Spin-orbit Angle Research Articles

WASP-167b/KELT-13b: joint discovery of a hot Jupiter transiting a rapidly rotating F1V star

We report the joint WASP/KELT discovery of WASP-167b/KELT-13b, a transiting hot Jupiter with a 2.02-d orbit around a $V$ = 10.5, F1V star with [Fe/H] = 0.1 $\pm$ 0.1. The 1.5 R$_{\rm Jup}$ planet was confirmed by Doppler tomography of the stellar line profiles during transit. We place a limit of $<$ 8 M$_{\rm Jup}$ on its mass. The planet is in a retrograde orbit with a sky-projected spin-orbit angle of $\lambda = -165^{\circ} \pm 5^{\circ}$. This is in agreement with the known tendency for orbits around hotter stars to be more likely to be misaligned. WASP-167/KELT-13 is one of the few systems where the stellar rotation period is less than the planetary orbital period. We find evidence of non-radial stellar pulsations in the host star, making it a $\delta$-Scuti or $\gamma$-Dor variable. The similarity to WASP-33, a previously known hot-Jupiter host with pulsations, adds to the suggestion that close-in planets might be able to excite stellar pulsations.

The GAPS Programme with HARPS-N at TNG

In the framework of the GAPS project, we are conducting an observational programme aimed at the determination of the orbital obliquity of known transiting exoplanets. The targets are selected to probe the obliquity against a wide range of stellar and planetary physical parameters. We exploit high-precision radial velocity (RV) measurements, delivered by the HARPS-N spectrograph at the 3.6m Telescopio Nazionale Galileo, to measure the Rossiter-McLaughlin (RM) effect in RV time-series bracketing planet transits, and to refine the orbital parameters determinations with out-of-transit RV data. We also analyse new transit light curves obtained with several 1-2m class telescopes to better constrain the physical fundamental parameters of the planets and parent stars. We report here on new transit spectroscopic observations for three very massive close-in giant planets: WASP43b, HATP20b and Qatar2b orbiting dwarf K-type stars with effective temperature well below 5000K. We find lambda = 3.5pm6.8 deg for WASP43b and lambda = -8.0pm6.9 deg for HATP20b, while for Qatar2, our faintest target, the RM effect is only marginally detected, though our best-fit value lambda = 15pm20 deg is in agreement with a previous determination. In combination with stellar rotational periods derived photometrically, we estimate the true spin-orbit angle, finding that WASP43b is aligned while the orbit of HATP20b presents a small but significant obliquity (Psi=36 _{-12}^{+10} deg). By analyzing the CaII H&K chromospheric emission lines for HATP20 and WASP43, we find evidence for an enhanced level of stellar activity which is possibly induced by star-planet interactions.

Tidal interactions in spin–orbit misaligned systems

Spin-orbit misalignments have been detected in exoplanetary systems and binary star systems. Tidal interactions may have played an important role in the evolution of the spin-orbit angle. In this study, we investigate the tidal interactions in spin-orbit misaligned systems. In particular, we focus on the tidal response of a rotating fluid body to the obliquity tide, which may be important for the evolution of the spin-orbit angle but hardly affects the orbital evolution. The obliquity tide also provides a torque for the mutual precession of the spin and orbital axes around the total angular momentum vector, which has not yet been considered in previous studies on the tidal interactions. In this paper, we first formulate a set of linearized equations describing the tidal response in spin-orbit misaligned systems, taking into account the precessional motion. Numerical solutions in a homogeneous fluid and in a polytrope of index 1 show that dissipative inertial waves can be excited on top of precession by the obliquity tide in the presence of a rigid core. The tidal quality factor associated with the obliquity tide $Q'_{210}$ can be several orders of magnitude smaller than those associated with other tidal components if their frequencies fall outside the frequency range of inertial waves. Therefore, it is possible that the spin-orbit misalignment undergoes much more rapid decay than the orbital decay in hot Jupiter systems owing to the enhanced dissipation of the obliquity tide.

