11institutetext: Observatoire Astronomique de l’Université de Genève, Chemin
Pegasi 51b, CH-1290 Versoix, Switzerland
11email<EMAIL_ADDRESS>
# DREAM II. The spin–orbit angle distribution of close-in exoplanets under the
lens of tides

O. Attia  V. Bourrier  J.-B. Delisle  P. Eggenberger

(Received XXX; accepted XXX)

The spin–orbit angle, or obliquity, is a powerful observational marker that
allows us to access the dynamical history of exoplanetary systems. For this
study, we have examined the distribution of spin–orbit angles for close-in
exoplanets and put it in a statistical context of tidal interactions between
planets and their host stars. We confirm the previously observed trends
between the obliquity and physical quantities directly connected to tides,
namely the stellar effective temperature, the planet-to-star mass ratio, and
the scaled orbital distance. We further devised a tidal efficiency factor
$\tau$ combining critical parameters that control the strength of tidal
effects and used it to corroborate the strong link between the spin–orbit
angle distribution and tidal interactions. In particular, we developed a
readily usable formula $\theta\left(\tau\right)$ to estimate the probability
that a system is misaligned, which will prove useful in global population
studies. By building a robust statistical framework, we reconstructed the
distribution of the three-dimensional spin–orbit angles, allowing for a sample
of nearly 200 true obliquities to be analyzed for the first time. This
realistic distribution maintains the sky-projected trends, and additionally
hints toward a striking pileup of truly aligned systems. In fact, we show that
the fraction of aligned orbits could be underestimated in classical analyses
of sky-projected obliquities due to an observational bias toward misaligned
systems. The comparison between the full population and a pristine subsample
unaffected by tidal interactions suggests that perpendicular architectures are
resilient toward tidal realignment, providing evidence that orbital
misalignments are sculpted by disruptive dynamical processes that
preferentially lead to polar orbits. On the other hand, star–planet
interactions seem to efficiently realign or quench the formation of any tilted
configuration other than for polar orbits, and in particular for antialigned
orbits. Observational and theoretical efforts focused on these pristine
systems are encouraged in order to study primordial mechanisms shaping orbital
architectures, which are unaltered by tidal effects.

###### Key Words.:

planet–star interactions – planets and satellites: dynamical evolution and
stability – methods: data analysis – methods: statistical

##  1 Introduction

The detection of the first exoplanet around a main sequence star (Mayor &
Queloz 1995) sparked a revolution in planetary science. Instead of orbiting
far from its star similar to gas giants in our Solar System, the discovered
“hot Jupiter” was found to orbit ten times closer than Mercury around the Sun,
challenging our understanding of planetary formation and evolution. Nearly 30
years later, a wealth of exoplanets evolving on very short orbits have been
discovered, enabling comprehensive studies on this still intriguing class of
distant worlds. As a matter of fact, the population of close-in exoplanets
shows a wider range of orbital characteristics that had been expected from our
initial interpretations of the Solar System (e.g., Winn & Fabrycky 2015).

One of the features illustrating this diversity is the “hot Neptune desert,”
with a striking paucity of intermediate-sized planets with short periods
($P<3$ d, e.g., Lecavelier des Etangs 2007; Beaugé & Nesvorný 2013; Lundkvist
et al. 2016, see also Fig. 1), as well as a “Neptune savanna,” identified as a
milder deficit at longer periods (Bourrier et al. 2023). It is thought that
orbital migration plays an important role in shaping the observed demographics
of close-in planets and the desert in particular (e.g., Mazeh et al. 2016;
Bourrier et al. 2018; Attia et al. 2021), although it remains to be explored
how the various migration processes shape the orbital distribution of
Neptunian worlds across the savanna and desert. Different mechanisms have
indeed been proposed to transport planets from their presumed birthplace to
their current location. Some of them would preserve the relative orientation
between the planetary orbit and the stellar equator (e.g., Lin et al. 1996;
Zhou et al. 2020; Mann et al. 2020), while others could disrupt this alignment
(e.g., Fabrycky & Tremaine 2007; Naoz et al. 2011; Nelson et al. 2017),
offering us the possibility to distinguish between different theories for the
origin of close-in systems.

For this reason, this so-called spin–orbit angle (or obliquity) represents a
crucial observational tracer for past dynamical histories. This angle can be
accessed when the probed planet transits its host star, creating detectable
anomalies in the spectral absorption lines of the eclipsed star originating
from the partial occultation of the rotating photosphere. This phenomenon,
dubbed the Rossiter–McLaughlin (RM) effect (Rossiter 1924; McLaughlin 1924),
has been a successful tool over the past decades to unveil a plethora of
surprisingly misaligned orbits (see Albrecht et al. 2022, for a recent
review). Yet, it is a difficult task to trace all the steps that occurred
during the past secular evolution of close-in planets, especially since their
short final orbits make them vulnerable to strong tidal interactions with
their host star, which progressively smooth out signs of disruptive dynamical
histories (e.g., Barker & Ogilvie 2010).

With the present work, we contribute to this global effort by conducting a
statistical study of the entire sample of known obliquity measurements. This
is the second article of the DREAM series, whose objectives are to better
understand the origins and evolution of close-in planets, particularly those
closely linked to the hot Neptune desert. In DREAM I (Bourrier et al. 2023),
we determined the orbital architecture of several planets spanning the borders
of the desert and savanna (Fig. 1) using the Rossiter–McLaughlin revolutions
(RMR) technique (Bourrier et al. 2021). For this study, we have put this
sample in a larger statistical framework, looked for possible trends, and
identified interesting links with star–planet interactions.

Figure 1: Distribution of close-in exoplanets as a function of their radius
and orbital period, featuring the Neptune desert and savanna. White diamonds
indicate exoplanets with measured spin–orbit angles. Blue stars highlight
planets in the survey described in DREAM I.

##  2 Observational trends: A link with tides

###  2.1 Constructing the obliquity sample

The survey described in detail in DREAM I (Bourrier et al. 2023) provides a
substantial revision to the systems with a characterized spin–orbit angle. The
sky-projected obliquity $\lambda$ was successfully measured for 11 targets.
Additionally, three host stars had known stellar inclinations, and it could be
constrained for four more, allowing us to derive the true obliquity $\psi$ in
seven systems. In this section, we put our sample in the context of the close-
in planet population, investigate known trends for spin–orbit angles, and
search for outliers from these trends that could hint at interesting dynamical
histories. To this effect, we collect spin–orbit angle measurements from the
TEPCat catalog (Southworth 2011) after a careful inspection of the accuracy of
all their sources, and combine the NASA Exoplanet
Archive111https://exoplanetarchive.ipac.caltech.edu/ and
Exoplanet.eu222http://exoplanet.eu/ for all other relevant parameters. In
total, our sample consists in 196 spin–orbit measurements, 42 of which are
three-dimensional (3D), the remaining 154 being sky-projected. All of the
obliquity measurements used in this work are summarized in App. A.

Being interested in the geometrical configuration of exoplanet systems, and
not the spin–orbit angle value per se, we fold the values to [0 ; 180]∘ (e.g.,
retrograde orbits are then grouped together irrespective of their
orientation). In doing so, we follow the same scheme as the state-of-the-art
review of Albrecht et al. (2022). Because of the possible biases on the
obliquity associated with the classical RM method, we choose to use values
derived through the RMR analysis for the sample studied in DREAM I, even when
values had already been published. In the following analysis, we define
misaligned orbits with $\lambda$ or $\psi>30^{\circ}$ as a conservative
threshold given the typical uncertainties on RM measurements. We plot
spin–orbit angle measurements as a function of various parameters to
investigate possible trends. On top of these figures, we draw a bar plot
counting the percentage of misaligned systems within each $x$-axis bin, and we
adapt the bin sizes to have roughly equivalent sample sizes in each of them.
In this section, we only use the obliquities of App. A coming from the
literature, in order to perform a reference-point analysis. We note again that
many spin–orbit angles in this sample only have sky-projected measurements,
which could bias the interpretation of their distribution. We address these
caveats in Sect. 4.

###  2.2 The Kraft break

We first look at the possible correlation between the spin–orbit angle and
stellar effective temperature ($T_{\rm eff}$, Fig. 2). A $T_{\rm eff}$ trend
was first noted by Winn et al. (2010a) for hot Jupiters alone, mainly because
this class of planets was the only one accessible to RM measurements. Even
though the sample size has nearly quintupled since this seminal study, our
analysis confirms this trend in line with Brown et al. (2017), Triaud (2018),
and Albrecht et al. (2022), for example. Planets around hot stars ($T_{\rm
eff}>6250$ K, or the Kraft break, Kraft 1967) tend to be more misaligned than
around cold stars. This is linked to the transition to F8 stars and the
disappearance of the stellar convective envelope at roughly that temperature
threshold (Kippenhahn et al. 2013). Indeed, cold stars with deep convective
envelopes undergo magnetic braking and rotate slower, which hastens the
effects of tidal dissipation and allows for a more efficient realignment of
the system (e.g., Hansen 2012; Valsecchi & Rasio 2014).

Figure 2: Distribution of spin–orbit angles as a function of the stellar
effective temperature. Circles represent projected obliquities $\lambda$ and
diamonds true obliquities $\psi$. Exoplanets from the survey described in
DREAM I are highlighted as blue symbols. The blue bars on top of the plots
count the percentage of misaligned systems ($\lambda$ or $\psi>30^{\circ}$)
within each $T_{\rm eff}$ bin. The top panel encompasses all systems with a
known spin–orbit angle, whereas the bottom panels are subsets with only hot
Jupiters (left) and sub-Saturns (right). The gray histogram on the right of
the top plot counts the number of measurements within each spin–orbit angle
bin. The color map represents a smoothed number-density of planets to guide
the eye. The Kraft break ($T_{\rm eff}=6250$ K) is shown as a magenta vertical
line.

Compared to Winn et al. (2010a), the larger sample of spin–orbit angles that
is now available allows us to draw a more global picture on their
distribution. In particular, one can see in Fig. 2 (right of top panel) that
it is far from uniform. Aligned systems seem to be dominant, but contrary to
what was suggested by Winn et al. (2010a), misaligned systems do not randomly
span the full range of obliquities. Instead, they seem to pile up on polar
($\sim 90^{\circ}$) orbits, a feature that was unveiled by Albrecht et al.
(2021) and deemed unlikely to be a statistical fluke. Our analysis agrees with
their results and further hints at the existence of a mechanism exciting
spin–orbit angles that has yet to be assessed, which preferentially leads to
polar orbits.

From the top panel of Fig. 2, one can also notice that despite the $\lambda-
T_{\rm eff}$ trend, many planets around cold stars are on substantially
misaligned orbits, primarily polar ones. In particular, this is the case for
all the planets of our survey that orbit cold stars, except for WASP-166 b.
These apparent outliers can in reality still be explained by tidal
interactions. Breaking down the sample into hot Jupiters and sub-Saturns
separately (Fig. 2, lower panels), the misalignment trend is even clearer for
the former while the latter shows a flat distribution. Because of their larger
mass, hot Jupiters raise more efficient tides, which is why the trend is all
the more accentuated for this subsample. Our apparent outliers turn out to be
part of the sub-Saturn group, for which no trend with $T_{\rm eff}$ is seen
since these lower-mass planets raise less efficient tides even around cold
stars.

An apparent tension might arise when comparing this last result to Morton &
Winn (2014), Mazeh et al. (2015), and Louden et al. (2021), who suggested that
the Kraft break persists even for smaller planets based on a statistical
argument regarding stellar inclinations of the Kepler sample. However, our
sub-Saturn sample still consists in large, and rather lone planets, thus
mainly affected by tides with the host star. On the other hand, their analyses
concerns Kepler planets, smaller in size and generally part of compact
multiplanet systems, for which gravitational interactions between the
different planets is dominant over tides. Hence, our study can be seen as
complementary to theirs, looking into exoplanets affected by a different class
of mechanisms.

###  2.3 Impact of mass and separation

To further investigate the impact of planetary mass in realigning the stellar
spin-axis, we show the spin–orbit angle distribution as a function of the
planet-to-star mass ratio ($M_{\rm p}/M_{\star}$, Fig. 3). Scrutinizing the
cold star sample in isolation, we see a downward trend of misalignment with
$M_{\rm p}/M_{\star}$. More massive planets tend to be more aligned, a trend
already noted by Hébrard et al. (2011a) with a smaller sample. This is
understood because the higher the mass of the planet relative to the mass of
the star, the faster its tides can align the stellar and orbital angular
momentum vectors (e.g., Barker & Ogilvie 2010; Dawson 2014). In contrast, hot
stars, for which tidal realignment is inefficient, expectedly showcase a flat
misalignment distribution.

