Spin-orbit alignment of exoplanet systems: ensemble analysis using asteroseismology T. L. Campante1,2 [email protected] 6 1 and 0 2 M. N. Lund1,2, J. S. Kuszlewicz1,2, G. R. Davies1,2, W. J. Chaplin1,2, S. Albrecht2, n a J. N. Winn3,4, T. R. Bedding5,2, O. Benomar6,7, D. Bossini1,2, R. Handberg2,1, J A. R. G. Santos8,1, V. Van Eylen2,4, S. Basu9, J. Christensen-Dalsgaard2, Y. P. Elsworth1,2, 2 2 S. Hekker10,2, T. Hirano11, D. Huber5,12, C. Karoff13,2, H. Kjeldsen2, M. S. Lundkvist2, T. S. H. North1,2, V. Silva Aguirre2, D. Stello5,2, T. R. White2,14 ] P E . h 1School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK p - 2Stellar Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus University, Ny o r Munkegade 120, DK-8000 Aarhus C, Denmark t s a 3Physics Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA [ 02139, USA 1 4MIT Kavli Institute for Astrophysics & Space Research, 70 Vassar Street, Cambridge, MA 02139, USA v 2 5Sydney Institute for Astronomy, School of Physics, University of Sydney, Sydney, Australia 5 0 6Department of Astronomy, The University of Tokyo, School of Science, 7-3-1 Hongo, Bunkyo-ku, Tokyo 6 0 113-0033, Japan . 1 7NYUAD Institute, Center for Space Science, New York University Abu Dhabi, PO Box 129188, Abu 0 Dhabi, UAE 6 1 8Instituto de Astrof´ısica e Ciˆencias do Espac¸o, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, : v Portugal i X 9Department of Astronomy, Yale University, New Haven, CT 06520, USA r a 10Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨ottingen, Germany 11DepartmentofEarthandPlanetarySciences,TokyoInstituteofTechnology,2-12-1Ookayama,Meguro- ku, Tokyo 152-8551, Japan 12SETI Institute, 189 Bernardo Avenue #100, Mountain View, CA 94043, USA 13Department of Geoscience, Aarhus University, Høegh-Guldbergs Gade 2, DK-8000 Aarhus C, Denmark 14InstitutfurAstrophysik,Georg-August-Universit¨atG¨ottingen,Friedrich-Hund-Platz1,37077G¨ottingen, Germany – 2 – ABSTRACT The angle ψ between a planet’s orbital axis and the spin axis of its parent star is an important diagnostic of planet formation, migration, and tidal evolution. We seek empirical constraints on ψ by measuring the stellar inclination i via s asteroseismology for an ensemble of 25 solar-type hosts observed with NASA’s Kepler satellite. Our results for i are consistent with alignment at the 2-σ s level for all stars in the sample, meaning that the system surrounding the red- giant star Kepler-56 remains as the only unambiguous misaligned multiple-planet system detected to date. The availability of a measurement of the projected spin-orbit angle λ for two of the systems allows us to estimate ψ. We find that the orbit of the hot-Jupiter HAT-P-7b is likely to be retrograde (ψ=116.4 +30.2), ◦ 14.7 − whereasthatofKepler-25cseemstobewellalignedwiththestellarspinaxis(ψ= 12.6 +6.7 ). While the latter result is in apparent contradiction with a statement ◦ 11.0 − made previously in the literature that the multi-transiting system Kepler-25 is misaligned, we show that the results are consistent, given the large associated uncertainties. Finally, we perform a hierarchical Bayesian analysis based on the asteroseismic sample in order to recover the underlying distribution of ψ. The ensemble analysis suggests that the directions of the stellar spin and planetary orbital axes are correlated, as conveyed by a tendency of the host stars to display large inclination values. Subject headings: asteroseismology — methods: statistical — planetary systems — planets and satellites: general — stars: solar-type — techniques: photometric 1. Introduction The angle ψ between the planetary orbital axis and the stellar spin axis (the true obliq- uity or spin-orbit angle) is a fundamental geometric property of planetary systems. Fur- thermore, it has been recognized as an important diagnostic of theories of planet formation, migration, and tidal evolution. For all these reasons, seeking empirical constraints on ψ is a matter of the utmost importance. In the case of