Draft version April 16, 2021 Typeset 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, Denmark 2Department of Astronomy & Astrophysics, Center for Exoplanets and Habitable Worlds,The Pennsylvania State University, University Park, PA 16802, USA 3Department 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 are a close encounter with another star (Heller 1993), a Since the earliest observations of sunspots by Fabri- torque resulting from motion of the protoplanetary disk cius, Scheiner, and Galileo it has been known that the through the interstellar medium Wijnen et al.(2017), a Sun’s equatorial plane is nearly aligned with the ecliptic torque from an undiscovered outer planet (Bailey et al. (Casanovas 1997). A modern measurement of the Sun’s 2016; Gomes et al. 2017; Lai 2016), an asymmetry of the obliquity, based on helioseismology, is 7.155 ± 0.002◦ solar wind (Spalding 2019), and the imprint of a nearby (Beck & Giles 2005). The low solar obliquity was part supernova Portegies Zwart et al.(2018). of the body of evidence that led Laplace to the “nebu- Exoplanetary systems have proven to show a wider lar theory” for the formation of the Solar System, which range of orbital characteristics than had been expected was incorrect but is remembered for the theoretical de- based on analyses of the Solar System (see, e.g., Winn but of the protoplanetary disk. The fact that the obliq- & Fabrycky 2015, for a review). Among these sur- uity is a little higher than the root-mean-squared mu- prises were close-orbiting giant planets (Mayor & Queloz tual inclination of 1.9◦ between the planetary orbits has 1995), high orbital eccentricities (Latham et al. 1989; also inspired theorists; among the proffered explanations 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- Corresponding author: Simon Albrecht ticle, large stellar obliquities (H´ebrardet al. 2008; Winn [email protected] et al. 2009). One of the main goals of exoplanetary sci- 2 Albrecht, Dawson, & Winn

Figure1 illustrates the angles that determine the ori- entation of a star (ˆn?) with respect to the line of sight (ˆz) and with respect to the orbital axis of a planet (ˆno). The obliquity ψ is the angle betweenn ˆ? andn ˆo. In the coordinate system shown in Figure1,

nˆo = sin io yˆ + cos io zˆ and (1)

nˆ? = sin i sin λ xˆ + sin i cos λ yˆ + cos i z,ˆ (2)

where we have chosen to orient they ˆ axis along the sky projection ofn ˆ?. 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 Figure 1. Coordinate system and angles that specify the orientation of the spin and orbital angular momentum vec- cos ψ =n ˆ? · nˆo = sin i cos λ sin io + cos i cos io . (3) tors (modeled after Perryman(2011)). The obliquity is ψ, the orbital inclination is io, and the inclination of the stellar Most of the observational methods do not measure rotation axis is i. ψ in one step. Instead, some techniques are capable of detecting differences between i? and io, leading to a ence is to understand the physical processes that are lower limit |io−i?| on the obliquity. Other techniques are sensitive to λ, which is a lower limit on ψ when |λ| < 90◦, responsible for this architectural diversity. ◦ Measuring the obliquity of an exoplanet host star is and an upper limit on ψ when |λ| > 90 . A sample of challenging, given that ordinary observations lack the stars with completely random orientations would show angular resolution to discern any details on the spa- a uniform distribution in the azimuthal angles λ and in tial scale of the stellar diameter. Nevertheless, using the cosines of the polar angles i, io, and ψ. an array of techniques, we now know the obliquities of For the statistical analysis of obliquity measurements, approximately 150 stars, and we have drawn statisti- two useful references are Fabrycky & Winn(2009) and cal inferences about the obliquity distribution of sam- Mu˜noz& Perets(2018). The former authors provided ples of ∼103 stars. Prograde, polar, and retrograde or- analytic formulas for the conditional probability densi- bits have been found, and a few patterns have emerged ties p(ψ|λ) and p(λ|ψ) under the assumption of random relating to stellar mass, planetary mass, and orbital orientations. They also showed how to use measure- distance. There is no unique interpretation of the re- ments of λ to model the obliquity distribution of a pop- ulation of stars as a von-Mises Fisher (vMF) distribu- sults. Misalignments might occur before, during, or af- 1 ter the epoch of planet formation. They may be linked tion , to specific dynamical events in a planet’s history such κ p(ψ) = exp(κ cos ψ) sin ψ. (4) as planet-planet scattering or high-eccentricity migra- 2 sinh κ tion, or they may be the outcome of general processes Mu˜noz& Perets(2018) extended this framework to in- affecting stars and protoplanetary disks irrespective of clude information about i in addition to λ. the planets that eventually form. This article is an attempt to review the current status 3. METHODS AND KEY FINDINGS of the observations and theories regarding the obliquities The main challenge in measuring any of the angles in of stars with planets. Section2 introduces the geometry Figure1 is that stars are almost always spatially un- and terminology that will be important throughout this resolved by our telescopes. We can only observe the article. Section3 describes the measurement techniques star’s flux and spectrum integrated over the star’s visi- and key findings. Section4 discusses the proposed phys- ble hemisphere. ical mechanisms that can excite or dampen obliquities and their success or failure in matching the observations. Section5 is a summary and a set of recommendations 1 The vMF distribution is a widely-used model in directional for future work in this area. statistics that resembles a two-dimensional Gaussian distribution wrapped around a sphere. For small values of the concentration parameter κ, the distribution becomes isotropic. For large val- ues of κ, the distribution approaches a Rayleigh distribution with 2. GEOMETRY width parameter σ = κ−1/2. Obliquity 3 ) R (

1

s 10 u i d a r

aligned y

r misaligned a Rossiter-McLaughlin t

e Asteroseismology n 100 a l Star Spots p Gravity Darkening 100 101 102 10000

) period (days) Interferometry Projected rotation rate

M 2 (

s s a m

r 1 a l l e t s 0 1 3 10 30 100 10000 period (days)

Figure 2. Parameter space of obliquity measurement methods. Each point represents an obliquity measurement reported 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); open symbols are for well-aligned stars or ambiguous cases. The RM, starspot, and gravity-darkening methods require observations during transits, making them less applicable to systems with smaller planets or longer periods. The gravity-darkening method requires 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 intensive observations conducted during transits, making them applicable to planets of all types. The asteroseismic method requires moderately rapid rotation and long-lived pulsation modes, which generally occur for stars somewhat more massive than the Sun. 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 orbital inclination. Also important, though not conveyed in this diagram, is that the methods differ in the achievable precision and parameter degeneracies.

Fortunately, some aspects of the disk-integrated flux These inclination-based methods have some im- and spectrum depend on the star’s inclination i with re- portant limitations. Because of the north/south symme- ◦ spect to the line of sight. One is the rotational Doppler try of the star, we cannot distinguish i? from 180 − i?, broadening of its spectral absorption lines, which is leading to a twofold degeneracy in obliquity constraints. proportional to v sin i where v is the rotation velocity In particular, we cannot tell whether a star has pro- (§ 3.4). Another is the star’s amplitude of photometric grade or retrograde rotation with respect to the line variability due to rotating starspots, which is expected of sight or the planetary orbit.2 Another problem is to vary roughly in proportion to sin i (§ 3.5). A third that sin i-based techniques are insensitive at high incli- type of data that bears information about orientation is nations. Even if sin i is constrained to be in the narrow the fine structure within the power spectrum of a star’s range from 0.9 to 1, the inclination can be any value in asteroseismic oscillations; the relative amplitudes of the the range from 64 to 116◦. This problem arises often modes within a rotationally-split multiplet depend on because high inclinations are common; 44% of the stars i (§ 3.2). When we also have knowledge of io (such as in a randomly-oriented population have sin i > 0.9. when the planet detected through the transit or astro- metric techniques), these types of data place constraints 2 on the stellar obliquity. For transiting planets, the same degeneracy afflicts measure- ments of io, although the geometrical requirement for transits im- ◦ plies that io is never far from 90 . 4 Albrecht, Dawson, & Winn

Figure 3. Geometry of the Rossiter-McLaughlin effect. The left panel illustrates a transit, with the planet crossing from 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 vectorsn ˆ? andn ˆo are separated by the angle λ, and the x-axis is perpendicular to the projected rotation axis. For the case of uniform rotation, the sub-planet radial velocity is (v sin i) x and the extrema of the RM signal occur at ingress (x1) and egress (x2). The relations between x1, x2, λ and the impact parameter b are indicated on the diagram. 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 ette divided by the area of the stellar disk. In practice, obliquity rely on a transiting planet to provide spa- this makes it very challenging to deploy these methods tially resolved information as its shadow scans across on planets smaller than Neptune around Sun-like stars. the stellar disk. A star’s intensity and emergent spec- Finally, there is a technique that is mainly sensitive trum vary across the stellar disk in a manner that de- to λ and does not require a transiting planet: optical in- pends on the star’s orientation. For example, stellar terferometry with high spectral resolution. For nearby rotation causes the radial velocity of the stellar disk to bright stars, interferometric observations can partially exhibit a gradient from the approaching side to the re- resolve the stellar disk and reveal the displacement on ceding side. When a transiting planet hides a portion of the sky between the redshifted and blueshifted halves of the stellar disk, the corresponding radial-velocity com- the rotating star (§ 3.3). This is still a highly special- ponent is absent from the disk-integrated stellar spec- ized technique, though, and leaves open the problem of trum, leading to line-profile distortions known as the determining the orientation of the planet’s orbit. Rossiter-McLaughlin effect (§ 3.1). Another technique Each technique works best in different circumstances. is based on detecting the glitches in the light curve when Figure2 illustrates the applicability of these different a transiting planet occults a starspot or other inhomo- techniques to systems with different stellar masses, plan- geneity on the stellar disk; the timings of such anomalies etary radii, and orbital periods. Below, we describe can sometimes be used to constrain the stellar obliquity these techniques in more detail, but not in the geometry- (§ 3.5). A third technique is based on gravity darkening: based order described here. Instead, we devote the most the equatorial zone of a rapidly rotating star is lifted to attention to the techniques that have delivered the most higher elevation, leading to a lower effective tempera- information. 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 3.1. The Rossiter-McLaughlin effect a relativistic effect known as rotational Doppler beaming In a letter the editor of the Sidereal Messenger, Holt (§ 3.7.1). (1893) pointed out that a star’s rotation rate could be These transit-based methods are usually more sen- measured by observing the time-variable distortions of sitive to λ than to i. Indeed, in the best cases, λ can its absorption spectrum during an eclipse. We have not ◦ be measured with a precision on the order of 1 . The been able to learn anything more about this insight- disadvantages of these methods are that they require ful correspondent, nor have we found any earlier ref- time-critical observations of transits, and the signals are erence to what is now called the Rossiter-McLaughlin generally proportional to the area of the planet’s silhou- effect. The name honors the work of Rossiter(1924) Obliquity 5

20 e 20 a 4 b c d

10 2 ] 10 − 1 ] ] − 1 − 1

0 0 0 RV [m s RV [m s −10 −2 RM effect [m s −10

−20 stellar rotation −4 turbulence+PSF differential rotation convective blueshift −20 −1.0−0.5 0.0 0.5 1.0 −1.0−0.5 0.0 0.5 1.0 −1.0−0.5 0.0 0.5 1.0 −1.0−0.5 0.0 0.5 1.0 distance [Rstar] distance [Rstar] distance [Rstar] distance [Rstar] −1.0 −0.5 0.0 0.5 1.0 distance [Rstar] 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) Instrumental −1 −1 broadening (taken to be 2.2 km s ) and macroturbulence (ζRT = 3 km s ) 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-like convective blueshift produces an anomalous velocity depending on distance from the center of the stellar disk. (e) The combined model 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 Alternatively, the line-profile distortions can be de- β Lyrae and Algol systems, respectively.3 tected and modeled directly without the intermediate One of the broadening mechanisms of stellar absorp- step of computing an anomalous radial velocity. Models tion lines is the variation in the rotational Doppler shift for this “Doppler shadow” have also been developed ex- between the two sides of the stellar disk. Due to ro- tensively, starting with a beautiful exposition by Struve tation, light from the approaching half of a star is & Elvey(1931) for the Algol system and continuing to blueshifted, light from the receding half is redshifted, the present (e.g. Albrecht et al. 2007; Collier Cameron and the disk-integrated spectrum shows a spread in et al. 2010; Albrecht et al. 2013a; Johnson et al. 2014; Doppler shifts. During an eclipse or transit, a a por- Cegla et al. 2016; Zhou et al. 2016; Johnson et al. 2017).4 tion of the stellar disk is hidden from view, weakening The RM effect has been the basis of most obliquity the corresponding radial-velocity components in an ab- measurements of individual planet-hosting stars (as op- sorption line. The character and time-evolution of this posed to statistical results from samples of stars). This spectral distortion depends chiefly on v sin i and λ. topic was reviewed recently by Triaud(2017). Below, Observers have detected and modeled the RM effect we described the two main methods for analyzing the in two different ways. When the spectral lines are not RM effect: as an anomalous radial velocity (§ 3.1.1) and well resolved, the line-profile distortions are manifested as a line-profile distortion (§ 3.1.2). Then, we review as shifts in the apparent central wavelength of the line. the key findings that have emerged from RM observa- When the blueshifted portion is eclipsed, the lines ex- tions (§ 3.1.3–3.1.10). Table1 gives an overview of these hibit an anomalous redshift, and vice versa. This is the trends and highlights particular systems. AppendixA manner in which Rossiter(1924) and McLaughlin(1924) describes the compilation of data that we assembled to displayed their data, as well as Queloz et al.(2000), who make the charts for this review. performed the first observations of the RM effect for a transiting planet. Parametric models for the “anoma- 3.1.1. The anomalous radial velocity lous radial velocity” and its relation to the positions Consider a transit of a planet of radius r across a and attributes of the two bodies have been developed uniformly-rotating star of radius R, equatorial rotation by many authors(e.g. Hosokawa 1953; Kopal 1959; Sato 1974; Ohta et al. 2005; Gim´enez 2006; Hirano et al. 2011; Shporer & Brown 2011). 4 The line-profile method has also been called “Doppler tomog- raphy,” a term we find confusing. The name originally belonged to line-profile analyses in which a star’s surface structure or a binary’s accretion geometry is reconstructed from spectral obser- 3 An earlier and less convincing detection as reported by vations obtained from many different viewing angles, as the star Schlesinger(1910) for the δ Lib system. rotates or the binary revolves all the way around. In the case of a planetary transit, though, the range of viewing angles is so narrow that there is no “tomographic” quality to the analysis. 6 Albrecht, Dawson, & Winn

