LETTERdoi:10.1038/nature11560

A primordial origin for misalignments betweenstellar spin axes and planetary orbitsKonstantin Batygin1,2

The existence of gaseous giant planets whose orbits lie close to theirhost stars (hot Jupiters) can largely be accounted for by planetarymigration associated with viscous evolution of proto-planetarynebulae1. Recently, observations of theRossiterMcLaughlin effect2

during planetary transits have revealed that a considerable fractionof hot Jupiters are on orbits that are misaligned with respect to thespin axes of their host stars3. This observation has cast doubt on theimportance of disk-drivenmigration as amechanism for producinghot Jupiters. Here I show that misaligned orbits can be a naturalconsequence of disk migration in binary systems whose orbitalplane is uncorrelated with the spin axes of the individual stars46.The gravitational torques arising from the dynamical evolution ofidealized proto-planetary disks under perturbations from massivedistant bodies act to misalign the orbital planes of the disks relativeto the spin poles of their host stars. As a result, I suggest that in theabsence of strong coupling between the angular momentum of thedisk and that of the host star, or of sufficient dissipation that acts torealign the stellar spin axis and the planetary orbits, the fraction ofplanetary systems (including systems of hot Neptunes and super-Earths) whose angular momentum vectors are misaligned withrespect to their host stars will be commensurate with the rate ofprimordial stellar multiplicity.The obliquities (angles between the planetary orbits and the stellar

spins) of detected planetary orbits range from almost perfectly alignedprograde to almost perfectly aligned retrograde systems7. Previously,the misalignment between planetary orbits and stellar spin axes hadbeen attributed to post-nebularmulti-body interactions.Most notably,Kozai cycles with tidal friction810, planetplanet scattering11,12, andchaotic secular excursions13 have been invoked as ameans of producingmisaligned planets. These mechanisms are probably responsible fora few specific examples (for example, the extreme eccentricity ofHD80606b is almost certainly due to Kozai resonance with the stellarcompanion HD806078). However, it is unlikely that they can explainmisalignedhot Jupiters as a population. For instance, theKozaimecha-nism can be stifled by forced apsidal precession in multi-planet sys-tems13. Likewise, within the context of planetplanet scattering andsecular chaos, the allowed parameter range is limited, because theproduction of close-in orbits requires the timescale for tidal captureto be considerably shorter than that for eccentricity growth12, whiledemanding the associated tidal heating to be small enough not to over-inflate the planet beyond its Roche lobe14. Additionally, the observedpresence of mean-motion resonances among giant planets on wideorbits (which rely on smooth, convergent migration to congregate15)provides furthermotivation for the development of a unifiedmodel fordisk migration that is capable of producing misaligned orbits.Thedynamics of self-gravitating proto-planetary disks under external

perturbations can be extremely complex, making precise quantitativemodelling computationally unfeasible. Consequently, here I shall con-centrate on characterization of the qualitative physical behaviour ofthe system and use classical perturbationmethods to obtain a solution.In the spirit of secular theory16, I model the proto-planetary disk as a

series of initially planar, circular, concentric, massive wires that inter-act gravitationally. Our model is based on the Gaussian averagingmethod17,18 and the gravitational potential is softened to partiallyaccount for the discrete representation of the disk. The effects of dissi-pative fluid forces within the disk are neglected. The perturbing body isalso modelled as a massive ring, but is eccentric (e95 0.5) and inclinedwith respect to the disk by an inclination i9. A description of the modeland its inherent assumptions is presented in ref. 19, and the details ofour implementation are stated in the Supplementary Information.A self-gravitating disk will preserve an untwisted structure and act

as a rigid body, provided that the characteristic timescale of the exter-nal perturbation greatly exceeds that of the disks self-interaction20.Mathematically, this amounts to a statement of adiabatic invarianceof the phase-space area occupied by a single secular cycle within thedisk21. If this condition is satisfied, the external perturbers sole effect isto induce a recession (that is, a retrograde drift) of the ascending nodeof the disk, as defined by the plane of the stellar orbit. The embeddedplanetary orbit will also adiabatically follow the disk.In the reference frame of the host star, the nodal recession of the disk

will appear as a cyclic excitation of inclination between the disk and thestellar spin axis (see Fig. 1), provided that the host stars angularmomentum vector does not adiabatically trail the disk. For this to holdtrue, the characteristic interaction timescale between the disk and the

