About

Venus completes one orbit around the Sun every 224.7 Earth days at a mean distance of 0.723 AU, with an eccentricity of only 0.0068 - the most circular planetary orbit in the Solar System. This simulator solves Kepler's equation numerically at each frame to compute the true anomaly ν, then derives the heliocentric radius r from the conic section equation. Both Venus and Earth are rendered with their real orbital parameters so you can observe synodic period relationships directly. The ratio of their periods produces the famous pentagram pattern when inferior conjunctions are connected over 8 Earth years.

Approximation note: this projection collapses the 3.4° orbital inclination of Venus into the ecliptic plane. Planetary radii are exaggerated for visibility. Gravitational perturbations from Jupiter and other bodies are not modeled; pure two-body Keplerian motion is assumed.

Formulas

The position of each planet is computed by solving Kepler's equation at every animation frame. Given elapsed simulation time, the mean anomaly M advances linearly:

M = 2πT ⋅ t

Kepler's equation relates mean anomaly to eccentric anomaly E:

M = E − e ⋅ sin(E)

This transcendental equation is solved iteratively via Newton-Raphson:

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

Finally, the orbital radius r and Cartesian coordinates:

r = a(1 − e²)1 + e ⋅ cos(ν)

x = r ⋅ cos(ν) y = r ⋅ sin(ν)

Where: a = semi-major axis AU, e = eccentricity, T = orbital period days, t = elapsed simulation time days, M = mean anomaly rad, E = eccentric anomaly rad, ν = true anomaly rad, r = heliocentric distance AU.

Reference Data

Parameter	Venus	Earth	Unit
Semi-major axis a	0.72333	1.00000	AU
Eccentricity e	0.00677	0.01671	-
Orbital period T	224.701	365.256	days
Perihelion distance	0.71844	0.98329	AU
Aphelion distance	0.72823	1.01671	AU
Mean orbital velocity	35.02	29.78	km/s
Orbital inclination	3.3947	0.0000	°
Longitude of ascending node	76.680	-11.261	°
Argument of perihelion	54.884	114.208	°
Synodic period	583.9		days
Mass	4.867 × 10²⁴	5.972 × 10²⁴	kg
Equatorial radius	6051.8	6371.0	km
Surface gravity	8.87	9.81	m/s²
Rotation period (sidereal)	−243.025	0.997	days
Axial tilt	177.36	23.44	°

Frequently Asked Questions

Venus orbits closer to the Sun at a mean distance of 0.723 AU, giving it a higher mean orbital velocity of 35.02 km/s versus Earth's 29.78 km/s. By Kepler's third law, closer planets have shorter periods. Venus completes a full orbit in 224.7 days while Earth takes 365.25 days. The angular speed difference is directly visible in the animation.

For Venus's eccentricity of 0.0068, the Newton-Raphson iteration converges in 2-3 steps to machine precision. The initial guess E₀ = M is already an excellent approximation because the correction term e ⋅ sin(E) is small. Convergence issues only arise for eccentricities approaching 1.0 (parabolic orbits), which is irrelevant for planetary simulations.

Venus and Earth reach inferior conjunction (closest approach) every 583.9 days (the synodic period). After 5 synodic periods (2919.6 days ≈ 8 Earth years), the conjunction points repeat at nearly the same ecliptic longitudes. Connecting these 5 points traces a regular pentagram inscribed in the zodiac. The ratio 13/8 (Venus orbits per 8 Earth years) approximates the golden ratio.

Venus's orbit is tilted 3.39° relative to the ecliptic. In a top-down 2D view, this tilt produces a maximum displacement of r ⋅ sin(3.39°) ≈ 0.043 AU perpendicular to the ecliptic - invisible at the scale of the animation. Including it would require a 3D projection or oblique view, adding complexity without visual benefit for understanding orbital mechanics.

By Kepler's second law, a planet sweeps equal areas in equal times. At perihelion (0.718 AU), Venus moves faster; at aphelion (0.728 AU), slower. The velocity ratio is (1+e)/(1−e) ≈ 1.014, meaning only a 1.4% speed variation. For comparison, Earth's eccentricity of 0.0167 produces a 3.4% variation. Venus's orbit is the most circular of all planets.

No. The simulation starts from an arbitrary epoch and does not use J2000 orbital elements with correct initial mean anomalies. It accurately represents the shapes, sizes, and speed ratios of the orbits, but the absolute angular positions do not correspond to real calendar dates. Predicting transits also requires modeling the 3.39° inclination, since transits only occur when Venus crosses the ecliptic plane near inferior conjunction.