Idealized Orbit

Earth System with Moon, showing the Moon's idealized orbit (File: 4.5 MB)

This page describes how we create the Moon's idealized orbit calculated from its orbital elements and how we are able to closely match the Moon's actual orbit. See the image above that illustrates this close match. The linked file provides a complete 3D plot that can be tilted and explored in detail. 

1. Time-Varying Orbital Elements

The Moon's orbit changes significantly over time due to gravitational perturbations. In idealized_orbits.py the function calculate_moon_orbital_elements accounts for this:

def calculate_moon_orbital_elements(date):

    # Calculate Julian centuries since J2000.0

    T = (date - j2000).total_seconds() / (36525.0 * 86400.0)

    # Node regression (retrograde motion) - 18.6 year cycle

    Omega = 125.08 - 0.0529538083 * (date - j2000).days

    # Apsidal precession - 8.85 year cycle  

    omega = 318.15 + 0.1643573223 * (date - j2000).days

The static version in idealized_orbits.py, planetary_params uses fixed values:

'Moon': {

    'a': 0.002570,   # Fixed semi-major axis

    'e': 0.0549,     # Fixed eccentricity  

    'i': 5.145,      # Fixed inclination

    'omega': 318.15, # Fixed argument of periapsis

    'Omega': 125.08  # Fixed longitude of ascending node

}

2. Major Perturbation Effects

The Moon experiences significant perturbations that the static model ignores:

Evection (largest perturbation):

# Mean anomalies for perturbation calculations

M_moon = (134.963 + 13.064993 * d) % 360  # Moon's mean anomaly

M_sun = (357.529 + 0.98560028 * d) % 360   # Sun's mean anomaly  

D = (297.850 + 12.190749 * d) % 360        # Mean elongation

# Evection - largest lunar perturbation (~1.3° amplitude)

e_evection = 0.01098 * np.cos(2*D_rad - M_moon_rad)

e = e_base + e_evection

Node Regression:

The Moon's ascending node moves backwards ~19.3°/year, completing a cycle in 18.6 years.

Apsidal Precession:

The Moon's line of apsides (perigee-apogee line) rotates ~40.7°/year, completing a cycle in 8.85 years.

3. Rotation Sequence Accuracy

Both functions use the same standard rotation sequence:

# 1. Argument of periapsis (ω) around z-axis

x_temp, y_temp, z_temp = rotate_points(x_orbit, y_orbit, z_orbit, omega_rad, 'z')

# 2. Inclination (i) around x-axis  

x_temp, y_temp, z_temp = rotate_points(x_temp, y_temp, z_temp, i_rad, 'x')

# 3. Longitude of ascending node (Ω) around z-axis

x_final, y_final, z_final = rotate_points(x_temp, y_temp, z_temp, Omega_rad, 'z')

The rotation mechanics are identical - the accuracy improvement comes from using the correct, time-varying orbital elements rather than any special rotation technique.

4. Why This Matters for the Moon

The Moon's proximity to Earth makes it subject to strong perturbations that cause rapid changes in its orbital elements:

• Node regression: ~19.3°/year change in Ω

• Apsidal precession: ~40.7°/year change in ω

• Evection: Periodic changes in eccentricity with ~31.8-day period

• Solar perturbations: Additional periodic variations

For comparison, planetary orbits change much more slowly, so static elements work reasonably well for short time periods.

5. Result

The time-varying approach produces an orbit that:

• Correctly shows the Moon's changing orientation in space

• Accounts for the precession of its orbital plane

• Includes the major periodic perturbations

• Matches JPL ephemeris data much more closely

This is why in idealized_orbits.py, plot_moon_ideal_orbit creates a "highly accurate ideal orbit" - it's using a sophisticated perturbation model rather than simple Keplerian elements.