Damping Obliquities of Hot Jupiter Hosts by Resonance Locking

When orbiting hotter stars, hot Jupiters are often highly inclined relative to their host star equator planes. By contrast, hot Jupiters orbiting cooler stars are more aligned. Prior attempts to explain this correlation between stellar obliquity and effective temperature have proven problematic. We show how resonance locking—the coupling of the planet's orbit to a stellar gravity mode (g-mode)—can solve this mystery. Cooler stars with their radiative cores are more likely to be found with g-mode frequencies increased substantially by core hydrogen burning. Strong frequency evolution in resonance lock drives strong tidal evolution; locking to an axisymmetric g-mode damps semimajor axes, eccentricities, and, as we show for the first time, obliquities. Around cooler stars, hot Jupiters evolve into spin–orbit alignment and may avoid engulfment. Hotter stars lack radiative cores and therefore preserve congenital spin–orbit misalignments. We focus on resonance locks with axisymmetric modes, supplementing our technical results with simple physical interpretations, and show that nonaxisymmetric modes also damp obliquity. Outstanding issues regarding the dissipation of tidally excited modes and the disabling of resonance locks are discussed quantitatively.

1. INTRODUCTION Winn et al. (2010) discovered that hot Jupiters orbiting cool stars have orbit normals that better align with stellar spin axes than hot Jupiters orbiting hot stars (see also e.g.Schlaufman 2010; Albrecht et al. 2012;Winn et al. 2017;Muñoz & Perets 2018;Albrecht et al. 2021;Hamer & Schlaufman 2022;Rice et al. 2022a,b;Siegel et al. 2023).Figure 1 displays this correlation between stellar obliquity  and stellar effective temperature  eff .Figure 2 shows that  eff seems to correlate also with orbital eccentricity : cooler stars host more circular hot Jupiters.
High-eccentricity migration is one way to form hot Jupiters (e.g.Dawson & Johnson 2018).One imagines that giant planets are delivered from afar onto high-, high-, low-periastron orbits by, e.g., planet-planet scatterings or Lidov-Kozai oscillations.Presumably this initial delivery unfolds similarly around cool and hot stars.Subsequent tidal interactions with the star shrink the orbit by damping  and orbital semi-major axis .To reproduce the observations, tidal damping of  and  would have to be, for some reason, more effective around cool stars than hot stars.In this paper we attempt to identify this reason.
The dividing line used in Figs. 1 and 2 to distinguish cool from hot stars is a stellar effective temperature  eff = 6100 K, a.k.a. the Kraft break below which stars have convective envelopes and radiative cores, and above which stars have radiative envelopes and convective cores.Damping of the equilibrium tide, and by extension obliquity and eccentricity, has been argued to be more effective in turbulent convective envelopes, potentially explaining the - eff trend (Winn et al. 2010;Albrecht et al. 2012).A problem with this idea is that convective eddy turnover times may be much too long compared to tidal forcing periods for turbulent viscosity to be significant (Goldreich & Nicholson 1977;Vidal & Barker 2020).Even if convective dissipation (or the dissipation associated with the interaction between tidal flows and convection according to Terquem 2021; but see Barker & Astoul 2021) were somehow more effective, another problem with the equilibrium tide is that it results in wholesale semimajor axis decay.By the time the obliquity damps from equilibrium tidal dissipation, the star engulfs the planet (see also Barker & Ogilvie 2009;Dawson 2014).Lai (2012) argued that dissipation of tidally excited inertial waves in convective zones could damp cool star obliquities while avoiding engulfment.One shortcoming of inertial wave dissipation is that it cannot take retrograde obliquities (which are observed for high-mass stars) and evolve them into prograde alignment (e.g.Rogers & Lin 2013;Valsecchi & Rasio 2014;Xue et al. 2014;Li & Winn 2016;Anderson et al. 2021;Spalding & Winn 2022).Another difficulty is that inertial wave dissipation scales with the host star's spin rate, which may be too slow on the main sequence to damp obliquities (e.g.Lin   Projected stellar obliquities  (as distinct from deprojected or true 3D obliquities ) vs. host star effective temperature  eff of hot Jupiter systems (selecting for planet masses  p ≥ 0.3 M J and semimajor axes  ≤ 10 ★ , where  ★ is the host star radius; data from Albrecht et al. 2022;Rice et al. 2022a,b;Siegel et al. 2023;Espinoza-Retamal et al. 2023;Sedaghati et al. 2023;Hu et al. 2024).Obliquities are larger for hot high-mass stars (stellar effective temperature  eff > 6100 K, red points) than for cool low-mass stars (blue points).This correlation (Winn et al. 2010) is statistically significant according to a two-sided Kolmogorov-Smirnov (KS) test, which yields a  = 1.6 × 10 −3 probability that the cumulative obliquity distributions for cool and hot stars are drawn from the same underlying distribution (right panel).
Projected obliquities vs. eccentricities for hot Jupiters (identical sample to Fig. 1).Obliquities and eccentricities are lower for cool stars than for hot stars.& Winn 2022).Dissipation rates are greater on the pre-main sequence, but observations indicate high- systems form late (Hamer & Schlaufman 2022), consistent with late-time dynamical instability in a high- migration scenario.Even if hot Jupiters formed early, high-mass stars have thick convective envelopes on the pre-main sequence, and inertial wave dissipation would predict their obliquities to be small, contrary to observation.
In this work we consider how  and the (true 3D) obliquity  evolve when the planet is resonantly locked with a stellar oscillation mode.The modes of interest are gravity modes (g modes), which exist in radiative (stably stratified) zones, not convective ones, and therefore behave differently between stars below and above the Kraft break.Gravity modes in the extensive radiative zones of stars have a dense frequency spectrum, and one can readily find a mode whose frequency matches (to within a low-integer factor) the planet's orbital frequency.In resonance lock (Witte & Savonije 1999, 2001;Savonije 2008), the frequency match is preserved as a star evolves and its internal structure changes.Typically g-mode frequencies increase from increasing stratification due to hydrogen burning, and the planet's orbital frequency follows suit -the planet migrates inward (Ma & Fuller 2021).Resonance locks and stellar evolution can change not only orbital semi-major axis , but also eccentricity  (e.g.Savonije 2008;Fuller 2017;Zanazzi & Wu 2021).We show for the first time here that obliquity  also evolves in resonance lock.
A full derivation of the equations governing , , and  is given in appendix A; a condensed version sketching our physical picture and listing the main results is provided in section 2. How g-mode frequencies evolve differently between cool and hot stars is explored with MESA stellar evolution calculations in section 3. Results for the time evolution of , , and  for proto-hot Jupiters in resonance lock are presented in section 4. Section 5 compares our theory with observations and offers some extensions, and section 6 concludes.In appendix B we consider the still-uncertain physics of non-linear g-mode energy dissipation, and estimate the largest orbital distance out to which our theory might apply.

