11institutetext:  Institute of Planetary Research, German Aerospace Center
(DLR), Rutherfordstrasse 2, 12489 Berlin 22institutetext:  ELKH-SZTE Stellar
Astrophysics Research Group, H-6500 Baja, Szegedi út Kt. 766, Hungary
33institutetext:  Zentrum für Astronomie und Astrophysik, Technische
Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
44institutetext:  Institut für Geologische Wissenschaften, Freie Universität
Berlin, 12249 Berlin, Germany 55institutetext:  Thüringer Landessternwarte
Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 66institutetext:
Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria,
1, 10125 Torino, Italy
66email<EMAIL_ADDRESS>

# Evidence of apsidal motion and a possible co-moving companion star detected
in the WASP-19 system

L. M. Bernabò ${}^{\href https://orcid.org/0000-0002-8035-1032}$ Evidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
systemEvidence of apsidal motion and a possible co-moving companion star
detected in the WASP-19 system Sz. Csizmadia ${}^{\href
https://orcid.org/0000-0001-6803-9698}$ Evidence of apsidal motion and a
possible co-moving companion star detected in the WASP-19 systemEvidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
systemEvidence of apsidal motion and a possible co-moving companion star
detected in the WASP-19 systemEvidence of apsidal motion and a possible co-
moving companion star detected in the WASP-19 system A. M. S. Smith${}^{\href
https://orcid.org/0000-0002-2386-4341}$ Evidence of apsidal motion and a
possible co-moving companion star detected in the WASP-19 systemEvidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
system H. Rauer${}^{\href https://orcid.org/0000-0002-6510-1828}$ Evidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
systemEvidence of apsidal motion and a possible co-moving companion star
detected in the WASP-19 systemEvidence of apsidal motion and a possible co-
moving companion star detected in the WASP-19 systemEvidence of apsidal motion
and a possible co-moving companion star detected in the WASP-19 systemEvidence
of apsidal motion and a possible co-moving companion star detected in the
WASP-19 systemEvidence of apsidal motion and a possible co-moving companion
star detected in the WASP-19 system A. Hatzes${}^{\href
https://orcid.org/0000-0002-3404-8358}$ Evidence of apsidal motion and a
possible co-moving companion star detected in the WASP-19 systemEvidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
system M. Esposito${}^{\href https://orcid.org/0000-0002-6893-4534}$ Evidence
of apsidal motion and a possible co-moving companion star detected in the
WASP-19 systemEvidence of apsidal motion and a possible co-moving companion
star detected in the WASP-19 system D. Gandolfi${}^{\href
https://orcid.org/0000-0001-8627-9628}$ Evidence of apsidal motion and a
possible co-moving companion star detected in the WASP-19 systemEvidence of
apsidal motion and a possible co-moving companion star detected in the WASP-19
system J. Cabrera${}^{\href https://orcid.org/0000-0001-6653-5487}$ Evidence
of apsidal motion and a possible co-moving companion star detected in the
WASP-19 systemEvidence of apsidal motion and a possible co-moving companion
star detected in the WASP-19 system

(Submitted 9 May 2023 ; accepted 8 January 2024)

###### Abstract

Context. Love numbers measure the reaction of a celestial body to perturbing
forces, such as the centrifugal force caused by rotation, or tidal forces
resulting from the interaction with a companion body. These parameters are
related to the interior density profile. The non-point mass nature of the host
star and a planet orbiting around each other contributes to the periastron
precession. The rate of this precession is characterized mainly by the second-
order Love number, which offers an opportunity to determine its value. When it
is known, the planetary interior structure can be studied with one additional
constraint beyond the mass, radius, and orbital parameters.

Aims. We aim to re-determine the orbital period, eccentricity, and argument of
the periastron for WASP-19Ab, along with a study of its periastron precession
rate. We calculated the planetary Love number from the observed periastron
precession rate, based on the assumption of the stellar Love number from
stellar evolutionary models.

Methods. We collected all available radial velocity (RV) data, along with the
transit and occultation times from the previous investigations of the system.
We supplemented the data set with 19 new RV data points of the host star
WASP-19A obtained by HARPS. Here, we summarize the technique for modeling the
RV observations and the photometric transit timing variations (TTVs) to
determine the rate of periastron precession in this system for the first time.

Results. We excluded the presence of a second possible planet up to a period
of $\sim$4200 d and with a radial velocity amplitude bigger than $\simeq$1 m
s-1. We show that a constant period is not able to reproduce the observed
radial velocities. We also investigated and excluded the possibility of tidal
decay and long-term acceleration in the system. However, the inclusion of a
small periastron precession term did indeed improve the quality of the fit. We
measured the periastron precession rate to be 233 ${}^{+25}_{-35}$
${}^{\prime\prime}d^{-1}$. By assuming synchronous rotation for the planet, it
indicates a $k_{2}$ Love number of 0.20 ${}^{+0.02}_{-0.03}$ for WASP-19Ab.

Conclusions. The derived $k_{\mathrm{{2,p}}}$ value of the planet has the same
order of magnitude as the estimated fluid Love number of other Jupiter-sized
exoplanets (WASP-18Ab, WASP-103b, and WASP-121b). A low value of
$k_{\mathrm{{2,p}}}$ indicates a higher concentration of mass toward the
planetary nucleus.

###### Key Words.:

##  1 Introduction

The understanding of planetary interiors constitutes one of the four major
challenges of contemporary exoplanetary science (Schneider, 2018). It is
crucial for assessing their potential habitability, formation, and evolution.
Interior processes play an essential role in creating and maintaining the
physical conditions that are required to support life (see, e.g., Van Hoolst
et al. 2019). They are useful for determining which planetary formation
process is most plausible; namely, whether it is core accretion (Pollack et
al. 1996 and references therein) or gravitational instability (Boss 1997 and
references therein). According to the former, the accretion of planetesimals
would result in the formation of a core comprised of heavy elements. On the
other hand, planets forming through disc instability can rapidly accrete gas.
These two scenarios lead to different core sizes. The size of the core is
related to the second order fluid $k_{\mathrm{{2,p}}}$ Love number (Love
1911), as explained in Becker & Batygin (2013). Knowledge of the interior of a
giant planet can help us distinguish between these two scenarios, in addition
to its Love number, as we describe in this work.
Love numbers can also be used to break the known mass-radius-composition
degeneracy of exoplanets (Baumeister et al., 2020). When an exoplanet is
characterizable both by transit and radial velocity techniques, we can derive
its radius and mass – and, thus, the planetary mean density. However, mass,
radius and mean density are degenerate when it comes to determining the
interior structure of bodies, such as the radial density, temperature,
pressure, and composition profiles. We can find multiple solutions for the
planetary interior producing the same planetary total mass and radius (see,
e.g., Valencia et al. 2007; Wagner et al. 2011; Damasso et al. 2018). This
kind of degeneracy can be decreased based on the strong assumption that the
planet and its host star share the same metallicity (Dorn et al., 2015).
For Solar System planets, the degeneracy between mean density and interior
structure can be reduced by in situ measurements of the gravitational field,
as well as by seismic measurements, such as InSight (Banerdt et al. 2020), and
by observations of moon motions. However, for exoplanets, gravitational
moments cannot be measured. Hence, Love numbers were proposed as further
observables in exoplanetary interior studies (see, e.g., Batygin et al. 2009;
Ragozzine & Wolf 2009 and references therein), as has already been applied to
eclipsing binary stars systems (see Russell 1928 and references therein).
These authors measured the deformations and mass re-distribution inside the
planet due to the tidal interaction with the host star. As shown by Baumeister
et al. (2020), thanks to the knowledge of the second-order fluid Love number
of the planet, we can better infer the distribution of possible thickness of
each interior layer.

In this work, we employ a technique used by Csizmadia et al. (2019), which we
further refined to determine $k_{\mathrm{{2,p}}}$ from periastron precession.
This precession is derived from radial velocity (RV) measurements and transit
and occultation mid-times. We apply it to the system WASP-19 based on an
approach that requires the calculation of the secular evolution of the orbital
elements, which is related to the mass distribution of the star and the planet
through their second-degree Love numbers.

This paper is organized as follows. Section 2 and Appendix A present the
theory of apsidal motion. Section 3 describes the system WASP-19 and provides
evidence of a possible companion star in the WASP-19 system. In Sect. 4, we
describe archival transit and occultation timing observations as well as the
archival radial velocity investigations of the system which are supplemented
by our recent RV study of the system. Section 5 contains our data analysis
along with the search for a second planetary companion. Our conclusions are
given in Sect. 6.

##  2 Apsidal motion

Apsidal motion, namely, the secular perturbation in the argument of
periastron, can be detected either in RV or in TTV datasets (or their
combination) to constrain the internal structure of planets. This type of
analysis on RVs was introduced by Kopal (1959, 1978). Here, we make use of
such equations, however, we have rewritten them in a way that allows them to
be directly applied to the observed data sets. We describe their derivations
more in detail in Appendix A.
The total rate of apsidal advance is the result of the three contributions:

$\dot{\omega}=\frac{d\omega}{dt}=\frac{d\omega_{\mathrm{GR}}}{dt}+\frac{d\omega_{\mathrm{tidal}}}{dt}+\frac{d\omega_{\mathrm{rot}}}{dt}.$
(1)

The general relativistic term $d\omega_{\mathrm{GR}}/dt$ (Einstein 1915) is:

$\frac{d\omega_{\mathrm{GR}}}{dt}=\frac{6\pi
GM_{\star}}{ac^{2}(1-e^{2})}\frac{1}{P_{\mathrm{a}}},$ (2)

where G is the gravitational constant, $M_{\mathrm{\star}}$ the mass of the
star, a and e the semi-major axis and eccentricity of the orbit, c the speed
of light in vacuum, and $P_{\mathrm{a}}$ the anomalistic period.
The secular perturbations of the tidal and rotational components
$d\omega_{\mathrm{tidal}}/dt$ and $d\omega_{\mathrm{rot}}/dt$ are derived in
Appendix A and can be expressed more explicitly if approximated to the second
order, as in Csizmadia et al. (2019):

$\begin{split}\frac{d\omega_{\mathrm{tidal}}}{dt}\simeq\frac{15}{{2}}\frac{n}{(1-e^{2})^{5}}\left[\frac{m_{\mathrm{p}}}{M_{\mathrm{\star}}}k_{\mathrm{{2,\star}}}\left(\frac{R_{\mathrm{\star}}}{a}\right)^{5}+\right.\\\
+\left.\frac{M_{\mathrm{\star}}}{m_{\mathrm{p}}}k_{\mathrm{{2,p}}}\left(\frac{R_{\mathrm{p}}}{a}\right)^{5}\right]\left(1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}\right),\end{split}$
(3)

and

$\displaystyle\begin{split}\frac{d\omega_{\mathrm{rot}}}{dt}\simeq\frac{1}{{2}}\frac{n}{(1-e^{2})^{2}}\left[\left(\frac{P_{\mathrm{orb}}}{P_{\mathrm{rot,\star}}}\right)^{2}k_{\mathrm{{2,\star}}}\left(\frac{R_{\mathrm{\star}}}{a}\right)^{5}\left(1+\frac{m_{\mathrm{p}}}{M_{\mathrm{\star}}}\right)+\right.\\\
\hskip
31.2982pt+\left.\left(\frac{P_{\mathrm{orb}}}{P_{\mathrm{rot,p}}}\right)^{2}k_{\mathrm{{2,p}}}\left(\frac{R_{\mathrm{p}}}{a}\right)^{5}\left(1+\frac{M_{\mathrm{\star}}}{m_{\mathrm{p}}}\right)\right].\end{split}$
(4)

In these equations, $n$ is the mean motion, $m_{\mathrm{p}}$ and
$R_{\mathrm{p}}$ the mass and radius of the planet, $R_{\mathrm{\star}}$ the
radius of the star, and $P_{\mathrm{rot,p}}$ and $P_{\mathrm{rot,\star}}$ are
the rotational periods of the two components. Also, $k_{\mathrm{{2,p}}}$ and
$k_{\mathrm{{2,\star}}}$ are their second-order fluid Love number
corresponding to the double the apsidal motion constant ((Csizmadia et al.,
2019)):

$k_{\mathrm{2,i}}=2k_{\mathrm{2,aps,i}},$ (5)

both for the star ($i=\star$) and the planet ($i=p$). These two equations were
obtained by Sterne (1939). Just as in Eq. (22), it is assumed that the
rotational axis of both components are perpendicular to the orbital plane.
Regardless of whether the stellar and planetary rotation is direct or
retrograde, the corresponding terms in Eq. (4) are positive, therefore they
cause the apsidal line to advance.

Assuming the values of the orbital parameters of WASP-19Ab from literature and
the Love number of the star from Claret (2004), we calculated the amplitude of
the general relativity term and the stellar tidal and rotational
contributions, as reported in Table 1. Moreover, a first estimate for the
planet can be calculated assuming synchronous rotation for the planet and
$k_{\mathrm{{2,p}}}$ between the values 0.01-1.5 (corresponding to the extreme
cases of a mass point and a homogeneous body, respectively). The total
contribution is between about 14.71 - 1689 ′′d${}^{-1}\simeq$ 0.0041 - 0.479 ∘
d-1, respectively when assuming $k_{\mathrm{{2,p}}}$=0.01 and
$k_{\mathrm{{2,p}}}$=1.5. The apsidal motion period $U$=2$\pi/\dot{\omega}$ is
therefore between $\simeq$ 2.10 yrs and $\simeq$ 241.25 yrs.

| $k_{\mathrm{{2,p}}}=0.01$ | $k_{\mathrm{{2,p}}}=1.5$
---|---|---
$\dot{\omega}_{GR}$ | 2.85 ′′ d-1 | 2.85 ′′ d-1
$\dot{\omega}_{tid,\star}$ | 0.47 ′′ d-1 | 0.47 ′′ d-1
$\dot{\omega}_{rot,\star}$ | 0.15 ′′ d-1 | 0.15 ′′ d-1
$\dot{\omega}_{tid,p}$ | 10.53 ′′ d-1 | 1580.16 ′′ d-1
$\dot{\omega}_{rot,p}$ | 0.70 ′′ d-1 | 105.46 ′′ d-1
Total $\dot{\omega}$ | 14.70 ′′ d-1 | 1689.10 ′′ d-1
Period | 241.25 yrs | 2.10 yrs
Table 1: Expected gravitational, rotational and tidal contribution to the
periastron precession in the extreme Love number cases (mass-point and
homogeneous planet) and the corresponding apsidal motion period, assuming
synchronous rotation for the planet.

