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Draft version April 16, 2021Typeset using LATEX twocolumn style in AASTeX62
Obliquities of exoplanet host stars
Simon Albrecht,1 Rebekah I. Dawson,2 and Joshua N. Winn3
1Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark2Department of Astronomy & Astrophysics, Center for Exoplanets and Habitable Worlds,The Pennsylvania State University, University
Park, PA 16802, USA3Department of Astrophysical Sciences, Peyton Hall, 4 Ivy Lane, Princeton, NJ 08540, USA
ABSTRACT
One of the surprises of exoplanetary science was that the rotation of a star need not be aligned with
the revolutions of its planets. Measurements of the stellar obliquity — the angle between a star’s spin
axis and the orbital axis of one or more of its planets — occupy the full range from nearly zero to
180, for reasons that remain unclear. Here, we review the measurement techniques and key findings,
along with theories for obliquity excitation and evolution.
The most precise individual measurements involve stars with short-period giant planets, which have
been found on prograde, polar, and retrograde orbits. It seems likely that dynamical processes such as
planet-planet scattering and secular perturbations are responsible for tilting the orbits of these planets,
just as these processes are implicated in exciting orbital eccentricities. The observed dependences of
the obliquity on the orbital separation, planet mass, and stellar mass suggest that in some cases, tidal
dissipation damps the obliquity within the star’s main-sequence lifetime.
The situation is not as clear for stars with smaller or wider-orbiting planets. Although the earliest
measurements tended to find low obliquities, some glaring exceptions are now known, in which the
star’s rotation is misaligned with respect to multiple coplanar planets. In addition, statistical analyses
of Kepler data suggest that high obliquities are widespread for stars hotter and more massive than the
Sun. This suggests it is no longer safe to assume that stars and their protoplanetary disks are aligned
— primordial misalignments might be produced by a neighboring star or more complex events that
occur during the epoch of planet formation.
Keywords: exoplanets, obliquities — planet formation — tides
1. INTRODUCTION
Since the earliest observations of sunspots by Fabri-
cius, Scheiner, and Galileo it has been known that the
Sun’s equatorial plane is nearly aligned with the ecliptic
(Casanovas 1997). A modern measurement of the Sun’s
obliquity, based on helioseismology, is 7.155 ± 0.002
(Beck & Giles 2005). The low solar obliquity was part
of the body of evidence that led Laplace to the “nebu-
lar theory” for the formation of the Solar System, which
was incorrect but is remembered for the theoretical de-
but of the protoplanetary disk. The fact that the obliq-
uity is a little higher than the root-mean-squared mu-
tual inclination of 1.9 between the planetary orbits has
also inspired theorists; among the proffered explanations
Corresponding author: Simon Albrecht
are a close encounter with another star (Heller 1993), a
torque resulting from motion of the protoplanetary disk
through the interstellar medium Wijnen et al. (2017), a
torque from an undiscovered outer planet (Bailey et al.
2016; Gomes et al. 2017; Lai 2016), an asymmetry of the
solar wind (Spalding 2019), and the imprint of a nearby
supernova Portegies Zwart et al. (2018).
Exoplanetary systems have proven to show a wider
range of orbital characteristics than had been expected
based on analyses of the Solar System (see, e.g., Winn
& Fabrycky 2015, for a review). Among these sur-
prises were close-orbiting giant planets (Mayor & Queloz
1995), high orbital eccentricities (Latham et al. 1989;
Marcy & Butler 1996), miniature systems of multiple
planets on tightly packed orbits (Lissauer et al. 2011;
Fabrycky et al. 2012) and, the reason for this review ar-
ticle, large stellar obliquities (Hebrard et al. 2008; Winn
et al. 2009). One of the main goals of exoplanetary sci-
2 Albrecht, Dawson, & Winn
Figure 1. Coordinate system and angles that specify theorientation of the spin and orbital angular momentum vec-tors (modeled after Perryman (2011)). The obliquity is ψ,the orbital inclination is io, and the inclination of the stellarrotation axis is i.
ence is to understand the physical processes that are
responsible for this architectural diversity.
Measuring the obliquity of an exoplanet host star is
challenging, given that ordinary observations lack the
angular resolution to discern any details on the spa-
tial scale of the stellar diameter. Nevertheless, using
an array of techniques, we now know the obliquities of
approximately 150 stars, and we have drawn statisti-
cal inferences about the obliquity distribution of sam-
ples of ∼103 stars. Prograde, polar, and retrograde or-
bits have been found, and a few patterns have emerged
relating to stellar mass, planetary mass, and orbital
distance. There is no unique interpretation of the re-
sults. Misalignments might occur before, during, or af-
ter the epoch of planet formation. They may be linked
to specific dynamical events in a planet’s history such
as planet-planet scattering or high-eccentricity migra-
tion, or they may be the outcome of general processes
affecting stars and protoplanetary disks irrespective of
the planets that eventually form.
This article is an attempt to review the current status
of the observations and theories regarding the obliquities
of stars with planets. Section 2 introduces the geometry
and terminology that will be important throughout this
article. Section 3 describes the measurement techniques
and key findings. Section 4 discusses the proposed phys-
ical mechanisms that can excite or dampen obliquities
and their success or failure in matching the observations.
Section 5 is a summary and a set of recommendations
for future work in this area.
2. GEOMETRY
Figure 1 illustrates the angles that determine the ori-
entation of a star (n?) with respect to the line of sight
(z) and with respect to the orbital axis of a planet (no).
The obliquity ψ is the angle between n? and no. In the
coordinate system shown in Figure 1,
no = sin io y + cos io z and (1)
n? = sin i sinλ x+ sin i cosλ y + cos i z, (2)
where we have chosen to orient the y axis along the
sky projection of n?. Here, i and io are the line-of-sight
inclinations of the stellar and orbital angular momentum
vectors, and λ is the position angle between the sky
projections of those two vectors. It follows that
cosψ = n? · no = sin i cosλ sin io + cos i cos io . (3)
Most of the observational methods do not measure
ψ in one step. Instead, some techniques are capable
of detecting differences between i? and io, leading to a
lower limit |io−i?| on the obliquity. Other techniques are
sensitive to λ, which is a lower limit on ψ when |λ| < 90,
and an upper limit on ψ when |λ| > 90. A sample of
stars with completely random orientations would show
a uniform distribution in the azimuthal angles λ and in
the cosines of the polar angles i, io, and ψ.
For the statistical analysis of obliquity measurements,
two useful references are Fabrycky & Winn (2009) and
Munoz & Perets (2018). The former authors provided
analytic formulas for the conditional probability densi-
ties p(ψ|λ) and p(λ|ψ) under the assumption of random
orientations. They also showed how to use measure-
ments of λ to model the obliquity distribution of a pop-
ulation of stars as a von-Mises Fisher (vMF) distribu-
tion1,
p(ψ) =κ
2 sinhκexp(κ cosψ) sinψ. (4)
Munoz & Perets (2018) extended this framework to in-
clude information about i in addition to λ.
3. METHODS AND KEY FINDINGS
The main challenge in measuring any of the angles in
Figure 1 is that stars are almost always spatially un-
resolved by our telescopes. We can only observe the
star’s flux and spectrum integrated over the star’s visi-
ble hemisphere.
1 The vMF distribution is a widely-used model in directionalstatistics that resembles a two-dimensional Gaussian distributionwrapped around a sphere. For small values of the concentrationparameter κ, the distribution becomes isotropic. For large val-ues of κ, the distribution approaches a Rayleigh distribution withwidth parameter σ = κ−1/2.
Obliquity 3
100 101 102
period (days)
100
101
plan
etar
y ra
dius
(R)
alignedmisaligned
10000
1 3 10 30 100period (days)
0
1
2
stel
lar m
ass (
M)
Rossiter-McLaughlinAsteroseismologyStar SpotsGravity DarkeningInterferometryProjected rotation rate
10000
Figure 2. Parameter space of obliquity measurement methods. Each point represents an obliquity measurementreported in the literature, with a location that specifies the orbital period and the planet’s radius (top panel) and stellar mass(bottom panel). The points are color coded by method. Solid symbols are for misaligned stars (by more than 3-σ); opensymbols are for well-aligned stars or ambiguous cases. The RM, starspot, and gravity-darkening methods require observationsduring transits, making them less applicable to systems with smaller planets or longer periods. The gravity-darkening methodrequires fast rotators, i.e., high-mass stars, while the starspot method is most applicable to lower-mass stars with large, long-lived starspots. The asteroseismic and projected rotation-rate methods require a transiting planet but do not require intensiveobservations conducted during transits, making them applicable to planets of all types. The asteroseismic method requiresmoderately rapid rotation and long-lived pulsation modes, which generally occur for stars somewhat more massive than theSun. Similarly, the projected rotation rate method needs moderately rapid rotation, which is found for more massive stars.The interferometric method requires very bright and rapidly rotating stars, as well as some constraint on the planetary orbitalinclination. Also important, though not conveyed in this diagram, is that the methods differ in the achievable precision andparameter degeneracies.
Fortunately, some aspects of the disk-integrated fluxand spectrum depend on the star’s inclination i with re-
spect to the line of sight. One is the rotational Doppler
broadening of its spectral absorption lines, which is
proportional to v sin i where v is the rotation velocity
(§ 3.4). Another is the star’s amplitude of photometric
variability due to rotating starspots, which is expected
to vary roughly in proportion to sin i (§ 3.5). A third
type of data that bears information about orientation is
the fine structure within the power spectrum of a star’s
asteroseismic oscillations; the relative amplitudes of the
modes within a rotationally-split multiplet depend on
i (§ 3.2). When we also have knowledge of io (such as
when the planet detected through the transit or astro-
metric techniques), these types of data place constraints
on the stellar obliquity.
These inclination-based methods have some im-
portant limitations. Because of the north/south symme-
try of the star, we cannot distinguish i? from 180 − i?,leading to a twofold degeneracy in obliquity constraints.
In particular, we cannot tell whether a star has pro-
grade or retrograde rotation with respect to the line
of sight or the planetary orbit.2 Another problem is
that sin i-based techniques are insensitive at high incli-
nations. Even if sin i is constrained to be in the narrow
range from 0.9 to 1, the inclination can be any value in
the range from 64 to 116. This problem arises often
because high inclinations are common; 44% of the stars
in a randomly-oriented population have sin i > 0.9.
2 For transiting planets, the same degeneracy afflicts measure-ments of io, although the geometrical requirement for transits im-plies that io is never far from 90.
4 Albrecht, Dawson, & Winn
Figure 3. Geometry of the Rossiter-McLaughlin effect. The left panel illustrates a transit, with the planet crossingfrom left to right. Due to stellar rotation the left side of the star is moving towards the observer and the right side is receding.The angle sky projections of the unit vectors n? and no are separated by the angle λ, and the x-axis is perpendicular to theprojected rotation axis. For the case of uniform rotation, the sub-planet radial velocity is (v sin i)x and the extrema of the RMsignal occur at ingress (x1) and egress (x2). The relations between x1, x2, λ and the impact parameter b are indicated on thediagram. The right panel shows the corresponding velocity of planet’s “Doppler shadow.” This figure is from Albrecht et al.(2011).
The other main class of methods for measuring the
obliquity rely on a transiting planet to provide spa-
tially resolved information as its shadow scans across
the stellar disk. A star’s intensity and emergent spec-
trum vary across the stellar disk in a manner that de-
pends on the star’s orientation. For example, stellar
rotation causes the radial velocity of the stellar disk to
exhibit a gradient from the approaching side to the re-
ceding side. When a transiting planet hides a portion of
the stellar disk, the corresponding radial-velocity com-
ponent is absent from the disk-integrated stellar spec-
trum, leading to line-profile distortions known as the
Rossiter-McLaughlin effect (§ 3.1). Another technique
is based on detecting the glitches in the light curve when
a transiting planet occults a starspot or other inhomo-
geneity on the stellar disk; the timings of such anomalies
can sometimes be used to constrain the stellar obliquity
(§ 3.5). A third technique is based on gravity darkening:
the equatorial zone of a rapidly rotating star is lifted to
higher elevation, leading to a lower effective tempera-
ture and a lower intensity than the polar regions. This
breaks the usual circular symmetry of the stellar disk,
which in turn causes a distortion of the transit light
curve (§ 3.7.2). The circular symmetry is also broken by
a relativistic effect known as rotational Doppler beaming
(§ 3.7.1).
These transit-based methods are usually more sen-
sitive to λ than to i. Indeed, in the best cases, λ can
be measured with a precision on the order of 1. The
disadvantages of these methods are that they require
time-critical observations of transits, and the signals are
generally proportional to the area of the planet’s silhou-
ette divided by the area of the stellar disk. In practice,
this makes it very challenging to deploy these methods
on planets smaller than Neptune around Sun-like stars.
Finally, there is a technique that is mainly sensitive
to λ and does not require a transiting planet: optical in-
terferometry with high spectral resolution. For nearby
bright stars, interferometric observations can partially
resolve the stellar disk and reveal the displacement on
the sky between the redshifted and blueshifted halves of
the rotating star (§ 3.3). This is still a highly special-
ized technique, though, and leaves open the problem of
determining the orientation of the planet’s orbit.
Each technique works best in different circumstances.
Figure 2 illustrates the applicability of these different
techniques to systems with different stellar masses, plan-
etary radii, and orbital periods. Below, we describe
these techniques in more detail, but not in the geometry-
based order described here. Instead, we devote the most
attention to the techniques that have delivered the most
information.
3.1. The Rossiter-McLaughlin effect
In a letter the editor of the Sidereal Messenger, Holt
(1893) pointed out that a star’s rotation rate could be
measured by observing the time-variable distortions of
its absorption spectrum during an eclipse. We have not
been able to learn anything more about this insight-
ful correspondent, nor have we found any earlier ref-
erence to what is now called the Rossiter-McLaughlin
effect. The name honors the work of Rossiter (1924)
Obliquity 5
−1.0−0.5 0.0 0.5 1.0distance [Rstar]
−20
−10
0
10
20
RV
[m
s−
1]
a
stellar rotation
−1.0−0.5 0.0 0.5 1.0distance [Rstar]
−4
−2
0
2
4
RV
[m
s−
1]
b
turbulence+PSF
−1.0−0.5 0.0 0.5 1.0distance [Rstar]
c
differential rotation
−1.0−0.5 0.0 0.5 1.0distance [Rstar]
d
convective blueshift
e
−1.0 −0.5 0.0 0.5 1.0distance [Rstar]
−20
−10
0
10
20
RM
effe
ct [m
s−
1]
Figure 4. Higher-order effects in the anomalous radial velocity, illustrated for the choices λ = 40, v sin i = 3 km s−1,r/R = 0.12 and b = 0.2. (a) Solar-like limb darkening “rounds off” the signal near ingress and egress. (b) Instrumentalbroadening (taken to be 2.2 km s−1) and macroturbulence (ζRT = 3 km s−1) acts oppositely to the rotational effect. (c) Solar-like differential rotation causes the effect to depend on the range of stellar latitudes crossed by the planet. (d) Solar-likeconvective blueshift produces an anomalous velocity depending on distance from the center of the stellar disk. (e) The combinedmodel including all aforementioned effects. The gray line is the model from panel (a). This figure is from Albrecht et al. (2012b).
and McLaughlin (1924), who observed the effect in the
β Lyrae and Algol systems, respectively.3
One of the broadening mechanisms of stellar absorp-
tion lines is the variation in the rotational Doppler shift
between the two sides of the stellar disk. Due to ro-
tation, light from the approaching half of a star is
blueshifted, light from the receding half is redshifted,
and the disk-integrated spectrum shows a spread in
Doppler shifts. During an eclipse or transit, a a por-
tion of the stellar disk is hidden from view, weakening
the corresponding radial-velocity components in an ab-
sorption line. The character and time-evolution of this
spectral distortion depends chiefly on v sin i and λ.
Observers have detected and modeled the RM effect
in two different ways. When the spectral lines are not
well resolved, the line-profile distortions are manifested
as shifts in the apparent central wavelength of the line.
When the blueshifted portion is eclipsed, the lines ex-
hibit an anomalous redshift, and vice versa. This is the
manner in which Rossiter (1924) and McLaughlin (1924)
displayed their data, as well as Queloz et al. (2000), who
performed the first observations of the RM effect for a
transiting planet. Parametric models for the “anoma-
lous radial velocity” and its relation to the positions
and attributes of the two bodies have been developed
by many authors(e.g. Hosokawa 1953; Kopal 1959; Sato
1974; Ohta et al. 2005; Gimenez 2006; Hirano et al. 2011;
Shporer & Brown 2011).
3 An earlier and less convincing detection as reported bySchlesinger (1910) for the δ Lib system.
Alternatively, the line-profile distortions can be de-
tected and modeled directly without the intermediate
step of computing an anomalous radial velocity. Models
for this “Doppler shadow” have also been developed ex-
tensively, starting with a beautiful exposition by Struve
& Elvey (1931) for the Algol system and continuing to
the present (e.g. Albrecht et al. 2007; Collier Cameron
et al. 2010; Albrecht et al. 2013a; Johnson et al. 2014;
Cegla et al. 2016; Zhou et al. 2016; Johnson et al. 2017).4
The RM effect has been the basis of most obliquity
measurements of individual planet-hosting stars (as op-
posed to statistical results from samples of stars). This
topic was reviewed recently by Triaud (2017). Below,
we described the two main methods for analyzing the
RM effect: as an anomalous radial velocity (§ 3.1.1) and
as a line-profile distortion (§ 3.1.2). Then, we review
the key findings that have emerged from RM observa-
tions (§ 3.1.3–3.1.10). Table 1 gives an overview of these
trends and highlights particular systems. Appendix A
describes the compilation of data that we assembled to
make the charts for this review.
3.1.1. The anomalous radial velocity
Consider a transit of a planet of radius r across a
uniformly-rotating star of radius R, equatorial rotation
4 The line-profile method has also been called “Doppler tomog-raphy,” a term we find confusing. The name originally belongedto line-profile analyses in which a star’s surface structure or abinary’s accretion geometry is reconstructed from spectral obser-vations obtained from many different viewing angles, as the starrotates or the binary revolves all the way around. In the case of aplanetary transit, though, the range of viewing angles is so narrowthat there is no “tomographic” quality to the analysis.
6 Albrecht, Dawson, & Winn
b)
d) Kepler-13, Johnson et al. (2014)
HAT-P-69, Zhou et al. (2020)
0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04Phase
14.80
14.78
14.76
14.74
14.72RV [
Km
/s]
MASCARA-2/KELT-20Hoeijmakers et al. (2020)
c)
a) HD 209458, Santos et al. (2020)
Figure 5. Illustrations of RM measurements taken in different systems using different visualisations. Left top:A measurement of the anomalous RVs due to the deformation of the stellar lines, as observed in the aligned (λ = 0.6 ± 0.4)HD 209458 system by Santos et al. (2020). During the first half of the transit blue shifted light is blocked from view, leadingto a redshift and therefore a positive RV excess on top of the orbital RVs of the host star. Red shifted light is blocked duringthe second half.Right top: This panel shows the deformation of the stellar lines ”planet shadow” in the prograde, misaligned(λ = 21.2+4.6
−3.6 deg) system HAT-P-69 with its fast rotating star host star(Zhou et al. 2019). The line deformation (dark stripe)does not reach the same absolute negative vp at the begin of the transit as at the end of the transit, a clear sign of misalignment.Panel (c) shows Fig. 3 from the work by Hoeijmakers et al. (2020) it illustrates the vp(t) in the aligned MASCARA-2 system.Finally panel (d) shows the stacked single of the RM deformation in the Kepler-13 system analyzed by Johnson et al. (2014).Here the line residuals (after subtraction of an out of transit line) are shifted and binned according to a particular vp(t) for eachobservation. The timeseries of different sub planet velocities relates to a given amplitude of the RM effect v14, and vcen whichrelates to the asymmetry of the signal, see § 3.1.
velocity v, and line-of-sight inclination i. During the
transit, the stellar absorption lines suffer a fractional
loss of light on the order of (r/R)2 associated with the
velocity component vp, where
vp(t) = (v sin i)x(t) (5)
is the velocity of the Doppler shadow (or the “sub-planet
velocity”), defined as the rotational radial velocity of
the point on the stellar disk directly behind the planet’s
center. Here, x(t) is the planet’s position in units of
the stellar radius along the coordinate axis running per-
pendicular to the star’s projected rotation axis, as in
Figure 1.