a) HD 209458, Santos et al. (2020) b) HAT-P-69, Zhou et al. (2020) 14.72

14.74

14.76

RV [Km/s] 14.78

14.80

0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 d) Kepler-13, Johnson et al. (2014) Phase MASCARA-2/KELT-20 c) Hoeijmakers 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, leading to a redshift and therefore a positive RV excess on top of the orbital RVs of the host star. Red shifted light is blocked during the second half.Right top: This panel shows the deformation of the stellar lines ”planet shadow” in the prograde, misaligned +4.6 (λ = 21.2−3.6 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 each observation. The timeseries of different sub planet velocities relates to a given amplitude of the RM effect v14, and vcen which relates to the asymmetry of the signal, see § 3.1. velocity v, and line-of-sight inclination i. During the traction algorithm will respond to the distortion by re- transit, the stellar absorption lines suffer a fractional porting an anomalous velocity on the order of 2 loss of light on the order of (r/R) associated with the  r 2 ∆V (t) ≈ − v (t). (6) velocity component vp, where R p There are corrections of order unity due to the effects of vp(t) = (v sin i) x(t) (5) limb darkening, turbulent and instrumental broadening, is the velocity of the Doppler shadow (or the “sub-planet and the details of the RV-extraction algorithm. For the velocity”), defined as the rotational radial velocity of case of a cross-correlation algorithm, an accurate for- the point on the stellar disk directly behind the planet’s mula was derived by Hirano et al.(2011), building on center. Here, x(t) is the planet’s position in units of work by Ohta et al.(2005). the stellar radius along the coordinate axis running per- If the radius ratio r/R and transit impact parame- pendicular to the star’s projected rotation axis, as in ter b are known, then observations of the time series Figure1. ∆V (t) can be used to determine λ and v sin i. Figure3 The effect on a spectral line is a distortion, not an illustrates the transit geometry and the corresponding overall Doppler shift. Nevertheless, a radial-velocity ex- ∆V (t). The extremes of the signal occur at ingress (x1) Obliquity 7 and egress (x2), with amplitudes 3.1.2. The Doppler Shadow

∆V1 = (v sin i) x1, ∆V2 = (v sin i) x2, (7) The line-profile distortions due to the RM effect can also be analyzed directly. Consider an idealized spectral and from the transit geometry, one can show line broadened only by rotation. When the planet is at p position x(t), the range of velocity components partially x = 1 − b2 cos λ − b sin λ, 1 blocked by the planet is (v sin i)(x ± r/R). Within this p 2 x2 = 1 − b cos λ + b sin λ. (8) velocity range, the fractional loss of light is equal to the area of the planet’s silhouette divided by the area of the We can recast these relationships as strip of the star within x ± r/R,

∆V2 − ∆V1 = 2(v sin i) sin λ × b, (9) p π r 1 ∆V + ∆V = 2(v sin i) cos λ × 1 − b2, (10) ∆LRM(t) ≈ − . (12) 2 1 8 R p1 − x(t)2 making it clear that the asymmetry of the signal depends on sin λ, while the total amplitude depends on cos λ. This is the intensity contrast of the “bump” that When both of these aspects of the signal are measured, would appear in the line profile — the planet’s Doppler and the impact parameter is known from other observa- shadow. Note that it scales in proportion to r/R, not tions, the preceding system of equations can be solved (r/R)2, making this technique potentially more sensitive for v sin i and λ. For more insight into the information to small planets than the anomalous-RV technique. In content of the RM signal, see Gaudi & Winn(2007). In practice, though, other line-broadening mechanisms will particular, those authors derived a formula to estimate reduce the contrast of the bump and at least partially the achievable precision in the measurements of λ, negate this advantage. Collier Cameron et al.(2010) presented a more realis- √ 1/2 σ / N  r −2 (1 − b2) sin2 λ + 3b2 cos2 λ tic analytic model for the distorted line profile, including σ = v λ v sin i R b2(1 − b2) limb darkening. Another approach is to create synthetic (11) line profiles by numerically integrating over a 2-d pix- based on N data points with independent Gaussian un- elated stellar disk, assigning intensities and velocities 5 certainties σv uniformly spanning the transit. Note to each pixel due to rotation, limb darkening, velocity that the uncertainty grows as b approaches 0 or 1. As fields, etc. The pixels hidden by the planet are simply b → 0, the asymmetry vanishes and there is not enough assigned zero intensity (e.g. Albrecht et al. 2007). In information to determine both λ and v sin i; in such the approach they called “RM Reloaded,” (Cegla et al. cases, an external constraint on v sin i is essential. As 2016) replaced the synthetic line profile with an empiri- b → 1, the transit signal itself vanishes. cal model based on spectra obtained outside of transits Figure4 shows some higher-order effects that were and used a parametric model only for the portion of the neglected in the preceding discussion. Limb darken- photosphere covered by the planet. ing weakens the RM effect near the ingress and egress Fig.5 compares four different representations of the phases. Differential stellar surface rotation causes vp to RM effect, drawn from the literature. The upper left be a function of both x and y, making the RM effect panel shows an anomalous radial velocity time series. sensitive to i in addition to λ (Gaudi & Winn 2007; The upper right panel shows the “Doppler shadow” as Cegla et al. 2016). Turbulence on the stellar surface a time series of residual line profiles derived from cross- also affects vp, as does the “convective blueshift” — correlation. Each row represents an observed line pro- the higher intensity of the hot, upwelling material com- file after subtracting the best-fitting model of an undis- pared to the sinking material (Shporer & Brown 2011; turbed line profile. As time progresses (upward, on the Cegla et al. 2016). Some other effects are usually ne- plot), the negative residual caused by the planet moves glected but may be important in special cases: the tidal from the blue end to the red end of the line profile. and rotational deformation of the star, the saturation or The lower left panel shows the time series of the sub- pressure-broadening of some lines, and the influence of planet velocity inferred with the RM Reloaded tech- star spots and pulsations. is normally ignored. 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- 5 The formula is only valid when enough data outside the tran- sit have been obtained for the RM signal to be isolated without ity at midtransit (vcen) and the difference in sub-planet ambiguity. It is best to obtain at least a few data points before velocities at ingress and egress (v14). Such a “data stack- and after the transit. ing” analysis can be useful in the presence of correlated 8 Albrecht, Dawson, & Winn

Table 1. Key results from obliquity measurements. The first column names the detected observational trend, the second column indicates the main measurement technique used. The section which discusses the particular trend is given in the last column 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 § 3.1.31 • Massive stars (M & 1.2M ) harboring HJs tend to have high obliquities v sin i § 3.62 • Massive planets tend to have low obliquities, low mass planets tend to have high obliquities RM § 3.1.43 • Planets traveling on large orbits tend to have large obliquities RM § 3.1.54 • 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 § 3.1.65 • 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 H´ebrardet al.(2011), 4 Albrecht et al.(2012b), 5 Stefansson et al. 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 Mu˜noz& Perets(2018), 13 Louden et al.(2021), 14 Wu & Murray(2003), 15 H´ebrard et al.(2010), 16 Huber et al.(2013), 17 Hjorth et al.(2021) noise (Johnson et al. 2014) or a low signal-to-noise ratio Figure6 displays projected obliquity and stellar rota- (Hjorth et al. 2021). tion measurements as function of the host star’s effec- Whether to analyze the data in terms of the anoma- tive temperature (Teff ). We highlight results for HJs. lous RV or the line-profile variations, or both, depends The trend reported by Winn et al.(2010) – that stars on the instrument and the system parameters. Roughly with Teff < 6250 K have projected obliquities consistent speaking, the larger the ratio with alignment and stars with Teff > 6250 K have a range of obliquities – exists in this significantly enlarged (v sin i)(r/R) α = , (13) sample. No host star with a HJ and Teff significantly p 2 2 2 σinst + σmic + σmac lower than the Kraft break has a spin-orbit misalign- ment. Out of the 56 HJ systems with T < 6250 K only the easier it will be to resolve the planet’s Doppler eff three (WASP-60, WASP-62 and WASP-94A), a fraction shadow in the line profiles. Here, σ is the instru- inst of 0.06 are misaligned.6 These three misaligned hosts mental broadening of the spectrograph and σ and mic have temperatures above 6100 K. 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- 6 When we discuss aligned/misaligned and circular/eccentric tion lines, besides rotation. For rapidly rotating stars, orbits then we define these via the following: An aligned system has a projected obliquity below 10 deg or a stellar inclination precise RV determination is difficult but the RM anoma- measurement above 80 deg. A misaligned system either excludes lies in the line profiles can reach depths of several percent 0 deg at the 3 − σ level and has a λ larger than 10 deg, or has of the overall line depth (e.g. Talens et al. 2018), making a stellar inclination measurement excluding 90 deg at 3 − σ and has a i measurement below 80 deg. We count an orbit as eccentric them relatively easy to detect. if the eccentricity is larger than 0.1 and an eccentricity of zero is excluded at a 3 − σ level. We describe a system as circular if its 3.1.3. Hot stars with hot Jupiters have high obliquities eccentricity measurement is below 0.1. Systems which do not fall Obliquity 9

180 150 120 90 60 30

proj. obliquity (deg) 0

) 3000 4000 5000 6000 7000 8000 9000 10000 1 102 s

m k (

e

t 1 10 HJ - cool host a r

HJ - hot host

n HJ - very hot host o i warm & cool Jupiters t

a sub-Saturns t 100 o multi-transiting R 3000 4000 5000 6000 7000 8000 9000 10000 Teff (K)

Figure 6. Projected obliquities and projected stellar rotation speeds of exoplanet host stars displayed over the effective temperature (Teff ). The upper panel shows projected obliquities (λ) and the lower panel shows projected stellar rotation speeds (v sin i). We color code different types of systems; Hot-Jupiter systems are systems with scaled orbital separations (a/R) below 10 and planet masses (or their upper limits, if only these are available) above 0.3 RJupiter. For these systems 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 prograde orbit, larger λ values indicate a retrograde orbit. As expected the host star v sin i does increase with stellar temperature in the range from K-A type host stars. The top panel also highlights that for HJ systems there is a clear increase in stellar obliquity from cool hosts (blue symbols), hotter stars (red symbols) which have a significant fraction of systems with large and retrograde stars, 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 p = 0.17. The second trend relates to v sin i, which no λ measurements grown, also the range of host star effec- longer increases with Teff for stars hotter than ∼ 7000 K tive temperatures has increased. In Figure6 we mark (Figure6 lower panel). This is consistent with other systems with stars above 7000 K with orange systems. samples presented in the literature, see e.g. Gray(2005). We are motivated to make this additional distinction This flatten out in the maximum v sin i is thought to be by two observational trends. The ratio of oblique ver- connected to the complete absence of a convective enve- sus well aligned systems raises from 1.4 (21 versus 15) lope above ∼ 7000 K (i.e., these stars have experienced in the range 6250 K < Teff < 7000 K to 3.7 (11 versus no convective braking). 3) above 7000 K. The Kolmogoro-Smirnov (KS) statistic indicates a p-value of 1.9×10−5 that the projected obliq- 3.1.4. High mass giant planets have low obliquity hosts uities are drawn from a uniform sample for stars with Figure7 displays projected obliquities as a function 6250 K < T < 7000 K. For very hot hosts the hypoth- eff of the planet-to-star mass ratio (m/M). Massive HJs esis that the projected obliquities are drawn from a uni- have low obliquity orbits, a trend observed earlier in an form sample can not be rejected with the data at hand, smaller sample (H´ebrardet al. 2011).There are cool host star systems with significant obliquities despite large into these categories e.g. they are formally misaligned/eccentric mass ratios. These are WJs (a/R > 10) and are in- but below a 3 − σ level then these are not counted. dicated by cyan symbols. The current sample indicates that the mass cut off for prograde orbits depends on 10 Albrecht, Dawson, & Winn

