Issue 
A&A
Volume 583, November 2015



Article Number  A50  
Number of page(s)  9  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201526673  
Published online  27 October 2015 
Online material
Appendix A: Flux received by the ring
Here, we detail the computation of the flux F_{r}(φ_{i}) received by the ring per unit area. The notation is the same as in Sect. 3.1 of the main text (see also Fig. 2). The general expression of the flux derived from Eqs. (10) and (11) is (A.1)where (A.2)The integrand in the expression of F_{r} is of order unity. We expand it at the first order in R_{0}/a ≪ 1. We get (A.3)We consider the most general case where each element of the ring only sees a fraction of the stellar disk (see Fig. A.1). This case happens when the tilt angle φ_{i} is less than the angular radius of the star . In this configuration, the visible surface is delimited by two curves: the arc CDE in the xyplane of the star and bounded by −π/ 2−ϕ_{a} ≤ ϕ_{0} ≤ π/ 2 + ϕ_{a}, and the arc EC, which is half of a circle of radius R_{0}cosϕ_{a} in the plane of the ring. For commodity, we recall the definition of the angle ϕ_{a} given in Eq. (14) To compute the surface integral (A.3), we make use of the Stockes theorem that transforms a surface integral over Σ into a closed integral over its boundary ∂Σ as (A.4)For this problem, we set (A.5)where (A.6)
Fig. A.1
Area enclosed by the thick curve CDEC is the visible part of the star seen from an element of the ring. 

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For the line integral CDE, we use r = (x,y,z) with (A.7)where ϕ_{0} goes from (− π/ 2−ϕ_{a}) to (π/ 2 + ϕ_{a}),. While for the line integral EC, we set r = (x,y,z) with (A.8)where ψ ranges from 0 to π. As a result, we get (A.9)In the case where φ_{i}>φ_{c}, Eq. (A.9) still holds if we set ϕ_{a} = π/ 2 so we get while for φ_{i} ≪ φ_{c}, ϕ_{a} ~ aφ_{i}/R_{0} and
Appendix B: Reflectivity
The reflectivity of the ring is computed by assuming an isotropic scattering. Furthermore, it is assumed that given an incoming flux F_{r}, only a fraction is reemitted in the visible spectrum. Thus, the luminous intensity I_{r} of the rings is uniform and such that (B.1)Besides this, the flux received on Earth from the disk is (B.2)where the ratio of the projected surface S_{proj} of the rings on the plane of the sky divided by the square of the distance D to the Earth represents the solid angle under which the rings are seen. As a result, we get (B.3)Moreover, the stellar flux F_{⋆} received on Earth is (B.4)
where we used F_{p} = πI_{s}(R_{0}/a)^{2}. Combining Eqs. (B.3) and (B.4), we get (B.5)with g_{r} a function representing the reflectivity of the rings given by (B.6)
© ESO, 2015
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