About

Saturn orbits the Sun at a mean distance of 9.537 AU with an orbital period of 29.457 Earth years and eccentricity e = 0.0541. A naive circular approximation introduces positional errors exceeding 0.5 AU at perihelion and aphelion. This tool solves Kepler's equation via Newton-Raphson iteration to produce accurate elliptical trajectories for Saturn and the inner reference planets. Ring geometry follows the Cassini Division structure with correct oblateness ratios. All orbital elements are sourced from JPL/NASA ephemeris data (J2000 epoch).

The animation renders Mercury through Saturn with proportional (logarithmically scaled) orbit sizes. Orbital speeds are time-accurate relative to each other. Note: planet sizes are exaggerated for visibility. Axial tilt of Saturn's rings is fixed at the approximate current value rather than precessing over the 29-year cycle. This tool approximates positions assuming two-body Keplerian motion. Perturbations from Jupiter and other planets are not modeled.

Formulas

Each planet's position is computed by solving Kepler's equation at every frame. The mean anomaly M advances linearly with time:

M = 2πT ⋅ t

where T = orbital period and t = elapsed time. The eccentric anomaly E is found by iterating:

E_n+1 = E_n − E_n − e ⋅ sin(E_n) − M1 − e ⋅ cos(E_n)

This Newton-Raphson iteration converges within 5 - 10 steps for planetary eccentricities. The true anomaly ν is then:

ν = 2 ⋅ atan2(√1 + e ⋅ sin(E÷2), √1 − e ⋅ cos(E÷2))

The radial distance r from the focus (Sun):

r = a ⋅ (1 − e ⋅ cos(E))

where a = semi-major axis (AU), e = orbital eccentricity, M = mean anomaly (rad), E = eccentric anomaly (rad), ν = true anomaly (rad), T = orbital period (yr), t = simulation time (yr).

Reference Data

Planet	Semi-Major Axis (AU)	Orbital Period (yr)	Eccentricity	Inclination (°)	Mean Velocity (km/s)	Perihelion (AU)	Aphelion (AU)
Mercury	0.387	0.241	0.2056	7.00	47.87	0.307	0.467
Venus	0.723	0.615	0.0068	3.39	35.02	0.718	0.728
Earth	1.000	1.000	0.0167	0.00	29.78	0.983	1.017
Mars	1.524	1.881	0.0934	1.85	24.07	1.381	1.666
Jupiter	5.203	11.862	0.0489	1.31	13.07	4.950	5.455
Saturn	9.537	29.457	0.0541	2.49	9.69	9.021	10.054
Uranus	19.191	84.011	0.0472	0.77	6.81	18.286	20.097
Neptune	30.069	164.79	0.0086	1.77	5.43	29.810	30.327
Pluto (dwarf)	39.482	247.92	0.2488	17.16	4.74	29.658	49.305
Ceres (dwarf)	2.767	4.600	0.0758	10.59	17.90	2.558	2.977
Halley's Comet	17.834	75.32	0.9671	162.26	Variable	0.586	35.082

Frequently Asked Questions

Orbital velocity is inversely related to the square root of the semi-major axis (Kepler's third law). Saturn at 9.537 AU orbits at approximately 9.69 km/s, while Earth at 1 AU travels at 29.78 km/s. The animation preserves these relative speeds accurately - Saturn completes one orbit every 29.457 Earth years.

The tool solves Kepler's equation using Newton-Raphson iteration with a convergence threshold of 1×10⁻⁸ radians. For Saturn's eccentricity of 0.0541, this converges in 3-4 iterations. The resulting positional accuracy is within 0.001% of the two-body analytical solution. Multi-body gravitational perturbations (primarily from Jupiter) are not modeled and can cause deviations of up to 0.01 AU over a full Saturnian year.

Saturn's equatorial radius is 58,232 km while its orbital radius is approximately 1.43 billion km - a ratio of about 1:24,500. Rendering planets at true scale would make them invisible at any zoom level that shows complete orbits. Planet radii are logarithmically exaggerated by a factor of roughly 500-2000× depending on the body.

Saturn's axial tilt is 26.73° relative to its orbital plane. The rings lie in Saturn's equatorial plane, so they appear at varying angles as Saturn orbits the Sun. This animation uses a fixed projected tilt for visual clarity. In reality, the ring opening angle as seen from Earth varies on a 29.5-year cycle, appearing edge-on approximately every 14.7 years (next edge-on: 2025).

The rendering approximates the ring structure: the B ring extends from about 1.526 to 1.951 Saturn radii, the Cassini Division gap spans 1.951 to 2.023 radii (a 4,700 km gap), and the A ring extends from 2.023 to 2.269 radii. These proportional ratios are maintained in the canvas rendering, though the gap width is slightly exaggerated for visibility at lower zoom levels.

At 1× speed, one second of real time equals one Earth day of simulation time. At maximum speed (365×), one second equals one Earth year. The simulation timestep is frame-rate independent - it uses the actual elapsed milliseconds between frames multiplied by the speed factor, preventing time drift on devices with variable frame rates.