This is a genuine sky-plane (bird's-eye) reconstruction, not a flattened 3D render: the star, planet and ring are drawn exactly as they project onto the two-dimensional plane of the sky, and the light curve underneath is integrated directly from that same 2D silhouette — this is how the "exoring" method (Barnes & Fortney 2004) actually works, since no telescope resolves the ring itself.
The ring is a flat annulus tilted by angle θ from face-on. Projected onto the sky it becomes an ellipse: semi-major axis r along the position-angle direction ψ, semi-minor axis r·cosθ perpendicular to it:
u = Δx·cosψ + Δy·sinψ
v = −Δx·sinψ + Δy·cosψ
inside ring ⇔ Rin² ≤ (u/1)² + (v/cosθ)² ≤ Rout² (r in planet radii)
The star's brightness is integrated with a genuinely independent numerical method from a 3D-rendered version of this same physics: a polar (concentric-ring) quadrature over the stellar disk — rings of radius r each split into enough angular wedges that every quadrature cell stays roughly square (arc length ≈ radial step) — weighted by quadratic limb darkening I(μ) = 1 − u₁(1−μ) − u₂(1−μ)². Each ring/wedge cell is tested against the ring ellipse (blocks a fraction τ) and the opaque planet disk as the planet sweeps across at impact parameter b:
F(x) = Σ wᵢ·block(sᵢ, x) / Σ wᵢ, block = 0 (planet), 1−τ (ring), 1 (clear)
wᵢ = I(μᵢ)·rᵢ·Δθᵢ (annulus-cell area element)
When the ring's position angle ψ is not 0° or 90°, the projected ellipse is not mirror-symmetric about the direction of motion, so the star-disk overlap area grows and shrinks at different rates on the way in versus the way out — an asymmetric light curve a spherical planet alone can never produce. The asymmetry readout compares the area under the dip on each side of the transit minimum (a sum over the whole curve, not a single crossing point), which stays numerically stable where a naive ingress/egress-time measurement does not.
- Outer radius / Tilt / Position angle — set the ring's true size and 3D orientation; watch the ellipse in the sky-plane view stretch and rotate to match.
- Optical depth — how much starlight the ring itself blocks (0 = transparent, 1 = opaque, like Saturn's B ring).
- Impact parameter b — how far off-center the planet's chord passes across the star.