180 HJ - cool host 150 warm & cool Jupiters 120 sub-Saturns 90 multi-transiting 60 30 0 180 HJ - hot host 150 warm & cool Jupiters M 120 / e multi-transiting n u

90 t p 60 e N

30 m 0 180 HJ - very hot host 150 warm & cool Jupiters 120 M projected obliquity (deg)

90 / r e t

60 i p u 30 J 0 m 10 4 10 3 10 2 planet/star mass ratio

Figure 7. Projected obliquities versus the planet to star mass ratio. Same color scheme as in figure6. Cool host stars show good alignment for massive planets (mass ratio above ≈ 0.0005) as long as these are not WJs. Hot hosts display misalignment 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 (Neveu-VanMalle et al. 2014) and the WASP-60 system stars with a planet to star mass ratio larger than 0.0005 (Mancini et al. 2018). Both have effective temperatures orbit prograde and are consistent with low obliquities. above 6100 K. For hot host stars (middle panel in Fig- For hot stars no retrograde systems are observed with ure8) four misaligned systems with a/R < 7 are known. ratios larger than 0.002 – a cut off four times larger than Additional observations will show if there is an increase that for cool stars. Also prograde systems with signif- of misaligned stars in this temperature bin for close in icant misalignment are observed for high m/M, in this giant planets. There is no obvious trend with a/R in temperature range. For the hottest host stars the mass the relatively small sample of very hot host star sys- ratios of close in planets cover a smaller parameter range tems. This is consistent with the tidal picture discussed and do not display any aparent trend.7 below, § 4.1. We note that the relatively small spread in host star masses - compared to the spread in planetary masses - 3.1.6. Aligned systems are very well aligned leads to similar correlations of the projected obliquity What is the dispersion in obliquities for systems which with mp and m/M?. are ”aligned”? The dispersion might be a useful diag- nostic in determining which process led to alignment, as 3.1.5. Planets with large separations have high obliquity dissipate processes would lead to a small overall value hosts with a small dispersion. In Figure8 top left panel we Figure8 displays projected obliquity measurements display all projected obliquity measurements (now rang- over the orbital separation, a/R. The correlation dis- ing from −180 to 180 deg) in systems with cool hosts cussed by (Albrecht et al. 2012a) – that close in (a/R . which systems which have prograde orbits and excellent 12) giant planets orbiting cool stars have aligned orbits measurement uncertainties of 2◦ or less. For guidance and further out systems have a large dispersion in obliq- we also display the (not projected) Solar Obliquity with uities – is present in the current data set. There are two respect to the invariable plane, 6.2◦. All these cool hosts exceptions, WASP-94Ab a HJ in a binary star system have projected obliquities with respect to their compan- ions well below the Solar value. In the sample with cool

7 hosts the mean projected obliquity of the cool HJ hosts Low mass stellar companions and double star systems are ◦ predominantly aligned for even hotter primaries, with notable ex- (a/R < 10 & m > 0.3MJupiter) sample is 0.23 while ceptions. However formation and evolution in such systems differs the standard deviation is 0.91◦, and the formal average and we therefore do not include them here. measurement uncertainty is 0.82◦. These values for the Obliquity 11

solar obliquity 5 180 0 HJ - cool host 150 warm & cool Jupiters sub Saturns 5 120 multi transiting solar90 obliquity 5 60 10 15 20 30 0 180 HJ - hot host 150 warm & cool Jupiters multi transiting 120 90 60 30

proj. obliquity 0 180 HJ - very hot host 150 warm & cool Jupiters 120 90 60 30 0 101 102 orbital separation (a/R )

Figure 8. Projected obliquities displayed over scaled separation a/R The inset highlights systems with cool hosts and measurement uncertainty below 2 deg. Among these, systems harboring a HJ display a mean projected obliquity of 0.2 deg and 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 3.1.7. Obliquities and stellar age 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 Figure9 displays the projected obliquities as function has reduced the obliquities. If confirmed by additional of stellar age. HJ systems with ages above ≈ 3 Gyr have high accuracy measurements of additional cool hosts or- projected obliquities consistent with alignment, as first bited by HJs and an careful ensemble study of aligned reported by Triaud(2011) for a smaller sample only in- systems with larger uncertainties de-convolving the un- cluding stars within a narrower mass range where stars derlying distribution from the measurement uncertain- evolve quickly (allowing for precise age estimates). As ties then this further indicates that HJs orbiting cool discussed by Albrecht et al.(2012a) this correlation does hosts on aligned orbits obtained this alignment through probably not represent a direct obliquity – time rela- tidal dissipation and that alignment might not be pri- tionship; rather this relationship might be connected to mordial. This will be discussed in more depth in the the the change in stellar structure during the MS lifetime forthcoming publication by Stefansson et al. in prep. It (i.e., stars cool and gain larger convective zones as they is also worth noticing that these measurements high- age), which then in turn might lead to tidal alignment. light that given high enough SNR RM measurements Recently Safsten et al.(2020) confirmed that the corre- and a careful analysis researchers are able to measure lation apparent in Figure9 is connected to the stellar projected obliquities to an accuracy below one degree, temperature and not the age of the the system, and as alleviate some of the concerns discussed earlier (§ ??). we will see below (§ 4.1) therefore most likely to tidal alignment. 12 Albrecht, Dawson, & Winn

HJ - cool host 180 180 data i data HJ - hot host HJ - very hot host warm & cool Jupiters HJ - very hot host ) 150 warm & cool Jupiters multi-transiting g

150 sub-Saturns e d (

120 i

120 90 90 o i

60 60 =

90 i 30 30

60 0 0 2 1 0 projected obliquity (deg) 10 10 10 30 Age (Gyr) Figure 10. Spin-orbit alignment in young systems. projected obliquity (deg) 0 This figure displays projected obliquity measurements (cir- 0 2 4 6 8 10 12 cles) and stellar inclination measurements (triangles) of sys- Age (Gyr) tems younger than billion years and with age uncertainties less than 300 million years. Figure 9. Projected obliquities displayed over system age ¯ for hosts stars. The color scheme of this plot is the same as for Figure6. As all other system parameters also the ages light curve modulations, presumably from spots. The are listed in tab.2. youngest system which appears to be misaligned is TOI- +0.037 9 811 (0.117−0.043 Gyr, Carmichael et al. 2020) . However 3.1.8. Very young systems with (close in) giant planets are the companion has a mass fully consistent with being a aligned +8.6 Brown Dwarf (m = 59.9−13 MJup) rather than having a mass in the planetary regime. The youngest plane- Recently RM observations as well as stellar inclina- tary mass object with an misaligned star is Kepler-63 b tion measurements via the v sin i method (§ 3.4) as well (0.210 ± 0.045 Gyr, Sanchis-Ojeda et al. 2013). It is as interferometric measurements (§ 3.3) have enabled worth noticing that the young planets on aligned orbits first obliquity measurements in very young systems with belong to the sub Saturn as well as WJ and CJ classes. (short) period giant planets. In Fig. 10 we show pro- These types of planets often travel on misaligned or- jected obliquities as well as inclination measurements bits when observed in older systems (see Figs.7 and8), for systems younger than 1 Gyr and age uncertainties yet these few younger systems are aligned. These few below 250 Myr.8 AU Mic b is a recently discovered observations suggest that giant planets which have ar- (Plavchan et al. 2020) transiting planet orbiting a young rived in the vicinity of their host stars at an early time (22 Myr) star which also hosts an edge on debris disk. (. 0.1 Gyr) did so by a process which does maintain or The inclinations of the planetary orbit and debri disk lead to a low obliquity. This would be consistent with are therefore consistent with alignment. A number of these younger planets arriving on their orbits via in situ authors (Addison et al. 2020; Hirano et al. 2020a; Palle formation or disk migration. This is also consistent with et al. 2020; Martioli et al. 2020) report good alignment large oblquities orginating from dynamical processes as for stellar spin and planetary orbit. Interferometry, dis- these tend to work on timescales often considered to be cussed below, allowed recently a measurement of the longer than a few Myr (§ 4). However see also Dawson projected obliquity in β Pic (Kraus et al. 2020). We & Johnson(2018) for a discussion on timesclaes. note that while this is also a young (26 Myr) system with a massive gas giant (and an edge on disk) this is 3.1.9. Stellar Obliquities and orbital Eccentricities not a compact system rather the planet travels on an decade long orbit. The well aligned host DS Tucanae A Wang et al. (in perp.) highlights that HJs orbiting (Zhou et al. 2020) has an age of 45 Myr. Additional cool stars travel not only on well aligned orbits (§ 3.1.5) information comes from inclination measurements via but these orbits also appear to be circular, while Jupiters the v sin i method, which is well suited for young stars orbiting hotter stars have eccentric orbits for smaller which often display fast rotation and large periodic separations, Figure 11. We note that this plots contains a number of biases, one of which is that planets orbit-

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 a 9 We use here the value from isochrone fitting for TOI-811 (as we young system, its age is given by ≈ 300 Myr (Gaudi et al. 2017), did for other systems when ever available) rather the value for from +61 it does not have a formal uncertainty. We omit it in this plot. gyrochronology, which however is fully consistent (93−29 Myr) Obliquity 13

circular & aligned or even smaller than in the Solar System. Dai et al. 100 misaligned eccentric (2018a) found that multi transiting systems harboring Ultra Short Period (USP) planets tend to have some- ◦ 30 what larger mutual inclinations & 7 . Recently Masuda R / et al.(2020) (see also the work by Herman et al. 2019) a found that systems harboring Cold Jupiters (CJs) and 10 close in super Earths have an inclination dispersion of ∼ 12◦ which further decreases with higher planet multi- 3 plicity. 3000 4000 5000 6000 7000 8000 9000 Albrecht et al.(2013b) concluded based on obliquity Teff (K) measurements in five compact multi transiting systems that these systems have low obliquities. To date pro- Figure 11. Orbital eccentricity and misalignment The figure display systems in the host star effective temper- jected obliquities or inclinations have been measured in ature and orbital separation planet. Systems which mea- 14 systems. Measurements in eleven systems are con- surements are consistent with aligned, circular orbits are in- sistent with low obliquities: Kepler-30 (Sanchis-Ojeda dicated by gray systems. If they have a secure (3 − σ) ec- et al. 2012), Kepler-50 & 65 (Chaplin et al. 2013), centricity measurement then a open green circle is added. Kepler-89 (Hirano et al. 2012), Kepler-25 (Albrecht et al. Securely misaligned systems have orange symbols. 2013b), WASP-47 (Sanchis-Ojeda et al. 2015), Kepler- 9 (Wang et al. 2018), HD 10635 (Zhou et al. 2018), ing hotter stars might have a good enough eccentricity TRAPPIST-1 Hirano et al.(2020b) HD 63433/TOI-1726 measurement to be included in this sample (σ < 0.3) (Mann et al. 2020; Dai et al. 2020), and TOI-451 New- 10 but still significant eccentricities can not be excluded ton et al.(2021). Two systems have large spin orbit for these systems. Nevertheless the plot does display angles, Kepler-56 (Huber et al. 2013) & HD 3167 (Dalal that for cooler stars and close in orbits both large ec- et al. 2019). K2-290 A a coplanar two planet system centricities as well as large misalignments are rare. This in a wide binary has a backward spinning star (Hjorth might suggest that not only obliquities are dampened et al. 2021). As we will discuss in the following section the reasons for the large obliquities in some of these sys- by tides raised by the planet on the star (§ 4.1), also some eccentricity damping occurs inside the star. We tems are not the same. More than one mechanism can note that using canonical values suggest that most of lead to large spin orbit angles in coplanar systems. the tidal energy is dissipated inside the planet, and that the planetary circularization timescale is shorter than 3.2. Asteroseismology the stellar circuilarization timescale (e.g. Schlaufman & Winn 2013). Also hot stars tend to be younger than If long duration, high cadence, high Signal-to-Noise their less massive cooler counterparts and this sample is time series (RV or photometric data) are available then no exception. Safsten et al.(2020) recently showed that stellar pulsation frequencies can be determined. By an- indeed the trend of circular orbits out to larger orbital alyzing the amplitudes, dispersion, and positions of fea- separations is connected to age. tures in frequency space inside information about the Given the small number of systems which eccentricity star can be obtained. Among such information is the and obliquities might not be affected by tidal circular- inclination of the stellar spin axis (Gough & Kosovichev ization and/or tidal alignment we postpone a discussion 1993; Gizon & Solanki 2003; Chaplin & Miglio 2013), about evidence for a dynamically hot (large obliquities see figure 12. and eccentricities) versus a dynamically cold (low eccen- In the non rotating frame of an observer azimuthal tricities and alignment). modes (m) of a pulsating star are separated in frequency as m 6= 0 modes either travel with or against the stellar 3.1.10. Obliquities and compact multi transiting planets: rotation. Therefore modes of radial order n and an- alignment with notable exceptions gular degree l are split into (2l + 1) modes. The new frequency (and therefore the separation) of the modes Systems in which multiple planets are transiting is in- ν does not only depend on the azimuthal order m, it teresting in the context of obliquity measurements, as nlm also depends on an average angular velocity of the star, 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 10 We exclude Kepler-410 (Van Eylen et al. 2014) here as the systems tend to have low mutual inclinations similar mutual inclination between the planets orbits is unknown. 14 Albrecht, Dawson, & Winn

a) b) i = 45◦ 0.5 i = 82.5◦ (best fit)

