11institutetext:  Observatoire de Genève, Département d’Astronomie, Université
de Genève, Chemin des Maillettes 51, 1290 Versoix, Switzerland
22institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
33institutetext: Institut d’astrophysique de Paris, UMR7095 CNRS, Université
Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France
44institutetext: Observatoire de Haute-Provence, 04670 Saint Michel
l’Observatoire, France 55institutetext: International Center for Advanced
Studies (ICAS) and ICIFI (CONICET), ECyT-UNSAM, Campus Miguelete, 25 de Mayo y
Francia, (1650) Buenos Aires, Argentina 66institutetext: IMCCE, UMR8028 CNRS,
Observatoire de Paris, PSL University, Sorbonne Univ., 77 av. Denfert-
Rochereau, 75014 Paris, France 77institutetext: CFisUC, Departamento de
Física, Universidade de Coimbra, 3004-516 Coimbra, Portugal 88institutetext:
Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 99institutetext: Department of Physics,
University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
1010institutetext: Astrophysics Group, Keele University, Staffordshire, ST5
5BG, United Kingdom 1111institutetext: Laboratoire d’Astrophysique de
Marseille, Univ. de Provence, UMR6110 CNRS, 38 r. F. Joliot Curie, 13388
Marseille cedex 13, France 1212institutetext: Instituto de Astrofísica de
Andalucía (IAA-CSIC), Glorieta de la Astronomía 3, 18008 Granada, Spain
1313institutetext: School of Physics and Astronomy, Physical Science Building,
North Haugh, St Andrews, United Kingdom 1414institutetext: NASA Ames Research
Center, Moffett Field, CA 94035, USA 1515institutetext: LESIA, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 92195
Meudon, France 1616institutetext: Institute of Astronomy, Faculty of Physics,
Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5,
87-100 Toruń, Poland 1717institutetext: Center for Space and Habitability,
University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
1818institutetext: Department of Physics and Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA 1919institutetext: Department of Earth, Atmospheric and Planetary
Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2020institutetext: Department of Aeronautics and Astronautics, MIT, 77
Massachusetts Avenue, Cambridge, MA 02139, USA 2121institutetext: Observatoire
Hubert-Reeves, 07320 Mars, France 2222institutetext: Department of
Astrophysical Sciences, Princeton University, NJ 08544, USA

# Photodynamical analysis of the nearly resonant planetary system WASP-148:

Accurate transit-timing variations and mutual orbital inclination

J.M. Almenara Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148: G. Hébrard Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148: R.F. Díaz Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148: J. Laskar Photodynamical analysis of the nearly
resonant planetary system WASP-148:Photodynamical analysis of the nearly
resonant planetary system WASP-148: A. C. M. Correia Photodynamical analysis
of the nearly resonant planetary system WASP-148:Photodynamical analysis of
the nearly resonant planetary system WASP-148:Photodynamical analysis of the
nearly resonant planetary system WASP-148:Photodynamical analysis of the
nearly resonant planetary system WASP-148: D. R. Anderson Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148: I. Boisse
Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148: X. Bonfils Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148: D. J. A. Brown Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148: V. Casanova Photodynamical analysis of the nearly
resonant planetary system WASP-148:Photodynamical analysis of the nearly
resonant planetary system WASP-148: A. Collier Cameron Photodynamical analysis
of the nearly resonant planetary system WASP-148:Photodynamical analysis of
the nearly resonant planetary system WASP-148: M. Fernández Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148: J.M. Jenkins
Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148: F. Kiefer Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148: A. Lecavelier des Étangs Photodynamical analysis of the
nearly resonant planetary system WASP-148:Photodynamical analysis of the
nearly resonant planetary system WASP-148: J.J Lissauer Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148: G. Maciejewski
Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148: J. McCormac Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148: H. Osborn Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148: D. Pollacco Photodynamical analysis of the nearly
resonant planetary system WASP-148:Photodynamical analysis of the nearly
resonant planetary system WASP-148:Photodynamical analysis of the nearly
resonant planetary system WASP-148:Photodynamical analysis of the nearly
resonant planetary system WASP-148: G. Ricker Photodynamical analysis of the
nearly resonant planetary system WASP-148:Photodynamical analysis of the
nearly resonant planetary system WASP-148: J. Sánchez Photodynamical analysis
of the nearly resonant planetary system WASP-148:Photodynamical analysis of
the nearly resonant planetary system WASP-148: S. Seager Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148:Photodynamical
analysis of the nearly resonant planetary system WASP-148: S. Udry
Photodynamical analysis of the nearly resonant planetary system
WASP-148:Photodynamical analysis of the nearly resonant planetary system
WASP-148: D. Verilhac Photodynamical analysis of the nearly resonant planetary
system WASP-148:Photodynamical analysis of the nearly resonant planetary
system WASP-148: J. Winn Photodynamical analysis of the nearly resonant
planetary system WASP-148:Photodynamical analysis of the nearly resonant
planetary system WASP-148:

WASP-148 is a recently announced extra-solar system harbouring at least two
giant planets. The inner planet transits its host star. The planets travel on
eccentric orbits and are near the 4:1 mean-motion resonance, which implies
significant mutual gravitational interactions. In particular, this causes
transit-timing variations of a few minutes, which were detected based on
ground-based photometry. This made WASP-148 one of the few cases where such a
phenomenon was detected without space-based photometry. Here, we present a
self-consistent model of WASP-148 that takes into account the gravitational
interactions between all known bodies in the system. Our analysis
simultaneously fits the available radial velocities and transit light curves.
In particular, we used the photometry secured by the Transiting Exoplanet
Survey Satellite (TESS) and made public after the WASP-148 discovery
announcement. The TESS data confirm the transit-timing variations, but only in
combination with previously measured transit times. The system parameters we
derived agree with those previously reported and have a significantly improved
precision, including the mass of the non-transiting planet. We found a
significant mutual inclination between the orbital planes of the two planets:
$I=41.0^{+6.2}_{-7.6}$° based on the modelling of the observations, although
we found $I=20.8\pm 4.6$° when we imposed a constraint on the model enforcing
long-term dynamical stability. When a third planet was added to the model –
based on a candidate signal in the radial velocity – the mutual inclination
between planets b and c changed significantly allowing solutions closer to
coplanar. We conclude that more data are needed to establish the true
architecture of the system. If the significant mutual inclination is
confirmed, WASP-148 would become one of the only few candidate non-coplanar
planetary systems. We discuss possible origins for this misalignment.

###### Key Words.:

stars: individual: WASP-148 – stars: planetary systems – techniques:
photometric – techniques: radial velocities

##  1 Introduction

While the orbit of a single planet around its host star is well reproduced by
a Keplerian model with a constant orbital period, multi-planetary systems have
more complex orbits. Indeed, the mutual gravitational interactions between
planets imply small deviations from the Keplerian orbits and in particular
slight variations of the orbital periods. Such effects are negligible for most
systems, but they are amplified when orbital periods are exactly or nearly
commensurable. When such a resonant or nearly resonant system includes
transiting planets, their orbital periods can be accurately measured, allowing
their variations to be detected. This was predicted in particular by Holman &
Murray (2005) and Agol et al. (2005), who showed how interactions in multi-
planetary systems might cause transit-timing variations (TTVs).

When they are measured, TTVs are a powerful tool to characterise planetary
systems. In particular, they allow constraints to be put on masses as well as
on the presence of additional planets. This was done on the first system in
which TTVs were detected (Holman et al. 2010) as well as on dozens of other
detections reported thereafter (e.g. Lissauer et al. 2011; Nesvorný et al.
2013; Wang et al. 2014; Hadden & Lithwick 2014; Gillon et al. 2017;
Freudenthal et al. 2019; Jontof-Hutter et al. 2021). Among different methods
employed to analyse TTV observations, the photodynamical modelling111A
photodynamical model is a light curve model for more than a two-body system
that includes the gravitational interactions between the assumed bodies in the
system, as the planet-planet interactions (and not just the planet-star
interaction) in a multi-planetary system. (Carter et al. 2011) of light curves
can be used together with radial velocities to constrain planetary system
parameters without using external inputs on the masses or radii of the host
stars often derived from stellar evolution models (Agol et al. 2005). One of
the strengths of this technique is that it fits the full transit light curves
(including transit durations) instead of only using the transit timings.
Photodynamical analyses have been used to characterise several TTV systems
(e.g. Almenara et al. 2018a, b). Here, we apply it to the system WASP-148
recently reported by Hébrard et al. (2020).

Most of the confirmed TTV detections to date have been discovered using light
curves obtained from space telescopes (namely Kepler or Spitzer). On the other
hand, the TTVs in the WASP-148 system were detected using ground-based
telescopes only. The WASP-148 system includes (at least) two giant planets.
The inner one, WASP-148 b, is a hot Saturn of $0.72\pm
0.06\leavevmode\nobreak\ \hbox{$\mathrm{R}_{\rm J}$}$ and $0.29\pm
0.03\leavevmode\nobreak\ \hbox{$\mathrm{M}_{\rm J}$}$ that transits its host
with an orbital period of 8.80 days. The outer planet, WASP-148 c, has an
orbital period of 34.5 days and a minimum mass of $0.40\pm
0.05\leavevmode\nobreak\ \hbox{$\mathrm{M}_{\rm J}$}$ and a true mass
$<0.60\leavevmode\nobreak\ \hbox{$\mathrm{M}_{\rm J}$}$. It was discovered and
characterised using radial velocities obtained with the SOPHIE spectrograph.
No transits of this planet were detected. The orbits of both planets have
significant eccentricities ($e_{b}=0.22\pm 0.06$ and $e_{c}=0.36\pm 0.09$) and
their orbital periods fall near the 1:4 mean-motion resonance. This particular
configuration induces amplified dynamical effects, and TTVs were detected with
an amplitude of about $\pm 15$ minutes and a period of roughly 460 days
(Hébrard et al. 2020).

Several analyses of the available data sets have been presented by Hébrard et
al. (2020). The first one simultaneously fitted transit light curves and
radial velocities, but it did not include any mutual interactions. The model
allowed for small, artificial, ad hoc shifts in the transit times in order to
reproduce the TTVs. The second model was fitted to the radial velocities
taking mutual interactions into account, constrained by the average period and
phase derived from the transit light curves. Both analyses gave similar
results and the derived system properties imply large TTVs for both planets,
whereas only those of WASP-148 b could actually be observed as no transits of
WASP-148 c were detected. However, Hébrard et al. (2020) did not present a
complete model fitted to both radial velocity and transit light curve data
sets that takes the gravitational interactions between the planets into
account. Finally, Hébrard et al. (2020) indicate that there is a hint of a
possible third planet in the system with a period near 150 days and a minimum
mass around $0.25\leavevmode\nobreak\ \hbox{$\mathrm{M}_{\rm J}$}$, without
confirming that however.

Following the detection and characterisation of the WASP-148 system by Hébrard
et al. (2020), observations of that star secured with the Transiting Exoplanet
Survey Satellite (TESS) were released. TESS provides continuous, high-quality
photometry over 28 days (Ricker et al. 2015). In the case of WASP-148, it
observed seven new transits of WASP-148 b, and provided a unique opportunity
to search for possible transits of WASP-148 c or potential additional planets
in the system. Maciejewski et al. (2020) published an analysis of the TTVs and
radial velocities of WASP-148 presented by Hébrard et al. (2020) in addition
to TESS data, as well as two new ground-based transit observations. They took
into account the gravitational interactions between the planets, but they used
only the transit timings instead of the whole transit light curves. Recently,
Wang et al. (2022) have shown the orbit of WASP-148 b is aligned and prograde.

Here, we present new analyses of the WASP-148 system, applying the full
photodynamical approach on the available transit light curves and radial
velocities. The article is organised as follows: In Section 2 we describe the
data used; in Section 3 we determine the stellar parameters; in Sections 4 and
5 we analyse the radial velocity and transits without accounting for the
gravitational interactions between the planets, respectively; in Section 6 we
detail the photodynamical modelling; and in Section 7 we present the results.
Finally, we discuss the results of our work in Section 8.

##  2 Observations

We used the ground-based photometry and SOPHIE radial velocities presented in
Hébrard et al. (2020). We converted the time of the photometry from BJDUTC to
BJDTDB222BJDTDB = BJDUTC \+ $\delta$, with $\delta$ being between 65.184 and
69.184 seconds depending on the date of the transit. (Eastman et al. 2010),
except for the Telescopio Carlos Sánchez transit, and NITES observation on
June 13, 2016, which were already in BJDTDB. In addition to the data in
Hébrard et al. (2020), we added the photometry from TESS and four transits
observed with the 1.5 m Ritchey-Chrétien Telescope at the Sierra Nevada
Observatory (OSN150), the first two of which have been presented in
Maciejewski et al. (2020).

###  2.1 TESS

TESS observed WASP-148 in sectors 24, 25, and 26, with a total time span of
79.3 days (TIC 115524421, TOI-2064). Two transits were lost during the
interruption of the observations that occur every TESS orbit (around 14 days)
at the middle of sectors 25 and 26. Each TESS sector lasts two orbits of the
satellite. Around the perigee of the TESS orbit, data collection is paused. In
total, seven new transits of WASP-148 b were observed. The photometry is
available in the full-frame images (FFIs) at 30-minute cadence. The FFIs were
calibrated by the Science Processing Operations Center (SPOC) at NASA Ames
Research Center (Jenkins et al. 2016). We used eleanor (Feinstein et al. 2019)
to extract the light curve of WASP-148 from the TESS FFIs. We chose the point
spread function photometry that has the lower dispersion for this object.
eleanor corrects the times for the object coordinates, which otherwise are set
to the centre of the CCD in FFIs. There is a star 4.5 mag fainter in the Gaia
G-band at 26″ of WASP-148 that partially contaminates the TESS
photometry333Assuming the flux of the contaminant star is completely inside
the TESS aperture and that the difference in magnitude in the Gaia band is the
same as in the TESS band, the dilution is about 1.6%, which can change the
planet to star radius ratio by 0.8%, and the final planet radius by
$\sim$0.4-$\sigma$.. This is taken into account in our modelling (Sections 5
and 6). The TESS data show no evidence for any transit of WASP-148 c, which is
in agreement with Maciejewski et al. (2020).

###  2.2 OSN150

Table 1: Details on the observing runs.

Date UT | UT start–end | $X$ | $N_{\rm{obs}}$ |  $t_{\rm{exp}}$ (s) | $\Gamma$ | pnr (ppth)
---|---|---|---|---|---|---
2021 March 22 | 00:20–05:34 | $1.70\rightarrow 1.01$ | 515 | 30 | 1.69 | 1.10
2021 June 26 | 21:19–03:00 | $1.09\rightarrow 1.01\rightarrow 1.01\rightarrow 1.47$ | 575 | 30 | 1.69 | 0.76
444Date UT is given for the beginning of an observing run. $X$ shows the
changes of the target’s airmass during a run. $N_{\rm{obs}}$ is the number of
useful scientific exposures. $t_{\rm{exp}}$ is the exposure time. $\Gamma$ is
the median number of exposures per minute. $pnr$ is the photometric noise rate
(Fulton et al. 2011) in parts per thousand (ppth) of the normalised flux per
minute of the observation.

Two new precise photometric time series for transits of WASP-148 b were
acquired in March and June 2021 using the 1.5 m Ritchey-Chrétien telescope
(OSN150) at the Sierra Nevada Observatory (OSN, Spain). The instrument was
equipped with a Roper Scientific VersArray 2048B CCD camera with a $2048\times
2048\times 13.5\,\mu\rm{m}$ back-illuminated matrix. The field of view was
$7\aas@@fstack{\prime}92\times 7\aas@@fstack{\prime}92$ with the pixel scale
of $0\aas@@fstack{\prime\prime}232$ per pixel. The instrument was mildly
defocussed to allow for longer exposure times and a lower fraction of time
lost for CCD readout. The observations were gathered without any filter to
increase the signal-to-noise ratio for transit-timing purposes. The observing
runs were scheduled following an ephemeris from Maciejewski et al. (2020). The
telescope was auto-guided to keep the star at the same position in the CCD
matrix. The details on the individual runs are given in Table 1. The
observations started about 90 minutes before a transit ingress and lasted
about 90 minutes after an egress. This out-of-transit monitoring was secured
for de-trending purposes. On March 22, 2021 the observations were stopped
about 12 minutes after the transit due to dawn.

