The gravitational interaction between planets on inclined orbits and protoplanetary disks as the origin of primordial spin--orbit misalignments
aa r X i v : . [ a s t r o - ph . E P ] D ec Draft version November 8, 2018
Preprint typeset using L A TEX style emulateapj v. 01/23/15
THE GRAVITATIONAL INTERACTION BETWEEN PLANETS ON INCLINED ORBITS ANDPROTOPLANETARY DISKS AS THE ORIGIN OF PRIMORDIAL SPIN–ORBIT MISALIGNMENTS
Titos Matsakos and Arieh K¨onigl
Department of Astronomy & Astrophysics and The Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA
Draft version November 8, 2018
ABSTRACTMany of the observed spin–orbit alignment properties of exoplanets can be explained in the contextof the primordial disk misalignment model, in which an initially aligned protoplanetary disk is torquedby a distant stellar companion on a misaligned orbit, resulting in a precessional motion that can leadto large-amplitude oscillations of the spin–orbit angle. We consider a variant of this model in whichthe companion is a giant planet with an orbital radius of a few au. Guided by the results of publishednumerical simulations, we model the dynamical evolution of this system by dividing the disk intoinner and outer parts—separated at the location of the planet—that behave as distinct, rigid disks.We show that the planet misaligns the inner disk even as the orientation of the outer disk remainsunchanged. In addition to the oscillations induced by the precessional motion, whose amplitude islarger the smaller the initial inner-disk-to-planet mass ratio, the spin–orbit angle also exhibits a seculargrowth in this case—driven by ongoing mass depletion from the disk—that becomes significant whenthe inner disk’s angular momentum drops below that of the planet. Altogether, these two effects canproduce significant misalignment angles for the inner disk, including retrograde configurations. Wediscuss these results within the framework of the Stranded Hot Jupiter scenario and consider theirimplications, including to the interpretation of the alignment properties of debris disks.
Subject headings: planet–disk interactions — planets and satellites: dynamical evolution and stability— protoplanetary disks — circumstellar matter INTRODUCTIONA major open question in the study of exoplanets is theorigin of their apparent obliquity properties—the distri-bution of the angle λ between the stellar spin and theplanet’s orbital angular momentum vectors as projectedon the sky (see, e.g., the review by Winn & Fabrycky2015). Measurements of the Rossiter–McLaughlin effectin hot Jupiters (HJs, defined here as planets with masses M p & . M J that have orbital periods P orb .
10 days)have indicated that λ spans the entire range from 0 ◦ to 180 ◦ , in stark contrast with the situation in the so-lar system (where the angle between the planets’ totalangular momentum vector and that of the Sun is only ∼ ◦ ). In addition, there is a marked difference in thedistribution of λ between G stars, where ∼ / λ < ◦ ) and the rest are spreadout roughly uniformly over the remainder of the λ range,and F stars of effective temperature T eff & M p < M J have apparent retrograde orbits( λ > ◦ ).Various explanations have been proposed to accountfor the broad range of observed obliquities, but the in-ferred dependences on T eff and M p provide strong con-straints on a viable model. In one scenario (Winn et al.2010; Albrecht et al. 2012), HJs arrive in the vicinity ofthe host star on a misaligned orbit and subsequently actto realign the host through a tidal interaction, which ismore effective in cool stars than in hot ones. In thispicture, HJs form at large radii and either migrate in- ward through their natal disk while maintaining nearlycircular orbits or are placed on a high-eccentricity orbitafter the gaseous disk dissipates—which enables themto approach the center and become tidally trapped bythe star (with their orbits getting circularized by tidalfriction; e.g., Ford & Rasio 2006). The processes thatinitiate high-eccentricity migration (HEM), which canbe either planet–planet scattering (e.g., Chatterjee et al.2008; Juri´c & Tremaine 2008; Beaug´e & Nesvorn´y 2012)or secular interactions that involve a stellar binary com-panion or one or more planetary companions (such asKozai-Lidov oscillations — e.g., Wu & Murray 2003;Fabrycky & Tremaine 2007; Naoz et al. 2011; Petrovich2015b—and secular chaos—e.g., Wu & Lithwick 2011;Lithwick & Wu 2014; Petrovich 2015a; Hamers et al.2017), all give rise to HJs with a distribution of mis-aligned orbits. In the case of classical disk migration,the observed obliquities can be attributed to a primor-dial misalignment of the natal disk that occurred dur-ing its initial assembly from a turbulent interstellar gas(e.g., Bate et al. 2010; Fielding et al. 2015) or as a re-sult of magnetic and/or gravitational torques induced,respectively, by a tilted stellar dipolar field and a mis-aligned companion (e.g., Lai et al. 2011; Batygin 2012;Batygin & Adams 2013; Lai 2014; Spalding & Batygin2014).The tidal realignment hypothesis that underlies theabove modeling framework was challenged by the re-sults of Mazeh et al. (2015), who examined the rotational The possibility of HJs forming at their observed locationshas also been considered in the literature (e.g., Boley et al. 2016;Batygin et al. 2016), but the likelihood of this scenario is still beingdebated.
Matsakos & K¨oniglphotometric modulations of a large number of
Kepler sources. Their analysis indicated that the common oc-currence of aligned systems around cool stars character-izes the general population of planets and not just HJs,and, moreover, that this property extends to orbital pe-riods as long as ∼
50 days, about an order of magnitudelarger than the maximum value of P orb for which tidal in-teraction with the star remains important. To reconcilethis finding with the above scenario, Matsakos & K¨onigl(2015) appealed to the results of planet formation andevolution models, which predict that giant planets formefficiently in protoplanetary disks and that most of themmigrate rapidly to the disk’s inner edge, where, if the ar-riving planet’s mass is not too high ( . M J ), it could re-main stranded near that radius for up to ∼ ∼
50% of planetary systems harboran SHJ with a typical mass of ∼ . M J . In this picture,the obliquity properties of currently observed HJs—andthe fact that they are consistent with those of lower-massand more distant planets—are most naturally explainedif most of the planets in a given system—including anySHJ that may have been present—are formed in, and mi-grate along the plane of, a primordially misaligned disk. This interpretation is compatible with the properties ofsystems like Kepler-56, in which two close-in planets have λ ≈ ◦ and yet are nearly coplanar (Huber et al. 2013),and 55 Cnc, a coplanar five-planet system with λ ≈ ◦ (e.g., Kaib et al. 2011; Bourrier & H´ebrard 2014). It isalso consistent with the apparent lack of a correlationbetween the obliquity properties of observed HJs andthe presence of a massive companion (e.g., Knutson et al.2014; Ngo et al. 2015; Piskorz et al. 2015).In this paper we explore a variant of the primordial diskmisalignment model first proposed by Batygin (2012),in which, instead of the tilting of the entire disk by adistant ( ∼
500 au) stellar companion on an inclined or-bit, we consider the gravitational torque exerted by amuch closer ( ∼ planetary companion on such an or-bit, which acts to misalign only the inner region of theprotoplanetary disk. This model is motivated by the in-ferences from radial velocity surveys and adaptive-opticsimaging data (Bryan et al. 2016; see also Knutson et al.2014) that ∼
70% of planetary systems harboring a tran-siting HJ have a companion with mass in the range 1– This explanation does not necessarily imply that all planetsthat reached the vicinity of the host star must have moved in byclassical migration, although SHJs evidently arrived in this way. Infact, Matsakos & K¨onigl (2016) inferred that most of the planetsthat delineate the boundary of the so-called sub-Jovian desert inthe orbital-period–planet-mass plane got in by a secular HEM pro-cess (one that, however, did not give rise to high orbital inclinationsrelative to the natal disk plane). The two-planet system KOI-89 (Ahlers et al. 2015) may beyet another example. M J and semimajor axis in the range 1–20 au, and that ∼
50% of systems harboring one or two planets detectedby the radial velocity method have a companion withmass in the range 1–20 M J and semimajor axis in therange 5–20 au. Further motivation is provided by thework of Li & Winn (2016), who re-examined the photo-metric data analyzed by Mazeh et al. (2015) and foundindications that the good-alignment property of planetsaround cool stars does not hold for large orbital peri-ods, with the obliquities of planets with P orb & daysappearing to tend toward a random distribution.One possible origin for a giant planet on an in-clined orbit with a semimajor axis a of a few auis planet–planet scattering in the natal disk. Cur-rent theories suggest that giant planets may form intightly packed configurations that can become dynam-ically unstable and undergo orbit crossing (see, e.g.,Davies et al. 2014 for a review). The instabilities startto develop before the gaseous disk component dissi-pates (e.g., Matsumura et al. 2010; Marzari et al. 2010),and it has been argued (Chatterjee et al. 2008) that theplanet–planet scattering process may, in fact, peak be-fore the disk is fully depleted of gas (see also Lega et al.2013). A close encounter between two giant planets islikely to result in a collision if the ratio ( M p /M ∗ )( a/R p )(the Safronov number) is < M ∗ is the stel-lar mass and R p is the planet’s radius), and in a scat-tering if this ratio is > ∼ –10 yr, which is compa-rable to the estimated formation time of Jupiter-massplanets at & M p & M J is placed on a high-inclination or-bit at a time t & Fig. 1.—
Schematic representation (not to scale) of the initialconfiguration of our model. See text for details. by the smooth-particle-hydrodynamics simulations car-ried out by Xiang-Gruess & Papaloizou (2013). Theyconsidered the interaction between a massive (1–6 M J )planet that is placed on an inclined, circular orbit of ra-dius 5 au and a low-mass (0 . M ∗ ) protoplanetary diskthat extends to 25 au. A key finding of these simulationswas that the disk develops a warped structure, with theregions interior and exterior to the planet’s radial loca-tion behaving as separate, rigid disks with distinct in-clinations; in particular, the inner disk was found to ex-hibit substantial misalignment with respect to its initialdirection when the planet’s mass was large enough andits initial inclination was intermediate between the lim-its of 0 ◦ and 90 ◦ at which no torque is exerted on thedisk. Motivated by these results, we construct an ana-lytic model for the gravitational interaction between theplanet and the two separate parts of the disk. The gen-eral effect of an interaction of this type between a planeton an inclined orbit and a rigid disk is to induce a preces-sion of the planet’s orbit about the total angular momen-tum vector. In contrast with Xiang-Gruess & Papaloizou(2013), whose simulations only extended over a fractionof a precession period, we consider the long-term evolu-tion of such systems. In particular, we use our analyticmodel to study how the ongoing depletion of the disk’smass affects the orbital orientations of the planet and ofthe disk’s two parts. We describe the model in Section 2and present our calculations in Section 3. We discuss theimplications of these results to planet obliquity measure-ments and to the alignment properties of debris disks inSection 4, and summarize in Section 5. MODELING APPROACH2.1.