RESONANCE LOCKING MODEL
We begin with a general set of equations relating the angular momentum of the planet's orbit with the angular momentum of the stellar spin.The star has mass  ★ , radius  ★ , spin frequency Ω ★ , moment of inertia  ★ =  ★  ★  2 ★ , and spin angular momentum  ★ =  ★ Ω ★ ĵ★ with ĵ★ the unit vector parallel to the stellar spin axis.The star is orbited by a planet of mass  p ≪  ★ , having orbital semi-major axis , eccentricity , mean-motion Ω = √︁   ★ / 3 , orbital energy  orb = −  ★  p /(2), and orbital angular momentum where  is the gravitational constant and ĵorb is the unit orbit normal.The stellar obliquity  is the angle between  ★ and  orb (cos  = ĵorb • ĵ★ ).We work in a Cartesian coordinate system such that ĵ★ always lies in the -direction, and ĵorb always lies in the - plane (i.e.ĵ★ × ĵorb defines the -direction).Energy is extracted from the orbit (from tidal dissipation) at a rate  orb < 0. Angular momentum is exchanged (by tides) between planet and star at rate  = −  orb = +  ★ ; we restrict consideration to the torque's  and -components,  =   ê +   ê , since   just causes ĵorb to precess about ĵ★ without changing .We thus have (Lai 2012): (4) See Figure 3 for an illustration of the coordinate system in which we are working.
Our theory is that  orb and  are driven predominantly by a particular oscillation mode in the star, resonantly excited by the planet.In resonance, an integer multiple  of the planet's orbital frequency matches the stellar oscillation frequency: where   is the mode oscillation frequency in the inertial frame,   is the oscillation frequency in the frame rotating with the star, and  is the mode's azimuthal number.For most of this paper, we restrict attention for simplicity to "zonal"  = 0 axisymmetric modes (Zanazzi & Wu 2021).
In resonance lock, oscillation and orbit track one another; the mode frequency evolves on the same timescale as the orbit evolves: where   = 3/2 for  = 0 (see appendix A and Fuller 2017 for the case  ≠ 0).In this paper, we assume the planet locks onto a stellar gravity mode (g mode), and compute how   /  evolves on the stellar main sequence (section 3).
In the inertial frame co-planar with the orbit, each term in the planet's tidal forcing potential varies as where  orb is the azimuthal coordinate measured in the orbital plane; note that  orb differs from the stellar mode azimuthal number  defined earlier.The dominant term in the forcing potential has  =  orb = 2. Figure 4 illustrates how  =  orb = 2 tidal forcing can excite an  = 0 mode in the star for  ≠ 0 (bottom right panel), and conversely how  = 0 represents a fixed point where an  = 0 mode is not excited (bottom left).Figure 5 illustrates how this  = 0 mode causes the obliquity to damp -the planet exerts a torque on the tidally lagged stellar bulge that always acts to bring the stellar spin vector into closer alignment with the orbit normal, at all orbital phases of the planet.Only when spin and orbit align does the torque vanish.Thus, the  = 0 fixed point is a stable attractor -the obliquity under many initial conditions damps to zero.We evaluate equations (1)-(4) for the case when planet and star are in resonance lock for {, ℓ, } = {2, 2, 0}, where ℓ is the angular degree of the spherical harmonic representing the stellar oscillation mode (for context, most other studies of tidally forced oscillations focus on ℓ = 2,  = 2).The full derivation is relegated to appendix A; the results are: d Note that  ★ does not change (  = 0; Fig. 5, right column) because we are considering only axisymmetric stellar modes (see section 5.7 and appendix A for a discussion of non-axisymmetric modes).The dimensionless torque  = 2Ω  /  is given by equation (A43), and depends on  through Hansen coefficients   ℓ, orb and  by a sum over Wigner- matrix coefficients (eq.A9).When  ≪ 1 and which vanishes as  → 0. Equations ( 8) and (10) then imply (1 −  2 ) ≃ constant, i.e. the orbit circularizes along a constant angular momentum track (Zanazzi & Wu 2021).In the opposite case when  The star stretches and compresses along its spin axis (-axis) twice for every orbit of the planet.The solid purple sphere marks the planet's location which is displaced ahead of the star's tidal bulge due to tidal friction, while the see-through purple sphere displays the planet's location in the absence of tidal friction.The top row shows different views of the same instant in time, when the star is maximally elongated but the planet is ahead at positive .The bottom row shows a different time, when the star is maximally compressed and similarly lagged in phase relative to the planet.The tidal bulge at position vector  feels a force from the planet  and by extension a torque  =  ×  that is positive in the -direction no matter the orbital phase (middle column).The torque   > 0 acts to bring the stellar spin direction ( ĵ★ ) toward the -axis, closer to the orbital angular momentum direction ( ĵorb ).Once spin and orbit are aligned, the torque vanishes (Fig. 4, bottom left panel).Because the oscillation is symmetric about the -axis (right column), there is no torque in the -direction.

