Issue 
A&A
Volume 557, September 2013



Article Number  A74  
Number of page(s)  13  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201321901  
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 righthanded reference frame S, whose origin coincides with the centre of the star. In this frame, the stellar rotation vector defines the + zdirection, while the yaxis 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/P_{rot} + φ_{0}, P_{rot} is the star’s rotational period, and φ_{0} is the longitude of the spot at some arbitrary time t = 0.
We defined a second righthanded reference frame S′ that shares the same origin and yaxis 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 lineofsight, i.e., the angle in the x′z′plane between the zaxis of frame S and the x′axis of frame S′. Coordinates in
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 yaxis (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 righthanded reference frame S′′ that shares the same origin as S and S′, and the same xaxis as frame S′ (i.e., x′′ ≡ x′), but has its y′′axis parallel to the transit chord. The skyprojected spinorbit 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 φvalues, ranging from 0 to 360° with 2° steps. If the conditions above (evaluated using the values for b and R_{p}/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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