SPIN–ORBIT MISALIGNMENT OF TWO-PLANET-SYSTEM KOI-89 VIA GRAVITY DARKENING

The Mikulski Archive for Space Telescopes Kepler Input Catalog (KIC) provided the Kepler photometry that we analyze for the KOI-89 system. We employ each of the 16 available quarters of KIC data, combining them into a single data set. We choose to only incorporate long cadence data (30 minute integrations) because both planets have transit durations of over 12 hr. Therefore ingress and egress are well-sampled by the 30 minutes time cadence, and inclusion of short (1 minute) cadence data would not provide additional constraints.

After concatenating all available long cadence photometry, we apply a median box filter of 44 hours (three times KOI-89.01's transit duration) to reduce long-term astrophysical and instrumental variability. Figure 1 displays the filtered time series. We then identify which transits correspond to which transiting body based on their KIC orbital periods, and separate them accordingly into individual data sets.

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We adjust the center-of-transit times of each transit light curve according to their measured transit timing variations (TTV; Rowe et al. 2014). We perform this adjustment for each measured TTV in Rowe et al. (2014), including thirteen transits for KOI-89.01 and five transits for KOI-89.02. These transits exclude the double transit identified in Figure 1.

With all of the TTV accounted for and the individual transits evenly separated by 84.69 and 207.58 days, respectively, we fold all KOI-89.01 transits on top of the epoch 34960800 ± 400 s transit and fold all KOI-89.02 transits on top of the epoch 25041400 ± 700 s transit. We then combine the two resulting light curves back into a single data set. With the Kepler photometry represented by a single light curve with two transit events, we bin the data at 15 minutes to improve the computation time of our fit. We determine the error bars of the binned data from the standard deviation of the flux values in the bin.

We build our work upon previous research of the KOI-89 system. We obtain the KOI-89 stellar mass, stellar temperature, and transit periods from the Community Follow-up Observation Program (CFOP). Figure 2 shows the spectroscopic determination of v sin(i). We list these and other relevant system parameters in Table 1.

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Rowe et al. (2014) confirmed 715 new systems—including KOI-89—via multiplicity. The two planets have a period ratio near the 5:2 mean-motion resonance (2.45). Follow-up observations of these phenomena could confirm/deny the existence of additional orbiting bodies, and could further constrain this system's formation and evolution.

The light curves of KOI-89.01 and KOI-89.02 (Figure 4) display unusual shapes. KOI-89.01 has the asymmetry expected of a misaligned body orbiting a fast-rotator (Barnes 2009). KOI-89.02 does not display this asymmetry, possibly due to lower photometric precision. Both transits show sloped ingresses/egresses and entirely non-constant transit depths, producing dominant V-shaped light curves. A typical light curve is symmetric with a steep ingress and egress with a relatively flat bottom, rounded only by limb darkening.

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KOI-89's V-shaped light curves can arise in one of two ways. First, planets only grazing their star during transit rather than fully eclipsing it block constantly changing sky-projected areas. This effect creates sloped ingresses/egresses. However, this situation is improbable for KOI-89 as both planets would require similar, high impact parameter values despite having significantly different semimajor axes.

The second way KOI-89 could generate V-shaped light-curve geometries is by having a gravity-darkened star with a very high stellar obliquity ψ—i.e., pole-on. In this case, the planets transit near a stellar pole and the gravity-darkened equator surrounds the outer edge of the star. The limb-darkening and gravity-darkening effects combine together to create a significant center-to-edge luminosity gradient. At ingress, the planet blocks a continuously increasing total flux as it moves closer toward the center of the star, and vice versa during egress. This produces a V-shaped transit light curve for each planet (Barnes 2009, Figure 4), consistent with the lack of the typical ingress-egress asymmetry expected in a misaligned gravity-darkened transit. We test for the possibility of grazing transits by fitting the system with impact parameters nearly at and slightly above 1.0. We find that we can not match the system's light curve with grazing transits: such an event can not reproduce the proper ingress-egress asymmetry seen in KOI-89.01.