GRAVITY MODE EVOLUTION BETWEEN LOW-MASS AND HIGH-MASS STARS
In the low-frequency limit, g-mode frequencies in the host star's rotating frame are given by where   ≫ 1 is the number of radial nodes, and the radial average of the Brunt-Väisälä frequency is given by with the integral being taken over the stably stratified radiative zone ( 2 > 0).As in Zanazzi & Wu (2021), we focus on zonal modes (azimuthal number  = 0), so  = , and  ev is given by ignoring Coriolis forces.
We define a fractional frequency change and use the stellar evolution code MESA (release r23.05.1) to compute ⟨ ()⟩ from  0 ≡ 1 Gyr (the subscript 0 denotes evaluation at time  0 ) to  max = min( MS , 12 Gyr), with the end of the main-sequence lifetime  MS defined by when the star's core has depleted its hydrogen.The MESA inlists we use are identical to those in Fuller (2017).How our results depend on  0 and  max is discussed in section 5.3.Over the star's main sequence lifetime, in response to the loss of gas pressure from the conversion of hydrogen into helium and the increase in mean molecular weight, the core contracts and becomes more dense and stratified.Thus ⟨⟩ tends to increase, the more so when the region of stable stratification extends down to the star's core, as it typically does for cool but not for hot stars.Figure 6 illustrates how the evolution of the star-averaged Brunt-Väisälä frequency differs Although high-mass stars (red curves) increase their frequencies dramatically quickly toward the end of their hydrogen-burning lives (as their cores switch from being unstably to stably stratified), the speed-up phase is short-lived and consequently unlikely to be observed.Sampled over the entirety of their main-sequence lives, Δ/⟨ 0 ⟩ is likely to be higher for low-mass stars, whose cores remain stably stratified throughout, than for high-mass stars (bottom panel showing median and ±1 intervals, calculated by evaluating Δ/ 0 at 1000 uniformly spaced times over the interval [ 0 ,  max ]).
between cool low-mass stars (blue curves) and hot high-mass stars (red curves).The sign of the difference can be subtle and varies with context.The top panel of Fig. 6 shows that for high-mass stars at the end of their main-sequence lifetimes, the mode evolution rate ⟨ ⟩/⟨⟩ actually exceeds that for low-mass stars -see how the red curves ultimately skyrocket above the blue curves.The last-minute speed up of mode evolution in high-mass stars, which occurs as they ascend the sub-giant branch and their cores switch from being convective to radiative, is so great that even though lowmass stars live longer than high-mass stars, the total change Δ accumulated over main-sequence lifetimes is comparable between low-mass and high-mass stars (middle panel of Fig. 6).Nevertheless, because the mode frequency speed-up for high-mass stars occurs over a small fraction of the highmass main-sequence lifetime, the time-averaged value for Δ is actually lower for high-mass as compared to low-mass stars (bottom panel of Fig. 6).The upshot is that given a lowmass main-sequence star and a high-mass main sequence star drawn randomly from the field, the low-mass star is more likely to have experienced a larger fractional change in its g-mode frequency.The expected change is on the order of unity for low-mass stars, and ≲ 10% for high-mass stars.This difference underpins our theory that stellar obliquities have evolved more for low-mass than high-mass stars.

RESULTS FOR OBLIQUITY AND ORBIT EVOLUTION
We integrate equations ( 8)-( 11) for a planet of mass  p = 1 M J in resonant lock with a stellar gravity mode.We start all calculations at time  0 = 1 Gyr, and evolve , , and  to  max = min( MS , 12 Gyr), where  MS is the time when the stellar core stops burning hydrogen.Note that if and when both ,  = 0, resonance lock with our assumed {, ℓ, } = {2, 2, 0} mode cannot be maintained (e.g.bottom left panel of Fig. 4), and , , and  cease to change.In section 5.2 and appendix B.2 we discuss more generally how resonance locks may actually break.Initial conditions vary over  0 = {0.035,0.07} AU,  0 = {0, 0.3, 0.6}, and  0 = {45 • , 90 • , 145 • } (variables subscripted 0 are evaluated at time  0 ).We integrate our equations using the Python routine solve ivp from scipy, setting rtol = 10 −6 and atol = 10 −11 , and evaluating  −1 ev by linear interpolation (using numpy interp) of the data in the top panel of Fig. 6.
Figures 7, 8, and 9 present results for initial conditions  0 = 0, 0.3, and 0.6, respectively.In all cases, obliquities  damp more for low-mass stars than high-mass stars.For low-mass stars, obliquities can start as high as 90 • -145 • and drop to zero or nearly so within main-sequence lifetimes.Once the obliquity vanishes, the planet can no longer drive the  = 0 mode that we are modeling; dissipation ceases and the planet stops migrating inward.For high-mass stars, obliquities hardly budge from their initial values.None of our planets is engulfed by its host star.
Larger eccentricities are seen to prolong obliquity damping and circularization times (compare Figs. 7,8,and 9).A starting eccentricity of 0.3 damps to zero within main-sequence lifetimes for both low-mass and high-mass stars (Fig. 8), about as fast as the obliquity damps.At  0 = 0.6, only the 1 and 0.8 M ⊙ models circularize and achieve spin-orbit alignment within 12 Gyr for  0 ∼ 45 • (Fig. 9).

DISCUSSION
We have shown that hot Jupiters locked in resonance with stellar g-modes can torque the stars into spin-orbit alignment and have their orbits circularized.The damping of obliquity and eccentricity is stronger for lower-mass stars because their g-mode frequencies evolve more on the main sequence.In the following subsections we delve deeper into the theory, and confront observations.

Obliquity and Stellar Effective Temperature: Theory vs. Observation
We construct a simple population synthesis to compare against observations.Host stellar masses  ★ are sampled uniformly from the set {0.6, 0.8, 1.0, 1.2, 1.4, 1.6} M ⊙ .Around each star we consider a planet of mass  p = 1 M J , initial semimajor axis  0 = 8 ★0 , eccentricity  0 = 0, and obliquity  0 drawn randomly from a uniform distribution in cos  0  Figure 8. Same as Fig. 7 except for initial eccentricities  0 = 0.3.For such initial conditions, orbits can circularize around cool stars on the same timescale as the obliquity damps, but not hot stars.The obliquity evolution is similar as for strictly circular orbits (compare with Fig. 7).
between −1 and 1 (for observational support of the latter distribution, see drawing  obs randomly from a uniform distribution between 0 and 2 (e.g.Fabrycky & Winn 2009).
Figure 10 compares our modeled projected obliquities  with those observed (top panel; observations from Rice et al. 2022a).For completeness, we also show our modeled obliquities  against the few deprojected obliquities that are available (bottom panel; data from Albrecht et al. 2022;Siegel et al. 2023).Broadly speaking, our model reproduces the dichotomy between hot oblique stars above the Kraft break, and cool aligned stars below.Obliquities of hot stars hardly change in our theory from their assumed initially isotropic distribution.Obliquities of cool stars are damped, all the way to zero if they start from  0 ≲ 120 • .Our model hot Jupiters have final semi-major axes  ≳ 2.2 ★ and thus avoid tidal disruption (e.g.Guillochon et al. 2011) and stellar engulfment (e.g.Barker & Ogilvie 2009;Winn et al. 2010;Dawson 2014).to become prograde but do not reach zero -see the cloud of open symbols from our model at  eff ≲ 5100 K, and contrast with the filled symbols at low obliquity from observations.Other shortcomings of our population synthesis include our neglect of initially non-zero eccentricities (which would lengthen obliquity damping times and worsen the discrepancy between theory and observation for cool stars) and our neglect of a distribution of initial semi-major axes.