The equations in Appendix A show that the angle of periastron $\omega$
oscillates over one orbital period but also shows a gradual long-term change
that can be retrieved in the RV observations. Due to the finite length of the
typical datasets of observations (12 years in the case of WASP-19Ab), in RV
datasets, the change in the angle of periastron $\omega$ can be retrieved in
the radial velocity of the star (see, e.g., Jackson et al. 2008 for more
details on the timescales). In long baselines of observations, all orbital
elements must be considered along with periastron precession. As described in
Kopal (1978), they include the change in semi-major axis (tidal decay),
eccentricity (circularization) and time of periastron passage. The semi-major
axis and its eccentricity would diminish in value and the time of periastron
passage is shifted in time.

##  3 WASP-19Ab

###  3.1 The system

The planet WASP-19Ab, hosted by the bright, active G-dwarf star ($V$=12.3,
Zacharias et al. 2012 and ${T_{\mathrm{eff,\star}}}\simeq$5568 K, Torres et
al. 2012), was discovered through the transit method by the Wide Angle Search
for Planets (WASP)-South observatory (Hebb et al. 2010). It is a hot Jupiter
with mass $m_{\mathrm{p}}$ = 1.139 $M_{\mathrm{\Jupiter}}$, inflated radius
$r_{\mathrm{p}}$ = 1.410 $R_{\Jupiter}$ (Mancini et al. 2013) and orbital
period $P_{\mathrm{orb}}\simeq$ 0.79 d.
So far, the system is among the best studied from the point of view of orbital
and planetary parameters (e.g., Lendl et al., 2013; Mancini et al., 2013;
Sedaghati et al., 2015, 2017) and atmosphere characterization (very pronounced
water absorption, Huitson et al. 2013; Sedaghati et al. 2017, and titanium
oxide detected in the transmission spectrum, Sedaghati et al. 2017). The
proximity of WASP-19Ab to its host star results in significant tidal
interaction that may govern the evolution of the planetary orbit, which makes
the system a good candidate for our study.
Based on transit observations, certain studies have suggested the possibility
of variations in the mid-transit time. Mancini et al. (2013) suggested a non-
linear ephemeris. However, Petrucci et al. (2020) discarded orbital decay with
74 complete transit light curves spanning over ten years of observations.
Moreover, as described more in detail in Sect. 5.1, Cortés-Zuleta et al.
(2020) used transit timing variations (TTVs) to infer an upper mass limit for
an additional candidate planetary companion in the system.

###  3.2 Companion star: WASP-19B

We searched the catalogue of the third Gaia data release (DR3) (Gaia
Collaboration et al., 2016, 2022) for possible co-moving companions to
WASP-19A. We searched all objects within $4^{\prime}$ of the target, which
corresponds to a physical separation of around 0.3 pc at the distance of
WASP-19A, for those with compatible parallax ($\theta$) and proper motions
(PM). Only one object was found to have parallax and proper motion values in
both the RA ($\alpha$) and Dec ($\delta$) directions that are compatible with
those of the target to within 5$\sigma$. This object (Gaia DR3 ID =
5411725145916602496), which is 8.8 magnitudes fainter in the $G$-band, has a
parallax that differs from that of WASP-19A by $1.6\,\sigma$ and
$\mathrm{PM_{\alpha}}$ and $\mathrm{PM_{\delta}}$ values differing by just
$0.1\,\sigma$ and $0.6\,\sigma$ respectively. This star is found in the
southernmost part of the south-east region of WASP-19A, at a separation of
$68^{\prime\prime}$. The magnitude and BP-RP colour of the companion suggests
that this star is a mid M-dwarf, with a best matching spectral type for the
magnitude of M6V and for the colour M2V, using the updated
table111https://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt
of Pecaut & Mamajek (2013).

To estimate the probability of this putative companion arising from a chance
matching of WASP-19 in parallax and proper motions, we searched a much larger
area of sky in $Gaia$ DR3, still centered on WASP-19, but with a radius of
$2^{\circ}$. This produced 77 objects (including the companion; hereafter:
‘WASP-19B’) that match WASP-19 within 5$\sigma$ in all of $\theta$,
$\mathrm{PM_{\alpha}}$, and $\mathrm{PM_{\delta}}$. This implies that there is
an 8.6% chance of finding a matching object within our initial search radius
of $4^{\prime}$, and just a 0.7% chance of finding an object as close as
WASP-19B. Relaxing the 5$\sigma$ limit to 3$\sigma$ changes these
probabilities to 2% and 0.16%, respectively. Furthermore, none of these
additional matches, all of which are more than ten times further from WASP-19A
than WASP-19B, demonstrate as good a match as WASP-19B. Mathematically, this
is expressed as WASP-19B having the lowest value of
$\sigma_{\mathrm{tot}}=\sigma_{\theta}+\sigma_{\mathrm{PM_{\alpha}}}+\sigma_{\mathrm{PM_{\delta}}}$,
where for each parameter, $X$ (where $X$ stands for $\theta$,
$\mathrm{PM}_{\alpha}$, $\mathrm{PM}_{\delta}$)
$\sigma_{X}=\frac{|X_{1}-X_{2}|}{\sqrt{\sigma^{2}_{X_{1}}+\sigma^{2}_{X_{2}}}}$,
where $X_{1}\pm\sigma_{X_{1}}$ and $X_{2}\pm\sigma_{X_{2}}$ are the values and
uncertainties of parameter $X$ for WASP-19 and its companion, respectively. We
recognise that the uncertainty on the parallax of WASP-19B is relatively
large. Even if WASP-19B is likely to be a true, gravitationally bound,
companion to WASP-19A, some doubt remains as to whether it truly is a bound
companion to WASP-19A. We adopted the planet name WASP-19Ab because it orbits
around the brighter primary star. Details of WASP-19B are given in Table 2.
With Eq. (9) of Csizmadia et al. (2019), we also determined that the effect of
the companion star on the radial velocity of star A is negligible. Star B may
be the cause of the presence of an eccentricity different from zero in the
orbit on the transiting planet. While tidal interaction leads to
circularization, the excitation of the orbit due to a distant companion might
prevent it, as discussed in detail in Appendix B.

Parameter | WASP-19A | WASP-19B
---|---|---
Gaia DR3 ID | 5411736896952029568 | 5411725145916602496
Apparent separation (″) | 0.0 | $68.0347\pm 0.0009$
Gaia $G$-magnitude | $12.1012$ | $20.8859$
Gaia $BP-RP$ colour | $0.953\pm 0.001$ | $2.213\pm 0.12$
pmRA (mas yr-1) | $-35.4571\pm 0.0089$ | $-35.33\pm 1.18$
pmDec (mas yr-1) | $17.378\pm 0.009$ | $18.22\pm 1.30$
Parallax (mas) | $3.7516\pm 0.0090$ | $1.92\pm 1.16$
Distance (pc)$*$ | $266.54\pm 0.64$ | $487\begin{subarray}{c}+564\\\ -184\end{subarray}$
Projected physical separation (AU) | 0.0 | $19976\pm 48$
Spectral type† | G8V | M2V - M6V

Table 2: WASP-19A and its stellar companion
* Assuming no extinction.
† Assuming that the companion is also a star on the main-sequence.

##  4 Observations

###  4.1 Transits and occultations

Mid-transit and occultation times used in this paper are listed in Tables C.1
\- C.4 in Appendix C. We collected publically available data, including some
amateur observations listed in the TRansiting ExoplanetS and CAndidates
(TRESCA)222http://var2.astro.cz/EN/tresca/index.php project database, part of
the Exoplanet Transit Database (Poddaný et al. 2010). In particular, we
included amateur data that were selected by Mancini et al. (2013) and other
high-quality data showing a clear light curve without gaps. As detailed in the
tables, we also re-analyzed some transits with the Transit and Light Curve
Modeller (TLCM, Csizmadia 2020).
The time unit of each mid-transit and mid-occultation time was carefully
checked and eventually converted into $\mathrm{BJD}_{\mathrm{TDB}}$
(barycentric Julian date, in the barycentric dynamical time system). The
difference between $\mathrm{BJD}_{\mathrm{UTC}}$ and
$\mathrm{BJD}_{\mathrm{TDB}}$ depends on the number of leap seconds (Eastman
et al., 2010). For our dataset, it is around one minute. If the time unit used
in the reference paper was not clear, the main author was contacted and the
issue was clarified; otherwise, the data were discarded and not reported here.
We used the estimated mid-times for each primary transit and occultation to
evaluate whether they exhibit deviations from a Keplerian orbit over a long
baseline of observations (see Sect. 5.2 for more details) as well as to
constrain the sidereal period of the planet.

###  4.2 RVs

We collected all available radial velocity measurements from 2008 onward and
we present new RV observations (see ID 8 below). Information on the RV
observations are listed in Table 3 and the complete datasets are shown in
Appendix D.

ID | $N_{\mathrm{obs}}$ | Instrument | Reference
---|---|---|---
| (out of transit) |  |
1 | 34 (30) | Coralie |  Hebb et al. (2010)
2 | 3 (2) | Coralie |  Hellier et al. (2011)
3 | 36 (21) | HARPS |  Hellier et al. (2011)
4 | 20 (6) | PFS |  Albrecht et al. (2012)
5 | 3 (3) | HIRES |  Knutson et al. (2014)
6, 7 | 88 (16, 6) | ESPRESSO |  Sedaghati et al. (2021)
8 | 19 (15) | HARPS | This work
Table 3: Radial velocity datasets used in this work.
RV datasets are described with ID number, number of points, instrument and
reference for the publication of the corresponding data and results. In
brackets, one can find the number of out-of-transit as those are the only RV
datapoints that are used in the fit.

Here, we list the main details on the datasets, with the labels corresponding
to the discussion in Appendix D:

  1. ID 1

Hebb et al. (2010) (discovery paper) reported 34 RV measurements obtained with
the CORALIE spectrograph on the 1.2 m Euler telescope between 2008 and 2009.
To the best of our knowledge and after contacting the authors of the paper,
the dataset reported in the paper is in $\mathrm{BJD_{UTC}}$. We present the
data transformed to $\mathrm{BJD}_{\mathrm{TDB}}$;

  2. ID 2

Hellier et al. (2011) reported 3 RV points from the CORALIE spectrograph on
the Swiss Euler 1.2 m telescope in 2009. The data were reported in
$\mathrm{BJD}_{\mathrm{UTC}}$ in Hellier et al. (2011), so they are
transformed here into $\mathrm{BJD}_{\mathrm{TDB}}$;

  3. ID 3

Hellier et al. (2011) reported 36 points during and around more than one
transit in 2010 using the HARPS spectrograph on the ESO 3.6 m at La Silla,
taken in order to measure the Rossiter-Mclaughlin effect. As well as for ID 2,
the data were reported in $\mathrm{BJD}_{\mathrm{UTC}}$ in the original paper,
so they have been transformed here into $\mathrm{BJD}_{\mathrm{TDB}}$;

  4. ID 4

Albrecht et al. (2012) reported 20 data points taken in 2010 as measurements
of the Rossiter-McLaughlin effect with the Magellan II (Clay) 6.5 m telescope
and the Planet Finder Spectrograph (PFS);

  5. ID 5

Knutson et al. (2014) reported 3 RV measurements obtained with the High
Resolution Echelle Spectrometer (HIRES) on the 10 m Keck I telescope between
2008 and 2013;

  6. IDs 6,7

Sedaghati et al. (2021) reported 88 RV points taken with VLT/ESPRESSO between
2019 and 2020, used to derive the Rossiter-McLaughlin effect and characterize
the atmosphere of the planet. The two different IDs are given because of some
setup changes in ESPRESSO, as described by Sedaghati et al. (2021), which lead
to the introduction of a small offset. Moreover, as clarified with the first
author of that paper, ID 6 was reported in $\mathrm{BJD}_{\mathrm{UTC}}$ in
the original paper, while ID 7 in $\mathrm{BJD}_{\mathrm{TDB}}$, therefore the
first one is transformed;

  7. ID 8

reported 19 RV points taken by us using the High Accuracy Radial velocity
Planet Searcher (HARPS, Mayor et al. 2003) at the ESO La Silla 3.6m telescope
under the programme 0104.C-0849 (PI: Sz. Csizmadia). The observations were
performed between 6 and 21 February 2020. We tried to obtain 1-3 points per
night and we planned the observations to have a good phase coverage. The
exposure time was 1200 seconds while the spectral resolution as 115,000. The
objAB observing mode of HARPS was used, covering the 378-691 nm wavelength
range. The measured signal-to-noise ratio (S/N) varied between 10.2 and 21.6
depending on weather and seeing conditions. The data were reduced by the
standard HARPS pipeline available at La Silla Observatory and the results with
their corresponding uncertainties are reported in Tables D.3 and D.4.

All RV observations are shown in Fig. 1. As described in Sect. 5.2, the
datasets have different offsets {which are subtracted in the figure. They are
fitted in our model by including the term $V_{\mathrm{{{i,instr}}}}$ in Eq.
(7). As for the transit and occultation mid-times, the time unit of the
observation was transformed and uniformized to $\mathrm{BJD}_{\mathrm{TDB}}$
when necessary.

Figure 1: Radial velocity observations of WASP-19Ab, time vs radial
velocities, with corresponding observational errorbars. The different data
sets (see Sect. 4) are represented by different colours. Between the datasets,
there are significant offsets between different instruments and studies. They
have been removed accordingly to the results in Table 4.

##  5 Data analysis

###  5.1 Long-term effects in RVs

In this section, we study the long-term effects affecting the RV curve. If
present, they must be identified and subtracted before modeling and estimating
the rate of periastron precession in Sect. 5.2.

First, we applied the $l_{1}$ periodogram (Hara et al. 2017) to the archive
and newly acquired RV data (ID 8, see Fig. 2) to search for further companions
in the system, in addition to the confirmed planet in the system. This method
was developed specifically to find companions in RV datasets and can be used
in a similar way to a Lomb–Scargle periodogram. To verify the significance of
the peaks, we followed the re-sampling approach suggested by Hara et al.
(2017): we randomly removed 10–20% of the data and re-computed the
periodogram. Some peaks did not appear after the re-sampling, therefore, we
did not consider them real. After the re-sampling, we identified only one peak
at 0.78 d, as shown in the top plot of Fig. 2. It was retrieved in
correspondence with the orbital period of WASP-19Ab, with a false alarm
probability (FAP) of $\mathrm{log_{10}(FAP)}$=-30.9 (FAP $\simeq 10^{-31}$).
The analysis shows no evidence of any additional periodic signal in the RV
data.
We also re-computed the periodogram on the RV residuals after the subtraction
of the signal of the first planet (see bottom plot in Fig. 2). After a re-
sampling of the data, only a few small peaks are left. However, their FAP is
$\geq$0.2, meaning that any peaks are due to artefacts.

