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 R₀/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 xy-plane of the star and bounded by −π/ 2−ϕ_a ≤ ϕ₀ ≤ π/ 2 + ϕ_a, and the arc EC, which is half of a circle of radius R₀cosϕ_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.
Open with DEXTER

For the line integral CDE, we use r = (x,y,z) with (A.7)where ϕ₀ 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/R₀ 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 re-emitted 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(R₀/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