About

The Moon follows an elliptical path around Earth with a semi-major axis of 384,400 km and eccentricity e = 0.0549. This simulation solves Kepler's equation iteratively to compute the true anomaly ν from mean anomaly M, producing accurate positional data at any simulation time. Orbital inclination of 5.145° relative to the ecliptic plane is modeled, causing the apparent vertical oscillation visible in the 3D view. Ignoring these parameters leads to incorrect eclipse predictions and tidal calculations.

The visualization renders Earth-Moon distance in real-time, ranging from perigee (356,500 km) to apogee (406,700 km). Moon phase is computed geometrically from the Sun-Earth-Moon angle, assuming a fixed Sun direction along the positive X-axis. This tool approximates a two-body problem; perturbations from the Sun and planets are not included. Pro tip: observe how distance varies as the Moon traverses its ellipse - closest approach occurs at the orbit's rightmost point (perigee).

Formulas

The radial distance r of the Moon from Earth at true anomaly ν follows the polar equation for an ellipse:

r = a(1 − e²)1 + e ⋅ cos(ν)

Where a = semi-major axis, e = eccentricity, ν = true anomaly (angle from perigee).

To find ν from time, we first compute mean anomaly M then solve Kepler's equation for eccentric anomaly E:

M = E − e ⋅ sin(E)

This transcendental equation is solved via Newton-Raphson iteration:

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

True anomaly is then derived from eccentric anomaly:

tan(ν2) = √1 + e1 − e ⋅ tan(E2)

3D Cartesian coordinates in the orbital plane are computed as:

x = r ⋅ cos(ν), y = r ⋅ sin(ν) ⋅ cos(i), z = r ⋅ sin(ν) ⋅ sin(i)

Where i = 5.145° is orbital inclination. Perspective projection onto 2D canvas uses focal length f:

x_screen = f ⋅ xz + d, y_screen = f ⋅ yz + d

Where d = camera distance from origin.

Reference Data

Parameter	Symbol	Value	Unit
Semi-major Axis	a	384,400	km
Eccentricity	e	0.0549	-
Orbital Period (Sidereal)	T	27.322	days
Orbital Period (Synodic)	T_syn	29.530	days
Inclination to Ecliptic	i	5.145	°
Mean Orbital Velocity	v	1.022	km/s
Perigee Distance	r_min	356,500	km
Apogee Distance	r_max	406,700	km
Moon Radius	R_Moon	1,737.4	km
Earth Radius	R_Earth	6,371	km
Argument of Perigee	ω	318.15	°
Longitude of Ascending Node	Ω	125.08	°
Mean Angular Motion	n	13.176	°/day
Escape Velocity (Moon)	v_esc	2.38	km/s
Surface Gravity (Moon)	g	1.62	m/s²
Mass Ratio (Moon/Earth)	μ	0.0123	-
Hill Sphere Radius	r_H	61,500	km
Orbital Energy	E	−3.8 × 10²⁸	J
Angular Momentum	L	2.88 × 10³⁴	kg⋅m²/s
Tidal Locking Ratio	-	1:1	-

Frequently Asked Questions

The Moon's orbit is elliptical with eccentricity e = 0.0549. At perigee (closest approach), the distance is approximately 356,500 km; at apogee (farthest point), it reaches 406,700 km. This ~50,000 km variation causes apparent size changes of about 14% and affects tidal forces by roughly 20%.

Kepler's second law states that equal areas are swept in equal times. The mean anomaly M increases linearly with time, but the true anomaly ν (actual angular position) varies non-linearly. Solving M = E − e·sin(E) via Newton-Raphson iteration yields eccentric anomaly E, from which ν is computed. Near perigee, the Moon moves faster (angular velocity peaks); near apogee, it slows.

The Moon's orbit is inclined 5.145° relative to Earth's ecliptic plane. This inclination is why lunar eclipses don't occur every full moon - the Moon usually passes above or below Earth's shadow. The 3D visualization applies rotation matrices to accurately represent this tilt.

Phase is computed geometrically assuming the Sun lies along the positive X-axis at infinite distance. The illumination fraction equals (1 + cos(θ))/2, where θ is the Sun-Earth-Moon angle. This yields 0% at new moon and 100% at full moon. Accuracy is within 1-2 days for phase identification; atmospheric refraction and lunar libration are not modeled.

No. This is a two-body approximation (Earth-Moon only). The Sun's gravitational influence causes apsidal precession (~8.85 years for perigee to complete one revolution) and nodal regression (~18.6 years). For multi-year predictions, an N-body integrator would be required. This tool is accurate for visualizing single-orbit geometry.

The sidereal period measures one complete orbit relative to distant stars. The synodic period (new moon to new moon) is longer because Earth also orbits the Sun; the Moon must travel an extra ~27° to realign with the Sun. The relationship is 1/T_syn = 1/T_sid − 1/T_Earth, where T_Earth ≈ 365.25 days.