Photometric data reduction was performed with the AstroImageJ software
(Collins et al. 2017) following a standard calibration procedure. The science
frames were de-biased and flat-field calibrated using sky flat frames. The
light curves were generated with the differential aperture photometry method.
The aperture size and a collection of comparison stars were set after a series
of test runs to minimise the data point scatter. The fluxes were de-trended
against airmass and time using out-of-transit data only and then they were
normalised to unity outside the transits. Timestamps were converted into
barycentric Julian dates in barycentric dynamical time $\rm{BJD_{TDB}}$. For
the homogeneity of our analysis, we also reprocessed the two transit light
curves from Maciejewski et al. (2020) following the same de-trending
procedure.

##  3 Stellar parameters

To determine the stellar parameters of WASP-148, we used the precise parallax
determination by Gaia (Gaia Collaboration et al. 2016, 2021), which was not
exploited by Hébrard et al. (2020). Stellar atmosphere models and stellar
evolution models are also required to model the observed spectral energy
distribution (SED). We constructed the SED of WASP-148 using the magnitudes
from Gaia Early Data Release 3 (Gaia EDR3, Riello et al. 2021), the 2-Micron
All-Sky Survey (2MASS, Skrutskie et al. 2006; Cutri et al. 2003), and the
Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010; Cutri & et al.
2013). The measurements are listed in Table 18. We modelled these magnitude
measurements using the procedure described by Díaz et al. (2014), with
informative priors for the effective temperature ($T_{\mathrm{eff}}$), surface
gravity ($\log g$), and metallicity ($[\rm{Fe/H}]$) from Hébrard et al.
(2020), and for the distance from Gaia EDR3 (Lindegren et al. 2021). We used
non-informative priors for the rest of parameters. The priors are listed in
Table 2.

We decided to use an additive jitter for each set of photometric bands (Gaia,
2MASS, and WISE), which had the effect of slightly broadening the posteriors
of the stellar parameters. We used the two stellar atmosphere models,
PHOENIX/BT-Settl (Allard et al. 2012) and ATLAS/Castelli & Kurucz (Castelli &
Kurucz 2003), and two stellar evolution models, Dartmouth (Dotter et al. 2008)
and PARSEC (Chen et al. 2014). We obtained posterior samples for the four
combinations of stellar atmosphere models and stellar evolution models using
the Markov Chain Monte Carlo (MCMC) algorithm from Díaz et al. (2014). The
posteriors of the stellar parameters of the individual combinations agree
within 1-$\sigma$ (see Figure 1). We merged the results assuming an equal
probability for each model combination (labelled as ’merged’ in Figure 1). The
posteriors’ median and 68.3% credible intervals (CI) for model parameters and
for derived physical quantities of interest are listed in Table 2. Those
results agree with the ones reported by Hébrard et al. (2020), but they are
more accurate, mainly due to the use of the parallax from Gaia.

The data with the maximum a posteriori (MAP) stellar atmosphere model is shown
in Figure 2. Before Gaia EDR3, we carried out the same analysis with Gaia DR2
(Gaia Collaboration et al. 2018), obtaining similar results. In particular, we
obtained the same stellar radius error, despite a factor $\sim$3 increase in
the precision of the parallax. Thus, at least for WASP-148, the precision in
the parallax is not the limiting factor to improve the stellar radius
determination with the SED technique.

We used the stellar rotation period, P${}_{\rm rot}=26.2\pm
1.3\leavevmode\nobreak\ $days, derived in Hébrard et al. (2020) and the
stellar mass derived in this section (Table 2) to estimate a gyrochronological
age, neglecting the influence of the planets, of 4.0${}^{+0.9}_{-0.7}$ Gyr
(Barnes 2010; Barnes & Kim 2010, using a P0 between 0.12 and 3.4 days), where
we added a systematic 10% error to the statistical one (Meibom et al. 2015).
The isochronal age (Table 2) agrees with the gyrochronological age within the
uncertainties, but it is less precise.

Table 2: Modelling of the spectral energy distribution.

Parameter |  | Prior | Posterior median
---|---|---|---
|  |  | and 68.3% CI
Effective temperature, $T_{\mathrm{eff}}$ | [K] |  $N$(5460, 130) |  5555 $\pm$ 90
Surface gravity, $\log g$ | [cgs] |  $N$(4.40, 0.15) |  4.490 ${}^{+0.027}_{-0.034}$
Metallicity, $[\rm{Fe/H}]$ | [dex] |  $N$(0.11, 0.08) |  0.099 $\pm$ 0.078
Distance | [pc] |  $N$(247.73, 0.60) |  247.73 $\pm$ 0.60
$E_{\mathrm{(B-V)}}$ | [mag] |  $U$(0, 3) |  0.026 ${}^{+0.041}_{-0.019}$
Jitter Gaia | [mag] |  $U$(0, 1) |  0.140 ${}^{+0.22}_{-0.080}$
Jitter 2MASS | [mag] |  $U$(0, 1) |  0.056 ${}^{+0.072}_{-0.030}$
Jitter WISE | [mag] |  $U$(0, 1) |  0.033 ${}^{+0.076}_{-0.024}$
Radius, $R_{\star}$ |  [R⊙] |  |  0.921 $\pm$ 0.016
Mass, $M_{\star}$ |  [M⊙] |  |  0.958 $\pm$ 0.048
Density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] |  |  1.73 $\pm$ 0.15
Isochronal age | [Gyr] |  |  3.6 ${}^{+4.0}_{-2.7}$
Luminosity |  [L⊙] |  |  0.726 $\pm$0.036
555$N$($\mu$,$\sigma$): Normal distribution prior with mean $\mu$, and
standard deviation $\sigma$. $U$(l,u): Uniform distribution prior in the range
[l, u].

Figure 1: Stellar parameters derived in this work compared with Hébrard et al.
(2020), the TESS Input Catalog (TIC) version 8 (Stassun et al. 2019), and Gaia
DR2 (Andrae et al. 2018). We finally adopt the values labelled as ’merged’.
Figure 2: Spectral energy distribution of WASP-148. Top panel: The solid line
is the MAP PHOENIX/BT-Settl interpolated synthetic spectrum, red circles are
the absolute photometric observations, and grey open circles are the result of
integrating the synthetic spectrum in the observed bandpasses. Bottom panel:
Residuals of the MAP model (the jitter has been added quadratically to the
data error bars).

##  4 SOPHIE-only analysis

When radial velocity and transit observations are modelled simultaneously (as
was done by Hébrard et al. 2020), both data sets constrain the eccentricity.
In this section, we aim for a planetary eccentricity determination using only
radial velocity data. We analysed the SOPHIE radial velocities with a two-
planet Keplerian model, without taking mutual gravitational interactions into
account and using juliet (Espinoza et al. 2019), radvel (Fulton et al. 2018),
and dynesty (Speagle 2020). We used normal priors from Hébrard et al. (2020)
for the period and the time of inferior conjunction of planet b, and non-
informative priors for the rest of the parameters. We included a jitter
parameter that was added quadratically to the velocity uncertainties. We
performed a second analysis using a Gaussian process (GP) regression model
with a quasi-periodic (QP) kernel to model the error terms. We employed the QP
kernel included in celerite (Foreman-Mackey et al. 2017), with non-informative
priors for the hyperparameters. We found an odds ratio of
$\ln\left(\mathcal{Z}_{\rm jitter}/\mathcal{Z}_{\rm jitter+QP}\right)=0.5\pm
0.4$. Neither of the models is strongly favoured over the other (Kass &
Raftery 1995), and we therefore adopted the simpler model of uncorrelated
error terms with an additive jitter. The posterior median and 68.3% credible
interval are shown in Table 3. They agree with the results presented by
Hébrard et al. (2020).

We investigated the significance of the main peak in the residuals of the two-
planet model (Fig. 3 in Hébrard et al. 2020), thus we repeated the analysis
adding a third Keplerian with a period of $\sim$150 days. We found a minimum
mass of $0.24\pm 0.05\leavevmode\nobreak\ \hbox{$\mathrm{M}_{\rm J}$}$,
similar to the 0.25 $\mathrm{M}_{\rm J}$ value reported by Hébrard et al.
(2020). The model comparison gives an odds ratio of $\ln\left(\mathcal{Z}_{\rm
3\leavevmode\nobreak\ planets}/\mathcal{Z}_{\rm 2\leavevmode\nobreak\
planets}\right)=1.6\pm 0.4$, which is positive evidence in favour of the
three-planet model but below the strong evidence cutoff to be preferred over
the two-planet model, so we continued with the adopted two-planet model.

Table 3: Two-Keplerian fit to the SOPHIE data.

Keplerian |  | WASP-148 b | WASP-148 c
---|---|---|---
P | [d] |  8.803809 $\pm$ 0.000043 |  34.524 $\pm$ 0.029
Tc | [BJD] |  2 457 957.4876 $\pm$ 0.0060 |  2 457 935.6 $\pm$ 1.2
e |  |  0.183 $\pm$ 0.070 |  0.352 $\pm$ 0.085
$\omega$ | [°] |  59 ${}^{+15}_{-20}$ |  9 ${}^{+17}_{-14}$
K |  [m s-1] |  28.8 $\pm$ 2.0 |  27.0 $\pm$ 2.9
$\gamma_{\rm SOPHIE}$ |  [ km s-1] |  -5.6174 $\pm$ 0.0014
jitter |  [m s-1] |  10.9 $\pm$ 1.3
666The parameters are: Orbital period, time of conjunction, eccentricity,
argument of pericentre, radial velocity semi-amplitude, systemic velocity, and
jitter.

##  5 Transit-only analysis

We derived the transit times of planet b without considering planet c or the
radial velocities. This modelling is independent of the two-planet system
hypothesis adopted in the photodynamical modelling (Section 6), and it can be
used to verify if it is an appropriate hypothesis given the data.

In Hébrard et al. (2020), the error term of the transit photometry data were
modelled using a simple additive jitter term. However, if systematics are not
correctly accounted for, the posteriors of the transit modelling can be biased
(Barros et al. 2013). This should be particularly severe for incomplete
transits. Here, we reanalyse the transits presented in Hébrard et al. (2020)
and the TESS observations using a more sophisticated error model.

To model the transits of planet b simultaneously with the systematics, we used
juliet (Espinoza et al. 2019), using batman (Kreidberg 2015) for the transit
model, and we chose the approximate Matern kernel GP included in celerite
(Foreman-Mackey et al. 2017). We used different GPs for each ground-based
transit, and for each TESS sector. The timing of each individual transit is a
free parameter. We used a prior for the stellar density from Section 3, and
for $\sqrt{e}\cos{\omega}$ and $\sqrt{e}\sin{\omega}$ from Hébrard et al.
(2020). Otherwise, we adopted non-informative or large priors for the rest of
the parameters. We oversampled (Kipping 2010a) the model of the MARS transit
by a factor 3, which has an 88 second cadence, and the TESS data by a factor
30, which have about a 30 minute cadence. The remaining transit observations
have cadences between 0.9 and 37 seconds, and we deemed them unnecessary to
oversample their model. We set a dilution factor for the wide-field
observations: WASP (one dilution for the three WASP transits), and TESS (one
dilution per sector). To reduce the number of free parameters, we made the
choice to use one set of limb darkening parameters for all transit
observations without a filter (’clear’), but they certainly do not correspond
to exactly the same instrument bandpass. In total, there are 76 free
parameters. To sample from the posterior, we used the nested sampling code
dynesty (Speagle 2020). The complete list of parameters, priors, and
posteriors are shown in Table 6. Figure 3 shows the data and the model
posterior. We compare the timing of the ground-based observations with the
results in Hébrard et al. (2020), and found differences within 1-$\sigma$,
except for the partial transit observed with MARS where it is $12.0\pm 2.5$
min later (see Figure 5). The transit normalisation can affect the transit-
timing determination (Barros et al. 2013). The MARS transit can be more
affected because it is a partial transit and the baseline is not well-defined.

We repeat this analysis for the four OSN150 transits, which were included at a
later stage in this work. The results are presented in Table 7 and Figure 4.

This analysis could be used to de-trend the transits for the photodynamical
modelling. However, in this process, one set of parameters of the transit
model needs to be fixed, including a transit time. Therefore, if the de-
trended transits are then modelled, the results can be biased. Instead, we
decided to include the GP in the photodynamical modelling.

Figure 3: Modelling of the planet b transits presented in Hébrard et al.
(2020) and observed by TESS (blue error bars). Each panel is labelled with the
observatory or instrument name and sequential night of observation if there
were more than one. Black lines and intervals in grey show the model median
and 68.3% credible interval computed with 1000 random samples of the
posterior. The mean of the predictive distribution of the kernel model is
shown in orange. On top of each panel, the transit-timing posterior is shown
for Hébrard et al. (2020) (red line histogram), the modelling with juliet
(Section 5, grey histogram), and the photodynamical modelling (blue points
with black error bars, although the errors are barely visible). Figure 4:
Idem. as Figure 3, but for the OSN150 transits. Figure 5: Comparison between
transit-time posteriors of Hébrard et al. (2020) (red line histograms), the
modelling with juliet (Section 5, grey histograms), and the photodynamical
modelling (Section 6, blue points with black error bars and blue histograms).

##  6 Photodynamical modelling

While Sections 4 and 5 do not take mutual interactions into account, here we
report our fits of the observed photometry and radial velocity measurements
accounting for the gravitational interactions between the three bodies known
in the system using a photodynamical model. Its positions and velocities in
time were obtained through an n-body integration. The sky-projected positions
were used to compute the light curve (Mandel & Agol 2002) using a quadratic
limb-darkening law (Manduca et al. 1977), which we parametrised following
Kipping (2013). To account for the integration time, the model was oversampled
by a factor of 30 and 3 for the TESS and MARS data, respectively, and then
binned back to match the cadence of the data points (Kipping 2010a). The line-
of-sight projected velocity of the star issued from the n-body integration was
used to model the radial velocity measurements. We used the n-body code
REBOUND (Rein & Liu 2012) with the WHFast integrator (Rein & Tamayo 2015) and
an integration step of 0.01 days, which results in a maximum error of $\sim$20
ppm for the photometric model. The light-time effect (Irwin 1952) is included,
although with an amplitude of $\sim$0.05 s in the transit timing; this is a
negligible effect for this system. The model was parametrised using the
stellar mass and radius, planet-to-star mass ratios, planet b-to-star radius
ratio, and Jacobi orbital elements (Table 4) at the reference time,
$t_{\mathrm{ref}}=2\;457\;957.48167$ BJDTDB, close to the RISE transit centre.
Due to the symmetry of the problem, we fixed the longitude of the ascending
node of the interior planet $\Omega_{\mathrm{b}}$ at $t_{\mathrm{ref}}$ to
180°, and we limited the inclination of the outer one $i_{c}>90$°. We used a
GP regression model, with an approximate Matern kernel (celerite, Foreman-
Mackey et al. 2017) for the model of the error terms of the transit light
curves. We used different kernel hyperparameters for each transit, except for
the TESS data for which we used different kernel hyperparameters for each
sector. We added one dilution factor777The definition of the dilution factor
and transit normalisation factor is different from the one in juliet. For the
photodynamical model, we used $M=f_{0}(f_{t}T+f_{c})/(f_{t}+f_{c})$, with $M$
being the model to be compared with the observations, $f_{0}$ being the
transit normalisation factor (the light curve flux level out of transit),
$f_{t}$ being the flux of the target star in the aperture, $T$ being the
transit model, and $f_{c}$ being the flux of the contaminant star in the
aperture. The dilution factor is defined as $d=f_{c}/(f_{t}+f_{c})$. for each
TESS sector, and another one for the WASP transits. For each photometric data
set, we added a transit normalisation factor and an additive jitter parameter.
For the radial velocity, we added a systemic radial velocity and an additive
jitter parameter. In total, the model has 90 free parameters. We used normal
priors for the stellar mass and radius from Section 3, a non-informative
sinusoidal prior for the orbital inclinations, and non-informative uniform
prior distributions for the rest of the parameters. The joint posterior
distribution was sampled using the emcee algorithm (Goodman & Weare 2010;
Foreman-Mackey et al. 2013) with 1 000 walkers with starting points based on
the results of Hébrard et al. (2020) and Section 5.