Figure 4 shows our best-fit light curve using grazing transits in blue. We apply grazing transits to both spherical and gravity-darkened models. We hold the stellar obliquity at zero in the gravity-darkened model to test the system for possible spin–orbit alignment. The poor fit of ${\chi }_{\mathrm{reduced}}^{2}=1.94$ (adjusted to account for holding the stellar obliquity constant) motivates us to investigate a model with a high stellar obliquity and rapid stellar rotation.

Using the Levenberg–Marqhardt χ² minimization technique, we fit for 13 parameters.

• 1.  
  The stellar equatorial radius (R_⋆).
• 2.  
  The stellar obliquity (ψ).
• 3.  
  The stellar normalized flux (F₀).
• 4.  
  The radii of KOI-89.01 and KOI-89.02 (${R}_{{p}_{1}}$, ${R}_{{p}_{2}}$).
• 5.  
  The inclinations of KOI-89.01 and KOI-89.02 (i₁, i₂).
• 6.  
  The sky-projected alignments (λ₁, λ₂).
• 7.  
  The two orbits' eccentricities (e₁ , e₂).
• 8.  
  The center-of-transit times (${T}_{{0}_{1}}$, ${T}_{{0}_{2}}$).

We display the best-fit light curve of our gravity-darkened model in Figure 4 as the red line.

We test the TTV ephemeris reported in Rowe et al. (2014) for systematic errors by fitting the KOI-89 series as two epochs. The first epoch is comprised of KOI-89.01's first seven transits and KOI-89.02's first two transits. The second epoch is comprised of KOI-89.01's remaining six transits and KOI-89.02's remaining three transits. For each half, we adjust all transits with respect to their TTV and fold the transits in the same fashion as described in Section 2.1.

We apply our gravity-darkening model to both epochs and find that the resulting parameters of each fitted data set have overlapping 1σ values with our best-fit values using the full time series (Table 2). We therefore detect no evidence of systematics in the TTV ephemeris listed in Rowe et al. (2014).

Table 2.  Best-fit Results for the KOI-89 System

Parameter	Best-fit Values
${\chi }_{\mathrm{reduced}}^{2}$	1.52
R⋆	2.3 ± 0.2 R⊙
ψ	69° ± 3°
c1 (fixed)	0.56
c2 (fixed)	−0.15
β (fixed)	0.25
F0	1.000009 ± 3 × 10−6
Prot (derived)	${8.81}_{-1.8}^{+1.9}\;\mathrm{hr}$
f⋆ (derived)	${0.19}_{-0.03}^{+0.04}$
${R}_{{{\rm{p}}}_{1}}$	0.45 ± 0.03 Rjup
${R}_{{{\rm{p}}}_{2}}$	0.43 ± 0.05 Rjup
e1 ≥	0.056 ± 0.019
e2 ≥	0.50 ± 0.09
i1	89340 ± 005
i2	9064 ± 006
b1 (derived)	${0.61}_{-0.07}^{+0.08}$
b2 (derived)	$-{0.57}_{-0.14}^{+0.12}$
${T}_{{0}_{1}}$	34960800 ± 400 s
${T}_{{0}_{2}}$	25041400 ± 700 s
λ1	−32° ± 11°
λ2	−32° ± 40°
φ1 (derived)	72° ± 3°
φ2 (derived)	${73}_{-5}^{^\circ +11}$

Note. We Calculated Stellar Period of Rotation P_rot from v sin(i), R_⋆, M_⋆, and ψ. We derived the stellar oblateness f from the Darwin–Radau relation. The impact parameters b₁ and b₂ were found using P₁ and P₂, i₁ and i₂, and R_⋆. We set our limb-darkening parameters c₁ = u₁ + u₂ and c₂ = u₁−u₂ according to Sing (2010).