Disabling Resonance Locks: 𝜓 unlock vs. 𝑎
As stellar obliquity decreases, tidal forcing of zonal gmodes weakens (see, e.g., the bottom left panel of Fig. 4).There are obliquity boundaries  unlock beyond which resonant locks fail.These boundaries depend on how tidal disturbances damp.
How tidally-driven modes damp is uncertain.We attempt two calculations, one using a "linear" damping rate for how waves lose energy to radiative diffusion when wave amplitudes are small, and another based on "non-linear" interactions that siphon energy away from tidally-driven modes to other stellar modes.Details are contained in appendix B. Results for  unlock , computed for a 4-Gyr-old solar-mass star, are presented in Figure 11.According to the linear theory (green bar), resonance locking can fully damp obliquities for / ★ < 4, but not for / ★ > 5. Non-linear damping can extend the reach of resonance locks out to / ★ ≈ 8-10 (purple curves).
Interestingly, as seen in Fig. 11, most measured projected obliquities  of cool stars appear small at semi-major axes as far out as / ★ ∼ 100 (see also Morgan et al. 2024).Our interpretation is that for stars beyond the resonance locking limit (/ ★ > 8 according to the non-linear damping theory), small obliquities are primordial: Jupiter-mass planets accreted from disks aligned with stellar equatorial planes.2024), for hot Jupiters ( p ≥ 0.3 M J , / ★ ≤ 10) above and below the Kraft break ( eff ≤ 6100 K in blue,  eff > 6100 K in red).In the model, obliquities are initially random (uniformly distributed in cos  0 between −1 and 1), and subsequently damp by resonance locking over Gyr timescales.Damping is seen to be negligible for most hot stars (excepting those with  0 ≲ 25 • which damp to zero), but significant for cool stars.Although roughly reproducing the break seen in the observations, the model fails to damp sufficiently cool star obliquities that start strongly retrograde; these evolve into the cloud of blue open symbols with obliquities between 10 • and 70 • sitting above the observations.This interpretation may also apply to the mostly small obliquities observed for hot stars with exo-Jupiters at large / ★ ∼ 20 (with large-obliquity outliers tending to be in stellar binaries; Rice et al. 2022b).By contrast, at / ★ < 8, hot stars are observed to have large obliquities, presumably the result of whatever process created hot Jupiters (e.g.planet-planet scatterings / high-eccentricity migration).Hot star obliquities hardly damp from resonance locking because the g-mode frequencies of hot stars, which lack radiative cores, change by ≲ 10% over most of their main sequence lives (sections 3 and 4).If hot Jupiters form around cool stars the same way they do around hot stars, they would be similarly initially misaligned.We have shown that cool star misalignments can be subsequently erased in resonance lock, as their radiative cores become more strongly stratified and g-mode frequencies increase.
We reiterate that the model curves for  unlock in Fig. 11 are specific to a 1 M ⊙ star, and uncertain as they depend on how tidally-excited modes damp.The damping mechanism is not merely a technical detail; its nature determines whether resonance locking is possible at all.Ma & Fuller (2021) did not expect hot Jupiters to resonantly lock to their host stars because giant planet perturbations may be sufficiently strong to trigger a parametric instability that spawns 'child' modes.When child modes take the form of standing waves and are fully excited, their energy dissipation rates are insensitive to distance from resonance (Ω −   ), and resonance locking is defeated (Essick & Weinberg 2016).In appendix B we review this argument and rebut it with the possibility that child modes may be so strong as to overturn, becoming traveling waves.Traveling waves dissipate qualitatively differently from standing waves, and we offer some exploratory calculations in the appendix showing how the resonant response of the star may be restored.The purple 'non-linear' curve for  nl unlock in Fig. 11 is computed under this tentative traveling-child picture.

Stellar Ages and Resonance Locking Duration
Our calculations of resonance locking start at  0 = 1 Gyr, when our higher-mass stars have only ∼1-3 Gyr left before they ascend the subgiant branch.We have checked that the calculations shown Figs.7-9 are robust to changes in  0 down to 0.1 Gyr.Starting times of 0.1-1 Gyr appear consistent with high-eccentricity migration, which takes of order such times to form hot Jupiters on initially misaligned orbits (e.g.Dawson & Johnson 2018;Wu 2018;Vick et al. 2019).Similarly long gestation times are indicated by the observation that high-mass, misaligned hot Jupiter hosts have higher Galactic velocity dispersions and are therefore older than aligned hosts (Hamer & Schlaufman 2022).
Late formation of misaligned hot Jupiters bypasses resonance locks that occur during the pre-main sequence (Zanazzi Locking to m = 0 not possible Charting the limits of resonance locking.Computed curves are for an axisymmetric ( = 0) g-mode of a 1 M ⊙ , 4-Gyr-old mainsequence star (see appendix B.2).The green bar ( lin unlock ) marks a transition region dividing the space where resonance locking can operate (to the left) from where it cannot (to the right), computed under the assumption that tidal disturbances are linear and damp by radiative diffusion.In purple are plotted the analogous boundaries for  nl unlock , computed assuming tidal disturbances are limited in amplitude by non-linear mode-mode couplings, which spawn traveling waves that break and dissipate (see appendix B).The non-linear model expands the regime where resonance locking is possible, out to / ★ ≈ 8. Overlaid for context are observed projected obliquities  (Albrecht et al. 2022;Rice et al. 2022a,b;Siegel et al. 2023;Espinoza-Retamal et al. 2023;Sedaghati et al. 2023;Hu et al. 2024).See section 5.2 for how we might interpret these data.& Wu 2021).Pre-main sequence locks would predict little difference in tidal evolution with stellar mass, as young stars have similar internal structures (mostly convective) irrespective of mass (e.g.Henyey et al. 1955).

Obliquity vs. Eccentricity
We have found that resonance locking damps eccentricity, more so for lower mass stars.Compared to obliquities, however, eccentricities damp more slowly (Figs. 8 and 9), suggesting that some hot Jupiters should be found at low  and high  -in conflict with the observed absence of such objects around cool stars (Fig. 2).Tidal dissipation in the planet, rather than the star, may be more effective at damping eccentricity (e.g.Rice et al. 2022a).