The effect on a spectral line is a distortion, not an
overall Doppler shift. Nevertheless, a radial-velocity ex-
traction algorithm will respond to the distortion by re-
porting an anomalous velocity on the order of
∆V (t) ≈ −( rR
)2
vp(t). (6)
There are corrections of order unity due to the effects of
limb darkening, turbulent and instrumental broadening,
and the details of the RV-extraction algorithm. For the
case of a cross-correlation algorithm, an accurate for-
mula was derived by Hirano et al. (2011), building on
work by Ohta et al. (2005).
If the radius ratio r/R and transit impact parame-
ter b are known, then observations of the time series
∆V (t) can be used to determine λ and v sin i. Figure 3
illustrates the transit geometry and the corresponding
∆V (t). The extremes of the signal occur at ingress (x1)
Obliquity 7
and egress (x2), with amplitudes
∆V1 = (v sin i)x1, ∆V2 = (v sin i)x2, (7)
and from the transit geometry, one can show
x1 =√
1− b2 cosλ− b sinλ,
x2 =√
1− b2 cosλ+ b sinλ. (8)
We can recast these relationships as
∆V2 −∆V1 = 2(v sin i) sinλ× b, (9)
∆V2 + ∆V1 = 2(v sin i) cosλ×√
1− b2, (10)
making it clear that the asymmetry of the signal depends
on sinλ, while the total amplitude depends on cosλ.
When both of these aspects of the signal are measured,
and the impact parameter is known from other observa-
tions, the preceding system of equations can be solved
for v sin i and λ. For more insight into the information
content of the RM signal, see Gaudi & Winn (2007). In
particular, those authors derived a formula to estimate
the achievable precision in the measurements of λ,
σλ =σv/√N
v sin i
( rR
)−2[
(1− b2) sin2 λ+ 3b2 cos2 λ
b2(1− b2)
]1/2
(11)
based on N data points with independent Gaussian un-
certainties σv uniformly spanning the transit.5 Note
that the uncertainty grows as b approaches 0 or 1. As
b→ 0, the asymmetry vanishes and there is not enough
information to determine both λ and v sin i; in such
cases, an external constraint on v sin i is essential. As
b→ 1, the transit signal itself vanishes.
Figure 4 shows some higher-order effects that were
neglected in the preceding discussion. Limb darken-
ing weakens the RM effect near the ingress and egressphases. Differential stellar surface rotation causes vp to
be a function of both x and y, making the RM effect
sensitive to i in addition to λ (Gaudi & Winn 2007;
Cegla et al. 2016). Turbulence on the stellar surface
also affects vp, as does the “convective blueshift” —
the higher intensity of the hot, upwelling material com-
pared to the sinking material (Shporer & Brown 2011;
Cegla et al. 2016). Some other effects are usually ne-
glected but may be important in special cases: the tidal
and rotational deformation of the star, the saturation or
pressure-broadening of some lines, and the influence of
star spots and pulsations. is normally ignored.
5 The formula is only valid when enough data outside the tran-sit have been obtained for the RM signal to be isolated withoutambiguity. It is best to obtain at least a few data points beforeand after the transit.
3.1.2. The Doppler Shadow
The line-profile distortions due to the RM effect can
also be analyzed directly. Consider an idealized spectral
line broadened only by rotation. When the planet is at
position x(t), the range of velocity components partially
blocked by the planet is (v sin i) (x± r/R). Within this
velocity range, the fractional loss of light is equal to the
area of the planet’s silhouette divided by the area of the
strip of the star within x± r/R,
∆LRM(t) ≈ −π8
r
R
1√1− x(t)2
. (12)
This is the intensity contrast of the “bump” that
would appear in the line profile — the planet’s Doppler
shadow. Note that it scales in proportion to r/R, not
(r/R)2, making this technique potentially more sensitive
to small planets than the anomalous-RV technique. In
practice, though, other line-broadening mechanisms will
reduce the contrast of the bump and at least partially
negate this advantage.
Collier Cameron et al. (2010) presented a more realis-
tic analytic model for the distorted line profile, including
limb darkening. Another approach is to create synthetic
line profiles by numerically integrating over a 2-d pix-
elated stellar disk, assigning intensities and velocities
to each pixel due to rotation, limb darkening, velocity
fields, etc. The pixels hidden by the planet are simply
assigned zero intensity (e.g. Albrecht et al. 2007). In
the approach they called “RM Reloaded,” (Cegla et al.
2016) replaced the synthetic line profile with an empiri-
cal model based on spectra obtained outside of transits
and used a parametric model only for the portion of the
photosphere covered by the planet.
Fig. 5 compares four different representations of the
RM effect, drawn from the literature. The upper left
panel shows an anomalous radial velocity time series.
The upper right panel shows the “Doppler shadow” as
a time series of residual line profiles derived from cross-
correlation. Each row represents an observed line pro-
file after subtracting the best-fitting model of an undis-
turbed line profile. As time progresses (upward, on the
plot), the negative residual caused by the planet moves
from the blue end to the red end of the line profile.
The lower left panel shows the time series of the sub-
planet velocity inferred with the RM Reloaded tech-
nique. In the lower right panel, the color scale indicates
the strength of line-profile residuals after shifting and
averaging them as a function of the sub-planet veloc-
ity at midtransit (vcen) and the difference in sub-planet
velocities at ingress and egress (v14). Such a “data stack-
ing” analysis can be useful in the presence of correlated
8 Albrecht, Dawson, & Winn
Table 1. Key results from obliquity measurements. The first column names the detected observational trend, the secondcolumn indicates the main measurement technique used. The section which discusses the particular trend is given in the lastcolumn together with the pointer to the main reference(s)
Observational trend/Key system Observational method Section Ref.
• Hot stars (Teff & 6250 K) harboring HJs
tend to have high obliquities RM § 3.1.3 1
• Massive stars (M & 1.2M) harboring HJs
tend to have high obliquities v sin i § 3.6 2
• Massive planets tend to have low obliquities,
low mass planets tend to have high obliquities RM § 3.1.4 3
• Planets traveling on large orbits
tend to have large obliquities RM § 3.1.5 4
• Very young systems tend to be aligned RM/v sin i/Interferometry § 3.1.8
• Aligned HJs orbiting cool stars are aligned to . 1 RM § 3.1.6 5
• Compact multi planet systems
tend to have low obliquities Spots/RM/Seismology § 3.1.10 6,7,8
• Cool exoplanet hosts are aligned Lightcurve variability § 3.6 9,10
• Systems with close in Neptune sized planets
tend to be aligned v sin i § 3.6 11,12
• Hot stars have large obliquities v sin i § 3.6 13
• HD 80606: prime example of KL-cycle caused by stellar companion RM § 4.3 14,15
• Kepler-56: orbits of inner planets precess, caused by outer giant planet Seismology § 3.1.10 16
• K2-290: retrograde coplanar orbits in wide double star system,
clear evidence for primordial disk misalignment RM/v sin i § 4.2 17
References—1 Winn et al. (2010), 2 Schlaufman (2010), 3 Hebrard et al. (2011), 4 Albrecht et al. (2012b), 5 Stefansson etal. in prep., 6 Albrecht et al. (2013a), 7 Morton & Winn (2014), 8 Campante et al. (2016), 9 Mazeh et al. (2015a), 10 Li &Winn (2016), 11 Winn et al. (2017), 12 Munoz & Perets (2018), 13 Louden et al. (2021), 14 Wu & Murray (2003), 15 Hebrardet al. (2010), 16 Huber et al. (2013), 17 Hjorth et al. (2021)
noise (Johnson et al. 2014) or a low signal-to-noise ratio
(Hjorth et al. 2021).
Whether to analyze the data in terms of the anoma-
lous RV or the line-profile variations, or both, depends
on the instrument and the system parameters. Roughly
speaking, the larger the ratio
α =(v sin i)(r/R)√
σ2inst + σ2
mic + σ2mac
, (13)
the easier it will be to resolve the planet’s Doppler
shadow in the line profiles. Here, σinst is the instru-
mental broadening of the spectrograph and σmic and
σmac are the magnitudes of micro- and macro-turbulence
(Gray 2005). These are the most important terms which
determine the shapes and widths of unsaturated absorp-
tion lines, besides rotation. For rapidly rotating stars,
precise RV determination is difficult but the RM anoma-
lies in the line profiles can reach depths of several percent
of the overall line depth (e.g. Talens et al. 2018), making
them relatively easy to detect.
3.1.3. Hot stars with hot Jupiters have high obliquities
Figure 6 displays projected obliquity and stellar rota-
tion measurements as function of the host star’s effec-
tive temperature (Teff). We highlight results for HJs.
The trend reported by Winn et al. (2010) – that stars
with Teff < 6250 K have projected obliquities consistent
with alignment and stars with Teff > 6250 K have a
range of obliquities – exists in this significantly enlarged
sample. No host star with a HJ and Teff significantly
lower than the Kraft break has a spin-orbit misalign-
ment. Out of the 56 HJ systems with Teff < 6250 K only
three (WASP-60, WASP-62 and WASP-94A), a fraction
of 0.06 are misaligned.6 These three misaligned hosts
have temperatures above 6100 K.
6 When we discuss aligned/misaligned and circular/eccentricorbits then we define these via the following: An aligned systemhas a projected obliquity below 10 deg or a stellar inclinationmeasurement above 80 deg. A misaligned system either excludes0 deg at the 3 − σ level and has a λ larger than 10 deg, or hasa stellar inclination measurement excluding 90 deg at 3 − σ andhas a i measurement below 80 deg. We count an orbit as eccentricif the eccentricity is larger than 0.1 and an eccentricity of zero isexcluded at a 3 − σ level. We describe a system as circular if itseccentricity measurement is below 0.1. Systems which do not fall
Obliquity 9
3000 4000 5000 6000 7000 8000 9000 100000
306090
120150180
proj
. obl
iqui
ty (d
eg)
3000 4000 5000 6000 7000 8000 9000 10000Teff (K)
100
101
102
Rota
tion
rate
(km
s1 )
HJ - cool hostHJ - hot hostHJ - very hot hostwarm & cool Jupiterssub-Saturnsmulti-transiting
Figure 6. Projected obliquities and projected stellar rotation speeds of exoplanet host stars displayed overthe effective temperature (Teff). The upper panel shows projected obliquities (λ) and the lower panel shows projectedstellar rotation speeds (v sin i). We color code different types of systems; Hot-Jupiter systems are systems with scaled orbitalseparations (a/R) below 10 and planet masses (or their upper limits, if only these are available) above 0.3 RJupiter. For thesesystems we also distinguish between ”cool hosts” (Teff < 6250, corresponding to a spectral class of G and lower), ”hot hosts”(6250 < Teff < 7000, F type stars), and ”very hot hosts”(7000 < Teff , A type stars). We label Jupiter mass planets as ”warm/cool Jupiters” if their a/R is larger than ten. Planets with masses less than approximately the mass of Saturn (0.3 MJupiter)are marked as ”Sub-Saturns”. We label all systems for which at least two different planets have been observed to transit as”multi transiting”. Each system is only counted once. An absolute projected obliquity |λ| value below 90 indicate a progradeorbit, larger λ values indicate a retrograde orbit. As expected the host star v sin i does increase with stellar temperature in therange from K-A type host stars. The top panel also highlights that for HJ systems there is a clear increase in stellar obliquityfrom cool hosts (blue symbols), hotter stars (red symbols) which have a significant fraction of systems with large and retrogradestars, until very hot hosts, which do in the current sample do not show any preference for alignment.
Since 2010, not only has the number of systems with
λ measurements grown, also the range of host star effec-tive temperatures has increased. In Figure 6 we mark
systems with stars above 7000 K with orange systems.
We are motivated to make this additional distinction
by two observational trends. The ratio of oblique ver-
sus well aligned systems raises from 1.4 (21 versus 15)
in the range 6250 K < Teff < 7000 K to 3.7 (11 versus
3) above 7000 K. The Kolmogoro-Smirnov (KS) statistic
indicates a p-value of 1.9×10−5 that the projected obliq-
uities are drawn from a uniform sample for stars with
6250 K < Teff < 7000 K. For very hot hosts the hypoth-
esis that the projected obliquities are drawn from a uni-
form sample can not be rejected with the data at hand,
into these categories e.g. they are formally misaligned/eccentricbut below a 3 − σ level then these are not counted.
p = 0.17. The second trend relates to v sin i, which no
longer increases with Teff for stars hotter than ∼ 7000 K(Figure 6 lower panel). This is consistent with other
samples presented in the literature, see e.g. Gray (2005).
This flatten out in the maximum v sin i is thought to be
connected to the complete absence of a convective enve-
lope above ∼ 7000 K (i.e., these stars have experienced
no convective braking).
3.1.4. High mass giant planets have low obliquity hosts
Figure 7 displays projected obliquities as a function
of the planet-to-star mass ratio (m/M). Massive HJs
have low obliquity orbits, a trend observed earlier in an
smaller sample (Hebrard et al. 2011).There are cool host
star systems with significant obliquities despite large
mass ratios. These are WJs (a/R > 10) and are in-
dicated by cyan symbols. The current sample indicates
that the mass cut off for prograde orbits depends on
10 Albrecht, Dawson, & Winn
10 4 10 3 10 2
planet/star mass ratio0
306090
120150180
mJu
pite
r/M
HJ - very hot hostwarm & cool Jupiters
0306090
120150180
proj
ecte
d ob
liqui
ty (d
eg)
mNe
ptun
e/M
HJ - hot hostwarm & cool Jupitersmulti-transiting
0306090
120150180 HJ - cool host
warm & cool Jupiterssub-Saturnsmulti-transiting
Figure 7. Projected obliquities versus the planet to star mass ratio. Same color scheme as in figure 6. Cool hoststars show good alignment for massive planets (mass ratio above ≈ 0.0005) as long as these are not WJs. Hot hosts displaymisalignment for all mass ratios but retrograde systems are absent for mass ratios above ≈ 0.002.
Teff as well. All close in planets orbiting cool host
stars with a planet to star mass ratio larger than 0.0005
orbit prograde and are consistent with low obliquities.
For hot stars no retrograde systems are observed with
ratios larger than 0.002 – a cut off four times larger than
that for cool stars. Also prograde systems with signif-
icant misalignment are observed for high m/M , in this
temperature range. For the hottest host stars the mass
ratios of close in planets cover a smaller parameter range
and do not display any aparent trend.7
We note that the relatively small spread in host star
masses - compared to the spread in planetary masses -
leads to similar correlations of the projected obliquity
with mp and m/M?.
3.1.5. Planets with large separations have high obliquityhosts
Figure 8 displays projected obliquity measurements
over the orbital separation, a/R. The correlation dis-
cussed by (Albrecht et al. 2012a) – that close in (a/R .12) giant planets orbiting cool stars have aligned orbits
and further out systems have a large dispersion in obliq-
uities – is present in the current data set. There are two
exceptions, WASP-94A b a HJ in a binary star system
7 Low mass stellar companions and double star systems arepredominantly aligned for even hotter primaries, with notable ex-ceptions. However formation and evolution in such systems differsand we therefore do not include them here.
(Neveu-VanMalle et al. 2014) and the WASP-60 system
(Mancini et al. 2018). Both have effective temperatures
above 6100 K. For hot host stars (middle panel in Fig-
ure 8) four misaligned systems with a/R < 7 are known.
Additional observations will show if there is an increase
of misaligned stars in this temperature bin for close in
giant planets. There is no obvious trend with a/R in
the relatively small sample of very hot host star sys-
tems. This is consistent with the tidal picture discussed
below, § 4.1.
3.1.6. Aligned systems are very well aligned
What is the dispersion in obliquities for systems which
are ”aligned”? The dispersion might be a useful diag-
nostic in determining which process led to alignment, as
dissipate processes would lead to a small overall value
with a small dispersion. In Figure 8 top left panel we
display all projected obliquity measurements (now rang-
ing from −180 to 180 deg) in systems with cool hosts
which systems which have prograde orbits and excellent
measurement uncertainties of 2 or less. For guidance
we also display the (not projected) Solar Obliquity with
respect to the invariable plane, 6.2. All these cool hosts
have projected obliquities with respect to their compan-
ions well below the Solar value. In the sample with cool
hosts the mean projected obliquity of the cool HJ hosts
(a/R < 10 & m > 0.3MJupiter) sample is 0.23 while
the standard deviation is 0.91, and the formal average
measurement uncertainty is 0.82. These values for the
Obliquity 11
101 102
orbital separation (a/R )0
306090
120150180 HJ - very hot host
warm & cool Jupiters
0306090
120150180
proj
. obl
iqui
ty HJ - hot hostwarm & cool Jupitersmulti transiting
0306090
120150180 HJ - cool host
warm & cool Jupiterssub Saturnsmulti transiting
5 10 15 20
5
0
5solar obliquity
solar obliquity
Figure 8. Projected obliquities displayed over scaled separation a/R The inset highlights systems with cool hostsand measurement uncertainty below 2 deg. Among these, systems harboring a HJ display a mean projected obliquity of 0.2 degand a spread of 0.9 deg. While there might be a trend towards a large fraction of alignment for close in systems with hot hosts,more data would be needed to confirm this. Olquities of A type host stars do not display any dependency on orbital separation.
dispersion and formal measurement accuracies indicate
that for these systems the measurements are fully consis-
tent with perfect alignment among this class of systems.
This is an indication that at some point during the for-
mation or evolution of the system a dissipative process
has reduced the obliquities. If confirmed by additional
high accuracy measurements of additional cool hosts or-
bited by HJs and an careful ensemble study of aligned
systems with larger uncertainties de-convolving the un-
derlying distribution from the measurement uncertain-
ties then this further indicates that HJs orbiting cool
hosts on aligned orbits obtained this alignment through
tidal dissipation and that alignment might not be pri-
mordial. This will be discussed in more depth in the the
forthcoming publication by Stefansson et al. in prep. It
is also worth noticing that these measurements high-
light that given high enough SNR RM measurements
and a careful analysis researchers are able to measure
projected obliquities to an accuracy below one degree,
alleviate some of the concerns discussed earlier (§ ??).
3.1.7. Obliquities and stellar age
Figure 9 displays the projected obliquities as function
of stellar age. HJ systems with ages above ≈ 3 Gyr have
projected obliquities consistent with alignment, as first
reported by Triaud (2011) for a smaller sample only in-
cluding stars within a narrower mass range where stars
evolve quickly (allowing for precise age estimates). As
discussed by Albrecht et al. (2012a) this correlation does
probably not represent a direct obliquity – time rela-
tionship; rather this relationship might be connected to
the change in stellar structure during the MS lifetime
(i.e., stars cool and gain larger convective zones as they
age), which then in turn might lead to tidal alignment.
Recently Safsten et al. (2020) confirmed that the corre-
lation apparent in Figure 9 is connected to the stellar
temperature and not the age of the the system, and as
we will see below (§ 4.1) therefore most likely to tidal
alignment.