Figure 3: Distribution of spin–orbit angles as a function of the planet-to-
star mass ratio, subdivided into systems with a cold ($T_{\rm eff}<6250$ K,
left) or hot ($T_{\rm eff}>6250$ K, right) host star. Same color, symbol, and
histogram schemes as in Fig. 2.

The major impact of tides is further confirmed by the distribution of
spin–orbit angles as a function of the scaled separation ($a/R_{\star}$, Fig.
4). Again, for cold stars, which efficiently realign their systems, there is
an upward misalignment trend with the semi-major axis (as broadly seen in
Anderson et al. 2015b). Systems with a short separation are more impacted by
tides (e.g., Barker & Ogilvie 2010; Dawson 2014) and get realigned more easily
while planets further out do not feel tidal effects and showcase a broader
spin–orbit angle distribution. We note the surprisingly aligned bin at
$a/R_{\star}\lesssim 30$ (Fig. 4, left panel), which we hypothesize could
trace the final orbit reached by planets that formed locally or underwent
disk-driven migration in a coplanar protoplanetary disk. On the other hand, we
expectedly see a flat distribution for hot stars irrespective of the orbital
distance.

Figure 4: Distribution of spin–orbit angles as a function of the semi-major
axis-to-stellar radius ratio, subdivided into systems with a cold ($T_{\rm
eff}<6250$ K, left) or hot ($T_{\rm eff}>6250$ K, right) host star. Same
color, symbol, and histogram schemes as in Fig. 2.

##  3 Quantifying tidal realignment

The previous investigated trends, although purely qualitative, show promising
links between the spin–orbit architecture of close-in planet systems and the
intensity of tidal interactions within them, paving the way for a more in-
depth analysis. In this section, we set the analytical framework that will
allow us to rigorously quantify the connection between the obliquity
distribution and tidal theory.

###  3.1 Tidal efficiency factor

Our aim here is to combine the various parameters involved in the strength of
tidal effects (Sect. 2) to design a global tidal efficiency factor. We employ
the formalism of equilibrium tides, descried in detail in Zahn (1966),
Alexander et al. (1976), Zahn (1977), and Zahn (1989). As seen in Zahn (1977),
for example, the characteristic timescale $t_{a}$ for a change in the semi-
major axis $a$ due to equilibrium tides is proportional to

$\frac{1}{t_{a}}=\left|\dot{a}/a\right|_{\mathrm{eq}}\;\propto\;\frac{M_{\mathrm{p}}}{M_{\star}}\left(1+\frac{M_{\mathrm{p}}}{M_{\star}}\right)\left(\frac{R_{\star}}{a}\right)^{8},$
(1)

The corresponding timescale $t_{\psi}$ for a change in obliquity is then given
by (e.g., Lai 2012)

$\frac{1}{t_{\psi}}\simeq\frac{L_{\mathrm{p}}}{2L_{\star}}\frac{1}{t_{a}},$
(2)

where $L_{\rm p}$ and $L_{\star}$ are the orbital angular momentum and the
stellar spin angular momentum, respectively. The latter quantities are
proportional to $L_{\rm p}\;\propto\;M_{\rm p}a^{2}/P$ and
$L_{\star}\;\propto\;M_{\star}R_{\star}^{2}/P_{\rm eq}$ respectively, where
$P$ is the orbital period and $P_{\rm eq}$ the stellar rotation period.
Consequently, linking this obliquity variation timescale to measurable
parameters yields

$\frac{1}{t_{\psi}}\;\propto\;\frac{P_{\mathrm{eq}}}{P}\left(\frac{M_{\mathrm{p}}}{M_{\star}}\right)^{2}\left(1+\frac{M_{\mathrm{p}}}{M_{\star}}\right)\left(\frac{a}{R_{\star}}\right)^{-6}.$
(3)

Our main objective is to quantify tidal dissipation efficiency for the targets
in our sample using Eq. (3), with order of magnitude comparison. Thus, the
$1+M_{\rm p}/M_{\star}$ term can safely be discarded as it stays close to
unity for planetary systems. Moreover, the $P_{\rm eq}/P$ term can be
neglected because these two periods usually remain within the same order of
magnitude, while the other ratios in Eq. (3) strongly vary from one system to
another, especially when raised to high powers. To quantify the variability in
orders of magnitude of the different factors intervening in Eq. (3), we
compute the standard deviation of their logarithms for all systems with
precise (relative error $<10\%$) measurements. We find
$\sigma\left(\log(P_{\rm eq}/P)\right)=0.78$, $\sigma\left(\log(M_{\rm
p}/M_{\star})^{2}\right)=2.16$, and
$\sigma\left(\log(a/R_{\star})^{-6}\right)=4.19$, confirming that $P_{\rm
eq}/P$ has much less influence on the inter-system variability of $t_{\psi}$
than the other factors.

Finally, the dissipation of these equilibrium tides is mainly provided by the
eddy viscosity in the convective envelope of the star. This introduces an
additional term related to the relative mass of this envelope (following e.g.,
Rasio et al. 1996; Privitera et al. 2016) in Eq. (3), which cannot be
neglected as the mass of the convective envelope can arbitrarily approach zero
for hot stars. As a result, the tidal efficiency factor used in the present
work, proportional to $1/t_{\psi}$, is given by

$\tau\equiv\frac{M_{\mathrm{conv}}}{M_{\star}}\left(\frac{M_{\mathrm{p}}}{M_{\star}}\right)^{2}\left(\frac{a}{R_{\star}}\right)^{-6}.$
(4)

Equation (4) is reminiscent of the empirical tidal efficiency factor devised
by Albrecht et al. (2012b). The latter authors emphasized that they could not
find a theoretical argument for the linear correlation they identified between
the convective mass and the spin–orbit angle distribution. However, this
dependence seems justified by invoking the relevant timescales (as seen in
Eqs. (1–4)) and by the fact that equilibrium tides are very efficiently
dissipated by turbulent friction in the external convective envelopes of stars
(Zahn 1977; Rasio et al. 1996; Privitera et al. 2016).

It is worth noting that Albrecht et al. (2012b) also considered tidal
dissipation within stellar radiative envelopes for stars hotter than the Kraft
break. As pointed out in their work, this approach can bias the results
because of the binary decision one has to make to decide whether a star is
cold and convective, or hot and radiative. It is particularly problematic for
the many stars with an effective temperature consistent with the Kraft break
(see Fig. 2), a caveat we want to avoid. Furthermore, we recall that
equilibrium tides are very efficiently dissipated by turbulent friction in the
external convective envelopes of stars, while this dissipation is much less
efficient in stably stratified radiative layers (Zahn 1977), adding more
weight as to why we only consider dissipation of equilibrium tides in
convective zones.

Dissipation of dynamical tides, resulting from the excitation of inertial
waves in convective stellar layers, can also play a role on top of equilibrium
tides. Nevertheless, dynamical tides in convective zones have a significant
impact on planetary orbits only for fast-rotating stars during their pre-main
sequence (Rao et al. 2018). Dissipation of dynamical tides can also occur in
stellar radiative zones as a result of the propagation of internal gravity
waves excited at the interface between convective and radiative regions. This
can lead to an efficient tidal dissipation, provided that these waves are
efficiently damped, as it is the case if wave breaking is able to occur in
stellar interiors. However, wave breaking only occurs when the mass of the
planet is higher than a given threshold. As shown by Barker (2020), in the
case of low-mass and solar-type stars, this critical planet mass becomes lower
than 1 $M_{\rm Jup}$ only very close to the end of the main sequence (a
typical value of about 3 $M_{\rm Jup}$ is found in the case of the Sun).
Consequently, for all but the most massive planets, tidal dissipation by
gravity waves is able to play a dominant role for the orbital evolution only
at the end of the main sequence and during the post-main sequence phase
(Barker 2020). This explains why we consider in the present work a tidal
efficiency factor $\tau\;\propto\;1/t_{\psi}$ as given by Eq. (4).

Table 1: Tabulated values of the mass of the stellar convective envelope, used
to compute the tidal efficiency factor, as a function of the stellar mass.

$M_{\star}$ [$M_{\odot}$] |  $\log T_{\rm eff}$ [K] | $M_{\rm conv}/M_{\star}$
---|---|---
0.5 | 3.60337 | 0.3508
0.6 | 3.62040 | 0.1842
0.7 | 3.64646 | 0.0991
0.8 | 3.68716 | 0.0667
0.9 | 3.72492 | 0.0437
1.0 | 3.75427 | 0.0257
1.1 | 3.77156 | 0.0107
1.2 | 3.79455 | 0.0031
1.3 | 3.81339 | 0.0003
1.4 | 3.83224 | 0.0000
1.5 | 3.85367 | 0.0000

  * Note: the stellar effective temperature, as generated by our model, is given for indicative purposes. In particular, one can see that $M_{\rm conv}/M_{\star}$ drops below 1% at roughly the temperature of the Kraft break.

###  3.2 Tidal efficiency trend

Figure 5: Distribution of spin–orbit angles as a function of the tidal
efficiency factor $\tau$. Same color, symbol, and histogram schemes as in Fig.
2. All systems with a zero tidal efficiency factor have been set to the
minimal value of the sample. The gray histogram on the left of the plot covers
only the first (lowest) tidal efficiency bin.

The following step is to compute $\tau$ for all the systems with a known
spin–orbit measurement to see if we confirm the trends we previously analyzed.
To this effect, $M_{\rm conv}$ is directly obtained from the computation of
stellar models for different masses using the Geneva stellar evolution code
GENEC (Eggenberger et al. 2008). We provide tabulated values of $M_{\rm conv}$
that can be interpolated for any useful purpose in Table 1. Because $M_{\rm
conv}$ and thus $\tau$ can be analytically null, all such systems are set to
the minimal nonzero value for $\tau$ within the full sample of investigated
planets, so as to still be able to compare orders of magnitude in log-space.
The main uncertainty while computing the values of the convective mass comes
from the uncertainty on the total stellar mass, since $M_{\rm conv}$ steeply
varies as $M_{\star}$ changes (see Table 1). We hence scale the error on
$M_{\rm conv}$ as the error on $M_{\star}$. This way, we set
$\sigma\left(M_{\rm conv}\right)/M_{\rm
conv}=\sigma\left(M_{\star}\right)/M_{\star}$, which allows us to define error
bars on the tidal efficiency factor as well.

The resulting plot is shown in Fig. 5, which shows a downward misalignment
trend, indicating that systems with a lower tidal efficiency tend to be more
misaligned. Figure 5 hence combines and confirms the trends seen in Sect. 2,
strengthening the role of tides in determining the distribution of known
spin–orbit angles. An interesting outlier from our survey is the polar orbit
of HAT-P-3 b, which should not have been affected by tidal effects during the
main sequence of the star according to Mancini et al. (2018). However, our own
estimates (Fig. 5) show that the system is in the range of tidal efficiency
where the orbit could have been realigned. In fact, it is the planet with the
highest $\tau$ value in our survey, which suggests that its polar orbit may
have been occasioned by a disruptive mechanism recently enough for the
realignment process to be still ongoing.

Despite the simplicity of our theoretical tidal framework, our analysis
provides evidence that spin–orbit angles are damped by tidal forces and that
strong star–planet interactions erase orbital histories by realigning systems.
Our tidal efficiency factor could be employed as a useful criterion for future
studies of close-in orbital architectures. Based on the trend in the
$\lambda-\tau$ bar plot of Fig. 5, we tentatively propose $\tau\sim 10^{-15}$
as a threshold when tidal realignment needs to be considered.

In this context, we construct a histogram of spin–orbit angles (left of Fig.
5) limited to systems within the lowest tidal efficiency bin, which can be
seen as pristine systems unaffected by stellar interactions. We recover the
aforementioned aligned and polar populations, and we unveil for the first time
a third, distinct group composed of antialigned systems. If confirmed, this
surprising distribution is indicative of disruptive processes that happened in
the dynamical past of the observed systems, and which were not damped by
star–planet interactions. This result suggests that in addition to the
“preponderance of perpendicular orbits” brought to light by Albrecht et al.
(2021), antialigned orbits unaffected by tides might also represent a
noteworthy feature in the global obliquity distribution. Interestingly, one
can see in Fig. 5 that there are nearly no planets on antialigned orbits
outside of the lowest $\tau$ bin. Even though we lack for now more
observational evidence to confirm these findings, we can still make two
hypotheses based on them. First, that the processes that form retrograde
orbits (e.g., Kozai–Lidov resonance following a spike in eccentricity, Naoz et
al. 2011, or a primordial flip of the protoplanetary disk, Zanazzi & Lai 2018)
do not lead to a uniform spin–orbit angle distribution over $90-180^{\circ}$,
but preferentially form perpendicular and antialigned orbits. Secondly, that
tides quench antialigned orbits back to a perpendicular configuration, or
prevent their formation in the first place. A sample of spin–orbit angles
covering a broader range of tidal efficiencies, and more detailed studies of
processes that can excite spin–orbit angles (e.g., Kozai–Lidov resonance,
Kozai 1962; Lidov 1962; Fabrycky & Tremaine 2007; Naoz et al. 2011, secular
resonance crossing, Petrovich et al. 2020, or magnetic warping, Foucart & Lai
2011; Romanova et al. 2021) are needed to investigate our hypothesis. In all
cases, systems within this low-$\tau$ bin, like the polar orbit of HAT-P-49 b,
are targets of choice for studying the lone impact of secular or primordial
dynamical processes since their orbital architectures are representative
pictures of these mechanisms, unaltered by tidal effects.