Figure 2:  $l_{1}$ periodogram of all RV data outside the transit (top). There
is only one significant peak with false alarm probability $\simeq 10^{-31}$.
The periodogram of the residuals after the subtraction of the contribution of
planet b (bottom). All residual peaks have false alarm probabilities of ¿20%.

We estimated the sensitivity of our search for perturbing bodies in the data
by performing a sensitivity study based on the transit timing variations. We
calculated the maximum mass of a perturber in the system as a function of the
maximum amplitude of the TTVs (see Eq. (2) in Holman & Murray 2005). Our
analysis shows that the maximum allowed mass of the perturber,
$M_{\mathrm{c}}$, as a function of its orbital period, $P_{\mathrm{c}}$,
assuming different values of the orbital eccentricity (e=0.0, 0.2, 0.4,0.6,
and 0.8). The analysis shows that the region above 4000 $M_{\Jupiter}$ and
period below 20 days is excluded by the TTV observations.
We also performed a sensitivity study based on the RVs. The maximum amplitude
of the RV residuals after the subtraction of the signal of the orbit of planet
b (with Sedaghati et al. 2021’s parameters) is around 52 m s-1. Through
Kepler’s third law, we estimated the maximum mass of the putative perturber.
With an orbital period equal to 12 yrs (the total length of the RV dataset),
we obtained $M_{\mathrm{c}}\simeq$ 3.9 $M_{\mathrm{\Jupiter}}$ and an orbital
distance of $\simeq$ 5 AU. We fitted the data with an additional Keplerian
signal (beyond the signal of planet b) mimicking a hypothetical candidate for
planet c. We injected a candidate and repeated the experiment two billion
times with different random eccentric orbital parameters and fitted the new
HARPS RV data. We varied the candidate parameters as follows:
$\sqrt{e}\cdot\sin\omega_{0}$ and $\sqrt{e}\cdot\cos\omega_{0}$ between -1.0
and 1.0, the RV semi-amplitude $K_{\mathrm{c}}$ between 0.1 and 100 m s-1 (to
widely investigate the amplitude of the RV residuals with errorbars) and the
orbital period between 0.1 and the length of the dataset ($\simeq$19 d for the
first attempt with the only HARPS dataset and $\simeq$ 12 yr for the second
attempt with the whole dataset). Moreover, from Kepler’s third law, we set the
following condition to ensure that the orbit would not collide with the star:

$P_{\mathrm{c}}\geq\frac{2\pi
R_{\mathrm{\star}}^{3/2}}{\sqrt{GM_{\mathrm{\star}}}},$ (6)

where $G$ is the gravitational constant, while $R_{\mathrm{\star}}$ and
$M_{\mathrm{\star}}$ are the radius and mass of the host star, respectively.
With these constraints and varying the orbital parameters in the
aforementioned ranges, the candidate mass range is found to be between
$\simeq$ 1.4 and $\simeq$ 400 M♁.
The results of our simulation (see Fig. 3) suggest scenarios that satisfy the
condition $\chi^{2}\leq 500$.The top plot shows the $\chi^{2}$ of the model
when including the proposed perturber, corresponding to five more free
parameters: transit epoch, period, RV semi-amplitude,
$\sqrt{e}\cdot\sin\omega_{0}$ and $\sqrt{e}\cdot\cos\omega_{0}$ (plotted as a
function of the orbital period). Many of these candidates are discarded
because the corresponding $\chi^{2}$ is too high or the amplitude of the
perturbing body on the RV curve would be over a dozen m s-1, so they would
definitely be visible in the RV data and in the periodogram. The red lines
underline the periods corresponding with the stable orbital regions present in
the stability map by Cortés-Zuleta et al. (2020). Their analysis allowed for
the presence of a second planet either in 1:2, 5:3, 2:1, 5:2, or 3:1 resonance
with planet b, corresponding to orbital periods of 0.39 d, 1.31 d, 1.58 d,
1.97 d, or 2.37, respectively. Some of these periods correspond with regions
in the figure that are densely populated by our best candidates. The bottom
plot shows the mass of the candidate plotted against its orbital period.
Again, we highlight the stable orbit by Cortés-Zuleta et al. (2020) with red
dashed lines.
The expected $\chi^{2}$ would be around the number of degrees of freedom (19
total datapoints for dataset 8, of which 15 were fitted as out-of-transit
points and 11 free parameters), while the lowest $\chi^{2}$ we obtained during
the simulations is $\simeq$ 134 (clearly visible in the top plot of Fig. 3 at
$\simeq$10 d orbital period). We fit the HARPS data with a one-planet
eccentric orbit model to compare it to the scenarios with a perturbed. Without
adding any RV jitter, the $\chi^{2}$ of this model is $\simeq$ 135\. Since the
two-planet model has four more free parameters, by applying the Bayesian
information criterion (BIC), we found no evidence of the presence of a
perturber in the system with a period within the length of the dataset with
this approach. For weak (respectively, strong) evidence (see Kass & Raftery
1995 for more details) that the two-planet model is preferred, a $\chi^{2}$
value below 115 (respectively, 98) is needed.

Figure 3: Properties of third bodies excluded by RV data. The plot shows the
results for the simulated scenarios including a putative companion planet c of
different masses at different periods. Each black dot represents one of the
around 20 000 selected scenarios (see the text for more details).
Top plot: $\chi^{2}$ value vs the orbital period of the perturber. The red
lines correspond to the period of the stable orbits predicted by Cortés-Zuleta
et al. (2020). The two horizontal dashed lines represent the $\chi^{2}$
required to have weak (upper line) or strong (lower line) evidence of a second
planet, as discussed in the text.
Bottom plot: Mass and orbital period of the selected candidates.

We also injected a candidate in the whole RV dataset (IDs from 1 to 8), but we
obtained an even higher $\chi^{2}$, resulting in an even more unlikely
presence of a perturbed. Our results exclude it up to an RV amplitude of
$\simeq$1 m s-1.
This simulation confirms the conclusion of Knutson et al. (2014) when looking
for the acceleration of the system due to possible long-period companions. The
cited paper does not find an acceleration that differs from zero by more than
3$\sigma$. In this way, we extended the orbital period range of investigation
in Knutson et al. (2014) from between 0 and 1702 d up to $\simeq$4200 d.

Since we did not find any additional companion and there is no evidence for
long-term trends in the data, we proceeded with our analysis by introducing
periastron precession in the fit.

###  5.2 Rate of apsidal motion

In this section, we describe the code used for the data analysis (by allowing
for non-zero periastron precession) with the fitted parameters and the
results.
We used the approach and idl code developed by Csizmadia et al. (2019) to
analyze the RV data in parallel with mid-transit and occultation times. Since
then, the code has been modified and updated. Now the Genetic Algorithm
minimization is followed by a differential-evolution Markov chain Monte Carlo
(DE-MCMC, Ter Braak 2006) algorithm to explore the parameter space. The values
of the fitted parameters (see Table 4 for the list of such parameters)
obtained by the Genetic Algorithm are used as a starting point for the DE-MCMC
analysis to derive the posterior probability distributions. The Genetic
Algorithm is initialized with a population set made of 200 individuals. In the
DE-MCMC we run 90 chains. The first 6$\cdot$104 steps of each chain are
discarded as part of the burn-in phase. They are followed by 104 more DE-MCMC
steps, which are used for the posterior distributions. The convergence is
checked by the Gelman - Rubin test applied on the steps that follow the burn-
in phase. At the end of the run, all parameters resulted in having R¡1.2
(Gelman & Rubin, 1992).
The priors that are used on the fitted parameters are set from the results in
the literature (mainly from Hellier et al. 2011) and noted in Table 4 along
with the results. The prior on the eccentricity is set as the weighted mean
and standard deviation of the results in the literature (Hebb et al., 2010;
Anderson et al., 2010, 2013; Hellier et al., 2011; Mandell et al., 2013; Lendl
et al., 2013; Zhou et al., 2014; Knutson et al., 2014). From this, the priors
on $\sqrt{e}\cdot\sin\omega_{0}$ and $\sqrt{e}\cdot\cos\omega_{0}$ are
calculated as a combination of the two parameters. The contribution on the
$\chi^{2}$ of the priors was calculated the same way as in Csizmadia et al.
(2019).

We modeled the radial velocity of the host star with a Keplerian motion but
with a time-variable $\omega$:

$V_{\mathrm{rad,i,j}}=V_{\mathrm{\gamma}}+V_{\mathrm{i,instr}}+K\cdot(e\cdot\cos\omega(t_{\mathrm{j}})+\cos(v_{\mathrm{j}}+\omega(t_{\mathrm{j}})))+\delta
V(t_{\mathrm{j}}),$ (7)

with a secular advance of the longitude of periastron, $\dot{\omega}$ =
$\mathrm{d\omega/dt,}$ caused by the tidal and rotational potential (as
expressed in Eq. (22)):

$\omega(t_{\mathrm{j}})=\omega_{0}+\dot{\omega}(t_{\mathrm{j}}-t_{0}),$ (8)

where $i$ is the index of the instrument used and $j$ is the index of the time
when the observation was taken; $V_{\mathrm{\gamma}}$ is the systematic
velocity of the barycenter of WASP-19Ab system relative to the barycenter of
the Solar System. Different datasets may have different instrumental zero-
point offsets, $V_{\mathrm{i,instr}}$. Finally, $K$ is the radial velocity
half-amplitude:

$K=\frac{2\pi a_{\mathrm{\star}}\sin i}{P_{\mathrm{a}}\sqrt{1-e^{2}}}.$ (9)

In this expression, $a_{\mathrm{\star}}$ is the semi-major axis of the star
around the common center of mass and $i$ is the inclination between the
orbital plane and the plane of the sky ( perpendicular to the line of sight).

Figure 4: Star-planet system geometry when tidal interactions occur. The
spherical shapes that star and the planet would have in the absence of tidal
interactions are drawn in red and dark brown. The tidal bulges are drawn in
orange and light brown, respectively. They arise on the stellar and planetary
surfaces (respectively in orange and light brown) and rotate with the star
itself and are elongated in the direction of the other body. The tidal lags
are also drawn. The image is not to scale.

Here, $\delta V$ is the non-orbital apparent contribution that arises from the
rotation of the star around its own axis. The stellar and planetary shapes are
distorted ellipsoids with the longest axis in the direction of the radius
vector connecting them, as shown in Fig. 4. Their apparent area varies
continuously during an orbital revolution, therefore the emitted light changes
(i.e., the ellipticity effect). Moreover, in close binary systems, part of the
radiation of each of the two components falls on the surface of the companion.
It is absorbed and re-emitted (or scattered) in all directions, varying during
the orbital phase (i.e., the reflection effect).
These two photometric effects in the RV data due to distortion are described
by Kopal (1959) as a non-orbital contribution to the RV curve:

$\displaystyle\begin{split}\delta V=-\frac{3}{2}K\hskip
2.84544pt\sqrt{1-e^{2}}\hskip
2.84544pt\frac{M_{\mathrm{\star}}+m_{\mathrm{p}}}{M_{\mathrm{\star}}}\hskip
2.84544pt\frac{P_{\mathrm{orb}}}{P_{\mathrm{rot,\star}}}\hskip
1.42271pt\left(\frac{R_{\mathrm{\star}}}{a}\right)^{4}\cdot\\\ \hskip
28.45274pt\cdot\left(1+\frac{k_{2,\star}}{2}\right)\sin i\cdot\sin\hskip
1.42271pt(2\hskip 1.42271pt(v+\omega))f_{2},\end{split}$ (10)

where the Love number of the star is assumed from theoretical calculations (we
take it from Claret 2004). Then, $P_{\mathrm{rot,\star}}$ is the rotational
period of the star around its axis and $f_{2}$ takes into account the limb
darkening. For more details on $f_{2}$ see Kopal (1959) and Csizmadia et al.
(2019).
Arras et al. (2012) already includes such calculations for precise RV analysis
of WASP-18A, but neglects the effect of stellar rotation. In other words, a
synchronous rotation of the star is assumed. However, for WASP-19A this is not
the case. It is known to be active, showing a rotational modulation with
$P_{\mathrm{rot,\star}}$ = 10.5 d (Hebb et al. 2010). Therefore, the term
$P_{\mathrm{orb}}$/ $P_{\mathrm{rot,\star}}$ is not expected to be 1, but
$\simeq$ 0.79 d/10.5 d $\simeq$ 0.08, introducing a factor of about 12
difference in the amplitude of such effect calculated by Arras et al. (2012)
and Kopal (1959). The amplitude of this contribution is around 0.19 m s-1, so
it is negligible for WASP-19A if compared to the typical observational
errorbars. The period of this effect is linked to the sine term of the true
anomaly in Eq. (10) and, therefore, it is half of the orbital period.

Moreover, we added jitter terms for each spectrograph to the RV model (see,
e.g., Baluev 2009 for a discussion of instrumental jitters). The jitter is
composed of an instrumental and a stellar contribution. It is not possible to
separate them. As a starting point, if present, we used the values proposed by
the corresponding literature. In particular, 13 m s-1 for Hebb et al. (2010)’s
data (ID 1, as in Anderson et al. 2010), 14.1 m s-1 and 6.9 m s-1 for Coralie
(ID 2) and HARPS (ID 3) Hellier et al. (2011)’s data, respectively, 20 m s-1
for Albrecht et al. (2012)’s (ID 4) and 17.8 m s-1 for Knutson et al. (2014)’s
(ID 5) RVs. Then, we refined all the jitter values by minimizing the
$\chi^{2}$ as:

$\chi^{2}_{\mathrm{jit,instr}}=\sum_{\mathrm{i=1}}^{n_{\mathrm{data}}}\frac{(O_{\mathrm{i}}-C_{\mathrm{i}})^{2}}{\sigma_{\mathrm{i}}^{2}+\sigma_{\mathrm{jit_{instr}}}^{2}}=n_{\mathrm{data}}-1,$
(11)

where $O_{\mathrm{i}}$ and $C_{\mathrm{i}}$ are the observed points (see
Tables D.1 and D.2) and the (calculated) expected values from the model,
respectively; $\sigma_{\mathrm{i}}$ represents the observed errors
corresponding to $O_{\mathrm{i}}$ and $n_{\mathrm{data}}$ is the number of
datapoints in the subset. This procedure was iteratively repeated until the
condition of Eq. (11) was fulfilled. Due to the highly inhomogeneous set of
data, this procedure allows for every RV dataset to have a weight in the fit
proportional to the number of its data points.
We also added a jitter value for transits and one for occultations to account
for inhomogeneities in the dataset and for a perturber with an orbital period
longer than 12 years, as described in Secion 5.1. The fitted values for the
jitter terms can be found in Table 4, along with all the fitted and derived
parameters in our model.

