About

Uranus possesses 27 known natural satellites, orbiting a planet tilted at 97.77° relative to its orbital plane. This axial tilt means the entire moon system orbits in a plane nearly perpendicular to the ecliptic. The five major moons - Miranda, Ariel, Umbriel, Titania, and Oberon - were discovered between 1787 and 1948. The remaining 22 were found via Voyager 2 flyby data or ground-based telescopes between 1986 and 2003. Orbital periods range from 0.335 days (Cordelia) to 1687.15 days (Ferdinand). This tool solves Kepler's equation numerically for each moon at every frame, projecting true orbital positions using real eccentricity and semi-major axis data from JPL ephemerides. Logarithmic radial scaling is applied because the ratio between the closest inner moon (49,752 km) and the most distant irregular (20,901,000 km) exceeds 400×.

Approximations: orbits are rendered as 2D projections of inclined ellipses. Mutual gravitational perturbations between moons are not modeled. Irregular moon orbital elements carry significant uncertainty due to limited observational arcs. The tool uses epoch-independent mean motion, so absolute positions are not tied to a specific date.

Formulas

Each moon's position is computed by solving Kepler's equation at every animation frame. Given the mean anomaly M advancing with time, the eccentric anomaly E is found iteratively:

M = E − e ⋅ sin(E)

This is solved via Newton-Raphson iteration:

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

The true anomaly ν is then derived from E:

tanν2 = √1 + e1 − e ⋅ tanE2

The orbital radius at true anomaly ν is:

r = a(1 − e²)1 + e ⋅ cos(ν)

Where a = semi-major axis, e = eccentricity, M = mean anomaly (= 2πT ⋅ t), T = orbital period, t = elapsed simulation time. Display coordinates use logarithmic radial scaling: r_display = log(r ÷ r_min) to compress the 400× distance ratio into a viewable canvas.

Reference Data

Moon	Group	Semi-major Axis (km)	Period (days)	Eccentricity	Inclination (°)	Radius (km)	Discovered
Cordelia	Inner	49,752	0.335	0.0003	0.08	20	1986
Ophelia	Inner	53,764	0.376	0.0099	0.10	21	1986
Bianca	Inner	59,166	0.435	0.0009	0.19	27	1986
Cressida	Inner	61,767	0.464	0.0004	0.01	41	1986
Desdemona	Inner	62,659	0.474	0.0001	0.11	35	1986
Juliet	Inner	64,358	0.493	0.0007	0.07	47	1986
Portia	Inner	66,097	0.513	0.0001	0.06	68	1986
Rosalind	Inner	69,927	0.558	0.0001	0.28	36	1986
Cupid	Inner	74,800	0.618	0.0013	0.10	9	2003
Belinda	Inner	75,255	0.624	0.0001	0.03	45	1986
Perdita	Inner	76,417	0.638	0.0116	0.47	15	1986
Puck	Inner	86,004	0.762	0.0001	0.32	81	1985
Mab	Inner	97,736	0.923	0.0025	0.13	12	2003
Miranda	Major	129,390	1.413	0.0013	4.34	236	1948
Ariel	Major	190,900	2.520	0.0012	0.04	579	1851
Umbriel	Major	266,000	4.144	0.0039	0.13	585	1851
Titania	Major	435,910	8.706	0.0011	0.08	789	1787
Oberon	Major	583,520	13.463	0.0014	0.07	761	1787
Francisco	Irregular	4,276,000	266.56	0.1459	145.2	11	2003
Caliban	Irregular	7,231,000	579.73	0.1587	141.5	36	1997
Stephano	Irregular	8,004,000	677.37	0.2292	144.1	16	1999
Trinculo	Irregular	8,504,000	749.24	0.2200	167.0	9	2001
Sycorax	Irregular	12,179,000	1283.4	0.5224	159.4	75	1997
Margaret	Irregular	14,345,000	1687.15	0.6608	57.4	10	2003
Prospero	Irregular	16,256,000	1978.29	0.4448	151.8	25	1999
Setebos	Irregular	17,418,000	2225.21	0.5914	158.2	24	1999
Ferdinand	Irregular	20,901,000	2887.21	0.3682	169.8	10	2003

Frequently Asked Questions

The ratio between the closest inner moon (Cordelia at 49,752 km) and the most distant irregular (Ferdinand at 20,901,000 km) exceeds 400×. A linear scale would render inner moons as a single pixel cluster. Logarithmic scaling compresses this range so all 27 moons remain visible simultaneously. Toggle the "Log Scale" option to see the effect.

Kepler's equation assumes a two-body problem (moon + Uranus). For the five major moons with eccentricities below 0.004, this is an excellent approximation. Irregular moons like Sycorax (e = 0.522) and Margaret (e = 0.661) experience significant solar perturbations that alter their elements over decades. The orbital elements used here are osculating elements valid near their discovery epoch, not propagated forward. Positional errors for irregulars accumulate at roughly 1-5° per century.

Nine of Uranus's moons have orbital inclinations exceeding 90° relative to Uranus's equatorial plane, making them retrograde. These are all irregular moons (Francisco, Caliban, Stephano, Trinculo, Sycorax, Prospero, Setebos, Ferdinand) likely captured from heliocentric orbits. Margaret (inclination 57.4°) is the only known prograde irregular moon of Uranus. In the animation, retrograde moons orbit clockwise when viewed from the north pole.

The leading hypothesis is a giant impact during the late stages of planetary formation, approximately 4 billion years ago. A body roughly 1-3 Earth masses struck proto-Uranus, tilting it onto its side. An alternative model proposes that gravitational resonance with a now-ejected satellite could have slowly torqued the planet over millions of years. The tilt means Uranus's ring and moon system orbits nearly perpendicular to the ecliptic.

For eccentricities below 0.3, convergence is typically achieved in 3-5 iterations. For Margaret (e = 0.661), convergence may require 8-12 iterations. The solver uses a maximum of 20 iterations with a tolerance of 1×10⁻⁸ radians. The initial guess is E₀ = M + e·sin(M), which accelerates convergence. Divergence has never been observed for any Uranian moon eccentricity.

Miranda has an orbital inclination of 4.34° relative to Uranus's equatorial plane, roughly 50-100 times greater than Ariel (0.04°) or Oberon (0.07°). This is likely a fossil signature of a past orbital resonance with Umbriel or Ariel. The tilt is exaggerated in the projection to make it visible, but the relative difference is accurate.