0.4 ] 2 0.3

0.2 oe [ppm Power

0.1

0.0 1920 1940 1960 1980 2000 2020 Frequency [µHz]

Figure 12. Limits on stellar inclination from light curves Power spectra obtained from light curves observed by the Kepler 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) mixed dipole modes, respectively. For Kepler-56, a subgiant, the modes are split into triplets by rotation and the m = ±1 and m = 0 modes can be clearly separated. The dispersion of the modes is lower than their separation in frequency space. From their near equal 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, an equatorial 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), Kepler data set (Campante et al. 2016), see also (Kami- aka et al. 2018). Evolved stars are favorable targets as mΩ νnlm = νnl + . (14) their lower surface gravity leads to large oscillation am- 2π plitudes. Compare panels ”a)” and panel ”b)” in Fig. 12. The amplitude of these different modes is expect to by If the star hosts transiting planet(s), then the incli- nearly equal. The measured amplitude ratio between the nation of the orbit(s) relative to the equatorial plane different azimuthal modes depend on the viewpoint of of the host star can be readily determined as i will be the observer. The visibility of m 6= 0 modes are maximal known. An advantage of this technique is that no ad- for an equatorial view (i = 90◦), while for a polar view ditional transits need to be observed, making planets (i = 0◦) the amplitude of the m = 0 mode is maximized. traveling on long period orbits, and importantly planets For the case of dipole (l = 1) multiplets the mode power with small planet/star radii ratios accessible to obliquity (E ) is given by equ. 12 & 13 in the work by Gizon & measurements. Solanki(2003), Asteroseismology was used by Chaplin et al.(2013) to determine the obliquities in the Kepler-50 and Kepler- 2 E1,0 = cos i, (15) 65 systems, multi transiting planet systems harboring 1 2 small Super Earth planets. The first measurement of E1,1 = 2 sin i. (16) a multi transiting planet system with co-aligned orbits Therefore if the m 6= 0 and the m = 0 modes can and a large stellar obliquity was achieved via seismic be measured in the power spectrum and their relative measurements of a sub-giant Kepler-56 (Huber et al. mode amplitude can be determined then i can be de- 2013). Van Eylen et al.(2014) found agreement in io rived. See for a detailed discussion of this mechanism and i for the eccentric orbit of a mini Neptune in a mul- Gizon & Solanki(2003); Ballot et al.(2006, 2008) and tiplanet system (Kepler-410). Recently a large obliquity Kuszlewicz et al.(2019) as well as references therein for was measured in Kepler-408, a system with a hot sub a discussion of best practise for the retrieval of incli- Earth-sized planet (Kamiaka et al. 2019). nation angles from seismic data. The successful sepa- A larger number of PLATO systems might be suitable ration of the m = ±1 (or even higher order azimuthal for determining stellar inclinations via asteroseismology. modes) and m = 0 modes and their amplitude mea- We note that also high SNR ground based time series of surements require a large ratio of the mode separation high resolution spectra could be used to determine i via over the width of the modes. The former quantity in- mode splitting. creases with faster rotation (equ. 14), while the later quantity increases with shorter mode lifetimes, which in 3.3. Interferometry turn decrease for larger Teff . This requirement limits A potential path towards overcoming our preoccupa- the number of main sequence planet host stars in the tion with transiting close in orbiting planets is inter- Obliquity 15

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),

B ρ = −2πp [rad]. (17) λwavelength 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 Figure 13. Spatially resolved Br γ absorption line of long orbit, Fig 13. β Pic Figures taken from Kraus et al.(2020). The top panel displays the flux measured in the Br γ line of β Pictoris in These studies have targeted bright fast rotating stars velocity space. The two lower panels displays the differen- and their pressure broadened line Brγ line as currently tial offset of the photocenters at for different wavelengths in there is no instrument available which can resolve iron 10−6 arcsec relative to the continuum flux along the North- lines in late type main sequence stars or obtain differ- South (middle panel) and East-West (bottom panel) axes as ential phase measurements on fainter targets. However derived from the interferometric measurements. preparations for high resolution instruments (with an resolution power of up to a few ten thousands) are cur- ferometry. Optical\NearIR Interferometric Long Base- rently underway at the CHARA array (Mourard et al. lines observations can (partially) resolve stellar surfaces 2018) and the VLT Interferometer (Kraus 2019). To fur- of main sequence stars in the solar neighborhood, solv- ther increase the magnitude of potential targets these in- ing the spatial resolution challenge without the need to struments will make use of fringe tracking, significantly resort to transits. If equipped with a spectrograph which increasing the integration time of the spectrographs con- can resolve stellar absorption lines then this allows for nected to the interferometer. example for the determination of the stellar rotation axis However this technique not only requires the combi- as projected on the sky plane (e.g. Albrecht et al. 2010). nation of high spatial and spectral observations. The A projected baseline11 oriented parallel to the stellar interpretation of the orientation of the stellar spin axes equator will resolve (partially) the stellar disk and the - with the sky plane as reference - in the context of stel- photo centers of the red and blue wings of stellar ab- lar obliquities in exoplanet systems (or double star sys- sorption lines, can be resolved. They will have different tems) does require knowledge of the orbital orientation interferometric phases, i.e. the position of the fringe on the sky plane as well. Specifically - the longitude of

12 11 The projection of the line connecting different telescopes in Fomalhaut appears to host a dispersing collision induced dust an array, as seen by the target. cloud and not an giant exoplanet as originally thought in its disk Gaspar & Rieke(2020) 16 Albrecht, Dawson, & Winn the ascending node (Ω) - not obtained by RV or transit to the stellar rotation period via the rotation of stellar measurements. The expected release13 of thousands of surface features e.g., spots in and out of view. exoplanet systems with astrometric orbits as measured We would like to highlight the results by Masuda & by the GAIA satellite (Perryman et al. 2014) will lead to Winn(2020). They highlight that care has to be taken a large pool of potential targets. GAIA will also deter- when deriving marginalized confidence intervals for i mine io and Ω for a number of known RV systems with and we refer to that work for details. They highlight giant planets on few year orbits. However a significant that v and i might not necessarily always independent number of these systems might be to faint to be studied e.g. in clusters. Currently more important was an often with this technique. Intererometers can in addition be made mistake, assuming that v and v sin i are indepen- used to search for (partial) alignment of stellar rotation dent variables, which they are not. By measuring one axes double star systems, in star forming regions and we gain some information about the other. A measure- stellar clusters. Thereby informing theories about the ment of v sin i gives knowledge on v (lower values of v initial conditions during star and planet formation, im- are disfavored) and vis versa. Using their equ. 10 when portant for the interpretation of obliquity measurements deriving uncertainty intervals – rather than simply and as discussed in section4. incorrectly applying equ. 18 when deriving posteriors For the fastest rotating stars departures from a purely – incorporates this dependency properly. We highlight spherical shape caused by centripetal forces can be used all single measurements using the procedure outlined by to learn about obliquities, without the need to spectrally Masuda & Winn(2020) in Figure2. resolving the stellar lines. Albeit this is currently only Using the v sin i technique Guthrie(1985) and Abt applicable to the very fastest rotators (e.g. Domiciano (2001) tested for, and did not find, a tendency for stars de Souza et al. 2003). to be preferentially aligned with the Galactic plane. In double star systems spin-spin alignment or orbit-spin 3.4. The v sin i technique alignment can be probed (e.g. Weis 1974; Hale 1994; As for seismology discussed above also for the v sin i Glebocki & Stawikowski 1997; Howe & Clarke 2009), but see also Justesen & Albrecht(2020) who showed technique only the information on io from the occurrence of transits is used, no transit observations are required. that the often quoted result that double stars with sep- A difficultly shared with the seismic determination of aration less than a few tens of au tend to be aligned, stellar inclinations is the flattening of the sine function can not be confirmed with the data at hand. Schlauf- near 90◦ as well as a degeneracy in i as mentioned at man(2010) was the first to use this technique for stars the begin of this section. Assuming solid body rotation hosting transiting planets, finding evidence that more we find, massive stars have high obliquities, coming to consis- tent result as Winn et al.(2010) using a different ap- v sin i  v sin i  proach. More recently this technique was used on very i = sin−1 = sin−1 . (18) vprior (2πR/Prot) young hosts of transiting planets. Such stars often have significant QPVs and rotation measurements and there- Therefore measurements of v sin i and prior information fore obtaining a measure of v is somewhat easier than on the rotation speed (vprior) could lead to an estima- for older main sequence stars. In addition these stars tion of i. Measuring the nominator in the above equa- tend to have a large v and therefore sin i thanks to their tion is challenging for slowly rotating stars as broaden- youth. Therefore both terms in equ. 18 can be deter- ing of stellar lines might not be dominated by rotational mined with some accuracy. In addition there are only Doppler shift. Obtaining the denominator may be done few obliquity measurements for the youngest systems, via a number of routes, see Maxted(2018) for a review. see § 3.1.8. The v sin istar technique was also used to Most commonly two paths are taken. One might esti- demonstrate that stellar spins in the open cluster NGC mate vprior assuming a particular dependency of v on 2516 have an isotropic distribution or at most moderate stellar mass and age e.g., square root brake down law alignment Healy & McCullough(2020). Skumanich(1972, 2019). Alternatively one might de- In the near future we might expect to obtain more in- termine R, and Prot. Rotation period measurements teresting results from the v sin i method (Quinn & White might for example be achieved via measurements Quasi 2016) based on TESS transiting systems for which pa- Periodic Variations (QPV) in long duration photomet- rameters appearing in equ. 18 should be obtained with ric time series. Periodic flux variations are associated higher accuracy and fidelity than before. GAIA data im- proves stellar radii measurements, very high resolution spectrographs (e.g. ESPRESSO, PFS, XPRES) allowing 13 https://www.cosmos.esa.int/web/gaia/release Obliquity 17

500 E = 3 14 1.000 seen equator on, everything else being equal. This 400 0.998 300 statement ”everything else being equal” might not be E = 4 0.996

Power 200 0.994 as easy to fulfill as hoped. For example stars for which

100 Relative flux + constant 0.992 Expected, for we can detect OPVs might be a particular subset of 0 ψ = 0 0 10 20 30 40 50 60 Period [days] −0.10 −0.05 0.00 0.05 0.10 stars, e.g. seen more equator on or of a particular stellar

E = 15 type. Therefore such studies should ideally encompass 1.000 Expected, for ψ = 0 1.005 0.998 two populations which are similar in as many aspects as E = 16 1.000 0.996 possible apart from the planet population. Mazeh et al.

