About

Jupiter completes one orbit around the Sun every 4,332.59 Earth days (11.86 years) along an ellipse with semi-major axis a = 5.2044 AU and eccentricity e = 0.0489. This simulator solves Kepler's equation at each frame using Newton-Raphson iteration to compute the Eccentric Anomaly E from the Mean Anomaly M, then derives the True Anomaly ν and radial distance r. Incorrect orbital period assumptions or circular-orbit approximations introduce positional errors exceeding 0.25 AU at aphelion. The inner planets (Mercury through Mars) are included for scale reference. All orbital elements use J2000 epoch values from JPL's planetary fact sheets. This tool approximates two-body Keplerian motion and does not model gravitational perturbations between planets.

Formulas

The position of a planet on its elliptical orbit is determined by solving Kepler's Equation, which relates the Mean Anomaly M (a linearly advancing angle) to the Eccentric Anomaly E (the geometric parameter of the ellipse):

M = E − e ⋅ sin(E)

This transcendental equation has no closed-form solution. The simulator uses Newton-Raphson iteration to converge on E:

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

Once E is found, the True Anomaly ν is computed:

ν = 2 ⋅ atan2(√1 + e ⋅ sin(E2), √1 − e ⋅ cos(E2))

The radial distance from the Sun is:

r = a ⋅ (1 − e ⋅ cos(E))

Cartesian coordinates in the orbital plane:

x = r ⋅ cos(ν), y = r ⋅ sin(ν)

Where a = semi-major axis (AU), e = eccentricity (dimensionless), M = mean anomaly (rad), E = eccentric anomaly (rad), ν = true anomaly (rad), r = heliocentric distance (AU).

Reference Data

Planet	Semi-Major Axis (AU)	Eccentricity	Orbital Period (days)	Orbital Period (years)	Inclination (°)	Mean Velocity (km/s)	Perihelion (AU)	Aphelion (AU)
Mercury	0.3871	0.2056	87.97	0.241	7.00	47.36	0.307	0.467
Venus	0.7233	0.0068	224.70	0.615	3.39	35.02	0.718	0.728
Earth	1.0000	0.0167	365.25	1.000	0.00	29.78	0.983	1.017
Mars	1.5237	0.0934	686.97	1.881	1.85	24.07	1.381	1.666
Jupiter	5.2044	0.0489	4,332.59	11.862	1.31	13.07	4.950	5.459
Saturn	9.5826	0.0565	10,759.22	29.457	2.49	9.68	9.041	10.124
Uranus	19.2184	0.0457	30,688.50	84.021	0.77	6.80	18.330	20.110
Neptune	30.1100	0.0113	60,182.00	164.800	1.77	5.43	29.770	30.440
Pluto (dwarf)	39.4821	0.2488	90,560.00	247.940	17.16	4.74	29.650	49.310
Ceres (dwarf)	2.7675	0.0758	1,681.63	4.604	10.59	17.90	2.558	2.977
Halley's Comet	17.8340	0.9671	27,507.00	75.320	162.26	1.32	0.586	35.082

Frequently Asked Questions

Kepler's equation M = E − e ⋅ sin(E) is transcendental. No closed-form algebraic solution exists for E given M. Newton-Raphson iteration converges to machine precision within 4-6 iterations for eccentricities below 0.1 (all major planets). For highly eccentric orbits like comets (e > 0.9), more iterations or Halley's method may be required.

The simulation uses J2000 epoch orbital elements from JPL. For short-term visualization (centuries), positional accuracy is within 0.01 AU for Jupiter. The model ignores gravitational perturbations from other planets, general relativistic precession, and non-gravitational forces. Over millennia, accumulated error in Jupiter's longitude of perihelion can exceed 1° due to Saturn's gravitational influence.

Kepler's Third Law states T² ∝ a³. Mercury's orbital period is 87.97 days versus Jupiter's 4,332.59 days. Mercury's mean orbital velocity is 47.36 km/s compared to Jupiter's 13.07 km/s. The ratio of approximately 3.6× in velocity combined with the much smaller orbit makes Mercury's angular speed roughly 49× faster than Jupiter's.

Mean Anomaly M is a fictitious angle that increases uniformly with time at rate 2π/T. Eccentric Anomaly E is the angle measured from the center of the ellipse to the projection of the planet onto the auxiliary circle. True Anomaly ν is the actual angle from perihelion as seen from the focus (the Sun). For circular orbits (e = 0), all three are identical. As eccentricity increases, the discrepancy grows. Jupiter's low eccentricity of 0.0489 produces a maximum difference of approximately 5.6° between M and ν.

No. This is a 2D projection onto the ecliptic plane (Earth's orbital plane, inclination 0°). Jupiter's inclination is only 1.31° relative to the ecliptic, so the visual error is negligible. Mercury's 7° inclination would produce a more noticeable out-of-plane displacement, but for a top-down view this is acceptable.

Jupiter's aphelion distance is 5.459 AU while Mercury's perihelion is 0.307 AU, a ratio of approximately 17.8×. The simulator maps the maximum orbital extent (Jupiter's aphelion plus a margin) to the canvas half-width. Planet radii are exaggerated with a minimum pixel size to ensure visibility. A zoom control allows focusing on inner or outer planets independently.