expected for a coherent signal. Its initial structure is also
confusing with a single sparse signal bifurcating into two strong signals with
periods 17.35 days and 23.47 d, where the 17.35 day signal can be explained as
the beat period occurring from the 8.97 days signal and the 5.92 days alias of
the transiting signal.

Figure 19: SBGLS periodograms for the 2008-9 and 2012 data subsets are given
in the left and right panels respectively.

The observations from 2012 (Haywood et al., 2014) returned a stacked BGLS
periodogram, as shown in the bottom panel of Figure 19. We noticed that the
transit signal does not appear to have the same period and stability as seen
from the previous figure. Instead, two other strong signals are visible at
0.78d and 0.88 days probably sharing the power from the real planet signal. We
can explain the former and the latter ones as the ${|\frac{1}{P}\pm 1|}^{-1}$
aliasing of the 0.85d signal and the 8.97 days signal, respectively.
Unsurprisingly, the 3.68 days signal again presents itself, with the strength
steadily increasing towards the maximum number of observations. The broadly
spread signal spanning from 6 to 13 days complicates the identification of the
8.97 days signal. The shorter observation baseline during the 2012 observing
run probably is insufficient to capture the 23.47 days signal. Mortier &
Collier Cameron (2017) also analysed the stacked BGLS periodograms for
different subsets and for the entire CoRoT-7 data from both RV campaigns, with
particular interest in investigating the planetary significance of the $\sim$9
days signal. They reported the origin of the $\sim$9 days signal to be more
likely from stellar activity than a third planet based on the unstable nature
recorded in the original dataset (Queloz et al., 2009), the follow-up campaign
(Haywood et al., 2014) and in the combined data set. In the present analysis,
we have the advantage of handling the scalpels-treated data, providing us with
less complicated, activity corrected, RV information spanning over a longer
baseline. Having effectively eliminated the short-term and long-term activity
and trends, we are left with well-defined signals whose nature is less
ambiguous.

It is also worth noting that most of the aliasing and beat period signals that
appeared in the individual data set analyses (Section E.1) are no more obvious
in the stacked BGLS periodogram for the entire data set.

## Appendix F

###  F.1 KIMA posteriors

Figure 20: Joint posterior distribution for the semi-amplitudes and
eccentricities along with the orbital periods in the x-axes

Figure 21: The FIP histograms showing the most probable periods for CoRoT-7c
and CoRoT-7d from the corresponding interference patterns. Figure 22:
Posterior distributions for the GP parameters and the extra white noise. The
samples for all values of Np were combined. The median of the posterior and
the uncertainties calculated from the 16% and 84% quantiles are given in
respective titles.