Volume 557, September 2013
|Number of page(s)||13|
|Section||Planets and planetary systems|
|Published online||03 September 2013|
The geometry of the system is shown in Fig. A.1. We started by considering a right-handed reference frame S, whose origin coincides with the centre of the star. In this frame, the stellar rotation vector defines the + z-direction, while the y-axis lies in the plane of the sky. If we set the stellar radius to unity, the coordinates of a starspot with latitude δ in this reference frame are simply where φ = 2πt/Prot + φ0, Prot is the star’s rotational period, and φ0 is the longitude of the spot at some arbitrary time t = 0.
We defined a second right-handed reference frame S′ that shares the same origin and y-axis as frame S (i.e., y′ ≡ y), but whose x′-axis is pointing towards the observer. The stellar inclination i⋆ is the angle between the stellar rotation axis and the line-of-sight, i.e., the angle in the x′z′-plane between the z-axis of frame S and the x′-axis of frame S′. Coordinates in
System geometry seen from the side (left) and from the observer’s standpoint (right), illustrating the two rotations needed to convert from frame S to frame S′′ (see text for details).
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frame S are transformed to frame S′ by performing a rotation by −(π/2 − i⋆) about the y-axis (or, equivalently, around the y′-axis; see Fig. A.1, left panel). This process is described in more detail in Appendix A of Aigrain et al. (2012).
We then considered a third right-handed reference frame S′′ that shares the same origin as S and S′, and the same x-axis as frame S′ (i.e., x′′ ≡ x′), but has its y′′-axis parallel to the transit chord. The sky-projected spin-orbit angle λ is defined as the angle in the plane of the sky between the projections of the orbital angular momentum and of the stellar spin axis, i.e., the angle in the y′′z′′-plane between the z′′-axis of frame S′′ and the z′-axis of frame S′. Coordinates in frame S′ are transformed to frame S′′ by performing a second rotation by −λ about the x′-axis (or, equivalently, around the x′′-axis; see Fig. A.1, right panel). The coordinates of the spot in frame S′′ are thus
If we observe the system for an infinite amount of time, a given spot that remains static relative to the rotating surface of the star will eventually be crossed by the planet during a transit, if, and only if, The absence of spot crossings therefore excludes a specific volume in (i⋆,λ,δ)-space that we explored by following the variations of x′′ and z′′ as φ varies. In the present study, this was done by stepping through a grid of values, ranging from 0 to 180° for i⋆ and λ, and from 0 to 90° for δ, with 2° steps. At each point of the grid, x′′ and z′′ were evaluated over a grid of φ-values, ranging from 0 to 360° with 2° steps. If the conditions above (evaluated using the values for b and Rp/R⋆ given in Table 3, i.e., 0.341 and 0.09924, respectively) were met for any value of φ, the corresponding cell in the (i⋆,λ,δ) grid was considered excluded.
© ESO, 2013
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