Obliquity vs. Planet Mass
We have focused in this work on Jupiter-mass planets as these have some of the most reliable obliquity measurements.Lower planet masses decrease obliquity damping rates (approximately linearly through the term  orb ∝  p in equation 11).Accordingly, we would not expect stars hosting hot sub-Neptunes to exhibit the same obliquity trends shown by stars hosting hot Jupiters.Observations appear to support this expectation (e.g.Hébrard et al. 2011;Albrecht et al. 2022), although Louden et al. (2021) found from rotational broadening measurements that cool host stars of sub-Neptunes may be more aligned than hot host stars.

Stellar Rotation
We draw stellar rotation rates from observations (e.g.Albrecht et al. 2021, Fig. 3): 20-day rotation periods for lowmass stars, and 5-day rotation periods for high-mass stars.The lower angular momenta of low-mass stars increases their rate of obliquity damping -see the term  orb / ★ in eq.14 -abetting the effects of core hydrogen burning (smaller  ev ).As stars spin down from magnetic braking, they should align faster.It may be possible to test, for a given stellar mass, whether slower rotating hosts are more aligned.

Non-Axisymmetric Modes
We have focussed in this work on the  = 0 axisymmetric stellar g-mode and how it damps obliquity.In appendix A we show that obliquity damps for a g-mode for all || ≤ ℓ, when forced into resonance lock by the {, ℓ} = {2, 2} component of the tidal potential of a circular planet ( = 0) with  orb >  ★ .The obliquity damps as (eq.A42) The parameter 0 <   ≤ 3/2 for typical hot Jupiter systems (eq.A37); if the planet mass is too large ( p ≳ few M J ), then   ≤ 0 and resonance locking is not possible.For  = 0, the torque coefficient  0 = 2/sin , while for  = ±1 and  = ±2, These torque coefficients are always positive, and for  orb >  ★ , they lead to the first term in ( 21) dominating the second term, giving d/d < 0 for all .
Calculating the detailed time history of  for  ≠ 0 is left for future work.Here we offer some comments and speculation.The likelihood of locking onto a mode of given  should be proportional to the amplitude of the tidal potential | 2 22 | (eq.A6).Whereas polar planets ( ≈ /2) are favored to lock onto zonal g-modes ( = 0), prograde obliquities ( < /2) will likely lock onto prograde modes ( > 0), and retrograde obliquities ( > /2) onto retrograde modes ( < 0).Because prograde locks have weaker alignment torques ( 1 ,  2 <  0 when  < /2) and spin up the host star (  ★ ∝  > 0, eq.A40), they will damp prograde obliquities more slowly than zonal locks.Sectoral ( = 2) locks may help explain the rapid rotation rates of hot Jupiter hosts (Penev et al. 2018;Ma & Fuller 2021), but also damp semi-major axes and obliquities at comparable rates (similar to equilibrium tides), thus raising the possibility of engulfment before alignment.Relative to zonal locks, retrograde locks may damp initially retrograde obliquities more slowly, but once the obliquity swings to prograde, damping may be faster because alignment torques are stronger ( −2 ,  −1 >  0 when  < /2), and because retrograde locks spin down the star (  ★ < 0).

SUMMARY
A planet can tidally force a stellar gravity mode (g mode), with consequences for the planet's orbit and the star's spin.Structural changes inside the star over the course of its evolution can change the companion's orbit so as to maintain a commensurability between the orbital frequency and the g-mode oscillation frequency.Such resonance locking can alter orbital semi-major axes and eccentricities, and stellar rotation rates (e.g.Zanazzi & Wu 2021;Ma & Fuller 2021).The direction of the stellar spin axis relative to the planet's orbit normal can also change in resonance lock, as we have established in this paper.
Stars that host hot Jupiters are known to have spin axes misaligned from their orbit normals.Large stellar obliquities are thought to be a relic of the gravitational scatterings and long-term dynamical perturbations that originally delivered hot Jupiters onto their close-in orbits (e.g.Dawson & Johnson 2018).Primordial spin-orbit misalignments appear to have been preserved for host stars with effective temperatures  eff ≳ 6100 K, but may have damped away for cooler stars with near-circular planets (Winn et al. 2010;Rice et al. 2022a).We have shown how obliquity damping goes hand-in-hand with semi-major axis and eccentricity damping when star and planet are resonantly locked.While on the main sequence, cooler stars experience much more damping than hotter stars.The dependence on  eff arises because stellar g-modes are sustained only in stably stratified radiative zones, and cool stars have radiative cores whose g-mode (Brunt-Väisälä) frequencies increase substantially from core hydrogen burning, thereby driving significant tidal evolution.Hotter stars lack such cores.Thus resonance locking can explain how cooler stars are aligned with hot Jupiters while hotter stars are not.
There are many areas where our work can be improved.As a simplification, we assumed throughout most of our study that the planet locks to an axisymmetric (zonal) gravity mode.The obliquity damps whether the planet locks to an axisymmetric or non-axisymmetric mode, but the efficiency of damping will vary with the azimuthal wavenumber , possibly significantly.More importantly, our work rests on an uncertain foundation.Resonance locking is not possible if tidallyexcited stellar oscillations decay into standing-wave 'child' modes whose energy dissipation does not depend on proximity to resonance (Essick & Weinberg 2016;Ma & Fuller 2021).We have proposed a way out, whereby child standing waves break to become traveling waves which damp qualitatively differently (appendix B).Our preliminary calculations in this regard appear promising but need further testing and development, probably from hydrodynamic simulations.

A. ENERGY AND ANGULAR MOMENTUM TRANSFER FROM AN INCLINED, ECCENTRIC PLANET TO A SPINNING STAR
This appendix considers the general case of an eccentric planet inclined to the star's equatorial plane, and calculates the energy and angular momentum exchange rate between the orbit of the planet, and a general stellar oscillation.We also calculate the orbital evolution which results from this angular momentum transfer.