12 Albrecht, Dawson, & Winn
0 2 4 6 8 10 12Age (Gyr)
0
30
60
90
120
150
180pr
ojec
ted
obliq
uity
(deg
) HJ - cool hostHJ - hot hostHJ - very hot host
Figure 9. P¯
rojected obliquities displayed over system agefor hosts stars. The color scheme of this plot is the same asfor Figure 6. As all other system parameters also the agesare listed in tab. 2.
3.1.8. Very young systems with (close in) giant planets arealigned
Recently RM observations as well as stellar inclina-
tion measurements via the v sin i method (§ 3.4) as well
as interferometric measurements (§ 3.3) have enabled
first obliquity measurements in very young systems with
(short) period giant planets. In Fig. 10 we show pro-
jected obliquities as well as inclination measurements
for systems younger than 1 Gyr and age uncertainties
below 250 Myr.8 AU Mic b is a recently discovered
(Plavchan et al. 2020) transiting planet orbiting a young
(22 Myr) star which also hosts an edge on debris disk.
The inclinations of the planetary orbit and debri disk
are therefore consistent with alignment. A number of
authors (Addison et al. 2020; Hirano et al. 2020a; Palleet al. 2020; Martioli et al. 2020) report good alignment
for stellar spin and planetary orbit. Interferometry, dis-
cussed below, allowed recently a measurement of the
projected obliquity in β Pic (Kraus et al. 2020). We
note that while this is also a young (26 Myr) system
with a massive gas giant (and an edge on disk) this is
not a compact system rather the planet travels on an
decade long orbit. The well aligned host DS Tucanae A
(Zhou et al. 2020) has an age of 45 Myr. Additional
information comes from inclination measurements via
the v sin i method, which is well suited for young stars
which often display fast rotation and large periodic
8 We note that KELT-9 has a large misalignment (λ = 85.01 ±0.23 deg, Gaudi et al. 2017). However while this appears to be ayoung system, its age is given by ≈ 300 Myr (Gaudi et al. 2017),it does not have a formal uncertainty. We omit it in this plot.
10 2 10 1 100
Age (Gyr)0
30
60
90
120
150
180
proj
ecte
d ob
liqui
ty (
deg)
dataHJ - very hot hostwarm & cool Jupiterssub-Saturns
0
30
60
90
i=i o
i (de
g)
i datawarm & cool Jupitersmulti-transiting
Figure 10. Spin-orbit alignment in young systems.This figure displays projected obliquity measurements (cir-cles) and stellar inclination measurements (triangles) of sys-tems younger than billion years and with age uncertaintiesless than 300 million years.
light curve modulations, presumably from spots. The
youngest system which appears to be misaligned is TOI-
811 (0.117+0.037−0.043 Gyr, Carmichael et al. 2020)9. However
the companion has a mass fully consistent with being a
Brown Dwarf (m = 59.9+8.6−13 MJup) rather than having
a mass in the planetary regime. The youngest plane-
tary mass object with an misaligned star is Kepler-63 b
(0.210 ± 0.045 Gyr, Sanchis-Ojeda et al. 2013). It is
worth noticing that the young planets on aligned orbits
belong to the sub Saturn as well as WJ and CJ classes.
These types of planets often travel on misaligned or-
bits when observed in older systems (see Figs. 7 and 8),
yet these few younger systems are aligned. These few
observations suggest that giant planets which have ar-
rived in the vicinity of their host stars at an early time
(. 0.1 Gyr) did so by a process which does maintain or
lead to a low obliquity. This would be consistent with
these younger planets arriving on their orbits via in situ
formation or disk migration. This is also consistent with
large oblquities orginating from dynamical processes as
these tend to work on timescales often considered to be
longer than a few Myr (§ 4). However see also Dawson
& Johnson (2018) for a discussion on timesclaes.
3.1.9. Stellar Obliquities and orbital Eccentricities
Wang et al. (in perp.) highlights that HJs orbiting
cool stars travel not only on well aligned orbits (§ 3.1.5)
but these orbits also appear to be circular, while Jupiters
orbiting hotter stars have eccentric orbits for smaller
separations, Figure 11. We note that this plots contains
a number of biases, one of which is that planets orbit-
9 We use here the value from isochrone fitting for TOI-811 (as wedid for other systems when ever available) rather the value for fromgyrochronology, which however is fully consistent (93+61
−29 Myr)
Obliquity 13
3000 4000 5000 6000 7000 8000 9000Teff (K)
3
10
30
100
a/R
circular & alignedmisalignedeccentric
Figure 11. Orbital eccentricity and misalignmentThe figure display systems in the host star effective temper-ature and orbital separation planet. Systems which mea-surements are consistent with aligned, circular orbits are in-dicated by gray systems. If they have a secure (3 − σ) ec-centricity measurement then a open green circle is added.Securely misaligned systems have orange symbols.
ing hotter stars might have a good enough eccentricity
measurement to be included in this sample (σ < 0.3)
but still significant eccentricities can not be excluded
for these systems. Nevertheless the plot does display
that for cooler stars and close in orbits both large ec-
centricities as well as large misalignments are rare. This
might suggest that not only obliquities are dampened
by tides raised by the planet on the star (§ 4.1), also
some eccentricity damping occurs inside the star. We
note that using canonical values suggest that most of
the tidal energy is dissipated inside the planet, and that
the planetary circularization timescale is shorter than
the stellar circuilarization timescale (e.g. Schlaufman &
Winn 2013). Also hot stars tend to be younger than
their less massive cooler counterparts and this sample is
no exception. Safsten et al. (2020) recently showed that
indeed the trend of circular orbits out to larger orbital
separations is connected to age.
Given the small number of systems which eccentricity
and obliquities might not be affected by tidal circular-
ization and/or tidal alignment we postpone a discussion
about evidence for a dynamically hot (large obliquities
and eccentricities) versus a dynamically cold (low eccen-
tricities and alignment).
3.1.10. Obliquities and compact multi transiting planets:alignment with notable exceptions
Systems in which multiple planets are transiting is in-
teresting in the context of obliquity measurements, as
the planets’ orbits have low mutual inclinations. Fab-
rycky et al. (2012, 2014); Xie et al. (2016); Herman et al.
(2019) determined that compact multi transiting planet
systems tend to have low mutual inclinations similar
or even smaller than in the Solar System. Dai et al.
(2018a) found that multi transiting systems harboring
Ultra Short Period (USP) planets tend to have some-
what larger mutual inclinations & 7. Recently Masuda
et al. (2020) (see also the work by Herman et al. 2019)
found that systems harboring Cold Jupiters (CJs) and
close in super Earths have an inclination dispersion of
∼ 12 which further decreases with higher planet multi-
plicity.
Albrecht et al. (2013b) concluded based on obliquity
measurements in five compact multi transiting systems
that these systems have low obliquities. To date pro-
jected obliquities or inclinations have been measured in
14 systems. Measurements in eleven systems are con-
sistent with low obliquities: Kepler-30 (Sanchis-Ojeda
et al. 2012), Kepler-50 & 65 (Chaplin et al. 2013),
Kepler-89 (Hirano et al. 2012), Kepler-25 (Albrecht et al.
2013b), WASP-47 (Sanchis-Ojeda et al. 2015), Kepler-
9 (Wang et al. 2018), HD 10635 (Zhou et al. 2018),
TRAPPIST-1 Hirano et al. (2020b) HD 63433/TOI-1726
(Mann et al. 2020; Dai et al. 2020), and TOI-451 New-
ton et al. (2021).10 Two systems have large spin orbit
angles, Kepler-56 (Huber et al. 2013) & HD 3167 (Dalal
et al. 2019). K2-290 A a coplanar two planet system
in a wide binary has a backward spinning star (Hjorth
et al. 2021). As we will discuss in the following section
the reasons for the large obliquities in some of these sys-
tems are not the same. More than one mechanism can
lead to large spin orbit angles in coplanar systems.
3.2. Asteroseismology
If long duration, high cadence, high Signal-to-Noise
time series (RV or photometric data) are available then
stellar pulsation frequencies can be determined. By an-
alyzing the amplitudes, dispersion, and positions of fea-
tures in frequency space inside information about the
star can be obtained. Among such information is the
inclination of the stellar spin axis (Gough & Kosovichev
1993; Gizon & Solanki 2003; Chaplin & Miglio 2013),
see figure 12.
In the non rotating frame of an observer azimuthal
modes (m) of a pulsating star are separated in frequency
as m 6= 0 modes either travel with or against the stellar
rotation. Therefore modes of radial order n and an-
gular degree l are split into (2l + 1) modes. The new
frequency (and therefore the separation) of the modes
νnlm does not only depend on the azimuthal order m, it
also depends on an average angular velocity of the star,
10 We exclude Kepler-410 (Van Eylen et al. 2014) here as themutual inclination between the planets orbits is unknown.
14 Albrecht, Dawson, & Winn
1920 1940 1960 1980 2000 2020
Frequency [µHz]
0.0
0.1
0.2
0.3
0.4
0.5
Pow
er[p
pm2
]
i = 45
i = 82.5 (best f i t )
a) b)
Figure 12. Limits on stellar inclination from light curves Power spectra obtained from light curves observed by theKepler spacecraft for the Kepler-56 and Kepler-410 host stars. The figures are taken from the work by Huber et al. (2013)and Van Eylen et al. (2014). panels a — Shows some gravity-dominated (top row) and pressure-dominated (lower row) mixeddipole modes, respectively. For Kepler-56, a subgiant, the modes are split into triplets by rotation and the m = ±1 and m = 0modes can be clearly separated. The dispersion of the modes is lower than their separation in frequency space. From their nearequal amplitudes an stellar inclinations of i = 47 ± 6 deg can be deduced. Panel b — Kepler-410, a hotter less evolved star,has azimuthal modes less clearly separated in the power spectrum. Even so a model with large amplitude of m 6= 0 modes, anequatorial view (red), gives a much better representation of the smoothed data (dark gray) than an inclined model (green).
Ω (Gizon et al. 2013, equ. 1),
νnlm = νnl +mΩ
2π. (14)
The amplitude of these different modes is expect to by
nearly equal. The measured amplitude ratio between the
different azimuthal modes depend on the viewpoint of
the observer. The visibility of m 6= 0 modes are maximal
for an equatorial view (i = 90), while for a polar view
(i = 0) the amplitude of the m = 0 mode is maximized.
For the case of dipole (l = 1) multiplets the mode power
(E ) is given by equ. 12 & 13 in the work by Gizon &
Solanki (2003),
E1,0 = cos2 i, (15)
E1,1 = 12 sin2 i. (16)
Therefore if the m 6= 0 and the m = 0 modes can
be measured in the power spectrum and their relative
mode amplitude can be determined then i can be de-
rived. See for a detailed discussion of this mechanism
Gizon & Solanki (2003); Ballot et al. (2006, 2008) and
Kuszlewicz et al. (2019) as well as references therein for
a discussion of best practise for the retrieval of incli-
nation angles from seismic data. The successful sepa-
ration of the m = ±1 (or even higher order azimuthal
modes) and m = 0 modes and their amplitude mea-
surements require a large ratio of the mode separation
over the width of the modes. The former quantity in-
creases with faster rotation (equ. 14), while the later
quantity increases with shorter mode lifetimes, which in
turn decrease for larger Teff . This requirement limits
the number of main sequence planet host stars in the
Kepler data set (Campante et al. 2016), see also (Kami-
aka et al. 2018). Evolved stars are favorable targets as
their lower surface gravity leads to large oscillation am-
plitudes. Compare panels ”a)” and panel ”b)” in Fig. 12.
If the star hosts transiting planet(s), then the incli-
nation of the orbit(s) relative to the equatorial plane
of the host star can be readily determined as i will be
known. An advantage of this technique is that no ad-
ditional transits need to be observed, making planets
traveling on long period orbits, and importantly planets
with small planet/star radii ratios accessible to obliquity
measurements.
Asteroseismology was used by Chaplin et al. (2013) to
determine the obliquities in the Kepler-50 and Kepler-
65 systems, multi transiting planet systems harboring
small Super Earth planets. The first measurement of
a multi transiting planet system with co-aligned orbits
and a large stellar obliquity was achieved via seismic
measurements of a sub-giant Kepler-56 (Huber et al.
2013). Van Eylen et al. (2014) found agreement in ioand i for the eccentric orbit of a mini Neptune in a mul-
tiplanet system (Kepler-410). Recently a large obliquity
was measured in Kepler-408, a system with a hot sub
Earth-sized planet (Kamiaka et al. 2019).
A larger number of PLATO systems might be suitable
for determining stellar inclinations via asteroseismology.
We note that also high SNR ground based time series of
high resolution spectra could be used to determine i via
mode splitting.
3.3. Interferometry
A potential path towards overcoming our preoccupa-
tion with transiting close in orbiting planets is inter-
Obliquity 15
Figure 13. Spatially resolved Br γ absorption line ofβ Pic Figures taken from Kraus et al. (2020). The top paneldisplays the flux measured in the Br γ line of β Pictoris invelocity space. The two lower panels displays the differen-tial offset of the photocenters at for different wavelengths in10−6 arcsec relative to the continuum flux along the North-South (middle panel) and East-West (bottom panel) axes asderived from the interferometric measurements.
ferometry. Optical\NearIR Interferometric Long Base-
lines observations can (partially) resolve stellar surfaces
of main sequence stars in the solar neighborhood, solv-
ing the spatial resolution challenge without the need to
resort to transits. If equipped with a spectrograph which
can resolve stellar absorption lines then this allows for
example for the determination of the stellar rotation axis
as projected on the sky plane (e.g. Albrecht et al. 2010).
A projected baseline11 oriented parallel to the stellar
equator will resolve (partially) the stellar disk and the
photo centers of the red and blue wings of stellar ab-
sorption lines, can be resolved. They will have different
interferometric phases, i.e. the position of the fringe
11 The projection of the line connecting different telescopes inan array, as seen by the target.
pattern will be shifted slightly, a small fraction of 2Π.
Conversely, if the baseline would be oriented parallel to
the stellar spin axis then the resolving power of the in-
terferometer along the stellar equator is reduced to the
resolving power of a single telescopes, the star remains a
point source and no phase shift between the red and blue
wings would be observed. See Petrov (1989) and Chelli
& Petrov (1995) for details. There is also the poten-
tial of measuring the stellar inclination along the LOS
(Domiciano de Souza et al. 2004) for solar like differ-
ential surface rotation. For marginally resolved targets
the differential phase can be calculated with the follow-
ing formula given by Lachaume (2003, thier equ. B.5)
and Le Bouquin et al. (2009),
ρ = −2πpB
λwavelength[rad]. (17)
The measured differential phase shift (ρ) between in-
terferometric fringes of two photo centers (i.e. the blue
and red shifted halves of the photosphere) depends on
their separation on the sky (p), the projected baseline
length between the telescopes (B), and the observing
wavelength (λwavelength). This technique has been used
in the debri disk system Fomalhaut12 (Le Bouquin et al.
2009), and more recently in the β Pictoris system (Kraus
et al. 2020). β Pictoris is a young (26 Myr) system with
an edge on disk and a massive gas giant on an decade
long orbit, Fig 13.
These studies have targeted bright fast rotating stars
and their pressure broadened line Brγ line as currently
there is no instrument available which can resolve iron
lines in late type main sequence stars or obtain differ-
ential phase measurements on fainter targets. However
preparations for high resolution instruments (with an
resolution power of up to a few ten thousands) are cur-
rently underway at the CHARA array (Mourard et al.
2018) and the VLT Interferometer (Kraus 2019). To fur-
ther increase the magnitude of potential targets these in-
struments will make use of fringe tracking, significantly
increasing the integration time of the spectrographs con-
nected to the interferometer.
However this technique not only requires the combi-
nation of high spatial and spectral observations. The
interpretation of the orientation of the stellar spin axes
- with the sky plane as reference - in the context of stel-
lar obliquities in exoplanet systems (or double star sys-
tems) does require knowledge of the orbital orientation
on the sky plane as well. Specifically - the longitude of
12 Fomalhaut appears to host a dispersing collision induced dustcloud and not an giant exoplanet as originally thought in its diskGaspar & Rieke (2020)
16 Albrecht, Dawson, & Winn
the ascending node (Ω) - not obtained by RV or transit
measurements. The expected release13 of thousands of
exoplanet systems with astrometric orbits as measured
by the GAIA satellite (Perryman et al. 2014) will lead to
a large pool of potential targets. GAIA will also deter-
mine io and Ω for a number of known RV systems with
giant planets on few year orbits. However a significant
number of these systems might be to faint to be studied
with this technique. Intererometers can in addition be
used to search for (partial) alignment of stellar rotation
axes double star systems, in star forming regions and
stellar clusters. Thereby informing theories about the
initial conditions during star and planet formation, im-
portant for the interpretation of obliquity measurements
as discussed in section 4.
For the fastest rotating stars departures from a purely
spherical shape caused by centripetal forces can be used
to learn about obliquities, without the need to spectrally
resolving the stellar lines. Albeit this is currently only
applicable to the very fastest rotators (e.g. Domiciano
de Souza et al. 2003).
3.4. The v sin i technique
As for seismology discussed above also for the v sin i
technique only the information on io from the occurrence
of transits is used, no transit observations are required.
A difficultly shared with the seismic determination of
stellar inclinations is the flattening of the sine function
near 90 as well as a degeneracy in i as mentioned at
the begin of this section. Assuming solid body rotation
we find,
i = sin−1
[v sin i
vprior
]= sin−1
[v sin i
(2πR/Prot)
]. (18)
Therefore measurements of v sin i and prior information
on the rotation speed (vprior) could lead to an estima-
tion of i. Measuring the nominator in the above equa-
tion is challenging for slowly rotating stars as broaden-
ing of stellar lines might not be dominated by rotational
Doppler shift. Obtaining the denominator may be done
via a number of routes, see Maxted (2018) for a review.
Most commonly two paths are taken. One might esti-
mate vprior assuming a particular dependency of v on
stellar mass and age e.g., square root brake down law
Skumanich (1972, 2019). Alternatively one might de-
termine R, and Prot. Rotation period measurements
might for example be achieved via measurements Quasi
Periodic Variations (QPV) in long duration photomet-
ric time series. Periodic flux variations are associated
13 https://www.cosmos.esa.int/web/gaia/release
to the stellar rotation period via the rotation of stellar
surface features e.g., spots in and out of view.
We would like to highlight the results by Masuda &
Winn (2020). They highlight that care has to be taken
when deriving marginalized confidence intervals for i
and we refer to that work for details. They highlight
that v and i might not necessarily always independent
e.g. in clusters. Currently more important was an often
made mistake, assuming that v and v sin i are indepen-
dent variables, which they are not. By measuring one
we gain some information about the other. A measure-
ment of v sin i gives knowledge on v (lower values of v
are disfavored) and vis versa. Using their equ. 10 when
deriving uncertainty intervals – rather than simply and
incorrectly applying equ. 18 when deriving posteriors
– incorporates this dependency properly. We highlight
all single measurements using the procedure outlined by
Masuda & Winn (2020) in Figure 2.
Using the v sin i technique Guthrie (1985) and Abt
(2001) tested for, and did not find, a tendency for stars
to be preferentially aligned with the Galactic plane. In
double star systems spin-spin alignment or orbit-spin
alignment can be probed (e.g. Weis 1974; Hale 1994;
Glebocki & Stawikowski 1997; Howe & Clarke 2009),
but see also Justesen & Albrecht (2020) who showed
that the often quoted result that double stars with sep-
aration less than a few tens of au tend to be aligned,
can not be confirmed with the data at hand. Schlauf-
man (2010) was the first to use this technique for stars
hosting transiting planets, finding evidence that more
massive stars have high obliquities, coming to consis-
tent result as Winn et al. (2010) using a different ap-
proach. More recently this technique was used on very
young hosts of transiting planets. Such stars often have
significant QPVs and rotation measurements and there-
fore obtaining a measure of v is somewhat easier thanfor older main sequence stars. In addition these stars
tend to have a large v and therefore sin i thanks to their
youth. Therefore both terms in equ. 18 can be deter-
mined with some accuracy. In addition there are only
few obliquity measurements for the youngest systems,
see § 3.1.8. The v sin istar technique was also used to
demonstrate that stellar spins in the open cluster NGC
2516 have an isotropic distribution or at most moderate
alignment Healy & McCullough (2020).