Relative flux 0.994 0.995 Relative flux + constant 0.992 (2015b) pioneered the usage of this geometric effect for

−0.4 −0.2 0.0 0.2 0.4 Rotation phase −0.10 −0.05 0.00 0.05 0.10 Time from midtransit [days] obliquity studies, see § 3.6. Figure 14. QPVs and spots crossing transits. Fig- 3.5.2. Starspot-tracking method ures taken from Sanchis-Ojeda & Winn(2011). The top If starspots are present then these might not only lead panel shows the Lomb-Scargle periodogram of Kepler pho- to QPV out of transits discussed above. During transits tometry taken of HAT-P-11, indicating a rotation period of +3.1 spots might be covered from view by the planet. This 30.5−3.2 days. The second panel from the top shows the out out-of-transit flux phase folded over this period, illustrating then results in an increased flux level for this part of the a Quasi Periodic Variability in the light-curve. The lower two transit light curve. A sequence of spot covering events panels show two pairs of consecutive transit epochs. Given during transits (or the absence of such a sequence) the orbital period (4.9 days) and rotation period a change can be used to deduce stellar obliquities (Sanchis-Ojeda of ≈ 60 deg in longitude between consecutive transits is ex- et al. 2011; D´esertet al. 2011). See figure 14. pected. An aligned orbit would lead to spot crossing events In addition phase information from the OPV and tran- in consecutive transits, as indicated by the red lines in the two lower panels. The data does not match a model with sit crossing events can be combined to drive information aligned spin and orbital axes (red line). 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 for finer sampling of late type stellar spectra obtained surface. Therefore spot coverage during the first half of with higher SNR on bright TESS host stars and a bet- a transit and decreasing stellar out of transit flux indi- ter calibration of stellar surface motion (e.g. Doyle et al. cates a prograde orbit and vise versa (Nutzman et al. 2014) might lead to improved v sin i measurements also 2011; Mazeh et al. 2015a; Holczer et al. 2015). for slower rotating stars. 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- 3.5. Starspots mination of alignment. 3.5.1. Quasi-Periodic Variation The first obliquity measurement in an multi transit- ing system (Kepler-30) - indicating good alignment - If star spots (or any other semi stationary stellar sur- was carried out by Sanchis-Ojeda et al.(2012) tracking face feature) are present then the flux received from a starspot coverings during transits as well as QPV out star varied with the stellar rotation frequency or multi- of transit. It is worth noticing that methods relying on ples thereof as the stellar rotation transports spots over star spots to deduce information on stellar obliquities the limb darkened stellar disk in and out of view. To- are complementary to the RM method (§ 3.1) as de- gether with the slow evolution of the spots themselves tectable spots are more prevalent in the photospheres this gives rise to out of transit Quasi-Periodic Variation of late type stars, for which the stellar rotation speed (QPV) in flux on the time scale of the stellar rotation is relatively slow leading to small RM amplitudes. The period. As mentioned in the above section periods de- TESS mission aims at detecting transiting systems with rived from QPVs can be used to estimate v and thereby low mass host stars. However the spot methods do ben- leading to an estimate of i via the v sin i method. How- efit from long time series. This makes TESS systems ever the amplitude of the QPVs itself can be used to detected near the elliptical poles more suitable for these obtain information on i. methods as TESS observes the elliptical poles for one The amplitude of the QPVs depends not only on con- year. 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 14 Higher mass stars might display polar spots. 18 Albrecht, Dawson, & Winn

3.6. Key results from ensemble studies on highly misaligned orbits (e.g., HAT-P-11., HAT-P- 18 and WASP-107) see also 3.1.9. • Developing and using the QPV approach Mazeh § et al.(2015b) found that host stars with effec- 3.7. Other methods tive temperatures below ∼ 5700 K tend to have good alignment with planets out to orbital periods 3.7.1. Rotational Doppler beaming of ≈ 50 days. Li & Winn(2016) reanalyzed the Conceptually related to the RM effect, Groot(2012) data and found that ”the evidence for alignment and Shporer et al.(2012) evaluated the potential of rel- becomes weaker for systems with an innermost ativistic beaming caused by the stellar rotation or the planet period & 10 days, and is consistent with photometric RM effect for obliquity measurements. The nearly random alignment for longer orbital peri- apparent brightening of the approaching and darken- ods (& 30days).” Mazeh et al.(2015b) also found ing of the receding stellar surface areas due to Doppler that hotter stars tend to be more misaligned. Im- beaming, will lead to a λ dependency of eclipse light portantly most of these stars do not harbor HJs curves. Shporer et al.(2012) give the following equation but smaller and further out planets. to estimate the photometric amplitude for this effect,

2 • Campante et al.(2016) employed asteroseismology v sin I r
A ≈ 10−5 R . (19) to study 24 Kepler Targets of Interest (KOI) with PRM 10 km s−1 0.1 planets and planet candidates in single transiting These authors concluded that due to the small am- and multi transiting systems with periods up to plitude of the effect obliquity measurements will be 180 days and sub Neptune sizes. These authors challenging. The most promising targets appear sys- found that their astronomic inclination measure- tems containing fast rotating early type stars and white ments are consistent with good alignment.15. dwarfs. For white dwarfs many of the other measure- • Also the v sin i method was further employed ment techniques available to measure ψ will not be ap- (Walkowicz & Basri 2013; Hirano et al. 2014; Mor- plicable. ton & Winn 2014). Winn et al.(2017) and Mu˜noz 3.7.2. Gravity darkening, fast rotators & Perets(2018) used data from the California- Kepler Survey (CKS, Petigura et al. 2017) sample. For rotating stars the effective local gravity near the They did find that their sample containing single stellar equator is reduced relative to the stellar poles, re- and multi transiting systems is consistent with sulting in a larger scale height of the photosphere. For good alignment, with the exception of HJ hosts. latitudes near 90◦ a specific optical depth is reached at lower temperatures than at latitudes closer to the pole. • Most recently Louden et al.(2021) analyzed a sub- This effect leads to increased brightness towards the stel- set of the Winn et al.(2017) sample. Improving lar poles. This is superimposed onto the radial symmet- on the former results with the use of a comparison ric center-to-limb brightness change due to stellar limb sample which has similar stellar properties to the darkening. The local temperature, Tl, can be described planet hosting sample but without transiting plan- by the von Ziepel theorem (Barnes 2009, and references ets. These authors find low obliquities for hosts therein), below 6250 K and a distribution consistent with β random orientation for hotter stars. This confirms gl Tl = Tp β . (20) the earlier result by Mazeh et al.(2015b), using a gp different technique. Here gl refers to the surface gravity. The indices l and p refer to the local quantities and polar quantities. The To summarize, these studies suggest that i) systems gravity darkening parameter β has a nominal value of with cool host stars have good alignment regards of 0.25 for radiative stars but varies with stellar type. For planetary orbit and planetary size/mass, and ii) hot the aligned and ani-aligned case (λ ≈ 0◦ or λ ≈ 180◦) host stars tend to have large obliquities, again regard- gravity darkening is challenging to detect in a single less of planet, size distance and multiplicity. We note band light curve as it will lead to an apparent decrease or that there is tension between these measurements and increase in the planet to star radii ratio, for low and high RM measurements of small planets orbiting cool stars impact parameters, respectively. For |λ| = 90◦ and sig- nificant gravity darkening, a symmetric light curve with 15 One of these systems, Kepler-408 was later found to be mis- apparent brightening of the photosphere at the limb is aligned observed, revealing the misalignment. Other projected Obliquity 19

Primordial Disk dispersal Post formation Inclined star or planetCyclic Secular (Kozai-Lidov) resonant excitation

Inclined star Secular chaos Spin Envelope ψ down ψ resonant Misalignment excitation during accretion

Magnetic breaking

Magnetic Planet- Warping planet scattering

Figure 15. Processes that create spin-orbit misalignments before (left) or after (right) planet formation. obliquities lead to asymmetric light curves around the Although tidal realignment may happen last (i.e., af- transit midpoint, see Barnes(2009). ter other processes create spin-orbit misalignments), we The successful observation of gravity darkening in discuss it first. There is compelling evidence that most transiting exoplanet systems require high signal-to-noise of the individual obliquities observed to date have been transit observations of fast rotating host stars reducing altered by tides, so we should not compare the predic- the gl near the equator (equ. 20). The first observations tions in subsequent sections to the observed obliquity of this effect have been made in the Kepler-13 system, distribution without taking tidal realignment into ac- (Barnes et al. 2011; Szab´oet al. 2011) for which the count. The strongest piece of evidence is sharp change in asymmetry in the light curve due to gravity darkening the obliquity distribution above the Kraft break stellar is of the order of 100 ppm. Other observations include effective temperature (Fig.6). The Kraft Break marks HAT-P-7 Masuda(2015), KOI 368 (Ahlers et al. 2014) a major difference in stars’ rotation rates and structure, as well as the more tentative measurements of alignment implicating tidal effects. In Section 4.1.1, we discuss the in KOI 2138 (Barnes et al. 2015) and misalignment in empirical consistency of a simplified tidal friction model the multi planet systems KOI-89 (Ahlers et al. 2015), with observed obliquity trends. In Section 4.1.2, we de- which however was shown to be spurious by Masuda & scribe the prospect for more complex and realistic tidal Tamayo(2020). More recently Gravity darkening was models to account for the observed trends. used to determine obliquities in TESS systems Ahlers et al.(2020a,b). 4.1.1. Simplified tidal friction model: empirical consistency 4. PROCESSES THAT INFLUENCE OBLIQUITIES with observed trends The observed obliquity distribution tests theories for In the theory of equilibrium tides, tidal friction occurs how stars and planets form and evolve, with a number of when the star rotates at a different rate and/or direction mechanisms proposed for altering the obliquity through- than the planet. Fluid elements of the star closer to the out the system’s history. Below we review how these planet feel a stronger gravitational force than those fur- theories’ predictions hold up against currently available ther away, stretching out the star and raising a bulge. If data and which measurements would further test each the planet orbits more quickly (slowly) than the star theory. We first discuss the theory that tidal realign- spins, the planet leads (lags) the bulge. The planet ment sometimes erases the obliquities established by the stretches out the star in different directions throughout other processes (Section 4.1). We then summarize the the orbit, dissipating energy in the star. Similarly, with theory of and evidence for primordial misalignment be- a spin-orbit misalignment, the bulge rotates away from fore the planet forms (Section 4.2), post-formation mis- the planet, and the planet has to stretch out the star alignment (Section 4.3), and changes in the stellar spin again and again. The bulge and planet exert a torque vector that are independent of the planet (Section 4.4). on each other that transfers angular momentum to syn- chronize and align the star. When the planet’s orbital 4.1. Tidal realignment period is shorter (longer) than the star’s spin period, 20 Albrecht, Dawson, & Winn the planet’s orbital angular momentum is transferred to (i.e., the observed alignment timescale must be shorter (from) the star’s spin angular momentum. than τeq,7). Both hot and cool stars have low obliquities In general, tidal interactions dissipate energy and ex- within a cut-off timescale and exhibit a range of obliq- change orbital and rotational angular momentum. Tides uities beyond. tend to circularize orbits, align rotational and orbital The realignment timescale spans many orders of mag- axes, and synchronize the rotational and orbital fre- nitude, making it difficult to detect obliquity time evo- quency. We refer to Zahn(2008); Mazeh(2008) and lution in a sample of main sequence stars. The apparent Ogilvie(2014) for reviews on tides in binary and exo- break with age seen in Figure9 at ∼ 3.5 Gyr is more planet systems. likely a manifestation of the temperature trend: in the Tidal friction can also be produced by dynamical current sample, HJ systems older than ∼ 3.5 Gyrs have tides, which involve exciting waves within – rather than host stars with Teff < 6250 K (Albrecht et al. 2012b; see raising a bulge on – a star. In the radiative zone of a Safsten et al. 2020 for a similar conclusion based on a star, tides generate gravity waves that are damped and Bayesian evidence odds ratio computation using hierar- dissipate energy (Zahn 1977). In Section 4.1.2, we will chical modeling of the temperature vs. age dependence). discuss the contribution of inertial waves. The hypothesis that tidal interactions have signifi- The observed trends between obliquity vs. planetary cantly sculpted the stellar obliquity distribution has a and stellar properties (Table1) are broadly consistent large, unresolved problem: a short period planet does with our expectations for tidal realignment (Winn et al. not have much orbital angular momentum to spare for 2010; Albrecht et al. 2012b). Stars orbited by more mas- realigning a star. The ratio of the planet’s orbital an- sive (Fig.7) and/or closer planets (Fig.8) – which exert gular momentum (Lorb) to the star’s spin angular mo- stronger tidal forces – are more likely to be aligned. Fur- mentum (S?) is of order unity: thermore, planets can more effectively align stars with 2 stronger tidal dissipation, that shed angular momentum Lorb mna = 2 through magnetic braking as they realign, and/or rotate S? k?MR Ω?      2   slowly enough that the planetary orbital frequency dom- 0.1 m/M a/R n/Ω? inates the tidal forcing frequency. These are the distinc- ∼ 2.5 (22) k? 0.001 5 10 tions between stars with stellar effective temperature where k is the stellar moment of inertia constant, Ω below and above the Kraft Break Teff ' 6250 K (Fig. ? ? 6). In fact, the closest HJs with highly accurate mea- is the stellar rotation angular frequency, and n is the surements orbiting cool stars are aligned to, and have a planet’s orbital angular frequency. Significantly altering dispersion in λ of, less than 1 deg (Fig.8). the magnitude and/or direction of the stellar spin typi- A simple realignment timescale that encapsulates cally requires shrinking the planet’s orbit to within the these scalings is tidal disruption limit. Furthermore, in order for us to catch all hot Jupiters orbiting cool stars in an aligned 1 1  q 2 a/R−α state, the ratio of the realignment timescale to the or- = . (21) L −3 bital decay timescale – which scales with orb – must τeq τeq,7 10 7 S? −3 be very small (. 10 ), which is not what we expect For the spin synchronization of double binary stars sys- from the simple tidal models above. More complex tidal 11 tems, the empirical calibrations are τeq,7 = 2.8 × 10 years models offer solutions to these problems. and α = 6 for stars where dissipation primarily occurs in the convective envelope via equilibrium tides and 4.1.2. Prospects for more complex, realistic tidal models to 15 5/6 enable realignment without complete decay τeq,7 = 4.6 × 10 (1 + q) years and α = 17/2 for stars that lack (or have insubstantial) convective Given the compelling evidence that tidal realignment envelopes (Zahn 1977) via dynamical tides. In Section has occurred in many observed systems, several theo- 4.1.2, we will discuss the use of this equation in more ries have been proposed to enable the planet to realign complex and realistic tidal evolution models. the star without tidal disruption and to account for the We plot the observed obliquities vs. tidal timescales observed trend with stellar effective temperature: in Fig. 16, using the Zahn(1977) parameters. The Planets with orbital angular momentum to data are consistent with the α (i.e., the a/R scaling) spare: For some individual systems — featuring mas- from Zahn(1977) but do not strongly constrain α or re- sive and/or widely separated planets and/or slowly ro- quire a different α for hot vs. cool stars. The data are tating (i.e., cool) stars — Lorb (Eqn. 22) is not unity but S? consistent with the relative τeq,7 for hot vs. cool stars 10 or more (e.g., Hansen 2012; Valsecchi & Rasio 2014), from Zahn(1977) but much shorter in absolute terms and the realignment timescale is shorter than the orbital Obliquity 21

