About

Uranus completes one orbit around the Sun every 84.01 Earth years at a mean distance of 19.19 AU (2.871 × 10⁹ km). Its orbit has an eccentricity of e = 0.0472, making it nearly circular but with perihelion at 18.29 AU and aphelion at 20.10 AU. The planet's axial tilt of 97.77° means it essentially rolls along its orbital path. This simulator solves Kepler's equation iteratively to compute true anomaly ν from mean anomaly M, then applies rotation matrices for orbital inclination (0.77° to the ecliptic). Incorrect orbital period assumptions propagate errors in mission planning and transit predictions. Note: this model assumes a two-body problem and ignores perturbations from Jupiter, Saturn, and Neptune.

Formulas

The position of Uranus in its orbit is computed using Keplerian orbital mechanics. First, compute the mean anomaly M as a function of time:

M = M 0 + n (t - t 0)

where n = 2πT is the mean motion (radians per unit time). Then solve Kepler's equation for eccentric anomaly E:

M = E - e \cdot sin (E)

This transcendental equation is solved via Newton-Raphson iteration. The true anomaly ν is then:

tan ν 2 = \sqrt 1 + e 1 - e \cdot tan E 2

The heliocentric distance r follows from the orbital equation:

r = a (1 - e 2) 1 + e \cdot cos (ν)

3D position is computed as (r ⋅ cosν, r ⋅ sinν, 0) in the orbital plane, then rotated by inclination i about the x-axis. Perspective projection maps 3D coordinates to 2D screen space using focal length f:

x screen = f \cdot x z + d

where a = semi-major axis (19.19 AU), e = eccentricity (0.0472), T = orbital period (84.01 years), i = inclination (0.77°), d = camera distance.

Reference Data

Parameter	Value	Unit	Notes
Semi-major axis (a)	19.19126	AU	Mean distance from Sun
Perihelion	18.2861	AU	Closest approach
Aphelion	20.0965	AU	Farthest distance
Eccentricity (e)	0.04717	-	Orbit shape (0 = circle)
Orbital period (T)	84.0205	years	30,688.5 Earth days
Mean orbital velocity	6.80	km/s	Average speed
Orbital inclination (i)	0.772	°	Relative to ecliptic
Longitude of ascending node (Ω)	73.989	°	J2000 epoch
Argument of perihelion (ω)	96.998	°	Perihelion position in orbit
Axial tilt	97.77	°	Extreme retrograde rotation
Rotation period	17.24	hours	Sidereal day
Equatorial radius	25,559	km	4.007 Earth radii
Mass	8.681 × 10²⁵	kg	14.54 Earth masses
Mean density	1.27	g/cm³	Ice giant composition
Surface gravity	8.69	m/s²	At 1 bar level
Escape velocity	21.3	km/s	From cloud tops
Known moons	28	-	As of 2024
Ring system	13 rings	-	Dark, narrow rings
Discovery date	1781	-	By William Herschel
Synodic period	369.66	days	Opposition to opposition

Frequently Asked Questions

Uranus has a low eccentricity of 0.0472, making its orbit nearly circular. When viewed face-on (camera perpendicular to orbital plane), the slight elliptical shape is visible. From edge-on views, the orbit appears as a line. The apparent shape depends entirely on the viewing angle relative to the orbital plane's inclination of 0.77°.

For Uranus's eccentricity of 0.0472, Newton-Raphson converges in 3-4 iterations to machine precision. For highly eccentric orbits (e > 0.9), convergence slows and may require 10+ iterations or alternative methods like Halley's method. This simulator uses a maximum of 50 iterations with tolerance 10⁻¹⁰.

Real-time animation at 1 second = 1 second would show no perceptible motion. The speed multiplier compresses time so that one simulated year passes in approximately 1 second at default speed. At maximum speed, the full 84-year orbit completes in about 8 seconds.

No. This is a pure two-body Keplerian model. Jupiter, Saturn, and Neptune cause orbital precession and periodic variations in Uranus's elements. For mission-critical calculations, use JPL's SPICE kernels or numerical integrators that include N-body perturbations. The error accumulation over centuries is measurable but negligible for visualization purposes.

The simulator uses manual matrix multiplication for rotation and a simple perspective divide: x_screen = f ⋅ x / (z + d). Rotation matrices Rₓ, Rᵧ, R_z are applied sequentially for camera orientation. This approach works well for single objects and simple scenes but lacks depth buffering for complex overlapping geometry.

The extreme axial tilt means Uranus's poles alternately point toward the Sun during its orbit. While this simulator focuses on orbital motion (not rotation), the tilt is represented visually in the planet's orientation. During solstices, one pole receives continuous sunlight for about 21 years while the other remains in darkness.