Assumptions
The initial configuration that we adopt is sketched inFigure 1. We consider a young star (subscript s) that issurrounded by a Keplerian accretion disk, and a Jupiter-mass planet (subscript p) on a circular orbit. The diskconsists of two parts: an inner disk (subscript d) thatextends between an inner radius r d , in and an outer ra-dius r d , out , and an outer disk (subscript h) that extendsbetween r h , in and r h , out ; they are separated by a narrowgap that is centered on the planet’s orbital radius a . Thetwo parts of the disk are initially coplanar, with their nor-mals aligned with the stellar angular momentum vector S , whereas the planet’s orbital angular momentum vec-tor P is initially inclined at an angle ψ p0 with respectto S (where the subscript 0 denotes the time t = t at which the planet is placed on the inclined orbit).We assume that, during the subsequent evolution, eachpart of the disk maintains a flat geometry and precessesas a rigid body. The rigidity approximation is com-monly adopted in this context and is attributed to effi-cient communication across the disk through the propa-gation of bending waves or the action of a viscous stress(e.g., Larwood et al. 1996; see also Lai 2014 and refer-ences therein). Based on the simulation results pre-sented in Xiang-Gruess & Papaloizou (2013), we conjec-ture that this communication is severed at the locationof the planet. This outcome is evidently the result ofthe planet’s opening up a gap in the disk, although itappears that the gap need not be fully evacuated forthis process to be effective. In fact, the most stronglywarped simulated disk configurations correspond to com-paratively high initial inclination angles, for which theplanet spends a relatively small fraction of the orbitaltime inside the disk, resulting in gaps that are less deepand wide than in the fully embedded case. Our calcu-lations indicate that, during the disk’s subsequent evo-lution, its inner and outer parts may actually detach asa result of the precessional oscillation of the inner disk.This oscillation is particularly strong in the case of highlymass-depleted disks on which we focus attention in thispaper: in the example shown in Figure 6 below, the ini-tial amplitude of this oscillation is ∼ ◦ .The planet’s orbital inclination is subject to damp-ing by dynamical friction (Xiang-Gruess & Papaloizou2013), although the damping rate is likely low for thehigh values of ψ p0 that are of particular interest to us(Bitsch et al. 2013). Furthermore, in cases where theprecessional oscillation of the inner disk causes the diskto split at the orbital radius of the planet, one can plau-sibly expect the local gas density to become too low fordynamical friction to continue to play a significant roleon timescales longer than the initial oscillation period( ∼ yr for the example shown in Figure 6). In light ofthese considerations, and in the interest of simplicity, wedo not include the effects of dynamical friction in any ofour presented models.As a further simplification, we assume that the planet’sorbit remains circular. The initial orbital eccentricity ofa planet ejected from the disk by either of the two mech-anisms mentioned in Section 1 may well have a nonneg-ligible eccentricity. However, the simulations performedby Bitsch et al. (2013) indicate that the dynamical fric-tion process damps eccentricities much faster than in-clinations, so that the orbit can potentially be circu-larized on a timescale that is shorter than the preces-sion time (i.e., before the two parts of the disk canbecome fully separated). On the other hand, even ifthe initial eccentricity is zero, it may be pumped upby the planet’s gravitational interaction with the outerdisk if ψ p0 is high enough ( & ◦ ; Teyssandier et al.2013). This is essentially the Kozai-Lidov effect, whereinthe eccentricity undergoes periodic oscillations in an-tiphase with the orbital inclination (Terquem & Ajmia2010). These oscillations were noticed in the numericalsimulations of Xiang-Gruess & Papaloizou (2013) and One should, however, bear in mind that real accretion disksare inherently fluid in nature and therefore cannot strictly obey therigid-body approximation; see, e.g., Rawiraswattana et al. (2016).
Matsakos & K¨oniglBitsch et al. (2013). Their period can be approximatedby τ KL ∼ ( r h , out /r h , in ) (2 π/ | Ω ph | ) (Terquem & Ajmia2010), where we used the expression for the preces-sion frequency Ω ph (Equation (A20)) that correspondsto the torque exerted by the outer disk on the misalignedplanet. For the parameters of the representative mass-depleted disk model shown in Figure 6, τ KL ∼ yr.This time is longer by a factor of ∼ than the initialprecession period of the inner disk in this example, im-plying that the Kozai-Lidov process will have little effecton the high-amplitude oscillations of ψ p . Kozai-Lidov os-cillations might, however, modify the details of the long-term behavior of the inner disk, since τ KL is comparableto the mass-depletion time τ (Equation (10)) that un-derlies the secular evolution of the system.Our model takes into account the tidal interaction ofthe spinning star with the inner and outer disks and withthe planet, which was not considered in the aforemen-tioned simulations. The inclusion of this interaction ismotivated by the finding (Batygin & Adams 2013; Lai2014; Spalding & Batygin 2014) that an evolving proto-planetary disk with a binary companion on an inclinedorbit can experience a resonance between the disk preces-sion frequency (driven by the companion) and the stellarprecession frequency (driven by the disk), and that thisresonance crossing can generate a strong misalignmentbetween the angular momentum vectors of the disk andthe star. As it turns out (see Section 3), in the case thatwe consider—in which the companion is a Jupiter-massplanet with an orbital radius of a few au rather than asolar-mass star at a distance of a few hundred au—thisresonance is not encountered. We also show that, evenin the case of a binary companion, the misalignment ef-fect associated with the resonance crossing is weaker thanthat inferred in the above works when one also takes intoaccount the torque that the star exerts on the inner disk(see Appendix C). 2.2. Equations
We model the dynamics of the system by followingthe temporal evolution of the angular momenta ( S , D , P , and H ) of the four constituents (the star, the innerdisk, the planet, and the outer disk, respectively) dueto their mutual gravitational torques. Given that theorbital period of the planet is much shorter than thecharacteristic precession time scales of the system, weapproximate the planet as a ring of uniform density, witha total mass equal to that of the planet and a radius equalto its semimajor axis.The evolution of the angular momentum L k of an ob-ject k under the influence of a torque T ik exerted by anobject i is given by d L k /dt = T ik . The set of equationsthat describes the temporal evolution of the four angularmomenta is thus d S dt = T ds + T ps + T hs , (1) d D dt = T sd + T pd + T hd , (2) d P dt = T sp + T dp + T hp , (3) d H dt = T sh + T dh + T ph , (4) where T ik = − T ki . The above equations can also beexpressed in terms of the precession frequencies Ω ik : d L k dt = X i T ik = X i Ω ik L i × L k J ik , (5)where J ik = | L i + L k | = ( L i + L k + 2 L i L k cos θ ik ) / andΩ ik = Ω ki . In Appendix A we derive analytic expressionsfor the torques T ik and the corresponding precession fre-quencies Ω ik . 2.3. Numerical Setup