Table 4: Inferred system parameters.

Parameter | Units | Prior | Median and 68.3% CI | Stable MAP | Stable median and 68.3% CI
---|---|---|---|---|---
_Star_ |  |  |  |  |
Stellar mass, $M_{\star}$ |  [$M_{\odot}$] | $N(0.958,0.048)$ |  0.967 $\pm$ 0.049 | 0.954 |  0.954 ${}^{+0.030}_{-0.052}$
Stellar radius, $R_{\star}$ |  [$\mathcal{R}^{\rm N}_{\odot}$] | $N(0.921,0.016)$ |  0.920 $\pm$ 0.016 | 0.913 |  0.912 $\pm$ 0.014
Stellar mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] |  |  1.76 $\pm$ 0.12 | 1.77 |  1.76 $\pm$ 0.12
Surface gravity, $\log g$ | [cgs] |  |  4.497 $\pm$ 0.025 | 4.497 |  4.496 ${}^{+0.022}_{-0.028}$
Kipping (2013) $q_{1},q_{2}$ CLEAR |  | $U(0,1)$ |  0.42 ${}^{+0.13}_{-0.11}$, 0.240 ${}^{+0.11}_{-0.094}$ | 0.304, 0.421 |  0.388 ${}^{+0.15}_{-0.083}$, 0.279 ${}^{+0.093}_{-0.16}$
Kipping (2013) $q_{1},q_{2}$ Johnson-R |  | $U(0,1)$ |  0.230 ${}^{+0.13}_{-0.096}$, 0.48 ${}^{+0.32}_{-0.26}$ | 0.106, 0.98 |  0.162 ${}^{+0.16}_{-0.056}$, 0.59 ${}^{+0.28}_{-0.32}$
Kipping (2013) $q_{1},q_{2}$ RISE |  | $U(0,1)$ |  0.80 $\pm$ 0.11, 0.269 $\pm$ 0.062 | 0.802, 0.272 |  0.802 ${}^{+0.091}_{-0.13}$, 0.278 ${}^{+0.056}_{-0.063}$
Kipping (2013) $q_{1},q_{2}$ TESS |  | $U(0,1)$ |  0.38 ${}^{+0.21}_{-0.15}$, 0.40 ${}^{+0.33}_{-0.24}$ | 0.26, 0.72 |  0.38 ${}^{+0.17}_{-0.12}$, 0.40 ${}^{+0.31}_{-0.20}$
_Planet b_ |  |  |  |  |
Semi-major axis, $a$ | [au] |  |  0.0825 $\pm$ 0.0014 | 0.08214 |  0.08215 ${}^{+0.00086}_{-0.0015}$
Eccentricity, $e$ |  |  |  0.214 ${}^{+0.021}_{-0.018}$ | 0.208 |  0.208 ${}^{+0.020}_{-0.025}$
Argument of pericentre, $\omega$ | [°] |  |  63.7 $\pm$ 8.3 | 62.6 |  60.6 ${}^{+5.0}_{-6.0}$
Inclination, $i$ | [°] | $S(0,180)$ |  89.61 ${}^{+0.21}_{-0.27}$ | 89.32 |  89.30 $\pm$ 0.24
Longitude of the ascending node, $\Omega$ | [°] |  |  180 (fixed at $t_{\mathrm{ref}}$) | 180 | 180
Mean anomaly, $M_{0}$ | [°] |  |  16.7 ${}^{+5.8}_{-5.1}$ | 17.8 |  19.2 ${}^{+4.0}_{-3.5}$
$\sqrt{e}\cos{\omega}$ |  | $U(-1,1)$ |  0.204 $\pm$ 0.056 | 0.210 |  0.219 $\pm$ 0.041
$\sqrt{e}\sin{\omega}$ |  | $U(-1,1)$ |  0.413 $\pm$ 0.039 | 0.405 |  0.394 ${}^{+0.028}_{-0.031}$
Mass ratio, $M_{\mathrm{p}}/M_{\star}$ |  | $U(0,1)$ |  0.000284 $\pm$ 0.000019 | 0.000309 |  0.000292 ${}^{+0.000012}_{-0.000014}$
Radius ratio, $R_{\mathrm{p}}/R_{\star}$ |  | $U(0,1)$ |  0.08436 $\pm$ 0.00058 | 0.08532 |  0.08498 ${}^{+0.00064}_{-0.00080}$
Scaled semi-major axis, $a/R_{\star}$ |  |  |  19.31 ${}^{+0.41}_{-0.46}$ | 19.34 |  19.33 ${}^{+0.39}_{-0.44}$
Impact parameter, $b$ |  |  |  0.105 ${}^{+0.073}_{-0.056}$ | 0.185 |  0.199 ${}^{+0.050}_{-0.070}$
$T_{0}^{\prime}$ \- 2 450 000 |  [BJDTDB] | $U(6957,8957)$ |  7957.48172 $\pm$ 0.00022 | 7957.48190 |  7957.48166 ${}^{+0.00025}_{-0.00015}$
$P^{\prime}$ | [d] | $U(0,1000)$ |  8.80369 ${}^{+0.00019}_{-0.00017}$ | 8.80366 |  8.80354 ${}^{+0.00011}_{-0.00016}$
$K^{\prime}$ |  [m s-1] |  |  29.7 $\pm$ 2.0 | 32.0 |  30.1 ${}^{+1.6}_{-1.5}$
Planet mass, $M_{\mathrm{p}}$ |  [$\mathrm{M}_{\rm J}$] |  |  0.288 $\pm$ 0.021 | 0.308 |  0.287 ${}^{+0.022}_{-0.016}$
Planet radius, $R_{\mathrm{p}}$ |  [$\mathcal{R}^{\rm N}_{e\rm J}$] |  |  0.756 $\pm$ 0.014 | 0.758 |  0.756 ${}^{+0.013}_{-0.017}$
Planet mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  |  0.829 $\pm$ 0.077 | 0.878 |  0.831 ${}^{+0.087}_{-0.069}$
Planet surface gravity, $\log$ $g_{\mathrm{p}}$ | [cgs] |  |  3.096 ${}^{+0.032}_{-0.036}$ | 3.124 |  3.098 ${}^{+0.037}_{-0.033}$
_Planet c_ |  |  |  |  |
Semi-major axis, $a$ | [au] |  |  0.2053 $\pm$ 0.0034 | 0.2044 |  0.2044 ${}^{+0.0021}_{-0.0038}$
Eccentricity, $e$ |  |  |  0.228 ${}^{+0.014}_{-0.019}$ | 0.1791 |  0.1809 ${}^{+0.018}_{-0.0072}$
Argument of pericentre, $\omega$ | [°] |  |  22.9 ${}^{+6.8}_{-5.1}$ | 25.7 |  26.1 $\pm$ 9.4
Inclination, $i$ | [°] | $S(90,180)$ |  120.6 $\pm$ 7.3 | 106.7 |  104.9 ${}^{+4.6}_{-7.3}$
Longitude of the ascending node, $\Omega$ | [°] | $U(90,270)$ |  207.4 ${}^{+4.4}_{-6.4}$ | 191.6 |  192.2 ${}^{+6.0}_{-2.7}$
Mean anomaly, $M_{0}$ | [°] |  |  260.5 $\pm$ 3.5 | 260.4 |  258.0 ${}^{+5.0}_{-4.0}$
$\sqrt{e}\cos{\omega}$ |  | $U(-1,1)$ |  0.437 ${}^{+0.025}_{-0.032}$ | 0.381 |  0.382 $\pm$ 0.030
$\sqrt{e}\sin{\omega}$ |  | $U(-1,1)$ |  0.185 ${}^{+0.051}_{-0.037}$ | 0.183 |  0.187 $\pm$ 0.062
Mass ratio, $M_{\mathrm{p}}/M_{\star}$ |  | $U(0,1)$ |  0.000417 $\pm$ 0.000040 | 0.000406 |  0.000394 $\pm$ 0.000027
Scaled semi-major axis, $a/R_{\star}$ |  |  |  48.0 $\pm$ 1.1 | 48.12 |  48.10 ${}^{+0.98}_{-1.1}$
Impact parameter, $b$ |  |  |  21.2 $\pm$ 4.5 | 12.4 |  11.0 ${}^{+3.4}_{-5.3}$
$T_{0}^{\prime}$ \- 2 450 000 |  [BJDTDB] | $U(6957,8957)$ |  7971.27 ${}^{+0.33}_{-0.28}$ | 7971.54 |  7971.62 ${}^{+0.31}_{-0.26}$
$P^{\prime}$ | [d] | $U(0,1000)$ |  34.5412 $\pm$ 0.0028 | 34.54626 |  34.54619 ${}^{+0.00094}_{-0.0016}$
$K^{\prime}$ |  [m s-1] |  |  23.7 $\pm$ 2.2 | 25.47 |  24.85 ${}^{+1.5}_{-0.97}$
Planet mass, $M_{\mathrm{p}}$ |  [$\mathrm{M}_{\rm J}$] |  |  0.424 $\pm$ 0.046 | 0.406 |  0.392 ${}^{+0.023}_{-0.027}$
$M_{\mathrm{p}}\sin{i}$ |  [$\mathrm{M}_{\rm J}$] |  |  0.361 $\pm$ 0.036 | 0.389 |  0.378 ${}^{+0.021}_{-0.024}$
Mutual inclination, $I$ | [°] |  |  41.0 ${}^{+6.2}_{-7.6}$ | 20.7 |  20.8 $\pm$ 4.6

888The table lists: Prior, posterior median, and 68.3% credible interval (CI)
for the photodynamical analysis (Section 6), maximum a posteriori (MAP) and
median, and 68.3% CI for the stable samples (Section 7.5). The Jacobi orbital
elements are given for the reference time
$t_{\mathrm{ref}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 2\;457\;957.48167$
BJDTDB. Additional parameters are in Table 8. The ’stable MAP’ solution is the
nominal solution used in Sections 7.5 and 7.7. We note that $P^{\prime}$ and
$T_{0}^{\prime}$ should not be confused with the period or the time of
conjunction, respectively, and they were only used to reduce the correlations
between jump parameters, replacing the semi-major axis and the mean anomaly at
$t_{\mathrm{ref}}$. $T^{\prime}_{0}\equiv
t_{\mathrm{ref}}-\frac{P^{\prime}}{2\pi}\left(M_{0}-E+e\sin{E}\right)$ with
$E=2\arctan{\left\\{\sqrt{\frac{1-e}{1+e}}\tan{\left[\frac{1}{2}\left(\frac{\pi}{2}-\omega\right)\right]}\right\\}}$,
$P^{\prime}\equiv\sqrt{\frac{4\pi^{2}a^{3}}{\mathcal{G}M_{\star}}}$,
$K^{\prime}\equiv\frac{M_{p}\sin{i}}{M_{\star}^{2/3}\sqrt{1-e^{2}}}\left(\frac{2\pi\mathcal{G}}{P^{\prime}}\right)^{1/3}$.
CODATA 2018: $\mathcal{G}$ = 6.674 30$\;\times 10^{-11}$
$\rm{m^{3}\;kg^{-1}\;s^{-2}}$. IAU 2012: au = 149 597 870 700 m . IAU 2015:
$\mathcal{R}^{\rm N}_{\odot}$= 6.957$\;\times 10^{8}$ m, $\mathcal{(GM)}^{\rm
N}_{\odot}$= 1.327 124 4$\;\times 10^{20}$ $\rm{m^{3}\;s^{-2}}$,
$\mathcal{R}^{\rm N}_{e\rm J}$ = 7.149 2$\;\times 10^{7}$ m,
$\mathcal{(GM)}^{\rm N}_{\rm J}$= 1.266 865 3$\;\times 10^{17}$
$\rm{m^{3}\;s^{-2}}$. $M_{\odot}$ = $\mathcal{(GM)}^{\rm
N}_{\odot}$/$\mathcal{G}$, $\mathrm{M}_{\rm J}$= $\mathcal{(GM)}^{\rm N}_{\rm
J}$/$\mathcal{G}$, $k^{2}$ = $\mathcal{(GM)}^{\rm
N}_{\odot}$$\;(86\;400\leavevmode\nobreak\ \rm{s})^{2}$/$\rm{au}^{3}$.
$N(\mu,\sigma)$: Normal distribution with mean $\mu$ and standard deviation
$\sigma$. $U(a,b)$: A uniform distribution defined between a lower $a$ and
upper $b$ limit. $S(a,b)$: A sinusoidal distribution defined between a lower
$a$ and upper $b$ limit.

##  7 Results

In Table 4 we list the prior, the median, and the 68% credible interval of the
inferred system parameters’ marginal distributions. The one- and two-
dimensional projections of the posterior sample are shown in Figure 15. The
MAP model is plotted in Figures 16 and 17. Figure 6 shows the posterior of the
planet orbits. With an inferred impact parameter of $21.2\pm 4.5$, the
transits of planet c are highly disfavoured, which is in agreement with the
null result of the transit search in Maciejewski et al. (2020). The derived
parameters agree with the ones reported by Hébrard et al. (2020), but they
have a significantly improved precision. A difference, however, is the
slightly larger radius ratio $R_{\mathrm{p}}/R_{\star}$. Instead of
$\rm{M_{p}\sin{i}}$, the photodynamical modelling allows the true mass of
planet c to be measured at $M_{\mathrm{c}}=0.424\pm 0.046$ $\mathrm{M}_{\rm
J}$, which is in agreement with the dynamical upper limit of 0.60
$\mathrm{M}_{\rm J}$ reported by Hébrard et al. (2020). Most of the posterior
samples have the apses of the orbits that librate around alignment. From tests
we determined that the improved precision in model posteriors over the earlier
study of Hébrard et al. (2020) is due to both the additional data and the
photodynamical modelling that account for the interactions between the
planets.

Figure 6: Orbital projections. Top: Orbital projections around the time of the
RISE transit for planet b (blue) and planet c (orange). The origin is the
system barycentre, and the orbits are projected in the sky plane seen by the
observer (X-Y, left) and X-Z plane (right, system top view, the movement is
clockwise, the positive Z-axis points towards the observer). A thousand random
orbits were drawn from the posterior samples, and the MAP is shown as a black
orbit. The black points mark the position of the star (size to scale) and the
planets at the central time of the RISE transit for the MAP (the size of
planet b is enlarged by a factor of 10, and the size of planet c is not
known). Bottom: Idem. as for 1000 stable solutions (Section 7.5).

###  7.1 Transit-timing variations

Figures 11 and 18 show the posterior TTVs of planet b, obtained from the time
of conjunction. The posterior transit times, which rely on the three-body
system hypothesis, agree with the individually derived transit times (Section
5). In addition, the periodicity of the TTVs agrees with the $\approx$450 days
super-period (Lithwick et al. 2012) for two planets near the 4:1 mean-motion
resonance999$P_{\rm sup}=\frac{P_{c}}{4|\Delta|}$,
$\Delta=\frac{P_{c}}{4P_{b}}-1$, with $P_{b,c}$ being the planet orbital
period of the corresponding subscript.. The periodicity of the TTVs also
agrees with the one reported by Hébrard et al. (2020).

The posterior timing from the photodynamical modelling is quite wide at some
epochs (lower panel of Figure 11 and 18). The uncertainty in the posterior
transit times is related with the knowledge of the system parameters. Future
transit observations should favour the epochs where the posterior transit
uncertainty is large to further improve the characterisation of the system.
There is room for improvement with $\sim$1 minute transit-timing precision
observations. Predictions of transit times up to 2026 are listed in Table 9.

If only the TESS observations are considered, TTVs would not have been
detected (Figure 11). TESS nearly continuous photometry observations of
WASP-148, with a time span of approximately nine planet b periods, are
insufficient to detect the TTVs of planet b. This is due to the particular
configuration of the TESS observations, which accidentally only cover a part
of increasing TTVs (see upper panel Figure 11). Thus, it is possible that a
similar situation is occurring in other systems on which TTVs are also missed.
The period derived using only the TESS data lasts $8.80604\pm 0.00014$ days,
and from all the observed transit times it is $8.8038083\pm 0.0000026$ days
(Section 7.2). This is a difference of $193\pm 12$ seconds. Using the TESS
ephemeris to predict future transit timing induces an offset of $\sim$2.2
hours/year.

The posterior TTVs’ evolution in the near future is shown in Figure 18. The
posterior transit duration variations are shown in Figure. 19.