In the near future we might expect to obtain more in-
teresting results from the v sin i method (Quinn & White
2016) based on TESS transiting systems for which pa-
rameters appearing in equ. 18 should be obtained with
higher accuracy and fidelity than before. GAIA data im-
proves stellar radii measurements, very high resolution
spectrographs (e.g. ESPRESSO, PFS, XPRES) allowing
Obliquity 17
0 10 20 30 40 50 60Period [days]
0
100
200
300
400
500
Pow
er
−0.4 −0.2 0.0 0.2 0.4Rotation phase
0.995
1.000
1.005
Rela
tive flu
x
−0.10 −0.05 0.00 0.05 0.10
0.992
0.994
0.996
0.998
1.000
Rela
tive flu
x +
consta
nt
E = 3
E = 4
Expected, forψ = 0
−0.10 −0.05 0.00 0.05 0.10Time from midtransit [days]
0.992
0.994
0.996
0.998
1.000
Rela
tive flu
x +
consta
nt
E = 15
E = 16
Expected, for ψ = 0
Figure 14. QPVs and spots crossing transits. Fig-ures taken from Sanchis-Ojeda & Winn (2011). The toppanel shows the Lomb-Scargle periodogram of Kepler pho-tometry taken of HAT-P-11, indicating a rotation period of30.5+3.1
−3.2 days. The second panel from the top shows the outout-of-transit flux phase folded over this period, illustratinga Quasi Periodic Variability in the light-curve. The lower twopanels show two pairs of consecutive transit epochs. Giventhe orbital period (4.9 days) and rotation period a changeof ≈ 60 deg in longitude between consecutive transits is ex-pected. An aligned orbit would lead to spot crossing eventsin consecutive transits, as indicated by the red lines in thetwo lower panels. The data does not match a model withaligned spin and orbital axes (red line).
for finer sampling of late type stellar spectra obtained
with higher SNR on bright TESS host stars and a bet-
ter calibration of stellar surface motion (e.g. Doyle et al.
2014) might lead to improved v sin i measurements also
for slower rotating stars.
3.5. Starspots
3.5.1. Quasi-Periodic Variation
If star spots (or any other semi stationary stellar sur-
face feature) are present then the flux received from a
star varied with the stellar rotation frequency or multi-
ples thereof as the stellar rotation transports spots over
the limb darkened stellar disk in and out of view. To-
gether with the slow evolution of the spots themselves
this gives rise to out of transit Quasi-Periodic Variation
(QPV) in flux on the time scale of the stellar rotation
period. As mentioned in the above section periods de-
rived from QPVs can be used to estimate v and thereby
leading to an estimate of i via the v sin i method. How-
ever the amplitude of the QPVs itself can be used to
obtain information on i.
The amplitude of the QPVs depends not only on con-
trast, distribution and occurrence rate of the surface fea-
tures but on i as well. Late type stars seen nearly pole-
on do display a lower photometric variability than stars
seen equator on, everything else being equal.14 This
statement ”everything else being equal” might not be
as easy to fulfill as hoped. For example stars for which
we can detect OPVs might be a particular subset of
stars, e.g. seen more equator on or of a particular stellar
type. Therefore such studies should ideally encompass
two populations which are similar in as many aspects as
possible apart from the planet population. Mazeh et al.
(2015b) pioneered the usage of this geometric effect for
obliquity studies, see § 3.6.
3.5.2. Starspot-tracking method
If starspots are present then these might not only lead
to QPV out of transits discussed above. During transits
spots might be covered from view by the planet. This
then results in an increased flux level for this part of the
transit light curve. A sequence of spot covering events
during transits (or the absence of such a sequence)
can be used to deduce stellar obliquities (Sanchis-Ojeda
et al. 2011; Desert et al. 2011). See figure 14.
In addition phase information from the OPV and tran-
sit crossing events can be combined to drive information
on obliquity. Stellar flux decreases while (the majority
of) spots are located on the approaching stellar surface,
and increases with spots located on the receding stellar
surface. Therefore spot coverage during the first half of
a transit and decreasing stellar out of transit flux indi-
cates a prograde orbit and vise versa (Nutzman et al.
2011; Mazeh et al. 2015a; Holczer et al. 2015).
For long time series but low SNR detections of in tran-
sit spot coverage Dai et al. (2018b) developed a statisti-
cal test for correlations between the anomalies observed
in a sequence of eclipses. This test allows for the deter-
mination of alignment.
The first obliquity measurement in an multi transit-
ing system (Kepler-30) - indicating good alignment -
was carried out by Sanchis-Ojeda et al. (2012) tracking
starspot coverings during transits as well as QPV out
of transit. It is worth noticing that methods relying on
star spots to deduce information on stellar obliquities
are complementary to the RM method (§ 3.1) as de-
tectable spots are more prevalent in the photospheres
of late type stars, for which the stellar rotation speed
is relatively slow leading to small RM amplitudes. The
TESS mission aims at detecting transiting systems with
low mass host stars. However the spot methods do ben-
efit from long time series. This makes TESS systems
detected near the elliptical poles more suitable for these
methods as TESS observes the elliptical poles for one
year.
14 Higher mass stars might display polar spots.
18 Albrecht, Dawson, & Winn
3.6. Key results from ensemble studies
• Developing and using the QPV approach Mazeh
et al. (2015b) found that host stars with effec-
tive temperatures below ∼ 5700 K tend to have
good alignment with planets out to orbital periods
of ≈ 50 days. Li & Winn (2016) reanalyzed the
data and found that ”the evidence for alignment
becomes weaker for systems with an innermost
planet period & 10 days, and is consistent with
nearly random alignment for longer orbital peri-
ods (& 30days).” Mazeh et al. (2015b) also found
that hotter stars tend to be more misaligned. Im-
portantly most of these stars do not harbor HJs
but smaller and further out planets.
• Campante et al. (2016) employed asteroseismology
to study 24 Kepler Targets of Interest (KOI) with
planets and planet candidates in single transiting
and multi transiting systems with periods up to
180 days and sub Neptune sizes. These authors
found that their astronomic inclination measure-
ments are consistent with good alignment.15.
• Also the v sin i method was further employed
(Walkowicz & Basri 2013; Hirano et al. 2014; Mor-
ton & Winn 2014). Winn et al. (2017) and Munoz
& Perets (2018) used data from the California-
Kepler Survey (CKS, Petigura et al. 2017) sample.
They did find that their sample containing single
and multi transiting systems is consistent with
good alignment, with the exception of HJ hosts.
• Most recently Louden et al. (2021) analyzed a sub-
set of the Winn et al. (2017) sample. Improving
on the former results with the use of a comparison
sample which has similar stellar properties to the
planet hosting sample but without transiting plan-
ets. These authors find low obliquities for hosts
below 6250 K and a distribution consistent with
random orientation for hotter stars. This confirms
the earlier result by Mazeh et al. (2015b), using a
different technique.
To summarize, these studies suggest that i) systems
with cool host stars have good alignment regards of
planetary orbit and planetary size/mass, and ii) hot
host stars tend to have large obliquities, again regard-
less of planet, size distance and multiplicity. We note
that there is tension between these measurements and
RM measurements of small planets orbiting cool stars
15 One of these systems, Kepler-408 was later found to be mis-aligned
on highly misaligned orbits (e.g., HAT-P-11., HAT-P-
18 and WASP-107) see also § 3.1.9.
3.7. Other methods
3.7.1. Rotational Doppler beaming
Conceptually related to the RM effect, Groot (2012)
and Shporer et al. (2012) evaluated the potential of rel-
ativistic beaming caused by the stellar rotation or the
photometric RM effect for obliquity measurements. The
apparent brightening of the approaching and darken-
ing of the receding stellar surface areas due to Doppler
beaming, will lead to a λ dependency of eclipse light
curves. Shporer et al. (2012) give the following equation
to estimate the photometric amplitude for this effect,
APRM ≈ 10−5 v sin I
10 km s−1
(rR
)20.1
. (19)
These authors concluded that due to the small am-
plitude of the effect obliquity measurements will be
challenging. The most promising targets appear sys-
tems containing fast rotating early type stars and white
dwarfs. For white dwarfs many of the other measure-
ment techniques available to measure ψ will not be ap-
plicable.
3.7.2. Gravity darkening, fast rotators
For rotating stars the effective local gravity near the
stellar equator is reduced relative to the stellar poles, re-
sulting in a larger scale height of the photosphere. For
latitudes near 90 a specific optical depth is reached at
lower temperatures than at latitudes closer to the pole.
This effect leads to increased brightness towards the stel-
lar poles. This is superimposed onto the radial symmet-
ric center-to-limb brightness change due to stellar limb
darkening. The local temperature, Tl, can be described
by the von Ziepel theorem (Barnes 2009, and references
therein),
Tl = Tpglβ
gpβ. (20)
Here gl refers to the surface gravity. The indices l and
p refer to the local quantities and polar quantities. The
gravity darkening parameter β has a nominal value of
0.25 for radiative stars but varies with stellar type. For
the aligned and ani-aligned case (λ ≈ 0 or λ ≈ 180)
gravity darkening is challenging to detect in a single
band light curve as it will lead to an apparent decrease or
increase in the planet to star radii ratio, for low and high
impact parameters, respectively. For |λ| = 90 and sig-
nificant gravity darkening, a symmetric light curve with
apparent brightening of the photosphere at the limb is
observed, revealing the misalignment. Other projected
Obliquity 19
Primordial
Envelope
Misalignment during accretion
Magnetic Warping
Disk dispersal
resonant excitation
Post formation
ψ
Cyclic Secular
(Kozai-Lidov)
Planet-planet
scattering
Inclined star
Inclined star or planet
Magnetic breaking
ψSpin down
resonant excitation
Secular chaos
Figure 15. Processes that create spin-orbit misalignments before (left) or after (right) planet formation.
obliquities lead to asymmetric light curves around the
transit midpoint, see Barnes (2009).
The successful observation of gravity darkening in
transiting exoplanet systems require high signal-to-noise
transit observations of fast rotating host stars reducing
the gl near the equator (equ. 20). The first observations
of this effect have been made in the Kepler-13 system,
(Barnes et al. 2011; Szabo et al. 2011) for which the
asymmetry in the light curve due to gravity darkening
is of the order of 100 ppm. Other observations include
HAT-P-7 Masuda (2015), KOI 368 (Ahlers et al. 2014)
as well as the more tentative measurements of alignment
in KOI 2138 (Barnes et al. 2015) and misalignment in
the multi planet systems KOI-89 (Ahlers et al. 2015),
which however was shown to be spurious by Masuda &
Tamayo (2020). More recently Gravity darkening was
used to determine obliquities in TESS systems Ahlers
et al. (2020a,b).
4. PROCESSES THAT INFLUENCE OBLIQUITIES
The observed obliquity distribution tests theories for
how stars and planets form and evolve, with a number of
mechanisms proposed for altering the obliquity through-
out the system’s history. Below we review how these
theories’ predictions hold up against currently available
data and which measurements would further test each
theory. We first discuss the theory that tidal realign-
ment sometimes erases the obliquities established by the
other processes (Section 4.1). We then summarize the
theory of and evidence for primordial misalignment be-
fore the planet forms (Section 4.2), post-formation mis-
alignment (Section 4.3), and changes in the stellar spin
vector that are independent of the planet (Section 4.4).
4.1. Tidal realignment
Although tidal realignment may happen last (i.e., af-
ter other processes create spin-orbit misalignments), we
discuss it first. There is compelling evidence that most
of the individual obliquities observed to date have been
altered by tides, so we should not compare the predic-
tions in subsequent sections to the observed obliquity
distribution without taking tidal realignment into ac-
count. The strongest piece of evidence is sharp change in
the obliquity distribution above the Kraft break stellar
effective temperature (Fig. 6). The Kraft Break marks
a major difference in stars’ rotation rates and structure,
implicating tidal effects. In Section 4.1.1, we discuss the
empirical consistency of a simplified tidal friction model
with observed obliquity trends. In Section 4.1.2, we de-
scribe the prospect for more complex and realistic tidal
models to account for the observed trends.
4.1.1. Simplified tidal friction model: empirical consistencywith observed trends
In the theory of equilibrium tides, tidal friction occurs
when the star rotates at a different rate and/or direction
than the planet. Fluid elements of the star closer to the
planet feel a stronger gravitational force than those fur-
ther away, stretching out the star and raising a bulge. If
the planet orbits more quickly (slowly) than the star
spins, the planet leads (lags) the bulge. The planet
stretches out the star in different directions throughout
the orbit, dissipating energy in the star. Similarly, with
a spin-orbit misalignment, the bulge rotates away from
the planet, and the planet has to stretch out the star
again and again. The bulge and planet exert a torque
on each other that transfers angular momentum to syn-
chronize and align the star. When the planet’s orbital
period is shorter (longer) than the star’s spin period,
20 Albrecht, Dawson, & Winn
the planet’s orbital angular momentum is transferred to
(from) the star’s spin angular momentum.
In general, tidal interactions dissipate energy and ex-
change orbital and rotational angular momentum. Tides
tend to circularize orbits, align rotational and orbital
axes, and synchronize the rotational and orbital fre-
quency. We refer to Zahn (2008); Mazeh (2008) and
Ogilvie (2014) for reviews on tides in binary and exo-
planet systems.
Tidal friction can also be produced by dynamical
tides, which involve exciting waves within – rather than
raising a bulge on – a star. In the radiative zone of a
star, tides generate gravity waves that are damped and
dissipate energy (Zahn 1977). In Section 4.1.2, we will
discuss the contribution of inertial waves.
The observed trends between obliquity vs. planetary
and stellar properties (Table 1) are broadly consistent
with our expectations for tidal realignment (Winn et al.
2010; Albrecht et al. 2012b). Stars orbited by more mas-
sive (Fig. 7) and/or closer planets (Fig. 8) – which exert
stronger tidal forces – are more likely to be aligned. Fur-
thermore, planets can more effectively align stars with
stronger tidal dissipation, that shed angular momentum
through magnetic braking as they realign, and/or rotate
slowly enough that the planetary orbital frequency dom-
inates the tidal forcing frequency. These are the distinc-
tions between stars with stellar effective temperature
below and above the Kraft Break Teff ' 6250 K (Fig.
6). In fact, the closest HJs with highly accurate mea-
surements orbiting cool stars are aligned to, and have a
dispersion in λ of, less than 1 deg (Fig. 8).
A simple realignment timescale that encapsulates
these scalings is
1
τeq=
1
τeq,7
( q
10−3
)2(a/R
7
)−α
. (21)
For the spin synchronization of double binary stars sys-
tems, the empirical calibrations are τeq,7 = 2.8× 1011 years
and α = 6 for stars where dissipation primarily occurs
in the convective envelope via equilibrium tides and
τeq,7 = 4.6 × 1015 (1 + q)5/6 years and α = 17/2
for stars that lack (or have insubstantial) convective
envelopes (Zahn 1977) via dynamical tides. In Section
4.1.2, we will discuss the use of this equation in more
complex and realistic tidal evolution models.
We plot the observed obliquities vs. tidal timescales
in Fig. 16, using the Zahn (1977) parameters. The
data are consistent with the α (i.e., the a/R scaling)
from Zahn (1977) but do not strongly constrain α or re-
quire a different α for hot vs. cool stars. The data are
consistent with the relative τeq,7 for hot vs. cool stars
from Zahn (1977) but much shorter in absolute terms
(i.e., the observed alignment timescale must be shorter
than τeq,7). Both hot and cool stars have low obliquities
within a cut-off timescale and exhibit a range of obliq-
uities beyond.
The realignment timescale spans many orders of mag-
nitude, making it difficult to detect obliquity time evo-
lution in a sample of main sequence stars. The apparent
break with age seen in Figure 9 at ∼ 3.5 Gyr is more
likely a manifestation of the temperature trend: in the
current sample, HJ systems older than ∼ 3.5 Gyrs have
host stars with Teff < 6250 K (Albrecht et al. 2012b; see
Safsten et al. 2020 for a similar conclusion based on a
Bayesian evidence odds ratio computation using hierar-
chical modeling of the temperature vs. age dependence).
The hypothesis that tidal interactions have signifi-
cantly sculpted the stellar obliquity distribution has a
large, unresolved problem: a short period planet does
not have much orbital angular momentum to spare for
realigning a star. The ratio of the planet’s orbital an-
gular momentum (Lorb) to the star’s spin angular mo-
mentum (S?) is of order unity:
Lorb
S?=
mna2
k?MR2Ω?
∼ 2.5
(0.1
k?
)(m/M
0.001
)(a/R
5
)2(n/Ω?
10
)(22)
where k? is the stellar moment of inertia constant, Ω?is the stellar rotation angular frequency, and n is the
planet’s orbital angular frequency. Significantly altering
the magnitude and/or direction of the stellar spin typi-
cally requires shrinking the planet’s orbit to within the
tidal disruption limit. Furthermore, in order for us to
catch all hot Jupiters orbiting cool stars in an aligned
state, the ratio of the realignment timescale to the or-
bital decay timescale – which scales with Lorb
S?– must
be very small (. 10−3), which is not what we expect
from the simple tidal models above. More complex tidal
models offer solutions to these problems.
4.1.2. Prospects for more complex, realistic tidal models toenable realignment without complete decay
Given the compelling evidence that tidal realignment
has occurred in many observed systems, several theo-
ries have been proposed to enable the planet to realign
the star without tidal disruption and to account for the
observed trend with stellar effective temperature:
Planets with orbital angular momentum to
spare: For some individual systems — featuring mas-
sive and/or widely separated planets and/or slowly ro-
tating (i.e., cool) stars — Lorb
S?(Eqn. 22) is not unity but
10 or more (e.g., Hansen 2012; Valsecchi & Rasio 2014),
and the realignment timescale is shorter than the orbital
Obliquity 21
100 102 104 106 108 1010 1012
tau (yr)
0
30
60
90
120
150
180|
| (de
g)HJ - cool hostHJ - hot hostHJ - very hot hostwarm & cool Jupiterssub Saturnsmulti transiting
Figure 16. Projected obliquities of exoplanet systems as function of a relative tidal-alignment timescale(Equation 21 with calibrations from Zahn 1977). Multi transiting planets are marked by black circles. The constantsin Equation 21 differ for host stars with temperatures lower than 6250 K (blue symbols) and hotter stars (red symbols). Notethat both timescales have been re-normalized by dividing by 5 · 109. We omit here the β Pictoris system, as the planet has adecade long orbital period and no meaningful tidal alignment occurs.
decay timescale. These planets may be able to realign
their stars without undergoing much orbital decay over
the star’s lifetime.
Inertial wave tidal dissipation: Tidal interactions
with planets can cause inertial waves driven in the con-
vective zone by Coriolis forces as the star rotates. For
misaligned systems, there are components of the tide
with forcing frequency Ω? that only affect the spin di-
rection and do not cause orbital decay (e.g., Lai 2012;
Damiani & Mathis 2018). Other components of the
tide that cause orbital decay, with forcing frequency
2(n−Ω?), are inactive when 2(n−Ω?) > 2Ω?, which is
usually the case for hot Jupiters orbiting cool stars. In-
ertial wave tidal dissipation drives the obliquity to equi-
libria at ψ = 0, 90, 180.