The host is assumed to be a protostar of mass M ∗ = M ⊙ , radius R ∗ = 2 R ⊙ , rotation rate Ω ∗ =0 . GM ∗ /R ∗ ) / , and angular momentum S = k ∗ M ∗ R ∗ Ω ∗ = 1 . × (6) × (cid:18) k ∗ . (cid:19) (cid:18) M ∗ M ⊙ (cid:19) (cid:18) R ∗ R ⊙ (cid:19) Ω ∗ . p GM ⊙ / (2 R ⊙ ) ! erg s , where k ∗ ≃ . n = 1 . M p = M J and R p = R J ,and a fixed semimajor axis, a = 5 au, so that its orbitalangular momentum is P = M p ( GM ∗ a ) / = 1 . × (7) × (cid:18) M p M J (cid:19) (cid:18) M ∗ M ⊙ (cid:19) / (cid:16) a (cid:17) / erg s . We consider two values for the total initial disk mass:(1) M t0 = 0 . M ∗ , corresponding to a comparativelymassive disk, and (2) M t0 = 0 . M ∗ , corresponding toa highly evolved system that has entered the transition-disk phase. In both cases we take the disk surface densityto scale with radius as r − . The inner disk extends from r d , in = 4 R ⊙ to r d , out = a , and initially has 10% of thetotal mass. Its angular momentum is D = 23 M d ( GM ∗ ) / r / , out − r / , in r d , out − r d , in (8) ≃ . × (cid:18) M d . M ⊙ (cid:19) (cid:18) M ∗ M ⊙ (cid:19) / (cid:16) a (cid:17) / erg s . The outer disk has edges at r h , in = a and r h , out = 50 au,and angular momentum H = 23 M h ( GM ∗ ) / r / , out − r / , in r h , out − r h , in (9) ≃ . × (cid:18) M h . M ⊙ (cid:19) (cid:18) M ∗ M ⊙ (cid:19) / (cid:16) r h , out
50 au (cid:17) / erg s . We model mass depletion in the disk using the expres-sion first employed in this context by Batygin & Adams(2013), M t ( t ) = M t ( t = 0)1 + t/τ , (10)where we adopt M t ( t = 0) = 0 . M ⊙ and τ = 0 . (cid:18) d D dt (cid:19) depl = − D τ (1 + t/τ ) ˆ D = − D τ + t . (11)For the outer disk we assume that the presence of theplanet inhibits efficient mass accretion, and we considerthe following limits: (1) the outer disk’s mass remainsconstant, and (2) the outer disk loses mass (e.g., throughphotoevaporation) at the rate given by Equation (10). We assume that any angular momentum lost by the diskis transported out of the system (for example, by a diskwind).We adopt a Cartesian coordinate system ( x, y, z ) asthe “lab” frame of reference (see Figure 1). Initially, theequatorial plane of the star and the planes of the innerand outer disks coincide with the x – y plane (i.e., ψ s0 = ψ d0 = ψ h0 = 0, where ψ k denotes the angle between L k and the z axis), and only the orbital plane of the planethas a finite initial inclination ( ψ p0 ). The x axis is chosento coincide with the initial line of nodes of the planet’sorbital plane.Table 1 presents the models we explore and summarizesthe relevant parameters. Specifically, column 1 containsthe models’ designations (with the letters M and m de-noting, respectively, high and low disk masses at time t = t ), columns 2–5 indicate which system componentsare being considered, columns 6–9 list the disk and planetmasses (with the arrow indicating active mass depletion),and columns 10 and 11 give the planet’s semimajor axisand initial misalignment angle, respectively. The lastlisted model ( binary ) does not correspond to a planetmisaligning the inner disk but rather to a binary startilting the entire disk. This case is considered for com-parison with the corresponding model in Lai (2014). RESULTSThe gravitational interactions among the differentcomponents of the system that we consider (star, in-ner disk, planet, and outer disk) can result in a highlynonlinear behavior. To gain insight into these interac-tions we start by analyzing a much simpler system, oneconsisting only of the inner disk and the (initially mis-aligned) planet. The relevant timescales that character-ize the evolution of this system are the precession period τ dp ≡ π/ Ω dp (Equation (A17)) and the mass depletiontimescale τ = 5 × yr (Equation (10)).Figure 2 shows the evolution of such a system for thecase (model DP-M ) where a Jupiter-mass planet on a mis-aligned orbit ( ψ p0 = 60 ◦ ) torques an inner disk of initialmass M d0 = 0 . M ∗ (corresponding to M t0 = 0 . M ∗ ,i.e., to t = 0 when M ∗ = M ⊙ ; see Equation (10)). The After the inner disk tilts away from the outer disk, the innerrim of the outer disk becomes exposed to the direct stellar radiationfield, which accelerates the evaporation process (Alexander et al.2006). According to current models, disk evaporation is in-duced primarily by X-ray and FUV photons and occurs at a rateof ∼ − –10 − M ⊙ yr − for typical stellar radiation fields (seeGorti et al. 2016 for a review). Even if the actual rate is nearthe lower end of this range, the outer disk in our low- M t0 modelswould be fully depleted of mass on a timescale of ∼
10 Myr; how-ever, a similar outcome for the high- M t0 models would require themass evaporation rate to be near the upper end of the estimatedrange. top left panel exhibits the angles ψ d and ψ p (blue: in-ner disk; red: planet) as a function of time. In this andthe subsequent figures, we show results for a total dura-tion of 10 Myr. This is long enough in comparison with τ to capture the secular evolution of the system, whichis driven by the mass depletion in the inner disk. Tocapture the details of the oscillatory behavior associatedwith the precession of the individual angular momentumvectors ( D and P ) about the total angular momentumvector J dp = D + P (subscript j)—which takes place onthe shorter timescale τ dp ( ≃ × yr at t = t )—wedisplay the initial 0 . D , ˆ P , and ˆ J dp inthe x – y plane during this time interval in the top rightpanel. Given that 0 . ≪ τ , the vectors ˆ D and ˆ P execute a circular motion about ˆ J dp with virtually con-stant inclinations with respect to the latter vector (givenby the angles θ jd and θ jp , respectively), and the orien-tation of ˆ J dp with respect to the z axis (given by theangle ψ j ) also remains essentially unchanged. (The pro-jection of ˆ J dp on the x – y plane is displaced from the cen-ter along the y axis, reflecting the fact that the planet’sinitial line of nodes coincides with the x axis.) As thevectors ˆ D and ˆ P precess about ˆ J dp , the angles ψ d and ψ p oscillate in the ranges | ψ j − θ jd | ≤ ψ d ≤ ψ j + θ jd and | ψ j − θ jp | ≤ ψ p ≤ ψ j + θ jp , respectively.A notable feature of the evolution of this system on atimescale & τ is the increase in the angle ψ d (blue line inthe top left panel)—indicating progressive misalignmentof the disk with respect to its initial orientation—as themagnitude of the angular momentum D decreases withthe loss of mass from the disk (blue line in the bottomright panel). At the same time, the orbital plane of theplanet (red line in the top left panel) tends toward align-ment with J dp . The magenta lines in the top left andbottom right panels indicate that the orientation of thevector J dp remains fixed even as its magnitude decreases(on a timescale & τ ) on account of the decrease in themagnitude of D . As we demonstrate analytically in Ap-pendix B, the constancy of ψ j is a consequence of theinequality τ dp ≪ τ .To better understand the evolution of the disk andplanet orientations, we consider the (small) variations in D and J dp that are induced by mass depletion over asmall fraction of the precession period. On the left-handside of Figure 3 we show a schematic sketch of the ori-entations of the vectors D , P , and J dp at some giventime (denoted by the subscript 1) and a short time later(subscript 2). During that time interval the vector J dp tilts slightly to the left, and as a result it moves awayfrom D and closer to P . The sketch on the right-handside of Figure 3 demonstrates that, if we were to con-sider the same evolution a half-cycle later, the same con-clusion would be reached: in this case the vector J dp3 moves slightly to the right (to become J dp4 ), with theangle between J dp and D again increasing even as theangle between J dp and P decreases. The angles betweenthe total angular momentum vector and the vectors D and P are thus seen to undergo a systematic, secularvariation. The sketch in Figure 3 also indicates that thevector J dp undergoes an oscillation over each precession Matsakos & K¨onigl TABLE 1Model parameters
Model
S D P H M d0 [ M ∗ ] M h0 [ M ∗ ] M t0 [ M ∗ ] M p a [au] ψ p0 [ ◦ ] DP-M – √ √ – 0 . ↓ – – M J DP-m – √ √ – 0 . ↓ – – M J all-M √ √ √ √ . ↓ . ↓ . M J all-m √ √ √ √ . ↓ . ↓ . M J all-Mx √ √ √ √ . ↓ .
