A bare planet produces a near-symmetric, roughly trapezoidal transit dip. A planet with an opaque ring system does not — the extra silhouette adds depth, stretches the ingress/egress, and — crucially — can make the dip asymmetric in time, which is the signature astronomers search for (the "exoring" method, after Barnes & Fortney 2004).
The ring is a flat annulus tilted out of the sky plane. Projected onto the sky, a circle of radius r tilted by angle θ from face-on becomes an ellipse with semi-major axis r (along the position-angle direction ψ) and semi-minor axis r·cos θ (perpendicular to it):
u = Δx·cosψ + Δy·sinψ
v = −Δx·sinψ + Δy·cosψ
inside ring ⇔ Rin² ≤ (u/1)² + (v/cosθ)² · 1 ≤ Rout² (r in planet radii)
Each frame the simulator samples thousands of points across the stellar disk (weighted by quadratic limb darkening, I(μ) = 1 − u₁(1−μ) − u₂(1−μ)²) and tests each one against the planet disk (fully opaque) and the ring ellipse (blocks a fraction τ, the optical depth) 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)
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 that a spherical planet alone can never produce. Real candidates such as 1SWASP J1407b's disk and several proposed "super-ring" Saturn analogues around close-in giants are searched for using exactly this deviation from a symmetric trapezoid.
- Outer radius / Tilt / Position angle — set the ring's true size and 3D orientation; watch the mesh in the viewport tilt 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.