Steeply frequency-dependent tidal dissipation:
The tidal dissipation efficiency could be a steep func-
tion of the tidal forcing frequency (Penev et al. 2018;
Anderson et al. 2021). If tidal dissipation is much more
efficient at longer orbital periods, the hot Jupiter can
realign and decay but stall aligned when it gets close to
the star. The temperature trend may be due to less effi-
ciency and/or a different frequency dependence for tidal
dissipation in hot stars.
Outer realignment: The planet could realign just
an outer layer of the star (e.g., Winn et al. 2010),
which would somehow remain decoupled from the in-
terior. Very hot stars lacking a convective outer layer
would not be realigned. Moderately hot stars would be
realigned less easily due to a lack of magnetic braking
(e.g., Dawson 2014), if their convective outer layers cou-
ple more strongly to the interior, and/ or if their tidal
dissipation is less efficient.
HJs misalign hot stars: Another possibility is that
instead of tidal interactions realigning cool stars, they
misalign hot stars. Cebron et al. (2013) suggest that hot
Jupiters could misalign stars through a hydrodynamic
instability known as the elliptical instability, in which
streamlines in a rotating fluid become tidally distorted,
causing turbulence and tilting the star. This instability
requires a stellar rotational period shorter than 3 times
the orbital period, leading to misalignments for systems
with around hot (rapidly rotating) stars and/or long or-
bital periods. Further work is needed to better under-
stand whether the dissipation is strong enough to cre-
ate a significant misalignment (e.g., Barker & Lithwick
2014) and the distribution the misalignments expected.
The first and second explanations have a firm basis
in theory but seem unable to fully account for the ob-
served trends. Figure 17 presents a toy model popula-
tion synthesis (described in detail in Appendix B) com-
paring the obliquity distributions resulting from the first
four explanations above to the observed population with
a/R < 10 and m > 0.5MJupiter. The top panel dis-
plays the projected spin-orbit alignment, and the bot-
tom panel displays v sin(i) as a proxy for stellar rotation
period. In each case, free parameters are tuned to pro-
vide the best match with the observed distribution.
With classical equilibrium tides, the most massive
planets can realign their stars but lower mass hot
22 Albrecht, Dawson, & Winn
Jupiters remain misaligned, even around cool stars (Col-
umn 3). Inertial wave tidal dissipation (column 4) can
very effectively realign cool stars but, even when equilib-
rium tides operate simultaneously (e.g., Xue et al. 2014;
Li & Winn 2016), result in a population stalled at the
ψ = 180 equilibrium not seen in the observations. A
related constraint is that there are no known ψ = 180
close double star systems, which we might expect to see
if inertial wave tidal dissipation is commonly at work.
We would expect fewer ψ = 180 planets if the initial
obliquity distribution has mostly prograde planets, but
such an initial distribution seems at odds with the ob-
served obliquities of hot stars. Obliquities can also stall
at the ψ = 90 equilibrium; however, in our example,
initially retrograde systems tend to evolve to and stall
at ψ = 180 because Lorb
S?> 1 and because the inertial
wave tide realignment timescale is much shorter than
equilibrium tide timescale (Xue et al. 2014).
The third and fourth explanations can account for the
observed temperature trend (Fig. 17, Column 5 and
6) but need more grounding in physical models. More
work on the theory of tidal dissipation is needed to de-
termine whether a steep dependence of tidal dissipation
efficiency on tidal forcing frequency is expected for the
relevant frequency range. The fourth explanation would
require very long timescales for the coupling of the outer
layer of the star to the interior and seems at odds with
the radially uniform rotation profile of the Sun. How-
ever, a decoupled outer layer could be analogous to our
Sun’s near-surface shear outer layer.
In summary, tidal alignment appears to play an im-
portant role in shaping the obliquity distribution of close
in, massive planets. Continuing work on the theory of
tides is needed to distinguish among hypotheses for how
planets realign their stars to ψ = 0 without tidal de-
struction.
4.2. Primordial misalignment
One might expect a star and its proto-planetary disk
to have aligned angular momenta, because they inherit
these from the same region of their parental molecu-
lar cloud and material is funneled via the disk onto
the young protostar. However, several processes have
been proposed that might create primordial misalign-
ment between the stellar equator and orbital mid plane
of the disk where planets are thought to form: misalign-
ment during accretion in chaotic star formation, mag-
netic warping, and tilt by a companion star (Fig. 15).
Misalignment during accretion might occur be-
cause stars form in a dense and chaotic environment,
causing the spin direction of the star and its disk to
change throughout the formation process. Late oblique
infall of material on the disk can warp the disk or tilt its
rotation relative to the axis of the star (Bate et al. 2010;
Thies et al. 2011; Fielding et al. 2015; Bate 2018). How-
ever, accretion from the disk onto the star can eliminate
such misalignments: therefore, by the planet forming
stage, the disk and star are likely aligned to within 20
degrees Takaishi et al. (2020).
Magnetic warping occurs when differential rotation
between a young star and the ionized inner disk twists
the magnetic field lines that link them, generating a
toroidal magnetic field that warps the disk (Foucart &
Lai 2011; Lai et al. 2011; see Romanova et al. 2013,
2020 for 3D MHD simulations). If the toroidal field is
sufficiently strong – and the realigning torques due to ac-
cretion onto the star, magnetic braking, disk winds, and
differential precession with the outer disk under high vis-
cosity are sufficiently weak– modest misalignments can
be generated. The misalignment may be suppressed if
the magnetic field becomes wrapped around the stel-
lar rotational axis (Romanova et al. 2020). A broader
distribution of alignment angles, including retrograde,
can be achieved through a simultaneous external distur-
bance to the outer disk, perhaps generated by a stellar
companion.
Inclined stellar or planetary companions can tilt
disks (e.g., Borderies et al. 1984; Lubow & Ogilvie 2000;
Batygin 2012; Matsakos & Konigl 2017). Although the
disk is coupled to the primary star, a misalignment can
be generated during resonance crossing of the stellar and
disk precession time scales (Batygin & Adams 2013; Lai
2014). The crossing occurs as the precession timescales
change due to disk evolution and mass loss (e.g., Spald-
ing et al. 2014). However, newly formed HJs are so
tightly coupled to host stars’ spin that they prevent
their host stars from becoming misaligned by this mech-
anism (Zanazzi & Lai 2018). Therefore companions tilt-ing disks through this resonance crossing mechanism are
unlikely to be responsible for most obliquities in the cur-
rent sample of individual system measurements, which
consists primarily of HJ hosts.
The direct route to measuring alignments between
stellar rotation and proto-planetary disks is blocked be-
cause the photospheres of protoplanetary disk hosting
stars are hidden from view. Some proto-planetary and
even embedded protostar disks exhibit misalignments
or warps between the inner and outer disk disk (e.g.,
Marino et al. 2015; Sakai et al. 2019; Ginski et al. 2021;
see Casassus 2016 for a review); however, the occurrence
rate of such misalignments is not yet known. These bro-
ken and internally misaligned disks might lead to in-
ner and outer planets orbiting with large mutual incli-
nations, setting the starting conditions for some of the
Obliquity 23
0
50
100
150|λ
| (deg
)
Observed
MJup:0.5-11-2.52.5-15
5000 6000Teff (K)
0.1
1.0
10.0
100.0
v s
in i
s (km
/s)
Sim: Initial
5000 6000Teff (K)
Sim: Equilibrium
5000 6000Teff (K)
Sim: Dynamic
5000 6000Teff (K)
Sim: Evolving Q
5000 6000Teff (K)
Sim: Decoupled
5000 6000Teff (K)
Figure 17. Observed (column 1) and modeled (column 2-5) projected obliquity distribution (top) and stellar rotationalvelocity (bottom). The projected stellar rotational velocity is examined as a proxy for stellar rotation period.
processes we discuss in the next section. For older debris
disks, researchers have found predominately – but not
exclusively – evidence for alignment between stars and
their disks using different variants of the v sin i method
to determine the stellar inclinations (Watson et al. 2011;
Greaves et al. 2014; Davies 2019).
Is primordial misalignment at work in HJ systems?
It may be, but if so, tidal realignment likely heavily
sculpts the resulting obliquity distribution: primordial
misalignment mechanisms alone do not seem to be able
to fully account for the observed obliquity trends. Pri-
mordial misalignments might vary with stellar mass – for
example, Spalding & Batygin (2015, 2016) propose thatlower mass young stars (< 1.2M) may be able to re-
align their disks – but in that case would more strongly
correlate with the initial main sequence effective tem-
perature than with present day effective temperature.
Primordial misalignment mechanisms also do not fully
account for correlations with mass ratio (§ 3.1.4) or or-
bital separation (§ 3.1.5).
Is primordial misalignment at work in non-HJ sys-
tems? There is growing evidence that the answer is yes.
If primordial misalignment is common, we expect to ob-
serve systems of coplanar planets that are misaligned
with their host star. Ensemble studies show indirect ev-
idence that hot stars are indeed misaligned with their
coplanar planetary systems (Section 3.6). This trend
with effective temperature is not expected, since most of
the systems are beyond the reach of tides, so future work
should probe whether it might actually be a trend with
stellar mass. Regarding individual systems, our sam-
ple of 14 compact, coplanar, multi-transiting systems
with obliquity measurements contains 11 well-aligned
systems of compact super-Earths and mini-Neptunes
(§ 3.1.10). Of the other three, HD 3167 (Dalal et al.
2019) does not show clear evidence for either a wider-
orbiting planet or a companion star. Kepler-56 (Hu-
ber et al. 2013) has a wider-orbiting third planet (Otor
et al. 2016) and its mass and distance are compatible
with tilting the orbital plane of the inner two planets
long after these planets have formed (Gratia & Fab-
rycky 2017). The third – K2-290 – features a pair of
planets – a warm Jupiter with an inner Neptune – on
retrograde yet coplanar orbits and a stellar companion
K2-290 B capable tilting the protoplantary disk (Hjorth
et al. 2021). K2-290 is therefore the first clear sign that
companion stars can generate obliquities by tilting the
disks planets form from. More generally, we know from
observations that disks with misaligned companions –
which could tilt disks – are present. In wide binary sys-
tems, proto-planetary disks can be misaligned from each
other and the binary orbit, as deduced from polarization
observations of disk jets (Monin et al. 2007, and refer-
ences therein) and ALMA/VLTI observations of proto-
planetary disks (e.g., HK Tauri Jensen & Akeson 2014).
Circumbinary debris disks can show misalignment with
the orbit as well, e.g., KH 15D (Winn et al. 2004; Chi-
ang & Murray-Clay 2004; Poon et al. 2021).
24 Albrecht, Dawson, & Winn
It is uncomfortable to tell two very different stories for
the obliquity-temperature trends of HJ hosts vs. other
hosts, but that is our current understanding. Our first
story is that most or all HJs are misaligned, not by a
companion tilting the disk they form from but possibly
magnetic warping or one of the post-formation mecha-
nisms described in the next section. They then tidally
realign cool host stars through a tidal mechanism that is
not well-understood. Our second story is that planetary
systems are primordially misaligned through a mecha-
nism that primarily operates around hot stars but is
not tidal; it could be – and, in the case of K2-290, very
likely is – a companion tilting the disk. These two sto-
ries must be reconciled to interpret stellar obliquities in
light of planets’ formation and evolution.
4.3. Post formation misalignment
After formation, gravitational interactions between
the planet and other bodies could alter the planet’s or-
bital plane, leading to misalignment with the host star’s
spin. These gravitational interactions may also lead to
high eccentricity tidal migration, in which a HJ forms
further from the star, is disturbed onto a highly ellipti-
cal orbit, and circularizes – due to tides raised on the
planet – to its present-day short period. When the first
misaligned hot Jupiters were first discovered, the stel-
lar obliquity was widely believed to primarily trace HJs’
dynamical history and to be driven by the same mecha-
nism(s) that led to its short orbital period (see Dawson
& Johnson 2018 for a review of hot Jupiters’ origins).
We thought that the obliquity distribution pointed to
either: a) a dynamical history that most commonly led
to aligned orbits but occasionally produced strongly mis-
aligned orbits, or b) two origins channels, one leading to
aligned orbits and the other to misaligned orbits (e.g.,
Fabrycky & Winn 2009). However, given the strong ev-
idence for tidal realignment of cool HJ hosts (Section
4.1), we now believe that a mechanism is operating that
produces a wide – possibly even isotropic – distribution
of obliquities for HJ hosts (Fig. 15). Some of these
mechanisms – as we will highlight below – can also ac-
count for the indirect evidence for misalignments of hot
stars hosting compact, coplanar systems (Section 3.6).
On the shortest timescales (as short as thousands of
years), planet-planet scattering can directly lead to
mutual inclinations among planets and misalignments
with the host star’s spin. Closely spaced and/or ellipti-
cal planets have close encounters that disturb their or-
bits, with eccentricities and mutual inclinations growing
as a random walk over many orbits. Planet-planet scat-
tering can take place shortly after the dissipation of the
gas disk when planets form close together, but may oc-
cur later when longer timescale chaotic evolution (see
below) or stellar flys (e.g., Malmberg et al. 2011) bring
planets together. Planet-planet scattering among plan-
ets that are low mass and/or close to their stars lead
to only small mutual inclinations because their close en-
counters lead to collisions rather than scattering (e.g.,
Goldreich et al. 2004). For giant planets further from
their star – which may become HJs through high eccen-
tricity tidal migration – the distribution of mutual in-
clinations produced by planet-planet scattering can be
broad but is still concentrated at low inclinations (e.g.,
Chatterjee et al. 2008). To get a range of obliquities
as broad as we observe, planet-planet scattering more
likely sets up the conditions for subsequent secular in-
teractions that lead to a broader obliquity distribution
(e.g., Nagasawa et al. 2008; Nagasawa & Ida 2011; in
Fig. 18 we compare the predicted obliquity distribution
from Nagasawa & Ida 2011, solid black, to predictions
from other mechanisms). A related mechanism that can
lead to a more isotropic distribution is direct disturbance
of a giant planet through a hyperbolic encounter with
a star in a very dense cluster environment, such as the
center of a globular cluster, where stars are approaching
at all angles (Hamers & Tremaine 2017). Future discov-
eries of HJs in globular clusters could test this theory.
Planets and stars exchange angular momentum over
longer timescales (typically thousands of orbits or more)
through cyclic secular interactions. Eccentricities and
mutual inclinations oscillate as bodies in the system
torque each other. In hierarchical (widely separated)
triple systems with large mutual inclinations and/or ec-
centricities, these variations are known as Kozai-Lidov
cycles (Kozai 1962; Lidov 1962) and can be driven by a
stellar or planetary companion (e.g., Wu & Murray 2003;
Fabrycky & Tremaine 2007; Naoz et al. 2011; see Naoz
2016 for a review). The timescale depends on the separa-
tion and mass of the perturbing companion, with typical
timescales of order millions of years. Although we often
model secular interactions after the gas disk stage, mu-
tual inclinations can also be excited during the gas disk
stage by secular interactions among the planet, disk,
and companion(s) (Picogna & Marzari 2015; Lubow &
Martin 2016; Franchini et al. 2020).
Resonant excitation of the stellar obliquity can
occur as the system evolves and a changing frequency
crosses the secular frequency. In a triple system when
the primary spins down due to magnetic braking, the ro-
tational oblateness precession frequency crosses the sec-
ular frequency, generating large misalignments (Ander-
son et al. 2018). However, we would expect this mecha-
nism to primarily operate for cool stars, the opposite of
the trends observed. It tends to produce primarily pro-
Obliquity 25
grade misalignments (Fig. 18, dashed gray line). In a
system with an outer planetary companion and dispers-
ing gas disk, the gas disk precession frequency can cross
the secular frequency, generating a large mutual inclina-
tion between the inner and outer planet and driving the
stellar obliquity to ψ = 90 (Petrovich et al. 2020). This
mechanism is most effective for close-in Neptune-mass
with outer Jupiter-mass companions, like the HAT-P-
11 system.
The resulting obliquity distribution from all these
types of secular interactions depends on the initial sys-
tem architecture, which may be established by earlier
evolution in the presence of a gas disk, planet-planet
scattering, and/or stellar fly bys (e.g., Hao et al. 2013).
Distant and/or circular companions driving Kozai-Lidov
cycles tend to produce a bimodal obliquity distribution
(Fig. 18, dashed red line) with peaks near 40 and 140
degrees and an absence of polar orbits (e.g., Fabrycky &
Tremaine 2007; Naoz et al. 2012), which may not be fully
consistent with the observed distribution of projected
obliquities of HJ hosts. Accounting for the host star’s
oblateness and spin evolution (which can sometimes
lead to chaotic variations in its spin vector, e.g., Storch
et al. 2014) further enhances this bimodality (Damiani
& Lanza 2015; Anderson et al. 2016), particularly for
cool stars. The fraction of planets on retrograde orbits is
larger and the distribution of obliquities is broader when
the companion is eccentric and/or nearby (Fig. 18, dot-
ted blue line) ; Naoz et al. 2011; Teyssandier et al. 2013;
Li et al. 2014b,a; Petrovich & Tremaine 2016), such as
companions that were engaged in planet scattering (Na-
gasawa et al. 2008; Nagasawa & Ida 2011). For Kozai-
Lidov cycles to significantly raise the mutual inclination,
the orbital precession caused by that companion must
dominate over precession from stellar oblateness, tides,
and general relativity. In compact systems where plan-
ets are more tightly coupled to each other than to an ex-
terior companion, the exterior companion can misalign
the entire interior system from its host star’s spin, as
observed for Kepler-56 (e.g., Takeda et al. 2008; Boue &
Fabrycky 2014; Li et al. 2014c; Gratia & Fabrycky 2017).
This explanation does not hold for K2-290 (the system
highlighted as an example of primordial misalignment in
Section 4.2), where the inner system is too tightly cou-
pled to the stellar spin by oblateness precession (Hjorth
et al. 2021).
Over many secular timescales – hundreds of mil-
lions of years or longer – mutual inclinations can grow
chaotically due to the overlap of secular frequencies
in multi-planet systems (Laskar 2008; Wu & Lithwick
2011; Hamers et al. 2017; Teyssandier et al. 2019) or
triple/quadruple star systems (e.g., Hamers 2017; Gr-
ishin et al. 2018), known as secular chaos. Similar to
cyclical secular interactions, the resulting obliquity dis-
tribution depends on the initial architecture; producing
planets on retrograde orbits requires eccentricities and
inclinations that are large to begin with (e.g., Lithwick &
Wu 2014), perhaps established by planet-planet scatter-
ing (Beauge & Nesvorny 2012). However, Teyssandier
et al. (2019) argue that this mechanism produces an in-
surmountable lack of retrograde planets (Fig. 18, solid
purple line) because the planet tends to circularize and
decouple from the companion before the obliquity grows
very large.
In summary, producing a broad obliquity distribution
with plenty of retrograde planets is the biggest chal-
lenge for these mechanisms, but the more complex and
multi-step dynamical histories – such as planet-planet
scattering followed by secular cycles – seem at at least
qualitatively consistent with the observed distribution
for HJs (Fig. 18). Producing fewer retrograde planets
would helpfully reduce the number of retrograde plan-
ets expected with ψ = 180 following inertial wave tidal
dissipation but seems at odds with the large number of
retrograde planets orbiting hot stars. One major uncer-
tainty in comparing the predictions of these mechanisms
to the observed obliquity distribution and even teasing
out the contributions of multiple mechanisms (e.g., Mor-
ton & Johnson 2011; Naoz et al. 2012) is that even the
obliquity distribution of hot stars hosting HJs may have
been altered by tides. Achieving an isotropic distribu-
tion post-formation for small, compact, coplanar planets
orbiting hot stars (Section 3.6) may be even more chal-
lenging and has not yet been demonstrated.