090 – 0 . M J all-mx √ √ √ √ . ↓ .
018 – 0 . M J retrograde √ √ √ √ . ↓ . ↓ . M J binary √ √ √ – – – 0 . ↓ M ⊙
300 10 [Myr]"DP-M": inner disk -- planett = 0 yr, M d0 = 0.01 M * < 10 M-r -6 -5 -4 -3 P r e c e ss o n f r e q u e n c - [ π / y r ] [Myr] Ω dp : inner disk -- planet −1 0 1x-component on unit sphere−101 y - c o m p o n e n t o n un i t s p h e r e t - t < 0.1 Myr ˆD ˆP ˆJ dp [Myr]10 A n g u l a r m o m e n t u m [ e r g s ] DPJ dp A n g l e [ d e g ] t - t < 0.1 Myr ψ d : inner disk -- z-axisψ p : planet -- z-axis θ dp : inner disk -- planetψ j : J dp -- z-axis Fig. 2.—
Time evolution of a “reduced” system, consisting of just a planet and an inner disk, for an initial disk mass M d0 = 0 . M ∗ (model DP-M ). Top left: the angles that the angular momentum vectors D , P and J dp form with the z axis (the initial direction of D ),as well as the angle between D and P . Top right: the projections of the angular momentum unit vectors onto the x – y plane. Bottomleft: the characteristic precession frequency. Bottom right: the magnitudes of the angular momentum vectors. In the left-hand panels, theinitial 0 . Fig. 3.—
Schematic sketch of the change in the total angularmomentum vector J dp that is induced by mass depletion from thedisk in the limit where the precession period τ dp is much shorterthan the characteristic depletion time τ . The two depicted config-urations are separated by 0 . τ dp . cycle. However, when τ dp ≪ τ and the fractional de-crease in M d over a precession period remains ≪
1, the amplitude of the oscillation is very small and J dp prac-tically maintains its initial direction (see Appendix Bfor a formal demonstration of this result). In the limitwhere the disk mass becomes highly depleted and D → J dp → P , i.e., the planet aligns with the initial directionof J dp ( θ jp → ψ p → ψ j ). The disk angular momen-tum vector then precesses about P , with its orientationangle ψ d (blue line in top left panel of Figure 2) oscil-lating between | ψ p − θ dp | and ψ p + θ dp . Note that theprecession frequency is also affected by the disk’s massdepletion and decreases with time (see Equation (A17));the time evolution of Ω dp is shown in the bottom leftpanel of Figure 2.Figure 4 shows the evolution of a similar system—model DP-m —in which the inner disk has a lower initialmass, M d0 = 0 . M ∗ (corresponding to M t0 = 0 . M ∗ ,i.e., to t = 2 Myr when M ∗ = M ⊙ ; see Equation (10)). The angle θ dp between D and P (cyan line in the top leftpanel of Figure 2) remains constant because there are no torquesthat can modify it. lanet-induced spin–orbit misalignments 7 [Myr]"DP-m": inner disk -- planett = 2 Myr, M d0 = 0.002 M * < 10 M-r -6 -5 -4 -3 P r e c e ss o n f r e q u e n c - [ π / y r ] [Myr] Ω dp : inner disk -- planet −1 0 1x-component on unit sphere−101 y - c o m p o n e n t o n un i t s p h e r e t - t < 0.1 Myr ˆD ˆP ˆJ dp [Myr]10 A n g u l a r m o m e n t u m [ e r g s ] DPJ dp A n g l e [ d e g ] t - t < 0.1 Myr ψ d : inner disk -- z-axisψ p : planet -- z-axis θ dp : inner disk -- planetψ j : J dp -- z-axis Fig. 4.—
Same as Figure 2, except that M d0 = 0 . M ∗ (model DP-m ). The initial oscillation frequency in this case is lower thanin model
DP-M , as expected from Equation (A17), but itattains the same asymptotic value (bottom left panel),corresponding to the limit J dp → P in which Ω dp be-comes independent of M d . The initial value of J dp /D is higher in the present model than in the model con-sidered in Figure 2 ( ≃ . ≃ .
1; see Equations (7)and (8)), which results in a higher value of ψ j (and, cor-respondingly, a higher initial value of θ jd and lower initialvalue of θ jp ). The higher value of ψ j is the reason why theoscillation amplitude of ψ d and the initial oscillation am-plitude of ψ p (top left panel) are larger in this case. Thehigher value of J dp /D in Figure 4 also accounts for thedifferences in the projection map shown in the top rightpanel (a larger y value for the projection of ˆ J dp , a largerarea encircled by the projection of ˆ D , and a smaller areaencircled by the projection of ˆ P ).We now consider the full system for two values ofthe total disk mass: M t0 = 0 . M ∗ (model all-M , cor-responding to t = 0; Figure 5) and M t0 = 0 . M ∗ (model all-m , corresponding to t = 2 Myr; Figure 6),assuming that both parts of the disk lose mass accordingto the relation given by Equation (10). The inner disksin these two cases correspond, respectively, to the diskmasses adopted in model DP-M (Figure 2) and model
DP-m (Figure 4). The merit of first considering the simplersystems described by the latter models becomes appar-ent from a comparison between the respective figures. Itis seen that the basic behavior of model all-M is similarto that of model
DP-M , and that the main differences be-tween model all-M and model all-m are captured by theway in which model
DP-m is distinct from model
DP-M .The physical basis for this correspondence is the central-ity of the torque exerted on the inner disk by the planet.According to Equation (5), the relative magnitudes ofthe torques acting on the disk at sufficiently late times (after D becomes smaller than the angular momentum ofeach of the other system components) are reflected in themagnitudes of the corresponding precession frequencies.The dominance of the planet’s contribution can thus beinferred from the plots in the bottom left panels of Fig-ures 5 and 6, which show that, after the contribution of D becomes unimportant (bottom right panels), the pre-cession frequency induced by the planet exceeds thoseinduced by the outer disk and by the star. While the basic disk misalignment mechanism is thesame as in the planet–inner-disk system, the detailed be-havior of the full system is understandably more com-plex. One difference that is apparent from a comparisonof the left-hand panels in Figures 5 and 2 is the higheroscillation frequency of ψ p and ψ d in the full model (withthe same frequency also seen in the timeline of ψ s ). Inthis case the planet–outer-disk precession frequency Ω ph (Equation (A20)) and the inner-disk–outer-disk preces-sion frequency Ω dh (Equation (A19)) are initially com-parable and larger than Ω dp , and Ω ph remains the dom-inant frequency throughout the system’s evolution. Thefact that the outer disk imposes a precession on both P and D has the effect of weakening the interaction be-tween the planet and the inner disk, which slows downthe disk misalignment process. Another difference is re-vealed by a comparison of the top right panels: in thefull system, ˆ J dp precesses on account of the torque in-duced by the outer disk, so it no longer corresponds tojust a single point in the x – y plane. This, in turn, in-creases the sizes of the regions traced in this plane by ˆ D and ˆ P . The behavior of the lower- M t0 model shown inFigure 6 is also more involved. In this case, in additionto the strong oscillations of the angles ψ i already man- The star–planet and star–outer-disk precession frequencies(Ω sp and Ω sh ; see Equations (A15) and (A16)) are not shown inthese figures because they are too low to fit in the plotted range. Matsakos & K¨onigl [Myr]"all-M": star, disks, planett = 0 yr, M d0 = 0.01 M * < 10 Myr -6 -5 -4 -3 P r e c e ss i o n f r e q u e n c y [ π / y r ] [Myr] Ω sd : star -- inner dis Ω dp : inner dis -- planet Ω dh : inner disk -- outer diskΩ ph : planet -- outer disk −1 0 1x-component on unit sphere−101 y - c o m p o n e n t o n un i t s p h e r e t - t < 0.1 Myr ˆD ˆP ˆJ dp [Myr]10 A n g u l a r m o m e n t u m [ e r g s ] SD PH A n g l e [ d e g ] t - t < 0.1 Myr ψ s : star -- z-axisψ d : inner disk -- z-axis ψ p : planet -- z-axisψ h : outer disk -- z-axis Fig. 5.—
Time evolution of the full system (star, inner disk, planet, outer disk) for an initial inner disk mass M d0 = 0 . M ∗ andinitial total disk mass M t0 = 0 . M ∗ (model all-M ). Panel arrangement is the same as in Figure 2, although the details of the displayedquantities—which are specified in each panel and now also include the angular momenta of the star ( S ) and the outer disk ( H )—aredifferent. [Myr]"all-m": star, disks, planett = 2 Myr, M d0 = 0.002 M * < 10 Myr -6 -5 -4 -3 P r e c e ss i o n f r e q u e n c y [ π / y r ] [Myr] Ω sd : star -- inner dis Ω dp : inner dis -- planet Ω dh : inner disk -- outer diskΩ ph : planet -- outer disk −1 0 1x-component on unit sphere−101 y - c o m p o n e n t o n un i t s p h e r e t - t < 0.1 Myr ˆD ˆP ˆJ dp [Myr]10 A n g u l a r m o m e n t u m [ e r g s ] SD PH A n g l e [ d e g ] t - t < 0.1 Myr ψ s : star -- z-axisψ d : inner disk -- z-axis ψ p : planet -- z-axisψ h : outer disk -- z-axis Fig. 6.—