THE GRAVITATIONAL INTERACTION BETWEEN PLANETS ON INCLINED ORBITS AND PROTOPLANETARY DISKS AS THE ORIGIN OF PRIMORDIAL SPIN–ORBIT MISALIGNMENTS

ABSTRACT Many of the observed spin–orbit alignment properties of exoplanets can be explained in the context of the primordial disk misalignment model, in which an initially aligned protoplanetary disk is torqued by a distant stellar companion on a misaligned orbit, resulting in a precessional motion that can lead to large-amplitude oscillations of the spin–orbit angle. We consider a variant of this model in which the companion is a giant planet with an orbital radius of a few astronomical units. Guided by the results of published numerical simulations, we model the dynamical evolution of this system by dividing the disk into inner 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 remains unchanged. In addition to the oscillations induced by the precessional motion, whose amplitude is larger the smaller the initial inner-disk-to-planet mass ratio, the spin–orbit angle also exhibits a secular growth in this case—driven by ongoing mass depletion from the disk—that becomes significant when the inner disk’s angular momentum drops below that of the planet. Altogether, these two effects can produce significant misalignment angles for the inner disk, including retrograde configurations. We discuss these results within the framework of the Stranded Hot Jupiter scenario and consider their implications, including the interpretation of the alignment properties of debris disks.

Dynamics of stellar spin driven by planets undergoing Lidov–Kozai migration: paths to spin–orbit misalignment

Many exoplanetary systems containing hot Jupiters (HJs) exhibit significant misalignment between the spin axes of the host stars and the orbital angular momentum axes of the planets ("spin-orbit misalignment"). High-eccentricity migration involving Lidov-Kozai oscillations of the planet's orbit induced by a distant perturber is a possible channel for producing such misaligned HJ systems. Previous works have shown that the dynamical evolution of the stellar spin axis during the high-$e$ migration plays a dominant role in generating the observed spin-orbit misalignment. Numerical studies have also revealed various patterns of the evolution of the stellar spin axis leading to the final misalignment. Here we develop an analytic theory to elucidate the evolution of spin-orbit misalignment during the Lidov-Kozai migration of planets in stellar binaries. Secular spin-orbit resonances play a key role in the misalignment evolution. We include the effects of short-range forces and tidal dissipation, and categorize the different possible paths to spin-orbit misalignment as a function of various physical parameters (e.g. planet mass and stellar rotation period). We identify five distinct spin-orbit evolution paths and outcomes, only two of which are capable of producing retrograde orbits. We show that these paths to misalignment and the outcomes depend only on two dimensionless parameters, which compare the stellar spin precession frequency with the rate of change of the planet's orbital axis, and the Lidov-Kozai oscillation frequency. Our analysis reveals a number of novel phenomena for the stellar spin evolution, ranging from bifurcation, adiabatic advection, to fully chaotic evolution of spin-orbit angles.

WARM JUPITERS FROM SECULAR PLANET–PLANET INTERACTIONS

ABSTRACT Most warm Jupiters (gas-giant planets with 0.1 au ≲ a ≲ 1 au) have pericenter distances that are too large for significant orbital migration by tidal friction. We study the possibility that the warm Jupiters are undergoing secular eccentricity oscillations excited by an outer companion (a planet or star) in an eccentric and/or mutually inclined orbit. In this model, the warm Jupiters migrate periodically, in the high-eccentricity phase of the oscillation, but are typically observed at lower eccentricities. We show that in this model the steady-state eccentricity distribution of the warm Jupiters is approximately flat, which is consistent with the observed distribution if we restrict the sample to warm Jupiters with detected outer planetary companions. The eccentricity distribution of warm Jupiters without companions exhibits a peak at e ≲ 0.2 that must be explained by a different formation mechanism. Based on a population synthesis study, we find that high-eccentricity migration excited by an outer planetary companion (1) can account for ∼ 20 % of the warm Jupiters and most of the warm Jupiters with e ≳ 0.4 ; and (2) can produce most of the observed population of hot Jupiters, with a semimajor axis distribution that matches the observations, but fails to account adequately for ∼ 60 % of hot Jupiters with projected obliquities ≲ 20 ° . Thus ∼ 20 % of the warm Jupiters and ∼ 60 % of the hot Jupiters can be produced by high-eccentricity migration. We also provide predictions for the expected mutual inclinations and spin-orbit angles of the planetary systems with hot and warm Jupiters produced by high-eccentricity migration.

Can stellar activity make a planet seem misaligned?