The orbital eccentricity is a key issue for Love number studies: in circular
orbits, apsidal motion is not observable. In addition, the argument of
periastron is hard to measure if eccentricity is not well known. When the
eccentricity is small, then its uncertainty range is often large and the
investigators find that the orbit is compatible with a circular one. However,
such results in the literature do not necessarily imply that the orbit is
circular, despite many investigators setting a zero eccentricity (and not
propagating the error bars of eccentricity to the corresponding result). It
only means that an upper limit exists for eccentricity.
Moreover, a certain dataset may indicate the presence of an eccentric orbit
with an accurate result. However, the sample size is not big enough or not
aptly distributed to get the uncertainty ranges accurate enough. The results
are imprecise and the real value of eccentricity is vague.
Although we collected more RV and TTV data than any previous study of the
WASP-19Ab system, the datasets come from various instruments with different
precisions and, thus, are non-homogeneous. The value of eccentricity may be
affected by the different data accuracy, as some of the data sets are diluted
by less precise measurement points. Because of these reasons, any previous
estimate of eccentricity can be accurate but imprecise.
Applying a prior on the eccentricity in the fitting procedure can be a valid
approach. The average and standard deviation of the previously reported
eccentricities can better represent reality than any single determination from
a limited dataset. Therefore, we calculated the weighted average of the
previous eccentricity measurement and their weighted standard error using
their uncertainty range as weights. Therefore, we performed a fit using this
mean eccentricity as a prior with their weighted standard error as a Gaussian
width.
In addition, in the calculation of the $\chi^{2}$ of the fit, priors also on
the stellar temperature, ${T_{\mathrm{eff,\star}}}$, and Love number,
$k_{\mathrm{{2,\star}}}$, the mass ratio, $q$, the orbital inclination, $i$,
the orbital over rotational period ratio, $\mathrm{F_{\star}}$, and the ratio,
$r_{\mathrm{p}}/R_{\mathrm{\star}}$, were added as described in Csizmadia et
al. (2019).

Parameter | Symbol | Fit | Value | Priors
---|---|---|---|---
Assumed parameters |  |  |  |
Stellar rotation | $V\sin(i)$ |  | 4.63 km s-1 †1 |
Stellar mass | $M_{\star}$ |  | 0.970 M☉ |
Planet-to-star radius ratio |  $r_{\mathrm{p}}$/$R_{\star}$ |  | 0.14248 |
Semi-major axis | $a$ |  | 0.01655 AU |
First transit epoch | $T_{0}$ |  | 2 454 775.3372 $\mathrm{BJD}_{\mathrm{TDB}}$ |
Adjusted parameters |  |  |  |
Jitters ID 1 | jit1 | RVs | 13.5 m s-1 |
ID 2 | jit2 | RVs | 35.0 m s-1 |
ID 3 | jit3 | RVs | 9.8 m s-1 |
ID 4 | jit4 | RVs | 11.5 m s-1 |
ID 5 | jit5 | RVs | 30.4 m s-1 |
ID 6 | jit6 | RVs | 13.1 m s-1 |
ID 7 | jit7 | RVs | 9.8 m s-1 |
ID 8 | jit8 | RVs | 22.3 m s-1 |
Transits | jittr | Tr | 0.00005 d |
Occultations | jitocc | Occ | 0.0001 d |
Fitted parameters |  |  |  |
Offset velocity of the system | $\mathrm{V_{\gamma}}$ | RVs | (20.7791 $\pm$ 0.0060) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
RV semi-amplitude | K | RVs | (253.48 $\pm$ 3.87 ) m s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
Offset instrumental velocity |  |  |  |
ID 2 | Vinstr,2 | RVs | (0.0258 $\pm$ 0.0071)km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 3 | Vinstr,3 | RVs | (0.0221 $\pm$ 0.0074 ) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 4 | Vinstr,4 | RVs | (-20.827 $\pm$ 0.016 ) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 5 | Vinstr,5 | RVs | (-20.712 $\pm$ 0.014 ) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 6 | Vinstr,6 | RVs | (-0.0919 $\pm$ 0.0092 ) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 7 | Vinstr,7 | RVs | (-0.1149 $\pm$ 0.0139 ) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
ID 8 | Vinstr,8 | RVs | (0.0750 $\pm$ 0.0090) km s-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
Anomalistic period | $P_{\mathrm{a}}$ | RVs and Tr/Occ | (0.788951 ${}^{+0.000012}_{-0.000017}$) d |  $\mathcal{U}$ [-$\infty$,+$\infty$]
| $\sqrt{e}\cdot\sin\omega_{0}$ | RVs and Tr/Occ | 0.03920 ${}^{+0.00458}_{-0.00682}$ |  $\mathcal{U}$ [-1,+1]
| $\sqrt{e}\cdot\cos\omega_{0}$ | RVs and Tr/Occ | -0.00056 ${}^{+0.01385}_{-0.01717}$ |  $\mathcal{U}$ [-1,+1]
Periastron precession rate | $d\omega/dt$ | RVs and Tr/Occ | (0.065 ${}^{+0.007}_{-0.010}$)∘ d-1 = |
|  |  | (233 ${}^{+25}_{-35}$)′′ d-1 |  $\mathcal{U}$ [-$\infty$,+$\infty$]
Stellar Love number | $k_{2,\star}$ | RVs | 0.0200 $\pm$ 0.0069 |  $\mathcal{N}$ (0.02;0.004)
Stellar temperature | ${T_{\mathrm{eff,\star}}}$ | RVs | (5498.65$\pm$ 172.86)K |  $\mathcal{N}$ (5500 K;100 K)
Planet-to-star mass ratio |  $q$ = $m_{\mathrm{p}}$/$M_{\star}$ | RVs | 0.001150 $\pm$ 0.000057 |  $\mathcal{N}$ (0.00115;3.3$\cdot$10-5)
Stellar radius over semi-major axis |  $R_{\star}$/$a$ | RVs | 0.2785$\pm$ 0.0070 |  $\mathcal{N}$ (0.2784;0.0040)
Orbital-to-rotational stellar period |  $F_{\star}$=$P_{\mathrm{orb}}$/$P_{\mathrm{rot,\star}}$ | RVs | 0.0729 $\pm$ 0.0074 |  $\mathcal{N}$ (0.0729;0.0043)
Orbital inclination | $i$ | RVs and Tr/Occ | (79.40 $\pm$ 0.69)∘ |  $\mathcal{N}$ (79.4;0.4)
Derived parameters |  |  |  |
Eccentricity | e |  | 0.00172 ${}^{+0.00035}_{-0.00033}$ |
Periastron angle | $\mathrm{\omega_{0}}$ |  | (89.71${}^{+24.68}_{-19.12}$ )∘ |
Sidereal period | $P_{s}$ |  | (0.788839267${}^{+0.000000018}_{-0.000000019}$ ) d |
Planetary Love number | $k_{\mathrm{{2,p}}}$ |  | 0.20 ${}^{+0.02}_{-0.03}$ |
Table 4: Parameters of the WASP-19Ab system. Assumed, adjusted – according to
Eq. (11) – fitted parameters and their priors, and derived parameters for our
model which fits RVs and transit/occultation mid-times in parallel. The third
column indicates if the parameter was fitted to model radial velocity (”RV”)
and/or mid-transit and mid-occultation times (”Tr/Occ”). The last column shows
the results of our analysis. The offset related to ID 1 ($V_{instr,1}$) was
not included because the first dataset was used to derive the relative
velocity between the observer and the center of mass of the system
($V_{gamma}$).
The solutions are calculated only from those steps of the DE-MCMC that fall
within the distribution around the highest peak of the posterior distribution
of $\dot{\omega}$ (see Fig. (6), solution B).
Errorbars are given with 1$\sigma$ confidence level.
The reported values and errorbars of the sidereal period $P_{s}$, $\omega_{0}$
and $e$ are not calculated - as
$P_{s}$=$P_{a}\cdot(1-P_{a}\cdot\dot{\omega}/n)$,
$\arctan(\sqrt{e}\cdot\sin\omega_{0}/\sqrt{e}\cdot\cos\omega_{0})$ and
$(\sqrt{e}\cdot\sin\omega_{0})^{2}$ \+ $(\sqrt{e}\cdot\cos\omega_{0})^{2}$
respectively - from the reported values of $P_{a}$, $\dot{\omega}$,
$\sqrt{e}\cdot\sin\omega_{0}$ and $\sqrt{e}\cdot\cos\omega_{0}$ but from the
median of the distribution resulting from the combination of the posterior
distributions of $\sqrt{e}\cdot\sin\omega_{0}$ and
$\sqrt{e}\cdot\cos\omega_{0}$.
†1 Hellier et al. 2011

Together with the RVs, we modeled mid-transit and mid-occultation times listed
in Tables C.1 \- C.4 mainly to refine the determination of the first transit
epoch and the period. To include periastron precession in the fit of predicted
mid-times, we followed the approach of Gimenez & Garcia-Pelayo (1983) and
Csizmadia et al. (2019).

###  5.3 Presence of apsidal motion

Because of the large parameter correlations between the mean motion and
periastron precession rate (see Csizmadia et al. 2019), as a first analysis,
we calculated 80 different models with fixed values of $\dot{\omega}$, the so-
called semi-grid analysis. This procedure also helps to get an initial range
for the fitted value of $\dot{\omega}$. The limiting values were
$\dot{\omega}=-0.25$ and $+0.45^{\circ}$d-1. We also fixed the epoch to 2 454
775.3372 $\mathrm{BJD}_{\mathrm{TDB}}$, while all other parameters were free
in the DE-MCMC analysis. The result is presented in Fig. (5), showing the
assumed value of the periastron precession rate and the corresponding
$\chi^{2}$ of the fit. The figure clearly shows that the joint fit of the RV
and TTV data is significantly worse if $\dot{\omega}=0$, namely, no periastron
precession is present. A small amount of periastron precession improves the
quality of the fit significantly. The figure shows that small negative
periastron precession rates also improve the fit but still result in a
statistically worse fit than the small positive precession values. From that
figure and the statistical analysis, we conclude that small positive
periastron precession rates are needed to have a good fit of the joint RV and
TTV data of WASP-19Ab, and we are able to give a lower and an upper limit of
$0.06\lesssim\dot{\omega}\lesssim 0.075^{\circ}$d-1. The result is refined in
the next section.

Figure 5: $\chi^{2}$ values of the fit when varying $\dot{\omega}$ in the
model. The horizontal dashed line separates the statistically worse models
(over the line) from the best models (below the dashed lines). The models
below the dashed lines are statistically equivalent within 1$\sigma$ (68%
confidence level for the red line) and 2$\sigma$ (95% confidence level for the
green line). Note: the significantly larger $\chi^{2}$ at $\dot{\omega}$=0.
The high amount of local minima is an indication of the complexity of the
problem.

##  6 Discussion of the results

###  6.1 Apsidal motion rate

In this section, we show and discuss the results of the analysis performed
with the model described in Sect. 5. Contrarily to the semi-grid method (Sect.
5.3), here we fit $\dot{\omega}$ along with the other parameters.

During the analysis, we noticed that different DE-MCMC chains converged to
different values of $\dot{\omega}$. In particular, as it is shown in Fig. 6
and in Table 5, during the run multiple chains explored various positive and
negative values of $\dot{\omega}$. Figure 6 clearly shows also the correlation
between the anomalistic period and the periastron precession rate, as
discussed in Csizmadia et al. (2019). It is due to the formula that links the
anomalistic period $P_{\mathrm{a}}$ and the sidereal period $P_{\mathrm{s}}$:

$P_{\mathrm{s}}=P_{\mathrm{a}}(1-\dot{\omega}/n).$ (12)

This formula that links the two quantities gives a between period and
periastron precession rate, but no unique solution. The added value of
including transit and occultation mid-times lies in the fact that they
constrain the sidereal period, and therefore a range of values for
$P_{\mathrm{a}}$ and $\dot{\omega}$ (as shown in Fig. 6), while avoiding some
other ranges.
A different chain convergence shows why it was important to run various
chains. Only one chain would have converged to one of the possible values,
excluding the other results or could have been trapped in a local minimum (the
same local minima found by the semi-grid method approach; see Fig. 5).
The highest peak of $\dot{\omega}$ (solution B) in the posterior distribution
results around $\sim$+0.065∘d-1 (summarized in Table 5). The positive
solutions C, D, and E can be discarded because of their high $\chi^{2}$
values. Moreover, solution E would lead to a planetary Love number higher than
1.5, which is unphysical.

Solution |  $\dot{\omega}$ [${}^{\degree}d^{-1}$] | $\chi^{2}$
---|---|---
A | -0.064 | 416
B | +0.065 | 408
C | +0.163 | 423
D | +0.261 | 424
E | +0.479 | 430
Table 5: Median values of the five posterior distributions shown in Fig. 6 and
corresponding $\chi^{2}$ (calculated before introducing the contribution of
the jitters to the uncertainty, as in Eq. (11)).  Figure 6:  Posterior
distributions and correlation between the anomalistic period and periastron
precession rate where the chains converged. The middle plot shows the strong
correlation between these two free parameters, as in Eq. (12). On the sides,
are the corresponding histograms of the posterior distributions. As can be
seen from the histograms on the sides, many chains converged to various
positive and negative peaks, as predicted by the semi-grid method and reported
in Table 5.

The negative periastron precession rate (solution A) would correspond either
to a retrograde orbit (of the planet with respect to the rotation of the star)
or to an unseen perturber. The retrograde orbit can be excluded from Rossiter-
McLaughlin effect studies and TESS light curves (Wong et al., 2020). They
confirm that the spin axis of the star is almost perpendicular to the orbital
plane.
We investigated in detail the presence of a perturber in Sect. 5.1 and no
evidence is found. Assuming a negative periastron precession amplitude as
observed of $\sim$-0.064∘d-1 = -229.68′′d-1, through Eqs. (12) and (16) in
Borkovits et al. (2011), we can estimate the period of the body producing such
a precession rate. We subtracted the general relativity contribution
(calculated in Sect. 5.2 as 2.85′′ d-1) from these solutions and derived the
orbital period of the perturber producing such an $\dot{\omega}$. For a
circular orbit, a body with a mass in the range 0.1 M♁ and 20 M♃ would have an
orbital period below 10 d, as shown in Fig. 7. Such a planet would be
retrievable in the RV data, while no long-term (over a 10-day orbital period)
companion in this mass range can produce a negative periastron precession.
Therefore the solutions yielding negative $\dot{\omega}$ values can be
discarded.