an exoplanet found through the radial-velocity (RV) method, no infor- mation is available about the degree of spin-orbit alignment. For transiting exoplanets, on the other hand, the Rossiter–McLaughlin (RM) effect has now been widely exploited (e.g., Queloz et al. 2000; Winn et al. 2005, 2009, 2010b, 2011; H´ebrard et al. 2008; Triaud et al. 2010; Hirano et al. 2011; Albrecht et al. 2012, 2013). This technique is sensitive to the angle – 3 – λ between the sky-projected orbital and spin axes (the projected spin-orbit angle). At the time of writing there are 87 planets with published measurements of the RM effect (see the online compilation1 by R. Heller), of which 36 ( 40%) show substantial misalignments ∼ according to at least one publication (see also fig. 7 of Xue et al. 2014). Other techniques that allow obliquity measurements of transiting systems include the analysis of planetary transits over starspots (e.g., D´esert et al. 2011; Nutzman et al. 2011; Sanchis-Ojeda et al. 2011, 2012), Doppler tomography (e.g., Collier Cameron et al. 2010; Gandolfi et al. 2012; Albrecht et al. 2013; Johnson et al. 2014), the analysis of the effect of gravity darkening on the transit light curve (e.g., Barnes 2009; Barnes et al. 2011; Ahlers et al. 2014), and the analysis of the photometric amplitude distribution of stellar rotation (Mazeh et al. 2015). Most obliquity measurements to date have been for systems harboring hot Jupiters, owingtothefactthattheRMeffectscalesastheplanet-to-stararearatioandtotheincreased opportunities for obtaining follow-up spectroscopic observations due to the frequent transit events. Empirical evidence has been found that the obliquities of hot-Jupiter systems are affected by tidal evolution (Schlaufman 2010; Winn et al. 2010a; Morton & Johnson 2011; Triaud 2011; Albrecht et al. 2012; although see Mazeh et al. 2015 for evidence against this theory): systems expected to undergo strong planet-star tidal interactions are preferentially found to have low obliquities, while systems with weaker tidal interactions display a wide range of obliquities that, besides well-aligned planets, also include planets in polar or even retrograde orbits. This suggests that the orbital plane has changed relative to the plane of the protoplanetary disk by the time hot Jupiters are formed and before tides have had any impact on shaping the system, which presumably happens due to the same mechanism responsible for their migration. The above discussion assumes that the protoplanetary disk is coplanar with the stellar equator. The possibility remains, however, that primordial star-disk misalignments are ubiq- uitous, meaning that large obliquities could be a generic feature of planetary systems and not specific to hot-Jupiter formation. This hypothesis may in principle be tested by measur- ing the obliquities of systems with multiple transiting planets, since the planetary orbits in these systems are nearly coplanar and presumably trace the plane of the protoplanetary disk (Lissauer et al. 2011; Fabrycky et al. 2014). Accordingly, if multi-transiting systems tend to have low obliquities, then the high obliquities observed for hot-Jupiter systems are likely to beassociatedwithplanetmigration. If, instead, theobliquitydistributionofmulti-transiting systems is similar to that of hot-Jupiter systems, then this would indicate more general pro- cesses of stellar and planet formation: processes such as chaotic star formation (e.g., Bate 1http://www.physics.mcmaster.ca/~rheller/ – 4 – et al. 2010; Thies et al. 2011; Fielding et al. 2015), magnetic star-disk interactions (e.g., Lai et al. 2011), torques due to internal gravity waves (e.g., Rogers et al. 2012), or torques due to neighboring stars (e.g., Batygin 2012). In order to study the dynamical histories of planetary systems across a comprehensive range of architectures and in a variety of environments, it is imperative to extend obliquity measurements to systems with smaller planets, longer-period planets, and