Same as Figure 5, except that M d0 = 0 . M ∗ and M t0 = 0 . M ∗ (model all-m ). ifested in Figure 4, the different precession frequenciesΩ ik also exhibit large-amplitude oscillations, reflectingtheir dependence on the angles θ ik between the angularmomentum vectors. In both of the full-system models,the strongest influence on the star is produced by itsinteraction with the inner disk, but the resulting preces-sion frequency (Ω sd ) remains low. Therefore, the stellarangular momentum vector essentially retains its original orientation, which implies that the angle ψ d is a goodproxy for the angle between the primordial stellar spinand the orbit of any planet that eventually forms in theinner disk.We repeated the calculations shown in Figures 5 and 6under the assumption that only the inner disk loses masswhile M h remains constant (models all-Mx and all-mx ;Figures 7 and 8, respectively). At the start of the evolu-lanet-induced spin–orbit misalignments 9 A n g l e [ d e g ] "all-Mx" t = 0 yrM d0 =0.01M * ψ s : star -- z-axisψ d : inner disk -- z-axis ψ p : planet -- z-axisψ h : outer disk -- z-axis [Myr]10 -6 -5 -4 -3 P r e c . f r e q u e n c i e s [ π / y r ] Ω dp : inner disk -- planetΩ dh : inner disk -- outer disk Ω ph : planet -- outer diskΩ sd : star -- inner disk Fig. 7.—
Time evolution of the full system in the limit whereonly the inner disk undergoes mass depletion and the mass of theouter disk remains unchanged, for the same initial conditions as inFigure 5 (model all-Mx ). The top and bottom panels correspond,respectively, to the top left and bottom left panels of Figure 5, butin this case the initial 0 . A n g l e [ d e g ] "all-mx" t = 2 MyrM d0 =0.002M * ψ s : star -- z-axisψ d : inner disk -- z-axis ψ p : planet -- z-axisψ h : outer disk -- z-axis [Myr]10 -6 -5 -4 -3 P r e c . f r e q u e n c i e s [ π / y r ] Ω dp : inner disk -- planetΩ dh : inner disk -- outer disk Ω ph : planet -- outer diskΩ sd : star -- inner disk Fig. 8.—
Same as Figure 7, but for the initial conditions ofFigure 6 (model all-mx ). tion, the frequencies Ω ph and Ω dh are ∝ M h , whereas Ω dp scales linearly (or, in the case of the lower- M d0 model,close to linearly) with M d (see Appendix A). In the casesconsidered in Figures 5 and 6 all these frequencies de-crease with time, so the relative magnitude of Ω dp re-mains comparatively large throughout the evolution. Incontrast, in the cases shown in Figures 7 and 8 the fre-quencies Ω ph and Ω dh remain constant and only Ω dp de-creases with time. As the difference between Ω dp and theother two frequencies starts to grow, the inner disk mis-alignment process is aborted, and thereafter the meanvalues of ψ d and ψ p remain constant. This behavior isconsistent with our conclusion about the central role thatthe torque exerted by the planet plays in misaligning theinner disk: when the fast precession that the outer disk induces in the orbital motions of both the planet andthe inner disk comes to dominate the system dynam-ics, the direct coupling between the planet and the innerdisk is effectively broken and the misalignment process ishalted. Note, however, from Figure 8 that, even in thiscase, the angle ψ d can attain a high value (as part of alarge-amplitude oscillation) when M t0 is small.To determine whether the proposed misalignmentmechanism can also account for disks (and, eventually,planets) on retrograde orbits, we consider a system inwhich the companion planet is placed on such an orbit(model retrograde , which is the same as model all-m except that ψ p0 is changed from 60 ◦ to 110 ◦ ). As Fig-ure 9 demonstrates, the disk in this case evolves to aretrograde configuration ( ψ d > ◦ ) at late times evenas the planet’s orbit reverts to prograde motion. A note-worthy feature of the plotted orbital evolution (shown inthe high-resolution portion of the figure) is the rapid in-crease in the value of ψ d (which is an adequate proxy for θ sd also in this case)—and corresponding fast decreasein the value of ψ p —that occurs when the planet’s orbittransitions from a retrograde to a prograde orientation.This behavior can be traced to the fact that cos θ ph van-ishes at essentially the same time that ψ p crosses 90 ◦ because the outer disk (which dominates the total an-gular momentum) remains well aligned with the z axis.This, in turn, implies (see Equation (A20)) that, at thetime of the retrograde-to-prograde transition, the planetbecomes dynamically decoupled from the outer disk andonly retains a coupling to the inner disk. Its evolutionis, however, different from that of a “reduced” system,in which only the planet and the inner disk interact, be-cause the inner disk remains dynamically “tethered” tothe outer disk ( θ dh = 90 ◦ ). As we verified by an explicitcalculation, the evolution of the reduced system remainssmooth when ψ p crosses 90 ◦ . The jump in ψ p exhibitedby the full system leads to a significant increase in thevalue of cos θ ph and hence of Ω ph , which, in turn, restores(and even enhances) the planet’s coupling to the outerdisk after its transition to retrograde motion (see bottompanel of Figure 9). The maximum value attained by θ sd in this example is ≃ ◦ , which, just as in the progradecase shown in Figure 6, exceeds the initial misalignmentangle of the planetary orbit (albeit to a much larger ex-tent in this case). It is, however, worth noting that not allmodel systems in which the planet is initially on a retro-grade orbit give rise to a retrograde inner disk at the endof the prescribed evolution time; in particular, we foundthat the outcome of the simulated evolution (which de-pends on whether ψ p drops below 90 ◦ ) is sensitive tothe value of the initial planetary misalignment angle ψ p0 (keeping all other model parameters unchanged).In concluding this section it is instructive to comparethe results obtained for our model with those foundfor the model originally proposed by Batygin (2012)(see Section 1 for references to additional work on thatmodel). We introduced our proposed scenario as a vari-ant of the latter model, with a close-by giant planet tak-ing the place of a distant stellar companion. In the origi-nal proposal the disk misalignment was attributed to theprecessional motion that is induced by the torque thatthe binary companion exerts on the disk. In this picturethe spin–orbit angle oscillates (on a timescale ∼ ◦ and roughly twice the0 Matsakos & K¨onigl < 5.6 Myr 5.6 Myr < t - t < 10 Myr [Myr] 6 7 8 9 10"retrograde": star, disks, planett = 2 Myr, M d0 = 0.002 M * A n g l e [ d e g ] t - t < 4.6 Myr ψ s : star -- z-axisψ d : inner disk -- z-axis ψ p : planet -- z-axisψ h : outer disk -- z-axis -7 -6 -5 -4 P r e c e ss i o n f r e q u e n c y [ π / y r ] star -- inner diskinner disk -- planet inner disk -- outer diskplanet -- outer disk Fig. 9.—