180 HJ - cool host HJ - hot host HJ - very hot host 150 warm & cool Jupiters sub Saturns ) 120 multi transiting g e

d 90 (

| 60 |

30

0 100 102 104 106 108 1010 1012 tau (yr) 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 constants in Equation 21 differ for host stars with temperatures lower than 6250 K (blue symbols) and hotter stars (red symbols). Note that both timescales have been re-normalized by dividing by 5 · 109. We omit here the β Pictoris system, as the planet has a decade long orbital period and no meaningful tidal alignment occurs. decay timescale. These planets may be able to realign (e.g., Dawson 2014), if their convective outer layers cou- their stars without undergoing much orbital decay over ple more strongly to the interior, and/ or if their tidal the star’s lifetime. dissipation is less efficient. Inertial wave tidal dissipation: Tidal interactions HJs misalign hot stars: Another possibility is that with planets can cause inertial waves driven in the con- instead of tidal interactions realigning cool stars, they vective zone by Coriolis forces as the star rotates. For misalign hot stars. C´ebron et al.(2013) suggest that hot misaligned systems, there are components of the tide Jupiters could misalign stars through a hydrodynamic with forcing frequency Ω? that only affect the spin di- instability known as the elliptical instability, in which rection and do not cause orbital decay (e.g., Lai 2012; streamlines in a rotating fluid become tidally distorted, Damiani & Mathis 2018). Other components of the causing turbulence and tilting the star. This instability tide that cause orbital decay, with forcing frequency requires a stellar rotational period shorter than 3 times 2(n − Ω?), are inactive when 2(n − Ω?) > 2Ω?, which is the orbital period, leading to misalignments for systems usually the case for hot Jupiters orbiting cool stars. In- with around hot (rapidly rotating) stars and/or long or- ertial wave tidal dissipation drives the obliquity to equi- bital periods. Further work is needed to better under- libria at ψ = 0, 90, 180◦. stand whether the dissipation is strong enough to cre- Steeply frequency-dependent tidal dissipation: ate a significant misalignment (e.g., Barker & Lithwick The tidal dissipation efficiency could be a steep func- 2014) and the distribution the misalignments expected. tion of the tidal forcing frequency (Penev et al. 2018; The first and second explanations have a firm basis Anderson et al. 2021). If tidal dissipation is much more in theory but seem unable to fully account for the ob- efficient at longer orbital periods, the hot Jupiter can served trends. Figure 17 presents a toy model popula- realign and decay but stall aligned when it gets close to tion synthesis (described in detail in AppendixB) com- the star. The temperature trend may be due to less effi- paring the obliquity distributions resulting from the first ciency and/or a different frequency dependence for tidal four explanations above to the observed population with dissipation in hot stars. a/R < 10 and m > 0.5MJupiter. The top panel dis- Outer realignment: The planet could realign just plays the projected spin-orbit alignment, and the bot- an outer layer of the star (e.g., Winn et al. 2010), tom panel displays v sin(i) as a proxy for stellar rotation which would somehow remain decoupled from the in- period. In each case, free parameters are tuned to pro- terior. Very hot stars lacking a convective outer layer vide the best match with the observed distribution. would not be realigned. Moderately hot stars would be With classical equilibrium tides, the most massive realigned less easily due to a lack of magnetic braking planets can realign their stars but lower mass hot 22 Albrecht, Dawson, & Winn

Jupiters remain misaligned, even around cool stars (Col- infall of material on the disk can warp the disk or tilt its umn 3). Inertial wave tidal dissipation (column 4) can rotation relative to the axis of the star (Bate et al. 2010; very effectively realign cool stars but, even when equilib- Thies et al. 2011; Fielding et al. 2015; Bate 2018). How- rium tides operate simultaneously (e.g., Xue et al. 2014; ever, accretion from the disk onto the star can eliminate Li & Winn 2016), result in a population stalled at the such misalignments: therefore, by the planet forming ψ = 180◦ equilibrium not seen in the observations. A stage, the disk and star are likely aligned to within 20 related constraint is that there are no known ψ = 180◦ degrees Takaishi et al.(2020). close double star systems, which we might expect to see Magnetic warping occurs when differential rotation if inertial wave tidal dissipation is commonly at work. between a young star and the ionized inner disk twists We would expect fewer ψ = 180◦ planets if the initial the magnetic field lines that link them, generating a obliquity distribution has mostly prograde planets, but toroidal magnetic field that warps the disk (Foucart & such an initial distribution seems at odds with the ob- Lai 2011; Lai et al. 2011; see Romanova et al. 2013, served obliquities of hot stars. Obliquities can also stall 2020 for 3D MHD simulations). If the toroidal field is at the ψ = 90◦ equilibrium; however, in our example, sufficiently strong – and the realigning torques due to ac- initially retrograde systems tend to evolve to and stall cretion onto the star, magnetic braking, disk winds, and at ψ = 180◦ because Lorb > 1 and because the inertial differential precession with the outer disk under high vis- S? wave tide realignment timescale is much shorter than cosity are sufficiently weak– modest misalignments can equilibrium tide timescale (Xue et al. 2014). be generated. The misalignment may be suppressed if The third and fourth explanations can account for the the magnetic field becomes wrapped around the stel- observed temperature trend (Fig. 17, Column 5 and lar rotational axis (Romanova et al. 2020). A broader 6) but need more grounding in physical models. More distribution of alignment angles, including retrograde, work on the theory of tidal dissipation is needed to de- can be achieved through a simultaneous external distur- termine whether a steep dependence of tidal dissipation bance to the outer disk, perhaps generated by a stellar efficiency on tidal forcing frequency is expected for the companion. relevant frequency range. The fourth explanation would Inclined stellar or planetary companions can tilt require very long timescales for the coupling of the outer disks (e.g., Borderies et al. 1984; Lubow & Ogilvie 2000; layer of the star to the interior and seems at odds with Batygin 2012; Matsakos & K¨onigl 2017). Although the the radially uniform rotation profile of the Sun. How- disk is coupled to the primary star, a misalignment can ever, a decoupled outer layer could be analogous to our be generated during resonance crossing of the stellar and Sun’s near-surface shear outer layer. disk precession time scales (Batygin & Adams 2013; Lai In summary, tidal alignment appears to play an im- 2014). The crossing occurs as the precession timescales portant role in shaping the obliquity distribution of close change due to disk evolution and mass loss (e.g., Spald- in, massive planets. Continuing work on the theory of ing et al. 2014). However, newly formed HJs are so tides is needed to distinguish among hypotheses for how tightly coupled to host stars’ spin that they prevent planets realign their stars to ψ = 0 without tidal de- their host stars from becoming misaligned by this mech- struction. anism (Zanazzi & Lai 2018). Therefore companions tilt- ing disks through this resonance crossing mechanism are 4.2. Primordial misalignment unlikely to be responsible for most obliquities in the cur- rent sample of individual system measurements, which One might expect a star and its proto-planetary disk consists primarily of HJ hosts. to have aligned angular momenta, because they inherit The direct route to measuring alignments between these from the same region of their parental molecu- stellar rotation and proto-planetary disks is blocked be- lar cloud and material is funneled via the disk onto cause the photospheres of protoplanetary disk hosting the young protostar. However, several processes have stars are hidden from view. Some proto-planetary and been proposed that might create primordial misalign- even embedded protostar disks exhibit misalignments ment between the stellar equator and orbital mid plane or warps between the inner and outer disk disk (e.g., of the disk where planets are thought to form: misalign- Marino et al. 2015; Sakai et al. 2019; Ginski et al. 2021; ment during accretion in chaotic star formation, mag- see Casassus 2016 for a review); however, the occurrence netic warping, and tilt by a companion star (Fig. 15). rate of such misalignments is not yet known. These bro- Misalignment during accretion might occur be- ken and internally misaligned disks might lead to in- cause stars form in a dense and chaotic environment, ner and outer planets orbiting with large mutual incli- causing the spin direction of the star and its disk to nations, setting the starting conditions for some of the change throughout the formation process. Late oblique Obliquity 23

Observed Sim: Initial Sim: Equilibrium Sim: Dynamic Sim: Evolving Q Sim: Decoupled

MJup: 150 0.5-1 1-2.5 2.5-15 100 | (deg) λ | 50

0 100.0

10.0 (km/s) s

1.0 v sin i

0.1 5000 6000 5000 6000 5000 6000 5000 6000 5000 6000 5000 6000 Teff (K) Teff (K) Teff (K) Teff (K) Teff (K) Teff (K)

Figure 17. Observed (column 1) and modeled (column 2-5) projected obliquity distribution (top) and stellar rotational velocity (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 should probe whether it might actually be a trend with disks, researchers have found predominately – but not stellar mass. Regarding individual systems, our sam- exclusively – evidence for alignment between stars and ple of 14 compact, coplanar, multi-transiting systems their disks using different variants of the v sin i method with obliquity measurements contains 11 well-aligned to determine the stellar inclinations (Watson et al. 2011; systems of compact super-Earths and mini-Neptunes Greaves et al. 2014; Davies 2019). (§ 3.1.10). Of the other three, HD 3167 (Dalal et al. Is primordial misalignment at work in HJ systems? 2019) does not show clear evidence for either a wider- It may be, but if so, tidal realignment likely heavily orbiting planet or a companion star. Kepler-56 (Hu- sculpts the resulting obliquity distribution: primordial ber et al. 2013) has a wider-orbiting third planet (Otor misalignment mechanisms alone do not seem to be able et al. 2016) and its mass and distance are compatible to fully account for the observed obliquity trends. Pri- with tilting the orbital plane of the inner two planets mordial misalignments might vary with stellar mass – for long after these planets have formed (Gratia & Fab- example, Spalding & Batygin(2015, 2016) propose that rycky 2017). The third – K2-290 – features a pair of lower mass young stars (< 1.2M ) may be able to re- planets – a warm Jupiter with an inner Neptune – on align their disks – but in that case would more strongly retrograde yet coplanar orbits and a stellar companion correlate with the initial main sequence effective tem- K2-290 B capable tilting the protoplantary disk (Hjorth perature than with present day effective temperature. et al. 2021). K2-290 is therefore the first clear sign that Primordial misalignment mechanisms also do not fully companion stars can generate obliquities by tilting the account for correlations with mass ratio (§ 3.1.4) or or- disks planets form from. More generally, we know from bital separation (§ 3.1.5). observations that disks with misaligned companions – Is primordial misalignment at work in non-HJ sys- which could tilt disks – are present. In wide binary sys- tems? There is growing evidence that the answer is yes. tems, proto-planetary disks can be misaligned from each If primordial misalignment is common, we expect to ob- other and the binary orbit, as deduced from polarization serve systems of coplanar planets that are misaligned observations of disk jets (Monin et al. 2007, and refer- with their host star. Ensemble studies show indirect ev- ences therein) and ALMA/VLTI observations of proto- idence that hot stars are indeed misaligned with their planetary disks (e.g., HK Tauri Jensen & Akeson 2014). coplanar planetary systems (Section 3.6). This trend Circumbinary debris disks can show misalignment with with effective temperature is not expected, since most of the orbit as well, e.g., KH 15D (Winn et al. 2004; Chi- the systems are beyond the reach of tides, so future work ang & Murray-Clay 2004; Poon et al. 2021). 24 Albrecht, Dawson, & Winn

