Saturn Rings (orbital resonance)

Math essence

Each particle obeys a_central = -M * r_hat / r^2 plus a tangential kick t_drift perpendicular to r_hat. Velocity is integrated and clamped to max_speed; position updates, with particles that leave the annulus being reborn on a fresh circular orbit at a new random radius. The integration preserves stable nearly-circular orbits while the per-particle tangential drift slowly smears each ring into a thin luminous band.

Parameters

Reference

First 60 lines of reference.py. Full source: library_external/hodgin_iquilezles/SaturnRings/reference.py

from __future__ import annotations

from typing import ClassVar

import numpy as np


class SaturnRings:
    """Saturn Rings — Robert Hodgin (flight404), inspired by *Saturn* / orbital systems.

    Hodgin built several orbital-resonance studies; the most famous is the
    *Saturn* sketch in which thousands of particles orbit a central mass in
    concentric rings, each ring at a slightly different resonance.  Particles
    are subject to inverse-square gravitational attraction toward a central
    mass plus a tangential impulse that keeps them in roughly circular
    orbits; small perturbations (drift + jitter) keep each ring shimmering.

    The "interesting" particles are the ones whose initial radius lies
    inside an allowed stable annulus.  Particles outside the annulus either
    spiral in or escape.
    """

    family: ClassVar[str] = "hodgin_iquilezles"
    sketch: ClassVar[str] = "saturnrings"

    def __init__(
        self,
        dim: int = 500,
        num: int = 4000,
        inner_radius: float = 90.0,
        outer_radius: float = 230.0,
        central_mass: float = 1.6,
        tangential_boost: float = 1.0,
        eccentricity: float = 0.06,
        radial_drift: float = 0.0006,
        ring_jitter: float = 0.08,
        max_speed: float = 12.0,
        trail_alpha: int = 18,
        ink_color: tuple[int, int, int] = (255, 240, 220),
        background: tuple[int, int, int] = (4, 4, 10),
        **_,
    ) -> None:
        self.dim = dim
        self.num = num
        self.inner_radius = inner_radius
        self.outer_radius = outer_radius
        self.central_mass = central_mass
        self.tangential_boost = tangential_boost
        self.eccentricity = eccentricity
        self.radial_drift = radial_drift
        self.ring_jitter = ring_jitter
        self.max_speed = max_speed
        self.trail_alpha = trail_alpha
        self.ink_color = np.array(ink_color, dtype=np.float32)
        self.bg_color = np.array(background, dtype=np.float32)
        self.reset()

    def reset(self, seed: int | None = None) -> None:
        self.rng = np.random.default_rng(seed)
        cx = self.dim * 0.5