##  4 Estimating the true distribution of obliquities

###  4.1 Reconstructing the obliquity sample

The largest limitation of this analysis is that we lack a measurement of the
true 3D spin–orbit angle $\psi$ for a substantial number of planets in the
sample. $\psi$ can be disentangled from $\lambda$ if one has knowledge on the
orbital $i_{\rm p}$ and stellar $i_{\star}$ inclinations (e.g., Winn et al.
2007)

$\cos\psi=\sin i_{\star}\sin i_{\mathrm{p}}\cos\lambda+\cos i_{\star}\cos
i_{\mathrm{p}},$ (5)

which poses the problem of acquiring a robust measurement of $i_{\star}$. The
most straightforward way to estimate it is assuming stellar solid body
rotation (e.g., Borucki & Summers 1984; Doyle et al. 1984)

$\sin i_{\star}=\frac{v_{\mathrm{eq}}\sin
i_{\star}}{v_{\mathrm{eq}}}=\frac{v_{\mathrm{eq}}\sin i_{\star}}{2\pi
R_{\star}/P_{\mathrm{eq}}},$ (6)

where $P_{\rm eq}$ is the stellar equatorial rotation period. It is important
to stress that directly using Eq. (6) to generate an $i_{\star}$ distribution
is hazardous, because $v_{\rm eq}\sin i_{\star}$ and $P_{\rm eq}$ are
correlated quantities. More generally, Masuda & Winn (2020) showed that
naively sampling a $\psi$ distribution using Eqs. (5, 6) could result in very
large biases because of the various correlations between the involved
parameters. An accurate determination of $i_{\star}$ is in reality even more
problematic because of the often large uncertainties in the different involved
quantities (Fabrycky & Winn 2009; Muñoz & Perets 2018).

Keeping these limitations in mind, we seek to recover true spin–orbit angles
$\psi$ for all close-in planets in order to assess whether our conclusions
stem from sky-projection effects or not. Regarding the 154 sky-projected
measurements in the sample, we need knowledge on the stellar inclination to
disentangle $\psi$ from $\lambda$ with Eq. (5). For the 28 systems of this
subsample that have a $P_{\rm eq}$ measurement, we propose the following
procedure to derive $i_{\star}$, so as to avoid the aforementioned biases
outlined in Masuda & Winn (2020). We fit for $v_{\rm eq}\sin i_{\star}$ with a
Markov chain Monte Carlo (MCMC, using the emcee package, Foreman-Mackey et al.
2013) based on the following formulation of Eq. (6)

$v_{\mathrm{eq}}\sin i_{\star}=\frac{2\pi
R_{\star}}{P_{\mathrm{eq}}}\sqrt{1-\cos^{2}i_{\star}},$ (7)

and using the literature value as a constraint for the computation of the
likelihood. We set $R_{\star}$, $P_{\rm eq}$, and $\cos i_{\star}$ as
independent jump parameters with measurement-informed Gaussian priors on
$R_{\star}$ and $P_{\rm eq}$, and a uniform prior on $\cos i_{\star}$ (i.e.,
assuming an isotropic stellar inclination distribution). We then derive from
the results the probability distribution function (PDF) of the stellar
inclination, which we use to generate $\psi$ using Eq. (5). The two degenerate
configurations corresponding to $i_{\star}$ and $\pi-i_{\star}$, assumed to be
equiprobable, are combined to yield the final PDF of $\psi$. Using this
procedure, based on Eq. (7), circumvents the interdependence problem that
Masuda & Winn (2020) warned about. Finally, for each of the remaining 126
systems, we sample $\psi$ through Eq. (5), using the observational Gaussian
distributions of $\lambda$ and $i_{\rm p}$, and a uniform distribution for
$\cos i_{\star}$. For all the systems for which we reconstruct the 3D
obliquity, we retain the median of the resulting distribution as an estimate
for $\psi$ and the 68% highest density interval (HDI) to define the error
bars.

We include in App. A the values of $\psi$ derived by fitting for $v_{\rm
eq}\sin i_{\star}$, as they can be readily used as accurate 3D spin–orbit
angles for the concerned systems. We choose not to include the values of
$\psi$ derived only using the isotropic stellar inclination assumption, as we
deem that this approach should not be employed for individual systems, but for
statistical studies like this one.

###  4.2 Analyzing the true distribution of obliquities

Figure 6: Histogram of 3D spin–orbit angles for the full (top) and pristine
(bottom) samples. The $\psi$ distributions for individual systems are
estimated using the procedure outlined in Sect. 4.1 and the histograms are
built following the approach of Sect. 4.2.

Following the approach of Sect. 4.1, we are able to estimate the 3D, true,
spin–orbit angles for all systems with an obliquity measurement, which calls
for analyzing their global distribution to see if ensemble features emerge. To
this end, it would be tempting to build an obliquity histogram by naively
counting the number of systems in each spin–orbit angle bin, as in Fig. 2 (top
panel, right of the figure). However, constructing such a histogram solely
based on the medians of the estimated $\psi$ distributions would lead to large
biases in the final results, since it does not capture the uncertainty
relative to falling into one of two adjacent bins. Plus, the computed $\psi$
distributions are not necessarily Gaussian and might be skewed, making the
median an unreliable estimator for such a procedure.

In order to construct a statistically more robust histogram that is informed
by the full PDFs of the $\psi$ distributions, and not just their medians, we
design a Bayesian framework with an MCMC implementation, setting the jump
parameters as the consecutive
$\boldsymbol{\theta}=\left(\theta_{1},...,\theta_{K}\right)$, which represent
the fraction of systems that fall in each of the $K$ obliquity bins. As these
fractions need to be positive and satisfy $\sum_{i=1}^{K}\theta_{i}=1$, we
impose a Dirichlet prior on their joint distribution, with a unitary parameter
in each dimension (i.e., a flat Dirichlet distribution). This can be seen as a
uniform, noninformative, prior over the open $K-1$ simplex. The computation of
the likelihood of the data $\mathcal{D}^{j}$ for one system $j$ can be derived
using a hierarchichal model

$p(\mathcal{D}^{j}\;|\;\boldsymbol{\theta})=\int
p(\mathcal{D}^{j}\;|\;\psi^{j})p(\psi^{j}\;|\;\boldsymbol{\theta})\;\mathrm{d}\psi^{j},$
(8)

where $p(\mathcal{D}^{j}\;|\;\psi^{j})$ can be drawn from the expression of
the posterior probability distribution of the spin–orbit angle for that system
$j$

$p(\psi^{j}\;|\;\mathcal{D}^{j})\;\propto\;p(\mathcal{D}^{j}\;|\;\psi^{j})\;p(\psi^{j}),$
(9)

with $p(\psi^{j})$ being the prior function on $\psi^{j}$, which leads to

$p(\mathcal{D}^{j}\;|\;\boldsymbol{\theta})\;\propto\;\int
p(\psi^{j}\;|\;\mathcal{D}^{j})\frac{p(\psi^{j}\;|\;\boldsymbol{\theta})}{p(\psi^{j})}\;\mathrm{d}\psi^{j}.$
(10)

The previous equation can be computed by means of a Monte Carlo integration,
based on importance sampling

$p(\mathcal{D}^{j}\;|\;\boldsymbol{\theta})\;\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}\;\sum_{i}\frac{p(\psi_{i}^{j}\;|\;\boldsymbol{\theta})}{p(\psi_{i}^{j})},$
(11)

where the various $\psi_{i}^{j}$ are sampled according to their previously
generated posterior distribution (Sect. 4.1) for the system $j$. The terms of
Eq. (11) can be easily evaluated, as
$p(\psi_{i}^{j}\;|\;\boldsymbol{\theta})=\theta_{k}$ if $\psi_{i}^{j}$ falls
into the obliquity bin $k$, and $p(\psi_{i}^{j})\;\propto\;\sin\psi_{i}$. The
latter point comes from an examination of the employed prior to derive the
$\psi$ posteriors, namely a uniform prior on $\cos\psi$. It can be seen as a
manifestation of the isotropic assumption, that we recover with Eq. (5) by
setting isotropic distributions for the other angles, that is to say uniform
on $\cos i_{\rm p}$, $\cos i_{\star}$, and $\lambda$ (and not $\cos\lambda$,
as it is the projection of an angle on a plane). Since the obliquity values of
different systems are independent from each other, the total log-likelihood
$p(\mathcal{D}\;|\;\boldsymbol{\theta})$ for one set of $\boldsymbol{\theta}$
is the sum of the log-likelihoods for the individual systems, which are
computed using Eq. (11).

Again, we make use of emcee to set up the MCMC simulation, adjusting the
number of walkers and the burn-in phase according to the convergence of the
chains. Figure 6 shows the results, the top panel featuring the entire sample,
and the bottom one being limited to systems within the lowest tidal efficiency
bin (as in Sect. 3.2). The height of the bars is set by the medians of the
$\theta_{i}$ marginal PDFs, and the error bars are defined by their 68% HDIs.

The first striking result is that our reconstruction of the 3D obliquity
histogram for the whole population (Fig. 6, top panel) showcases an
significant peak around $90^{\circ}$. The preponderance of perpendicular
orbits spotlighted by Albrecht et al. (2021) seems to be a robust feature,
especially considering the fact that our sample is nearly four times larger.
It is worth emphasizing that this prominence of perpendicular systems cannot
be an artifact of our choice of prior function on $\psi$, as polar
architectures are actually penalized by the likelihood function (Eq. (11)). It
is also remarkable that the fraction of polar orbits within the entire
population on the one hand, and the pristine population (Fig. 6, bottom panel)
on the other hand, is roughly equivalent ($\sim 15$ %). This noteworthy
feature suggests that the mechanisms generating polar systems, either
primordial or evolutionary, are resilient against tidal realignment. These
mechanisms could be the result of spin–orbit resonances leading to such orbits
in a dynamically stable configuration.

Contrastingly, the fraction of aligned systems is significantly different
between the two samples ($\sim 3\sigma$), stressing again the role of tidal
effects in realigning planetary systems. The substantially larger number of
truly aligned systems compared to the low-efficiency distribution implies that
many systems end up aligned as a result of tidal interactions rather than
through formation processes. In fact, the relatively small fraction of aligned
orbits in a distribution of pristine systems unaffected by star–planet
interactions ($\sim 40$ %) somewhat reverses the classical picture that the
majority of planets smoothly form in the disk, keeping alignments intact. This
classical picture is further challenged by the very existence of antialigned
orbits (that might surprisingly be as numerous as 10 % of the pristine sample,
within $1\sigma$). The absence of flipped orbits in the entire population
indicates that tidal interactions play a definite role in quenching
antialigned architectures.

###  4.3 Reassessing the tidal efficiency trend

Figure 7: Same as Fig. 5, but with 3D obliquities for all systems. $\psi$ is
estimated for systems that only have a sky-projected measurement using the
procedure outlined in Sect. 4.1. We note the different vertical range for the
blue bar plot on the top of the figure as compared to Fig. 5. The bar plot is
built according to the procedure described in Sect. 4.3. The orange dashed
line on the bar plot is the best linear fit between the misalignment fraction
and the tidal efficiency, with the orange shaded area around it shown as its
$1\sigma$ envelope. The bar plot, along with its linear fit, is also
separately shown in Fig. 8 for clarity. Figure 8: Misalignment fraction as a
function of the tidal efficiency factor for all systems in our sample. Top:
misalignment fraction bar plot, reported from the top of Fig. 7. The best
linear fit relation (Eq. (12)) can be used as an estimator for the probability
that a system lies in a misaligned configuration. Bottom: residuals between
the medians of the misalignment fractions and the linear fit. Figure 9:
Estimates of $\psi$ as a function of $\lambda$ under the isotropic stellar
inclination assumption, for three different values of the orbital inclination.
Distributions for $\psi$ are computed as described in the main text, assuming
Gaussian distributions for $\lambda$ and $i_{\rm p}$ with respective standard
deviations of $5^{\circ}$ and $1^{\circ}$. Plotted values are the medians of
the $\psi$ distributions, with error bars set to their 68% HDIs.