Figure 7: Mass and orbital period of a second body that would cause a negative
periastron precession rate of $\sim$-0.0638∘d-1 (corresponding to the negative
solution of the DE-MCMC analysis) on planet b, as deduced from Borkovits et
al. (2011). See text for more details.

The results for the fitted parameters of the $\dot{\omega}$-positive solution
(solution B) with lowest $\chi^{2}$ are shown in Table 4, along with their
errorbars. With 99 RV points (outside the transit, 203 in total), 162 mid-
transit and 11 mid-occultation points, and 19 free parameters, the reduced
$\chi^{2}$ is 1.02. The observed value of the total contribution
$\dot{\omega}$ from the DE-MCMC analysis of the data is
(233${}^{+25}_{-35}$)′′d-1 (see Table 4), corresponding to an apsidal motion
period of $U$=$2\pi/\dot{\omega}$ $\simeq$ 15 years.
Figure 8 shows the residuals of the transit and occultation mid-times with
such a fit. Figure 9 shows the phase-folded RV curve and the residuals of our
periastron precession fit.
We subtract the General Relativity term and the tidal and rotational
components relative to the star from the observed $\dot{\omega}$. What is left
in Eq. (1) are the tidal and rotational components due to the planet. They are
related to the interior of the planet and show clearly the dependence on Love
number (see Eqs. (3) and (4)). Then, $k_{\mathrm{{2,p}}}$ can then be
estimated by inverting these equations.
The only unknown parameter is the rotational period $P_{\mathrm{rot,p}}$ of
the planet itself. Since there is no measurement of the rotational period of
the planet, we can only assume a value for $P_{\mathrm{rot,p}}$. The two
extreme cases are presented when:

  1. -

tidal forces lead to the disruption of the planet ($P_{\mathrm{rot,p}}$=0.14
d), then we obtain $k_{\mathrm{{2,p}}}$= 0.07;

  2. -

the planet is non-rotating ($P_{\mathrm{orb}}/P_{\mathrm{rot,p}}\rightarrow$
0), then we obtain $k_{\mathrm{{2,p}}}$=0.22.

The measured Love number of the planet must lie in between these two values.
We can suppose that tidal forces synchronize the orbital and rotational
periods quite fast. This means that $P_{\mathrm{orb}}$ = $P_{\mathrm{rot,p}}$.
We can then derive $k_{\mathrm{{2,p}}}$ from Eqs. (16) and (21):
$k_{\mathrm{{2,p}}}$=0.20 ${}^{+0.02}_{-0.03}$, with a precision of 15$\%$.

For comparison, the most up-to-date values of the tidal Love number for
planets and satellites in the Solar System and exoplanets are listed in Table
6. The value obtained for the fluid Love number of WASP-19Ab has the same
order of magnitude as the other values derived for exoplanets.
The lower the value, the more matter is condensed toward the center of the
body. The low Love number obtained, $k_{\mathrm{{2,p}}}$=0.20, is an
indication of a high-mass core. This has to be tested and verified with
simulations of planetary interior structure through the equations of state,
however, this is beyond the scope of this paper.
Moreover, if hydrostatic equilibrium is assumed (whereby the surface of the
body can be described as an equipotential surface of the tidal and rotational
potentials), the second-order fluid Love number $h_{\mathrm{2p}}$ can also be
calculated. This assumption is close to reality if the body is a giant gaseous
planet. In this case, $h_{\mathrm{2,p}}$=$k_{\mathrm{{2,p}}}$+1 (Munk &
MacDonald 1960), therefore $h_{\mathrm{2,p}}$=1.20 for synchronous rotation.
This causes a small-shaped deformation that can be hard to observe
photometrically (Hellard et al., 2019).

During the analysis, we noticed that the main contribution to the uncertainty
of the fit is related to the transit fit. It is due to underestimated
errorbars on the mid-transit times, which compromise the reliability of the
results, increasing the $\chi^{2}$ of the fit. This was already suggested by
Mancini et al. (2013) regarding the 54 transits they analysed. Therefore, we
increased the errorbars to 0.0005 d (approximately 43 s) if the reported
uncertainties were lower than this threshold.
We also underline the importance of acquiring new light curves. Precise
transit and, mostly, occultation mid-times are needed to break the degeneracy
between periastron precession and the change of orbital period due to for
instance companions or orbital decay.

Figure 8:  Residuals of the periastron precession fit for transits and
occultation mid-times, calculated with the sidereal period. Transit mid-times
are plotted as blue circles, occultation mid-times as red circles. The
errorbars include the jitters, as in Table 4.
Figure 9:  Data with errorbars, phase-folded with our fit including periastron
precession, shown in the upper plot. The colours represent different datasets,
as in Fig. (1) (see also Sect. 4.2 for more details).
Residuals of the periastron precession rate fit are shown in the lower plot.
The errorbars include the jitters, as in Table 4.
In both plots, the in-transit points are not shown as they were not used in
the fit.

###  6.2 Alternative explanations

We investigated alternative scenarios to the presence of periastron
precession. In particular, we searched for distant massive companions with the
approach of Knutson et al. (2014). The long-term acceleration of the system
was fitted by adding a $\gamma$ acceleration to the RV Eq. (7) as:

$V_{\mathrm{acc}}(t_{\mathrm{j}})=\dot{V_{\gamma}}\cdot(t_{\mathrm{j}}-t_{0}).$
(13)

We ran the DE-MCMC algorithm in the case of a circular and an eccentric orbit.
The posterior distribution of $\dot{V_{\gamma}}$ has median of (7.37 $\pm$
5.02)$\cdot 10^{-7}$m s-2 for the circular case and (7.43 $\pm$ 4.89)$\cdot
10^{-7}$ m s-2 for the eccentric one. Both results are compatible with zero
and reproduce the results of Knutson et al. (2014) of (7.52 $\pm$ 3.94) $\cdot
10^{-7}$ m s-2. Our results increase the confidence level of a non-detection
of long-term acceleration from Knutson et al. (2014)’s 1.9$\sigma$ to
1.5$\sigma$.

We also investigated the tidal decay scenario, as it has been analysed and
confirmed in the system WASP-12b (see respectively Patra et al. 2017; Yee et
al. 2020; Turner et al. 2021; Wong et al. 2022). The observed TTVs can be due
to tidal decay or periastron precession. We fitted the RV semi-amplitude $K$
in Eq. (9) as a time-varying parameter:

$K(t_{\mathrm{j}})=K_{\mathrm{0}}+\dot{K}(t_{\mathrm{j}}-t_{\mathrm{0}}).$
(14)

Deriving Kepler’s laws with respect to time, we can see that a change in the
RV semi-amplitude results in an opposite change in the semi-major axis.
For the circular orbit case, we obtained $\dot{K}$=(-2.77 $\pm$
2.80)$\cdot$10-8 m s-2 while for the eccentric case (-2.60 $\pm$2.73)
$\cdot$10-8 m s-2. As Rosário et al. (2022), we did not find significant
evidence of orbital decay. Both in the case of a long-term velocity and of
tidal decay, the results are compatible with zero and therefore considered as
constant.

Body | Fluid Love number | Method | Reference
---|---|---|---
| $k_{\mathrm{{2,p}}}$ |  |
WASP-19Ab | 0.20 ${}^{+0.02}_{-0.03}$ | RVs and TTVs: periastron precession | This work
WASP-18Ab | 0.62${}^{+0.55}_{-0.19}$ | RVs and TTVs: periastron precession |  Csizmadia et al. (2019)
WASP-103b† | 0.59 ${}^{+0.23}_{-0.27}$ | LCs: shape deformation |  Barros et al. (2022)
WASP-121b† | 0.39 ${}_{-0.81}^{+0.71}$ | LCs: shape deformation |  Hellard et al. (2020)
Jupiter | 0.565 $\pm$ 0.006 | Juno: Doppler shift measurements |  Durante et al. (2020)
Saturn | 0.390 $\pm$ 0.024 | Cassini: gravitational perturbations on Enceladus, Tethys, Dione and Rhea |  Lainey et al. (2017)
Enceladus | 0.9896 $\pm 0.0103$ | Cassini: quadrupole gravity field and its hemispherical asymmetry |  Taubner et al. (2016)
Neptune | 0.392 | Orbital perturbations on Triton and Nereid |  Jacobson (2009)
Body | Tidal Love number | Method | Reference
HAT-P-13b | 0.31 ${}^{+0.08}_{-0.05}$ | RVs and TTVs: $\dot{e}$ and $\dot{\omega}$ |  Buhler et al. (2016)
” ” | 0.81 $\pm$ 0.10 | RVs and TTVs: $\dot{e}$ and $\dot{\omega}$ |  Hardy et al. (2017)
Mercury | 0.53 $\pm$ 0.03 | MESSENGER: Topography |  Konopliv et al. (2020)
Venus | 0.295 $\pm$ 0.066 | Magellan and Pioneer Venus Orbiter: Doppler tracking |  Konopliv & Yoder (1996)
Moon | 0.02416 $\pm$ 0.00022 | GRAIL and LLR: mapping of the gravitational field |  Williams et al. (2014)
Mars | 0.183 $\pm$ 0.009 | Mars Express: Doppler tracking and characterization of the orbits of Phobos and Deimos |  Jacobson & Lainey (2014)
Titan | 0.637 $\pm$ 0.112 | Cassini: response to Saturn’s variable tidal field |  Iess et al. (2012)

Table 6: Fluid and tidal Love numbers measured for Solar System planets and
minor bodies and exoplanets. In the first half of the table, we show fluid
Love numbers, which are comparable with our result. In the second half, we
report tidal Love numbers. They refer to different quantities, so they are not
comparable, except at longer periods when elastic stresses relax.
† Calculated from $h_{2}$ assuming hydrostatic equilibrium

##  7 Conclusions

In this study, we review the theory on tidal interaction, as described by
Kopal (1978) and reported with modern notation. Using assumptions on the Love
number of the star, based on well-established theoretical models of its
evolution and interior structure, the apsidal motion rate allows us to
constrain the Love number $k_{\mathrm{{2,p}}}$ of the planet. We applied the
results to a case study, namely, the system WASP-19Ab, by using archival mid-
transit times, occultations, and radial velocity data, including previously
unpublished RV observations made by HARPS. This dataset covers twelve years in
total. First, we ruled out previous claims of additional planetary companions
in the system as possible perturbers in the radial velocity curve that could
hide the small contribution of the periastron precession motion. Moreover, we
investigated the presence of a long-term acceleration in the RV dataset and of
tidal decay. Both these scenarios have been rejected with respect to being
compatible with a zero trend. Then, we used a model that accounts for an
eccentric orbit and apsidal motion in order to fit the data. We detected the
presence of apsidal motion in the planetary orbit and linked it to the Love
number of the planet. By assuming synchronous rotation for the planet, we
derived $k_{\mathrm{{2,p}}}$=0.20${}^{+0.02}_{-0.03}$. The planetary fluid
Love number is constrained in this system for the first time. There are few
systems for which it has been measured. We take the opportunity to compare
WASP-19Ab to them. Table 6 shows the fluid and tidal Love numbers of Solar
system planets and bodies and exoplanets which are reported in the literature.
The derived value of the fluid Love number $k_{\mathrm{{2,p}}}$ of WASP-19Ab
is on the same order of magnitude as the one of other Jupiter-like planets
(WASP-4b, WASP-18Ab, WASP-103b and WASP-121b). A low value suggests the
presence of a dense core in the planet, similar to the case of Saturn,
Neptune, and WASP-121b. Note that the Love numbers of WASP-12b and WASP-4b
(derived by Campo et al. 2011 and Bouma et al. 2019, respectively) are
intentionally left out of the table. In the first case, the TTVs might be
explained by tidal decay (Yee et al. (2020) and reference therein). In the
second case, the presence of a second, long-period planet in the system
(Turner et al. 2022) further complicates the TTV analysis. The results of
Bouma et al. (2020) were discussed further by Harre & Smith (2023) in the
light of this second planet, making the determination of $k_{\mathrm{{2,p}}}$
uncertain.
No general conclusion on planetary physics can be drawn from this set of
measured Love numbers because the sample is too small to be representative of
the existing exoplanet population. We need observations of the Love number of
more exoplanets.

We also characterize, for the first time, the stellar multiplicity of the
system. We find a stellar companion with an apparent separation of 68′′,
corresponding to a physical distance of almost 20 000 AU, from the system
under analysis. However, its expected effect on the RV of WASP-19A is
negligible.

Acknowledgments We acknowledge the support of DFG Research Unit 2440: ”Matter
Under Planetary Interior Conditions: High Pressure, Planetary, and Plasma
Physics” and of DFG grants RA 714/14-1 within the DFG Schwerpunkt SPP 1992,
Exploring the Diversity of Extrasolar Planets. L. M. B. would also like to
thank Carl Knight for providing us with two transits of WASP-19Ab observed by
the Ngileah Observatory, New Zealand, in May 2022. L. M. B. would also like to
acknowledge H. Hußmann and F. Sohl for the useful discussion on the different
kinds of Love numbers.
The work is based on observations collected at the European Southern
Observatory under ESO programme 0104.C-0849.
This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
This work presents results from the European Space Agency (ESA) space mission
$Gaia$ (https://www.cosmos.esa.int/gaia). $Gaia$ data are being processed by
the $Gaia$ Data Processing and Analysis Consortium (DPAC) (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC is
provided by national institutions, in particular the institutions
participating in the Gaia MultiLateral Agreement (MLA).