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Limb-darkening has traditionally been problematic for the the gravity-darkening technique. Via Doppler Tomography, Johnson et al. (2014) measured the sky-projected alignment of KOI-13.01 and found it to differ significantly from the gravity-darkening measurement performed in Barnes et al. (2011). Masuda (2015) proposed a solution to this discrepancy by demonstrating that the gravity-darkening model produces concurring measurements with Doppler Tomography when using a nonzero second quadratic limb-darkening term (c₂). This limb-darkening term is also a possible explanation for KOI-368's different spin–orbit misalignment values measured in Zhou & Huang (2013) and Ahlers et al. (2014). These works motivated us to update our gravity-darkening model to include both quadratic terms, c₁ and c₂.

KOI-89's very high stellar obliquity brings about an additional challenge in resolving limb-darkening. With the stellar pole near the center of the sky-projected stellar disk, the gravity-darkening and limb-darkening luminosity gradients behave nearly identically and are essentially additive. The resulting combined effects on a transit light curve are therefore degenerate with a stellar obliquity near 90°.

KOI-89's spectroscopically determined effective temperature of 7717 ± 225 K corresponds to an approximate range of 0.55 to 0.57 for c₁ and −0.165 to −0.135 for c₂ (Sing 2010). We test the robustness of our assumed limb-darkening values by refitting using (0.55, −0.135) and (0.57, −0.165) for c₁ and c₂, respectively.

Applying the assumed (c₁, c₂) values (0.55, −0.135), we measure a slight increase in KOI-89.02's impact parameter; however, this increase vanishes when adjusting the gravity-darkening value β to match Altair's value of 0.19 (Monnier et al. 2007). We detect no significant changes in our best-fit results when employing the limb-darkening values (0.57, −0.165). We cannot resolve the accuracy of our limb-darkening parameters without higher precision data, and therefore elect to apply the assumed values of β = 0.25 (Von Zeipel 1924) and (c₁, c₂) values of (0.56, −0.15) (Sing 2010).

We constrain the lower limits of eccentricity to 0.056 ± 0.019 and 0.50 ± 0.09, respectively. We address the degeneracy between eccentricity and argument of periapsis following Price et al. (2015). Our eccentricities do not vary significantly for $| \omega | \leqslant 150^\circ$ away from center-of-transit, consistent with Price et al. (2015) and Barnes (2007). To find the lower limit for eccentricity, we set the center-of-transit at periapsis for both planets fit for the eccentricities using our gravity-darkened model.we analyze the plausibility of our eccentricity values in Section 5.2.

Our gravity-darkened model results in a degeneracy in the sky-projected alignments between the values λ and 180 − λ(Ahlers et al. 2014). We assume a prograde orbit for KOI-89.01, constraining λ to a single value. This allows us to produce single, nondegenerate values for the obliquity (ψ) and each planet's inclination (i) in our best-fit model, which we allow to float in the full range of 0°–360°.

We find that KOI-89 is highly misaligned with a stellar obliquity ψ of 69° ± 3°, inclinations i₁ and i₂ of 89340 ± 005 and 9064 ± 006 respectively, and sky-projected alignments λ₁ and λ₂ of −32° ± 11° and −32° ± 40° respectively. The high uncertainty of λ₂ is due to the apparent lack of asymmetry in KOI-89.02's light curve because of its photometrically imprecise data. We show our constraint of λ₂ in Section 5, which removes prograde/retrograde degeneracy via dynamic stability tests.

With these constraints we calculate the true spin–orbit misalignments φ₁ and φ₂ from the equation (Winn et al. 2007),

$$\begin{eqnarray}&&\mathrm{cos}({\varphi }_{i})=\mathrm{sin}(\psi )\mathrm{cos}({i}_{i})+\mathrm{cos}(\psi )\mathrm{sin}({i}_{i})\mathrm{cos}({\lambda }_{i})\end{eqnarray} \tag{ 1 }$$

modified for our parameter definitions. We calculate spin–orbit alignment angles of 72° ± 3° and ${73}_{-5}^{^\circ +11}$ for the two planets respectively. Table 2 lists all of KOI-89's parameter constraints.