Equipped with the estimated $\psi$ distributions (Sect. 4.1), we reconstruct
in Fig. 7 the tidal efficiency plot using 3D obliquities. In particular, we
reanalyze the fraction of misaligned systems (bar plot, top of Fig. 7, also
separately shown in Fig. 8 for clarity), which we determine this time using
the same procedure as for constructing the histogram in Fig. 6. We run a joint
MCMC simulation for all of the $\tau$ bins at once, the jump parameters being
the fraction of misaligned systems within each bin. In addition to using the
information of the full $\psi$ PDFs and alleviate the possible biases
resulting from a naive count of misaligned systems based on the $\psi$ medians
alone, this procedure takes into account the error bars on the $\tau$ values
as well, making our analysis more robust against the potential uncertainty
relative to falling into one of two adjacent tidal efficiency bins. The
heights of the blue bars are hence defined by the medians of the misalignment
fraction PDFs, and their 68% HDIs set the error bars.

First, we see that the number of misaligned systems is substantially higher in
all $\tau$ bins as compared to Fig. 5. This can be understood by examining our
procedure to estimate the true spin–orbit angle under the assumption of
isotropic stellar inclination, which may have revealed an important
observational bias. In Fig. 9, we show values of $\psi$ estimated using this
approach as a function of $\lambda$, for three different values of $i_{\rm
p}$. The true and sky-projected spin–orbit angles are similar around 90∘, but
the estimated $\psi$ deviates from $\lambda$ when the latter goes toward lower
or higher values, up to a bias of about 90∘ for $\lambda\sim 0^{\circ}$ or
$180^{\circ}$. This explains the global increase in misaligned systems in Fig.
7, as systems considered as “aligned” based on their sky-projected spin–orbit
angle ($\lambda<30^{\circ}$ in Fig. 5) are more likely to be misaligned based
on their estimated true spin–orbit angle (where $\psi\gtrsim 30^{\circ}$ for
transiting systems, Fig 9). This is exactly the case for HAT-P-33 b, for which
the sky-projected obliquity $\lambda$ could be interpreted as a surprisingly
aligned orbit (Fig. 5) despite the shallow stellar convective envelope (Fig.
2), but may be more expectedly misaligned in reality (Fig. 7). This bias is
actually even more accentuated for nontransiting systems ($i_{\rm
p}=10^{\circ}$ in Fig. 9), for which the true spin–orbit angle is almost
always estimated as polar whatever the value of $\lambda$. In a sense, Fig. 9
shows that all sky-projected methods for determining the spin–orbit angle
might be biased toward misaligned systems, even the ones not limited to
transiting systems (e.g., interferometry, Kraus et al. 2022).

Another important feature in Fig. 7 is the very high occurence rate of
misaligned orbits in the DREAM I sample. Noticeably, two thirds of the
surveyed systems are consistent with a polar architecture within $<2\sigma$,
strongly contrasting with the $\sim 15\%$ fraction of polar orbits we find for
the entire population of exoplanets with an obliquity measurement (Fig. 6).
Even though the latter value should be considered with caution because of the
various selection biases (e.g., hot Jupiters are by far the most represented
category of planets with a characterized spin–orbit angle, Fig. 1), this
dichotomy might be indicative of disruptive dynamical processes that
particularly affects planets spanning the edge of the desert and the breadth
of the savanna. Consequently, high-eccentricity migration scenarios, which
naturally lead to highly misaligned configurations, may be the predominant
process to bring these planets from their birthplace to their current close-in
orbit.

Furthermore, we notice that the downward misalignment trend with increasing
tidal efficiencies identified in Sect. 3.2 persists. Motivated by the visible
robustness of this result, we quantify this trend using the Spearman and
Pearson $r$ statistical tests, which respectively measure rank and linear
correlations. To this end, misalignment fractions are randomly sampled from
their PDFs and used to compute a distribution of Spearman and Pearson
coefficients. The results are presented in Fig. 10, which shows a clear
anticorrelation for both coefficients. The misalignment fraction not only
decreases monotonically with an increasing log-tidal efficiency (Spearman
$r=-0.79^{+0.07}_{-0.18}$), but the two quantities are moreover linearly
anticorrelated (Pearson $r=-0.78^{+0.09}_{-0.13}$). The two coefficients are
in fact consistent with $-1$ (perfect anticorrelation) within $<2\sigma$.
Defining the null hypothesis as the absence of, or a positive correlation
($r\geqslant 0$), the respective $p$-values are $0.080\%$ and $0.003\%$,
highlighting again the robustness of the result. The best linear fit between
the misalignment fraction and the tidal efficiency factor, along with its
$1\sigma$ uncertainty envelope, is shown on top of Fig. 7, for which we
provide here an explicit expression

$\theta\left(\tau\right)=\alpha\times\log_{10}\left(\tau\right)+\beta,$ (12)
$\alpha=-0.0417^{+0.0093}_{-0.0098},\;\beta=-0.0706^{+0.1550}_{-0.1712},$

$\theta$ being the misalignment fraction. The $\theta\left(\tau\right)$ bar
plot, as well as its linear fit, can be seen in isolation in Fig. 8, which
further shows residuals satisfactorily consistent with zero within $1\sigma$
for all tidal efficiency bins. Equation (12) can be conveniently used for
future studies, as $\theta\left(\tau\right)$ can be interpreted as the
probability that a system with a tidal efficiency factor $\tau$ is misaligned.
We note that this relationship is satisfyingly consistent with the threshold
we set for considering tidal realignment processes (Sect. 3.2), since
$\theta\left(\tau=10^{-15}\right)\simeq 50\%$.

We highlight that the conclusions we put forward in this section do not depend
on the threshold of $30^{\circ}$ we set to define misaligned orbits. By
conducting the same analysis for thresholds of $20^{\circ}$ and then
$10^{\circ}$, we expectedly see increasingly higher misalignment fractions
within all the tidal efficiency bins. The downward misalignment trend remains
nevertheless intact, with Spearman and Pearson $r$ coefficients still
consistent with a perfect anticorrelation within $\sim 1.5\sigma$ and slightly
higher $p$-values (yet below 0.5%). The only substantial change while
considering a different misalignment threshold is the value of $\beta$ in Eq.
(12), but not its slope. The values of $\alpha$ corresponding to the different
thresholds are in fact consistent with each other within their error bars. We
thus recommend keeping in mind our conservative definition of a misaligned
orbit in future works, should Eq. (12) be employed.

Figure 10: Distribution of Pearson (blue) and Spearman (orange) $r$
correlation coefficients between misalignment fractions (blue bars on top of
Fig. 7) and log-tidal efficiencies (Eq. (4)). Misalignment fractions are
randomly sampled from their PDFs to generate this distribution (Sect. 4.3).

##  5 Summary and conclusion

In this work, we conducted a global analysis of the distribution of known
spin–orbit angles. We collected a reliable database of exoplanet parameters
after a critical review of the sources, and complemented it with the obliquity
measurements derived in DREAM I (Bourrier et al. 2023). The resulting large
sample of 196 spin–orbit angles (App. A) allowed us to draw a detailed picture
of the orbital architectures of close-in exoplanets. We confirmed the strong
correlation between misalignment fractions and the weakness of star–planet
tidal interactions by refining trends with individual parameters controlling
the intensity of tides like the stellar effective temperature, the planet-to-
star mass ratio, or the scaled separation, in line with Triaud (2018) and
Albrecht et al. (2022), for example.

Going a step further, we devised a tidal efficiency parameter $\tau$ combining
several pertinent planetary, stellar, and orbital quantities (Eq. (4),
inspired by Albrecht et al. 2012b) and made use of it to corroborate our
conclusions. While the aforementioned trends with individual parameters can
only be seen by examining relevant sub-populations in isolation, the
$\lambda-\tau$ correlation is explicit for the whole exoplanet sample
irrespective of their individual specificities: systems where tidal forces are
weaker tend to be more misaligned and vice-versa, strongly suggesting that
tidal realignment drives the global distribution of spin–orbit angles. We can
thus propose for future studies the use of this tidal efficiency parameter as
a criterion to assess the impact of tidal realignment, calculating it based on
the explicit recipe we lay out (Eq. (4), Table 1).

Being aware that our conclusions might be biased by sky-projection effects, we
reconstructed the distribution of true, 3D, obliquities, under the assumption
of isotropic stellar inclinations. In doing so, the anticorrelation of
misalignment fractions $\theta$ with increasing tidal efficiencies $\tau$
persists, proving the robustness of this result. Indeed, statistical tests
strongly support a linear anticorrelation between the two quantities, and we
provide an explicit formula for the best fit ($\theta\left(\tau\right)$, Eq.
(12)). This expression can be conveniently used for estimating the probability
that a system is misaligned, which will prove useful in particular for setting
the distribution of misalignments in global population studies of close-in
exoplanets.

We also showed that misaligned orbits do not randomly span the entire possible
range of obliquities. Rather, the distribution of $\psi$ favors polar
architectures, a result first identified by Albrecht et al. (2021), which we
confirm with a sample nearly four times larger. On the other hand, pristine
systems that do not feel tidal interactions may showcase a flatter
distribution around $90^{\circ}$ than the entire population, implying that
polar orbits are a stable outcome of dynamical processes that can hardly be
aligned back by tides.

Our results suggest that exoplanet systems may be born with a broad, non-
unifrom, range of spin–orbit angles, only to be realigned thanks to tides if
the physical properties of the system allow it. This interpretation is
substantiated by comparing the obliquity distribution for all systems and for
the pristine ones unaffected by tides, the former showing a stronger
preference for aligned orbits. It is all the more interesting to put these
conclusions in perspective of the Neptune desert, for which evidence has been
put forward that it may be shaped by secular disruptive processes (e.g., Attia
et al. 2021). Such mechanisms, like the Kozai–Lidov resonance, naturally
produce highly misaligned orbits that can only be damped if tidal forces are
strong enough. In fact, the surveyed sample presented in DREAM I, cherry-
picked to span the rims of the desert and the breadth of the savanna,
showcases a high occurence rate of considerably misaligned orbits, polar or
retrograde for most of them, further hinting that this class of planets may be
particularly affected by these disruptive processes. Additional observational
efforts on these exciting regions of the parameter space will be pivotal to
shed light on the possible history of the planets populating them. Besides
expanding the current obliquity sample, we recommend parallel searches for
outer companions that could have induced high-eccentricity migration, using
both radial velocities and direct imaging.

###### Acknowledgements.

We thank A.H.M.J. Triaud for bringing the question of sky-projection effects
into our mind and inciting us to look deeper into this matter. This work has
been carried out within the framework of the NCCR PlanetS supported by the
Swiss National Science Foundation under grants 51NF40$\\_$182901 and
51NF40$\\_$205606\. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (project Spice Dune, grant agreement No 947634; grant
agreement No 730890). This material reflects only the authors’ views and the
Commission is not liable for any use that may be made of the information
contained therein.