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##  Appendix A Equations of apsidal motion

In this Appendix, we rewrite the equations derived by Kopal (1959, 1978) with
modern notation.
First, we define the stellar and planetary disturbing accelerations
$\mathbb{R}_{\mathrm{i}}$ and $\mathbb{S}_{\mathrm{i}}$, where the subscript
$i$ represents either the star ($i=\star$) or the planet ($i=p$).
$\mathbb{R}_{\mathrm{i}}$ acts in the direction of the radius vector and
$\mathbb{S}_{\mathrm{i}}$ in the plane of the orbit in the direction
perpendicular to the radius vector and positive in the direction of motion.
The radial perturbation accelerations can be expressed as:

$\mathbb{R}_{\star}=-\frac{3}{4}G(M_{\mathrm{\star}}+m_{\mathrm{p}})\frac{m_{\mathrm{p}}}{M_{\mathrm{\star}}}k_{\mathrm{{2,\star}}}\frac{R_{\mathrm{\star}}^{5}}{r^{4}\hskip
2.84544ptr_{\mathrm{\epsilon_{\star}}}^{3}}(3\cos^{2}\epsilon_{\mathrm{\star}}-1),$
(15)

and

$\mathbb{R}_{\mathrm{p}}=-\frac{3}{4}G(M_{\mathrm{\star}}+m_{\mathrm{p}})\frac{M_{\mathrm{\star}}}{m_{\mathrm{p}}}k_{\mathrm{{2,p}}}\frac{r_{\mathrm{p}}^{5}}{r^{4}\hskip
2.84544ptr_{\mathrm{{\epsilon_{p}}}}^{3}}(3\cos^{2}\epsilon_{\mathrm{p}}-1),$
(16)

where $G$ is the Gravitational constant, $M_{\mathrm{\star}}$ and
$m_{\mathrm{p}}$ are the masses of the star and the planet, respectively,
$R_{\mathrm{\star}}$ and $r_{\mathrm{p}}$ their radii and
$k_{\mathrm{{2,\star}}}$ and $k_{\mathrm{{2,p}}}$ their second-order fluid
Love number.
Here, $r$ is the radius vector:

$r=\frac{a(1-e^{2})}{1+e\cdot\cos v},$ (17)

where $a$ is the semi-major axis of the orbit, $e$ is the orbital eccentricity
and $v$ is the true anomaly of the planetary orbit, as shown in Fig. A.1.

Figure A.1: Geometry of the orbital system in the simple case of orbital
inclination of $i$ =90∘. The star symbol and the open dot denote the positions
of the star and the planet, respectively; $\omega$ is the angle of periastron,
$\theta$ is the phase from superior conjunction, and $v$ is the true anomaly;
$l.o.s.$ denotes the line of sight. In this configuration, $\theta$ equals 0
for transits and $\pi$ for occultations.

A tidal lag in latitude arises if the spin axis of the two bodies is not
perpendicular to the orbital plane and in general is small or even zero. On
the other hand, a lag in longitude arises from asynchronism between spin and
orbital rotation. The angles $\epsilon_{\star}$ and $\epsilon_{\mathrm{p}}$
represent the tidal lag of the star or the planet in astrocentric longitude
(Efroimsky & Makarov 2013) as:

$\epsilon_{\mathrm{i}}=\frac{1}{2}\arcsin\left(-\frac{1}{Q_{i}}\operatorname{sign}{[P_{\mathrm{orb}}/P_{\mathrm{rot,i}}-1]}\right),$
(18)

where $P_{\mathrm{orb}}$ is the orbital period and $P_{\mathrm{rot,i}}$ is the
rotational period of the star or the planet. The function $sign$ takes the
value of +1, 0 or -1, depending on the sign of the expression in the square
brackets. $Q_{\mathrm{i}}$ is the tidal quality factor. A high $Q$ factor
corresponds to a weak dissipation and vice-versa (see Kaula (1964) and
reference therein).
The radius vector, $r_{\mathrm{\epsilon}}$, is defined as:

$r_{\mathrm{\epsilon,i}}=\frac{a(1-e^{2})}{1+e\cdot\cos(v-\epsilon_{\mathrm{i}})}.$
(19)

The difference between $r$ and $r_{\mathrm{\epsilon_{i}}}$ is due to a finite
speed of progression of the radial tidal waves in viscous matter. The
amplitude of tidal distortion is not maximum when the bodies are closest to
each other, but at a later time. We consider a fluid case, which corresponds
to $\epsilon_{\mathrm{i}}$=0.

In addition, we define the tangential perturbation accelerations:

$\mathbb{S}_{\star}=\frac{3}{4}G(M_{\mathrm{\star}}+m_{\mathrm{p}})\frac{m_{\mathrm{p}}}{M_{\mathrm{\star}}}k_{\mathrm{{2,\star}}}\frac{{R_{\mathrm{\star}}}^{5}}{r^{4}r_{\epsilon_{\star}}^{3}}\sin(2\epsilon_{\star}),$
(20)

for the star and

$\mathbb{S}_{\mathrm{p}}=\frac{3}{4}G(M_{\mathrm{\star}}+m_{\mathrm{p}})\frac{M_{\mathrm{\star}}}{m_{\mathrm{p}}}k_{\mathrm{{2,p}}}\frac{r_{\mathrm{p}}^{5}}{r^{4}r_{\mathrm{{\epsilon_{p}}}}^{3}}\sin(2\epsilon_{\mathrm{p}}),$
(21)

for the planet. Finally, $\mathbb{R}$ = $\mathbb{R}_{\mathrm{\star}}$ \+
$\mathbb{R}_{\mathrm{p}}$ and $\mathbb{S}$ = $\mathbb{S}_{\mathrm{\star}}$ \+
$\mathbb{S}_{\mathrm{p}}$.

There is a third perturbing acceleration component beyond $\mathbb{S}$ and
$\mathbb{R}$. It is normal to the plane of the orbit and positive toward the
north pole. However, its contribution is zero as it is proportional to the
latitude component of the angle of tidal lag. It can arise only if the orbital
plane is inclined with respect to the equatorial plane of the body (star or
planet). We neglect this term because the Rossiter-McLaughlin measurements of
WASP-19A show an equatorial orbit (see, e.g., Sedaghati et al. 2021).
With the introduction of the terms, $\mathbb{S}$ and $\mathbb{R}$, the
perturbation equations can be easily written following Kopal (1959, 1978). The
temporal change of the argument of periastron measured from the ascending node
is:

$\frac{d\omega}{dt}=\frac{\sqrt{1-e^{2}}}{nae}\left[-\mathbb{R}\hskip
2.84544pt\cos v+\mathbb{S}\left(1+\frac{r}{a(1-e^{2})}\right)\sin v\right],$
(22)

where $n$ is the mean motion that can be defined through Kepler’s law or
through the orbital period $P_{\mathrm{orb}}$.
Figure A.2 shows the effect of the periodical and secular perturbations on
$\omega$, as in Eq. (22). The plots were made setting the physical and orbital
parameters to those of the exoplanetary system WASP-19Ab (see Sect. 3) with
the Love number of WASP-18Ab (Csizmadia et al. 2019) as a first guess. We
present how it evolves over 1, 10, 50, 100, and 500 orbits.

Figure A.2: Temporal change of the osculating orbital element $\omega(t)$ due
to tidal and rotational potentials, as described by Eq. (22), assuming the
orbital properties of the system WASP-19 (see Sect. 3.1) and the Love number
of WASP-18Ab Csizmadia et al. (2019).

##  Appendix B Planetary eccentricity driven by WASP-19B

Here, we show that a companion star on a very eccentric orbit is able to cause
an eccentric orbit of the planet in the WASP-19 system.

Borkovits et al. (2011) gave analytic expressions for the variations of the
orbital elements up to the sixth order of the inner eccentricity for any
arbitrary orientation of the orbits. However, we do not know the spatial
orientation of the orbit of the companion star B nor the longitude of the node
of the planet b. Therefore, we assume co-planar orbits for the planet and star
B, and we also assume that their semi-major axes are aligned. In this specific
case, the integration of Eq. (9) of Borkovits et al. (2011) gives the simple
expression for the eccentricity of the planet.

$e_{b}=\sqrt{\frac{e^{A_{L}/2\cdot f(v_{B})}}{1+e^{A_{L}/2\cdot f(v_{B})}}},$
(23)

where $f(v_{B})$ is

$f(v_{B})=\frac{1}{2}\cos 2v_{B}+\frac{4e_{B}}{3}\cos^{3}v_{B},$ (24)

and, $A_{L},$

$A_{L}=\frac{15}{8}\frac{P_{b}}{P_{B}}\frac{M_{B}}{M_{A}+M_{B}+M_{b}}\frac{1}{(1-e_{B})^{3/2}}.$
(25)

Here $v_{B}$ is the true anomaly of star B and it holds the time-dependence,
$P$ and $M$ are the orbital periods around star A and the masses of the bodies
and $e$ is the eccentricity. Indices $A$ and $B$ stand for stars A and B
respectively, while index $b$ denotes the planet.

The lower period limit of star B around star A can be estimated assuming that
$P_{B}$ is minimum if star B is in apoastron. In this case, the observed
separation between stars A and B corresponds to a distance equal to
$a_{B}(1+e_{B})$. For different values of $e_{B}$, we can determine the semi-
major axis that yields the orbital period, $P_{B}$, via Kepler’s third law. In
case of $e_{B}=0.995$ $a_{B}\approx 10^{4}$ AU and $P_{B}\approx 1$ Myrs
(note: the periastron point is still located at 50 AU from star A).

Since we do not know the exact orbit of star B, we just estimate the planetary
eccentricity. Taking the Taylor series of the exponential function in Eq. (23)
and neglecting the time dependence, we obtain

$e_{b}\approx\sqrt{A_{L}/2\times\left(\frac{1}{2}+\frac{4e_{B}}{3}\right)}.$
(26)

Using the aforementioned numerical values and assuming $M_{B}\sim
0.4M_{\odot}$, we get $e_{b}\approx 0.0011$. This calculation leads to an
order of magnitude estimation of whether any (or a specific) combination of
orbital elements can be responsible for the observed eccentricity. A deviation
from this estimate can be due to the unknown spatial orientation of the
different orbits.

##  Appendix C Tables of transits and occultations

Cycle number |  Tmid [days] |  Uncertainty [days] | Observations | Reference
---|---|---|---|---
N |  $\mathrm{BJD}_{\mathrm{TDB}}$ \- 2 450 000 |  |  |
0 | 4775.33720 | 0.00150 | WASP-South, 2.0m - Faulkes Telescope South (FTS) |  Hebb et al. (2010)
2 | 4776.91566 | 0.00019 | HAWK-I (VLT) |  Anderson et al. (2010)
53 | 4817.14633 | 0.00021 | 2.0m - Faulkes Telescope South (FTS) |  Lendl et al. (2013), from Hebb et al. (2010)
499 | 5168.96879 | 0.00009 |  EFOSC on NTT, WASP-South and Hebb et al. (2010)’s LCs |  Hellier et al. (2011)
517 | 5183.167890 | 0.000068 | IRAC on Spitzer |  Anderson et al. (2013)
538 | 5199.73343 | 0.000134 |  0.25m - Observatorio Astronómico Beta Orionis†1 | F. Tifner (TRESCA)
604 | 5251.79657 | 0.00014 | EFOSC2 on NTT |  Tregloan-Reed et al. (2013)
605 | 5252.58544 | 0.00010 | EFOSC2 on NTT |  Tregloan-Reed et al. (2013)
609 | 5255.74077 | 0.00012 | EFOSC2 on NTT |  Tregloan-Reed et al. (2013)
609 | 5255.741050 | 0.000145 | EFOSC on NTT |  Dragomir et al. (2011), from Hellier et al. (2011)
614 | 5259.68459 | 0.000359 |  0.40m - Observatorio Cerro Armazones†2 |  Colque (TRESCA) re-analysed by Petrucci et al. (2020)
632 | 5273.88282 | 0.00062 |  0.25m - Rarotonga Observatory†3 | P. Evans (TRESCA)
664 | 5299.12768 | 0.00055 |  0.24m - YSVP Observatory†4 | I. Curtis (TRESCA)
687 | 5317.27076 | 0.00006 | ASTEP 400 Telescope |  Abe et al. (2013)
709 | 5334.62540 | 0.00021 | DFOSC on 1.54m - Danish Telescope |  Mancini et al. (2013)
714 | 5338.56929 | 0.00023 | TRAPPIST |  Lendl et al. (2013)
733 | 5353.55659 | 0.00024 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
746 | 5363.81131 | 0.00041 |  0.30m - MAGS Observatory†5 | G. Milne (TRESCA)
752 | 5368.54285 | 0.00212 | DFOSC on 1.54m - Danish Telescope |  Mancini et al. (2013)
969 | 5539.72329 | 0.00030 | EulerCam on Euler-Swiss |  Lendl et al. (2013)
1007 | 5569.69827 | 0.00036 | TRAPPIST |  Lendl et al. (2013)
1021 | 5580.74124 | 0.00058 |  1.0m - CTIO†8 |  Dragomir et al. (2011)
1024 | 5583.10979 | 0.00089 |  0.24m - YSVP Observatory†4 | I. Curtis (TRESCA)
1026 | 5584.68693 | 0.00024 | TRAPPIST |  Lendl et al. (2013)
1026 | 5584.68685 | 0.00019 | EulerCam on Euler-Swiss |  Lendl et al. (2013)
1038 | 5594.15188 | 0.00168 |  0.30m - PEST†6 |  TG Tan re-analysed by Mancini et al. (2013)
1047 | 5601.25164 | 0.00071 |  D0.30m - PEST†6 |  TG Tan (TRESCA) re-analysed by Mancini et al. (2013)
1049 | 5602.83138 | 0.00046 | TRAPPIST |  Lendl et al. (2013)
1052 | 5605.19414 | 0.00180 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analysed by Mancini et al. (2013)
1054 | 5606.77464 | 0.00022 | EulerCam on Euler-Swiss |  Lendl et al. (2013)
1055 | 5607.56241 | 0.00033 | TRAPPIST |  Lendl et al. (2013)
1074 | 5622.55057 | 0.00026 | TRAPPIST |  Lendl et al. (2013)
1076 | 5624.12787 | 0.00142 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
1087 | 5632.80612 | 0.00025 | EulerCam on Euler-Swiss |  Lendl et al. (2013)
1116 | 5655.68222 | 0.00045 | TRAPPIST |  Lendl et al. (2013)
1135 | 5670.66976 | 0.00064 | TRAPPIST |  Lendl et al. (2013)
1144 | 5677.77038 | 0.00195 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1158 | 5688.81201 | 0.00333 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1159 | 5689.60276 | 0.00030 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
1163 | 5692.75674 | 0.00255 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1164 | 5693.54639 | 0.00013 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
1177 | 5703.79933 | 0.00411 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1178 | 5704.59078 | 0.00034 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
1183 | 5708.53495 | 0.00015 | DFOSC on 1.54m Danish Telescope |  Mancini et al. (2013)
1228 | 5744.032190 | 0.000040 | WFC3 on HST |  Mandell et al. (2013)
1254 | 5764.54014 | 0.00125 |  CTIO 1m†8 |  Cortés-Zuleta et al. (2020), analyzed by TLCM
1270 | 5777.16366 | 0.00022 | IRAC on Spitzer |  Wong et al. (2016)
1283 | 5787.41863 | 0.00023 | IRAC on Spitzer |  Wong et al. (2016)
1409 | 5886.81234 | 0.00208 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1421 | 5896.27611 | 0.00210 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1445 | 5915.20980 | 0.00065 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1450 | 5919.15485 | 0.00103 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1454 | 5922.30966 | 0.00555 |  0.30m - PEST†6 |  TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)
1552 | 5999.616301 | 0.00007 | MMIRS on the Magellan II Telescope |  Bean et al. (2013)
1580 | 6021.703740 | 0.000085 | MMIRS on the Magellan II Telescope |  Bean et al. (2013)
1595 | 6033.53645 | 0.00014 |  GROND†7 |  Mancini et al. (2013)
1595 | 6033.53651 | 0.00007 |  GROND†7 |  Mancini et al. (2013)
1595 | 6033.53643 | 0.00007 |  GROND†7 |  Mancini et al. (2013)
1595 | 6033.53652 | 0.00009 |  GROND†7 |  Mancini et al. (2013)
1590 | 6029.59251 | 0.00035 | EulerCam on Euler-Swiss |  Lendl et al. (2013)
1633 | 6063.51175 | 0.00030 | TRAPPIST |  Lendl et al. (2013)
1977 | 6334.87208 | 0.00053 |  0.25m - Rarotonga Observatory†3 | P. Evans (TRESCA)
1996 | 6349.86037 | 0.00081 |  0.25m - Rarotonga Observatory†3 | P. Evans (TRESCA)
2016 | 6365.63794 | 0.00010 | DFOSC on 1.54m Danish Telescope |  Cortés-Zuleta et al. (2020), analyzed by TLCM
2048 | 6390.87994 | 0.00043 |  0.25m - Rarotonga Observatory†3 | P. Evans (TRESCA)
2064 | 6403.50164 | 0.00022 | DFOSC on 1.54m Danish Telescope |  Cortés-Zuleta et al. (2020), analyzed by TLCM
2465 | 6719.82642 | 0.00050 |  0.25m - Rarotonga Observatory†3 | P. Evans (TRESCA)
2490 | 6739.551364 | 0.000046 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
2490 | 6739.54756 | 0.00029 |  0.30m - FRAM†9 | M. Mašek, K. Hoňková, J. Juryšek (TRESCA)