Figure 7: Posterior TTV predictions of planet b computed relative to a linear
ephemeris111111Computed using the posterior median of the mid-transit times
between the first and last transit observation, 2457957.493798 + 8.803808
$\times$ Epoch [BJDTDB] are shown in blue. A thousand random draws from the
posterior distribution were used to estimate the TTV median value and its
uncertainty (68.3% credible interval). In the upper panel, the median TTV
values are shown and compared with individual transit-time determinations
(Hébrard et al. 2020 in red, and Section 5 in black). The thick grey line
represents a linear ephemeris computed using only the transits observed by
TESS (whose residuals are shown in the small panel in the upper right). In the
lower panel, the posterior median transit-timing value was subtracted to
visualise the uncertainty of the distribution. The posterior median transit
time was also subtracted from each observed epoch for the individual transit-
time determinations to allow for better comparison with the photodynamical
modelling. The orange curve in the upper panel represents the variation in the
times of inferior conjunction for planet c.

###  7.2 Model-independent linear ephemeris

The predictions issued from the photodynamical model and presented in Table 9
have the disadvantage of having been obtained under the assumption that the
two-planet model is correct. The results from the transit-only analysis
presented in Section 5 can be used to provide a model independent ephemeris
which should provide valid predictions, albeit less precise ones, even if the
system is discovered to contain additional planets in the future.

A value of a mean period and time of transit can be straightforwardly produced
by fitting a slope to the transit times in Table 6. However, in the presence
of transit-timing variations, a slope is not a flexible enough model to
describe the transit times. Therefore, the parameters inferred from such a
model are likely biased (e.g. Bishop 2007). Hébrard et al. (2020) dealt with
this by inflating the error bars of the individual transit times to reach
$\chi^{2}\sim 1$. Here we decided to use a non-parametric model to describe
the variation of the transit times over the linear ephemeris model.

More precisely, we chose a GP regression model whose mean function was
specified to be a linear function with parameters –slope and intercept– to be
inferred from the data (see Rasmussen & Williams 2005, Section 2.7). If normal
priors are chosen for these parameters, the computation of the marginal
likelihood can be performed analytically. We chose diffuse priors with widths
of 100 days, centred at 8.804 days for the slope (period) and at the observed
time of the second TESS transit observed in sector 25 for the intercept.

We tried several kernel functions to define the covariance function of the GP.
We used a modified version of the implementation in scikit-learn (Pedregosa et
al. 2011) and optimised the hyperparameters using the L-BFGS-B (Byrd et al.
1995; Zhu et al. 1997) and the Sequential Least SQuares Programming (SLSQP)
algorithms (Kraft 1988), implemented in the scipy package (Virtanen et al.
2020). All tested kernels produced almost identical results for the parameters
of the linear ephemeris. We discuss below the results issued from the kernel
choice that provided the largest value of optimised marginal likelihood.

The kernel function, $k(x,x^{\prime})$, producing the largest marginal
likelihood value was a exp-sine-squared (ESS) kernel, without exponential
decay:

$\begin{array}[]{rl}k(x,x^{\prime})=&A_{\mathrm{ESS}}\exp{\left\\{-\frac{2\sin^{2}{\left[\pi(x-x^{\prime})^{2}/P\right]}}{\epsilon^{2}}\right\\}},\end{array}$
(1)

where $P$ and $\epsilon$ are the kernel hyperparameters that correspond to the
period and length-scale, respectively, and $A_{\mathrm{ESS}}$ is the amplitude
of the covariance function. The posterior mean and 68.3% intervals for the
TTVs are shown in Fig. 8. A kernel function with an additional decay term
produced similar results, with a decay timescale far exceeding the time span
of the observations.

Figure 8: Model-independent analysis of the timings of the observed transits
(black circles). The black line is the mean prediction of the posterior GP
model, and the blue band is the 68.3% credible interval.

Under the assumption of normal priors, the linear model parameter posterior is
also normal. We find the posterior ephemeris:

$\begin{array}[]{lcl}P&=&8.8038083\pm 0.0000026\;\text{d}\\\
T_{c}&=&2459005.148\pm
0.016\;\left[\mathrm{BJD}_{\mathrm{TDB}}\right]\;\;,\end{array}$

with covariance term, $\mathrm{cov}(P,T_{c})=6.2\times 10^{-10}$. The large
uncertainties in $T_{c}$ are misleading. The predicted transit times issued
from this model have a covariance which includes additional terms coming from
the interaction between the mean function and the non-parametric part of the
model. In fact, the predictions of the transit times remain precise to better
than 10 minutes for over 2640 transits of planet b, that is to say over 60
years. Comparison with the transit times reported in Table 9 shows that this
method predicts transit times that are in agreement with the fully
photodynamical one to better than 5 minutes up to the end of 2026.

In comparison, an ordinary least squares (OLS) fit to the transit times, with
uncertainties scaled to have a reduced $\chi^{2}$ of one as presented in
Hébrard et al. (2020), produces a period of $P=8.803824\pm 0.000029$ days,
which is in agreement with the result from above and with the values presented
in Hébrard et al. (2020). The OLS intercept estimator has a standard deviation
six times smaller than the one in our model, which is probably unrealistic.
However, the predictions from the OLS model remain precise to better than 10
minutes for less than four years.

The optimised covariance amplitude is $A_{\mathrm{ESS}}=25.3$ minutes, and the
period $P$ is 49.7 orbits, corresponding to 437 days. The length-scale
$\epsilon$ was fixed to 2.0. The fact that the model with the largest marginal
likelihood does not include a long-term evolution of the transit times means
that with the current data, such a trend is not detected.

###  7.3 Mutual inclination

We inferred a mutual inclination between the planets121212$\cos I=\cos
i_{b}\cos i_{c}+\sin i_{b}\sin i_{c}\cos\left({\Omega_{b}-\Omega_{c}}\right)$,
$I$, at $t_{\mathrm{ref}}$ of 41.0${}^{+6.2}_{-7.6}$ ° (median and 68.3%
credible interval), and a 95% highest density interval (HDI) of [22.8°, 54.8°]
(Figure 9). With these values, a coplanar system, as assumed in Maciejewski et
al. (2020), was discarded. Maciejewski et al. (2020) tried a non-coplanar
model and found a best fit solution (with
$\Omega_{c}-\Omega_{b}\approx-17\degree$ and $i_{c}=47\degree$, or
$\Omega_{c}-\Omega_{b}\approx+17\degree$ and $i_{c}=133\degree$) which
corresponds to a mutual inclination of $\sim$46°, although they also found
that the Bayesian information criterion (BIC) disfavours the non-coplanar
solution. However, for high-dimensional models such as these ones, the BIC is
known to provide unreliable results (Díaz et al. 2016; Nelson et al. 2020).
The coplanar solution in Maciejewski et al. (2020) has a lower eccentricity
for planet b (Figure 10). From a stability analysis of the orbital solution
they derived, Hébrard et al. (2020) found an upper limit of 35° for the mutual
inclination. We note that the reduced-$\chi^{2}$ level curves of the Newtonian
fit plotted in Fig. 9 of Hébrard et al. (2020) favoured mutual inclinations
around 30°.

We tried to investigate which observable favours the significant mutual
inclination. For this, we repeated the photo-dynamical analysis (Section 6)
assuming coplanar orbits; we fixed the longitude of the ascending node of both
planets to the same value, and we matched the orbital inclination of planet c
to the one of planet b, with the latter still being a free parameter. The
results are presented in Table 10, and Figures 10, 18, and 19. The TTVs’
posteriors of coplanar and inclined orbits are roughly the same for the
observed transits (Figure 18). On the other hand, the transit duration
posteriors are different, the transit duration of the observed transits
remains almost constant for the coplanar model, whereas it decreased for the
model with inclined orbits (Figure 19). However, we found that the precision
of individually determined transit durations131313We analysed each transit
observation individually to determine the transit duration (with the
techniques described in Section 5), excluding partial transits, as well as
transits observed by SuperWASP (poor precision) and TESS (30-minute cadence).
The transit duration was computed numerically from 1,000 models from the
posterior predictive distribution sampled to one-second cadence. This values
differs from the transit duration computed in Hébrard et al. (2020) using the
approximations in Tingley & Sackett (2005), whose limitations were discussed
by Kipping (2010b). is not enough to confirm the results of the modelling with
inclined orbits independently. The photodynamical modelling is in principle
more sensible than the analysis on individual transits (Almenara et al. 2015),
but the heterogeneity of the transit observations analysed in this work call
for caution. Small variations in transit shape or timing intrinsic to the
different observations could be wrongly interpreted by photodynamical
modelling as an evolution in the orbital parameters141414For example, transit
depth variations, which have not been discussed in this work, can be due to
changes in the impact parameter, but also due to a different type of
contamination in the photometric aperture.. If not correctly taken into
account, the limb darkening dependence with the observation bandpass could be
misinterpreted as changes in the impact parameter, and therefore high mutual
inclinations between the planets. However, no correlation is seen between the
mutual inclination and the limb-darkening parameters. In addition, the
posterior distributions of the limb-darkening parameters agree with those
expected from theoretical computation (Claret & Bloemen 2011; Claret 2017),
although they are much wider.

In addition, models starting with a co-planar configuration were run. We found
that the co-planar region of parameter space is left quickly by the MCMC
walkers. The detection of a non-negative mutual inclination does not seem to
be produced by an inadequate exploration of a space parameter, or by bad
mixing or lack of convergence of the MCMCs. Future observations focussed on
distinguishing between the increasingly and differently predicted transit
duration could conclude about the mutual inclination.

Figure 9: Posterior of the mutual inclination at $t_{\mathrm{ref}}$ from the
photodynamical modelling (Section 6), the stable samples (Section 7.5, the
stable MAP value is marked by a vertical grey line), and the three-planet
model (Section 7.8). Figure 10: Planet b and c eccentricity, as well as the
mutual inclination for the different analysis presented in this work, and in
previous literature. Maciejewski et al. (2020) used a
$t_{\mathrm{ref}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 2\;459\;048.747$
BJDTDB, which is different from the one used in this work, but the difference
in the inferred orbital parameters cannot be explained by their evolution
between the two epochs (Figure 20). We adopted the values labelled as
’stable’.

###  7.4 Photodynamical modelling without stellar priors

The photodynamical modelling of photometry and radial velocity data allows one
to measure the absolute radius and mass in multi-planetary systems (Agol et
al. 2005; Almenara et al. 2015). To test the precision of this determination
in this system, we ran the same analysis as in Section 6, but with non-
informative priors for the stellar mass and radius. Masses, radii, densities,
and orbital parameters are listed in Table 11. The absolute parameters’
precision is poor. The radii were determined with a precision of 32% relative
uncertainty for the star and planet b. The bulk densities of the star and
planet b were determined with a precision of 20% and 39%, respectively. The
masses of the star, planet b, and planet c were determined with a precision of
89%, 58%, and 82%, respectively. The precision on planet masses outperform the
one on the star (Almenara et al. 2018b). The posterior eccentricities are also
more precise for the photodynamical modelling with the stellar priors (Figure
10). This means that the photoeccentric effect (Dawson & Johnson 2012) puts
constraints on the eccentricities in addition to the ones coming from the
TTVs. The same is true for the mutual inclination.

###  7.5 Stability analysis

The dynamical analysis of the WASP-148 orbital solution reported in Hébrard et
al. (2020) has shown that the system is stable, despite significant mutual
gravitational interactions between the planets. For this study, we repeated a
similar stability study, but instead we performed a global frequency analysis
(Laskar 1990, 1993) on 50,000 samples of the posterior distribution obtained
from our new photodynamical model. We used the symplectic integrator SABA1064
of Farrés et al. (2013), with a step size of $5\times 10^{-3}$ yr and general
relativity corrections. Each initial condition was integrated over 50 kyr, and
a stability indicator $D$ was derived with the frequency analysis of the mean
longitude, that is the variation in the measured mean motion over the two
consecutive 25 kyr intervals of time (for more details, see Couetdic et al.
2010). For regular motion there is no significant variation in the mean motion
along the trajectory, while it can significantly vary for chaotic
trajectories. In Figure 11 we show the distribution of the 50,000 samples
projected in a $(\Omega_{c},i_{c})$ diagram, which corresponds to the two less
constrained parameters. The colour index gives the value of $D$ for each
solution. The values of $\log_{10}D<-6$ for both WASP-148 b and c correspond
to stable systems on scales of billions of years (Correia et al. 2010). Only a
small region from this diagram is stable for $\Omega_{c}<205^{\circ}$ and
$i_{c}<120^{\circ}$, corresponding to 1239 solutions (2.5% of the total). In
Figure 9 we show the probability density function (PDF) of this stable subset
of solutions. The solutions cluster around a mutual inclination of $20.8\pm
4.6$ °. We hence conclude from that three-body analysis that WASP-148 can only
be stable for mutual inclinations below about $30^{\circ}$. In the last two
columns in Table 4, we provide the system parameters corresponding to the MAP,
the median, and the 68% credible interval of the stable solutions.

Figure 11: Stability analysis of 50,000 samples of the posterior distribution
obtained from the photodynamical modelling (Section 6), projected in a
$(\Omega_{c},i_{c})$ diagram. The colour scale corresponds to values between
–8.5 (black) and 0.9 (red) for the decimal logarithm of the stability index
$D$ used in Correia et al. (2010). The red zones correspond to highly unstable
orbits, while the dark blue region can be assumed to be stable on a billion-
year timescale.

As in Hébrard et al. (2020), we also explored the stability around the stable
MAP solution (Table 4), which we refer to as the nominal solution
henceforward. As expected, in the $(a_{c},e_{c})$ domain, we confirm that it
lies in a stable area, with the orbits close to the 4:1 mean-motion resonance.
In the $(i_{c},\Omega_{c})$ domain (Figure 12), we confirm that stable orbits
must have a mutual inclination of $I\lesssim 30\degree$. In comparison to the
stability map shown in Fig. 9 in Hébrard et al. (2020) with $i_{c}<90\degree$,
here we only zoom into the most stable regions, and imposed $i_{c}>90\degree$:
this is equivalent from a dynamical point of view, but now the stable
solutions are centred around $\Omega_{c}=180\degree$ rather than
$\Omega_{c}=0\degree$. The main difference in the new analysis is the fact
that the nodes and the inclination are now constrained by the observations.

The two stability analyses presented above use different approaches: Fig. 11
presents a stability analysis carried out on different solutions allowing all
parameters to vary, whereas Fig. 12 studies the stability with only two free
parameters ($i_{c}$, $\Omega_{c}$) and the other parameters are fixed at their
MAP values. We show here that they provide similar results in terms of the
derived parameters for stability, which supports the reliability of both
approaches and their results.

Figure 12: Global stability analysis of the WASP-148 planetary system. We
fixed all orbital parameters of the stable MAP solution (Table 4), and we
varied the inclination, $i_{c}$, and the longitude of the ascending node,
$\Omega_{c}$, of planet-$c$. The step size is $0.25\degree$ in both axes. For
each initial condition, the system was integrated over 50 kyr and a stability
criterion was derived with the frequency analysis of the mean longitude. White
dashed curves give the isolines of constant mutual inclination $I=10\degree$,
$20\degree$, $30\degree$, and $40\degree$. The white dot marks the position of
the stable MAP solution from Table 4. The colour bar corresponds to the one in
Figure 11.

###  7.6 Orbital evolution

To explore the dynamics of the system, we analyse 1000 stable samples from the
joint parameter posterior distribution of the photodynamical model and
performed numerical integrations151515With the same n-body integrator and time
step used in Section 6. for 1 kyr after $t_{\mathrm{ref}}$. The results for
the selected parameters are plotted in Figure 20 and 21. The posterior median
and 68.3% credible interval of the mutual inclination over 1 kyr is $20.3\pm
4.5\degree$, and for the eccentricities it is $e_{b}=0.126\pm 0.076$ and
$e_{c}=0.217^{+0.018}_{-0.027}$. The mutual inclination remains above
$\sim$10° at the 95.4% credible interval. Over the 1 kyr integration, the
orbital inclination of planet c is too low for transits to occur for most of
the samples. Interestingly, planet b only transits for a small fraction of
that time (Figure 20); in particular, it will not transit anymore after about
200 years, then it will transit again in about 600 years.