Several studies have shown that the occultation of stellar active regions by the transiting planet can generate anomalies in the high-precision transit light curves, and these anomalies may lead to an inaccurate estimate of the planetary parameters (e.g., the planet radius). Since the physics and geometry behind the transit light curve and the Rossiter- McLaughlin effect (spectroscopic transit) are the same, the Rossiter-McLaughlin observations are expected to be affected by the occultation of stellar active regions in a similar way. In this paper we perform a fundamental test on the spin-orbit angles as derived by Rossiter-McLaughlin measurements, and we examine the impact of the occultation of stellar active regions by the transiting planet on the spin-orbit angle estimations. Our results show that the inaccurate estimation on the spin-orbit angle due to stellar activity can be quite significant (up to 30 degrees), particularly for the edge-on, aligned, and small transiting planets. Therefore, our results suggest that the aligned transiting planets are the ones that can be easily misinterpreted as misaligned owing to the stellar activity. In other words, the biases introduced by ignoring stellar activity are unlikely to be the culprit for the highly misaligned systems.

Interplay between resonant tunneling and spin precession oscillations in all-electric all-semiconductor spin transistors

We investigate the transmission properties of a spin transistor coupled to two quantum point contacts acting as spin injector and detector. In the Fabry-Perot regime, transport is mediated by quasibound states formed between tunnel barriers. Interestingly, the spin-orbit interaction of the Rashba type can be tuned in such a way that nonuniform Rashba fields can point along distinct directions in different points of the sample. We discuss both spin-conserving and spin-flipping transitions as the spin-orbit angle of orientation increases from parallel to antiparallel configurations. Spin precession oscillations are clearly seen as a function of the length of the central channel. Remarkably, we find that these oscillations combine with the Fabry-Perot motion giving rise to quasiperiodic transmissions in the purely one-dimensional case. Furthermore, we consider the more realistic case of a finite width in the transverse direction and find that the coherent oscillations become deteriorated for moderate values of the spin-orbit strength. Our results then determine the precise role of the Rashba intersubband coupling potential in the Fabry-Perot-Datta-Das intermixed oscillations.

Spin–orbit alignment for KELT-7b and HAT-P-56b via Doppler tomography with TRES

We present Doppler tomographic analyses for the spectroscopic transits of KELT-7b and HAT-P-56b, two hot-Jupiters orbiting rapidly rotating F-dwarf host stars. These include analyses of archival TRES observations for KELT-7b, and a new TRES transit observation of HAT-P-56b. We report spin-orbit aligned geometries for KELT-7b (2.7 +/- 0.6 deg) and HAT-P-56b (8 +/- 2 deg). The host stars KELT-7 and HAT-P-56 are among some of the most rapidly rotating planet-hosting stars known. We examine the tidal re-alignment model for the evolution of the spin-orbit angle in the context of the spin rates of these stars. We find no evidence that the rotation rates of KELT-7 and HAT-P-56 have been modified by star-planet tidal interactions, suggesting that the spin-orbit angle of systems around these hot stars may represent their primordial configuration. In fact, KELT-7 and HAT-P-56 are two of three systems in super-synchronous, spin-orbit aligned states, where the rotation periods of the host stars are faster than the orbital periods of the planets.

SPIN–ORBIT ALIGNMENT FOR THREE TRANSITING HOT JUPITERS: WASP-103b, WASP-87b, and WASP-66b†

ABSTRACT We have measured the sky-projected spin–orbit alignments for three transiting hot Jupiters, WASP-103b, WASP-87b, and WASP-66b, using spectroscopic measurements of the Rossiter–McLaughlin effect, with the CYCLOPS2 optical fiber bundle system feeding the UCLES spectrograph on the Anglo-Australian Telescope. The resulting sky-projected spin–orbit angles of λ = 3° ± 33°, λ = −8° ± 11°, and λ = −4° ± 22° for WASP-103b, WASP-87b, and WASP-66b, respectively, suggest that these three planets are likely on nearly aligned orbits with respect to their host star’s spin axis. WASP-103 is a particularly interesting system as its orbital distance is only 20% larger than its host star’s Roche radius and the planet likely experiences strong tidal effects. WASP-87 and WASP-66 are hot (T eff = 6450 ± 120 K and T eff = 6600 ± 150 K, respectively) mid-F stars, making them similar to the majority of stars hosting planets on high-obliquity orbits. Moderate spin–orbit misalignments for WASP-103b and WASP-66b are consistent with our data, but polar and retrograde orbits are not favored for these systems.