1DivisionofGeological andPlanetarySciences, California Institute of Technology, Pasadena, California91125,USA. 2Institute for Theory andComputation,Harvard-SmithsonianCenter for Astrophysics, 60Garden Street, Cambridge, Massachusetts 02138, USA.

Nodal recession of the diskforced by the stellar companion

Orbital plane of thestellar companion

Shrinkingplanetary orbit

Stellar spin angular-momentum

vector direction

Disk angular-momentum vector direction

Stellar binary orbit angular-momentum

vector direction

Figure 1 | Geometrical set-up of the problem. This figure depicts a schematicrepresentation of the production of misaligned close-in planets through disk-driven migration in binary systems. The adiabatic response of a self-gravitatingdisk to long-term perturbations by a stellar companion is the recession of itsascending node, as defined by the orbital plane of the stellar companion. Therecession of the disks angular momentum vector about the stellar binaryorbital angular momentum vector appears to be an excitation of themisalignment angle y between the stellar spin axis and the disk in the starsreference frame.

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stellar spin-axis, T star, must exceed the disks nodal recession time-scale, T disk, by a considerable amount (that is, angular momentumcoupling between the disk and the host star must be non-adiabatic)9.The former can be estimated by modelling the stellar rotational bulgeas an inertially equivalent orbiting ring, effectively reducing the cha-racteristic interaction timescale to the forced nodal recession period ofthe ring.Observations suggest that rotational periods of TTauri pre-main

sequence stars, whose masses exceed Mw0:25M8, where M8 is themass of the Sun, form a bimodal distribution where fast and slow rota-tors are centred around 2days and 8days respectively, with a preferencefor slow rotation at highermasses22. Thus, for typical pre-main-sequencestars, we obtain T star

tend to have small spin-orbit angles3. This transition has been attri-buted to tidal re-alignment of cooler stars owing to the increased size oftheir convective zones28 and thus appears to be in good agreementwithour model.The consistency arguments presented above indicate that disk-

drivenmigration in binary systems is a favourable origin ofmisalignedhot Jupiters. This model can be tested as follows. Although multipleexplanations exist for the origins of hot Jupiters, short-period multi-planet systems, which are typically less massive, have almost certainlyundergone disk-drivenmigration. Theoretically, disk-drivenmigrationtends to maintain coplanarity and near-resonances in the planetarysystems29, both of which have been spectacularly confirmed by theKepler mission30. Within the context of the model proposed here,proto-planetary disks become misaligned with the spin axes of theirhost stars irrespectively of the masses and orbital radii of the newlyformed planets. I predict that future observations of the RossiterMcLaughlin effect will reveal that systems of close-in coplanar sub-giant planets can be misaligned with respect to the spin axes of theirhost stars in the absence of significant dissipative processes. Similarly,the presence of distant, nearly circular resonant planet pairs in systemsthat host hot Jupiters on oblique orbits would also point to disk-torqueas the likely mechanism for the origin of spin-orbit misalignment.High-precision radial velocity monitoring of transiting systems shoulddirectly test this prediction in the near future.

Received 2 November 2011; accepted 24 August 2012.

1. Lin, D.N. C., Bodenheimer, P.&Richardson, D. C.Orbitalmigrationof the planetarycompanion of 51 Pegasi to its present location. Nature 380, 606607 (1996).