###  7.7 Tidal evolution

The semi-major axis of the innermost planet is only 0.082 au, which means that
the planet is close enough to the star to undergo some tidal evolution. As a
result, the eccentricity can be damped and the inner orbit circularised (e.g.
Hut 1981). Adopting a value identical to Jupiter’s value (Lainey et al. 2009)
for the tidal quality factor $Q=10^{5}$, we get a characteristic timescale
$\sim$10 Gyr (e.g. Correia & Laskar 2010) for the circularisation, which is
comparable to the lifetime of the system. Therefore, it is not surprising that
at present the innermost planet still shows a significant eccentricity.
Moreover, the orbits of the two planets strongly interact, and secular or
resonant effects can also excite the eccentricity of the innermost planet
(e.g. Correia et al. 2012, 2013). To check this scenario, using a direct
three-body model with linear tides (Correia 2018), and general relativity
corrections, we ran a simulation over 1 Gyr, starting with the initial
conditions of the nominal solution from Table 4. We adopted a Love number of
$k_{2}=0.5$ and a time lag of $\Delta t=1$ s (equivalent to $Q\sim 10^{5}$).
We observed that the eccentricity of the innermost planet undergoes large
oscillations owing to secular interactions. Moreover, after 130 Myr, the
system crosses a resonance which pumps the inner planet eccentricity and damps
the mutual inclination (Figure 13). As a result, at the end of the simulation,
the average inner planet eccentricity is higher than the initial one. We
conclude that the presently observed non-zero value is compatible with the
tidal evolution of the WASP-148 system. As expected, the semi-major axis of
the innermost planet also slightly decreases, and we have $a_{b}\approx 0.073$
au after 1 Gyr. Since the age of the star at present is already 4 Gyr, we can
assume that the initial semi-major axis was slightly larger than the present
value, and so the two planets could even be trapped in the 4:1 mean motion
resonance. This is an interesting formation scenario that deserves more
attention in future work on the system.

Figure 13: Tidal evolution of the WASP-148 system. Starting with the nominal
solution from Table 4, we show the evolution of the mutual inclination (top),
the eccentricities (middle), and semi-major axes (bottom) over 1 Gyr. We used
a direct three-body model with linear tides (Correia 2018) with general
relativity corrections, and adopted $k_{2}=0.5$ and $\Delta t=1$ s for both
planets.

###  7.8 Three-planet model

All these results rely on the three-body system hypothesis. If the dynamics of
planet b is affected by additional planets other than planet c, increasing the
mutual inclination could provide the additional variability in the model
required to fit the data. The photodynamical model is flexible and could
overfit the data. The argument of an inaccurately specified model is discussed
by Petit et al. (2020) in the context of the determination of the eccentricity
of the K2-19 planets.

To test the influence in the system parameters of an unaccounted for
additional planet in the system, we repeated the photodynamical analysis
(Section 6) with a third planet, which we started at the period of 150 days,
which is the peak in the residuals of a 2 Keplerian fit of the radial
velocities (Sect. 4.1 and Fig. 3 in Hébrard et al. 2020). The results are
presented in Table 12 and Figures 9, 10, 18, and 19. The third planet
converges to a period of $151.2^{+2.1}_{-1.7}$ days and a mass of
$0.262^{+0.10}_{-0.076}$ $\mathrm{M}_{\rm J}$, which is in agreement with the
estimations reported in Hébrard et al. (2020). Its mutual inclination relative
to the planets b and c is not well-constrained. The mutual inclination between
planet b and c is reduced from $41.0^{+6.2}_{-7.6}$ ° in the two-planet model
(without imposing long-term stability) to $21\pm 12$ ° in the three-planet
model. The masses of planets b and c as well as the radius of planet b are
compatible within $1\leavevmode\nobreak\ \sigma$ to the two-planet model. The
transit timing and duration variations do not allow one to distinguish between
the two- and three-planet models with the current data (Figures 18 and 19).
More data are needed to assert the presence of additional planets in the
system.

##  8 Discussion

The WASP-148 system is composed of a G5V star orbited by a hot Saturn
($0.287^{+0.022}_{-0.016}$ $\mathrm{M}_{\rm J}$, $0.756^{+0.013}_{-0.017}$
$\mathrm{R}_{\rm J}$, and $0.08215^{+0.00086}_{-0.0015}$ au) and a warm
Jupiter ($0.392^{+0.023}_{-0.027}$ $\mathrm{M}_{\rm J}$and
$0.2044^{+0.0021}_{-0.0038}$ au) near a 4:1 mean-motion resonance161616The
period ratio at $t_{\mathrm{ref}}$ is $3.92411^{+0.00014}_{-0.00017}$ and
computed as $(a_{c}/a_{b})^{3/2}$, with
$a_{c}/a_{b}=2.487868^{+0.000061}_{-0.000073}$. For the stable MAP solution,
the period ratio (computed over a 1000 orbits of the planet c) is 3.922.. The
planets have eccentricities $e_{b}=0.208^{+0.020}_{-0.025}$ and
$e_{c}=0.1809^{+0.018}_{-0.0072}$, and a mass ratio of
$M_{c}/M_{b}=1.351^{+0.11}_{-0.076}$. Also, assuming only two planets in the
system, their orbits have a mutual inclination of $20.8\pm 4.6$ °.

Figure 14: Planet mass versus orbital semi-major axis in logarithmic scale.
Error bars mark the position of planet b (blue) and planet c (orange). Grey
dots are planets listed in the NASA Exoplanet Archive accessed through DACE
API (dace.unige.ch). A Gaussian kernel density estimate is shown in different
intensities of green. Solar System planets (black points, from left to right:
Venus, Earth, Jupiter, Saturn, Uranus, and Neptune) from NASA.

Both planets are located in a scarcely populated part of the mass-separation
diagram (Figure 14), which cannot be fully explained by a detection
bias171717While the detection of systems similar to WASP-148 is accessible for
high-precision stable radial velocities’ instruments, they are difficult to
detect with transit searches. Due to the mutually inclined orbits and the
secular oscillation of the orbital inclination, the probability that both
planets transit for a given observer is low (Section 7.6)., thus indicating a
low probability outcome for the planetary formation process. The scarcity of
planets such as WASP-148 b can be partially explained by the low efficiency of
the Lidov-Kozai mechanism (Lidov 1962; Kozai 1962) to form hot Saturns because
most migrating planets are tidally disrupted (Anderson et al. 2016). WASP-148
c was placed in the ’period valley’ (Udry et al. 2003) between the hot and
warm Jupiters clumps.

The WASP-148 giant planets are expected to form in a proto-planetary disk
beyond the snow line, which is located at a few au (Lecar et al. 2006), and
then migrate inwards (Goldreich & Tremaine 1980). Convergent migration leads
to planets being captured in mean-motion resonances, but usually of first
order. High order resonances such as the 4:1 one require high initial orbital
eccentricities (Rein & Papaloizou 2010). However, disk migration does not
favour significant orbital eccentricity excitation because planet-disk
interactions tend to damp eccentricities (Bitsch et al. 2013; Dunhill et al.
2013). After the disk disappears, planet-planet scattering can cause high
eccentricity migration for the inner planet (Rasio & Ford 1996), or the
eccentricities can be excited by chaotic secular interactions (Wu & Lithwick
2011). Both mechanisms are compatible with the non-coplanarity of the observed
orbits.

Another possibility that does not require planet-planet scattering or secular
chaos is explored in Lee & Peale (2002): the planets are captured in the mean-
motion resonance and are massive enough to open gaps in the disk. With the
mechanism for damping the eccentricity being reduced, the orbits can become
elliptical during the inward migration within the resonance (Artymowicz 1992).
Capture in the 4:1 mean-motion resonance is then able to produce some
inclination excitation given that the inner planet is not too massive (Thommes
& Lissauer 2003; Libert & Tsiganis 2009).

Finally, at a later stage, tidal interactions with the star should shrink the
inner planet orbit afterwards and pull the system out of resonance. Tidal
interactions usually also damp the eccentricity, but the presence of the outer
planet delays this process due to secular and resonant interactions.

There are only a few other planetary systems for which a large mutual
inclination has been reported: 102.3${}^{+7.4}_{-8.0}$° in HD 3167 (Dalal et
al. 2019; Bourrier et al. 2021), 30$\pm$1° in $\upsilon$ Andromedae (McArthur
et al. 2010), 24${}^{+11}_{-8}$° in Kepler-108 (Mills & Fabrycky 2017), [34.5,
140.6]° (95% credible interval) in $\pi$ Mensae (De Rosa et al. 2020), and
some systems with ultra short period planets with mutual inclinations larger
than 10° reported in Dai et al. (2018). HD 3167 is a system with three small-
sized planets, including two transiting that allowed Rossiter-McLaughlin
measurements. $\pi$ Mensae is a hierarchical system with a period ratio of
$\sim$330, and a large difference in planet masses. $\upsilon$ Andromedae is
composed of three planets, and the mutual inclination is measured for the
outer pair of super Jovian planets with a period ratio of 5.3. Kepler-108,
composed of two Saturn-mass planets with a period ratio of 3.88, is the one
that resembles WASP-148 the most, but with longer period planets in lower
eccentricity orbits. Interestingly, the period ratio is also close to the 4:1
mean-motion resonance.

As shown in Section 7.8, additional planets not accounted for in the modelling
can affect the determination of the system parameters, in particular the
mutual inclination. Thus more data are needed to conclude on the true
architecture of the system.

###### Acknowledgements.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. Resources supporting this
work were provided by the NASA High-End Computing (HEC) Program through the
NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the
production of the SPOC data products. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Simulations in this paper
made use of the REBOUND code which can be downloaded freely at
http://github.com/hannorein/rebound. Part of these simulations have been run
on the Lesta cluster kindly provided by the Observatoire de Genève. 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.
AC acknowledges support by CFisUC projects (UIDB/04564/2020 and
UIDP/04564/2020), GRAVITY (PTDC/FIS-AST/7002/2020), ENGAGE SKA
(POCI-01-0145-FEDER-022217), and PHOBOS (POCI-01-0145-FEDER-029932), funded by
COMPETE 2020 and FCT, Portugal. GM acknowledges the financial support from the
National Science Centre, Poland through grant no. 2016/23/B/ST9/00579. MF
acknowledges financial support from grant
PID2019-109522GB-C5X/AEI/10.13039/501100011033 of the Spanish Ministry of
Science and Innovation (MICINN). MF, VC and JS acknowledge financial support
from the State Agency for Research of the Spanish MCIU through the Center of
Excellence Severo Ochoa award to the Instituto de Astrofísica de Andalucía
(SEV-2017-0709). JMA and XB acknowledges funding from the European Research
Council under the ERC Grant Agreement n. 337591-ExTrA. We thank L. Kreidberg
for her Mandel & Agol code.

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##  Appendix A Additional figures and tables

Table 5: Photometric measurements used for the SED analysis of WASP-148.

Filter | Magnitude | $\pm 1\sigma$
---|---|---
Gaia-G | 12.0790 | 0.0028
Gaia-BP | 12.4636 | 0.0030
Gaia-RP | 11.5315 | 0.0029
2MASS-J | 10.938 | 0.024
2MASS-H | 10.585 | 0.018
2MASS-Ks | 10.506 | 0.017
WISE-W1 | 10.466 | 0.022
WISE-W2 | 10.519 | 0.020
WISE-W3(†) | 10.476 | 0.064
181818(†) Not covered by the Castelli & Kurucz models.

Table 6: Inferred system parameters for the transit-only analysis.