KOI-89.01 and KOI-89.02 simultaneously transit halfway through 2011, causing a significantly larger transit depth. We indicate this event with the arrow in Figure 1 and show the double transit and its synthetic light curve in Figure 5. Our best-fit parameters produce a synthetic light curve that adequately models this event. We do not find evidence of a mutual event in the Kepler data set.

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We test the feasibility of KOI-89's oblateness value of ${0.19}_{-0.03}^{+0.04}$ by analyzing its breakup rotation period,

$$\begin{eqnarray}&&{P}_{{rot}}=2\pi \sqrt{\displaystyle \frac{{R}_{\star }^{3}}{{{GM}}_{\star }}}.\end{eqnarray} \tag{ 2 }$$

We find that the star is rotating at 55%–76% of its breakup speed by calculating its rotation period, listed in Table 2:

$$\begin{eqnarray}&&{P}_{\star }=\displaystyle \frac{2\pi {R}_{\star }\mathrm{cos}(\psi )}{v\mathrm{sin}(i)}.\end{eqnarray} \tag{ 3 }$$

This explains KOI-89's highly oblate shape and its gravity-darkened gradient. This star's oblateness is comparable to the fast-rotator Achernar with oblateness ∼0.36 (Carciofi et al. 2008) or other well-known oblate stars such as Altair (∼0.2; Monnier et al. 2007). Hence, our model produces physically plausible stellar parameters.

Equation (1) gives a planet's spin–orbit alignment φ_i dependence on the sky-projected alignment λ_i. We fit for KOI-89's λ_i in our gravity-darkening model, but are unable to resolve λ₂ due to its low photometric resolution and the host star's high stellar obliquity ψ. With gravity-darkening-driven asymmetry absent in KOI-89.02's light curve, we could not fully constrain its transit geometry.

To estimate KOI-89.02's sky-projected alignment, we tested the system for dynamic stability for various transit geometries. Using our gravity darkening model, we constrained KOI-89's orbital elements with various assumed λ₂ values. We then used the orbit integrator Mercury from Chambers (1999) to test each orbit geometry for dynamic stability.

Using Mercury, we perform mixed-variable symplectic (MVS) integrations of KOI-89 over 10⁸ years using 0.5 day timesteps. Using a spherical star allows for physically sound integrations that obey the conservation of angular momentum with minimal sacrifice; the stellar J2 ∼ 10⁻⁴ value (calculated following Murray & Dermont 2008), coupled with the planet's long orbit periods, cause nodal precession on a timescale that would not significantly affect the system's stability. We assume ice-giant densities of ρ = 1.64 g cm⁻³ for both planets.

We define an angle α of coalignment between the two orbits, defined relative to their angular momentum vectors:

$$\begin{eqnarray}&&\alpha \equiv {\mathrm{cos}}^{-1}\left(\displaystyle \frac{{{\boldsymbol{L}}}_{1}\cdot {{\boldsymbol{L}}}_{2}}{| {{\boldsymbol{L}}}_{1}| | {{\boldsymbol{L}}}_{2}| }\right).\end{eqnarray} \tag{ 4 }$$

Figure 6 shows the survival time of KOI-89 as a function of the coalignment angle α and conjuction longitude. By varying KOI-89.02's longitude of periapsis, we vary the conjuction longitude between the two planets. We define system instability as a planet ejection or collision event. None of our 360 simulations produced a stable orbit for 10⁸ years, indicating we have not found a physically viable system yet. In general, survival times are longer for lower α, but the longest-lived architectures are non-planar.

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The coalignment angle α is approximately the difference between the two planets' sky-projected alignment angles. Using the results of our orbital integrations, we estimate the difference between the sky-projected alignment angles $\alpha \approx | {\lambda }_{2}-{\lambda }_{1}|$ to be 20° ± 20°. This is a conservative estimate based on our results in Figure 6; follow-up observations would provide a much better calculation of this parameter.