## References

  * Addison et al. (2014) Addison, B. C., Tinney, C. G., Wright, D. J., & Bayliss, D. 2014, ApJ, 792, 112
  * Addison et al. (2016) Addison, B. C., Tinney, C. G., Wright, D. J., & Bayliss, D. 2016, ApJ, 823, 29
  * Addison et al. (2018) Addison, B. C., Wang, S., Johnson, M. C., et al. 2018, AJ, 156, 197
  * Ahlers et al. (2015) Ahlers, J. P., Barnes, J. W., & Barnes, R. 2015, ApJ, 814, 67
  * Ahlers et al. (2020) Ahlers, J. P., Kruse, E., Colón, K. D., et al. 2020, ApJ, 888, 63
  * Ahlers et al. (2014) Ahlers, J. P., Seubert, S. A., & Barnes, J. W. 2014, ApJ, 786, 131
  * Albrecht et al. (2012a) Albrecht, S., Winn, J. N., Butler, R. P., et al. 2012a, ApJ, 744, 189
  * Albrecht et al. (2011) Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2011, ApJ, 738, 50
  * Albrecht et al. (2012b) Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012b, ApJ, 757, 18
  * Albrecht et al. (2013) Albrecht, S., Winn, J. N., Marcy, G. W., et al. 2013, ApJ, 771, 11
  * Albrecht et al. (2022) Albrecht, S. H., Dawson, R. I., & Winn, J. N. 2022, PASP, 134, 082001
  * Albrecht et al. (2021) Albrecht, S. H., Marcussen, M. L., Winn, J. N., Dawson, R. I., & Knudstrup, E. 2021, ApJ, 916, L1
  * Alexander et al. (1976) Alexander, M. E., Chau, W. Y., & Henriksen, R. N. 1976, ApJ, 204, 879
  * Allart et al. (2020) Allart, R., Pino, L., Lovis, C., et al. 2020, A&A, 644, A155
  * Anderson et al. (2018a) Anderson, D. R., Bouchy, F., Brown, D. J. A., et al. 2018a, arXiv e-prints, arXiv:1809.07709
  * Anderson et al. (2018b) Anderson, D. R., Bouchy, F., Brown, D. J. A., et al. 2018b, arXiv e-prints, arXiv:1812.09264
  * Anderson et al. (2014) Anderson, D. R., Brown, D. J. A., Collier Cameron, A., et al. 2014, arXiv e-prints, arXiv:1410.3449
  * Anderson et al. (2011) Anderson, D. R., Collier Cameron, A., Gillon, M., et al. 2011, A&A, 534, A16
  * Anderson et al. (2015a) Anderson, D. R., Collier Cameron, A., Hellier, C., et al. 2015a, A&A, 575, A61
  * Anderson et al. (2015b) Anderson, D. R., Triaud, A. H. M. J., Turner, O. D., et al. 2015b, ApJ, 800, L9
  * Attia et al. (2021) Attia, O., Bourrier, V., Eggenberger, P., et al. 2021, A&A, 647, A40
  * Barker (2020) Barker, A. J. 2020, MNRAS, 498, 2270
  * Barker & Ogilvie (2010) Barker, A. J. & Ogilvie, G. I. 2010, MNRAS, 404, 1849
  * Barnes et al. (2015) Barnes, J. W., Ahlers, J. P., Seubert, S. A., & Relles, H. M. 2015, ApJ, 808, L38
  * Beaugé & Nesvorný (2013) Beaugé, C. & Nesvorný, D. 2013, ApJ, 763, 12
  * Bello-Arufe et al. (2022) Bello-Arufe, A., Cabot, S. H. C., Mendonça, J. M., Buchhave, L. A., & Rathcke, A. D. 2022, AJ, 163, 96
  * Benomar et al. (2014) Benomar, O., Masuda, K., Shibahashi, H., & Suto, Y. 2014, PASJ, 66, 94
  * Borucki & Summers (1984) Borucki, W. J. & Summers, A. L. 1984, Icarus, 58, 121
  * Bouchy et al. (2008) Bouchy, F., Queloz, D., Deleuil, M., et al. 2008, A&A, 482, L25
  * Bourrier et al. (2023) Bourrier, V., Attia, O., Mallonn, M., et al. 2023, A&A, 669, A63
  * Bourrier et al. (2017) Bourrier, V., Cegla, H. M., Lovis, C., & Wyttenbach, A. 2017, A&A, 599, A33
  * Bourrier et al. (2020) Bourrier, V., Ehrenreich, D., Lendl, M., et al. 2020, A&A, 635, A205
  * Bourrier & Hébrard (2014) Bourrier, V. & Hébrard, G. 2014, A&A, 569, A65
  * Bourrier et al. (2018) Bourrier, V., Lovis, C., Beust, H., et al. 2018, Nature, 553, 477
  * Bourrier et al. (2021) Bourrier, V., Lovis, C., Cretignier, M., et al. 2021, A&A, 654, A152
  * Bourrier et al. (2022) Bourrier, V., Zapatero Osorio, M. R., Allart, R., et al. 2022, A&A, 663, A160
  * Brothwell et al. (2014) Brothwell, R. D., Watson, C. A., Hébrard, G., et al. 2014, MNRAS, 440, 3392
  * Brown et al. (2012a) Brown, D. J. A., Cameron, A. C., Anderson, D. R., et al. 2012a, MNRAS, 423, 1503
  * Brown et al. (2012b) Brown, D. J. A., Collier Cameron, A., Díaz, R. F., et al. 2012b, ApJ, 760, 139
  * Brown et al. (2017) Brown, D. J. A., Triaud, A. H. M. J., Doyle, A. P., et al. 2017, MNRAS, 464, 810
  * Cabot et al. (2021) Cabot, S. H. C., Bello-Arufe, A., Mendonça, J. M., et al. 2021, AJ, 162, 218
  * Carone et al. (2021) Carone, L., Mollière, P., Zhou, Y., et al. 2021, A&A, 646, A168
  * Casasayas-Barris et al. (2017) Casasayas-Barris, N., Palle, E., Nowak, G., et al. 2017, A&A, 608, A135
  * Casasayas-Barris et al. (2021) Casasayas-Barris, N., Palle, E., Stangret, M., et al. 2021, A&A, 647, A26
  * Cegla et al. (2016) Cegla, H. M., Lovis, C., Bourrier, V., et al. 2016, A&A, 588, A127
  * Chen et al. (2020a) Chen, G., Casasayas-Barris, N., Pallé, E., et al. 2020a, A&A, 642, A54
  * Chen et al. (2020b) Chen, G., Casasayas-Barris, N., Pallé, E., et al. 2020b, A&A, 635, A171
  * Covino et al. (2013) Covino, E., Esposito, M., Barbieri, M., et al. 2013, A&A, 554, A28
  * Crouzet et al. (2017) Crouzet, N., McCullough, P. R., Long, D., et al. 2017, AJ, 153, 94
  * Dai et al. (2020) Dai, F., Roy, A., Fulton, B., et al. 2020, AJ, 160, 193
  * Damasso et al. (2015a) Damasso, M., Biazzo, K., Bonomo, A. S., et al. 2015a, A&A, 575, A111
  * Damasso et al. (2015b) Damasso, M., Esposito, M., Nascimbeni, V., et al. 2015b, A&A, 581, L6
  * Dawson (2014) Dawson, R. I. 2014, ApJ, 790, L31
  * Deline et al. (2022) Deline, A., Hooton, M. J., Lendl, M., et al. 2022, A&A, 659, A74
  * Désert et al. (2011) Désert, J.-M., Charbonneau, D., Demory, B.-O., et al. 2011, ApJS, 197, 14
  * Dong et al. (2022) Dong, J., Huang, C. X., Zhou, G., et al. 2022, ApJ, 926, L7
  * Dorval et al. (2020) Dorval, P., Talens, G. J. J., Otten, G. P. P. L., et al. 2020, A&A, 635, A60
  * Doyle et al. (1984) Doyle, L. R., Wilcox, T. J., & Lorre, J. J. 1984, ApJ, 287, 307
  * Eastman et al. (2016) Eastman, J. D., Beatty, T. G., Siverd, R. J., et al. 2016, AJ, 151, 45
  * Eggenberger et al. (2008) Eggenberger, P., Meynet, G., Maeder, A., et al. 2008, Ap&SS, 316, 43
  * Ehrenreich et al. (2020) Ehrenreich, D., Lovis, C., Allart, R., et al. 2020, Nature, 580, 597
  * Esposito et al. (2017) Esposito, M., Covino, E., Desidera, S., et al. 2017, A&A, 601, A53
  * Esposito et al. (2014) Esposito, M., Covino, E., Mancini, L., et al. 2014, A&A, 564, L13
  * Fabrycky & Tremaine (2007) Fabrycky, D. & Tremaine, S. 2007, ApJ, 669, 1298
  * Fabrycky & Winn (2009) Fabrycky, D. C. & Winn, J. N. 2009, ApJ, 696, 1230
  * Feinstein et al. (2021) Feinstein, A. D., Montet, B. T., Johnson, M. C., et al. 2021, AJ, 162, 213
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Foucart & Lai (2011) Foucart, F. & Lai, D. 2011, MNRAS, 412, 2799
  * Fulton et al. (2013) Fulton, B. J., Howard, A. W., Winn, J. N., et al. 2013, ApJ, 772, 80
  * Gandolfi et al. (2012) Gandolfi, D., Collier Cameron, A., Endl, M., et al. 2012, A&A, 543, L5
  * Guenther et al. (2012) Guenther, E. W., Díaz, R. F., Gazzano, J. C., et al. 2012, A&A, 537, A136
  * Hansen (2012) Hansen, B. M. S. 2012, ApJ, 757, 6
  * Hébrard et al. (2010) Hébrard, G., Désert, J.-M., Díaz, R. F., et al. 2010, A&A, 516, A95
  * Hébrard et al. (2011a) Hébrard, G., Ehrenreich, D., Bouchy, F., et al. 2011a, A&A, 527, L11
  * Hébrard et al. (2011b) Hébrard, G., Evans, T. M., Alonso, R., et al. 2011b, A&A, 533, A130
  * Heitzmann et al. (2021) Heitzmann, A., Zhou, G., Quinn, S. N., et al. 2021, ApJ, 922, L1
  * Hirano et al. (2020a) Hirano, T., Gaidos, E., Winn, J. N., et al. 2020a, ApJ, 890, L27
  * Hirano et al. (2020b) Hirano, T., Krishnamurthy, V., Gaidos, E., et al. 2020b, ApJ, 899, L13
  * Hirano et al. (2012) Hirano, T., Narita, N., Sato, B., et al. 2012, ApJ, 759, L36
  * Hirano et al. (2011) Hirano, T., Narita, N., Sato, B., et al. 2011, PASJ, 63, L57
  * Hirano et al. (2016) Hirano, T., Nowak, G., Kuzuhara, M., et al. 2016, ApJ, 825, 53
  * Hjorth et al. (2021) Hjorth, M., Albrecht, S., Hirano, T., et al. 2021, Proceedings of the National Academy of Science, 118, 2017418118
  * Hooton et al. (2022) Hooton, M. J., Hoyer, S., Kitzmann, D., et al. 2022, A&A, 658, A75
  * Howarth & Morello (2017) Howarth, I. D. & Morello, G. 2017, MNRAS, 470, 932
  * Johnson et al. (2009) Johnson, J. A., Winn, J. N., Albrecht, S., et al. 2009, PASP, 121, 1104
  * Johnson et al. (2011) Johnson, J. A., Winn, J. N., Bakos, G. Á., et al. 2011, ApJ, 735, 24
  * Johnson et al. (2008) Johnson, J. A., Winn, J. N., Narita, N., et al. 2008, ApJ, 686, 649
  * Johnson et al. (2017) Johnson, M. C., Cochran, W. D., Addison, B. C., Tinney, C. G., & Wright, D. J. 2017, AJ, 154, 137
  * Johnson et al. (2022) Johnson, M. C., David, T. J., Petigura, E. A., et al. 2022, AJ, 163, 247
  * Kippenhahn et al. (2013) Kippenhahn, R., Weigert, A., & Weiss, A. 2013, Stellar Structure and Evolution
  * Knudstrup & Albrecht (2022) Knudstrup, E. & Albrecht, S. H. 2022, A&A, 660, A99
  * Knudstrup et al. (2022) Knudstrup, E., Serrano, L. M., Gandolfi, D., et al. 2022, A&A, 667, A22