Parameter | Units | Prior | Posterior median and 68.3% CI
---|---|---|---
_Star_ |  |  |
Stellar mean density |  [$\rm{g\;cm^{-3}}$] |  $N$(1.75, 0.15) |  1.77 $\pm$ 0.13
Kipping (2013) $q_{1}$ for Clear, Johnson-R, RISE, TESS |  |  $U$(0, 1) |  0.38${}^{+0.25}_{-0.18}$, 0.204${}^{+0.13}_{-0.090}$, 0.78 $\pm$ 0.16, 0.35${}^{+0.22}_{-0.15}$
Kipping (2013) $q_{2}$ for Clear, Johnson-R, RISE, TESS |  |  $U$(0, 1) |  0.29${}^{+0.26}_{-0.19}$, 0.57 $\pm$ 0.28, 0.40${}^{+0.14}_{-0.12}$, 0.39${}^{+0.29}_{-0.23}$
_Planet b_ |  |  |
Espinoza (2018) $r_{1}$ |  |  $U$(0, 1) |  0.432${}^{+0.080}_{-0.065}$
Espinoza (2018) $r_{2}$ |  |  $U$(0, 1) |  0.0854 $\pm$ 0.0013
$\sqrt{e}\cos{\omega}$ |  |  $N$(0.24, 0.09) |  0.243 $\pm$ 0.080
$\sqrt{e}\sin{\omega}$ |  |  $N$(0.40, 0.12) |  0.384${}^{+0.043}_{-0.049}$
_Transit timings (instrument, band, epoch)_ |  |  |
WASP1 | Clear | -377 |  [BJDTDB] |  $U$(2454638.265, 2454638.773) |  2454638.461${}^{+0.21}_{-0.097}$
WASP2 | Clear | -296 |  [BJDTDB] |  $U$(2455351.288, 2455351.835) |  2455351.517${}^{+0.033}_{-0.041}$
WASP3 | Clear | -255 |  [BJDTDB] |  $U$(2455712.42, 2455712.65) |  2455712.5208${}^{+0.0080}_{-0.0067}$
NITES1 | Clear | -122 |  [BJDTDB] |  $U$(2456883.440, 2456883.449) |  2456883.4442 $\pm$ 0.0010
SANCHEZ | Johnson-R | -87 |  [BJDTDB] |  $U$(2457191.55, 2457191.56) |  2457191.5555 $\pm$ 0.0010
NITES2 | Johnson-R | -46 |  [BJDTDB] |  $U$(2457552.5005, 2457552.5082) |  2457552.50421 $\pm$ 0.00088
MARS | Clear | -11 |  [BJDTDB] |  $U$(2457860.648, 2457860.660) |  2457860.6550 $\pm$ 0.0012
RISE | RISE | 0 |  [BJDTDB] |  $U$(2457957.480, 2457957.484) |  2457957.48202 $\pm$ 0.00042
TESS sector 24 | TESS | 114 |  [BJDTDB] |  $U$(2458961.1150, 2458961.1331) |  2458961.1226 $\pm$ 0.0019
TESS sector 24 | TESS | 115 |  [BJDTDB] |  $U$(2458969.9220, 2458969.9358) |  2458969.9285 $\pm$ 0.0015
TESS sector 24 | TESS | 116 |  [BJDTDB] |  $U$(2458978.7245, 2458978.7438) |  2458978.7346 $\pm$ 0.0022
TESS sector 25 | TESS | 117 |  [BJDTDB] |  $U$(2458987.531, 2458987.5476) |  2458987.5403 $\pm$ 0.0019
TESS sector 25 | TESS | 119 |  [BJDTDB] |  $U$(2459005.1466, 2459005.1581) |  2459005.1521 $\pm$ 0.0016
TESS sector 26 | TESS | 120 |  [BJDTDB] |  $U$(2459013.9516, 2459013.970) |  2459013.9600 $\pm$ 0.0021
TESS sector 26 | TESS | 122 |  [BJDTDB] |  $U$(2459031.5623, 2459031.5792) |  2459031.5701 $\pm$ 0.0018
_Instruments_ |  |  |
Dilution factor for WASP |  |  $TN$(1.0, 0.1, 0.0, 1.0) |  0.923${}^{+0.047}_{-0.069}$
Dilution factor for TESS sector 24, 25, 26 |  |  $TN$(1.0, 0.1, 0.0, 1.0) |  0.949 $\pm$ 0.033, 0.936 $\pm$ 0.035, 0.945${}^{+0.032}_{-0.041}$
Offset relative flux for WASP1, WASP2, WASP3 | [Relative flux] |  $N$(0.0, 0.1) |  0.0030${}^{+0.0038}_{-0.0033}$, -0.0006${}^{+0.0037}_{-0.0030}$, 0.0008 $\pm$ 0.0022
Offset relative flux for NITES1, NITES2 | [Relative flux] |  $N$(0.0, 0.1) |  0.00061${}^{+0.00058}_{-0.00086}$, -0.00015${}^{+0.0016}_{-0.00068}$
Offset relative flux for MARS | [Relative flux] |  $N$(0.0, 0.1) |  -0.0019${}^{+0.0062}_{-0.0060}$
Offset relative flux for SANCHEZ | [Relative flux] |  $N$(0.0, 0.1) |  -0.00201${}^{+0.00081}_{-0.00090}$
Offset relative flux for RISE | [Relative flux] |  $N$(0.0, 0.1) |  0.0034${}^{+0.035}_{-0.0059}$
Offset relative flux for TESS sector 24, 25, 26 | [Relative flux] |  $N$(0.0, 0.1) |  0.0004 $\pm$ 0.0013, 0.0005 $\pm$ 0.0019, -0.0003 $\pm$ 0.0014
Additive jitter for WASP1, WASP2, WASP3 | [ppm] |  $J$(1, 10000) |  82${}^{+980}_{-76}$, 61${}^{+630}_{-56}$, 210${}^{+2800}_{-200}$
Additive jitter for NITES1, NITES2 | [ppm] |  $J$(1, 4000) |  18${}^{+100}_{-15}$, 3290 $\pm$ 130
Additive jitter for MARS | [ppm] |  $J$(1, 1600) |  860 $\pm$ 120
Additive jitter for SANCHEZ | [ppm] |  $J$(1, 1600) |  27${}^{+150}_{-24}$
Additive jitter for RISE | [ppm] |  $J$(1, 300) |  10.5${}^{+36}_{-8.1}$
Additive jitter for TESS sector 24, 25, 26 | [ppm] |  $J$(1, 900) |  27${}^{+110}_{-23}$, 94${}^{+120}_{-85}$, 744 $\pm$ 27
Timescale of the GP for WASP1, WASP2, WASP3 | [days] |  $J$(0.001, 1000) |  1.0${}^{+91}_{-1.0}$, 1.8${}^{+75}_{-1.8}$, 0.021${}^{+0.034}_{-0.012}$
Timescale of the GP for NITES1, NITES2 | [days] |  $J$(0.001, 1000) |  6.0${}^{+160}_{-5.9}$, 12${}^{+220}_{-12}$
Timescale of the GP for MARS | [days] |  $J$(0.001, 1000) |  1.3${}^{+15}_{-1.1}$
Timescale of the GP for SANCHEZ | [days] |  $J$(0.001, 1000) |  0.0233${}^{+0.016}_{-0.0098}$
Timescale of the GP for RISE | [days] |  $J$(0.001, 1000) |  0.34${}^{+0.75}_{-0.22}$
Timescale of the GP for TESS sector 24, 25, 26 | [days] |  $J$(0.001, 1000) |  1.81${}^{+0.38}_{-0.30}$, 5.1${}^{+2.4}_{-1.4}$, 1.04${}^{+0.16}_{-0.13}$
Amplitude of the GP for WASP1, WASP2, WASP3 | [Relative flux] |  $J$(10-6, 1) |  0.00022${}^{+0.0046}_{-0.00021}$, 0.00072${}^{+0.013}_{-0.00070}$, 0.0040${}^{+0.0020}_{-0.0013}$
Amplitude of the GP for NITES1, NITES2 | [Relative flux] |  $J$(10-6, 1) |  0.00020${}^{+0.0044}_{-0.00019}$, 0.00040${}^{+0.0088}_{-0.00038}$
Amplitude of the GP for MARS | [Relative flux] |  $J$(10-6, 1) |  0.014${}^{+0.10}_{-0.012}$
Amplitude of the GP for SANCHEZ | [Relative flux] |  $J$(10-6, 1) |  0.00177${}^{+0.00085}_{-0.00045}$
Amplitude of the GP for RISE | [Relative flux] |  $J$(10-6, 1) |  0.0104${}^{+0.041}_{-0.0074}$
Amplitude of the GP for TESS sector 24, 25, 26 | [Relative flux] |  $J$(10-6, 1) |  0.00312${}^{+0.00077}_{-0.00050}$, 0.00285${}^{+0.0019}_{-0.00090}$, 0.00430${}^{+0.00077}_{-0.00058}$
_Derived_ |  |  |
Radius ratio, $R_{\mathrm{p}}/R_{\star}$ |  |  |  0.0854 $\pm$ 0.0013
Impact parameter |  |  |  0.148${}^{+0.12}_{-0.098}$
Eccentricity, $e$ |  |  |  0.213 $\pm$ 0.035
Argument of pericentre, $\omega$ | [°] |  |  58 $\pm$ 11
191919The table lists: Priors and posterior median as well as 68.3% CI for the
transit-only analysis with juliet (Section 5). $N(\mu,\sigma)$: Normal
distribution with mean $\mu$ and standard deviation $\sigma$.
$TN(\mu,\sigma,a,b)$: Normal distribution with mean $\mu$ and standard
deviation $\sigma$, truncated between a lower $a$ and upper $b$ limit.
$U(a,b)$: A uniform distribution defined between a lower $a$ and upper $b$
limit. $J(a,b)$: Jeffreys (or log-uniform) distribution defined between a
lower $a$ and upper $b$ limit.

Table 7: Idem. as Figure 3, but for the analysis of OSN150 transits.

Parameter | Units | Prior | Posterior median and 68.3% CI
---|---|---|---
_Star_ |  |  |
Stellar mean density |  [$\rm{g\;cm^{-3}}$] |  $N$(1.75, 0.15) |  1.73 $\pm$ 0.15
Kipping (2013) $q_{1}$ for Clear |  |  $U$(0, 1) |  0.45 ${}^{+0.25}_{-0.16}$
Kipping (2013) $q_{2}$ for Clear |  |  $U$(0, 1) |  0.17 ${}^{+0.17}_{-0.11}$
_Planet b_ |  |  |
Espinoza (2018) $r_{1}$ |  |  $U$(0, 1) |  0.550 ${}^{+0.064}_{-0.11}$
Espinoza (2018) $r_{2}$ |  |  $U$(0, 1) |  0.0851 ${}^{+0.0020}_{-0.0016}$
$\sqrt{e}\cos{\omega}$ |  |  $N$(0.24, 0.09) |  0.238 $\pm$ 0.091
$\sqrt{e}\sin{\omega}$ |  |  $N$(0.40, 0.12) |  0.366${}^{+0.053}_{-0.062}$
_Transit timings (instrument, band, epoch)_ |  |  |
OSN1 | Clear | 123 |  [BJDTDB] |  $U$(2459040.365914, 2459040.49543) |  2459040.37337 ${}^{+0.00052}_{-0.00047}$
OSN2 | Clear | 128 |  [BJDTDB] |  $U$(2459084.327292, 2459084.50194) |  2459084.39721 ${}^{+0.00058}_{-0.00067}$
OSN3 | Clear | 152 |  [BJDTDB] |  $U$(2459295.515766, 2459295.733722) |  2459295.65861 ${}^{+0.00053}_{-0.00053}$
OSN4 | Clear | 163 |  [BJDTDB] |  $U$(2459392.390943, 2459392.628101) |  2459392.51001 ${}^{+0.00036}_{-0.00036}$
_Instruments_ |  |  |
Offset relative flux for OSN1 | [Relative flux] |  $U$(-0.001, 0.001) |  -0.00035 ${}^{+0.00067}_{-0.00041}$
Offset relative flux for OSN2 | [Relative flux] |  $U$(-0.001, 0.001) |  0.00037 ${}^{+0.00043}_{-0.00068}$
Offset relative flux for OSN3 | [Relative flux] |  $U$(-0.001, 0.001) |  0.00040 ${}^{+0.00042}_{-0.00066}$
Offset relative flux for OSN4 | [Relative flux] |  $U$(-0.001, 0.001) |  0.00006 ${}^{+0.00047}_{-0.00055}$
Additive jitter for OSN1 | [ppm] |  $J$(1, 2000) |  979 $\pm$ 63
Additive jitter for OSN2 | [ppm] |  $J$(1, 2000) |  1374 $\pm$ 77
Additive jitter for OSN3 | [ppm] |  $J$(1, 2000) |  1051 $\pm$ 56
Additive jitter for OSN4 | [ppm] |  $J$(1, 2000) |  19 ${}^{+130}_{-17}$
Timescale of the GP for OSN1 | [days] |  $J$(0.001, 1000) |  0.118 ${}^{+0.33}_{-0.075}$
Timescale of the GP for OSN2 | [days] |  $J$(0.001, 1000) |  0.063 ${}^{+0.068}_{-0.032}$
Timescale of the GP for OSN3 | [days] |  $J$(0.001, 1000) |  0.048 ${}^{+0.060}_{-0.024}$
Timescale of the GP for OSN4 | [days] |  $J$(0.001, 1000) |  0.066 ${}^{+0.076}_{-0.035}$
Amplitude of the GP for OSN1 | [Relative flux] |  $J$(10-6, 0.05) |  0.00106 ${}^{+0.0023}_{-0.00056}$
Amplitude of the GP for OSN2 | [Relative flux] |  $J$(10-6, 0.05) |  0.00120 ${}^{+0.0010}_{-0.00056}$
Amplitude of the GP for OSN3 | [Relative flux] |  $J$(10-6, 0.05) |  0.00117 ${}^{+0.0010}_{-0.00044}$
Amplitude of the GP for OSN4 | [Relative flux] |  $J$(10-6, 0.05) |  0.00090 ${}^{+0.00095}_{-0.00034}$
_Derived_ |  |  |
Radius ratio, $R_{\mathrm{p}}/R_{\star}$ |  |  |  0.0851 ${}^{+0.0020}_{-0.0016}$
Impact parameter |  |  |  0.324 ${}^{+0.096}_{-0.17}$
Eccentricity, $e$ |  |  |  0.197 $\pm$ 0.046
Argument of pericentre, $\omega$ | [°] |  |  57 $\pm$ 13
202020$N(\mu,\sigma)$: Normal distribution with mean $\mu$ and standard
deviation $\sigma$. $TN(\mu,\sigma,a,b)$: Normal distribution with mean $\mu$
and standard deviation $\sigma$, truncated between a lower $a$ and upper $b$
limit. $U(a,b)$: A uniform distribution defined between a lower $a$ and upper
$b$ limit. $J(a,b)$: Jeffreys (or log-uniform) distribution defined between a
lower $a$ and upper $b$ limit.

Table 8: Continuation of Table 4 for the photodynamical modelling.

Parameter | Units | Prior | Median and 68.3% CI
---|---|---|---
SOPHIE offset |  [ km s-1] |  $U$(-100, 100) |  5.6179 $\pm$ 0.0014
SOPHIE multiplicative jitter |  |  $U$(0, 10) |  1.53 ${}^{+0.12}_{-0.10}$
Dilution factor for WASP |  |  $U$(0, 1) |  0.53 ${}^{+0.25}_{-0.31}$
Dilution factor for TESS sector 24 |  |  $U$(0, 1) |  0.104 ${}^{+0.035}_{-0.041}$
Dilution factor for TESS sector 25 |  |  $U$(0, 1) |  0.122 ${}^{+0.037}_{-0.044}$
Dilution factor for TESS sector 26 |  |  $U$(0, 1) |  0.116 $\pm$ 0.053
Relative flux for WASP1 | [Relative flux] |  $U$(0.9, 1.1) |  1.0002 ${}^{+0.0023}_{-0.0027}$
Relative flux for WASP2 | [Relative flux] |  $U$(0.9, 1.1) |  1.0022 $\pm$ 0.0015
Relative flux for WASP3 | [Relative flux] |  $U$(0.9, 1.1) |  0.9995 $\pm$ 0.0011
Relative flux for NITES1 | [Relative flux] |  $U$(0.9, 1.1) |  1.00014 $\pm$ 0.00018
Relative flux for NITES2 | [Relative flux] |  $U$(0.9, 1.1) |  0.99994 $\pm$ 0.00016
Relative flux for MARS | [Relative flux] |  $U$(0.9, 1.1) |  0.99994 $\pm$ 0.00013
Relative flux for SANCHEZ | [Relative flux] |  $U$(0.9, 1.1) |  1.00006 ${}^{+0.00015}_{-0.00013}$
Relative flux for RISE | [Relative flux] |  $U$(0.9, 1.1) |  1.000018 ${}^{+0.000065}_{-0.000057}$
Relative flux for TESS sector 24 | [Relative flux] |  $U$(0.9, 1.1) |  0.999921 $\pm$ 0.00011
Relative flux for TESS sector 25 | [Relative flux] |  $U$(0.9, 1.1) |  1.00001 $\pm$ 0.00012
Relative flux for TESS sector 26 | [Relative flux] |  $U$(0.9, 1.1) |  0.99991 $\pm$ 0.00014
Relative flux for OSN1 | [Relative flux] |  $U$(0.9, 1.1) |  1.000054 $\pm$ 0.000086
Relative flux for OSN2 | [Relative flux] |  $U$(0.9, 1.1) |  1.000009 $\pm$ 0.000097
Relative flux for OSN3 | [Relative flux] |  $U$(0.9, 1.1) |  0.999996 $\pm$ 0.000072
Relative flux for OSN4 | [Relative flux] |  $U$(0.9, 1.1) |  0.999998 $\pm$ 0.000052
Additive jitter for WASP1 | [ppm] |  $J$(1, 10000) |  69 ${}^{+1000}_{-65}$
Additive jitter for WASP2 | [ppm] |  $J$(1, 10000) |  53 ${}^{+540}_{-49}$
Additive jitter for WASP3 | [ppm] |  $J$(1, 10000) |  320 ${}^{+2500}_{-310}$
Additive jitter for NITES1 | [ppm] |  $J$(1, 10000) |  25 ${}^{+130}_{-22}$
Additive jitter for NITES2 | [ppm] |  $J$(1, 10000) |  3300 $\pm$ 140
Additive jitter for MARS | [ppm] |  $J$(1, 10000) |  870 $\pm$ 140
Additive jitter for SANCHEZ | [ppm] |  $J$(1, 10000) |  31 ${}^{+180}_{-29}$
Additive jitter for RISE | [ppm] |  $J$(1, 10000) |  15 ${}^{+43}_{-12}$
Additive jitter for TESS sector 24 | [ppm] |  $J$(1, 10000) |  26 ${}^{+100}_{-23}$
Additive jitter for TESS sector 25 | [ppm] |  $J$(1, 10000) |  110 ${}^{+120}_{-100}$
Additive jitter for TESS sector 26 | [ppm] |  $J$(1, 10000) |  745 $\pm$ 27
Additive jitter for OSN1 | [ppm] |  $J$(1, 10000) |  972 $\pm$ 64
Additive jitter for OSN2 | [ppm] |  $J$(1, 10000) |  1372 $\pm$ 77
Additive jitter for OSN3 | [ppm] |  $J$(1, 10000) |  1052 $\pm$ 61
Additive jitter for OSN4 | [ppm] |  $J$(1, 10000) |  20 ${}^{+110}_{-18}$
Timescale of the GP for WASP1 | [days] |  $J$(0.001, 1000) |  3.0 ${}^{+100}_{-3.0}$
Timescale of the GP for WASP2 | [days] |  $J$(0.001, 1000) |  2.0 ${}^{+63}_{-1.9}$
Timescale of the GP for WASP3 | [days] |  $J$(0.001, 1000) |  0.040 ${}^{+0.061}_{-0.024}$
Timescale of the GP for NITES1 | [days] |  $J$(0.001, 1000) |  2.7 ${}^{+72}_{-2.7}$
Timescale of the GP for NITES2 | [days] |  $J$(0.001, 1000) |  13 ${}^{+160}_{-13}$
Timescale of the GP for MARS | [days] |  $J$(0.001, 1000) |  1.2 ${}^{+12}_{-1.1}$
Timescale of the GP for SANCHEZ | [days] |  $J$(0.001, 1000) |  0.0230 ${}^{+0.017}_{-0.0076}$
Timescale of the GP for RISE | [days] |  $J$(0.001, 1000) |  1.03 ${}^{+1.5}_{-0.91}$
Timescale of the GP for TESS sector 24 | [days] |  $J$(0.001, 1000) |  1.79 ${}^{+0.40}_{-0.30}$
Timescale of the GP for TESS sector 25 | [days] |  $J$(0.001, 1000) |  4.7 ${}^{+2.4}_{-1.3}$
Timescale of the GP for TESS sector 26 | [days] |  $J$(0.001, 1000) |  1.05 ${}^{+0.17}_{-0.14}$
Timescale of the GP for OSN1 | [days] |  $J$(0.001, 1000) |  0.130 ${}^{+0.16}_{-0.083}$
Timescale of the GP for OSN2 | [days] |  $J$(0.001, 1000) |  0.081 ${}^{+3.6}_{-0.041}$
Timescale of the GP for OSN3 | [days] |  $J$(0.001, 1000) |  0.062 ${}^{+0.080}_{-0.028}$
Timescale of the GP for OSN4 | [days] |  $J$(0.001, 1000) |  0.046 ${}^{+0.040}_{-0.018}$
Amplitude of the GP for WASP1 | [Relative flux] |  $J$(10-6, 1) |  0.00036 ${}^{+0.0050}_{-0.00035}$
Amplitude of the GP for WASP2 | [Relative flux] |  $J$(10-6, 1) |  0.0019 ${}^{+0.0053}_{-0.0019}$
Amplitude of the GP for WASP3 | [Relative flux] |  $J$(10-6, 1) |  0.0058 ${}^{+0.0038}_{-0.0024}$
Amplitude of the GP for NITES1 | [Relative flux] |  $J$(10-6, 1) |  0.00051 ${}^{+0.0011}_{-0.00049}$
Amplitude of the GP for NITES2 | [Relative flux] |  $J$(10-6, 1) |  0.00023 ${}^{+0.0049}_{-0.00022}$
Amplitude of the GP for MARS | [Relative flux] |  $J$(10-6, 1) |  0.0052 ${}^{+0.0053}_{-0.0040}$
Amplitude of the GP for SANCHEZ | [Relative flux] |  $J$(10-6, 1) |  0.00181 ${}^{+0.00082}_{-0.00040}$
Amplitude of the GP for RISE | [Relative flux] |  $J$(10-6, 1) |  0.051 ${}^{+0.11}_{-0.049}$
Amplitude of the GP for TESS sector 24 | [Relative flux] |  $J$(10-6, 1) |  0.00312 ${}^{+0.00084}_{-0.00056}$
Amplitude of the GP for TESS sector 25 | [Relative flux] |  $J$(10-6, 1) |  0.00266 ${}^{+0.0018}_{-0.00087}$
Amplitude of the GP for TESS sector 26 | [Relative flux] |  $J$(10-6, 1) |  0.00436 ${}^{+0.00076}_{-0.00062}$
Amplitude of the GP for OSN1 | [Relative flux] |  $J$(10-6, 1) |  0.0019 ${}^{+0.0039}_{-0.0013}$
Amplitude of the GP for OSN2 | [Relative flux] |  $J$(10-6, 1) |  0.00108 ${}^{+0.0013}_{-0.00037}$
Amplitude of the GP for OSN3 | [Relative flux] |  $J$(10-6, 1) |  0.00102 ${}^{+0.00076}_{-0.00022}$
Amplitude of the GP for OSN4 | [Relative flux] |  $J$(10-6, 1) |  0.00076 ${}^{+0.00058}_{-0.00022}$
212121$U(a,b)$: A uniform distribution defined between a lower $a$ and upper
$b$ limit. $J(a,b)$: Jeffreys (or log-uniform) distribution defined between a
lower $a$ and upper $b$ limit.