Our 1332 orbit integrations resulted in a maximum survival time of 5.1 × 10⁷ years. The lack of stable configurations suggests that this system is not yet fully understood. If the system is in resonance and is non-planar, it may evolve chaotically (Barnes et al. 2015b), and hence long-lived configurations may only exist in small "islands" of parameter space. Alternatively, KOI-89's stability could be brought about by unknown additional bodies in the system. We show in Section 5.2 that KOI-89 could be stable if KOI-89.02's eccentricity is lower than our best-fit value of 0.50 ± 0.09. A better characterization of this system's stability could be understood via TTV analysis or Rossiter–McLaughlin measurements, but such work is outside the scope of this project.

In addition to our coalignment/mean longitude stability tests, we also test the stability of KOI-89.02's eccentricity of 0.50 ± 0.09 in a coplanar configuration. Van Eylen & Albrecht (2015) demonstrated that, in general, multiplanet systems have low eccentricities, making KOI-89 a potential exception to the rule. See Section 4.4 for an explanation of our treatment of longitude of periapsis.

We perform a series of integrations in Mercury (Chambers 1999) using assumed e₂ values ranging from 0.0 to 0.95 and a 0.05 step size. All e₂ ≤ 0.35 are stable, roughly consistent with Petrovich (2015). We show the results of these integrations in Figure 7.

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Not surprisingly, lower e₂ values yield longer lifetimes and overall higher stability. This result suggests three possibilities. The first (and least likely) possibility is that this system is in fact not coplanar. If the fitted e₂ value of 0.50 ± 0.09 is correct, then perhaps higher stabilities are found in slightly non-coplanar orbits. While higher stability in such a configuration is counterintuitive, it does at least reduce the odds of a close encounter between the two planets, limiting the chances of a violent collision/ejection event.

The second possibility is that our eccentricity measurement contains systematics. A grazing transit would reduce the transit duration time similarly to an eccentric orbit transiting near periapsis, and could produce a V-shaped light curve like we see in Figure 4. KOI-89.02's low signal-to-noise ratio, coupled with the degeneracy between impact parameter and planet radius that arises in all grazing transits, prevents us from resolving whether KOI-89.02 is in fact fully eclipsing its host star. KOI-89.02's TTVs could also drive up our eccentricity measurement if they are not fully accounted for (Van Eylen & Albrecht 2015). High-precision follow-up photometry could better determine KOI-89.02's orbit parameters, including its eccentricity.

The third possibility is that unknown bodies in the system provide stability to these orbits. Antoniadou & Voyatzis (2015) demonstrated that highly eccentric orbits in or near mean-motion resonance can exhibit long-term stability. Additional bodies could help stabilize KOI-89.01 and KOI-89.02, explaining why our best-fit parameters do not display dynamic stability through 10⁸ years in our orbit integrations.

Batygin (2012) first showed that a stellar companion could warp a star's protoplanetary disk into misalignment. Planets could then form in the plane of the disk, resulting in primordial spin–orbit misalignment (Batygin 2012; Lai 2014; Xiang-Gruess & Papaloizou 2014). This mechanism requires an unknown binary star in the KOI-89 system, but fundamentally agrees with our results in that it could produce highly misaligned, coplanar orbits.

A planet in the potential of a warped protoplanetary disk can be driven to very high misalignment values (Terquem 2013). Teyssandier et al. (2013) found that Jupiter-mass planets misaligned from a warped disk experience dynamic friction that realigns the planet on timescales shorter than the lifetime of the disk. However, Neptune-mass planets can remain misaligned and have their eccentricities driven up by orbital perturbations from the disk's gravitational potential. This mechanism has only been applied to single-planet systems, so criterion 3 is inconclusive. However, this mechanism agrees with the first two criteria and cannot be ruled out based on our results.

Our results cannot entirely rule out planet–planet scattering, which is orbit migration due to close encounters between high-mass objects (e.g., Ford et al. 2005; Chatterjee et al. 2008; Nagasawa et al. 2008; Raymond et al. 2008). With KOI-89's net orbital angular momentum highly misaligned from the star's spin angular momentum, conservation of angular momentum would require additional planet(s) to scatter the known two planets. In this scenario it is highly unlikely that the two planets would end up near mutual alignment. However, if a sufficiently large unknown body exists in this system, then our orbit integrations are unsound and our coalignment constraint for this system is invalid. We therefore deem this mechanism consistent as a possible cause of KOI-89's misalignment. Further studies of KOI-89's TTVs could confirm the existence of additional planets.