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591
  * Kraft (1967) Kraft, R. P. 1967, ApJ, 150, 551
  * Kraus et al. (2022) Kraus, S., Mortimer, D., Chhabra, S., et al. 2022, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 12183, Optical and Infrared Interferometry and Imaging VIII, ed. A. Mérand, S. Sallum, & J. Sanchez-Bermudez, 121831S
  * Kunovac Hodžić et al. (2021) Kunovac Hodžić, V., Triaud, A. H. M. J., Cegla, H. M., Chaplin, W. J., & Davies, G. R. 2021, MNRAS, 502, 2893
  * Lai (2012) Lai, D. 2012, MNRAS, 423, 486
  * Lecavelier des Etangs (2007) Lecavelier des Etangs, A. 2007, A&A, 461, 1185
  * Lidov (1962) Lidov, M. L. 1962, Planet. Space Sci., 9, 719
  * Lin et al. (1996) Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996, Nature, 380, 606
  * Loeillet et al. (2008) Loeillet, B., Shporer, A., Bouchy, F., et al. 2008, A&A, 481, 529
  * Louden et al. (2021) Louden, E. M., Winn, J. N., Petigura, E. A., et al. 2021, AJ, 161, 68
  * Lund et al. (2017) Lund, M. B., Rodriguez, J. E., Zhou, G., et al. 2017, AJ, 154, 194
  * Lundkvist et al. (2016) Lundkvist, M. S., Kjeldsen, H., Albrecht, S., et al. 2016, Nature Communications, 7, 11201
  * Luque et al. (2020) Luque, R., Casasayas-Barris, N., Parviainen, H., et al. 2020, A&A, 642, A50
  * Mancini et al. (2015) Mancini, L., Esposito, M., Covino, E., et al. 2015, A&A, 579, A136
  * Mancini et al. (2018) Mancini, L., Esposito, M., Covino, E., et al. 2018, A&A, 613, A41
  * Mann et al. (2020) Mann, A. W., Johnson, M. C., Vanderburg, A., et al. 2020, AJ, 160, 179
  * Masuda & Winn (2020) Masuda, K. & Winn, J. N. 2020, AJ, 159, 81
  * Mayor & Queloz (1995) Mayor, M. & Queloz, D. 1995, Nature, 378, 355
  * Mazeh et al. (2016) Mazeh, T., Holczer, T., & Faigler, S. 2016, A&A, 589, A75
  * Mazeh et al. (2015) Mazeh, T., Perets, H. B., McQuillan, A., & Goldstein, E. S. 2015, ApJ, 801, 3
  * McLaughlin (1924) McLaughlin, D. B. 1924, ApJ, 60, 22
  * Miller et al. (2010) Miller, G. R. M., Collier Cameron, A., Simpson, E. K., et al. 2010, A&A, 523, A52
  * Mohler-Fischer et al. (2013) Mohler-Fischer, M., Mancini, L., Hartman, J. D., et al. 2013, A&A, 558, A55
  * Morton & Winn (2014) Morton, T. D. & Winn, J. N. 2014, ApJ, 796, 47
  * Mounzer et al. (2022) Mounzer, D., Lovis, C., Seidel, J. V., et al. 2022, A&A, 668, A1
  * Moutou et al. (2011) Moutou, C., Díaz, R. F., Udry, S., et al. 2011, A&A, 533, A113
  * Močnik et al. (2017) Močnik, T., Hellier, C., Anderson, D. R., Clark, B. J. M., & Southworth, J. 2017, MNRAS, 469, 1622
  * Muñoz & Perets (2018) Muñoz, D. J. & Perets, H. B. 2018, AJ, 156, 253
  * Naoz et al. (2011) Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. 2011, Nature, 473, 187
  * Narita et al. (2007) Narita, N., Enya, K., Sato, B., et al. 2007, PASJ, 59, 763
  * Narita et al. (2010a) Narita, N., Hirano, T., Sanchis-Ojeda, R., et al. 2010a, PASJ, 62, L61
  * Narita et al. (2009) Narita, N., Hirano, T., Sato, B., et al. 2009, PASJ, 61, 991
  * Narita et al. (2010b) Narita, N., Sato, B., Hirano, T., et al. 2010b, PASJ, 62, 653
  * Nelson et al. (2017) Nelson, B. E., Ford, E. B., & Rasio, F. A. 2017, AJ, 154, 106
  * Nesvorný et al. (2013) Nesvorný, D., Kipping, D., Terrell, D., et al. 2013, ApJ, 777, 3
  * Neveu-VanMalle et al. (2014) Neveu-VanMalle, M., Queloz, D., Anderson, D. R., et al. 2014, A&A, 572, A49
  * Petrovich et al. (2020) Petrovich, C., Muñoz, D. J., Kratter, K. M., & Malhotra, R. 2020, ApJ, 902, L5
  * Pont et al. (2010) Pont, F., Endl, M., Cochran, W. D., et al. 2010, MNRAS, 402, L1
  * Privitera et al. (2016) Privitera, G., Meynet, G., Eggenberger, P., et al. 2016, A&A, 593, A128
  * Rao et al. (2018) Rao, S., Meynet, G., Eggenberger, P., et al. 2018, A&A, 618, A18
  * Rasio et al. (1996) Rasio, F. A., Tout, C. A., Lubow, S. H., & Livio, M. 1996, ApJ, 470, 1187
  * Rice et al. (2021) Rice, M., Wang, S., Howard, A. W., et al. 2021, AJ, 162, 182
  * Rodriguez et al. (2019) Rodriguez, J. E., Eastman, J. D., Zhou, G., et al. 2019, AJ, 158, 197
  * Rodríguez Martínez et al. (2020) Rodríguez Martínez, R., Gaudi, B. S., Rodriguez, J. E., et al. 2020, AJ, 160, 111
  * Romanova et al. (2021) Romanova, M. M., Koldoba, A. V., Ustyugova, G. V., et al. 2021, MNRAS, 506, 372
  * Rossiter (1924) Rossiter, R. A. 1924, ApJ, 60, 15
  * Sanchis-Ojeda et al. (2012) Sanchis-Ojeda, R., Fabrycky, D. C., Winn, J. N., et al. 2012, Nature, 487, 449
  * Sanchis-Ojeda et al. (2015) Sanchis-Ojeda, R., Winn, J. N., Dai, F., et al. 2015, ApJ, 812, L11
  * Santerne et al. (2014) Santerne, A., Hébrard, G., Deleuil, M., et al. 2014, A&A, 571, A37
  * Santerne et al. (2016) Santerne, A., Hébrard, G., Lillo-Box, J., et al. 2016, ApJ, 824, 55
  * Sedaghati et al. (2021) Sedaghati, E., MacDonald, R. J., Casasayas-Barris, N., et al. 2021, MNRAS, 505, 435
  * Simpson et al. (2011) Simpson, E. K., Pollacco, D., Cameron, A. C., et al. 2011, MNRAS, 414, 3023
  * Siverd et al. (2012) Siverd, R. J., Beatty, T. G., Pepper, J., et al. 2012, ApJ, 761, 123
  * Siverd et al. (2018) Siverd, R. J., Collins, K. A., Zhou, G., et al. 2018, AJ, 155, 35
  * Southworth (2011) Southworth, J. 2011, MNRAS, 417, 2166
  * Southworth et al. (2016) Southworth, J., Tregloan-Reed, J., Andersen, M. I., et al. 2016, MNRAS, 457, 4205
  * Stangret et al. (2021) Stangret, M., Pallé, E., Casasayas-Barris, N., et al. 2021, A&A, 654, A73
  * Stefansson et al. (2020) Stefansson, G., Mahadevan, S., Maney, M., et al. 2020, AJ, 160, 192
  * Stefànsson et al. (2022) Stefànsson, G., Mahadevan, S., Petrovich, C., et al. 2022, ApJ, 931, L15
  * Stephan et al. (2022) Stephan, A. P., Wang, J., Cauley, P. W., et al. 2022, ApJ, 931, 111
  * Temple et al. (2017) Temple, L. Y., Hellier, C., Albrow, M. D., et al. 2017, MNRAS, 471, 2743
  * Temple et al. (2018) Temple, L. Y., Hellier, C., Almleaky, Y., et al. 2018, MNRAS, 480, 5307
  * Temple et al. (2019a) Temple, L. Y., Hellier, C., Almleaky, Y., et al. 2019a, AJ, 157, 141
  * Temple et al. (2019b) Temple, L. Y., Hellier, C., Anderson, D. R., et al. 2019b, MNRAS, 490, 2467
  * Tregloan-Reed et al. (2015) Tregloan-Reed, J., Southworth, J., Burgdorf, M., et al. 2015, MNRAS, 450, 1760
  * Triaud (2018) Triaud, A. H. M. J. 2018, The Rossiter-McLaughlin Effect in Exoplanet Research, 2
  * Triaud et al. (2010) Triaud, A. H. M. J., Collier Cameron, A., Queloz, D., et al. 2010, A&A, 524, A25
  * Triaud et al. (2015) Triaud, A. H. M. J., Gillon, M., Ehrenreich, D., et al. 2015, MNRAS, 450, 2279
  * Triaud et al. (2017) Triaud, A. H. M. J., Neveu-VanMalle, M., Lendl, M., et al. 2017, MNRAS, 467, 1714
  * Triaud et al. (2009) Triaud, A. H. M. J., Queloz, D., Bouchy, F., et al. 2009, A&A, 506, 377
  * Valsecchi & Rasio (2014) Valsecchi, F. & Rasio, F. A. 2014, ApJ, 786, 102
  * Wang et al. (2018) Wang, S., Addison, B., Fischer, D. A., et al. 2018, AJ, 155, 70
  * Wang et al. (2021) Wang, S., Winn, J. N., Addison, B. C., et al. 2021, AJ, 162, 50
  * Wang et al. (2022) Wang, X.-Y., Rice, M., Wang, S., et al. 2022, ApJ, 926, L8
  * Watanabe et al. (2022) Watanabe, N., Narita, N., Palle, E., et al. 2022, MNRAS, 512, 4404
  * Winn et al. (2010a) Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010a, ApJ, 718, L145
  * Winn & Fabrycky (2015) Winn, J. N. & Fabrycky, D. C. 2015, ARA&A, 53, 409
  * Winn et al. (2007) Winn, J. N., Holman, M. J., Henry, G. W., et al. 2007, AJ, 133, 1828
  * Winn et al. (2011) Winn, J. N., Howard, A. W., Johnson, J. A., et al. 2011, AJ, 141, 63
  * Winn et al. (2010b) Winn, J. N., Johnson, J. A., Howard, A. W., et al. 2010b, ApJ, 718, 575
  * Winn et al. (2008) Winn, J. N., Johnson, J. A., Narita, N., et al. 2008, ApJ, 682, 1283
  * Wirth et al. (2021) Wirth, C. P., Zhou, G., Quinn, S. N., et al. 2021, ApJ, 917, L34
  * Wong et al. (2021) Wong, I., Shporer, A., Zhou, G., et al. 2021, AJ, 162, 256
  * Wyttenbach et al. (2017) Wyttenbach, A., Lovis, C., Ehrenreich, D., et al. 2017, A&A, 602, A36
  * Yu et al. (2018) Yu, L., Rodriguez, J. E., Eastman, J. D., et al. 2018, AJ, 156, 127
  * Zahn (1966) Zahn, J. P. 1966, Annales d’Astrophysique, 29, 489
  * Zahn (1977) Zahn, J. P. 1977, A&A, 57, 383
  * Zahn (1989) Zahn, J. P. 1989, A&A, 220, 112
  * Zanazzi & Lai (2018) Zanazzi, J. J. & Lai, D. 2018, MNRAS, 478, 835
  * Zhou et al. (2019a) Zhou, G., Bakos, G. Á., Bayliss, D., et al. 2019a, AJ, 157, 31
  * Zhou et al. (2015) Zhou, G., Bayliss, D., Hartman, J. D., et al. 2015, ApJ, 814, L16
  * Zhou et al. (2019b) Zhou, G., Huang, C. X., Bakos, G. Á., et al. 2019b, AJ, 158, 141
  * Zhou et al. (2016a) Zhou, G., Latham, D. W., Bieryla, A., et al. 2016a, MNRAS, 460, 3376
  * Zhou et al. (2016b) Zhou, G., Rodriguez, J. E., Collins, K. A., et al. 2016b, AJ, 152, 136
  * Zhou et al. (2020) Zhou, G., Winn, J. N., Newton, E. R., et al. 2020, ApJ, 892, L21