Table C.1: Transits found in the literature: cycle number, referred to in Hebb
et al. (2010) \- mid-transit point in $\mathrm{BJD}_{\mathrm{TDB}}$, its
error, details on the observations, and reference. The time unit of each mid-
transit time was checked and uniformized to $\mathrm{BJD}_{\mathrm{TDB}}$.
Some digits are scientifically insignificant, but are left the same as in the
reference papers.
†1 Beta Orionis Observatory, Argentina
†2 Cerro Armazones Observatory, Argentina
†3 Rarotonga Observatory, Cook Islands, New Zealand
†4 YSVP Observatory, South Australia, Australia
†5 Mount Albert Grammar School Observatory, New Zealand
†6 Perth Exoplanet Survey Telescope, West Australia, Australia
†7 Gamma Ray Burst Optical and Near-Infrared Detector on MPG/ESO, La Silla,
Chile
†8 Cerro Tololo Inter-American Observatory, Chile
†9 F/Photometric Robotic Atmospheric Monitor telescope, Pierre Auger
Observatory, Argentina, Ebr et al. (2014)

Cycle number |  Tmid [days] |  Uncertainty  [days] | Observations | Reference
---|---|---|---|---
N |  $\mathrm{BJD}_{\mathrm{TDB}}$ \- 2 450 000 |  |  |
2532 | 6772.67789 | 0.00052 | TRAPPIST-South |  Patra et al. (2020)
2537 | 6776.625805 | 0.000066 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
2792 | 6977.77653 | 0.00019 | FORS2 on VLT |  Sedaghati et al. (2015)
2930 | 7086.63573 | 0.00020 | DFOSC on 1.54m Danish Telescope |  Cortés-Zuleta et al. (2020), analyzed by TLCM
3044 | 7176.565524 | 0.000094 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
3351 | 7418.736819 | 0.000234 | FORS2 on VLT |  Sedaghati et al. (2017) re-analyzed by Petrucci et al. (2020)
3389 | 7448.712917 | 0.0000766 | FORS2 on VLT |  Sedaghati et al. (2017) re-analyzed by Petrucci et al. (2020)
3446 | 7493.67653 | 0.00024 | DFOSC on 1.54m - Danish Telescope |  Cortés-Zuleta et al. (2020), analyzed by TLCM
3477 | 7518.131796 | 0.000380 | 0.30m - PEST†6 |  Ivshina & Winn (2022) from Petrucci et al. (2020), TG Tan (TRESCA)
3513 | 7546.53025 | 0.00038 | TRAPPIST-South |  Patra et al. (2020)
3513 | 7546.52975 | 0.0002759 | 1.54m - EABA†10 |  Ivshina & Winn (2022) from Petrucci et al. (2020)
3518 | 7550.472701 | 0.0004477 | 1.54m - EABA†10 |  Ivshina & Winn (2022) from Petrucci et al. (2020)
3551 | 7576.504698 | 0.0007447 | 0.30m - FRAM†9 | M. Mašek, K. Hoňková, J. Juryšek (TRESCA),
|  |  |  | re-analyzed by Ivshina & Winn (2022) from Petrucci et al. (2020)
3739 | 7724.807825 | 0.000828 | 2.15m - CASLEO†11 | E. Fernández-Lajús, R. P. Di Sisto (TRESCA), re-analysed by Petrucci et al. (2020)
3758 | 7739.794151 | 0.000537 | 1.54m - EABA†10 |  Petrucci et al. (2020)
3806 | 7777.657426 | 0.001404 | 2.15m - CASLEO†11 |  Petrucci et al. (2020)
3815 | 7784.758917 | 0.000176 | 1.54m - EABA†10 |  Petrucci et al. (2020)
3825 | 7792.647526 | 0.000759 | 0.30m -FRAM†9 | M. Mašek, K. Hoňková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)
3830 | 7796.591396 | 0.000316 | 0.30m - FRAM†9 | M. Mašek, K. Hoňková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)
3830 | 7796.592241 | 0.000061 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
3834 | 7799.74616 | 0.00013 | DFOSC on 1.54m - Danish Telescope |  Cortés-Zuleta et al. (2020), analysed by TLCM
3896 | 7848.655592 | 0.000068 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
3901 | 7852.59809 | 0.00028 | DFOSC on 1.54m - Danish Telescope |  Cortés-Zuleta et al. (2020), analysed by TLCM
3906 | 7856.543843 | 0.000111 | IMACS on Magellan I Telescope |  Espinoza et al. (2019)
3906 | 7856.542283 | 0.000524 | 0.30m - FRAM†9 | M. Mašek, K. Hoňková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)
4127 | 8030.87615 | 0.00213 | DFOSC on 1.54m - Danish Telescope |  Cortés-Zuleta et al. (2020), analysed by TLCM
4189 | 8079.785673 | 0.0002142 | 1.54m - EABA†10 |  Petrucci et al. (2020)
4227 | 8109.763837 | 0.001527 | 2.15m - CASLEO†11 |  Petrucci et al. (2020)
4359 | 8213.887473 | 0.000243 | 0.30m - Ngileah Observatory†12 |  Petrucci et al. (2020) from C. Knight (TRESCA)
4757 | 8527.845759 | 0.000453 | 2.15m - CASLEO†11 |  Petrucci et al. (2020)
4778 | 8544.41145 | 0.0004479 | TESS |  Ivshina & Winn (2022)
4779 | 8545.19907 | 0.0003870 | TESS |  Ivshina & Winn (2022)
4780 | 8545.98902 | 0.0004050 | TESS |  Ivshina & Winn (2022)
4781 | 8546.77731 | 0.0004406 | TESS |  Ivshina & Winn (2022)
4782 | 8547.56698 | 0.0004338 | TESS |  Ivshina & Winn (2022)
4783 | 8548.35489 | 0.0004466 | TESS |  Ivshina & Winn (2022)
4784 | 8549.14427 | 0.0005019 | TESS |  Ivshina & Winn (2022)
4784 | 8549.144717 | 0.000534 | 0.32m - Bathurst Observatory†13 |  Petrucci et al. (2020) from A. Wünsche (TRESCA)
4785 | 8549.93223 | 0.0004417 | TESS |  Ivshina & Winn (2022)
4786 | 8550.72120 | 0.0004841 | TESS |  Ivshina & Winn (2022)
4787 | 8551.50999 | 0.0004451 | TESS |  Ivshina & Winn (2022)
4788 | 8552.29907 | 0.0004410 | TESS |  Ivshina & Winn (2022)
4789 | 8553.08782 | 0.0005204 | TESS |  Ivshina & Winn (2022)
4790 | 8553.87694 | 0.0004185 | TESS |  Ivshina & Winn (2022)
4791 | 8554.66562 | 0.0004163 | TESS |  Ivshina & Winn (2022)
4792 | 8555.45470 | 0.00011 | TESS |  Wong et al. (2020)
4795 | 8557.82154 | 0.0005118 | TESS |  Ivshina & Winn (2022)
4796 | 8558.60951 | 0.0005083 | TESS |  Ivshina & Winn (2022)
4797 | 8559.39917 | 0.0004697 | TESS |  Ivshina & Winn (2022)
4798 | 8560.18675 | 0.0004762 | TESS |  Ivshina & Winn (2022)
4799 | 8560.97736 | 0.0004474 | TESS |  Ivshina & Winn (2022)
4800 | 8561.76555 | 0.0004190 | TESS |  Ivshina & Winn (2022)
4801 | 8562.55365 | 0.0004069 | TESS |  Ivshina & Winn (2022)
4802 | 8563.34279 | 0.0004283 | TESS |  Ivshina & Winn (2022)
4803 | 8564.13199 | 0.0004019 | TESS |  Ivshina & Winn (2022)
4804 | 8564.92114 | 0.0005022 | TESS |  Ivshina & Winn (2022)
4805 | 8565.70943 | 0.0004585 | TESS |  Ivshina & Winn (2022)
4806 | 8566.49870 | 0.0003981 | TESS |  Ivshina & Winn (2022)
4807 | 8567.28776 | 0.0004885 | TESS |  Ivshina & Winn (2022)
4808 | 8568.07637 | 0.0004006 | TESS |  Ivshina & Winn (2022)
5714 | 9282.76417 | 0.0003798 | TESS |  Ivshina & Winn (2022)
5715 | 9283.55339 | 0.0004562 | TESS |  Ivshina & Winn (2022)
5716 | 9284.34253 | 0.0005261 | TESS |  Ivshina & Winn (2022)
5717 | 9285.13026 | 0.0004526 | TESS |  Ivshina & Winn (2022)
5718 | 9285.91981 | 0.0004843 | TESS |  Ivshina & Winn (2022)
5719 | 9286.70858 | 0.0004234 | TESS |  Ivshina & Winn (2022)
5720 | 9287.49712 | 0.0004700 | TESS |  Ivshina & Winn (2022)
5721 | 9288.28588 | 0.0004688 | TESS |  Ivshina & Winn (2022)
5722 | 9289.07500 | 0.0004201 | TESS |  Ivshina & Winn (2022)
5723 | 9289.86356 | 0.0003895 | TESS |  Ivshina & Winn (2022)
5724 | 9290.65264 | 0.0004974 | TESS |  Ivshina & Winn (2022)
5725 | 9291.44097 | 0.0004855 | TESS |  Ivshina & Winn (2022)
5726 | 9292.23008 | 0.0004141 | TESS |  Ivshina & Winn (2022)

Table C.2: Continuation of Table C.1.
†9 F/Photometric Robotic Atmospheric Monitor telescope, Pierre Auger
Observatory, Argentina
†10 Estación Astrofísica de Bosque Alegre, Argentina
†11 Complejo Astronómico El Leoncito, Argentina
†12 Ngileah Observatory, New Zealand
†13 Bathurst Observatory, New South Wales, Australia

Cycle number |  Tmid [days] |  Uncertainty  [days] | Observations | Reference
---|---|---|---|---
N |  $\mathrm{BJD}_{\mathrm{TDB}}$ \- 2.450.000 |  |  |
4824 | 8580.69724 | 0.00040 | TRAPPIST-South |  Patra et al. (2020)
4829 | 8584.641667 | 0.000268 | 1.54m - EABA†9 |  Petrucci et al. (2020)
5730 | 9295.38506316 | 0.0005686 | TESS |  Rosário et al. (2022)
5731 | 9296.17539 | 0.0004310 | TESS |  Ivshina & Winn (2022)
5732 | 9296.96307 | 0.0004743 | TESS |  Ivshina & Winn (2022)
5733 | 9297.75141 | 0.0005115 | TESS |  Ivshina & Winn (2022)
5734 | 9298.54007 | 0.0004491 | TESS |  Ivshina & Winn (2022)
5735 | 9299.32928 | 0.0004109 | TESS |  Ivshina & Winn (2022)
5736 | 9300.11829 | 0.0004992 | TESS |  Ivshina & Winn (2022)
5737 | 9300.90761 | 0.0004934 | TESS |  Ivshina & Winn (2022)
5738 | 9301.69654 | 0.0004158 | TESS |  Ivshina & Winn (2022)
5739 | 9302.48453 | 0.0004775 | TESS |  Ivshina & Winn (2022)
5740 | 9303.27451 | 0.0004810 | TESS |  Ivshina & Winn (2022)
5741 | 9304.06354 | 0.0004313 | TESS |  Ivshina & Winn (2022)
5742 | 9304.85186 | 0.0004499 | TESS |  Ivshina & Winn (2022)
6258 | 9711.89359 | 0.00034 | 0.30m - Ngileah Observatory†12 | C. Knight (TRESCA), analysed by TLCM
6272 | 9722.93627 | 0.00084 | 0.30m - Ngileah Observatory†12 | C. Knight (TRESCA), analysed by TLCM
6505 | 9906.734941 | 0.00058 | 0.43m - Deep Sky, Chile | Y. Jongen (TRESCA)
6141 | 9619.60147 | 0.00039 | 0.43m - Deep Sky, Chile | J.-P. Vignes (TRESCA)
6145 | 9622.756750 | 0.00042 | 0.43m - Deep Sky, Chile | J.-P. Vignes (TRESCA)
6557 | 9947.75209 | 0.00114 | 0.15m - Observatorio Vuelta por el Universo†14 | M. Anzola (TRESCA)

Table C.3: Continuation of Table C.1.
†10 Estación Astrofísica de Bosque Alegre, Argentina
†12 Ngileah Observatory, New Zealand
†14 Observatorio Vuelta por el Universo, Argentina

Cycle number |  Tmid [days] |  Uncertainty  [days] | Observations | Reference
---|---|---|---|---
N |  $\mathrm{BJD}_{\mathrm{TDB}}$ \- 2 450 000 |  |  |
2 | 4777.313572 | ${}^{+0.00013}_{-0.00014}$ | HAWK-I (VLT) |  Anderson et al. (2010)
185 | 4921.66849 | 0.00190 | HAWK-I (VLT) |  Gibson et al. (2010)
1018 | 5578.7684 | 0.0039 | ULTRACAM on NTT |  Burton et al. (2012)
1199 | 5721.5508 | ${}^{+0.0028}_{-0.0031}$ | DFOSC on 1.54m - Danish Telescope |  Mancini et al. (2013)
1269 | 5776.77020 | 0.00083 | IRAC on Spitzer |  Wong et al. (2016)
1270 | 5777.55733 | 0.00092 | IRAC on Spitzer |  Wong et al. (2016)
1282 | 5787.02383 | 0.00077 | IRAC on Spitzer |  Wong et al. (2016)
1283 | 5787.81228 | 0.00076 | IRAC on Spitzer |  Wong et al. (2016)
1549 | 5997.6440 | 0.001 | MMIRS on Magellan II telescope |  Bean et al. (2013)
1582 | 6023.6763 | 0.001 | MMIRS on Magellan II telescope |  Bean et al. (2013)

Table C.4: Occultations found in the literature. The cycle number is referred
to the previous transit (half phase before the occultation). Light travel time
across the system (around 16.3 s) is subtracted from all mid-occultation
times.