multiple planets (note that, according to the current state of knowledge, hot Jupiters rarely have nearby planetary companions and may thus occur preferentially as single planets; Steffen et al. 2012). For long-period planets, however, the opportunities to observe transits occur less frequently, which limits the possibility of obtaining follow-up observations or identifying the transit geometry from starspot crossings. Furthermore, application of the RM technique becomes increasingly more challenging when dealing with multiple-planet systems, since these systems also tend to involve smaller planets (e.g., Latham et al. 2011). An alternative technique for measuring the obliquities of planetary systems, one that does not depend on the signal-to-noise ratio (S/N) of the transit data and hence on planet size, makes use of a combination of the photometric stellar rotation period, P , and the rot spectroscopically-determined projected rotational velocity, vsini , and stellar radius, R , to s s determinethesineoftheanglei betweenthestellarspinaxisandthelineofsight(thestellar s inclination angle). This technique evolved from the technique originally developed by Herbst et al. (1986) and Hendry et al. (1993), having been revisited more recently by a number of authors (Jackson & Jeffries 2010; Schlaufman 2010), including a series of applications (e.g., Hirano et al. 2012, 2014; Walkowicz & Basri 2013; Morton & Winn 2014) to transiting systems observed with the NASA Kepler mission (Borucki et al. 2010; Koch et al. 2010). Finally, asteroseismologyprovidesanindependentmeansofdirectlydeterminingi . The s asteroseismic estimation of i rests on our ability to resolve and extract signatures of rotation s in the power spectra of non-radial modes of oscillation (e.g., Gizon & Solanki 2003; Ballot et al. 2006, 2008; Benomar et al. 2009; Campante et al. 2011). The asteroseismic technique requires bright targets and long-duration time series to attain the desired S/N and frequency resolution. The applicability of this technique depends entirely on the stellar properties and notontheplanetaryororbitalparameters, whichisanadvantagewhenmeasuringobliquities ofsystemswithsmalland/orlong-periodplanets. Followingitsapplicationtohoststarswith single, non-transiting large planets discovered using the RV method (Wright et al. 2011; Gizon et al. 2013), the asteroseismic technique has been applied to several Kepler Sun-like hosts (Chaplin et al. 2013; Benomar et al. 2014; Lund et al. 2014b; Van Eylen et al. 2014). In addition, Huber et al. (2013b) used asteroseismology to measure a large obliquity for Kepler-56, a red giant hosting two transiting coplanar planets, thus showing that spin-orbit – 5 – misalignments are not confined to hot-Jupiter systems. Another instance of an asteroseismic obliquity measurement of an evolved host is that of Kepler-432 (Quinn et al. 2015). Recently, the stellar inclination angles of the solar analogs 16 Cyg A and B (the B component hosts a Jovian planet) were determined using asteroseismology (Davies et al. 2015a). Herewepresentthefirstanalysisofanensembleofasteroseismicobliquitymeasurements obtained for solar-type stars with transiting planets. The asteroseismic sample consists of 25 Kepler planet-candidate host stars (also designated as Kepler Objects of Interest or KOIs), of which 20 are confirmed hosts. The host stars are distributed along the main sequence and subgiant branch, and all exhibit solar-like oscillations. The rest of the paper is organized as follows. In Sect. 2 we present a recap of the spin-orbit geometry. A detailed asteroseismic analysis of the individual planetary-system hosts follows in Sect. 3. In Sect. 4 we place statistical constraints on the spin-orbit alignment. Finally, a discussion of the results and conclusions make up Sect. 5. 