Although the mechanisms discussed here operate on
a range of timescales – from during the gas disk stage
to throughout the star’s lifetime – they generally re-
quire that the HJ form further from its star. Planet-
planet scattering generally fails to generate large mis-
alignments very close to the star and secular mechanisms
require very nearby planets that can overcome the cou-
pling of the HJ to the star, which most HJs lack (see
below). Therefore, under the theories discussed here, we
expect HJs have their eccentricity excited by the same
process that misaligns them, undergo high eccentricity
tidal migration via tides raised on the planet, and ar-
rive misaligned. The tidal migration timescale is uncer-
tain and very sensitive to the eccentric planet’s periapse
distance and thus can span many orders of magnitude.
More work on misalignment theories is needed to ex-
plore how the obliquity distribution changes over time
and whether we expect young HJs to be just as mis-
aligned as older ones (Section 3.1.8). However, Beauge
& Nesvorny (2012) do predict that retrograde planets
26 Albrecht, Dawson, & Winn
0 30 60 90 120 150 180 ψ (deg)
Re
lative
nu
mn
be
r
Resonant (A18)
Sec Chaos (T19)
Planet Kozai (P16)
Stellar Kozai (A16)
Scatter/Sec (N11)
Figure 18. Example population synthesis (unpro-jected) obliquity distributions from studies of differ-ent misalignment mechanisms: resonance crossing (Andersonet al. 2018), secular chaos (Teyssandier et al. 2019), planet-planet Kozai-Lidov cycles (Petrovich & Tremaine 2016), star-planet Kozai-Lidov for a 1 MJup HJ orbiting an F star (An-derson et al. 2016), and planet-planet scattering with secularcycles (Nagasawa & Ida 2011).
will also tend to have closer periapses and are thus more
likely to raise tides on the star that drive orbital decay;
they predict that retrograde planets should be system-
atically younger.
One avenue to test the secular cycle hypothesis in
particular is to search for companions capable of driv-
ing Kozai-Lidov cycles. A prime example would be the
HD 80606 double star system with its highly eccentric
warm Jupiter (Wu & Murray 2003) on an oblique or-
bit (Hebrard et al. 2010), which may be in the midst
of Kozai-Lidov driven high eccentricity tidal migration.
The Friends of Hot Jupiters survey (Knutson et al. 2014;
Ngo et al. 2015; Piskorz et al. 2015; Bryan et al. 2016;
Ngo et al. 2016) found that most hot Jupiters lack a
capable stellar companion but that many have a po-
1 10semi-major axis ratio/25
0.1
1.0
10.0
100.0
mass r
atio
1 10
0.1
1.0
10.0
100.01 10
HJ co
upled to
frie
nd
Figure 19. Planet-planet coupling. A handful of HJswith low obliquities orbiting cool stars (blue symbols in theorange region) are strongly coupled to an nearby companion,preventing tidal realignment. Lines represent companionsthat are detected as radial velocity trends (for which massand semi-major axis are degenerate).
tentially suitable planetary companion. Gaia measure-
ments will probe whether these companions have suffi-
cient mutual inclinations.
HJ companions can also shed light on the tidal re-
alignment hypothesis. If a mutually inclined companion
that would cause misalignment is massive and nearby
enough (Becker et al. 2017) to overcome the HJ’s stel-
lar oblateness coupling (Lai et al. 2018) – i.e., a giant
planet companion interior to ∼ 1 au – it can continue
to drive secular cycles and prevent the HJ from tidally
realigning its star. Several observed systems have strong
coupling between the HJ and its outer companion but
low obliquities (Fig. 19); these systems cannot be ex-
plained by Kozai-Lidov cycle misalignment followed by
tidal realignment. However, the majority of known HJ
companions are not sufficiently coupled, and the com-
panion would not interfere with the HJ tidally realigning
its star.
4.4. Altering the stellar spin vector
The processes discussed so far involve changing the
orbital plane of the planet(s) or changing the spin of
the star as a response to an external force. Rogers et al.
(2012, 2013) showed that for hotter stars with convective
cores and radiative envelopes, Internal Gravity Waves
(IGW) can lead to a tilt of the photosphere relative to
the total angular momentum, on timescales of 104 yrs or
less. Changes in λ and v sin i over time in systems with
hot host stars would indicate that IGW are at work but
are not easily observable16 with the current short time
baseline. Radial differential rotation, which could be
16 Precession due to spin-orbit coupling observed in exoplanethosts (e.g. WASP-33 Johnson et al. 2014) as well as in close doublestars (Albrecht et al. 2014) can complicate our interpretation ofsuch changes.
Obliquity 27
detected via asteroseismology (Christensen-Dalsgaard &
Thompson 2011), would also be a hallmark of IGW.
IGWs can account for some but not all observed
trends. Although IGWs can account for the higher
obliquities of hot stars (§ 3.1.3), they cannot account
for correlations of obliquity with mass ratio (§ 3.1.4) or
orbital separation (§ 3.1.5). Furthermore, we would ex-
pect coplanar systems to be misaligned with hot stars.
Ensemble studies indirectly suggest that they are (Sec-
tion 3.6); however, of the three known coplanar systems
orbiting hot stars with individual obliquity measure-
ments (HD 106315, K2-290, and Kepler-25), the only
misaligned one is K2-290, for which the stellar compan-
ion is believed to be responsible for the retrograde orbit
(Hjorth et al. 2021), rather than IGWs. We can also
test this mechanism using binaries with separations be-
yond the reach of tides containing one low mass star and
one high mass star. If IGWs are at work, we expect to
more often observe the high mass star misaligned with
the binary’s orbit and the low mass star aligned. Cur-
rently there are no suitable binary systems to perform
this test.
5. SUMMARY AND DISCUSSION
Available evidence points towards two pathways to-
wards spin orbit misalignment, primordial and dynami-
cal processes after planet formation. Furthermore tidal
interactions between the star and planet are important
for the observed population of exoplanet systems.
While tides are not fully understood there are several
indications that the problem of giant planet destruction
during realignment might not be as severe as originally
feared. Tides are also most successful in explaining ob-
servational trends with stellar structure, orbital separa-
tion, planetary mass, and that HJs orbiting cool stars
which have well measured obliquities (σλ < 2) show
alignment and dispersion both below 1.
There is indirect evidence, from observations of jets
and disks in young stellar systems, that primordial star
disk misalignment does at least occasionally occur dur-
ing the early systems evolution, before the protoplan-
etary disk is dispersed. While the stellar spin is un-
known in these systems HK Tauri is one of the clear-
est examples that not all vectors, stellar spin, orbital
spin, and disk spin, can be aligned in such systems.
However theoretical work, observations of debri disks,
obliquity measurements in a small number of young ex-
oplanet systems, and in compact transiting multi planet
systems with cool host stars suggest that such misalign-
ments when present might not always survive the final
stages of the systems formation. Nevertheless one plan-
etary system, K2-290 A, part of a wide binary features
retrograde coplanar orbiting planets. This configura-
tion is a result of primordial misalignment caused by
the companion. That we see not more such systems in
the current sample might be a result of selection biases
as K2-290 A is the only multi transiting exoplanet sys-
tem in a wide binary for which the obliquity has been
measured. In addition ensemble studies of transiting
planets indicate that opposite to cool stars which tend
to be aligned in population studies also involving non HJ
systems, hot stars in general have planets on misaligned
orbits, not only HJs. This suggests that misalignment
mechanism(s) can operates independently of the plan-
etary parameter range observed. Primordial misalign-
ment would be such a mechanism. Together these lines
of evidence illustrate that the textbook example of a star
with an aligned protoplanetary disk does not encompass
all important aspects of planetary formation.
Large obliquities in systems with close in giant plan-
ets seem to be best explained by dynamical interactions
which occurred after planet formation. The strongest
observational support comes from the difference in obliq-
uity distribution between single and coplanar systems
and the increased (planetary) companionship to such
systems. The observations of alignment in a small num-
ber of young (. 100 Myr) systems with (close in or-
biting) giant planets further suggests that if compact
systems are misaligned then this is caused by dynamical
interactions followed by high eccentricity migration, as
this process can occur on similar or longer times than
the life times of the systems.
We do not yet know which of the proposed dynami-
cal processes has a dominate role, if any. Poster child
systems for KL-cycles (caused by stellar companion) do
exist (e.g. HD 80606). Among post formation scenarios
KL-cycles can most easily generate retrograde orbits.
However surveys have not been able to clearly identify
the necessary companions (stellar or planetary) with
the required parameters to drive KL-cycles to the HJ
sample, though there seem to be a suitable number of
planetary companions if they have the necessary mu-
tual inclination. Different post formation mechanisms
scenarios do lead to different obliquity distributions and
this can in principle be used to differentiate between
the different post formation processes. However tidal
alignment and our lack of quantitative understanding
thereof blurs the observational distention between dif-
ferent mechanisms.
We should remind ourselves that the current sample of
systems with obliquity measurements is heavily biased
in a number of ways, most noticeably towards close in
giant planets orbiting main sequence stars, mainly F-K
28 Albrecht, Dawson, & Winn
type. As these planetary systems do not present the
complete spectrum of planetary systems, so might their
obliquity distribution.
A number of new missions have the potential to
change this preoccupation with a small subset of sys-
tems: Bright and well characterized TESS systems allow
for more precise RM and v sin i measurements in a more
diverse planetary population. GAIA will enable mea-
surements of mutual inclinations, (less precise) v sin i
measurements in a larger sample as well as interfero-
metric obliquity measurements in a smaller brighter sub-
set of systems. PLATO, while also enabling RM mea-
surements, will be more crucial for seismic, spot, and
v sin i measurements in new types of systems. Combing
these new samples with the availability of new or soon to
be operational spectrographs and intererometers should
lead to new insights, among them:
• Bright well studied TESS systems harboring close
in giant planets will allow for precise (σλ . 2)
RM measurements employing new ground based
spectrographs. This will lead to an increased un-
derstanding of tidal alignment, crucial to better in-
terpret existing and upcoming obliquity measure-
ments.
• Primordial misalignment can be tested in young
systems, planets with large orbital separations,
systems with multi transiting systems (if compan-
ionship is known), and via star debri disk align-
ments. This can be achieved via RM measure-
ments in some systems, and via v sin i, spot, and
seismic measurements in populations, and interfer-
ometric obliquity measurements in systems with
multi year periods, i.e. astrometric orbits (GAIA).
• Observations of misalignments and warps between
inner and outer disks, alignments of disks in wide
forming double stars, as well as measurements of
debri disk alignments will inform theories of pri-
mordial misalignment.
• The increasing number of known transiting bright
systems (TESS and later PLATO) allows for more
precise obliquity & eccentricity measurements in
systems with longer orbits and smaller planets. It
also allows for a more complete characterization
of companionship. Armed with a better under-
standing of tidal alignment (first point above) this
will allow for a meaningful comparison between
the measured obliquity distribution and predic-
tions by post formation misalignment mechanisms.
This should also lead to a significant improvement
of our understanding of the formation of gas giant
planets inside one au.
• Finally obliquity measurements in (wide) double
star and multiple star systems will determine the
coherence length scale of the angular momentum
distribution, which in turn determines for which
kind of systems certain types of primordial disk
misalignment mechanisms and stellar KL-cycles
could be important. These samples could also
serve to better test the role of IGW.
The obliquity of a body (star, planet, moon) is a fun-
damental orbital parameter and should be considered
an important observable worth measuring if a system is
studied in detail.
SA acknowledges the support from the Danish Coun-
cil for Independent Research through the DFF Sapere
Aude Starting Grant No. 4181-00487B, and the Stellar
Astrophysics Centre which funding is provided by The
Danish National Research Foundation (Grant agreement
no.: DNRF106). RID acknowledge supports from grant
NNX16AB50G awarded by the NASA Exoplanets Re-
search Program and the Alfred P. Sloan Foundation’s
Sloan Research Fellowship. The Center for Exoplanets
and Habitable Worlds is supported by the Pennsylvania
State University, the Eberly College of Science, and the
Pennsylvania Space Grant Consortium.
We thank J.J. Zanazzi for helpful comments and sug-
gestions.
Obliquity 29
APPENDIX
A. SYSTEMS
Here we describe the sources of the system parameters and what vetting we have carried out. We started by
downloading data from the TEPCAT catalog on January 5th 2021) curated by John Southworth available here:
TEPCat Southworth (2011). We added the following obliquity measurements in β Pictoris (Kraus et al. 2020),
HD 332231 (Knudstrup et al. in prep), K2-290 (Hjorth et al. 2021), a measurement of the second planet in HD 63433
Dai et al. (2020). We also included stellar inclination measurements obtained with the method outlined in Masuda &
Winn (2020). These are TOI-251 & TOI-942 (Zhou et al. 2021), TOI-451 (Newton et al. 2021), TOI-811 & TOI-852
(Carmichael et al. 2020), and TOI-1333 (Rodriguez et al. 2021). For a number of systems more than one measurement
of the stellar inclination or projected obliquity does exist. We chose the same preferred measurements as indicated
by TEPCAT for all systems but the following (We note that this selection does not have any influence of any of
the conclusions we make in the paper.): For HAT-P-7 we chose the ”solution 1” from Masuda (2015), for HAT-P-16
we chose the result by Moutou et al. (2011), Kepler-25 (Albrecht et al. 2013b), MASCARA-4 Dorval et al. (2020),
WASP-18 & WASP-31 Albrecht et al. (2012b), WASP-33 the ”2014” data Johnson et al. (2015). We then folded the
measurements of projected obliquity reported in this catalog onto a half circle ranging from 0 to 180. (The only
exception is one panel in Fig. 8.)
We obtained orbital eccentricity data either from the detection papers or when available from the comprehensive
work by Bonomo et al. (2017). For Kepler-448 we use the eccentricity obtained by Masuda (2017) We further extracted
information on companionship from the ”Friends of hot Jupiters” paper series by Knutson et al. (2014); Piskorz et al.
(2015); Ngo et al. (2016). .
We excluded systems with uncertainties in the projected obliquity larger than 50 deg, specifically HAT-P-27 (Brown
et al. 2012) and Wasp-49 Wyttenbach et al. (2017). We also excluded some other specific systems: The hot Jupiter
system CoRoT-1 has two RM datasets, one indicating good alignment Bouchy et al. (2008), and one indicating strong
misalignment Pont (2009). Guenther et al. (2012) found a projected obliquity of −52−22+27 deg for CoRoT-19.
However no post-egress data were obtained and the Rossiter-McLaughlin effect was detected at an 2.3σ level only.
Zhou et al. (2015) report for HATS-14 a misaligned orbit (|λ| = 76+4−5 deg). However there is no post egress data and
as highlighted by the authors making different assumptions about the orbital semi-amplitude does lead to different
conclusions about the obliquity. WASP-134 b (Anderson et al. 2018) is also excluded from our analysis for reasons
similar to HATS-14. WASP-23 has a low impact parameter and a low v sin i preventing Triaud et al. (2011) from
concluding more than that the orbit is prograde. We further exclude the WASP-1 and WASP-2 (Triaud et al. 2010)
as discussed in detail by Albrecht et al. (2011) and Triaud (2017). Bourrier & Hebrard (2014) claimed a significant
misalignment in the 55 Cnc system, which was proven to be spurious by Lopez-Morales et al. (2014). There is also the
tentative detection of misalignment in KOI-89 (Ahlers et al. 2015), but a recent reanalysis of the Kepler data showed
that the obliquity is unconstrained by the data (Masuda & Tamayo 2020).
Most of the quoted v sin i measurements have been obtained from RM measurements. In particular for lower v sin i
values and large impact parameter is large these values can be more precise. However for some systems where the RM
data is of low SNR (e.g. Qatar-2, (Esposito et al. 2017) we opted to quote the spectroscopic value. It is also worth
noticing that the spectroscopic value is a disk integrated value whereas the value obtained from RM studies connects
to the surface motion under the planets path over the stellar disk).
30 Albrecht, Dawson, & Winn
Table
2.
Lis
ting
of
the
syst
ems
and
som
ehost
star
para
met
ers
consi
der
edin
this
revie
w.
The
orb
ital
para
met
ers
of
thei
rpla
net
sare
list
edin
table
3.
Num
ber
Syst
emλ
vsi
ni
Teff
M?
R?
age
Com
panio
nR
efer
ence
s
()
(km
s−1)
(K)
(M
)(R
)
(Gyr)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Note—
-
Obliquity 31
Table
3.
Lis
ting
of
the
pla
net
sfo
rw
hic
hλ
was
det
erm
ined
.T
he
stel
lar
para
met
ers
of
thei
rhost
stars
are
list
edin
table
2
Num
ber
Pla
net
Per
iod
a/R
Mp
Rp
eR
efer
ence
s
(day
s)(M
Jupit
er)
(RJupit
er)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Note—
-
32 Albrecht, Dawson, & Winn
B. POPULATION SYNTHESIS SIMULATIONS
The simulations follow Dawson (2014), with updates
to incorporate options for intertial wave tidal dissipa-
tion and a frequency dependent tidal dissipation effi-
ciency. We numerically integrate the planet’s specific
orbital angular momentum vector ~h and the host star’s
spin angular frequency vector, assuming a circular or-
bit. The equations here correspond to Barker & Ogilvie
(2009), Eqn. A7 and A12 with the eccentricity vector
~e = 0.
(~h)eq
= − 1
τeq~h+
1
τeq
Ω?2n
(~Ω? · ~hΩ? h
· ~h+h
Ω?~Ω?
)(~Ω?
)eq,α
= − m
k?,effMR2~heq − α brakeΩ2
?~Ω?
(B1)
for which
τeq =Q
6kL
M
R5(M +m)8G7
M
mh13
= τeq,0
(h
h0
)130.5M Jup
m(B2)
is an orbital decay timescale, kL is the Love number,
Q is the tidal quality factor, k?,eff is the effective con-
stant of the stellar moment of inertia participating in
the tidal realignment, α brake is a braking constant, and
h0 =√a0G(M +m) is the initial specific angular mo-
mentum. By default, we use k?,effMR2 = 0.08MR2
for cool stars, k?,effMR2 = 0.08(1.2M)(1.4R)2 for hot
stars, and Ωs,0 = 800 AU2yr−1. We use α = 3 × 10−16
for hot stars, α = 1.4× 10−14 for cool stars, τeq,0 = 500
Gyr, and h0 = 1.33 au2yr−1. For the pure equilibrium
tides simulation, we use h0 = 1.68 au2yr−1. For the
frequency-dependent Q simulations, we use τeq,0 = 10
Gyr, and h0 = 1.85 au2yr−1. For the decoupled outer
envelope simulations, we use α = ×10−13 for cool stars
For intertial wave tidal dissipation, the tidal forcing
component that excites inertial waves exerts a torque.
Here we follow Lai (2012) to compute its affects on ~h and~Ω?. One component is parallel to the stellar spin, i.e., in
the ~Ω? direction. A second component is perpendicular
to both ~h and ~Ω?, i.e., in the ~Ωs × ~h direction, and is
ignored because it does not affect the alignment. The
third component is perpendicular to the other two and
thus we compute its unit vector as:
x = (~h× ~Ωs)×~Ωs×
Ω2? h sinψ
(B3)
where
cosψ =~Ω? · ~hΩ? h
sinψ =| ~Ω? × ~h|
Ω? h
(B4)
We add the following terms to Eqn. B1.(~Ω?
)dy
=− 1
τdy
(1− τ0,dy
τ0,eq
)[(sinψ cosψ)
2 ~Ω? − sinψ cosψ3Ω?x]
(~h)dy
=−k?,effMR2
m
(~Ω?