Time evolution with the same initial conditions as in Figure 6, except that the planet is initially on a retrograde orbit ( ψ p0 is changed from 60 ◦ to 110 ◦ ; model retrograde ). The display format is the same as in Figure 7, but in this case the panels also show azoomed-in version of the evolution around the time of the jumps in ψ p and ψ d . The dashed line in the top panel marks the transitionbetween prograde and retrograde orientations (90 ◦ ). binary orbital inclination, so it can be large if observedat the “right” time. Our model retains this feature ofthe earlier proposal, particularly in cases where the com-panion planet is placed on a high-inclination orbit afterthe disk has already lost much of its initial mass, but italso exhibits a novel feature that gives rise to a secular(rather than oscillatory) change in the spin–orbit angle(which can potentially lead to a substantial increase inthis angle). This new behavior represents an “exchangeof orientations” between the planet and the inner diskthat is driven by the mass loss from the inner disk andcorresponds to a decrease of the inner disk’s angular mo-mentum from a value higher than that of the planet to alower value (with the two remaining within an order ofmagnitude of each other for representative parameters).This behavior is not found in a binary system becauseof the large mismatch between the angular momenta ofthe companion and the disk in that case (and, in fact, itis also suppressed in the case of a planetary companionwhen the mass of the outer disk is not depleted).As we already noted in Section 2.1, Batygin & Adams(2013) suggested that the disk misalignment in a bi-nary system can be significantly increased due to a reso-nance between the star–disk and binary–disk precessionfrequencies. (We can use Equations (A14) and (A17),respectively, to evaluate these frequencies, plugging invalues for the outer disk radius, companion orbital ra-dius, and companion mass that are appropriate for thebinary case.) Lai (2014) clarified the effect of this res-onance and emphasized that, for plausible system pa-rameters, it can be expected to be crossed as the diskbecomes depleted of mass. However, for the planetary-companion systems considered in this paper the ratio | Ω sd / Ω dp | remains < sd is initially ∝ M d , so it decreases dur-ing the early evolution. The same scaling also character-izes Ω dp in the planetary case, which explains why thecorresponding curves do not cross. In contrast, in thebinary case (for which the sum of the disk and compan-ion angular momenta is dominated by the companion’scontribution) the frequency Ω dp does not scale with thedisk mass and it thus remains nearly constant, whichmakes it possible for the corresponding curves to cross(see Figure 12 in Appendix C). Since our formalism alsoencompasses the binary case, we examined one such sys-tem (model binary )—using the parameters adopted infigure 3 of Lai (2014)—for comparison with the resultsof that work. Our findings are presented in Appendix C. DISCUSSIONThe model considered in this paper represents a variantof the primordial disk misalignment scenario of Batygin(2012) in which the companion is a nearby planet ratherthan a distant star and only the inner region of the pro-toplanetary disk (interior to the planet’s orbit) becomesinclined. In this section we assess whether this modelprovides a viable framework for interpreting the relevantobservations.The first—and most basic—question that needs to beaddressed is whether the proposed misalignment mech-anism is compatible with the broad range of apparentspin–orbit angles indicated by the data. In Section 3 weshowed that the spin–orbit angle θ sd can deviate fromits initial value of 0 ◦ either because of the precessionalmotion that is induced by the planet’s torque on the diskor on account of the secular variation that is driven bythe mass depletion process. In the “reduced” disk–planetmodel considered in Figures 2 and 4, for which the an-gle ψ d is taken as a proxy for the intrinsic spin–orbitlanet-induced spin–orbit misalignments 11angle, the latter mechanism increases θ sd to ∼ ◦ –50 ◦ on a timescale of 10 Myr for an initial planetary incli-nation ψ p0 = 60 ◦ . The maximum disk misalignment is,however, increased above this value by the precessionaloscillation, whose amplitude is higher the lower the ini-tial mass of the disk. Based on the heuristic discussiongiven in connection with Figure 3, the maximum possi-ble value of ψ d (corresponding to the limit J dp → P ) isgiven by ψ d , max = arccos D + P cos ψ p0 ( D + P + 2 D P cos ψ p0 ) / + ψ p0 . (12)For the parameters of Figure 4, ψ d , max ≈ . ◦ , whichcan be compared with the actual maximum value ( ≃ ◦ )attained over the course of the 10-Myr evolution depictedin this figure. Although the behavior of the full sys-tem (which includes also the outer disk and the star) ismore complicated, we found (see Figures 5 and 6) that,if the outer disk also loses mass, the maximum value at-tained by θ sd ( ≃ ◦ ) is not much smaller than in thesimplified model. Note that in the original primordial-misalignment scenario the maximum value of θ sd ( ≃ ψ p0 ) would have been considerably higher ( ≃ ◦ )for the parameters employed in our example. However,as indicated by Equation (12), the maximum value pre-dicted by our model depends on the ratio P/D and canin principle exceed the binary-companion limit if D issmall and P is sufficiently large. Repeating the calcula-tions shown in Figure 6 for higher values of M p , we foundthat the maximum value of θ sd is ∼ ◦ , 104 ◦ and 125 ◦ when M p /M J increases from 1 to 2, 3, and 4, respectively.These results further demonstrate that the disk can betilted to a retrograde configuration even when ψ p0 < ◦ if the planet is sufficiently massive, although a retrogradedisk orientation can also be attained (including in thecase of M p . M J ) if the planet’s orbit is initially retro-grade (see Figure 9). A low initial value of the disk angu-lar momentum D arises naturally in the leading scenariosfor placing planets in inclined orbits, which favor compar-atively low disk masses (see Section 1). The distributionof ψ p0 as well as those of the occurrence rate, mass, andorbital radius of planets on inclined orbits are requiredfor determining the predicted distribution of primordialinner-disk misalignment angles in this scenario, for com-parison with observations. However, this information,as well as data on the relevant values of M d0 , are not yetavailable, so our results for θ sd are only a first step (a The intrinsic spin–orbit angle is not directly measurable, so itsvalue must be inferred from that of the apparent (projected) mis-alignment angle λ (Fabrycky & Winn 2009). In the special case of aplanet whose orbital plane contains the line of sight—an excellentapproximation for planets observed by the transits method—theapparent obliquity cannot exceed the associated intrinsic misalign-ment angle (i.e., λ ≤ θ sd ). D , the magnitude of the initial angular momentum of the in-ner disk, cannot be much smaller than the value adopted in mod-els DP-m and all-m in view of the minimum value of M d0 thatis needed to account for the observed misaligned planets in theprimordial-disk-misalignment scenario (and also for the no-longer-present HJ in the SHJ picture). Matsakos & K¨onigl (2015) were able to reproduce the ob-served obliquity distributions of HJs around G and F stars withinthe framework of the SHJ model under the assumption that theintrinsic spin–orbit angle has a random distribution (correspondingto a flat distribution of λ ; see Fabrycky & Winn 2009). proof of concept) toward validating this interpretation ofthe measured planet obliquities.Our proposed misalignment mechanism is most effec-tive when the disk mass within the planetary orbit dropsto ∼ M p . In the example demonstrating this fact (Fig-ure 6), M d0 ≈ M J . In the primordial disk misalign-ment scenario, M d0 includes the mass that would even-tually be detected in the form of an HJ (or a lower-mass planet) moving around the central star on a mis-aligned orbit. Furthermore, if the ingestion of an HJon a misaligned orbit is as ubiquitous as inferred in theSHJ picture, that mass, too, must be included in thetally. These requirements are consistent with the factthat the typical disk misalignment time in our model(a few Myr) is comparable to the expected giant-planetformation time, but this similarity also raises the ques-tion of whether the torque exerted by the initially mis-aligned planet has the same effect on the gaseous innerdisk and on a giant planet embedded within it. Thisquestion was considered by several authors in the contextof a binary companion (e.g., Xiang-Gruess & Papaloizou2014; Picogna & Marzari 2015; Martin et al. 2016). Auseful gauge of the outcome of this dynamical inter-action is the ratio of the precession frequency inducedin the embedded planet (which we label Ω pp ) to Ω dp (Picogna & Marzari 2015). We derive an expression forΩ pp by approximating the inclined and embedded plan-ets as two rings with radii a and a < a , respectively(see Appendix A), and evaluate Ω dp under the assump-tion that the disk mass has been sufficiently depleted forthe planetary contribution ( P ) to dominate J dp . Thisleads to Ω pp / Ω dp ≃ a /r d , out ) / , which is the sameas the estimate obtained by Picogna & Marzari (2015)for a binary system. In the latter case, this ratio is small( . .