2. McLaughlin, D. B. Some results of a spectrographic study of the Algol system.Astrophys. J. 60, 2231 (1924).

3. Winn, J. N., Fabrycky, D., Albrecht, S. & Johnson, J. A. Hot stars with hot Jupitershave high obliquities. Astrophys. J. 718, L145L149 (2010).

4. Ghez, A. M., Neugebauer, G. & Matthews, K. Themultiplicity of T Tauri stars in thestar forming regions Taurus-Auriga and Ophiuchus-Scorpius: a 2.2 micronspeckle imaging survey. Astron. J. 106, 20052023 (1993).

5. Kraus, A. L., Ireland, M. J., Martinache, F. &Hillenbrand, L. A. Mapping the shores ofthe brown dwarf desert. II. Multiple star formation in Taurus-Auriga. Astrophys. J.731, 8 (2011).

6. Marks, M. & Kroupa, P. Inverse dynamical population synthesis: constraining theinitial conditions of young stellar clusters by studying their binary populations.Astron. Astrophys. 543, A8 (2012).

7. Hebrard, G. 20 colleagues The retrograde orbit of the HAT-P-6b exoplanet. Astron.Astrophys. 527, L11 (2011).

8. Wu, Y. & Murray, N. Planet migration and binary companions: the case of HD80606b. Astrophys. J. 589, 605614 (2003).

9. Fabrycky, D. & Tremaine, S. Shrinking binary and planetary orbits by Kozai cycleswith tidal friction. Astrophys. J. 669, 12981315 (2007).

10. Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A. & Teyssandier, J. Hot Jupiters fromsecular planetplanet interactions. Nature 473, 187189 (2011).

11. Ford, E. B. & Rasio, F. A. Origins of eccentric extrasolar planets: testing the planet-planet scattering model. Astrophys. J. 686, 621636 (2008).

12. Nagasawa, M., Ida, S. & Bessho, T. Formation of hot planets by a combination ofplanet scattering, tidal circularization, and theKozaimechanism.Astrophys. J.678,498508 (2008).

13. Wu, Y. & Lithwick, Y. Secular chaos and the production of hot Jupiters. Astrophys. J.735, 109 (2011).

14. Guillochon, J., Ramirez-Ruiz, E. & Lin, D. Consequences of the ejection anddisruption of giant planets. Astrophys. J. 732, 74 (2011).

15. Morbidelli, A. & Crida, A. The dynamics of Jupiter and Saturn in the gaseousprotoplanetary disk. Icarus 191, 158171 (2007).

16. Marquis de Laplace, P.-S. Traite de Mecanique Celeste, 1 and 2 Vol. 1, Ch. VII,569634, http://ia600303.us.archive.org/19/items/mcaniquecles01laplrich/mcaniquecles01laplrich.pdf (Hillard, Gray, Little and Wilkins, 1799).

17. Gauss, C. F. Theoria Motus Corporum Coelestium in Sectionibus Conicis SolemAmbientium (eds Perthes, F. & Besser, I. H.) (Sumtibus, 1809); inWerke Vol. 3,331355, http://resolver.sub.uni-goettingen.de/purl?PPN235999628(Dieterich, Gottinger, 1866).

18. Murray, C. D. & Dermott, S. F. Solar System Dynamics Ch. 7 (Cambridge UniversityPress, 1999).

19. Touma, J. R., Tremaine, S. & Kazandjian, M. V. Gausss method for seculardynamics, softened.Mon. Not. R. Astron. Soc. 394, 10851108 (2009).

20. Batygin, K., Morbidelli, A. & Tsiganis, K. Formation and evolution of planetarysystems inpresenceofhighly inclinedstellarperturbers.Astron. Astrophys.533,A7(2011).

21. Morbidelli, A.Modern Celestial Mechanics: Aspects of Solar System Dynamics Chs 4and 8 (Taylor and Francis, 2002).

22. Herbst, W., Bailer-Jones, C. A. L., Mundt, R., Meisenheimer, K. & Wackermann, R.Stellar rotation and variability in the Orion nebula cluster. Astron. Astrophys. 396,513532 (2002).