##  Appendix A Spin–orbit angles used in this work

Table 2: Spin–orbit angle values (projected in the sky plane and 3D) used in
our analysis along with their sources.

Planet name | Projected spin–orbit angle | 3D spin–orbit angle
---|---|---
|  $\lambda$ (deg) | Source |  $\psi$ (deg) | Source
55 Cnc e | $72.4^{+12.7}_{-11.5}$ | Bourrier & Hébrard (2014) | $72.7^{+12.2}_{-12.7}$ | This work
AU Mic b | $-4.7^{+6.8}_{-6.4}$ | Hirano et al. (2020b) | $9.2^{+4.3}_{-5.3}$ | This work
CoRoT-1 b | $77.0^{+11.0}_{-11.0}$ | Pont et al. (2010) | — | —
CoRoT-11 b | $0.1^{+2.6}_{-2.6}$ | Gandolfi et al. (2012) | — | —
CoRoT-18 b | $-10.0^{+20.0}_{-20.0}$ | Hébrard et al. (2011b) | $20.0^{+20.0}_{-20.0}$ | Hébrard et al. (2011b)
CoRoT-19 b | $-52.0^{+27.0}_{-22.0}$ | Guenther et al. (2012) | — | —
CoRoT-2 b | $7.2^{+4.5}_{-4.5}$ | Bouchy et al. (2008) | $10.7^{+4.3}_{-5.1}$ | This work
CoRoT-3 b | $-37.6^{+22.3}_{-10.0}$ | Triaud et al. (2009) | — | —
DS Tuc b | $2.9^{+0.9}_{-0.9}$ | Zhou et al. (2020) | $14.2^{+2.9}_{-3.6}$ | This work
GJ 3470 b | $101.0^{+29.0}_{-14.0}$ | Stefànsson et al. (2022) | $97.0^{+16.0}_{-11.0}$ | Stefànsson et al. (2022)
GJ 436 b | $114.0^{+23.0}_{-17.0}$ | Bourrier et al. (2022) | $103.0^{+13.0}_{-12.0}$ | Bourrier et al. (2022)
HAT-P-1 b | $3.7^{+2.1}_{-2.1}$ | Johnson et al. (2008) | — | —
HAT-P-11 b | $133.9^{+7.1}_{-8.3}$ | Bourrier et al. (2023) | $104.9^{+8.6}_{-9.1}$ | Bourrier et al. (2023)
HAT-P-12 b | $-54.0^{+41.0}_{-13.0}$ | Mancini et al. (2018) | — | —
HAT-P-13 b | $1.9^{+8.6}_{-8.6}$ | Winn et al. (2010b) | — | —
HAT-P-14 b | $189.1^{+5.1}_{-5.1}$ | Winn et al. (2011) | — | —
HAT-P-16 b | $-2.0^{+55.0}_{-46.0}$ | Albrecht et al. (2012b) | — | —
HAT-P-17 b | $19.0^{+14.0}_{-16.0}$ | Fulton et al. (2013) | — | —
HAT-P-18 b | $132.0^{+15.0}_{-15.0}$ | Esposito et al. (2014) | — | —
HAT-P-2 b | $0.2^{+12.2}_{-12.5}$ | Loeillet et al. (2008) | — | —
HAT-P-20 b | $-8.0^{+6.9}_{-6.9}$ | Esposito et al. (2017) | $36.0^{+10.0}_{-12.0}$ | Esposito et al. (2017)
HAT-P-22 b | $-2.1^{+3.0}_{-3.0}$ | Mancini et al. (2018) | $24.0^{+18.0}_{-18.0}$ | Mancini et al. (2018)
HAT-P-23 b | $15.0^{+22.0}_{-22.0}$ | Moutou et al. (2011) | — | —
HAT-P-24 b | $20.0^{+16.0}_{-16.0}$ | Albrecht et al. (2012b) | — | —
HAT-P-27 b | $24.2^{+76.0}_{-44.5}$ | Brown et al. (2012b) | — | —
HAT-P-3 b | $-25.3^{+29.4}_{-22.8}$ | Bourrier et al. (2023) | $75.7^{+8.5}_{-7.9}$ | Bourrier et al. (2023)
HAT-P-30 b | $73.5^{+9.0}_{-9.0}$ | Johnson et al. (2011) | — | —
HAT-P-32 b | $85.0^{+1.5}_{-1.5}$ | Albrecht et al. (2012b) | — | —
HAT-P-33 b | $-5.9^{+4.1}_{-4.1}$ | Bourrier et al. (2023) | — | —
HAT-P-34 b | $0.0^{+14.0}_{-14.0}$ | Albrecht et al. (2012b) | — | —
HAT-P-36 b | $-14.0^{+18.0}_{-18.0}$ | Mancini et al. (2015) | $0.0^{+63.0}_{-63.0}$ | Mancini et al. (2015)
HAT-P-4 b | $-4.9^{+11.9}_{-11.9}$ | Winn et al. (2011) | — | —
HAT-P-41 b | $-22.1^{+0.8}_{-6.0}$ | Johnson et al. (2017) | — | —
HAT-P-49 b | $-97.7^{+1.8}_{-1.8}$ | Bourrier et al. (2023) | — | —
HAT-P-56 b | $8.0^{+2.0}_{-2.0}$ | Zhou et al. (2016a) | — | —
HAT-P-6 b | $165.0^{+6.0}_{-6.0}$ | Albrecht et al. (2012b) | — | —
HAT-P-69 b | $30.3^{+6.1}_{-7.3}$ | Zhou et al. (2019b) | — | —
HAT-P-7 b | $220.3^{+8.2}_{-9.3}$ | Benomar et al. (2014) | $115.0^{+19.0}_{-16.0}$ | Benomar et al. (2014)
HAT-P-70 b | $107.9^{+2.0}_{-1.7}$ | Bello-Arufe et al. (2022) | — | —
HAT-P-8 b | $-17.0^{+9.2}_{-11.5}$ | Moutou et al. (2011) | — | —
HAT-P-9 b | $-16.0^{+8.0}_{-8.0}$ | Moutou et al. (2011) | — | —
HATS-14 b | $76.0^{+4.0}_{-5.0}$ | Zhou et al. (2015) | — | —
HATS-2 b | $8.0^{+8.0}_{-8.0}$ | Mohler-Fischer et al. (2013) | $28.5^{+14.0}_{-18.0}$ | This work
HATS-3 b | $3.0^{+25.0}_{-25.0}$ | Addison et al. (2014) | — | —
HATS-70 b | $8.9^{+5.6}_{-4.5}$ | Zhou et al. (2019a) | $13.2^{+6.4}_{-5.9}$ | Zhou et al. (2019a)
HD 106315 c | $-2.7^{+2.7}_{-2.6}$ | Bourrier et al. (2023) | — | —
HD 149026 b | $12.0^{+7.0}_{-7.0}$ | Albrecht et al. (2012b) | — | —
HD 17156 b | $10.0^{+5.1}_{-5.1}$ | Narita et al. (2009) | $61.3^{+6.0}_{-6.8}$ | This work
HD 189733 b | $-0.4^{+0.2}_{-0.2}$ | Cegla et al. (2016) | $7.0^{+12.0}_{-4.0}$ | Cegla et al. (2016)
HD 209458 b | $1.6^{+0.1}_{-0.1}$ | Casasayas-Barris et al. (2021) | $37.3^{+8.8}_{-5.4}$ | This work
HD 3167 b | $-6.6^{+6.6}_{-7.9}$ | Bourrier et al. (2021) | $29.5^{+7.2}_{-9.4}$ | Bourrier et al. (2021)
HD 3167 c | $-108.9^{+5.4}_{-5.5}$ | Bourrier et al. (2021) | $107.7^{+5.1}_{-4.9}$ | Bourrier et al. (2021)
HD 332231 b | $-2.0^{+6.0}_{-6.0}$ | Knudstrup & Albrecht (2022) | — | —

  * Note: as for the 3D spin–orbit angles, we include the values coming for the literature, plus the ones we derive in our study by fitting for $v_{\rm eq}\sin i_{\star}$ (as described in Sect. 4.1). The latter 3D obliquities are referenced as “This work” and can be readily used in future studies. We do not include the 3D obliquities we derive based on an isotropic stellar inclination distribution (Sect. 4.1) because of the underlying assumptions. We recommend using the latter approach only in ensemble studies and not for individual systems.

