About

Mercury follows the most eccentric orbit of all planets in our solar system, with an eccentricity of e = 0.2056. This means its distance from the Sun varies from 46.0 million km at perihelion to 69.8 million km at aphelion. A miscalculation of Mercury's position contributed to anomalies that Newtonian mechanics could not explain. General relativity resolved this with a perihelion precession of 43 arcseconds per century. This simulator solves Kepler's equation numerically each frame to compute true anomaly ν from mean anomaly M, producing an accurate depiction of orbital speed variation governed by Kepler's second law.

The tool approximates Mercury's orbit as a fixed 2D ellipse and does not model relativistic precession or gravitational perturbations from other planets. Orbital elements are sourced from NASA JPL epoch J2000. Velocity shading on the trail illustrates the conservation of angular momentum: Mercury moves fastest at perihelion and slowest at aphelion.

Formulas

The position of Mercury at any time is found by solving Kepler's Equation, which relates mean anomaly M (linear in time) to eccentric anomaly E (geometric angle on the auxiliary circle):

M = E − e ⋅ sin(E)

This transcendental equation is solved iteratively using 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:

tan(ν2) = √1 + e1 − e ⋅ tan(E2)

The radial distance from the Sun is computed from the orbit equation:

r = a(1 − e²)1 + e ⋅ cos(ν)

Cartesian coordinates follow from rotation by the argument of perihelion ω:

x = r ⋅ cos(ν + ω)

y = r ⋅ sin(ν + ω)

Where: a = semi-major axis (57.909 ×10⁶ km), e = eccentricity (0.20563), M = mean anomaly (advances 4.092°/day), E = eccentric anomaly, ν = true anomaly, ω = argument of perihelion (29.124°), r = heliocentric distance.

Reference Data

Parameter	Symbol	Value	Unit
Semi-major axis	a	57.909	×10⁶ km
Semi-minor axis	b	56.672	×10⁶ km
Eccentricity	e	0.20563	-
Orbital period	T	87.969	Earth days
Perihelion distance	q	46.002	×10⁶ km
Aphelion distance	Q	69.817	×10⁶ km
Mean orbital velocity	v_mean	47.36	km/s
Max orbital velocity (perihelion)	v_max	58.98	km/s
Min orbital velocity (aphelion)	v_min	38.86	km/s
Argument of perihelion	ω	29.124	°
Inclination to ecliptic	i	7.005	°
Longitude of ascending node	Ω	48.331	°
Mass	m	3.3011 × 10²³	kg
Equatorial radius	R	2439.7	km
Surface gravity	g	3.7	m/s²
Rotational period	-	58.646	Earth days
Solar day (synodic)	-	175.942	Earth days
Perihelion precession (GR)	-	43	arcsec/century
Hill sphere radius	-	175,300	km
Escape velocity	-	4.25	km/s

Frequently Asked Questions

Kepler's second law states that a line joining a planet to the Sun sweeps equal areas in equal time intervals. At perihelion, the radial distance r is smallest, so the tangential velocity must increase to maintain constant areal velocity. For Mercury, velocity ranges from 58.98 km/s at perihelion to 38.86 km/s at aphelion - a 52% variation, the largest among planets due to its high eccentricity of 0.2056.

The tool uses Newton-Raphson iteration starting from the initial guess E₀ = M. Each step refines via E_{n+1} = E_n − (E_n − e·sin(E_n) − M) / (1 − e·cos(E_n)). For Mercury's eccentricity of 0.2056, convergence to 10⁻¹⁰ radians precision typically requires 3-5 iterations. The method is unconditionally convergent for e < 1.

No. This simulation models the classical two-body Keplerian orbit with fixed orbital elements. Mercury's general-relativistic precession of 43 arcseconds per century (0.012° per century) is far below visual resolution at this timescale. Including it would require post-Newtonian corrections to the equation of motion.

The Sun sits at one focus of the ellipse, not its geometric center. The distance between the center and the focus is c = a·e = 57.909 × 0.2056 ≈ 11.906 million km. This offset is physically real and directly responsible for the perihelion-aphelion distance asymmetry.

All elements are from the NASA JPL DE440 ephemeris at epoch J2000.0 (January 1.5, 2000 TDB). They are valid for visualization purposes over several centuries. For mission-critical trajectory planning, osculating elements with perturbation models (N-body, solar oblateness, Yarkovsky) must be used instead.

As e → 1, the orbit becomes increasingly elongated and Newton-Raphson convergence slows. The tool caps eccentricity at 0.95 to maintain numerical stability. At e = 1, the orbit becomes parabolic (escape trajectory), which requires a different mathematical framework using Barker's equation rather than Kepler's.