23. Calvet, N.et al.Diskevolution in theOrionOB1association.Astron. J.129,935946(2005).

24. Kaib, N. A., Raymond, S. N. &Duncan,M. J. 55Cancri: a coplanar planetary systemthat is likely misaligned with its star. Astrophys. J. 742, L24 (2011).

25. Kraus, A. L. & Hillenbrand, L. A. The role of mass and environment inmultiple-starformation: a 2MASS survey of wide multiplicity in three young associations.Astrophys. J. 662, 413430 (2007).

26. Adams, F. C. The birth environment of the Solar System. Annu. Rev. Astron.Astrophys. 48, 4785 (2010).

27. Bate, M. R., Lodato, G. & Pringle, J. E. Chaotic star formation and the alignment ofstellar rotation with disc and planetary orbital axes. Mon. Not. R. Astron. Soc. 401,15051513 (2010).

28. Lai, D. Tidal dissipation in planet-hosting stars: damping of spin-orbitmisalignment and survival of hot Jupiters.Mon. Not. R. Astron. Soc. 423, 486492(2011).

29. Terquem, C. & Papaloizou, J. C. B. Migration and the formation of systems of hotsuper-Earths and Neptunes. Astrophys. J. 654, 11101120 (2007).

30. Lissauer, J. J. et al. Architecture and dynamics of Keplers candidate multipletransiting planet systems. Astrophys. J. Suppl. Ser. 197, 8 (2011).

Supplementary Information is available in the online version of the paper.

Acknowledgements I thank A. Morbidelli, P. Golreich, H. Knutson, S. Tremaine, J. Winnand F. Adams for numerous conversations and D. Stevenson and G. Laughlin forcarefully reading themanuscript. I am greatly indebted toM. Kazandjian and J. Toumafor providingmewith the softenedanalyticalGaussianaveraging algorithmused in thiswork and help with implementation. Finally, I am grateful to D. Fabrycky for carefulexamination of the paper and numerous suggestions, which resulted in a substantialimprovement of the manuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The author declares no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to K.B. (kbatygin@gps.caltech.edu).

Trailing of stellar spin axisPrecession period disk lifetime

100 200 400 800a (AU)

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od =

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/M *)

Myr

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Mc

osi(

1 +

3e

/2)

(M*)

mdisk = 102M*

ain = 0.05 AUaout = 50 AU

Figure 3 | Timescales for excitation of spin-orbit misalignment. Thecharacteristic nodal recession period of a disk withmdisk 5 10

22Mis shown asa function of binary mass and orbital properties. The disk considered has anouter edge at aout5 50 AU. Increasing aout will result in an approximately linearincrease of the precession frequency. The period is expressed as a scaling law inthemass of the host star. In the region of parameter space where the precessionperiod greatly exceeds the disk lifetime, only small misalignment anglesbetween the disk and the host star can be excited. In principle, however, if thestellar companion does not get stripped away, the ascending node of theinvariable plane of the formed planetary system can also recess. However, thedegree to which this can affect a planet on a close-in orbit is sensitive to theparticular architecture of the system. In the regionof the parameter spacewheredynamics ceases to be adiabatic, misalignment is certainly attainable, but morequantitatively precise (magnetohydrodynamical) modelling is required for itscharacterization. Finally, at the extreme high-mass/small-orbital-separationend of parameter space, one could envision a scenario where a newly formeddisk becomes severely twisted and eventually gets disrupted as a result of strongexternal perturbations. Driven by viscous dissipation, however, such a structurewould probably re-collapse into a new protoplanetary disk, whose orbital planewill be close to the Laplace plane of the stellar binary orbit.

RESEARCH LETTER

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TitleAuthorsAbstractReferencesFigure 1 Geometrical set-up of the problem.Figure 2 Excitation of disk-star misalignment.Figure 3 Timescales for excitation of spin-orbit misalignment.