Abstract
In Sect. 3.4, we discussed the importance of individual obliquity measurements for long-period planets around hot stars. Such measurements may be possible by applying the gravity-darkening method (Sect. 2.3.2) to existing data of transiting systems obtained by the Kepler space telescope. The methodology, however, is not fully established, given the discrepancy between this method and the spectroscopic one recently reported for the hot Jupiter system Kepler-13A. In this chapter, we discuss the origin of the discrepancy and present a possible solution. In addition, we show that the solution can be tested by future follow-up observations, on the basis of dynamical modeling of transit variations observed in this system. The revised methodology is then applied for the first time to the HAT-P-7 system, providing a useful cross-check between the gravity-darkening result and the measurement made in Chap. 4. The results presented in this chapter clarify the validity and limitation of the gravity-darkening method, and also demonstrate the potential of space-based photometry data to characterize exoplanets and their host stars.
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- 1.
In this chapter, \(R_\star \) denotes the equatorial radius of the star.
- 2.
- 3.
Although this vector \(-\varvec{g}_\mathrm{eff}\) is not exactly parallel to the surface normal of the spheroid we assume, the difference is \(\mathcal {O}(\gamma ^2)\) and thus negligible.
- 4.
Use of the quadratic polynomial helps the better removal of flux variation not due to the transit, i.e., planetary light, ellipsoidal variation, and Doppler beaming.
- 5.
Only in Sect. 5.4.1, \(\rho _\star \), \(c_1\), \(c_2\), \(R_\mathrm{p}/R_\star \), and \(f_\mathrm{rot}\) are also included.
- 6.
We do not examine the dependence on \(\beta \) here because B11 have already shown that a different choice of \(\beta =0.19\), suggested by the interferometric observation of Altair (Monnier et al. 2007), does not change the result significantly.
- 7.
For reference, we find \(c_2=0.1-0.2\) if we adopt the model without gravity darkening (Mandel and Agol 2002), which suggests that the choice of \(c_2=0\) is not indispensable.
- 8.
Note that, in Szabó et al. (2012), the degeneracy between \(a/R_\star \) (or \(\rho _\star \)) and \(\cos i_\mathrm{orb}\) was not solved.
- 9.
If either of the angular momenta of the stellar spin or the orbital motion dominates, \(i_\mathrm{orb}\) or \(i_\star \) is almost constant. In the Kepler-13A system, the two angular momenta have comparable magnitudes and so all three angles modulate due to the precession. A similar case, the PTFO 8-8695 system, has been studied by Barnes et al. (2013) and Kamiaka et al. (2015).
- 10.
- 11.
The existence of the two solutions in this case should be distinguished from the degeneracy intrinsic to the gravity-darkening method. For each of the two solution listed here, there additionally exists the model that yields exactly the same light curve, where \(\cos i_\mathrm{orb}\) is replaced with \(-\cos i_\mathrm{orb}\) and \(\lambda \) with \(\pi -\lambda \). These intrinsically-degenerate solutions are not discussed here because they are in any case rejected in the joint solution, where \(\lambda \) is constrained by the prior. This is the same logic as used in the last paragraph of Sect. 5.3.1.
- 12.
We also applied the residual permutation method described in Winn et al. (2009b) for another estimate of the parameter uncertainties, and confirmed that they are not significantly affected by the correlated noise component.
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Masuda, K. (2018). Spin–Orbit Misalignments of Kepler-13Ab and HAT-P-7b from Gravity-Darkened Transit Light Curves. In: Exploring the Architecture of Transiting Exoplanetary Systems with High-Precision Photometry. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-8453-9_5
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