)dy
(B5)
where
τdy =τ0,dyτ0,eq
Ω?Ω?,0
h0
hτeq. (B6)
We setτ0,dyτ0,eq
= 10−5 for Fig. 17.
For the frequency-dependent tidal dissipation effi-
ciency model (Penev et al. 2018), we use Eqn. B1 with
a modified value of teq:
teq,f = teqMax[106/Ptide[days]3.1, 105]
Max[106/Ptide0[days]3.1, 105](B7)
where Ptide = π/(n− Ωs) is in units of days.
To generate the populations for Fig. 17, we selecta uniform random 4800 < Teff < 6800 K, a log-uniform
0.5M Jupiter < m < 15M Jupiter, ψ from an isotropic dis-
tribution, a uniform random evolution time 0 < t? < 10
Gyr for cool stars (T < 6250K) or 0 < t? < 4 Gyr for hot
stars, and a uniform random longitude of ascending node
0 < Ω < 2π. Then we integrate the momentum equa-
tions above for t?. We compute λ = tan−1 (tanψ sin Ω)
(Fabrycky & Winn 2009, Eqn. 11; Column 2 of our
Fig. 17 shows the initial distribution of λ), sin i =√1− (sinψ cos Ω)2, and v sin (i) /R = Ω? sin i.
REFERENCES
Abt, H. A. 2001, AJ, 122, 2008, doi: 10.1086/323300
Addison, B. C., Horner, J., Wittenmyer, R. A., et al. 2020,
arXiv e-prints, arXiv:2006.13675.
https://arxiv.org/abs/2006.13675
Ahlers, J. P., Barnes, J. W., & Barnes, R. 2015, ApJ, 814,
67, doi: 10.1088/0004-637X/814/1/67
Ahlers, J. P., Seubert, S. A., & Barnes, J. W. 2014, ApJ,
786, 131, doi: 10.1088/0004-637X/786/2/131
Obliquity 33
Ahlers, J. P., Kruse, E., Colon, K. D., et al. 2020a, ApJ,
888, 63, doi: 10.3847/1538-4357/ab59d0
Ahlers, J. P., Johnson, M. C., Stassun, K. G., et al. 2020b,
AJ, 160, 4, doi: 10.3847/1538-3881/ab8fa3
Albrecht, S., Quirrenbach, A., Tubbs, R. N., & Vink, R.
2010, Experimental Astronomy, 27, 157,
doi: 10.1007/s10686-009-9181-6
Albrecht, S., Reffert, S., Snellen, I., Quirrenbach, A., &
Mitchell, D. S. 2007, A&A, 474, 565,
doi: 10.1051/0004-6361:20077953
Albrecht, S., Setiawan, J., Torres, G., Fabrycky, D. C., &
Winn, J. N. 2013a, ApJ, 767, 32,
doi: 10.1088/0004-637X/767/1/32
Albrecht, S., Winn, J. N., Butler, R. P., et al. 2012a, ApJ,
744, 189, doi: 10.1088/0004-637X/744/2/189
Albrecht, S., Winn, J. N., Marcy, G. W., et al. 2013b, ApJ,
771, 11, doi: 10.1088/0004-637X/771/1/11
Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2011, ApJ,
738, 50, doi: 10.1088/0004-637X/738/1/50
—. 2012b, ApJ, 757, 18, doi: 10.1088/0004-637X/757/1/18
Albrecht, S., Winn, J. N., Torres, G., et al. 2014, ApJ, 785,
83, doi: 10.1088/0004-637X/785/2/83
Anderson, D. R., Bouchy, F., Brown, D. J. A., et al. 2018,
arXiv e-prints, arXiv:1812.09264.
https://arxiv.org/abs/1812.09264
Anderson, K. R., Storch, N. I., & Lai, D. 2016, MNRAS,
456, 3671, doi: 10.1093/mnras/stv2906
Anderson, K. R., Winn, J. N., & Penev, K. 2021, arXiv
e-prints, arXiv:2102.01081.
https://arxiv.org/abs/2102.01081
Bailey, E., Batygin, K., & Brown, M. E. 2016, AJ, 152, 126,
doi: 10.3847/0004-6256/152/5/126
Ballot, J., Appourchaux, T., Toutain, T., & Guittet, M.
2008, A&A, 486, 867, doi: 10.1051/0004-6361:20079343
Ballot, J., Garcıa, R. A., & Lambert, P. 2006, MNRAS,
369, 1281, doi: 10.1111/j.1365-2966.2006.10375.x
Barker, A. J., & Lithwick, Y. 2014, MNRAS, 437, 305,
doi: 10.1093/mnras/stt1884
Barker, A. J., & Ogilvie, G. I. 2009, MNRAS, 395, 2268,
doi: 10.1111/j.1365-2966.2009.14694.x
Barnes, J. W. 2009, ApJ, 705, 683,
doi: 10.1088/0004-637X/705/1/683
Barnes, J. W., Ahlers, J. P., Seubert, S. A., & Relles, H. M.
2015, ApJL, 808, L38, doi: 10.1088/2041-8205/808/2/L38
Barnes, J. W., Linscott, E., & Shporer, A. 2011, ApJS, 197,
10, doi: 10.1088/0067-0049/197/1/10
Bate, M. R. 2018, MNRAS, 475, 5618,
doi: 10.1093/mnras/sty169
Bate, M. R., Lodato, G., & Pringle, J. E. 2010, MNRAS,
401, 1505, doi: 10.1111/j.1365-2966.2009.15773.x
Batygin, K. 2012, Nature, doi: 10.1038/nature11560
Batygin, K., & Adams, F. C. 2013, ApJ, 778, 169,
doi: 10.1088/0004-637X/778/2/169
Beauge, C., & Nesvorny, D. 2012, ApJ, 751, 119,
doi: 10.1088/0004-637X/751/2/119
Beck, J. G., & Giles, P. 2005, ApJL, 621, L153,
doi: 10.1086/429224
Becker, J. C., Vanderburg, A., Adams, F. C., Khain, T., &
Bryan, M. 2017, AJ, 154, 230,
doi: 10.3847/1538-3881/aa9176
Bonomo, A. S., Desidera, S., Benatti, S., et al. 2017, A&A,
602, A107, doi: 10.1051/0004-6361/201629882
Borderies, N., Goldreich, P., & Tremaine, S. 1984, ApJ,
284, 429, doi: 10.1086/162423
Bouchy, F., Queloz, D., Deleuil, M., et al. 2008, A&A, 482,
L25, doi: 10.1051/0004-6361:200809433
Boue, G., & Fabrycky, D. C. 2014, ApJ, 789, 110,
doi: 10.1088/0004-637X/789/2/110
Bourrier, V., & Hebrard, G. 2014, A&A, 569, A65,
doi: 10.1051/0004-6361/201424266
Brown, D. J. A., Cameron, A. C., Anderson, D. R., et al.
2012, MNRAS, 423, 1503,
doi: 10.1111/j.1365-2966.2012.20973.x
Bryan, M. L., Knutson, H. A., Howard, A. W., et al. 2016,
ApJ, 821, 89, doi: 10.3847/0004-637X/821/2/89
Campante, T. L., Lund, M. N., Kuszlewicz, J. S., et al.
2016, ApJ, 819, 85, doi: 10.3847/0004-637X/819/1/85
Carmichael, T. W., Quinn, S. N., Mustill, A. J., et al. 2020,
AJ, 160, 53, doi: 10.3847/1538-3881/ab9b84
Casanovas, J. 1997, in Astronomical Society of the Pacific
Conference Series, Vol. 118, 1st Advances in Solar Physics
Euroconference. Advances in Physics of Sunspots, ed.
B. Schmieder, J. C. del Toro Iniesta, & M. Vazquez, 3
Casassus, S. 2016, Publications of the Astronomical Society
of Australia, 33, e013, doi: 10.1017/pasa.2016.7
Cebron, D., Le Bars, M., Le Gal, P., et al. 2013, Icarus,
226, 1642, doi: 10.1016/j.icarus.2012.12.017
Cegla, H. M., Lovis, C., Bourrier, V., et al. 2016, A&A,
588, A127, doi: 10.1051/0004-6361/201527794
Chaplin, W. J., & Miglio, A. 2013, ARA&A, 51, 353,
doi: 10.1146/annurev-astro-082812-140938
Chaplin, W. J., Sanchis-Ojeda, R., Campante, T. L., et al.
2013, ApJ, 766, 101, doi: 10.1088/0004-637X/766/2/101
Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A.
2008, ApJ, 686, 580, doi: 10.1086/590227
Chelli, A., & Petrov, R. G. 1995, A&AS, 109, 401
Chiang, E. I., & Murray-Clay, R. A. 2004, ApJ, 607, 913,
doi: 10.1086/383522
34 Albrecht, Dawson, & Winn
Christensen-Dalsgaard, J., & Thompson, M. J. 2011, in
IAU Symposium, Vol. 271, Astrophysical Dynamics:
From Stars to Galaxies, ed. N. H. Brummell, A. S. Brun,
M. S. Miesch, & Y. Ponty, 32–61
Collier Cameron, A., Bruce, V. A., Miller, G. R. M.,
Triaud, A. H. M. J., & Queloz, D. 2010, MNRAS, 403,
151, doi: 10.1111/j.1365-2966.2009.16131.x
Dai, F., Masuda, K., & Winn, J. N. 2018a, ApJL, 864, L38,
doi: 10.3847/2041-8213/aadd4f
Dai, F., Winn, J. N., Berta-Thompson, Z., Sanchis-Ojeda,
R., & Albrecht, S. 2018b, AJ, 155, 177,
doi: 10.3847/1538-3881/aab618
Dai, F., Roy, A., Fulton, B., et al. 2020, arXiv e-prints,
arXiv:2008.12397. https://arxiv.org/abs/2008.12397
Dalal, S., Hebrard, G., Lecavelier des Etangs, A., et al.
2019, A&A, 631, A28, doi: 10.1051/0004-6361/201935944
Damiani, C., & Lanza, A. F. 2015, A&A, 574, A39,
doi: 10.1051/0004-6361/201424318
Damiani, C., & Mathis, S. 2018, A&A, 618, A90,
doi: 10.1051/0004-6361/201732538
Davies, C. L. 2019, MNRAS, doi: 10.1093/mnras/stz086
Dawson, R. I. 2014, ApJL, 790, L31,
doi: 10.1088/2041-8205/790/2/L31
Dawson, R. I., & Johnson, J. A. 2018, ArXiv e-prints.
https://arxiv.org/abs/1801.06117
Desert, J.-M., Charbonneau, D., Demory, B.-O., et al.
2011, ApJS, 197, 14, doi: 10.1088/0067-0049/197/1/14
Domiciano de Souza, A., Kervella, P., Jankov, S., et al.
2003, A&A, 407, L47, doi: 10.1051/0004-6361:20030786
Domiciano de Souza, A., Zorec, J., Jankov, S., et al. 2004,
A&A, 418, 781, doi: 10.1051/0004-6361:20040051
Dorval, P., Talens, G. J. J., Otten, G. P. P. L., et al. 2020,
A&A, 635, A60, doi: 10.1051/0004-6361/201935611
Doyle, A. P., Davies, G. R., Smalley, B., Chaplin, W. J., &
Elsworth, Y. 2014, MNRAS, 444, 3592,
doi: 10.1093/mnras/stu1692
Esposito, M., Covino, E., Desidera, S., et al. 2017, A&A,
601, A53, doi: 10.1051/0004-6361/201629720
Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298,
doi: 10.1086/521702
Fabrycky, D. C., & Winn, J. N. 2009, ApJ, 696, 1230,
doi: 10.1088/0004-637X/696/2/1230
Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2012,
ArXiv. https://arxiv.org/abs/1202.6328
—. 2014, ApJ, 790, 146, doi: 10.1088/0004-637X/790/2/146
Fielding, D. B., McKee, C. F., Socrates, A., Cunningham,
A. J., & Klein, R. I. 2015, MNRAS, 450, 3306,
doi: 10.1093/mnras/stv836
Foucart, F., & Lai, D. 2011, MNRAS, 412, 2799,
doi: 10.1111/j.1365-2966.2010.18176.x
Franchini, A., Martin, R. G., & Lubow, S. H. 2020,
MNRAS, 491, 5351, doi: 10.1093/mnras/stz3175
Gaspar, A., & Rieke, G. 2020, Proceedings of the National
Academy of Science, 117, 9712,
doi: 10.1073/pnas.1912506117
Gaudi, B. S., & Winn, J. N. 2007, ApJ, 655, 550,
doi: 10.1086/509910
Gaudi, B. S., Stassun, K. G., Collins, K. A., et al. 2017,
Nature, 546, 514, doi: 10.1038/nature22392
Gimenez, A. 2006, ApJ, 650, 408, doi: 10.1086/507021
Ginski, C., Facchini, S., Huang, J., et al. 2021, ApJL, 908,
L25, doi: 10.3847/2041-8213/abdf57
Gizon, L., & Solanki, S. K. 2003, ApJ, 1009,
doi: 10.1086/374715
Gizon, L., Ballot, J., Michel, E., et al. 2013, Proceedings of
the National Academy of Science, 110, 13267,
doi: 10.1073/pnas.1303291110
Glebocki, R., & Stawikowski, A. 1997, A&A, 328, 579
Goldreich, P., Lithwick, Y., & Sari, R. 2004, ARA&A, 42,
549, doi: 10.1146/annurev.astro.42.053102.134004
Gomes, R., Deienno, R., & Morbidelli, A. 2017, AJ, 153,
27, doi: 10.3847/1538-3881/153/1/27
Gough, D. O., & Kosovichev, A. G. 1993, in Astronomical
Society of the Pacific Conference Series, Vol. 40, IAU
Colloq. 137: Inside the Stars, ed. W. W. Weiss &
A. Baglin, 566
Gratia, P., & Fabrycky, D. 2017, MNRAS, 464, 1709,
doi: 10.1093/mnras/stw2180
Gray, D. F. 2005, The Observation and Analysis of Stellar
Photospheres, 3rd Ed. (ISBN 0521851866, Cambridge
University Press)
Greaves, J. S., Kennedy, G. M., Thureau, N., et al. 2014,
MNRAS, 438, L31, doi: 10.1093/mnrasl/slt153
Grishin, E., Lai, D., & Perets, H. B. 2018, MNRAS, 474,
3547, doi: 10.1093/mnras/stx3005
Groot, P. J. 2012, ApJ, 745, 55,
doi: 10.1088/0004-637X/745/1/55
Guenther, E. W., Dıaz, R. F., Gazzano, J.-C., et al. 2012,
A&A, 537, A136, doi: 10.1051/0004-6361/201117706
Guthrie, B. N. G. 1985, MNRAS, 215, 545
Hale, A. 1994, AJ, 107, 306, doi: 10.1086/116855
Hamers, A. S. 2017, MNRAS, 466, 4107,
doi: 10.1093/mnras/stx035
Hamers, A. S., Antonini, F., Lithwick, Y., Perets, H. B., &
Portegies Zwart, S. F. 2017, MNRAS, 464, 688,
doi: 10.1093/mnras/stw2370
Hamers, A. S., & Tremaine, S. 2017, AJ, 154, 272,
doi: 10.3847/1538-3881/aa9926
Hansen, B. M. S. 2012, ApJ, 757, 6,
doi: 10.1088/0004-637X/757/1/6
Obliquity 35
Hao, W., Kouwenhoven, M. B. N., & Spurzem, R. 2013,
MNRAS, 433, 867, doi: 10.1093/mnras/stt771
Healy, B. F., & McCullough, P. R. 2020, ApJ, 903, 99,
doi: 10.3847/1538-4357/abbc03
Hebrard, G., Bouchy, F., Pont, F., et al. 2008, A&A, 488,
763, doi: 10.1051/0004-6361:200810056
Hebrard, G., Desert, J.-M., Dıaz, R. F., et al. 2010, A&A,
516, A95, doi: 10.1051/0004-6361/201014327
Hebrard, G., Evans, T. M., Alonso, R., et al. 2011, A&A,
533, A130, doi: 10.1051/0004-6361/201117192
Heller, C. H. 1993, ApJ, 408, 337, doi: 10.1086/172591
Herman, M. K., Zhu, W., & Wu, Y. 2019, arXiv e-prints,
arXiv:1901.01974. https://arxiv.org/abs/1901.01974
Hirano, T., Sanchis-Ojeda, R., Takeda, Y., et al. 2012, ApJ,
756, 66, doi: 10.1088/0004-637X/756/1/66
—. 2014, ApJ, 783, 9, doi: 10.1088/0004-637X/783/1/9
Hirano, T., Suto, Y., Winn, J. N., et al. 2011, ApJ, 742, 69,
doi: 10.1088/0004-637X/742/2/69
Hirano, T., Krishnamurthy, V., Gaidos, E., et al. 2020a,
arXiv e-prints, arXiv:2006.13243.
https://arxiv.org/abs/2006.13243
Hirano, T., Gaidos, E., Winn, J. N., et al. 2020b, arXiv
e-prints, arXiv:2002.05892.
https://arxiv.org/abs/2002.05892
Hjorth, M., Albrecht, S., Hirano, T., et al. 2021, arXiv
e-prints, arXiv:2102.07677.
https://arxiv.org/abs/2102.07677
Hoeijmakers, H. J., Cabot, S. H. C., Zhao, L., et al. 2020,
A&A, 641, A120, doi: 10.1051/0004-6361/202037437
Holczer, T., Shporer, A., Mazeh, T., et al. 2015, ApJ, 807,
170, doi: 10.1088/0004-637X/807/2/170
Holt, J. R. 1893, A&A, 12, 646
Hosokawa, Y. 1953, PASJ, 5, 88
Howe, K. S., & Clarke, C. J. 2009, MNRAS, 392, 448,
doi: 10.1111/j.1365-2966.2008.14073.x
Huber, D., Carter, J. A., Barbieri, M., et al. 2013, Science,
342, 331, doi: 10.1126/science.1242066
Jensen, E. L. N., & Akeson, R. 2014, Nature, 511, 567,
doi: 10.1038/nature13521
Johnson, M. C., Cochran, W. D., Addison, B. C., Tinney,
C. G., & Wright, D. J. 2017, AJ, 154, 137,
doi: 10.3847/1538-3881/aa8462
Johnson, M. C., Cochran, W. D., Albrecht, S., et al. 2014,
ApJ, 790, 30, doi: 10.1088/0004-637X/790/1/30
Johnson, M. C., Cochran, W. D., Collier Cameron, A., &
Bayliss, D. 2015, ApJL, 810, L23,
doi: 10.1088/2041-8205/810/2/L23
Justesen, A. B., & Albrecht, S. 2020, arXiv e-prints,
arXiv:2008.12068. https://arxiv.org/abs/2008.12068
Kamiaka, S., Benomar, O., & Suto, Y. 2018, MNRAS, 479,
391, doi: 10.1093/mnras/sty1358
Kamiaka, S., Benomar, O., Suto, Y., et al. 2019, AJ, 157,
137, doi: 10.3847/1538-3881/ab04a9
Knutson, H. A., Fulton, B. J., Montet, B. T., et al. 2014,
ApJ, 785, 126, doi: 10.1088/0004-637X/785/2/126
Kopal, Z. 1959, Close binary systems (The International
Astrophysics Series, London: Chapman & Hall, 1959)
Kozai, Y. 1962, AJ, 67, 591, doi: 10.1086/108790
Kraus, S. 2019, in The Very Large Telescope in 2030, 36
Kraus, S., Le Bouquin, J.-B., Kreplin, A., et al. 2020,
ApJL, 897, L8, doi: 10.3847/2041-8213/ab9d27
Kuszlewicz, J. S., Chaplin, W. J., North, T. S. H., et al.