Figure 15: Two-parameter joint posterior distributions for the most relevant
model parameters from the photodynamical modelling (Section 6). The 39.3,
86.5, and 98.9% two-variable joint confidence regions are denoted by three
different grey levels; in the case of a Gaussian posterior, these regions
project on to the one-dimensional 1, 2, and 3 $\sigma$ intervals. The
histogram of the marginal distribution for each parameter is shown at the top
of each column, except for the parameter on the last line, which is shown at
the end of the line. Units are the same as in Table 4.

Figure 16: Transits of WASP-148 b (blue points) and the MAP model (black line)
from the photodynamical modelling (Section 6). Each panel is centred at the
linear ephemeris (indicated by the vertical grey lines, and reported in the
caption of Fig. 11). For the SANCHEZ transit, 90-second binned data are shown
in grey in addition to the observed data points. Each panel is labelled with
the epoch; zero is the transit at $t_{\mathrm{ref}}$. In the lower part of
each panel, the residuals after subtracting the MAP model are shown. Figure
17: SOPHIE radial velocities of WASP-148 (green error bars) and the MAP model
(black line) from the photodynamical modelling (Section 6). In the lower
panel, the residuals after subtracting the MAP model are shown.

Table 9: Planet b transit-time predictions from 2021 to 2026.

Date | Mid-transit time | 68.3% CI
---|---|---
|  [BJDTDB] | [days] | [days]
2021-01-01 | 2459216.4351 | -0.0014 | +0.0012
2021-01-10 | 2459225.2363 | -0.0012 | +0.0011
2021-01-19 | 2459234.0382 | -0.0011 | +0.0011
2021-01-28 | 2459242.8399 | -0.0012 | +0.0011
2021-02-06 | 2459251.6440 | -0.0010 | +0.0009
2021-02-14 | 2459260.4460 | -0.0008 | +0.0007
2021-02-23 | 2459269.2488 | -0.0006 | +0.0007
2021-03-04 | 2459278.0509 | -0.0006 | +0.0006
2021-03-13 | 2459286.8557 | -0.0005 | +0.0005
2021-03-22 | 2459295.6588 | -0.0004 | +0.0004
2021-03-30 | 2459304.4627 | -0.0004 | +0.0004
2021-04-08 | 2459313.2654 | -0.0004 | +0.0005
2021-04-17 | 2459322.0709 | -0.0004 | +0.0004
2021-04-26 | 2459330.8749 | -0.0004 | +0.0005
2021-05-05 | 2459339.6799 | -0.0004 | +0.0005
2021-05-13 | 2459348.4830 | -0.0004 | +0.0005
2021-05-22 | 2459357.2891 | -0.0005 | +0.0004
2021-05-31 | 2459366.0938 | -0.0004 | +0.0004
2021-06-09 | 2459374.8997 | -0.0004 | +0.0004
2021-06-18 | 2459383.7032 | -0.0004 | +0.0004
2021-06-27 | 2459392.5095 | -0.0003 | +0.0003
2021-07-05 | 2459401.3150 | -0.0003 | +0.0003
2021-07-14 | 2459410.1215 | -0.0003 | +0.0004
2021-07-23 | 2459418.9253 | -0.0004 | +0.0004
2021-08-01 | 2459427.7316 | -0.0004 | +0.0004
2021-08-10 | 2459436.5372 | -0.0004 | +0.0004
2021-08-18 | 2459445.3440 | -0.0004 | +0.0005
2021-08-27 | 2459454.1481 | -0.0004 | +0.0005
2021-09-05 | 2459462.9537 | -0.0004 | +0.0004
2021-09-14 | 2459471.7591 | -0.0004 | +0.0005
2021-09-23 | 2459480.5656 | -0.0004 | +0.0004
2021-10-01 | 2459489.3695 | -0.0004 | +0.0005
2021-10-10 | 2459498.1741 | -0.0004 | +0.0005
2021-10-19 | 2459506.9785 | -0.0005 | +0.0005
2021-10-28 | 2459515.7840 | -0.0005 | +0.0005
2021-11-06 | 2459524.5876 | -0.0005 | +0.0005
2021-11-14 | 2459533.3907 | -0.0005 | +0.0005
2021-11-23 | 2459542.1940 | -0.0006 | +0.0006
2021-12-02 | 2459550.9982 | -0.0008 | +0.0008
2021-12-11 | 2459559.8014 | -0.0008 | +0.0008
2021-12-20 | 2459568.6032 | -0.0009 | +0.0009
2021-12-28 | 2459577.4054 | -0.0011 | +0.0010
2022-01-06 | 2459586.2083 | -0.0013 | +0.0012
2022-01-15 | 2459595.0113 | -0.0014 | +0.0012
2022-01-24 | 2459603.8124 | -0.0014 | +0.0013
2022-02-02 | 2459612.6139 | -0.0015 | +0.0014
2022-02-10 | 2459621.4159 | -0.0018 | +0.0016
2022-02-19 | 2459630.2190 | -0.0017 | +0.0015
2022-02-28 | 2459639.0200 | -0.0017 | +0.0015
2022-03-09 | 2459647.8215 | -0.0017 | +0.0015
2022-03-18 | 2459656.6231 | -0.0018 | +0.0017
2022-03-26 | 2459665.4267 | -0.0016 | +0.0015
2022-04-04 | 2459674.2281 | -0.0014 | +0.0014
2022-04-13 | 2459683.0302 | -0.0012 | +0.0013
2022-04-22 | 2459691.8320 | -0.0013 | +0.0013
2022-05-01 | 2459700.6363 | -0.0010 | +0.0012
2022-05-09 | 2459709.4387 | -0.0008 | +0.0009
2022-05-18 | 2459718.2418 | -0.0007 | +0.0008
2022-05-27 | 2459727.0441 | -0.0007 | +0.0008
2022-06-05 | 2459735.8492 | -0.0006 | +0.0006
2022-06-14 | 2459744.6525 | -0.0005 | +0.0005
2022-06-22 | 2459753.4568 | -0.0005 | +0.0005
2022-07-01 | 2459762.2596 | -0.0005 | +0.0006
2022-07-10 | 2459771.0653 | -0.0005 | +0.0005
2022-07-19 | 2459779.8695 | -0.0005 | +0.0005
2022-07-28 | 2459788.6748 | -0.0005 | +0.0005
2022-08-05 | 2459797.4781 | -0.0005 | +0.0005
2022-08-14 | 2459806.2842 | -0.0005 | +0.0004
2022-08-23 | 2459815.0892 | -0.0005 | +0.0005
2022-09-01 | 2459823.8953 | -0.0004 | +0.0004
2022-09-10 | 2459832.6990 | -0.0004 | +0.0005
2022-09-19 | 2459841.5053 | -0.0004 | +0.0004
2022-09-27 | 2459850.3109 | -0.0004 | +0.0004
2022-10-06 | 2459859.1176 | -0.0004 | +0.0005
2022-10-15 | 2459867.9216 | -0.0005 | +0.0006
2022-10-24 | 2459876.7277 | -0.0005 | +0.0005
2022-11-02 | 2459885.5334 | -0.0005 | +0.0005
2022-11-10 | 2459894.3402 | -0.0005 | +0.0006
2022-11-19 | 2459903.1443 | -0.0006 | +0.0006
2022-11-28 | 2459911.9496 | -0.0005 | +0.0006
2022-12-07 | 2459920.7548 | -0.0005 | +0.0006
2022-12-16 | 2459929.5610 | -0.0005 | +0.0005
2022-12-24 | 2459938.3649 | -0.0005 | +0.0006

Date | Mid-transit time | 68.3% CI
---|---|---
|  [BJDTDB] | [days] | [days]
2023-01-02 | 2459947.1690 | -0.0005 | +0.0005
2023-01-11 | 2459955.9730 | -0.0006 | +0.0006
2023-01-20 | 2459964.7781 | -0.0006 | +0.0006
2023-01-29 | 2459973.5816 | -0.0007 | +0.0006
2023-02-06 | 2459982.3843 | -0.0008 | +0.0007
2023-02-15 | 2459991.1872 | -0.0009 | +0.0009
2023-02-24 | 2459999.9909 | -0.0011 | +0.0011
2023-03-05 | 2460008.7940 | -0.0012 | +0.0011
2023-03-14 | 2460017.5955 | -0.0014 | +0.0012
2023-03-22 | 2460026.3974 | -0.0016 | +0.0014
2023-03-31 | 2460035.1999 | -0.0018 | +0.0017
2023-04-09 | 2460044.0029 | -0.0018 | +0.0017
2023-04-18 | 2460052.8039 | -0.0019 | +0.0017
2023-04-27 | 2460061.6053 | -0.0020 | +0.0018
2023-05-05 | 2460070.4070 | -0.0022 | +0.0020
2023-05-14 | 2460079.2103 | -0.0021 | +0.0019
2023-05-23 | 2460088.0113 | -0.0020 | +0.0019
2023-06-01 | 2460096.8130 | -0.0019 | +0.0019
2023-06-10 | 2460105.6146 | -0.0020 | +0.0020
2023-06-18 | 2460114.4184 | -0.0017 | +0.0018
2023-06-27 | 2460123.2200 | -0.0015 | +0.0016
2023-07-06 | 2460132.0224 | -0.0013 | +0.0016
2023-07-15 | 2460140.8243 | -0.0013 | +0.0015
2023-07-24 | 2460149.6289 | -0.0011 | +0.0013
2023-08-01 | 2460158.4316 | -0.0009 | +0.0010
2023-08-10 | 2460167.2351 | -0.0009 | +0.0009
2023-08-19 | 2460176.0375 | -0.0009 | +0.0009
2023-08-28 | 2460184.8428 | -0.0007 | +0.0007
2023-09-06 | 2460193.6464 | -0.0006 | +0.0006
2023-09-14 | 2460202.4510 | -0.0006 | +0.0006
2023-09-23 | 2460211.2539 | -0.0007 | +0.0007
2023-10-02 | 2460220.0598 | -0.0006 | +0.0006
2023-10-11 | 2460228.8643 | -0.0006 | +0.0006
2023-10-20 | 2460237.6699 | -0.0006 | +0.0006
2023-10-28 | 2460246.4732 | -0.0007 | +0.0007
2023-11-06 | 2460255.2795 | -0.0006 | +0.0006
2023-11-15 | 2460264.0848 | -0.0006 | +0.0006
2023-11-24 | 2460272.8911 | -0.0006 | +0.0006
2023-12-03 | 2460281.6949 | -0.0006 | +0.0007
2023-12-12 | 2460290.5013 | -0.0006 | +0.0006
2023-12-20 | 2460299.3070 | -0.0006 | +0.0007
2023-12-29 | 2460308.1138 | -0.0006 | +0.0007
2024-01-07 | 2460316.9178 | -0.0006 | +0.0008
2024-01-16 | 2460325.7238 | -0.0006 | +0.0007
2024-01-25 | 2460334.5294 | -0.0007 | +0.0007
2024-02-02 | 2460343.3361 | -0.0007 | +0.0007
2024-02-11 | 2460352.1402 | -0.0007 | +0.0008
2024-02-20 | 2460360.9452 | -0.0007 | +0.0007
2024-02-29 | 2460369.7500 | -0.0007 | +0.0007
2024-03-09 | 2460378.5560 | -0.0006 | +0.0006
2024-03-17 | 2460387.3598 | -0.0007 | +0.0007
2024-03-26 | 2460396.1633 | -0.0007 | +0.0007
2024-04-04 | 2460404.9670 | -0.0008 | +0.0007
2024-04-13 | 2460413.7717 | -0.0009 | +0.0009
2024-04-22 | 2460422.5750 | -0.0010 | +0.0009
2024-04-30 | 2460431.3772 | -0.0011 | +0.0011
2024-05-09 | 2460440.1798 | -0.0014 | +0.0013
2024-05-18 | 2460448.9831 | -0.0017 | +0.0016
2024-05-27 | 2460457.7861 | -0.0017 | +0.0016
2024-06-05 | 2460466.5873 | -0.0019 | +0.0017
2024-06-13 | 2460475.3890 | -0.0021 | +0.0020
2024-06-22 | 2460484.1912 | -0.0024 | +0.0022
2024-07-01 | 2460492.9942 | -0.0023 | +0.0021
2024-07-10 | 2460501.7951 | -0.0023 | +0.0022
2024-07-19 | 2460510.5965 | -0.0024 | +0.0023
2024-07-27 | 2460519.3981 | -0.0025 | +0.0026
2024-08-05 | 2460528.2014 | -0.0023 | +0.0024
2024-08-14 | 2460537.0027 | -0.0022 | +0.0023
2024-08-23 | 2460545.8045 | -0.0020 | +0.0022
2024-09-01 | 2460554.6061 | -0.0021 | +0.0023
2024-09-09 | 2460563.4101 | -0.0018 | +0.0021
2024-09-18 | 2460572.2121 | -0.0017 | +0.0018
2024-09-27 | 2460581.0148 | -0.0014 | +0.0017
2024-10-06 | 2460589.8169 | -0.0014 | +0.0016
2024-10-15 | 2460598.6217 | -0.0013 | +0.0014
2024-10-23 | 2460607.4246 | -0.0011 | +0.0011
2024-11-01 | 2460616.2285 | -0.0010 | +0.0010
2024-11-10 | 2460625.0311 | -0.0010 | +0.0010
2024-11-19 | 2460633.8365 | -0.0009 | +0.0009
2024-11-28 | 2460642.6405 | -0.0008 | +0.0008
2024-12-06 | 2460651.4453 | -0.0008 | +0.0008
2024-12-15 | 2460660.2484 | -0.0009 | +0.0009
2024-12-24 | 2460669.0544 | -0.0008 | +0.0007