On turbulence driven by axial precession and tidal evolution of the spin–orbit angle of close-in giant planets

The spin axis of a rotationally deformed planet is forced to precess about its orbital angular momentum vector, due to the tidal gravity of its host star, if these directions are misaligned. This induces internal fluid motions inside the planet that are subject to a hydrodynamic instability. We study the turbulent damping of precessional fluid motions, as a result of this instability, in the simplest local computational model of a giant planet (or star), with and without a weak internal magnetic field. Our aim is to determine the outcome of this instability, and its importance in driving tidal evolution of the spin-orbit angle in precessing planets (and stars). We find that this instability produces turbulent dissipation that is sufficiently strong that it could drive significant tidal evolution of the spin-orbit angle for hot Jupiters with orbital periods shorter than about 10-18 days. If this mechanism acts in isolation, this evolution would be towards alignment or anti-alignment, depending on the initial angle, but the ultimate evolution (if other tidal mechanisms also contribute) is expected to be towards alignment. The turbulent dissipation is proportional to the cube of the precession frequency, so it leads to much slower damping of stellar spin-orbit angles, implying that this instability is unlikely to drive evolution of the spin-orbit angle in stars (either in planetary or close binary systems). We also find that the instability-driven flow can act as a system-scale dynamo, which may play a role in producing magnetic fields in short-period planets.

DIFFICULTY IN THE FORMATION OF COUNTER-ORBITING HOT JUPITERS FROM NEAR-COPLANAR HIERARCHICAL TRIPLE SYSTEMS: A SUB-STELLAR PERTURBER

ABSTRACT Among 100 transiting planets with a measured projected spin–orbit angle λ, several systems are suggested to be counter-orbiting. While these cases may be due to the projection effect, the mechanism that produces a counter-orbiting planet has not been established. A promising scenario for counter-orbiting planets is the extreme eccentricity evolution in near-coplanar hierarchical triple systems with eccentric inner and outer orbits. We examine this scenario in detail by performing a series of systematic numerical simulations, and consider the possibility of forming hot Jupiters (HJs), especially a counter-orbiting one under this mechanism with a distant sub-stellar perturber. We incorporate quadrupole and octupole secular gravitational interaction between the two orbits, and also short-range forces (correction for general relativity, star and inner planetary tide, and rotational distortion) simultaneously. We find that most systems are tidally disrupted and that a small fraction of the surviving planets turn out to be prograde. The formation of counter-orbiting HJs in this scenario is possible only in a very restricted parameter region, and thus is very unlikely in practice.

WASP-121 b: a hot Jupiter close to tidal disruption transiting an active F star

We present the discovery by the WASP-South survey of WASP-121 b, a new remarkable short-period transiting hot Jupiter. The planet has a mass of |$1.183_{-0.062}^{+0.064}$|MJup, a radius of 1.865 ± 0.044 RJup, and transits every |$1.274\,9255_{-0.000\,0025}^{+0.000\,0020}$| days an active F6-type main-sequence star (V = 10.4, |$1.353_{-0.079}^{+0.080}$| M⊙, 1.458 ± 0.030 R⊙, Teff = 6460 ± 140 K). A notable property of WASP-121 b is that its orbital semimajor axis is only ∼1.15 times larger than its Roche limit, which suggests that the planet is close to tidal disruption. Furthermore, its large size and extreme irradiation (∼7.1 109 erg s−1 cm−2) make it an excellent target for atmospheric studies via secondary eclipse observations. Using the TRAnsiting Planets and PlanetesImals Small Telescope, we indeed detect its emission in the z′-band at better than ∼4σ, the measured occultation depth being 603 ± 130 ppm. Finally, from a measurement of the Rossiter–McLaughlin effect with the CORALIE spectrograph, we infer a sky-projected spin-orbit angle of |$257{^{\circ}_{.}} 8_{-5{^{\circ}_{.}} 5}^{+5{^{\circ}_{.}} 3}$|⁠. This result may suggest a significant misalignment between the spin axis of the host star and the orbital plane of the planet. If confirmed, this high misalignment would favour a migration of the planet involving strong dynamical events with a third body.

SPIN–ORBIT ALIGNMENT OF EXOPLANET SYSTEMS: ENSEMBLE ANALYSIS USING ASTEROSEISMOLOGY

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 s via asteroseismology for an ensemble of 25 solar-type hosts observed with NASA’s Kepler satellite. Our results for i s are consistent with alignment at the 2σ 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 ( &psgr; = 116 &fdg; 4 &minus; 14.7 &plus; 30.2 ?> ), whereas that of Kepler-25c seems to be well aligned with the stellar spin axis ( &psgr; = 12 &fdg; 6 &minus; 11.0 &plus; 6.7 ?> ). While the latter result is in apparent contradiction with a statement 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 values of inclination.