2. Spin-orbit geometry Figure 1 shows an observer-oriented coordinate system (left panel) and an orbit-oriented coordinate system (right panel). In the former, the orbital angular momentum unit vector, n , lies on the yz plane and is solely determined by the angle i between the planetary orbital o o axis and the line of sight (the orbital inclination angle). The angle i can be measured for a o transiting planet via transit photometry (e.g., Charbonneau et al. 2000; Henry et al. 2000), in which case one necessarily has sini 1 (i.e., an edge-on orbit). Determination of the o ≈ stellar rotation angular momentum unit vector, n , requires a polar and an azimuthal angle, s respectively i and λ. To avoid degeneracies, we restrict i and i to the interval [0,π/2], s o s while λ is allowed to vary in the interval [ π,π]. In the orbit-oriented coordinate system, n s − is determined by the polar and azimuthal angles ψ and φ. The spin-orbit angle ψ is the angle between n and n , taking values in the interval [0,π]. Values of ψ lower (greater) than π/2 o s correspond to prograde (retrograde) orbits. The specific cases of a parallel, an antiparallel, and a polar orbit then correspond to ψ=0, ψ=π, and ψ=π/2, respectively. The azimuthal angle φ is allowed to vary in the range [ π,π] and takes the value π along the line of sight. − The various angles are related according to the following equations (for a derivation see, e.g., Fabrycky & Winn 2009): sini sinλ = sinψsinφ, (1a) s cosψ = sini cosλsini +cosi cosi , (1b) s o s o sinψcosφ = sini cosλcosi cosi sini . (1c) s o s o − – 6 – y y y0 no no ψ ns λ φ io ns is z x x0≡x z z0 Fig. 1.— Spin-orbit geometry. Left panel: Observer-oriented coordinate system. Here the z axis points toward the observer, the x axis points along the line of nodes, the y axis completes a right- handed triad, and the xy plane is the plane of the sky. Right panel: Orbit-oriented coordinate system (obtained from the observer-oriented system by a counterclockwise rotation of π/2 i o − about the x x axis). Here the y axis is the planetary orbital axis and the xz plane is the orbital (cid:48) (cid:48) (cid:48) (cid:48) ≡ plane. The unit vectors n and n respectively denote the orbital and stellar rotation angular o s momentum unit vectors. All depicted angles are introduced in the text. Equation (1b) is of particular interest, as it allows determination of the spin-orbit angle ψ providedthatmeasurementsofi ,i ,andλareavailable. Ajointanalysisofasteroseismology, o s the transit light curve, and the RM effect made it possible to determine ψ for the hot-Jupiter system HAT-P-7 (Kepler-2; Benomar et al. 2014; Lund et al. 2014b) and the multi-transiting system Kepler-25 (Benomar et al. 2014). Both these systems are revisited in this work. For an individual transiting system, we would still expect to place mild constraints on ψ even when lacking a measurement of λ. In Fig. 2 we show the posterior probability distribution (after normalization) for ψ conditioned on i and i , p(ψ i ,i ) (see Appendix s o s o | A for a derivation of the analytical expression). We have assumed an edge-on orbit (i.e., i =90 ), having selected three error-free values for the stellar inclination angle (i =30 , o ◦ s ◦ i =60 , and i =85 ). The main conclusions to be drawn from this simple exercise follow s ◦ s ◦ immediately from an inspection of Fig. 2: For a transiting system, a small value of i implies s a spin-orbit misalignment. The converse is not true, since a large value of i is consistent s with, but does not necessarily imply, a spin-orbit alignment. The interpretation of the spin- – 7 – Fig. 2.— Posterior probability distribution for the spin-orbit angle ψ conditioned on i and i , s o p(ψ i ,i ). We have assumed an edge-on orbit (i.e., i =90 ), having selected three error-free values s o o ◦ | for i (i =30 , i =60 , and i =85 ). The vertical dashed lines are placed at the asymptotes s s ◦ s ◦ s ◦ ψ= i i and ψ=i +i . The vertical dotted line at ψ=π/2 marks the transition between a o s o s | − | prograde and a retrograde orbit. orbit alignment in terms of the measured i can thus be ambiguous. This provides one of s the main motivations for this work, namely that, in order to draw general inferences about spin-orbit alignment, a statistical analysis of an ensemble of such measurements is needed. 