A.1. Tidal Potential
In a spherical coordinate system (,  orb ,  orb ) centered on the star, with the  orb -axis parallel to the planet's orbit normal and the  orb -axis pointing towards pericenter, the tidal potential is given by (e.g.Zanazzi & Wu 2021) Here  is the orbital harmonic, ℓ is the angular degree,  orb is the azimuthal number of the tidal potential in a coordinate system aligned with the planet's orbital plane,  ℓ orb is given by equation ( 24) of Press & Teukolsky (1977), and are Hansen coefficients, with  the true anomaly.Notice the eccentricity  and exponential e differ typographically.For ℓ = 2, the non-zero  ℓ orb values are  2,±2 = √︁ 3/10 and  20 = − √︁ /5.For  ≪ 1 (Weinberg et al. 2012) Figure 12 plots the coefficients | 2 2, orb | relevant for our obliquity integrations.For  ≲ 0.4, the  orb = 2 Hansen coefficient plotted in Figure 4 has the largest magnitude.At larger , the  orb = 0 Hansen coefficient dominates.
We transform ( orb ,  orb ,  orb ) to a coordinate system with the -axis parallel to the stellar spin axis ĵ★ , and the -axis parallel to ĵorb × ĵ★ (the orbit's node), co-rotating with the stellar spin.After rotating  ℓ orb ( orb ,  orb ) by the Euler angles in the rotating frame of the star, which transforms to in the inertial frame, where Notice that when   ≈   , a single oscillation  forced by the tidal component {, ℓ, } has an amplitude much larger than other oscillations.For this reason, we will consider a single oscillation  forced by a fixed component of the tidal potential {, ℓ, }, and drop subscripts {, , ℓ} on all quantities for the remainder of appendix A (but keep {,  orb }).Also, we will work solely in the inertial frame, and drop the subscript "inert" on equation (A15).

A.3. Energy and Angular Momentum Transfer
Equation (A5) shows each tidal component {, ℓ, } is forced by multiple orbital azimuthal numbers  orb , each of which differs in phase by  orb   .Because we expect short-range forces to cause   to precess over a timescale much shorter than ∼1 Gyr (e.g.Liu et al. 2015), we will average over the planet's argument of pericenter, on top of azimuthal and time averages.Every orbital azimuthal number  orb from an eccentric planet's tidal potential, forcing a g-mode with azimuthal number , is included in our calculation.

A.3.1. Energy Transfer
The energy transferred from the planet's orbit to the star is (e.g.Lai 2012): Here, ℜ( ) denotes the real part of , and we have added  to its complex conjugate (e.g.Schenk et al. 2001;Burkart et al. 2014).This latter procedure is for bookkeeping the complex addends in expansion (A10), since we focus on modes with eigenfrequencies   > 0. Inserting equations (A5) and (A15) after averaging over time, Also averaging over   , this reduces to A  The torque in the -direction is (e.g.Lai 2012) where is the -component of the angular momentum operator.Inserting equation (A5) and averaging over one orbital period, we find Also averaging over   , this reduces to where   = 2/  is the oscillation period of the parent.Tidal dissipation saps the parent's energy at the rate where ℑ( ) denotes the imaginary part of  (Schenk et al. 2001;Weinberg et al. 2012;Burkart et al. 2014).
When the child modes are absent or only weakly excited (|  |, |  ′ | ≪ |  ℓ orb  ℓ /(2  ′ )| 1/2 ),  tide is dominated by 'linear' dissipation from the parent: where Δ  =   −   is the parent's de-tuning frequency.Equation (B47) describes the Lorentzian response of the parent which makes resonance locking possible.Parametric instability occurs when the parent's energy exceeds a certain threshold where Δ  ′ =   +   ′ −   is the child modes' de-tuning frequency, and  sw  ,  sw  ′ are the child standing-wave damping rates (Wu & Goldreich 2001;Weinberg et al. 2012;Essick & Weinberg 2016).Parametrically unstable child modes can grow in amplitude and sap energy from the parent via non-linear interactions.If  lin  ≫  thr,sw  , the child mode energies   and  ′  reach respectively (Weinberg et al. 2012), while the parent energy is limited to   ≈  thr,sw  , which does not depend on Δ  .To demonstrate how parametric instability and the spawning of child modes can suppress the resonant response of the parent, we solve equations (B44) using the following model.We consider tidal forcing from the {, , ,  orb } = {2, 2, 0, 2} component of  of a Jupiter-mass planet on a circular and polar orbit with period  orb .The parent mode oscillation period is taken to be   = 1.5 days, and the child oscillations are assumed to be 'twins,' with nearly equal frequencies in perfect parametric resonance with the parent (  =   ′ =   /2,   =   ′ , Δ  ′ = 0).Because twin standing waves minimize  thr,sw  , they are the child modes most likely to be parametrically excited (Wu & Goldreich 2001;Essick & Weinberg 2016).The stellar oscillation code GYRE is used to calculate the parent modes of a 4-Gyr-old, solar-mass star modeled with MESA, with parent radiative damping rates   evaluated using the GYRE inlists in Fuller (2017).The damping rates of standing child modes are approximately  sw  ,  sw  ′ ≈ 4  (for angular degrees ℓ  , ℓ  ′ = 2; Weinberg et al. 2012).Numerical solutions to equations (B44) are shown in Figure 13, using odeint from scipy with rtol = 10 −12 , atol = 10 −16 (also getting rid of fast oscillations through a variable transform, see e.g.Essick & Weinberg 2016, for details), whose left and middle panels illustrate how child mode energies eventually dominate parent mode energies, and how both energies are independent of the frequency detuning.Resonance locking would appear to be impossible when standing-wave child modes are fully excited.
What is neglected in the above discussion is the possibility that a mode amplitude can become large enough to render the background medium, which is ordinarily stably stratified, convectively unstable.When this occurs, the mode is said to 'overturn' or 'break', and the oscillation reduces to a single travelling wave, rather than a standing wave (= oppositely directed travelling waves trapped in a resonant cavity  .Parametric instability assuming the child modes behave as standing waves that do not break.We integrate equations (B44) using the  2 202 component of the gravitational potential of a Jupiter-mass planet on a circular ( = 0), polar ( = 90 • ) orbit of semi-major axis .The parent stellar g mode, of oscillation period   = 1.5 d, is computed from GYRE using a 4-Gyr-old, solar-mass star of radius  ★ modeled with MESA.Equations (B44) are integrated to a time  end = 1.1 × 10 10  −1  , with   (0) set by eq. ( A16), and   (0) = 10 −6   (0).Left: Energies of the parent mode and one of the twin child modes vs. time, for frequency de-tuning Δ  = 3.3 × 10 −4   .Parent and child energies evolve to their equilibrium standing-wave energies  thr,sw  (eq.B48) and  sw  (eq.B49), respectively.Middle: Parent and child energies (crosses) at times  >  end /2, vs. forcing frequency.Both parent and child energies are independent of Δ  , in contrast to the Lorentzian resonant response of the parent mode neglecting child modes ( lin  , eq.B47).Right: The child mode energy  sw  exceeds the child breaking energy  brk  (eq.B51) for values of / ★ relevant to hot Jupiter systems.Breaking of standing child modes might help restore the resonant response of the parent mode-see Figs. 14 and  Travelling wave dissipation Figure 14.Same as Fig. 13, except we replace   with  −1 gr, (eq.B52) once   >  brk  and child modes become traveling waves.Left: Energies of the parent (solid green line) and child (solid purple line) vs. time for Δ  = 3.3 × 10 −4   .For these parameters, the parent's energy stays below the threshold  thr,tw  (eq.B53) for maintaining a traveling child wave.Thus the child mode disappears as soon as it breaks, and the parent energy remains close to  lin  .Middle: Same as left but for Δ  = 4.7 × 10 −5   (closer to exact resonance).Here the parent energy exceeds  thr,tw  and can sustain a traveling child mode.The parent and child energies asymptote to  thr,tw  and  tw  (eq.B54), respectively.Right: Equilibrium parent and child energies (evaluated at  >  end /2) vs. forcing frequency.Exact resonance is located at  orb /  = 2. Away from exact resonance, child energies are too low to be plotted, and leave parent energies near  lin  .Closer to exact resonance, child energies grow to  tw  , causing the parent's energy to flat-line at  thr,tw  , and preventing the parent from breaking.
Because   ,   ′ <   , the child breaking energy is always smaller than that of the parent.In the right panel of Figure 13, we see that for our standing-wave twin-child model,  sw  >  brk  and  sw  ′ >  brk  ′ (albeit only by factors of several), for the entire range of / ★ tested.The breaking of standing child waves opens the possibility of restoring the resonant response of the parent.
Once broken, child modes become traveling waves, and may no longer be able to sap the parent of energy as effectively as standing waves do.Travelling waves rapidly deplete their energy over their group crossing time  gr ≃   (B52) (e.g.Zanazzi & Wu 2021;Ma & Fuller 2021), where  denotes the number of radial nodes.The parent-child coupling coefficient   ′ is largely the same whether the child mode is standing or travelling (Weinberg et al. 2012).However, to parametrically excite a travelling child, the parent's energy must now exceed a much higher threshold  thr,tw ★ (B53) (Wu & Goldreich 2001), where the child travelling wave damping rate is  tw  ,  tw  ′ ≈  −1 gr, ,  −1 gr, ′ ≈ (4 gr,  ) −1 (for ℓ  , ℓ  ′ = 2).The number of parent radial nodes can be calculated from GYRE.
Figure 14 presents new solutions to equations (B44), now taking   =  tw  when   >  brk  .For  orb /  = 2.00066, we have  thr,tw  >   >  thr,sw  : standing child modes are parametrically excited but die upon breaking (Fig. 14, left panel).Closer to exact resonance ( orb /  = 2.00009), we have   >  thr,tw  : now each child traveling mode grows to an energy while the parent depletes in energy to  thr,tw  (Fig. 14, middle panel).The resonant response of the parent mode is restored to an extent (Fig. 14, right panel); moreover, while the child mode breaks, the parent mode does not (although the margin of safety is only a factor of a few between  thr,tw  and  brk  ).Comparing  thr,tw  to  brk  over a broad range of / ★ , parent modes might excite travelling children, without the parents breaking themselves (Fig. 15, right panel; with a factor of a few between  thr,tw  and  brk  ).In the next subsection B.2, we will continue to study this 'traveling child' scenario and its implications for resonance locking.Essick & Weinberg (2016) argued that child mode breaking can be avoided by accounting for 'grandchild' modes, since child modes are themselves parametrically unstable.We question this argument.From their Fig. 7, the typical energies of grandchild modes ( grandchild ∼ 10 −11  ★ for / ★ ≈ 4, and  grandchild ∼ 10 −15  ★ for / ★ ≈ 9) appear ∼10 2 times larger than their respective breaking energies (estimated using  brk  from the right panel of our Fig.13, and the scaling  brk grandchild ∼  brk  /2 6 from eq.B51).Thus grandchild modes would seem even more prone to breaking than child modes.