Kozai resonance in the KOI-89 system requires an unknown body that is significantly misaligned with its known orbital plane (Libert & Tsiganis 2009; Payne et al. 2010; Thies et al. 2011). Such an event would likely not produce coplanar orbits for KOI-89.01 and KOI-89.02. However, Kaib et al. (2011) suggests that coplanar, inclined orbits might arise as a result of this mechanism. We therefore deem this method consistent.

Rogers et al. (2012) showed that angular momentum transport between the convective interior and radiative exterior of hot, early-type stars can change the observed stellar spin axis, resulting in spin–orbit misalignment. This misalignment mechanism happens independently of orbiting bodies and does not affect coplanarity. The 2D simulations performed in Rogers et al. (2012) found that this mechanism can occur on a timescale as short as tens of years, and can explain retrograde orbits. Whether this mechanism can produce spin–orbit misalignments near 90° is still under investigation.

Planet–embryo collisions can occur in any standard formation model, and they can drive migration in various ways (Levison et al. 1998; Charnoz et al. 2001). However, this mechanism can produce large spin–orbit misalignment angles only for small rocky bodies and does not apply to the highly misaligned giant planets KOI-89.01 and KOI-89.02. Additionally, this mechanism likely could not produce coplanar misaligned orbits because the collisions driving this mechanism are unique to each planet.

Storch et al. (2014) and Valsecchi & Rasio (2014) demonstrated that strong tidal dissipation can cause chaotic evolution of the stellar spin. This mechanism requires hot Jupiters with periods ≲3 days. Such a body in the KOI-89 system would have to be drastically misaligned from the plane of the other two orbits; therefore, this mechanism cannot be the standalone cause of misalignment because some other mechanism would have to misalign the hot Jupiter. If there was a non-transiting hot Jupiter that was initially misaligned, it could torque the star into misalignment with the other planets. If said hot Jupiter fell into its host star because of tidal decay, it could change both KOI-89's rotation axis and rotation rate (Jackson et al. 2009). However, early-type stars such as KOI-89 have weak tidal interactions in general (Ogilvie & Lin 2007).

Magnetic torquing between a stellar magnetic field and a protoplanetary disk can cause misalignment by torquing the disk away from the star's equatorial plane (Lai et al. 2011; Spalding & Batygin 2014). KOI-89 is an early-type star with a weak magnetic field (Bagnulo et al. 2002), so this mechanism could not cause KOI-89's high misalignment. We note that the magnetic fields of fast-rotators are still under investigation (Ibañez-Mejia & Braithwaite 2015); a better understanding of these magnetic fields may reveal this to be a possible misalignment mechanism for KOI-89.

Coplanar high-eccentricity migration can occur in mutually aligned multiplanet systems with at least one highly eccentric orbit. Secular gravitational effects excite the inner planet's eccentricity to very high values, and planetary tidal dissipation during periapsis reduces the orbit's semimajor axis. This mechanism occurs primarily in the planets' orbital plane, predominantly maintaining the system's original spin–orbit alignment angles (Petrovich 2014).

If the planets are in resonance and possess a mutual inclination, then the orbital inclinations can be driven to very large values (Barnes et al. 2015b). In that case we may expect to find at least one planet in a misaligned orbit. This phenomenon can also produce very large eccentricities. However, this mechanism depends on stellar torquing from tidal interactions for both planets to be discovered in a misaligned state. Such tidal interaction is weak around early-type stars (Ogilvie & Lin 2007). While this is a possible cause of KOI-89's extreme misalignment, it requires an external mechanism to bring about an initial mutual inclination. More work is needed to understand if this scenario is possible and could apply to KOI-89.