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tokens: lines) are shown. The colours represent different values of the orbital periastron argument (ω) in Fig. 1 . Lower panel: Relationship between the bright time ( t bright ) and the eccentricity ( e ) given by equation ( 5 ). In the lower panel of Fig. 5 , we analyse how ω affects the relationship between t bright and e (equation 5 ). As expected, t bright is minimum for the toy model (ω = 0°) and reaches a maximum value when the summer solstice occurs at quadrature (ω = 90°, −90°). Since an optimal orbital configuration was assumed in the toy model, equation ( 8 ) and the observed t bright will underestimate the true eccentricity of the orbit (see the dashed curve in the lower panel of Fig. 5 ). However, even in a more realistic situation, equation ( 8 ) is still valid as a formula for constraining e : Also, an unknown orbital inclination contributes to either increasing or reducing F dark and F max with respect to their corresponding values in the toy model (equations 9 and 11 , respectively). In the case of F dark , ω tends to increase its true value, while i may either reduce it or increase it, thereby compensating the effect of ω. On the other hand, the constraint given by equation ( 10 ) will not be substantially modified (at least within a first-order estimation of A planet ). By comparison with the toy model, both ω and i lower the measured value of F max . This implies that equation ( 13 ) would remain valid, although the minimum A ring will probably be underestimated. To find fiducial parameters of real systems, a more general and complex model for fitting stellar light curves needs to be developed. To that end, we should include second-order effects such as the shadow/shine of the planet’s surface and atmosphere, forward scattering, the opacity of the rings, the polarization and reflection of light by dusty/icy ring particles, etc. For this optical complexity , we propose (and intend) to create a scattering model centred on the ring (most models are centred on the star), where the properties of the scattered light are computed in the sky above the planetary rings after describing the astronomical position of the star, the planet, and the observer. However, such a model is the subject of an ongoing research that will be introduced in a forthcoming paper. 5 SUMMARY AND CONCLUSIONS The first-order semi-analytical model presented here demonstrates that rings may produce detectable and distinguishable photometric signatures on the light curve of non-transiting exoplanets. We have quantified the phase-dependent amount of light that a tilted ringed planet can reflect towards a distant observer, and the model is described through four main morphological features of the so-called bright curve that help us derive some analytical constraints to orbital and physical parameters of the system. This model can be useful to detect (ringed) planets with large ring surfaces lying in face-on orbits, whose survival time-scale is ∼1 Myr (Schlichting Chang