3. Asteroseismic analysis 3.1. Sample characterization Our asteroseismic sample consists of 25 solar-type KOIs, of which 20 are confirmed hosts and thus have been assigned a Kepler ID (see Table 1 for a complete list). At the time of writing, all planets in the systems awaiting confirmation have been designated as – 8 – ‘CANDIDATE’ in the cumulative table of the NASA Exoplanet Archive2 (Akeson et al. 2013). Fundamental properties (e.g., surface gravity, radius, mass, and age) from a detailed asteroseismic analysis are available for all the KOIs in the sample (Silva Aguirre et al. 2015). A systematic study of Kepler planet-candidate hosts using asteroseismology was pre- sented by Huber et al. (2013a), in which fundamental properties were determined for 66 host stars based on their average asteroseismic parameters. This raised the number of Kepler hosts with asteroseismic solutions to nearly 80 stars. Whether or not a given host star is in- cluded in the present sample was determined by our ability to resolve and extract signatures of rotation in the oscillation spectrum, which required relatively bright targets (Kepler-band magnitude m (cid:46)12) and multi-quarter observations. The intrinsic stellar properties have Kep also played a crucial role in this regard, since it is well known that the signatures of rotation tend to be substantially blended in the power spectra of main-sequence solar-like oscillators hotter than about 6400 K (e.g., Appourchaux et al. 2012). Figure 3 displays the sample on a logg vs. T diagram. They are predominantly positioned along the main sequence eff and range in spectral type from early K to late F (i.e., 5000K(cid:46)T (cid:46)6500K). A number eff of stars in the sample seem to have evolved slightly beyond the main-sequence phase, one example being the bright subgiant Kepler-21 (Howell et al. 2012). There is also varying level of evidence of mixed (e.g., Osaki 1975; Aizenman et al. 1977) quadrupole modes in the power spectra of Kepler-36, Kepler-100, Kepler-128, and Kepler-129, an indication that these stars may have already left the main sequence. The fact that central hydrogen has been depleted in models of these stars (Silva Aguirre et al. 2015) supports this scenario. The sample contains 16 multiple-planet systems, of which all except Kepler-93 (Ballard et al. 2014) and Kepler-410 A (Van Eylen et al. 2014) are also multi-transiting systems3. Moreover, anon-transitingplanetwasrevealedbyRVmeasurementsorbitingbeyondthetwo transiting planets in the Kepler-25 (Marcy et al. 2014) and Kepler-68 (Gilliland et al. 2013) systems. Most of the multi-transiting systems in our sample have had the eccentricities of their planets measured from transit photometry (Van Eylen & Albrecht 2015), which revealed a tendency toward low eccentricities that are consistent with nearly circular orbits. Oftheremaining9single-planetsystems, onlyone(HAT-P-7;Pa´letal.2008)isahot-Jupiter system. From Table 1, we also see a clear prevalence of systems that contain small planets (i.e., R 4R ) and long-period planets (i.e., P >10d). p o ≤ ⊕ 2http://exoplanetarchive.ipac.caltech.edu/ 3Although being a single-transiting system, transit-timing variations (TTVs) suggest the presence of at least one additional (non-transiting) planet in the Kepler-410 A system. – 9 – 3.4 Multiple-planetsystems Single-planetsystems 3.6 3.8 1.6M⊙ 1.4M⊙ 1.2M⊙ ex] 4 1.0M⊙ [d 0.8M⊙ g g o l 4.2 4.4 ⊕⊕⊕⊕ 4.6 7000 6500 6000 5500 5000 4500 Teff [K] Fig. 3.