1) for typical parameters, implying that the embed-ded planet cannot keep up with the disk precession andhence that its orbit develops a significant tilt with respectto the disk’s plane. However, when the companion is aplanet, the above ratio equals ( a /a ) / and may be con-siderably larger ( . Kepler sources (Mazeh et al. 2015) be-comes weaker (with the inferred orientations possiblytending toward a nearly random distribution) at largeorbital periods ( P orb & days). The interpretationof these results in our picture is that the outer planetsremain aligned with the original stellar-spin direction,whereas the inner planets—and, according to the SHJmodel, also the stellar spin in ∼
50% of sources—assumethe orientation of the misaligned inner disk (which sam-ples a broad range of angles with respect to the initialspin direction). Further observations and analysis arerequired to corroborate and refine these findings so thatthey can be used to place tighter constrains on the mod-2 Matsakos & K¨oniglels.The result reported by Li & Winn (2016) is seeminglyat odds with another set of observational findings—thediscovery that the orbital planes of debris disks (on scales & au) are by and large well aligned with the spin axisof the central star (Watson et al. 2011; Greaves et al.2014). This inferred alignment also seemingly rules outany interpretation of the obliquity properties of exoplan-ets (including the SHJ model) that appeals to a tidal re-alignment of the host star by a misaligned HJ. These ap-parent difficulties can, however, be alleviated in the con-text of the SHJ scenario and our present model. Specif-ically, in the SHJ picture the realignment of the hoststar occurs on a relatively long timescale ( . ∼ ∼ ∼
50% of sys-tems ingest an SHJ and should exhibit spin–orbit align-ment to within 20 ◦ , with the rest remaining misaligned.Thus, the probability of observing an aligned debris diskin an older system is ∼ /
2, implying that the chance ofdetecting 2 out of 2 such systems is ∼ /
4. It is, however,worth noting that the two aforementioned systems maynot actually be well aligned: based on the formal mea-surement uncertainties quoted in Greaves et al. (2014),the misalignment angle could be as large as 36 ◦ in 10 CVnand 31 ◦ in 61 Vir. Further measurements that target oldsystems might be able to test the proposed explanation,although one should bear in mind that additional factorsmay affect the observational findings. For example, inthe tidal-downsizing scenario of planet formation, debrisdisks are less likely to exist around stars that host giantplanets (see Fletcher & Nayakshin 2016). CONCLUSIONIn this paper we conduct a proof-of-concept study ofa variant of the primordial disk misalignment model ofBatygin (2012). In that model, a binary companion withan orbital radius of a few hundred au exerts a gravita-tional torque on a protoplanetary disk that causes itsplane to precess and leads to a large-amplitude oscilla-tion of the spin–orbit angle θ sd (the angle between theangular momentum vectors of the disk and the centralstar). Motivated by recent observations, we explore analternative model in which the role of the distant binaryis taken by a giant planet with an orbital radius of justa few au. Such a companion likely resided originally inthe disk, and its orbit most probably became inclinedaway from the disk’s plane through a gravitational inter-action with other planets (involving either scattering orresonant excitation).Our model setup is guided by indications from numer- ical simulations (Xiang-Gruess & Papaloizou 2013) that,in the presence of the misaligned planet, the disk sepa-rates at the planet’s orbital radius into inner and outerparts that exhibit distinct dynamical behaviors even aseach can still be well approximated as a rigid body. Weintegrate the secular dynamical evolution equations inthe quadrupole approximation for a system consisting ofthe inclined planet, the two disk parts, and the spin-ning star, with the disk assumed to undergo continuousmass depletion. We show that this model can give riseto a broad range of values for the angle between the an-gular momentum vectors of the inner disk and the star(including values of θ sd in excess of 90 ◦ ), but that the ori-entation of the outer disk remains virtually unchanged.We demonstrate that the misalignment is induced by thetorque that the planet exerts on the inner disk and that itis suppressed when the mass depletion time in the outerdisk is much longer than in the inner disk, so that theouter disk remains comparatively massive and the fastprecession that it induces in the motions of the inner diskand the planet effectively breaks the dynamical couplingbetween the latter two. Our calculations reveal that thelargest misalignments are attained when the initial diskmass is low (on the order of that of observed systems atthe onset of the transition-disk phase). We argued that,when the misalignment angle is large, the inner and outerparts of the disk become fully detached and damping ofthe planet’s orbital inclination by dynamical friction ef-fectively ceases. This suggests a consistent primordialmisalignment scenario: the inner region of a protoplan-etary disk can be strongly misaligned by a giant planeton a high-inclination orbit if the disk’s mass is low (i.e.,late in the disk’s evolution); in turn, the planet’s orbitalinclination is least susceptible to damping in a disk thatundergoes a strong misalignment.We find that, in addition to the precession-relatedoscillations seen in the binary-companion model, thespin–orbit angle also exhibits a secular growth in theplanetary-companion case, corresponding to a monotonicincrease in the angle between the inner disk’s and the to-tal (inner disk plus planet) angular momentum vectors(accompanied by a monotonic decrease in the angle be-tween the planet’s and the total angular momentum vec-tors). This behavior arises when the magnitude of theinner disk’s angular momentum is initially comparable tothat of the planet but drops below it as a result of massdepletion (on a timescale that is long in comparison withthe precession period). This does not happen when thecompanion is a binary, since in that case the companion’sangular momentum far exceeds that of the inner disk atall times. On the other hand, in the binary case the massdepletion process can drive the system to a resonance be-tween the disk–planet and star–disk precession frequen-cies, which has the potential of significantly increasingthe maximum value of θ sd (e.g., Batygin & Adams 2013;Lai 2014). We show that this resonance is not encoun-tered when the companion is a nearby planet because—in contrast with the binary-companion case, in whichthe disk–binary precession frequency remains constant—both of these precession frequencies decrease with timein the planetary-companion case. However, we also showthat when the torque that the star exerts on the diskis taken into account (and not just that exerted by thecompanion, as in previous treatments), the misalignmentlanet-induced spin–orbit misalignments 13effect of the resonance crossing in the binary case is mea-surably weaker.A key underlying assumption of the primordial disk-misalignment model is that the planets embedded in thedisk remain confined to its plane as the disk’s orienta-tion shifts, so that their orbits become misaligned to thesame extent as that of the gaseous disk. However, theprecession frequency that a binary companion inducesin the disk can be significantly higher than the one in-duced by its direct interaction with an embedded planet,which would lead to the planet’s orbital plane separatingfrom that of the disk: this argument was used to cri-tique the original version of the primordial misalignmentmodel (e.g., Picogna & Marzari 2015). However, this po-tential difficulty is mitigated in the planetary-companionscenario, where the ratio of these two frequencies is typ-ically substantially smaller.The apparent difference in the obliquity properties ofHJs around cool and hot stars can be attributed tothe tidal realignment of a cool host star by an initiallymisaligned HJ (e.g., Albrecht et al. 2012). The finding(Mazeh et al. 2015) that this dichotomy is exhibited alsoby lower-mass planets and extends to orbital distanceswhere tidal interactions with the star are very weakmotivated the SHJ proposal (Matsakos & K¨onigl 2015),which postulates that ∼
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Fig. 10.—
Basic configuration for the torque calculation. TheCartesian coordinate system is defined so that ring k lies in the x – y plane, with the plane containing ring i intersecting it along the x axis at an angle θ ik . The two rings are centered at O and haveradii r k and r i , respectively, and mass elements dm k and dm i . APPENDIX
CALCULATION OF THE TORQUES ANDPRECESSION FREQUENCIES
Torques