It is uncomfortable to tell two very different stories for cur later when longer timescale chaotic evolution (see the obliquity-temperature trends of HJ hosts vs. other below) or stellar flys (e.g., Malmberg et al. 2011) bring hosts, but that is our current understanding. Our first planets together. Planet-planet scattering among plan- story is that most or all HJs are misaligned, not by a ets that are low mass and/or close to their stars lead companion tilting the disk they form from but possibly to only small mutual inclinations because their close en- magnetic warping or one of the post-formation mecha- counters lead to collisions rather than scattering (e.g., nisms described in the next section. They then tidally Goldreich et al. 2004). For giant planets further from realign cool host stars through a tidal mechanism that is their star – which may become HJs through high eccen- not well-understood. Our second story is that planetary tricity tidal migration – the distribution of mutual in- systems are primordially misaligned through a mecha- clinations produced by planet-planet scattering can be nism that primarily operates around hot stars but is broad but is still concentrated at low inclinations (e.g., not tidal; it could be – and, in the case of K2-290, very Chatterjee et al. 2008). To get a range of obliquities likely is – a companion tilting the disk. These two sto- as broad as we observe, planet-planet scattering more ries must be reconciled to interpret stellar obliquities in likely sets up the conditions for subsequent secular in- light of planets’ formation and evolution. teractions that lead to a broader obliquity distribution (e.g., Nagasawa et al. 2008; Nagasawa & Ida 2011; in 4.3. Post formation misalignment Fig. 18 we compare the predicted obliquity distribution from Nagasawa & Ida 2011, solid black, to predictions After formation, gravitational interactions between from other mechanisms). A related mechanism that can the planet and other bodies could alter the planet’s or- lead to a more isotropic distribution is direct disturbance bital plane, leading to misalignment with the host star’s of a giant planet through a hyperbolic encounter with spin. These gravitational interactions may also lead to a star in a very dense cluster environment, such as the high eccentricity tidal migration, in which a HJ forms center of a globular cluster, where stars are approaching further from the star, is disturbed onto a highly ellipti- at all angles (Hamers & Tremaine 2017). Future discov- cal orbit, and circularizes – due to tides raised on the eries of HJs in globular clusters could test this theory. planet – to its present-day short period. When the first Planets and stars exchange angular momentum over misaligned hot Jupiters were first discovered, the stel- longer timescales (typically thousands of orbits or more) lar obliquity was widely believed to primarily trace HJs’ through cyclic secular interactions. Eccentricities and dynamical history and to be driven by the same mecha- mutual inclinations oscillate as bodies in the system nism(s) that led to its short orbital period (see Dawson torque each other. In hierarchical (widely separated) & Johnson 2018 for a review of hot Jupiters’ origins). triple systems with large mutual inclinations and/or ec- We thought that the obliquity distribution pointed to centricities, these variations are known as Kozai-Lidov either: a) a dynamical history that most commonly led cycles (Kozai 1962; Lidov 1962) and can be driven by a to aligned orbits but occasionally produced strongly mis- stellar or planetary companion (e.g., Wu & Murray 2003; aligned orbits, or b) two origins channels, one leading to Fabrycky & Tremaine 2007; Naoz et al. 2011; see Naoz aligned orbits and the other to misaligned orbits (e.g., 2016 for a review). The timescale depends on the separa- Fabrycky & Winn 2009). However, given the strong ev- tion and mass of the perturbing companion, with typical idence for tidal realignment of cool HJ hosts (Section timescales of order millions of years. Although we often 4.1), we now believe that a mechanism is operating that model secular interactions after the gas disk stage, mu- produces a wide – possibly even isotropic – distribution tual inclinations can also be excited during the gas disk of obliquities for HJ hosts (Fig. 15). Some of these stage by secular interactions among the planet, disk, mechanisms – as we will highlight below – can also ac- and companion(s) (Picogna & Marzari 2015; Lubow & count for the indirect evidence for misalignments of hot Martin 2016; Franchini et al. 2020). stars hosting compact, coplanar systems (Section 3.6). Resonant excitation of the stellar obliquity can On the shortest timescales (as short as thousands of occur as the system evolves and a changing frequency years), planet-planet scattering can directly lead to crosses the secular frequency. In a triple system when mutual inclinations among planets and misalignments the primary spins down due to magnetic braking, the ro- with the host star’s spin. Closely spaced and/or ellipti- tational oblateness precession frequency crosses the sec- cal planets have close encounters that disturb their or- ular frequency, generating large misalignments (Ander- bits, with eccentricities and mutual inclinations growing son et al. 2018). However, we would expect this mecha- as a random walk over many orbits. Planet-planet scat- nism to primarily operate for cool stars, the opposite of tering can take place shortly after the dissipation of the the trends observed. It tends to produce primarily pro- gas disk when planets form close together, but may oc- Obliquity 25 grade misalignments (Fig. 18, dashed gray line). In a ishin et al. 2018), known as secular chaos. Similar to system with an outer planetary companion and dispers- cyclical secular interactions, the resulting obliquity dis- ing gas disk, the gas disk precession frequency can cross tribution depends on the initial architecture; producing the secular frequency, generating a large mutual inclina- planets on retrograde orbits requires eccentricities and tion between the inner and outer planet and driving the inclinations that are large to begin with (e.g., Lithwick & stellar obliquity to ψ = 90◦ (Petrovich et al. 2020). This Wu 2014), perhaps established by planet-planet scatter- mechanism is most effective for close-in Neptune-mass ing (Beaug´e& Nesvorn´y 2012). However, Teyssandier with outer Jupiter-mass companions, like the HAT-P- et al.(2019) argue that this mechanism produces an in- 11 system. surmountable lack of retrograde planets (Fig. 18, solid The resulting obliquity distribution from all these purple line) because the planet tends to circularize and types of secular interactions depends on the initial sys- decouple from the companion before the obliquity grows tem architecture, which may be established by earlier very large. evolution in the presence of a gas disk, planet-planet In summary, producing a broad obliquity distribution scattering, and/or stellar fly bys (e.g., Hao et al. 2013). with plenty of retrograde planets is the biggest chal- Distant and/or circular companions driving Kozai-Lidov lenge for these mechanisms, but the more complex and cycles tend to produce a bimodal obliquity distribution multi-step dynamical histories – such as planet-planet (Fig. 18, dashed red line) with peaks near 40 and 140 scattering followed by secular cycles – seem at at least degrees and an absence of polar orbits (e.g., Fabrycky & qualitatively consistent with the observed distribution Tremaine 2007; Naoz et al. 2012), which may not be fully for HJs (Fig. 18). Producing fewer retrograde planets consistent with the observed distribution of projected would helpfully reduce the number of retrograde plan- obliquities of HJ hosts. Accounting for the host star’s ets expected with ψ = 180◦ following inertial wave tidal oblateness and spin evolution (which can sometimes dissipation but seems at odds with the large number of lead to chaotic variations in its spin vector, e.g., Storch retrograde planets orbiting hot stars. One major uncer- et al. 2014) further enhances this bimodality (Damiani tainty in comparing the predictions of these mechanisms & Lanza 2015; Anderson et al. 2016), particularly for to the observed obliquity distribution and even teasing cool stars. The fraction of planets on retrograde orbits is out the contributions of multiple mechanisms (e.g., Mor- larger and the distribution of obliquities is broader when ton & Johnson 2011; Naoz et al. 2012) is that even the the companion is eccentric and/or nearby (Fig. 18, dot- obliquity distribution of hot stars hosting HJs may have ted blue line) ; Naoz et al. 2011; Teyssandier et al. 2013; been altered by tides. Achieving an isotropic distribu- Li et al. 2014b,a; Petrovich & Tremaine 2016), such as tion post-formation for small, compact, coplanar planets companions that were engaged in planet scattering (Na- orbiting hot stars (Section 3.6) may be even more chal- gasawa et al. 2008; Nagasawa & Ida 2011). For Kozai- lenging and has not yet been demonstrated. Lidov cycles to significantly raise the mutual inclination, Although the mechanisms discussed here operate on the orbital precession caused by that companion must a range of timescales – from during the gas disk stage dominate over precession from stellar oblateness, tides, to throughout the star’s lifetime – they generally re- and general relativity. In compact systems where plan- quire that the HJ form further from its star. Planet- ets are more tightly coupled to each other than to an ex- planet scattering generally fails to generate large mis- terior companion, the exterior companion can misalign alignments very close to the star and secular mechanisms the entire interior system from its host star’s spin, as require very nearby planets that can overcome the cou- observed for Kepler-56 (e.g., Takeda et al. 2008; Bou´e& pling of the HJ to the star, which most HJs lack (see Fabrycky 2014; Li et al. 2014c; Gratia & Fabrycky 2017). below). Therefore, under the theories discussed here, we This explanation does not hold for K2-290 (the system expect HJs have their eccentricity excited by the same highlighted as an example of primordial misalignment in process that misaligns them, undergo high eccentricity Section 4.2), where the inner system is too tightly cou- tidal migration via tides raised on the planet, and ar- pled to the stellar spin by oblateness precession (Hjorth rive misaligned. The tidal migration timescale is uncer- et al. 2021). tain and very sensitive to the eccentric planet’s periapse Over many secular timescales – hundreds of mil- distance and thus can span many orders of magnitude. lions of years or longer – mutual inclinations can grow More work on misalignment theories is needed to ex- chaotically due to the overlap of secular frequencies plore how the obliquity distribution changes over time in multi-planet systems (Laskar 2008; Wu & Lithwick and whether we expect young HJs to be just as mis- 2011; Hamers et al. 2017; Teyssandier et al. 2019) or aligned as older ones (Section 3.1.8). However, Beaug´e triple/quadruple star systems (e.g., Hamers 2017; Gr- & Nesvorn´y(2012) do predict that retrograde planets 26 Albrecht, Dawson, & Winn 1 10 100.0

Resonant (A18) 10.0 Sec Chaos (T19) Planet Kozai (P16) 1.0

Stellar Kozai (A16) mass ratio HJ coupled to friend Scatter/Sec (N11) 0.1 1 10 semi-major axis ratio/25 Figure 19. Planet-planet coupling. A handful of HJs with low obliquities orbiting cool stars (blue symbols in the orange region) are strongly coupled to an nearby companion, preventing tidal realignment. Lines represent companions that are detected as radial velocity trends (for which mass and semi-major axis are degenerate).