2019, MNRAS, 488, 572, doi: 10.1093/mnras/stz1689
Lachaume, R. 2003, A&A, 400, 795,
doi: 10.1051/0004-6361:20030072
Lai, D. 2012, MNRAS, 423, 486,
doi: 10.1111/j.1365-2966.2012.20893.x
—. 2014, MNRAS, 440, 3532, doi: 10.1093/mnras/stu485
—. 2016, AJ, 152, 215, doi: 10.3847/0004-6256/152/6/215
Lai, D., Anderson, K. R., & Pu, B. 2018, MNRAS, 475,
5231, doi: 10.1093/mnras/sty133
Lai, D., Foucart, F., & Lin, D. N. C. 2011, MNRAS, 412,
2790, doi: 10.1111/j.1365-2966.2010.18127.x
Laskar, J. 2008, Icarus, 196, 1,
doi: 10.1016/j.icarus.2008.02.017
Latham, D. W., Mazeh, T., Stefanik, R. P., Mayor, M., &
Burki, G. 1989, Nature, 339, 38, doi: 10.1038/339038a0
Le Bouquin, J., Absil, O., Benisty, M., et al. 2009, A&A,
498, L41, doi: 10.1051/0004-6361/200911854
Li, G., Naoz, S., Holman, M., & Loeb, A. 2014a, ApJ, 791,
86, doi: 10.1088/0004-637X/791/2/86
Li, G., Naoz, S., Kocsis, B., & Loeb, A. 2014b, ApJ, 785,
116, doi: 10.1088/0004-637X/785/2/116
Li, G., Naoz, S., Valsecchi, F., Johnson, J. A., & Rasio,
F. A. 2014c, ApJ, 794, 131,
doi: 10.1088/0004-637X/794/2/131
Li, G., & Winn, J. N. 2016, ApJ, 818, 5,
doi: 10.3847/0004-637X/818/1/5
Lidov, M. L. 1962, Planet. Space Sci., 9, 719,
doi: 10.1016/0032-0633(62)90129-0
Lissauer, J. J., Fabrycky, D. C., Ford, E. B., et al. 2011,
Nature, 470, 53, doi: 10.1038/nature09760
Lithwick, Y., & Wu, Y. 2014, Proceedings of the National
Academy of Science, 111, 12610,
doi: 10.1073/pnas.1308261110
Lopez-Morales, M., Triaud, A. H. M. J., Rodler, F., et al.
2014, ApJL, 792, L31, doi: 10.1088/2041-8205/792/2/L31
Louden, E. M., Winn, J. N., Petigura, E. A., et al. 2021,
AJ, 161, 68, doi: 10.3847/1538-3881/abcebd
36 Albrecht, Dawson, & Winn
Lubow, S. H., & Martin, R. G. 2016, ApJ, 817, 30,
doi: 10.3847/0004-637X/817/1/30
Lubow, S. H., & Ogilvie, G. I. 2000, ApJ, 538, 326,
doi: 10.1086/309101
Malmberg, D., Davies, M. B., & Heggie, D. C. 2011,
MNRAS, 411, 859, doi: 10.1111/j.1365-2966.2010.17730.x
Mancini, L., Esposito, M., Covino, E., et al. 2018, ArXiv
e-prints. https://arxiv.org/abs/1802.03859
Mann, A. W., Johnson, M. C., Vanderburg, A., et al. 2020,
arXiv e-prints, arXiv:2005.00047.
https://arxiv.org/abs/2005.00047
Marcy, G. W., & Butler, R. P. 1996, ApJL, 464, L147,
doi: 10.1086/310096
Marino, S., Perez, S., & Casassus, S. 2015, ApJ, 798, L44,
doi: 10.1088/2041-8205/798/2/L44
Martioli, E., Hebrard, G., Moutou, C., et al. 2020, arXiv
e-prints, arXiv:2006.13269.
https://arxiv.org/abs/2006.13269
Masuda, K. 2015, ApJ, 805, 28,
doi: 10.1088/0004-637X/805/1/28
—. 2017, AJ, 154, 64, doi: 10.3847/1538-3881/aa7aeb
Masuda, K., & Tamayo, D. 2020, arXiv e-prints,
arXiv:2009.06850. https://arxiv.org/abs/2009.06850
Masuda, K., & Winn, J. N. 2020, AJ, 159, 81,
doi: 10.3847/1538-3881/ab65be
Masuda, K., Winn, J. N., & Kawahara, H. 2020, AJ, 159,
38, doi: 10.3847/1538-3881/ab5c1d
Matsakos, T., & Konigl, A. 2017, AJ, 153, 60,
doi: 10.3847/1538-3881/153/2/60
Maxted, P. F. L. 2018, Rotation of Planet-Hosting Stars,
ed. H. J. Deeg & J. A. Belmonte, 18
Mayor, M., & Queloz, D. 1995, Nature, 378, 355,
doi: 10.1038/378355a0
Mazeh, T. 2008, in EAS Publications Series, Vol. 29, EAS
Publications Series, ed. M.-J. Goupil & J.-P. Zahn, 1–65
Mazeh, T., Holczer, T., & Shporer, A. 2015a, ApJ, 800,
142, doi: 10.1088/0004-637X/800/2/142
Mazeh, T., Perets, H. B., McQuillan, A., & Goldstein, E. S.
2015b, ApJ, 801, 3, doi: 10.1088/0004-637X/801/1/3
McLaughlin, D. B. 1924, ApJ, 60, 22, doi: 10.1086/142826
Monin, J.-L., Clarke, C. J., Prato, L., & McCabe, C. 2007,
Protostars and Planets V, 395
Morton, T. D., & Johnson, J. A. 2011, ApJ, 729, 138,
doi: 10.1088/0004-637X/729/2/138
Morton, T. D., & Winn, J. N. 2014, ApJ, 796, 47,
doi: 10.1088/0004-637X/796/1/47
Mourard, D., Nardetto, N., ten Brummelaar, T., et al.
2018, in Society of Photo-Optical Instrumentation
Engineers (SPIE) Conference Series, Vol. 10701, Optical
and Infrared Interferometry and Imaging VI, ed. M. J.
Creech-Eakman, P. G. Tuthill, & A. Merand, 1070120
Moutou, C., Dıaz, R. F., Udry, S., et al. 2011, A&A, 533,
A113, doi: 10.1051/0004-6361/201116760
Munoz, D. J., & Perets, H. B. 2018, ArXiv e-prints.
https://arxiv.org/abs/1805.03654
Nagasawa, M., & Ida, S. 2011, ApJ, 742, 72,
doi: 10.1088/0004-637X/742/2/72
Nagasawa, M., Ida, S., & Bessho, T. 2008, ApJ, 678, 498,
doi: 10.1086/529369
Naoz, S. 2016, ARA&A, 54, 441,
doi: 10.1146/annurev-astro-081915-023315
Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., &
Teyssandier, J. 2011, Nature, 473, 187,
doi: 10.1038/nature10076
Naoz, S., Farr, W. M., & Rasio, F. A. 2012, ApJL, 754,
L36, doi: 10.1088/2041-8205/754/2/L36
Neveu-VanMalle, M., Queloz, D., Anderson, D. R., et al.
2014, A&A, 572, A49, doi: 10.1051/0004-6361/201424744
Newton, E. R., Mann, A. W., Kraus, A. L., et al. 2021, AJ,
161, 65, doi: 10.3847/1538-3881/abccc6
Ngo, H., Knutson, H. A., Hinkley, S., et al. 2015, ApJ, 800,
138, doi: 10.1088/0004-637X/800/2/138
—. 2016, ApJ, 827, 8, doi: 10.3847/0004-637X/827/1/8
Nutzman, P. A., Fabrycky, D. C., & Fortney, J. J. 2011,
ApJL, 740, L10, doi: 10.1088/2041-8205/740/1/L10
Ogilvie, G. I. 2014, ARA&A, 52, 171,
doi: 10.1146/annurev-astro-081913-035941
Ohta, Y., Taruya, A., & Suto, Y. 2005, ApJ, 622, 1118,
doi: 10.1086/428344
Otor, O. J., Montet, B. T., Johnson, J. A., et al. 2016, AJ,
152, 165, doi: 10.3847/0004-6256/152/6/165
Palle, E., Oshagh, M., Casasayas-Barris, N., et al. 2020,
arXiv e-prints, arXiv:2006.13609.
https://arxiv.org/abs/2006.13609
Penev, K., Bouma, L. G., Winn, J. N., & Hartman, J. D.
2018, AJ, 155, 165, doi: 10.3847/1538-3881/aaaf71
Perryman, M. 2011, The Exoplanet Handbook, ed.
Perryman, M.
Perryman, M., Hartman, J., Bakos, G. A., & Lindegren, L.
2014, ApJ, 797, 14, doi: 10.1088/0004-637X/797/1/14
Petigura, E. A., Howard, A. W., Marcy, G. W., et al. 2017,
AJ, 154, 107, doi: 10.3847/1538-3881/aa80de
Petrov, R. G. 1989, in NATO ASIC Proc. 274:
Diffraction-Limited Imaging with Very Large Telescopes,
ed. D. M. Alloin & J.-M. Mariotti, 249
Obliquity 37
Petrovich, C., Munoz, D. J., Kratter, K. M., & Malhotra,
R. 2020, ApJL, 902, L5, doi: 10.3847/2041-8213/abb952
Petrovich, C., & Tremaine, S. 2016, ApJ, 829, 132,
doi: 10.3847/0004-637X/829/2/132
Picogna, G., & Marzari, F. 2015, A&A, 583, A133,
doi: 10.1051/0004-6361/201526162
Piskorz, D., Knutson, H. A., Ngo, H., et al. 2015, ApJ, 814,
148, doi: 10.1088/0004-637X/814/2/148
Plavchan, P., Barclay, T., Gagne, J., et al. 2020, Nature,
582, 497, doi: 10.1038/s41586-020-2400-z
Pont, F. 2009, MNRAS, 396, 1789,
doi: 10.1111/j.1365-2966.2009.14868.x
Poon, M., Zanazzi, J. J., & Zhu, W. 2021, MNRAS, 503,
1599, doi: 10.1093/mnras/stab575
Portegies Zwart, S., Pelupessy, I., van Elteren, A., Wijnen,
T. P. G., & Lugaro, M. 2018, A&A, 616, A85,
doi: 10.1051/0004-6361/201732060
Queloz, D., Eggenberger, A., Mayor, M., et al. 2000, A&A,
359, L13
Quinn, S. N., & White, R. J. 2016, ApJ, 833, 173,
doi: 10.3847/1538-4357/833/2/173
Rodriguez, J. E., Quinn, S. N., Zhou, G., et al. 2021, arXiv
e-prints, arXiv:2101.01726.
https://arxiv.org/abs/2101.01726
Rogers, T. M., Lin, D. N. C., & Lau, H. H. B. 2012, ApJL,
758, L6, doi: 10.1088/2041-8205/758/1/L6
Rogers, T. M., Lin, D. N. C., McElwaine, J. N., & Lau,
H. H. B. 2013, ApJ, 772, 21,
doi: 10.1088/0004-637X/772/1/21
Romanova, M. M., Koldoba, A. V., Ustyugova, G. V., et al.
2020, arXiv e-prints, arXiv:2012.10826.
https://arxiv.org/abs/2012.10826
Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., &
Lovelace, R. V. E. 2013, MNRAS, 430, 699,
doi: 10.1093/mnras/sts670
Rossiter, R. A. 1924, ApJ, 60, 15, doi: 10.1086/142825
Safsten, E. D., Dawson, R. I., & Wolfgang, A. 2020, arXiv
e-prints, arXiv:2009.02357.
https://arxiv.org/abs/2009.02357
Sakai, N., Hanawa, T., Zhang, Y., et al. 2019, Nature, 565,
206, doi: 10.1038/s41586-018-0819-2
Sanchis-Ojeda, R., & Winn, J. N. 2011, ApJ, 743, 61,
doi: 10.1088/0004-637X/743/1/61
Sanchis-Ojeda, R., Winn, J. N., Holman, M. J., et al. 2011,
ApJ, 733, 127, doi: 10.1088/0004-637X/733/2/127
Sanchis-Ojeda, R., Fabrycky, D. C., Winn, J. N., et al.
2012, Nature, 487, 449, doi: 10.1038/nature11301
Sanchis-Ojeda, R., Winn, J. N., Marcy, G. W., et al. 2013,
ApJ, 775, 54, doi: 10.1088/0004-637X/775/1/54
Sanchis-Ojeda, R., Winn, J. N., Dai, F., et al. 2015, ApJL,
812, L11, doi: 10.1088/2041-8205/812/1/L11
Santos, N. C., Cristo, E., Demangeon, O., et al. 2020,
A&A, 644, A51, doi: 10.1051/0004-6361/202039454
Sato, K. 1974, PASJ, 26, 65
Schlaufman, K. C. 2010, ApJ, 719, 602,
doi: 10.1088/0004-637X/719/1/602
Schlaufman, K. C., & Winn, J. N. 2013, ApJ, 772, 143,
doi: 10.1088/0004-637X/772/2/143
Schlesinger, F. 1910, Publications of the Allegheny
Observatory of the University of Pittsburgh, 1, 123
Shporer, A., & Brown, T. 2011, ApJ, 733, 30,
doi: 10.1088/0004-637X/733/1/30
Shporer, A., Brown, T., Mazeh, T., & Zucker, S. 2012, New
Astronomy, 17, 309, doi: 10.1016/j.newast.2011.08.006
Skumanich, A. 1972, ApJ, 171, 565, doi: 10.1086/151310
—. 2019, ApJ, 878, 35, doi: 10.3847/1538-4357/ab1b24
Southworth, J. 2011, MNRAS, 417, 2166,
doi: 10.1111/j.1365-2966.2011.19399.x
Spalding, C. 2019, ApJ, 879, 12,
doi: 10.3847/1538-4357/ab23f5
Spalding, C., & Batygin, K. 2015, ApJ, 811, 82,
doi: 10.1088/0004-637X/811/2/82
—. 2016, ApJ, 830, 5, doi: 10.3847/0004-637X/830/1/5
Spalding, C., Batygin, K., & Adams, F. C. 2014, ApJL,
797, L29, doi: 10.1088/2041-8205/797/2/L29
Storch, N. I., Anderson, K. R., & Lai, D. 2014, Science,
345, 1317, doi: 10.1126/science.1254358
Struve, O., & Elvey, C. T. 1931, MNRAS, 91, 663
Szabo, G. M., Szabo, R., Benko, J. M., et al. 2011, ApJL,
736, L4, doi: 10.1088/2041-8205/736/1/L4
Takaishi, D., Tsukamoto, Y., & Suto, Y. 2020, arXiv
e-prints, arXiv:2001.05456.
https://arxiv.org/abs/2001.05456
Takeda, G., Kita, R., & Rasio, F. A. 2008, ApJ, 683, 1063,
doi: 10.1086/589852
Talens, G. J. J., Justesen, A. B., Albrecht, S., et al. 2018,
A&A, 612, A57, doi: 10.1051/0004-6361/201731512
Teyssandier, J., Lai, D., & Vick, M. 2019, MNRAS, 486,
2265, doi: 10.1093/mnras/stz1011
Teyssandier, J., Naoz, S., Lizarraga, I., & Rasio, F. A.
2013, ApJ, 779, 166, doi: 10.1088/0004-637X/779/2/166
Thies, I., Kroupa, P., Goodwin, S. P., Stamatellos, D., &
Whitworth, A. P. 2011, MNRAS, 417, 1817,
doi: 10.1111/j.1365-2966.2011.19390.x
Triaud, A. H. M. J. 2011, A&A, 534, L6,
doi: 10.1051/0004-6361/201117713
—. 2017, The Rossiter-McLaughlin Effect in Exoplanet
Research, 2
38 Albrecht, Dawson, & Winn
Triaud, A. H. M. J., Collier Cameron, A., Queloz, D., et al.
2010, A&A, 524, A25, doi: 10.1051/0004-6361/201014525
Triaud, A. H. M. J., Queloz, D., Hellier, C., et al. 2011,
A&A, 531, A24, doi: 10.1051/0004-6361/201016367
Valsecchi, F., & Rasio, F. A. 2014, ApJ, 786, 102,
doi: 10.1088/0004-637X/786/2/102
Van Eylen, V., Lund, M. N., Silva Aguirre, V., et al. 2014,
ApJ, 782, 14, doi: 10.1088/0004-637X/782/1/14
Walkowicz, L. M., & Basri, G. S. 2013, MNRAS, 436, 1883,
doi: 10.1093/mnras/stt1700
Wang, S., Addison, B., Fischer, D. A., et al. 2018, AJ, 155,
70, doi: 10.3847/1538-3881/aaa2fb
Watson, C. A., Littlefair, S. P., Diamond, C., et al. 2011,
MNRAS, 413, L71, doi: 10.1111/j.1745-3933.2011.01036.x
Weis, E. W. 1974, ApJ, 190, 331, doi: 10.1086/152881
Wijnen, T. P. G., Pelupessy, F. I., Pols, O. R., & Portegies
Zwart, S. 2017, A&A, 604, A88,
doi: 10.1051/0004-6361/201730793
Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A.
2010, ApJL, 718, L145,
doi: 10.1088/2041-8205/718/2/L145
Winn, J. N., & Fabrycky, D. C. 2015, ARA&A, 53, 409,
doi: 10.1146/annurev-astro-082214-122246
Winn, J. N., Holman, M. J., Johnson, J. A., Stanek, K. Z.,
& Garnavich, P. M. 2004, ApJL, 603, L45,
doi: 10.1086/383089
Winn, J. N., Johnson, J. A., Fabrycky, D., et al. 2009, ApJ,
700, 302, doi: 10.1088/0004-637X/700/1/302
Winn, J. N., Petigura, E. A., Morton, T. D., et al. 2017,
AJ, 154, 270, doi: 10.3847/1538-3881/aa93e3
Wu, Y., & Lithwick, Y. 2011, ApJ, 735, 109,
doi: 10.1088/0004-637X/735/2/109
Wu, Y., & Murray, N. 2003, ApJ, 589, 605,
doi: 10.1086/374598
Wyttenbach, A., Lovis, C., Ehrenreich, D., et al. 2017,
A&A, 602, A36, doi: 10.1051/0004-6361/201630063
Xie, J.-W., Dong, S., Zhu, Z., et al. 2016, Proceedings of
the National Academy of Science, 113, 11431,
doi: 10.1073/pnas.1604692113
Xue, Y., Suto, Y., Taruya, A., et al. 2014, ApJ, 784, 66,
doi: 10.1088/0004-637X/784/1/66
Zahn, J.-P. 1977, A&A, 57, 383
Zahn, J. P. 2008, in EAS Publications Series, Vol. 29, EAS
Publications Series, ed. M. J. Goupil & J. P. Zahn, 67–90
Zanazzi, J. J., & Lai, D. 2018, MNRAS, 478, 835,
doi: 10.1093/mnras/sty1075
Zhou, G., Latham, D. W., Bieryla, A., et al. 2016, MNRAS,
460, 3376, doi: 10.1093/mnras/stw1107
Zhou, G., Bayliss, D., Hartman, J. D., et al. 2015, ApJL,
814, L16, doi: 10.1088/2041-8205/814/1/L16
Zhou, G., Rodriguez, J. E., Vanderburg, A., et al. 2018, AJ,
156, 93, doi: 10.3847/1538-3881/aad085
Zhou, G., Huang, C. X., Bakos, G. A., et al. 2019, AJ, 158,
141, doi: 10.3847/1538-3881/ab36b5
Zhou, G., Winn, J. N., Newton, E. R., et al. 2020, ApJL,
892, L21, doi: 10.3847/2041-8213/ab7d3c
Zhou, G., Quinn, S. N., Irwin, J., et al. 2021, AJ, 161, 2,
doi: 10.3847/1538-3881/abba22