The GAPS programme with HARPS-N at TNG

Aims. For more than 1.5 years we monitored spectroscopically the star KELT-6 (BD+312447), known to host the transiting hot Saturn KELT-6b, because a previously observed long-term trend in radial velocity time series suggested the existence of an outer companion. Methods. We collected a total of 93 new spectra with the HARPS-N and TRES spectrographs. A spectroscopic transit of KELT-6b was observed with HARPS-N, and simultaneous photometry was obtained with the IAC-80 telescope. Results. We proved the existence of an outer planet with a mininum mass M$_{\rm p}$sini=3.71$\pm$0.21 M$_{\rm Jup}$ and a moderately eccentric orbit ($e=0.21_{-0.036}^{+0.039}$) of period P$\sim$3.5 years. We improved the orbital solution of KELT-6b and obtained the first measurement of the Rossiter-McLaughlin effect, showing that the planet has a likely circular, prograde, and slightly misaligned orbit, with a projected spin-orbit angle $\lambda$=$-$36$\pm$11 degrees. We improved the KELT-6b transit ephemeris from photometry, and we provided new measurements of the stellar parameters. KELT-6 appears as an interesting case to study the formation and evolution of multi-planet systems.

The GAPS programme with HARPS-N at TNG

XO-2 is the first confirmed wide stellar binary system where the almost twin components XO-2N and XO-2S have planets. This stimulated a detailed characterization study of the stellar and planetary components based on new observations. We collected high-resolution spectra with the HARPS-N spectrograph and multi-band light curves. Spectral analysis led to an accurate determination of the stellar atmospheric parameters and characterization of the stellar activity. We collected 14 transit light curves of XO-2Nb used to improve the transit parameters. Photometry provided accurate magnitude differences between the stars and a measure of their rotation periods. The iron abundance of XO-2N was found to be +0.054 dex greater, within more than 3-sigma, than that of XO-2S. We confirm a long-term variation in the radial velocities of XO-2N, and we detected a turn-over with respect to previous measurements. We suggest the presence of a second massive companion in an outer orbit or the stellar activity cycle as possible causes of the observed acceleration. The latter explanation seems more plausible with the present dataset. We obtained an accurate value of the projected spin-orbit angle for the XO-2N system (lambda=7+/-11 degrees), and estimated the real 3-D spin-orbit angle (psi=27 +12/-27 degrees). We measured the XO-2 rotation periods, and found a value of P=41.6 days in the case of XO-2N, in excellent agreement with the predictions. The period of XO-2S appears shorter, with an ambiguity between 26 and 34.5 days that we cannot solve with the present dataset alone. XO-2N appears to be more active than the companion, and this could be due to the fact that we sampled different phases of their activity cycle, or to an interaction between XO-2N and its hot Jupiter that we could not confirm.

THE WELL-ALIGNED ORBIT OF WASP-84b: EVIDENCE FOR DISK MIGRATION OF A HOT JUPITER

We report the sky-projected orbital obliquity (spin-orbit angle) of WASP-84b, a 0.70-$M_{\rm Jup}$ planet in a 8.52-day orbit around a G9V/K0V star, to be $\lambda = 0.3 \pm 1.7^\circ$. We obtain a true obliquity of $\psi = 14.8 \pm 8.0^\circ$ from a measurement of the inclination of the stellar spin axis with respect to the sky plane. Due to the young age and the weak tidal forcing of the system, we suggest that the orbit of WASP-84b is unlikely to have both realigned and circularised from the misaligned and/or eccentric orbit likely to have arisen from high-eccentricity migration. Therefore we conclude that the planet probably migrated via interaction with the protoplanetary disc. This would make it the first short-orbit, giant planet to have been shown to have migrated via this pathway. Further, we argue that the distribution of obliquities for planets orbiting cool stars ($T_{\rm eff}$ < 6250 K) suggests that high-eccentricity migration is an important pathway for the formation of short-orbit, giant planets.