Figure 10 shows two concentric rings in a Cartesiancoordinate system, oriented so that their mutual gravi-tational torques induce a rotation about the x axis. Be-cause of the configuration’s symmetry, the only nonzerocomponent of the torque that ring i exerts on ring k isthat along the x axis:[ T ik ] x = − Z k Z i (cid:20) y k × (cid:18) G dm k dm i w ˆ w (cid:19)(cid:21) x , (A1)where y k = r k sin φ k ˆ y , (A2)ˆ w = w w = r i − r k w , (A3) r k = r k cos φ k ˆ x + r k sin φ k ˆ y , (A4) r i = r i cos φ i ˆ x + r i sin φ i cos θ ik ˆ y − r i sin φ i sin θ ik ˆ z , (A5) w = (cid:2) r k + r i − r k r i (cos φ k cos φ i + sin φ k sin φ i cos θ ik )] / , (A6)[ y k × w ] x = r k r i sin φ k sin φ i sin θ ik ˆ x , (A7)and R k , R i denote integrals over the masses m k and m i . These expressions can be readily generalized to a“continuum of rings”—i.e., a disk—with inner and outerradii of r in and r out , respectively. In the case of a ring dm = λr dφ , where λ = m/ πr is the linear mass den-sity, whereas in the case of a disk dm = Σ r drdφ , whereΣ is the surface density. Adopting Σ = Σ ( r /r ) (as inBatygin 2012 and Lai 2014), where Σ , r are constants,gives m = 2 π Σ r ( r out − r in ). Therefore, dm can beexpressed as dm = m π dφ for a ring ,m π ( r out − r in ) drdφ for a disk . (A8)For r i ≫ r k one can approximate1 w ≃ r i + 3 r k r i cos φ k cos φ i + 3 r k r i sin φ k sin φ i cos θ ik , (A9)and thus the torque becomes[ T ik ] x ≃ − A i B k sin θ ik cos θ ik == − (cid:18) G Z i sin φ i r i dm i (cid:19)(cid:18)Z k r k sin φ k dm k (cid:19) sin θ ik cos θ ik (A10)(the other terms integrate to zero), where A = Gm r for a ring , Gm ( r out + r in )4 r r for a disk , (A11)lanet-induced spin–orbit misalignments 15and B = mr ,m ( r − r )6( r out − r in ) for a disk . (A12)The torque that k exerts on i is [ T ki ] x = − [ T ik ] x . Equa-tion (A10) can also be used when object k is a star bysetting B k = k q M ∗ R ∗ Ω ∗ / ( GM ∗ /R ∗ ) and, in the case of aprotostar, using k q ≃ . m = M J , and we set θ ik = 30 ◦ .Each point in the figure (representing a superposed pairof + and × symbols) corresponds to a different system,characterized by its relevant parameters (the radius ofthe ring or the inner and outer radii of the disk). For thering–ring system, ring 1 has radius r = 1 au, and thedifferent cases correspond to r ∈ [0 . ,
10] au. For thering–disk system, the ring has radius r = 2 . r ∈ [0 . , r ∈ [0 . , , M t = 0 . M ∗ , Batygin (2012) verified by an explicit cal-culation that these approximations are well justified evenif the binary moves on an eccentric orbit and self-gravityis the only mode of internal interaction in the disk. Precession Frequencies
By combining Equations (5) and (A10) we obtain ananalytic expression for the precession frequencies:Ω ik = Ω ki ≃ − A i B k J ik L i L k cos θ ik , (A13) Note that the value of [ T ki ] x cannot be calculated from Equa-tion (A10), which only holds for r i ≫ r k , and instead has to beevaluated from [ T ik ] x using Newton’s third law. where, again, object i is taken to be “outside of” object k ( r i ≫ r k ). The six characteristic frequencies areΩ sd ≃ − . × − (cid:18) k q k ∗ (cid:19) (cid:18) M d . M ∗ (cid:19) × (cid:18) R ∗ R ⊙ (cid:19) (cid:18) r d , in R ∗ (cid:19) − (cid:16) r d , out (cid:17) − J sd D × Ω ∗ . p GM ⊙ / (2 R ⊙ ) ! cos θ sd π yr , (A14)Ω sp ≃ − . × − (cid:18) k q k ∗ (cid:19) (cid:18) M p M J (cid:19) (cid:18) M ∗ M ⊙ (cid:19) − × (cid:18) R ∗ R ⊙ (cid:19) (cid:16) a (cid:17) − J sp P × Ω ∗ . p GM ⊙ / (2 R ⊙ ) ! cos θ sp π yr , (A15)Ω sh ≃ − . × − (cid:18) k q k ∗ (cid:19) (cid:18) M h . M ∗ (cid:19) × (cid:18) R ∗ R ⊙ (cid:19) (cid:16) r h , in (cid:17) − (cid:16) r h , out
50 au (cid:17) − J sh H × Ω ∗ . p GM ⊙ / (2 R ⊙ ) ! cos θ sh π yr , (A16)Ω dp ≃ − . × − (cid:18) M d . M ∗ (cid:19) × (cid:16) r d , out a (cid:17) J dp D × Ω p p GM ⊙ / (5 au) ! cos θ dp π yr (A17) ≃ − . × − (cid:18) M p M J (cid:19) (cid:18) M ∗ M ⊙ (cid:19) − × (cid:16) r d , out a (cid:17) / J dp P × Ω p p GM ⊙ / (5 au) ! cos θ dp π yr , (A18)Ω dh ≃ − . × − (cid:18) M h . M ∗ (cid:19) × (cid:18) r d , out r h , in (cid:19) (cid:18) r d , out r h , out (cid:19) J dh H × GM ∗ /r , out GM ⊙ / (5 au) ! / cos θ dh π yr , (A19)6 Matsakos & K¨onigl -1 r /r T o r q u e [ e r g ] ring--ring |T | Eq. (A1)|T | Eq. (A1)|T | Eq. (A10)|T | Eq. (A10) -1 r /r -2 -1 r /r T o r q u e [ e r g ] ring--disk |T | Eq. (A1)|T | Eq. (A1)|T | Eq. (A10)|T | Eq. (A10) -1 r /r -4 -3 -2 -1 r /r T o r q u e [ e r g ] disk--disk |T | Eq. (A1)|T | Eq. (A1)|T | Eq. (A10)|T | Eq. (A10) -2 -1 r /r Fig. 11.—
Comparison of the exact torque (Equation (A1); points) with the quadrupole approximation (Equation (A10); lines) for threegeneric configurations: ring–ring (left-hand panel), ring–disk (middle panel), and disk–disk (right-hand panel). Each pair of symbols (+and × , corresponding to T and T , respectively) represents a different system, with the ratio(s) of their defining radii shown on the topand bottom horizontal axes. and Ω ph ≃ − . × − (cid:18) M h . M ∗ (cid:19) × (cid:18) ar h , in (cid:19) (cid:18) ar h , out (cid:19) J ph H × (cid:18) GM ∗ /a GM ⊙ / (5 au) (cid:19) / cos θ ph π yr . (A20) ANGULAR MOMENTUM OF THE INNERDISK AND PLANET
To obtain an expression for the time evolution of J dp = D + P , we write d D dt = T pd + (cid:18) d D dt (cid:19) depl , (B1) d P dt = − T pd , (B2)and take their sum using Equation (11): d J dp dt = (cid:18) dDdt (cid:19) depl (cos φ ′ sin θ ′ ˆ x ′ + sin φ ′ sin θ ′ ˆ y ′ + cos θ ′ ˆ z ′ ) , (B3)where we expressed ˆ D in a cartesian coordinate system( x ′ , y ′ , z ′ ) with ˆ z ′ = ˆ J dp and with θ ′ , φ ′ the sphericalpolar angles. Since the precession period is much shorterthan the depletion time (for example, for the parame-ters that characterize model DP-M , the initial value of τ dp /τ is ≃ . dD/dt ) depl as a constant over one precession pe-riod. Averaging over φ therefore gives (cid:28) d J dp dt (cid:29) ≃ (cid:18) dDdt (cid:19) depl ˆ J dp , (B4)where the angle brackets denote an average over a pre-cession period. This implies that h J dp i decreases in mag-nitude without changing its direction. The oscillation of ψ j during a single precession period—described in Fig-ure 3—is in practice so small (its amplitude is ≃ τ dp /τ for t ≪ τ ) that it cannot be picked out in Figures 2 and 4. RESONANCE CROSSING INSTAR–DISK–BINARY SYSTEMS
The formalism employed in this paper can also be usedto treat the original variant of the primordial disk mis-alignment model, in which the companion is a distantstar rather than a nearby giant planet. To validate ourcode, we consider one such system (model binary ; seeTable 1), which corresponds to the example presented infigure 3 of Lai (2014). In that work, the evolution of asystem consisting of a star, a binary star (subscript b),and a disk that undergoes mass depletion (according tothe prescription given by Equation (10)) was studied byintegrating the equations d S dt = T ds (C1)and d D dt = T bd . (C2)The results of solving these two equation with our nu-merical scheme are presented in the top two panels ofFigure 12. These results are identical to those obtainedby Lai (2014) and indicate that even a system withsmall initial misalignments between the star and thedisk ( θ sd = 5 ◦ ) and between the disk and the binary( θ db = 5 ◦ , with θ sb = 10 ◦ ) can attain a large final spin–orbit angle if a resonance between the precessions fre-quencies Ω sd and Ω db is crossed. In this case the pre-cession frequency that the torque exerted by the diskinduces in the stellar angular momentum vector is ini-tially high enough (Ω sd > Ω db ) for the star–disk pair toremain coupled as the disk precesses under the influenceof the binary. However, as the mass of the disk becomesdepleted, Ω sd decreases and eventually crosses Ω db . Be-yond that point, the stellar angular momentum can nolonger follow the precession of the disk’s angular mo-mentum, and the motion of these two vectors decouples.At resonance the star–disk system may attain a largelanet-induced spin–orbit misalignments 17 A n g l e s [ d e g ] star, disks, planetM d = 0. 01M ∗ θ sd : star -- diskθ sb : star -- binary P r e c . f r e q s . [ M y r − ] Ω sd : star -- diskΩ db : disk -- binary A n g l e s [ d e g ] star, disks, planetM d = 0. 01M ∗ θ sd : star -- diskθ sb : star -- binary P r e c . f r e q s . [ M y r − ] Ω sd : star -- diskΩ db : disk -- binary Fig. 12.—
Time evolution of a star–disk–binary system (model binary ) for the parameters used in figure 3 of Lai (2014). The toptwo panels present the results obtained by neglecting the torquethat the star exerts on the disk, whereas in the bottom two panelsthe effect of this torque is included. misalignment, which, in the absence of strong star–diskcoupling, remains “frozen” during the ensuing evolution.In the model presented in Lai (2014) the torque thatthe star exerts on the disk is neglected. We now usethe more general formulation employed in this work toextend that model by including also the torques exertedby the star. Thus, instead of Equations (C1) and (C2),we integrate d S dt = T ds + T bs (C3)and d D dt = T sd + T bd . (C4)In practice, only the torque that the star exerts on thedisk plays a role, with the effect of T bsbs