Kepler - Aarhus Universitet

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

[email protected]

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-

http://orcid.org/0000-0003-1762-8235

http://orcid.org/0000-0002-4265-047X

mailto: [email protected]

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.


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.


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


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 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.


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.


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)


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

https://www.cosmos.esa.int/web/gaia/release

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.


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,


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


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).


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


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


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).

http://www.astro.keele.ac.uk/jkt/tepcat/tepcat.html


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—

-


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.

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