B.2. Resonant Dissipation from Travelling Wave Child Modes
For a hot Jupiter to resonantly lock onto a stellar mode, the tidal dissipation time  diss = | orb |/|  tide | must be shorter than the stellar evolution time 3 ev /2.Following appendix B.1, when traveling wave child modes are excited, they dominate the dissipation rate near exact resonance at  orb = 2  (Fig. 15, middle panel, solid circles near figure center).Though the parent energy exceeds that of the child modes (Fig. 14, right panel), the timescale  gr, over which child traveling waves dissipate is much shorter than the dissipation time 1/  for the parent; consequently, the energy dissipation rate from the two child modes exceeds that from the parent (i.e., 2 × 2 −1 gr,  tw  > 2   thr,tw

𝛼
).Thus the tidal dissipation time (superscript "nl" for non-linear, following the main text) is given by Comparing  nl diss (evaluated for a hot Jupiter on a polar orbit  = 90 • ) to 3 ev /2 in the right panel of Fig. 15, we see that resonance locks are possible for semi-major axes inside (/ ★ ) crit ∼ 8 − 10.
Travelling wave child energies Figure1.Projected stellar obliquities  (as distinct from deprojected or true 3D obliquities ) vs. host star effective temperature  eff of hot Jupiter systems (selecting for planet masses  p ≥ 0.3 M J and semimajor axes  ≤ 10 ★ , where  ★ is the host star radius; data fromAlbrecht et al. 2022; Rice et al. 2022a,b;Siegel et al. 2023;Espinoza-Retamal et al. 2023;Sedaghati et al. 2023;Hu et al. 2024).Obliquities are larger for hot high-mass stars (stellar effective temperature  eff > 6100 K, red points) than for cool low-mass stars (blue points).This correlation(Winn et al. 2010) is statistically significant according to a two-sided Kolmogorov-Smirnov (KS) test, which yields a  = 1.6 × 10 −3 probability that the cumulative obliquity distributions for cool and hot stars are drawn from the same underlying distribution (right panel).