— Surface gravity vs. effective temperature for the KOIs in the asteroseismic sample. Filled blue circles represent multiple-planet systems, while open red circles represent single-planet systems. Symbol size scales linearly with planetary size (for multiple-planet systems, the smallest planet is considered). For reference, a hypothetical solar twin harboring an Earth-size planet is represented by ‘ ’. Solar-calibrated evolutionary tracks spanning the mass range 0.8–1.6M (in ⊕ (cid:12) stepsof0.2M )areshownascontinuouslines. ThesetrackshavebeencomputedusingtheModules (cid:12) for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011, 2013) evolution code. 3.2. Data preparation Raw pixel data (Jenkins et al. 2010) were downloaded from the Kepler Asteroseismic Science Operations Center4 (KASOC) database and subsequently run through the homony- mous filter (Handberg & Lund 2014). The KASOC filter has been specifically designed to automaticallycarryoutthepreparationofKeplerphotometrictimeseriesofplanet-candidate hosts for asteroseismic analysis. The time series used in this work were acquired in short- cadence mode (∆t 58.85s) to allow investigation of solar-like oscillations in main-sequence ∼ stars, whose dominant periods are typically several minutes. A weighted least-squares sine- wave-fitting method was then used to compute rms-scaled power spectra of the time series for further analysis (Kjeldsen 1992; Frandsen et al. 1995). 4http://kasoc.phys.au.dk – 10 – 3.3. Estimation of the stellar inclination angle 3.3.1. Principle Solar-like oscillations are predominantly acoustic global standing waves. Commonly called p modes, owing to the fact that pressure plays the role of the restoring force, these modes are intrinsically damped and stochastically excited by near-surface convection (for a review see, e.g., Christensen-Dalsgaard 2004; Cunha et al. 2007; Chaplin & Miglio 2013). The oscillation modes are characterized by the radial order n, the spherical degree l, and the azimuthal order m. Radial modes have l = 0, whereas non-radial modes have l > 0. Values of m range from l to l, meaning that there are 2l + 1 azimuthal components for − a given multiplet of degree l. Observed oscillation modes are typically high-order modes of low spherical degree, with the associated power spectrum showing a pattern of peaks with near-regular frequency separations (Vandakurov 1967; Tassoul 1980). The asteroseismic estimation of i rests on our ability to resolve and extract signatures s of rotation in the power spectra of non-radial modes of oscillation. Rotation lifts the de- generacy in the frequencies, ν , of non-radial modes and introduces a dependence of the nl oscillation frequencies on m, with prograde (retrograde) modes (with m>0 and m<0, re- spectively) having frequencies slightly higher (lower) than the axisymmetric mode (m=0) in the observer’s frame of reference. For the fairly modest values of the stellar angular velocity Ω that are typical of solar-like oscillators, the effect of rotation can be treated following a perturbative analysis (e.g., Reese et al. 2006) and the star is generally assumed to rotate as a solid body (i.e., Ω=const.). In the limit of solid-body rotation, the frequency ν of a nlm mode, as observed in an inertial frame, can be expressed to first order as (Ledoux 1951): Ω ν = ν +m (1 C ) ν +mν . (2) nlm nl0 nl nl0 s 2π − ≈ The kinematic splitting mΩ/(2π) is corrected for the effect of the Coriolis force through the dimensionless Ledoux constant, C . For high-order p modes, C 1, with the rotational nl nl (cid:28) splitting being dominated by advection and given approximately by the angular velocity, i.e., ν Ω/(2π) (e.g., Lund et al. 2014a; Davies et al. 2015a). s ≈ To a second order of approximation, centrifugal effects that disrupt the equilibrium structure of the star can be taken into account through an additional frequency perturbation (e.g.,Ballot2010). Thisperturbationscalesastheratioofthecentrifugaltothegravitational forces at the stellar surface, i.e., Ω2 R3/(GM ), where G is the gravitational constant. We surf s s madeuseoftheavailablevaluesofP inTable1tocomputetheratiosofthesurfaceangular rot (cid:112) velocity to the Keplerian break-up rotation rate, i.e., Ω /Ω 2π/(P GM /R3). We surf K ≡ rot s s obtained (Ω /Ω )2 (cid:46) 9 10 4 for the stars in the asteroseismic sample and have thus surf K − ×