Relative numnber 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 0 30 60 90 120 150 180 to drive secular cycles and prevent the HJ from tidally ψ (deg) realigning its star. Several observed systems have strong coupling between the HJ and its outer companion but Figure 18. Example population synthesis (unpro- low obliquities (Fig. 19); these systems cannot be ex- jected) obliquity distributions from studies of differ- plained by Kozai-Lidov cycle misalignment followed by ent misalignment mechanisms: resonance crossing (Anderson tidal realignment. However, the majority of known HJ et al. 2018), secular chaos (Teyssandier et al. 2019), planet- companions are not sufficiently coupled, and the com- planet Kozai-Lidov cycles (Petrovich & Tremaine 2016), star- panion would not interfere with the HJ tidally realigning planet Kozai-Lidov for a 1 MJup HJ orbiting an F star (An- its star. derson et al. 2016), and planet-planet scattering with secular cycles (Nagasawa & Ida 2011). 4.4. Altering the stellar spin vector will also tend to have closer periapses and are thus more The processes discussed so far involve changing the likely to raise tides on the star that drive orbital decay; orbital plane of the planet(s) or changing the spin of they predict that retrograde planets should be system- the star as a response to an external force. Rogers et al. atically younger. (2012, 2013) showed that for hotter stars with convective One avenue to test the secular cycle hypothesis in cores and radiative envelopes, Internal Gravity Waves particular is to search for companions capable of driv- (IGW) can lead to a tilt of the photosphere relative to 4 ing Kozai-Lidov cycles. A prime example would be the the total angular momentum, on timescales of 10 yrs or HD 80606 double star system with its highly eccentric less. Changes in λ and v sin i over time in systems with warm Jupiter (Wu & Murray 2003) on an oblique or- hot host stars would indicate that IGW are at work but 16 bit (H´ebrardet al. 2010), which may be in the midst are not easily observable with the current short time of Kozai-Lidov driven high eccentricity tidal migration. baseline. Radial differential rotation, which could be The Friends of Hot Jupiters survey (Knutson et al. 2014; Ngo et al. 2015; Piskorz et al. 2015; Bryan et al. 2016; 16 Precession due to spin-orbit coupling observed in exoplanet Ngo et al. 2016) found that most hot Jupiters lack a hosts (e.g. WASP-33 Johnson et al. 2014) as well as in close double capable stellar companion but that many have a po- stars (Albrecht et al. 2014) can complicate our interpretation of such changes. Obliquity 27 detected via asteroseismology (Christensen-Dalsgaard & retrograde coplanar orbiting planets. This configura- Thompson 2011), would also be a hallmark of IGW. tion is a result of primordial misalignment caused by IGWs can account for some but not all observed the companion. That we see not more such systems in trends. Although IGWs can account for the higher the current sample might be a result of selection biases obliquities of hot stars (§ 3.1.3), they cannot account as K2-290 A is the only multi transiting exoplanet sys- for correlations of obliquity with mass ratio (§ 3.1.4) or tem in a wide binary for which the obliquity has been orbital separation (§ 3.1.5). Furthermore, we would ex- measured. In addition ensemble studies of transiting pect coplanar systems to be misaligned with hot stars. planets indicate that opposite to cool stars which tend Ensemble studies indirectly suggest that they are (Sec- to be aligned in population studies also involving non HJ tion 3.6); however, of the three known coplanar systems systems, hot stars in general have planets on misaligned orbiting hot stars with individual obliquity measure- orbits, not only HJs. This suggests that misalignment ments (HD 106315, K2-290, and Kepler-25), the only mechanism(s) can operates independently of the plan- misaligned one is K2-290, for which the stellar compan- etary parameter range observed. Primordial misalign- ion is believed to be responsible for the retrograde orbit ment would be such a mechanism. Together these lines (Hjorth et al. 2021), rather than IGWs. We can also of evidence illustrate that the textbook example of a star test this mechanism using binaries with separations be- with an aligned protoplanetary disk does not encompass yond the reach of tides containing one low mass star and all important aspects of planetary formation. one high mass star. If IGWs are at work, we expect to Large obliquities in systems with close in giant plan- more often observe the high mass star misaligned with ets seem to be best explained by dynamical interactions the binary’s orbit and the low mass star aligned. Cur- which occurred after planet formation. The strongest rently there are no suitable binary systems to perform observational support comes from the difference in obliq- this test. uity distribution between single and coplanar systems and the increased (planetary) companionship to such 5. SUMMARY AND DISCUSSION systems. The observations of alignment in a small num- ber of young ( 100 Myr) systems with (close in or- Available evidence points towards two pathways to- . biting) giant planets further suggests that if compact wards spin orbit misalignment, primordial and dynami- systems are misaligned then this is caused by dynamical cal processes after planet formation. Furthermore tidal interactions followed by high eccentricity migration, as interactions between the star and planet are important this process can occur on similar or longer times than for the observed population of exoplanet systems. the life times of the systems. While tides are not fully understood there are several We do not yet know which of the proposed dynami- indications that the problem of giant planet destruction cal processes has a dominate role, if any. Poster child during realignment might not be as severe as originally systems for KL-cycles (caused by stellar companion) do feared. Tides are also most successful in explaining ob- exist (e.g. HD 80606). Among post formation scenarios servational trends with stellar structure, orbital separa- KL-cycles can most easily generate retrograde orbits. tion, planetary mass, and that HJs orbiting cool stars However surveys have not been able to clearly identify which have well measured obliquities (σ < 2◦) show λ the necessary companions (stellar or planetary) with alignment and dispersion both below 1◦. the required parameters to drive KL-cycles to the HJ There is indirect evidence, from observations of jets sample, though there seem to be a suitable number of and disks in young stellar systems, that primordial star planetary companions if they have the necessary mu- disk misalignment does at least occasionally occur dur- tual inclination. Different post formation mechanisms ing the early systems evolution, before the protoplan- scenarios do lead to different obliquity distributions and etary disk is dispersed. While the stellar spin is un- this can in principle be used to differentiate between known in these systems HK Tauri is one of the clear- the different post formation processes. However tidal est examples that not all vectors, stellar spin, orbital alignment and our lack of quantitative understanding spin, and disk spin, can be aligned in such systems. thereof blurs the observational distention between dif- However theoretical work, observations of debri disks, ferent mechanisms. obliquity measurements in a small number of young ex- oplanet systems, and in compact transiting multi planet We should remind ourselves that the current sample of systems with cool host stars suggest that such misalign- systems with obliquity measurements is heavily biased ments when present might not always survive the final in a number of ways, most noticeably towards close in stages of the systems formation. Nevertheless one plan- giant planets orbiting main sequence stars, mainly F-K etary system, K2-290 A, part of a wide binary features 28 Albrecht, Dawson, & Winn type. As these planetary systems do not present the of our understanding of the formation of gas giant complete spectrum of planetary systems, so might their planets inside one au. obliquity distribution. A number of new missions have the potential to • Finally obliquity measurements in (wide) double change this preoccupation with a small subset of sys- star and multiple star systems will determine the tems: Bright and well characterized TESS systems allow coherence length scale of the angular momentum for more precise RM and v sin i measurements in a more distribution, which in turn determines for which diverse planetary population. GAIA will enable mea- kind of systems certain types of primordial disk surements of mutual inclinations, (less precise) v sin i misalignment mechanisms and stellar KL-cycles measurements in a larger sample as well as interfero- could be important. These samples could also metric obliquity measurements in a smaller brighter sub- serve to better test the role of IGW. set of systems. PLATO, while also enabling RM mea- The obliquity of a body (star, planet, moon) is a fun- surements, will be more crucial for seismic, spot, and damental orbital parameter and should be considered v sin i measurements in new types of systems. Combing an important observable worth measuring if a system is these new samples with the availability of new or soon to studied in detail. be operational spectrographs and intererometers should lead to new insights, among them: SA acknowledges the support from the Danish Coun- • Bright well studied TESS systems harboring close cil for Independent Research through the DFF Sapere ◦ Aude Starting Grant No. 4181-00487B, and the Stellar in giant planets will allow for precise (σλ . 2 ) RM measurements employing new ground based Astrophysics Centre which funding is provided by The spectrographs. This will lead to an increased un- Danish National Research Foundation (Grant agreement derstanding of tidal alignment, crucial to better in- no.: DNRF106). RID acknowledge supports from grant terpret existing and upcoming obliquity measure- NNX16AB50G awarded by the NASA Exoplanets Re- ments. search Program and the Alfred P. Sloan Foundation’s Sloan Research Fellowship. The Center for Exoplanets • Primordial misalignment can be tested in young and Habitable Worlds is supported by the Pennsylvania systems, planets with large orbital separations, State University, the Eberly College of Science, and the systems with multi transiting systems (if compan- Pennsylvania Space Grant Consortium. ionship is known), and via star debri disk align- We thank J.J. Zanazzi for helpful comments and sug- ments. This can be achieved via RM measure- gestions. 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 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 +27 misalignment Pont(2009). Guenther et al.(2012) found a projected obliquity of −52−22 deg for CoRoT-19. However no post-egress data were obtained and the Rossiter-McLaughlin effect was detected at an 2.3σ level only. +4 Zhou et al.(2015) report for HATS-14 a misaligned orbit ( |λ| = 76−5 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 & H´ebrard(2014) claimed a significant misalignment in the 55 Cnc system, which was proven to be spurious by L´opez-Morales et al.(2014). There is also the tentative detection of misalignment in KOI-89 (Ahlers et al. 2015), but a recent reanalysis of the Kepler data showed that the obliquity is unconstrained by the data (Masuda & Tamayo 2020). Most of the quoted v sin i measurements have been obtained from RM measurements. In particular for lower v sin i values and large impact parameter is large these values can be more precise. However for some systems where the RM data is of low SNR (e.g. Qatar-2, (Esposito et al. 2017) we opted to quote the spectroscopic value. It is also worth noticing that the spectroscopic value is a disk integrated value whereas the value obtained from RM studies connects to the surface motion under the planets path over the stellar disk). 30 Albrecht, Dawson, & Winn age Companion References ) (Gyr) ?

R ) (R ?

M eff T ) (K) (M 1 i − sin ) (km s ◦ λ v ( . Listing of the systems and some host star parameters considered in this review. The orbital (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Number System —- Table 2 parameters of their planets are listed in table 3 . Note Obliquity 31 References e ) p R Jupiter ) (R p was determined. The stellar parameters M Jupiter λ a/R (days) (M . Listing of the planets for which (1) (2) (3) (4) (5) (6) (7) (8) —- Number Planet Period Table 3 Note of their host stars are listed in table 2 32 Albrecht, Dawson, & Winn

B. POPULATION SYNTHESIS SIMULATIONS ignored because it does not affect the alignment. The The simulations follow Dawson(2014), with updates third component is perpendicular to the other two and to incorporate options for intertial wave tidal dissipa- thus we compute its unit vector as: tion and a frequency dependent tidal dissipation effi- Ω~ × ciency. We numerically integrate the planet’s specific xˆ = (~h × Ω~ ) × s (B3) s Ω2 h sin ψ orbital angular momentum vector ~h and the host star’s where ? spin angular frequency vector, assuming a circular or- Ω~ · ~h bit. The equations here correspond to Barker & Ogilvie cos ψ = ? (2009), Eqn. A7 and A12 with the eccentricity vector Ω? h ~e = 0. |Ω~ × ~h| sin ψ = ? Ω? h (B4) ~ ~ ! ~˙  1 ~ 1 Ω? Ω? · h ~ h h = − h + · h + Ω~? eq τeq τeq 2n Ω? h Ω? We add the following terms to Eqn. B1.

 ˙  m ˙ 2   ~ ~ ~  ˙  1 τ0,dy Ω? = − 2 heq − α brakeΩ?Ω? ~ eq,α k?,eff MR Ω? = − 1 − dy τdy τ0,eq (B1) h 2 3 i (sin ψ cos ψ) Ω~? − sin ψ cos ψ Ω?xˆ for which 2 ~˙  k?,eff MR  ˙  h = − Ω~? dy m dy Q M M 13 τeq = 5 8 7 h (B5) 6kL R (M + m) G m  13 h 0.5M Jup where = τeq,0 (B2) τ Ω h h0 m 0,dy ? 0 τdy = τeq. (B6) τ0,eq Ω?,0 h is an orbital decay timescale, k is the Love number, L We set τ0,dy = 10−5 for Fig. 17. Q is the tidal quality factor, k is the effective con- τ0,eq ?,eff For the frequency-dependent tidal dissipation effi- stant of the stellar moment of inertia participating in ciency model (Penev et al. 2018), we use Eqn. B1 with the tidal realignment, α brake is a braking constant, and p a modified value of teq: h0 = a0G(M + m) is the initial specific angular mo- 2 2 mentum. By default, we use k?,eff MR = 0.08M R Max[106/Ptide[days]3.1, 105] 2 2 t = t (B7) for cool stars, k?,eff MR = 0.08(1.2M )(1.4R ) for hot eq,f eq Max[106/Ptide0[days]3.1, 105] 2 −1 −16 stars, and Ωs,0 = 800 AU yr . We use α = 3 × 10 −14 for hot stars, α = 1.4 × 10 for cool stars, τeq,0 = 500 where P tide = π/(n − Ωs) is in units of days. 2 −1 Gyr, and h0 = 1.33 au yr . For the pure equilibrium To generate the populations for Fig. 17, we select 2 −1 tides simulation, we use h0 = 1.68 au yr . For the a uniform random 4800 < Teff < 6800 K, a log-uniform frequency-dependent Q simulations, we use τeq,0 = 10 0.5M Jupiter < m < 15M Jupiter, ψ from an isotropic dis- 2 −1 Gyr, and h0 = 1.85 au yr . For the decoupled outer tribution, a uniform random evolution time 0 < t? < 10 −13 envelope simulations, we use α = ×10 for cool stars Gyr for cool stars (T < 6250K) or 0 < t? < 4 Gyr for hot For intertial wave tidal dissipation, the tidal forcing stars, and a uniform random longitude of ascending node component that excites inertial waves exerts a torque. 0 < Ω < 2π. Then we integrate the momentum equa- ~ −1 Here we follow Lai(2012) to compute its affects on h and tions above for t?. We compute λ = tan (tan ψ sin Ω) Ω~?. One component is parallel to the stellar spin, i.e., in (Fabrycky & Winn 2009, Eqn. 11; Column 2 of our the Ω~? direction. A second component is perpendicular Fig. 17 shows the initial distribution of λ), sin i = ~ ~ p 2 to both h and Ω~?, i.e., in the Ω~s × h direction, and is 1 − (sin ψ cos Ω) , and v sin (i) /R = Ω? sin i.

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