##  Appendix D Tables of RVs

Instr. ID | $\mathrm{BJD}_{\mathrm{TDB}}$ | $RV_{obs}$ | Uncertainty
---|---|---|---
| $-2\leavevmode\nobreak\ 450\leavevmode\nobreak\ 000.0$ | [km s-1] | [km s-1]
1 | 4616.46326 | 20.965 | 0.0218
1 | 4623.46713 | 20.712 | 0.0289
1 | 4624.46191 | 21.019 | 0.0220
1 | 4652.46587 | 20.512 | 0.0210
1 | 4653.46543 | 20.770 | 0.0231
1 | 4654.46530 | 21.007 | 0.0219
1 | 4656.47564 | 20.511 | 0.0193
1 | 4657.46805 | 20.902 | 0.0386
1 | 4658.46376 | 21.008 | 0.0302
1 | 4660.46652 | 20.604 | 0.0400
1 | 4661.46434 | 20.974 | 0.0217
1 | 4662.46549 | 20.922 | 0.0191
1 | 4663.46607 | 20.547 | 0.0188
1 | 4664.46573 | 20.645 | 0.0362
1 | 4665.46709 | 21.077 | 0.0305
1 | 4827.74752 | 20.683 | 0.0188
1 | 4832.74362 | 21.050 | 0.0242
1 | 4833.66844 | 20.860 | 0.0212
1 | 4834.67693 | 20.548 | 0.0191
1 | 4837.67024 | 20.726 | 0.0227
1 | 4838.68496 | 20.562 | 0.0244
1 | 4839.70064 | 20.936 | 0.0185
1 | 4890.61491 | 20.637 | 0.0187
1 | 4894.68743 | 20.518 | 0.0290
1 | 4895.69238 | 20.894 | 0.0179
1 | 4896.66180 | 21.041 | 0.0145
1 | 4897.65715 | 20.674 | 0.0175
1 | 4898.66015 | 20.544 | 0.0197
1 | 4939.53373 | 20.578 | 0.0164
1 | 4940.52747 | 20.642 | 0.0154
1 | 4941.53928 | 20.996 | 0.0156
1 | 4942.52421 | 20.899 | 0.0169
1 | 4943.53594 | 20.559 | 0.0191
1 | 4944.52955 | 20.798 | 0.0155
2 | 4972.4990 | 20.8874 | 0.0176
2 | 4973.4656 | 20.6042 | 0.0182
2 | 4999.4865 | 20.5571 | 0.0341
3 | 5242.6892 | 20.7469 | 0.0036
3 | 5242.8465 | 21.0203 | 0.0050
3 | 5243.6598 | 21.0391 | 0.0042
3 | 5243.8250 | 20.9547 | 0.0043
3 | 5244.7229 | 20.7355 | 0.0031
3 | 5247.7159 | 21.0254 | 0.0088
3 | 5272.5453 | 20.5741 | 0.0044
3 | 5272.7784 | 20.9778 | 0.0052
3 | 5274.5336 | 21.0391 | 0.0073
3 | 5274.6155 | 20.9318 | 0.0124
3 | 5274.6232 | 20.9129 | 0.0112
3 | 5274.6307 | 20.8819 | 0.0094
3 | 5274.6383 | 20.8842 | 0.0088
3 | 5274.6459 | 20.8711 | 0.0084
3 | 5274.6533 | 20.8914 | 0.0086
3 | 5274.6610 | 20.8473 | 0.0094
3 | 5274.6687 | 20.8214 | 0.0100
3 | 5274.6762 | 20.7819 | 0.0108
3 | 5274.6838 | 20.7568 | 0.0118
3 | 5274.6915 | 20.7315 | 0.0108
3 | 5274.6991 | 20.7138 | 0.0103

Table D.1: Radial velocity of WASP-19Ab system. The indices of the
observations are the same as in Fig. 1 and Sect. 4.

Instr. ID | $\mathrm{BJD}_{\mathrm{TDB}}$ | $RV_{obs}$ | Uncertainty
---|---|---|---
| $-2\leavevmode\nobreak\ 450\leavevmode\nobreak\ 000.0$ | [km s-1] | [km s-1]
3 | 5274.7066 | 20.7262 | 0.0106
3 | 5274.7142 | 20.7093 | 0.0104
3 | 5274.7218 | 20.7151 | 0.0118
3 | 5274.7296 | 20.7056 | 0.0120
3 | 5274.7371 | 20.6582 | 0.0116
3 | 5274.7446 | 20.6687 | 0.0104
3 | 5274.7522 | 20.6387 | 0.0119
3 | 5274.7599 | 20.6452 | 0.0113
3 | 5274.7675 | 20.6136 | 0.0105
3 | 5274.7748 | 20.6079 | 0.0106
3 | 5274.7826 | 20.5771 | 0.0123
3 | 5274.8224 | 20.5707 | 0.0069
3 | 5275.5229 | 20.6841 | 0.0052
3 | 5275.7903 | 20.6841 | 0.0077
3 | 5276.5154 | 20.5850 | 0.0062
4 | 5338.47873 | 0.13439 | 0.01522
4 | 5338.48498 | 0.11220 | 0.01214
4 | 5338.49280 | 0.09687 | 0.01226
4 | 5338.50068 | 0.06777 | 0.01130
4 | 5338.50859 | 0.05913 | 0.01108
4 | 5338.51623 | 0.04541 | 0.01068
4 | 5338.52418 | 0.00975 | 0.01071
4 | 5338.53215 | 0.00564 | 0.01140
4 | 5338.53993 | 0.00808 | 0.01054
4 | 5338.54787 | 0.06322 | 0.01216
4 | 5338.55585 | -0.00877 | 0.01121
4 | 5338.56351 | -0.01221 | 0.01281
4 | 5338.57167 | -0.05589 | 0.01214
5 | 6018.8117983 | -0.168663 | 0.003077
5 | 6290.1405875 | -0.137606 | 0.003032
5 | 6318.9215917 | 0.306274 | 0.002318
6 | 8498.59068874 | 20.82646 | 0.00818
6 | 8498.60180095 | 20.7993 | 0.00716
6 | 8498.61279395 | 20.7785 | 0.00819
6 | 8498.62459247 | 20.76231 | 0.00651
6 | 8498.63454273 | 20.76321 | 0.0145
6 | 8498.63972473 | 20.77791 | 0.01274
6 | 8498.64464221 | 20.79349 | 0.01338
6 | 8498.64956116 | 20.73967 | 0.01424
6 | 8498.65436899 | 20.69906 | 0.01462
6 | 8498.65942065 | 20.69273 | 0.01402
6 | 8498.66430660 | 20.67341 | 0.01475
6 | 8498.66926300 | 20.63671 | 0.01324
6 | 8498.67428582 | 20.61285 | 0.01011
6 | 8498.67907061 | 20.60608 | 0.00899
6 | 8498.68407176 | 20.61570 | 0.00897
6 | 8498.68899962 | 20.61782 | 0.00781
6 | 8498.69387378 | 20.62464 | 0.00745
6 | 8498.70087685 | 20.61316 | 0.00457
6 | 8498.71044096 | 20.60040 | 0.00456
6 | 8498.71956719 | 20.59894 | 0.00435
6 | 8498.72920549 | 20.56458 | 0.00439
6 | 8498.73862103 | 20.55429 | 0.00399
6 | 8546.70109917 | 20.81404 | 0.00542
6 | 8546.71810796 | 20.77644 | 0.00864
6 | 8546.73102276 | 20.74880 | 0.00911
6 | 8546.74715133 | 20.74357 | 0.00749
6 | 8546.75677225 | 20.74175 | 0.01771
6 | 8546.76244128 | 20.70465 | 0.00183
6 | 8546.76773966 | 20.74182 | 0.02089
6 | 8546.77334046 | 20.66661 | 0.01599
6 | 8546.77859110 | 20.65585 | 0.01502

Table D.2: Continuation of Table D.1.

Instr. ID | $\mathrm{BJD}_{\mathrm{TDB}}$ | $RV_{obs}$ | Uncertainty
---|---|---|---
| $-2\leavevmode\nobreak\ 450\leavevmode\nobreak\ 000.0$ | [km s-1] | [km s-1]
6 | 8546.78395751 | 20.66390 | 0.01514
6 | 8546.78909247 | 20.57827 | 0.02002
6 | 8546.79475258 | 20.59739 | 0.01890
6 | 8546.80036398 | 20.55623 | 0.02008
6 | 8546.80553459 | 20.63033 | 0.01792
6 | 8546.81107313 | 20.62332 | 0.01776
6 | 8546.81638670 | 20.59751 | 0.01665
6 | 8546.82327218 | 20.61843 | 0.01500
6 | 8546.83186421 | 20.50721 | 0.01337
6 | 8546.84018927 | 20.57332 | 0.01128
6 | 8546.84855560 | 20.55209 | 0.01556
6 | 8546.85730223 | 20.51568 | 0.01358
6 | 8565.63006930 | 20.85050 | 0.00419
6 | 8565.64050392 | 20.83263 | 0.00415
6 | 8565.65068188 | 20.81295 | 0.00392
6 | 8565.66089612 | 20.79194 | 0.00431
6 | 8565.67121537 | 20.76472 | 0.00420
6 | 8565.68117929 | 20.77662 | 0.00364
6 | 8565.68892203 | 20.78292 | 0.00850
6 | 8565.69398195 | 20.76429 | 0.00868
6 | 8565.69899674 | 20.74261 | 0.00836
6 | 8565.70399522 | 20.72989 | 0.00813
6 | 8565.70895259 | 20.68067 | 0.00844
6 | 8565.71404306 | 20.66203 | 0.00896
6 | 8565.71903842 | 20.64335 | 0.00830
6 | 8565.72397488 | 20.61127 | 0.00858
6 | 8565.72902717 | 20.61737 | 0.00865
6 | 8565.73403920 | 20.61288 | 0.00769
6 | 8565.73901729 | 20.62037 | 0.00740
6 | 8565.74406099 | 20.61360 | 0.00733
6 | 8565.75100246 | 20.60935 | 0.00391
6 | 8565.75992839 | 20.59612 | 0.00389
6 | 8565.76886384 | 20.57339 | 0.00399
6 | 8565.77776409 | 20.56269 | 0.00403
6 | 8565.78359563 | 20.54291 | 0.01537
7 | 8860.65868965 | 20.80633 | 0.00247
7 | 8860.66953784 | 20.78622 | 0.00219
7 | 8860.68086427 | 20.77083 | 0.00224
7 | 8860.69232959 | 20.75252 | 0.00205
7 | 8860.70329540 | 20.73853 | 0.00201
7 | 8860.71235224 | 20.76021 | 0.00306
7 | 8860.71764990 | 20.75988 | 0.00329
7 | 8860.72296232 | 20.74915 | 0.00298
7 | 8860.72842374 | 20.72062 | 0.00296
7 | 8860.73374936 | 20.68917 | 0.00305
7 | 8860.73902683 | 20.65960 | 0.00289
7 | 8860.74434671 | 20.62669 | 0.00289
7 | 8860.74960873 | 20.60702 | 0.00300
7 | 8860.75500031 | 20.59131 | 0.00293
7 | 8860.76033814 | 20.59485 | 0.00285
7 | 8860.76571960 | 20.60629 | 0.00280
7 | 8860.77096557 | 20.60589 | 0.00288
7 | 8860.77847463 | 20.59486 | 0.00195
7 | 8860.78808536 | 20.57681 | 0.00200
7 | 8860.79770303 | 20.56175 | 0.00208
7 | 8860.80730381 | 20.54011 | 0.00213
7 | 8860.81678339 | 20.52403 | 0.00211
8 | 8885.81831 | 21.05010 | 0.00590
8 | 8886.63492 | 21.04724 | 0.00588
8 | 8886.74637 | 20.91119 | 0.00495

Table D.3: Continuation of Table D.2.

Instr. ID | $\mathrm{BJD}_{\mathrm{TDB}}$ | $RV_{obs}$ | Uncertainty
---|---|---|---
| $-2\leavevmode\nobreak\ 450\leavevmode\nobreak\ 000.0$ | [km s-1] | [km s-1]
8 | 8887.70434 | 20.62644 | 0.01004
8 | 8887.78381 | 20.61007 | 0.00596
8 | 8888.61415 | 20.65507 | 0.00593
8 | 8888.65498 | 20.69769 | 0.00576
8 | 8889.67038 | 21.11750 | 0.00694
8 | 8889.82412 | 21.04723 | 0.00663
8 | 8893.66960 | 21.10599 | 0.01208
8 | 8893.76426 | 21.04400 | 0.00549
8 | 8893.83745 | 20.91112 | 0.00615
8 | 8894.63550 | 20.90340 | 0.00821
8 | 8894.69107 | 20.75424 | 0.00781
8 | 8894.77058 | 20.62550 | 0.00732
8 | 8897.68716 | 21.06761 | 0.00712
8 | 8897.75751 | 20.97271 | 0.00668
8 | 8899.85008 | 21.01457 | 0.00592

Table D.4: Continuation of Table D.3.