Table 3: Table 2 continued. Planet name | Projected spin–orbit angle | 3D spin–orbit angle
---|---|---
|  $\lambda$ (deg) | Source |  $\psi$ (deg) | Source
HD 63433 b | $8.0^{+33.0}_{-45.0}$ | Mann et al. (2020) | $33.1^{+17.0}_{-26.2}$ | This work
HD 63433 c | $-1.0^{+35.0}_{-32.0}$ | Dai et al. (2020) | $26.4^{+13.0}_{-20.3}$ | This work
HD 80606 b | $42.0^{+8.0}_{-8.0}$ | Hébrard et al. (2010) | — | —
HD 85628 A b | $-115.1^{+2.7}_{-3.6}$ | Dorval et al. (2020) | — | —
HD 89345 b | $74.2^{+33.6}_{-32.5}$ | Bourrier et al. (2023) | $80.1^{+22.3}_{-23.1}$ | Bourrier et al. (2023)
HIP 67522 b | $5.8^{+2.5}_{-3.7}$ | Heitzmann et al. (2021) | $20.2^{+10.3}_{-8.7}$ | Heitzmann et al. (2021)
K2-105 b | $-81.0^{+50.0}_{-47.0}$ | Bourrier et al. (2023) | — | —
K2-140 b | $0.5^{+9.7}_{-9.7}$ | Rice et al. (2021) | $24.0^{+10.9}_{-18.5}$ | This work
K2-232 b | $-11.1^{+6.6}_{-6.6}$ | Wang et al. (2021) | — | —
K2-25 b | $3.0^{+16.0}_{-16.0}$ | Stefansson et al. (2020) | $17.0^{+11.0}_{-8.0}$ | Stefansson et al. (2020)
K2-267 b | $-1.5^{+0.8}_{-0.8}$ | Yu et al. (2018) | $3.4^{+3.5}_{-1.6}$ | Yu et al. (2018)
K2-29 b | $1.5^{+8.7}_{-8.7}$ | Santerne et al. (2016) | $21.6^{+10.3}_{-13.3}$ | This work
K2-290 b | $173.0^{+45.0}_{-53.0}$ | Hjorth et al. (2021) | — | —
K2-290 c | $153.0^{+8.0}_{-8.0}$ | Hjorth et al. (2021) | $124.0^{+6.0}_{-6.0}$ | Hjorth et al. (2021)
K2-34 b | $-1.0^{+10.0}_{-9.0}$ | Hirano et al. (2016) | — | —
KELT-1 b | $2.0^{+16.0}_{-16.0}$ | Siverd et al. (2012) | — | —
KELT-11 b | $-77.9^{+2.4}_{-2.3}$ | Mounzer et al. (2022) | — | —
KELT-17 b | $-115.9^{+4.1}_{-4.1}$ | Zhou et al. (2016b) | $116.0^{+4.0}_{-4.0}$ | Zhou et al. (2016b)
KELT-19 b | $-179.7^{+3.7}_{-3.8}$ | Siverd et al. (2018) | — | —
KELT-20 b | $3.4^{+2.1}_{-2.1}$ | Lund et al. (2017) | $35.6^{+34.2}_{-34.0}$ | Lund et al. (2017)
KELT-21 b | $-5.6^{+1.7}_{-1.9}$ | Johnson et al. (2017) | — | —
KELT-24 b | $2.6^{+5.1}_{-3.6}$ | Rodriguez et al. (2019) | — | —
KELT-25 b | $23.4^{+3.2}_{-2.3}$ | Rodríguez Martínez et al. (2020) | — | —
KELT-4 A b | $14.0^{+100.0}_{-64.0}$ | Eastman et al. (2016) | — | —
KELT-6 b | $-36.0^{+11.0}_{-11.0}$ | Damasso et al. (2015b) | — | —
KELT-7 b | $2.7^{+0.6}_{-0.6}$ | Zhou et al. (2016a) | — | —
KELT-9 b | $-84.8^{+0.3}_{-0.3}$ | Stephan et al. (2022) | $87.5^{+0.2}_{-0.2}$ | Stephan et al. (2022)
KOI-142 b | $6.4^{+0.1}_{-0.1}$ | Nesvorný et al. (2013) | — | —
KOI-142 c | $-107.1^{+0.9}_{-0.6}$ | Nesvorný et al. (2013) | — | —
KOI-368 b | $10.0^{+2.0}_{-2.0}$ | Ahlers et al. (2014) | — | —
KOI-94 d | $-11.0^{+11.0}_{-11.0}$ | Albrecht et al. (2013) | — | —
Kepler-1115 b | $1.0^{+13.0}_{-13.0}$ | Barnes et al. (2015) | $-4.0^{+60.0}_{-60.0}$ | Barnes et al. (2015)
Kepler-13 b | $59.2^{+0.1}_{-0.1}$ | Howarth & Morello (2017) | $60.2^{+0.1}_{-0.1}$ | Howarth & Morello (2017)
Kepler-17 b | $0.0^{+15.0}_{-15.0}$ | Désert et al. (2011) | $0.0^{+15.0}_{-15.0}$ | Désert et al. (2011)
Kepler-25 c | $-0.9^{+7.7}_{-6.4}$ | Bourrier et al. (2023) | $24.1^{+9.2}_{-9.3}$ | Bourrier et al. (2023)
Kepler-30 b | $4.0^{+10.0}_{-10.0}$ | Sanchis-Ojeda et al. (2012) | — | —
Kepler-420 b | $74.0^{+32.0}_{-46.0}$ | Santerne et al. (2014) | — | —
Kepler-448 b | $-7.1^{+4.2}_{-2.8}$ | Johnson et al. (2017) | — | —
Kepler-462 b | $-32.0^{+11.0}_{-11.0}$ | Ahlers et al. (2015) | $72.0^{+3.0}_{-3.0}$ | This work
Kepler-462 c | $-32.0^{+40.0}_{-40.0}$ | Ahlers et al. (2015) | $72.2^{+7.5}_{-9.9}$ | This work
Kepler-63 b | $-135.0^{+21.2}_{-26.8}$ | Bourrier et al. (2023) | $114.6^{+16.6}_{-12.5}$ | Bourrier et al. (2023)
Kepler-8 b | $5.0^{+7.0}_{-7.0}$ | Albrecht et al. (2012b) | — | —
Kepler-89 d | $-6.0^{+13.0}_{-11.0}$ | Hirano et al. (2012) | — | —
Kepler-9 b | $-13.0^{+16.0}_{-16.0}$ | Wang et al. (2018) | — | —
MASCARA-1 b | $69.2^{+3.1}_{-3.4}$ | Hooton et al. (2022) | $72.1^{+2.5}_{-2.4}$ | Hooton et al. (2022)
MASCARA-4 b | $244.0^{+15.0}_{-15.0}$ | Ahlers et al. (2020) | $104.0^{+7.0}_{-13.0}$ | Ahlers et al. (2020)
NGTS-2 b | $-11.3^{+4.8}_{-4.8}$ | Anderson et al. (2018a) | — | —
Qatar-1 b | $-8.4^{+7.1}_{-7.1}$ | Covino et al. (2013) | $18.0^{+7.7}_{-10.8}$ | This work
Qatar-2 b | $0.0^{+8.0}_{-8.0}$ | Močnik et al. (2017) | — | —
TOI-1268 b | $40.0^{+7.2}_{-9.9}$ | Dong et al. (2022) | — | —
TOI-1431 b | $-155.0^{+20.0}_{-10.0}$ | Stangret et al. (2021) | — | —
TOI-1518 b | $240.3^{+0.9}_{-1.0}$ | Cabot et al. (2021) | — | —
TOI-2025 b | $9.0^{+33.0}_{-31.0}$ | Knudstrup et al. (2022) | — | —
TOI-2109 b | $1.7^{+1.7}_{-1.7}$ | Wong et al. (2021) | — | —
TOI-942 b | $1.0^{+41.0}_{-33.0}$ | Wirth et al. (2021) | $2.0^{+27.0}_{-23.0}$ | Wirth et al. (2021)
TRAPPIST-1 b | $15.0^{+26.0}_{-30.0}$ | Hirano et al. (2020a) | — | —
TRAPPIST-1 e | $9.0^{+45.0}_{-51.0}$ | Hirano et al. (2020a) | — | —
TRAPPIST-1 f | $21.0^{+32.0}_{-32.0}$ | Hirano et al. (2020a) | — | —
Table 4: Table 2 continued. TrES-1 b | $30.0^{+21.0}_{-21.0}$ | Narita et al. (2007) | — | —
---|---|---|---|---
TrES-2 b | $-9.0^{+12.0}_{-12.0}$ | Winn et al. (2008) | — | —
TrES-4 b | $6.3^{+4.7}_{-4.7}$ | Narita et al. (2010b) | — | —
V1298 Tau b | $4.0^{+7.0}_{-10.0}$ | Johnson et al. (2022) | $8.0^{+4.0}_{-7.0}$ | Johnson et al. (2022)
V1298 Tau c | $4.9^{+15.0}_{-15.1}$ | Feinstein et al. (2021) | $21.3^{+10.5}_{-10.3}$ | This work
WASP-1 b | $-59.0^{+99.0}_{-26.0}$ | Albrecht et al. (2011) | — | —
WASP-100 b | $79.0^{+19.0}_{-10.0}$ | Addison et al. (2018) | $82.0^{+14.9}_{-15.3}$ | This work
WASP-103 b | $3.0^{+33.0}_{-33.0}$ | Addison et al. (2016) | — | —
WASP-107 b | $-158.0^{+15.2}_{-18.5}$ | Bourrier et al. (2023) | $103.5^{+1.7}_{-1.8}$ | Bourrier et al. (2023)
WASP-109 b | $99.0^{+10.0}_{-9.0}$ | Addison et al. (2018) | — | —
WASP-11 b | $7.0^{+5.0}_{-5.0}$ | Mancini et al. (2015) | — | —
WASP-111 b | $-5.0^{+16.0}_{-16.0}$ | Anderson et al. (2014) | — | —
WASP-117 b | $-46.9^{+5.5}_{-4.8}$ | Carone et al. (2021) | $69.6^{+4.7}_{-4.1}$ | Carone et al. (2021)
WASP-12 b | $59.0^{+15.0}_{-20.0}$ | Albrecht et al. (2012b) | — | —
WASP-121 b | $87.2^{+0.4}_{-0.5}$ | Bourrier et al. (2020) | $88.1^{+0.2}_{-0.2}$ | Bourrier et al. (2020)
WASP-127 b | $-128.4^{+5.6}_{-5.5}$ | Allart et al. (2020) | — | —
WASP-13 b | $8.0^{+13.0}_{-12.0}$ | Brothwell et al. (2014) | — | —
WASP-134 b | $-43.7^{+9.9}_{-9.9}$ | Anderson et al. (2018b) | — | —
WASP-14 b | $-33.1^{+7.4}_{-7.4}$ | Johnson et al. (2009) | — | —
WASP-148 b | $-8.2^{+8.7}_{-9.7}$ | Wang et al. (2022) | $29.8^{+13.2}_{-23.4}$ | This work
WASP-15 b | $-139.6^{+5.2}_{-4.3}$ | Triaud et al. (2010) | — | —
WASP-156 b | $105.7^{+14.0}_{-14.4}$ | Bourrier et al. (2023) | — | —
WASP-16 b | $-4.2^{+11.0}_{-13.9}$ | Brown et al. (2012a) | — | —
WASP-166 b | $-0.7^{+1.6}_{-1.6}$ | Bourrier et al. (2023) | $14.2^{+6.0}_{-12.8}$ | This work
WASP-167 b | $-165.0^{+5.0}_{-5.0}$ | Temple et al. (2017) | $145.6^{+17.6}_{-11.6}$ | This work
WASP-17 b | $-148.5^{+5.1}_{-4.2}$ | Triaud et al. (2010) | — | —
WASP-174 b | $31.0^{+1.0}_{-1.0}$ | Temple et al. (2018) | — | —
WASP-178 b | $91.3^{+6.5}_{-6.3}$ | Rodríguez Martínez et al. (2020) | — | —
WASP-18 b | $4.0^{+5.0}_{-5.0}$ | Triaud et al. (2010) | — | —
WASP-180 b | $-162.0^{+5.0}_{-5.0}$ | Temple et al. (2019b) | $148.9^{+9.6}_{-7.8}$ | This work
WASP-189 b | $91.7^{+1.2}_{-1.2}$ | Deline et al. (2022) | $89.6^{+1.2}_{-1.2}$ | Deline et al. (2022)
WASP-19 b | $-1.9^{+1.1}_{-1.1}$ | Sedaghati et al. (2021) | $14.2^{+7.8}_{-12.9}$ | This work
WASP-190 b | $21.0^{+6.0}_{-6.0}$ | Temple et al. (2019a) | — | —
WASP-2 b | $-153.0^{+11.0}_{-15.0}$ | Triaud et al. (2010) | — | —
WASP-20 b | $12.7^{+4.2}_{-4.2}$ | Anderson et al. (2015a) | — | —
WASP-21 b | $8.0^{+26.0}_{-27.0}$ | Chen et al. (2020a) | — | —
WASP-22 b | $22.0^{+16.0}_{-16.0}$ | Anderson et al. (2011) | $30.5^{+13.6}_{-14.3}$ | This work
WASP-24 b | $-4.7^{+4.0}_{-4.0}$ | Simpson et al. (2011) | — | —
WASP-25 b | $14.6^{+6.7}_{-6.7}$ | Brown et al. (2012a) | — | —
WASP-26 b | $-34.0^{+36.0}_{-26.0}$ | Albrecht et al. (2012b) | — | —
WASP-28 b | $8.0^{+18.0}_{-18.0}$ | Anderson et al. (2015a) | — | —
WASP-3 b | $5.0^{+6.0}_{-5.0}$ | Miller et al. (2010) | — | —
WASP-31 b | $2.8^{+3.1}_{-3.1}$ | Brown et al. (2012a) | — | —
WASP-32 b | $10.5^{+6.4}_{-6.5}$ | Brown et al. (2012b) | $2.0^{+16.0}_{-16.0}$ | Brown et al. (2012b)
WASP-33 b | $-111.6^{+0.3}_{-0.3}$ | Watanabe et al. (2022) | $108.2^{+0.9}_{-1.0}$ | Watanabe et al. (2022)
WASP-38 b | $7.5^{+4.7}_{-6.1}$ | Brown et al. (2012b) | — | —
WASP-39 b | $0.0^{+11.0}_{-11.0}$ | Mancini et al. (2018) | — | —
WASP-4 b | $4.0^{+43.0}_{-34.0}$ | Triaud et al. (2010) | $39.3^{+16.8}_{-24.2}$ | This work
WASP-41 b | $6.0^{+11.0}_{-11.0}$ | Southworth et al. (2016) | $16.7^{+7.9}_{-9.3}$ | This work
WASP-43 b | $3.5^{+6.8}_{-6.8}$ | Esposito et al. (2017) | — | —
WASP-47 b | $0.0^{+24.0}_{-24.0}$ | Sanchis-Ojeda et al. (2015) | $29.2^{+11.1}_{-13.3}$ | Bourrier et al. (2023)
WASP-49 b | $54.0^{+79.0}_{-58.0}$ | Wyttenbach et al. (2017) | — | —
WASP-5 b | $12.1^{+10.0}_{-8.0}$ | Triaud et al. (2010) | $26.6^{+12.5}_{-14.9}$ | This work
WASP-52 b | $1.1^{+1.1}_{-1.1}$ | Chen et al. (2020b) | — | —
WASP-53 b | $-1.0^{+12.0}_{-12.0}$ | Triaud et al. (2017) | — | —
WASP-6 b | $7.2^{+3.7}_{-3.7}$ | Tregloan-Reed et al. (2015) | $29.6^{+13.7}_{-22.7}$ | This work
WASP-60 b | $-129.0^{+17.0}_{-17.0}$ | Mancini et al. (2018) | — | —
WASP-61 b | $4.0^{+17.1}_{-18.4}$ | Brown et al. (2017) | — | —
WASP-62 b | $19.4^{+5.1}_{-4.9}$ | Brown et al. (2017) | — | —
Table 5: Table 2 continued. WASP-66 b | $-4.0^{+22.0}_{-22.0}$ | Addison et al. (2016) | — | —
---|---|---|---|---
WASP-69 b | $0.4^{+2.0}_{-1.9}$ | Casasayas-Barris et al. (2017) | $15.6^{+7.3}_{-14.3}$ | This work
WASP-7 b | $86.0^{+6.0}_{-6.0}$ | Albrecht et al. (2012a) | — | —
WASP-71 b | $-1.9^{+7.1}_{-7.5}$ | Brown et al. (2017) | — | —
WASP-72 b | $-7.0^{+11.0}_{-12.0}$ | Addison et al. (2018) | — | —
WASP-74 b | $0.8^{+1.0}_{-1.0}$ | Luque et al. (2020) | — | —
WASP-76 b | $61.3^{+7.6}_{-5.1}$ | Ehrenreich et al. (2020) | — | —
WASP-78 b | $-6.4^{+5.9}_{-5.9}$ | Brown et al. (2017) | — | —
WASP-79 b | $-99.1^{+4.1}_{-3.9}$ | Johnson et al. (2017) | — | —
WASP-8 b | $-143.0^{+1.6}_{-1.5}$ | Bourrier et al. (2017) | — | —
WASP-80 b | $-14.0^{+14.0}_{-14.0}$ | Triaud et al. (2015) | — | —
WASP-84 b | $-0.3^{+1.7}_{-1.7}$ | Anderson et al. (2015b) | $17.3^{+7.7}_{-7.7}$ | Anderson et al. (2015b)
WASP-87 b | $-8.0^{+11.0}_{-11.0}$ | Addison et al. (2016) | — | —
WASP-94 b | $151.0^{+16.0}_{-23.0}$ | Neveu-VanMalle et al. (2014) | — | —
XO-2 b | $7.0^{+11.0}_{-11.0}$ | Damasso et al. (2015a) | $27.0^{+12.0}_{-27.0}$ | Damasso et al. (2015a)
XO-3 b | $37.3^{+3.0}_{-3.0}$ | Hirano et al. (2011) | — | —
XO-4 b | $-46.7^{+8.1}_{-6.1}$ | Narita et al. (2010a) | — | —
XO-6 b | $-20.7^{+2.3}_{-2.3}$ | Crouzet et al. (2017) | — | —
pi Men c | $-24.0^{+4.1}_{-4.1}$ | Kunovac Hodžić et al. (2021) | $26.9^{+5.8}_{-4.7}$ | Kunovac Hodžić et al. (2021)