Date | Mid-transit time | 68.3% CI
---|---|---
|  [BJDTDB] | [days] | [days]
2025-01-02 | 2460677.8592 | -0.0008 | +0.0007
2025-01-11 | 2460686.6651 | -0.0007 | +0.0008
2025-01-19 | 2460695.4686 | -0.0008 | +0.0008
2025-01-28 | 2460704.2750 | -0.0008 | +0.0008
2025-02-06 | 2460713.0804 | -0.0008 | +0.0008
2025-02-15 | 2460721.8870 | -0.0008 | +0.0008
2025-02-24 | 2460730.6909 | -0.0008 | +0.0009
2025-03-04 | 2460739.4972 | -0.0008 | +0.0008
2025-03-13 | 2460748.3030 | -0.0008 | +0.0009
2025-03-22 | 2460757.1099 | -0.0008 | +0.0009
2025-03-31 | 2460765.9140 | -0.0009 | +0.0010
2025-04-09 | 2460774.7198 | -0.0008 | +0.0009
2025-04-18 | 2460783.5252 | -0.0008 | +0.0009
2025-04-26 | 2460792.3318 | -0.0008 | +0.0008
2025-05-05 | 2460801.1358 | -0.0008 | +0.0009
2025-05-14 | 2460809.9403 | -0.0008 | +0.0008
2025-05-23 | 2460818.7448 | -0.0008 | +0.0008
2025-06-01 | 2460827.5504 | -0.0007 | +0.0008
2025-06-09 | 2460836.3541 | -0.0008 | +0.0008
2025-06-18 | 2460845.1572 | -0.0009 | +0.0009
2025-06-27 | 2460853.9604 | -0.0011 | +0.0010
2025-07-06 | 2460862.7646 | -0.0013 | +0.0013
2025-07-15 | 2460871.5678 | -0.0014 | +0.0014
2025-07-23 | 2460880.3697 | -0.0016 | +0.0016
2025-08-01 | 2460889.1719 | -0.0019 | +0.0018
2025-08-10 | 2460897.9747 | -0.0023 | +0.0021
2025-08-19 | 2460906.7777 | -0.0023 | +0.0022
2025-08-28 | 2460915.5787 | -0.0024 | +0.0023
2025-09-05 | 2460924.3803 | -0.0026 | +0.0025
2025-09-14 | 2460933.1821 | -0.0029 | +0.0029
2025-09-23 | 2460941.9852 | -0.0028 | +0.0028
2025-10-02 | 2460950.7861 | -0.0027 | +0.0028
2025-10-11 | 2460959.5876 | -0.0027 | +0.0028
2025-10-19 | 2460968.3892 | -0.0028 | +0.0030
2025-10-28 | 2460977.1927 | -0.0026 | +0.0029
2025-11-06 | 2460985.9941 | -0.0024 | +0.0026
2025-11-15 | 2460994.7961 | -0.0021 | +0.0025
2025-11-24 | 2461003.5978 | -0.0022 | +0.0026
2025-12-02 | 2461012.4021 | -0.0020 | +0.0023
2025-12-11 | 2461021.2043 | -0.0018 | +0.0020
2025-12-20 | 2461030.0074 | -0.0016 | +0.0018
2025-12-29 | 2461038.8096 | -0.0015 | +0.0018
2026-01-07 | 2461047.6146 | -0.0014 | +0.0015
2026-01-15 | 2461056.4179 | -0.0013 | +0.0013
2026-01-24 | 2461065.2221 | -0.0012 | +0.0012
2026-02-02 | 2461074.0248 | -0.0012 | +0.0012
2026-02-11 | 2461082.8305 | -0.0011 | +0.0010
2026-02-20 | 2461091.6347 | -0.0010 | +0.0010
2026-02-28 | 2461100.4399 | -0.0010 | +0.0010
2026-03-09 | 2461109.2431 | -0.0011 | +0.0010
2026-03-18 | 2461118.0493 | -0.0010 | +0.0010
2026-03-27 | 2461126.8543 | -0.0010 | +0.0010
2026-04-05 | 2461135.6604 | -0.0009 | +0.0010
2026-04-13 | 2461144.4641 | -0.0010 | +0.0011
2026-04-22 | 2461153.2706 | -0.0010 | +0.0010
2026-05-01 | 2461162.0762 | -0.0010 | +0.0010
2026-05-10 | 2461170.8830 | -0.0010 | +0.0010
2026-05-19 | 2461179.6870 | -0.0011 | +0.0012
2026-05-27 | 2461188.4932 | -0.0010 | +0.0011
2026-06-05 | 2461197.2990 | -0.0011 | +0.0011
2026-06-14 | 2461206.1059 | -0.0010 | +0.0011
2026-06-23 | 2461214.9101 | -0.0011 | +0.0012
2026-07-02 | 2461223.7155 | -0.0010 | +0.0010
2026-07-11 | 2461232.5207 | -0.0010 | +0.0010
2026-07-19 | 2461241.3270 | -0.0009 | +0.0009
2026-07-28 | 2461250.1309 | -0.0010 | +0.0009
2026-08-06 | 2461258.9350 | -0.0009 | +0.0009
2026-08-15 | 2461267.7391 | -0.0009 | +0.0009
2026-08-24 | 2461276.5442 | -0.0011 | +0.0010
2026-09-01 | 2461285.3478 | -0.0012 | +0.0010
2026-09-10 | 2461294.1504 | -0.0013 | +0.0013
2026-09-19 | 2461302.9533 | -0.0016 | +0.0015
2026-09-28 | 2461311.7569 | -0.0019 | +0.0019
2026-10-07 | 2461320.5600 | -0.0020 | +0.0020
2026-10-15 | 2461329.3616 | -0.0022 | +0.0022
2026-10-24 | 2461338.1634 | -0.0025 | +0.0025
2026-11-02 | 2461346.9659 | -0.0029 | +0.0029
2026-11-11 | 2461355.7689 | -0.0029 | +0.0028
2026-11-20 | 2461364.5698 | -0.0030 | +0.0029
2026-11-28 | 2461373.3712 | -0.0030 | +0.0032
2026-12-07 | 2461382.1729 | -0.0033 | +0.0035
2026-12-16 | 2461390.9761 | -0.0032 | +0.0034
2026-12-25 | 2461399.7771 | -0.0030 | +0.0033
222222The table lists: Calendar UT date (YYYY-MM-DD), mid-transit time, and
68.3% CI.

Figure 18: Comparison of TTVs for different models. Left: Idem. as Figure 11,
but up to the year 2100 with the two-planet photodynamical modelling (Section
6), three-planet photodynamical modelling (Section 7.8), and coplanar two-
planet photodynamical modelling (Section 7.3). Right: Zoom of the plot on the
left.

Figure 19: Comparison of transit duration variations for different models.
Left: Posterior transit duration variations of planet b up to the end of the
year 2100 from 1,000 random draws from the posterior distribution of the two-
planet photodynamical modelling (Section 6), three-planet photodynamical
modelling (Section 7.8), and coplanar two-planet photodynamical modelling
(Section 7.3). The colours correspond with the ones used in Figure 18. The
points with error bars are the durations measured on individual transits
(Section 7.3). Right: Zoom of the plot on the left. Table 10: Inferred system
parameters with the photodynamical modelling assuming coplanar orbits.

Parameter |  | Median
---|---|---
|  | and 68.3% CI
_Star_ |  |
Mass, $M_{\star}$ |  ($M_{\odot}$) |  0.968 $\pm$ 0.050
Radius, $R_{\star}$ |  ($\mathcal{R}^{\rm N}_{\odot}$) |  0.918 $\pm$ 0.017
Stellar mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] |  1.76 ${}^{+0.14}_{-0.12}$
_Planet b_ |  |
Semi-major axis, $a$ | [au] |  0.0825 $\pm$ 0.0015
Eccentricity, $e$ |  |  0.194 $\pm$ 0.035
Argument of pericentre, $\omega$ | [°] |  61.4 $\pm$ 7.2
Inclination, $i$ | [°] |  89.06 ${}^{+0.43}_{-0.31}$
Longitude of the ascending node, $\Omega$ | [°] |  180 (fixed at $t_{\mathrm{ref}}$)
Mean anomaly, $M_{0}$ | [°] |  19.0 ${}^{+6.1}_{-5.0}$
Impact parameter, $b$ |  |  0.259 ${}^{+0.096}_{-0.12}$
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  0.289 $\pm$ 0.023
Radius, $R_{\mathrm{p}}$ |  ($\mathcal{R}^{\rm N}_{e\rm J}$) |  0.756 $\pm$ 0.017
Planet mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  0.825 ${}^{+0.094}_{-0.078}$
_Planet c_ |  |
Semi-major axis, $a$ | [au] |  0.2054 $\pm$ 0.0036
Eccentricity, $e$ |  |  0.168 ${}^{+0.016}_{-0.010}$
Argument of pericentre, $\omega$ | [°] |  26.3 $\pm$ 6.0
Inclination, $i$ | [°] |  $\equiv$ Planet b inclination
Longitude of the ascending node, $\Omega$ | [°] |  180 (fixed at $t_{\mathrm{ref}}$)
Mean anomaly, $M_{0}$ | [°] |  258.1 $\pm$ 3.6
Impact parameter, $b$ |  |  0.71 ${}^{+0.23}_{-0.33}$
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  0.375 $\pm$ 0.040
232323The table lists: Posterior median and 68.3% CI for the photodynamical
modelling assuming coplanar orbits (Section 7.3). Only masses, radii,
densities, and orbital parameters (at $t_{\mathrm{ref}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 2\;457\;957.48167$ BJDTDB) are listed.

Table 11: Inferred system parameters with the photodynamical modelling without
stellar priors.

Parameter |  | Median
---|---|---
|  | and 68.3% CI
_Star_ |  |
Mass, $M_{\star}$ |  ($M_{\odot}$) |  9.7 ${}^{+12}_{-5.3}$
Radius, $R_{\star}$ |  ($\mathcal{R}^{\rm N}_{\odot}$) |  2.14 ${}^{+0.69}_{-0.54}$
Stellar mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] |  1.41 ${}^{+0.29}_{-0.22}$
_Planet b_ |  |
Semi-major axis, $a$ | [au] |  0.178 ${}^{+0.053}_{-0.041}$
Eccentricity, $e$ |  |  0.254 $\pm$ 0.045
Argument of pericentre, $\omega$ | [°] |  65.6 $\pm$ 9.6
Inclination, $i$ | [°] |  88.90 ${}^{+0.56}_{-0.45}$
Longitude of the ascending node, $\Omega$ | [°] |  180 (fixed at $t_{\mathrm{ref}}$)
Mean anomaly, $M_{0}$ | [°] |  13.9 ${}^{+7.0}_{-5.6}$
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  1.32 ${}^{+0.88}_{-0.53}$
Radius, $R_{\mathrm{p}}$ |  ($\mathcal{R}^{\rm N}_{e\rm J}$) |  1.77 ${}^{+0.58}_{-0.45}$
Planet mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  0.293 ${}^{+0.14}_{-0.089}$
_Planet c_ |  |
Semi-major axis, $a$ | [au] |  0.44 ${}^{+0.13}_{-0.10}$
Eccentricity, $e$ |  |  0.280 $\pm$ 0.023
Argument of pericentre, $\omega$ | [°] |  23.6 $\pm$ 7.0
Inclination, $i$ | [°] |  129 $\pm$ 13
Longitude of the ascending node, $\Omega$ | [°] |  214.4 $\pm$ 8.2
Mean anomaly, $M_{0}$ | [°] |  262.1 $\pm$ 4.4
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  2.4 ${}^{+2.1}_{-1.2}$
Mutual inclination, $I$ | [°] |  52 ${}^{+10}_{-13}$
242424The table lists: Posterior median and 68.3% CI for the photodynamical
modelling without stellar priors (Section 7.4). Only masses, radii, densities,
and orbital parameters (at $t_{\mathrm{ref}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 2\;457\;957.48167$ BJDTDB) are listed.

Table 12: Inferred system parameters with the photodynamical modelling with
three planets.

Parameter |  | Median
---|---|---
|  | and 68.3% CI
_Star_ |  |
Mass, $M_{\star}$ |  ($M_{\odot}$) |  0.976 $\pm$ 0.053
Radius, $R_{\star}$ |  ($\mathcal{R}^{\rm N}_{\odot}$) |  0.918 $\pm$ 0.017
Stellar mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] |  1.79 $\pm$ 0.13
_Planet b_ |  |
Semi-major axis, $a$ | [au] |  0.0828 $\pm$ 0.0015
Eccentricity, $e$ |  |  0.183 ${}^{+0.028}_{-0.031}$
Argument of pericentre, $\omega$ | [°] |  50.7 ${}^{+9.8}_{-8.6}$
Inclination, $i$ | [°] |  88.87 $\pm$ 0.21
Longitude of the ascending node, $\Omega$ | [°] |  180 (fixed at $t_{\mathrm{ref}}$)
Mean anomaly, $M_{0}$ | [°] |  27.3 ${}^{+6.4}_{-7.1}$
Impact parameter, $b$ |  |  0.323 $\pm$ 0.064
$T_{0}^{\prime}$ \- 2 450 000 |  [BJDTDB] |  7957.48176 $\pm$ 0.00025
$P^{\prime}$ | [d] |  8.80358 $\pm$ 0.00018
$K^{\prime}$ |  [m s-1] |  29.1 $\pm$ 1.9
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  0.284 ${}^{+0.021}_{-0.018}$
Radius, $R_{\mathrm{p}}$ |  ($\mathcal{R}^{\rm N}_{e\rm J}$) |  0.765 $\pm$ 0.016
Planet mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  0.788 $\pm$ 0.071
_Planet c_ |  |
Semi-major axis, $a$ | [au] |  0.2059 $\pm$ 0.0037
Eccentricity, $e$ |  |  0.196 ${}^{+0.023}_{-0.019}$
Argument of pericentre, $\omega$ | [°] |  25.4 $\pm$ 5.6
Inclination, $i$ | [°] |  108 ${}^{+11}_{-14}$
Longitude of the ascending node, $\Omega$ | [°] |  188.4 ${}^{+4.5}_{-2.9}$
Mean anomaly, $M_{0}$ | [°] |  259.0 $\pm$ 3.8
Impact parameter, $b$ |  |  13.3 ${}^{+7.5}_{-9.6}$
$T_{0}^{\prime}$ \- 2 450 000 |  [BJDTDB] |  7971.56 $\pm$ 0.31
$P^{\prime}$ | [d] |  34.5441 $\pm$ 0.0026
$K^{\prime}$ |  [m s-1] |  25.0 ${}^{+2.5}_{-2.2}$
Mass, $M_{\mathrm{p}}$ |  ($\mathrm{M}_{\rm J}$) |  0.411 ${}^{+0.054}_{-0.044}$
_Planet (d)_ |  |
Semi-major axis, $a$ | [au] |  0.551 ${}^{+0.010}_{-0.012}$
Eccentricity, $e$ |  |  0.19 ${}^{+0.19}_{-0.13}$
Argument of pericentre, $\omega$ | [°] |  148 $\pm$ 96
Inclination, $i$ | [°] |  82 ${}^{+39}_{-33}$
Longitude of the ascending node, $\Omega$ | [°] |  199 ${}^{+110}_{-150}$
Mean anomaly, $M_{0}$ | [°] |  134 ${}^{+110}_{-85}$
Impact parameter, $b$ |  |  48 ${}^{+40}_{-34}$
$T_{0}^{\prime}$ \- 2 450 000 |  [BJDTDB] |  7891 ${}^{+20}_{-11}$