Free Access
Volume 557, September 2013
Article Number A74
Number of page(s) 13
Section Planets and planetary systems
Published online 03 September 2013

Online material

Appendix A: Tracking a rotating starspot with respect to the transit chord

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 xz′-plane between the z-axis of frame S and the x′-axis of frame S′. Coordinates in

thumbnail Fig. A.1

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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