Figure 3 .
Figure 3. Coordinate system used to analyze the coupling between stellar spin and planet orbit.The -axis is chosen to always point in the direction of the stellar spin ĵ★ , and the -axis is chosen to always point normal to ĵ★ and ĵorb .The obliquity  is the angle between ĵ★ and ĵorb .In this paper we consider only the components of the tidal torque  which act to change , i.e.only the  and -components.

Figure 4 .
Figure 4.An axisymmetric  = 0 oscillation on the host star (top left) can be excited by the dominant  =  orb = 2 component of the planet's tidal potential (top right) when  > 0 (bottom right showing that the radial tidal force integrated over the stellar equator is net outward).When  = 0, tidal forces cancel over stellar azimuth and the  = 0 mode cannot be excited (bottom left).

Figure 5 .
Figure5.How a planet can tidally torque the host star into spin-orbit alignment, by way of an axisymmetric  = 0 g-mode oscillation forced by the planet's  =  orb = 2 potential.The star stretches and compresses along its spin axis (-axis) twice for every orbit of the planet.The solid purple sphere marks the planet's location which is displaced ahead of the star's tidal bulge due to tidal friction, while the see-through purple sphere displays the planet's location in the absence of tidal friction.The top row shows different views of the same instant in time, when the star is maximally elongated but the planet is ahead at positive .The bottom row shows a different time, when the star is maximally compressed and similarly lagged in phase relative to the planet.The tidal bulge at position vector  feels a force from the planet  and by extension a torque  =  ×  that is positive in the -direction no matter the orbital phase (middle column).The torque   > 0 acts to bring the stellar spin direction ( ĵ★ ) toward the -axis, closer to the orbital angular momentum direction ( ĵorb ).Once spin and orbit are aligned, the torque vanishes (Fig.4, bottom left panel).Because the oscillation is symmetric about the -axis (right column), there is no torque in the -direction.

Figure 6 .
Figure 6.Rate of Brunt-Väisälä frequency evolution (top panel, eq.16) and accumulated frequency change (middle panel, eq.18) vs. main-sequence age, for stars of different masses as indicated.Although high-mass stars (red curves) increase their frequencies dramatically quickly toward the end of their hydrogen-burning lives (as their cores switch from being unstably to stably stratified), the speed-up phase is short-lived and consequently unlikely to be observed.Sampled over the entirety of their main-sequence lives, Δ/⟨ 0 ⟩ is likely to be higher for low-mass stars, whose cores remain stably stratified throughout, than for high-mass stars (bottom panel showing median and ±1 intervals, calculated by evaluating Δ/ 0 at 1000 uniformly spaced times over the interval [ 0 ,  max ]).
Figure 10 compares our modeled projected obliquities  with those observed (top panel; observations fromRice et al. 2022a).For completeness, we also show our modeled obliquities  against the few deprojected obliquities that are available (bottom panel; data fromAlbrecht et al. 2022;Siegel et al. 2023).Broadly speaking, our model reproduces the dichotomy between hot oblique stars above the Kraft break, and cool aligned stars below.Obliquities of hot stars hardly change in our theory from their assumed initially isotropic distribution.Obliquities of cool stars are damped, all the way to zero if they start from  0 ≲ 120 • .Our model hot Jupiters have final semi-major axes  ≳ 2.2 ★ and thus avoid tidal disruption (e.g.Guillochon et al. 2011) and stellar engulfment (e.g.Barker & Ogilvie 2009;Winn et al. 2010;Dawson 2014).Fig.10alsoshows that our theory may over-predict the obliquities of the coolest (K-type) stars.If the initial obliquities are retrograde ( 0 ≳ 120 • for 0.6 M ⊙ stars) they evolve Figure 10.A population synthesis based on resonance locking (open symbols) compared against observations (filled symbols).Data for projected obliquities  (top) and obliquities  (bottom) are taken fromAlbrecht et al. (2022),Rice et al. (2022a,b),Siegel et al. (2023), Espinoza-Retamal et al. (2023),Sedaghati et al. (2023), andHu et al. (2024), for hot Jupiters ( p ≥ 0.3 M J , / ★ ≤ 10) above and below the Kraft break ( eff ≤ 6100 K in blue,  eff > 6100 K in red).In the model, obliquities are initially random (uniformly distributed in cos  0 between −1 and 1), and subsequently damp by resonance locking over Gyr timescales.Damping is seen to be negligible for most hot stars (excepting those with  0 ≲ 25 • which damp to zero), but significant for cool stars.Although roughly reproducing the break seen in the observations, the model fails to damp sufficiently cool star obliquities that start strongly retrograde; these evolve into the cloud of blue open symbols with obliquities between 10 • and 70 • sitting above the observations.
Financial support was provided by a 51 Pegasi b Heising-Simons Fellowship awarded to JJZ.JWD was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference #CITA 490888-16].EC acknowledges support from a Simons Investigator award and NSF AST grant 2205500.
Figure13.Parametric instability assuming the child modes behave as standing waves that do not break.We integrate equations (B44) using the  2 202 component of the gravitational potential of a Jupiter-mass planet on a circular ( = 0), polar ( = 90 • ) orbit of semi-major axis .The parent stellar g mode, of oscillation period   = 1.5 d, is computed from GYRE using a 4-Gyr-old, solar-mass star of radius  ★ modeled with MESA.Equations (B44) are integrated to a time  end = 1.1 × 10 10  −1  , with   (0) set by eq.(A16), and   (0) = 10 −6   (0).Left: Energies of the parent mode and one of the twin child modes vs. time, for frequency de-tuning Δ  = 3.3 × 10 −4   .Parent and child energies evolve to their equilibrium standing-wave energies  thr,sw ≪ 1 and  ≫ , Siegel et al. 2023 andDong &Foreman- Mackey 2023).Our equations of motion are then integrated from  0 = 1 Gyr to  max (either 12 Gyr or the main-sequence lifetime, whichever comes first), and the final values for  extracted for comparison with observations.Since the observations typically trace stellar effective temperature  eff instead of stellar mass, we map our set of 6  ★ values to the tan  ≃ tan  sin  obs , Same as Figs.7 and 8 except for initial conditions { 0 ,  0 } = {0.6,0.07 AU}.At these larger eccentricities, damping of , , and  takes longer, but the equilibrium fixed point of ,  = 0 can still be reached within main-sequence lifetimes for cool stars whose obliquities are not too high.Hot star obliquities remain near their initial values regardless.