WASP-94 A and B planets: hot-Jupiter cousins in a twin-star system

We report the discovery of two hot-Jupiter planets, each orbiting one of the stars of a wide binary system. WASP-94A (2MASS 20550794-3408079) is an F8 type star hosting a transiting planet with a radius of 1.72 +/- 0.06 R_Jup, a mass of 0.452 +/- 0.034 M_Jup, and an orbital period of 3.95 days. The Rossiter-McLaughlin effect is clearly detected, and the measured projected spin-orbit angle indicates that the planet occupies a retrograde orbit. WASP-94B (2MASS 20550915-3408078) is an F9 stellar companion at an angular separation of 15" (projected separation 2700 au), hosting a gas giant with a minimum mass of 0.618 +/- 0.028 M_Jup with a period of 2.008 days, detected by Doppler measurements. The orbital planes of the two planets are inclined relative to each other, indicating that at least one of them is inclined relative to the plane of the stellar binary. These hot Jupiters in a binary system bring new insights into the formation of close-in giant planets and the role of stellar multiplicity.

DERIVING STELLAR INCLINATION OF SLOW ROTATORS USING STELLAR ACTIVITY

Stellar inclination is an important parameter for many astrophysical studies. Although different techniques allow us to estimate stellar inclinationt for fast rotators, it becomes much more difficult when stars are rotating slower than $\sim2$-2.5 \kms. By using the new activity simulation SOAP 2.0 that can reproduce the photometric and spectroscopic variations induced by stellar activity, we are able to fit observations of solar-type stars and derive their inclination. For HD189733, we estimate the stellar inclination to be $i=84^{+6}_{-20}$ degrees, which implies a star-planet obliquity of $\psi=4^{+18}_{-4}$ considering previous measurements of the spin-orbit angle. For $\alpha$ Cen B, we derive an inclination of $i=45^{+9}_{-19}$, which implies that the rotational spin of the star is not aligned with the orbital spin of the $\alpha$ Cen binary system. In addition, assuming that $\alpha$ Cen Bb is aligned with its host star, no transit would occur. The inclination of $\alpha$ Cen B can be measured using 40 radial-velocity measurements, which is remarkable given that the projected rotational velocity of the star is smaller than $1.15\,km\,s^{-1}$.

Determination of three-dimensional spin–orbit angle with joint analysis of asteroseismology, transit lightcurve, and the Rossiter–McLaughlin effect: Cases of HAT-P-7 and Kepler-25

Abstract We develop a detailed methodology of determining three-dimensionally the angle between the stellar spin and the planetary orbit axis vectors, ψ, for transiting planetary systems. The determination of ψ requires the independent estimates of the inclination angles of the stellar spin axis and of the planetary orbital axis with respect to the line of sight, i⋆ and iorb, and the projection of the spin–orbit angle on to the plane of the sky, λ. These are mainly derived from asteroseismology, transit lightcurve, and the Rossiter–McLaughlin effect, respectively. The detailed joint analysis of those three datasets enables an accurate and precise determination of the numerous parameters characterizing the planetary system, in addition to ψ. We demonstrate the power of the joint analysis for the two specific systems HAT-P-7 and Kepler-25. HAT-P-7b is the first exoplanet suspected to be a retrograde (or polar) planet because of the significant misalignment λ ≈ 180°. Our joint analysis indicates i⋆ ≈ 30° and ψ ≈ 120°, suggesting that the planetary orbit is closer to polar rather than retrograde. Kepler-25 is one of the few multi-transiting planetary systems with measured λ, and hosts two short-period transiting planets and one outer non-transiting planet. The projected spin–orbit angle of the larger transiting planet, Kepler-25c, has been measured to be λ ≈ 0°, implying that the system is well aligned. With the help of the tight constraint from asteroseismology, however, we obtain $i_\star ={65{{^{\circ}_{.}}}4}^{+{10{{\kern.4pt.\vphantom{|}}^{\kern-2pt\circ}}6}}_{-{6{{\kern.4pt.\vphantom{|}}^{\kern-2pt\circ}}4}}$ and $\psi ={26{{^{\circ}_{.}}}9}^{+{7{{\kern.4pt.\vphantom{|}}^{\kern-2pt\circ}}0}}_{-{9{{\kern.4pt.\vphantom{|}}^{\kern-2pt\circ}}2}}$, and thus find that the system is actually mildly misaligned. This is the first detection of the spin–orbit misalignment for the multiple planetary system with a main-sequence host star, and points to mechanisms that tilt a stellar spin axis relative to its protoplanetary disk.