Planetary Data

Planets

┌──────────┬────────────────┬──────────┬────────┬───────┬───────┬────────┬─────────┬──────────┬─────────┬────────┐ │ Body │ Mass (kg) │ Radius │ a │ e │ inc │ Ω │ ω │ Period │ Day │ M │ │ │ │ (km) │(AU) │ │ (°) │ (°) │ (°) │ (days) │ (hours) │ (°) │ ├──────────┼────────────────┼──────────┼────────┼───────┼───────┼────────┼─────────┼──────────┼─────────┼────────┤ │ Venus │ 4.87×10²⁴ │ 6,052 │ 0.723 │ 0.007 │ 3.39 │ 76.68 │ 54.92 │ 224.7 │ 2,802.0 │ 50.38 │ │ Earth │ 5.97×10²⁴ │ 6,378 │ 1.000 │ 0.017 │ 0.00 │ 0.00 │ 102.94 │ 365.2 │ 24.0 │ −2.47 │ │ Mars │ 6.42×10²³ │ 3,396 │ 1.524 │ 0.093 │ 1.85 │ 49.56 │ 286.50 │ 687.0 │ 24.7 │ 19.39 │ │ Jupiter │ 1.898×10²⁷ │71,492 │ 5.203 │ 0.049 │ 1.31 │100.47 │ 274.25 │ 4,331 │ 9.9 │ 19.67 │ │ Saturn │ 5.68×10²⁶ │60,268 │ 9.537 │ 0.057 │ 2.49 │113.66 │ 338.94 │10,747 │ 10.7 │ −42.64 │ │ Uranus │ 8.68×10²⁵ │25,559 │19.19 │ 0.046 │ 0.77 │ 74.02 │ 96.94 │30,589 │ 17.2 │ 142.28 │ │ Neptune │ 1.02×10²⁶ │24,764 │30.07 │ 0.010 │ 1.77 │131.78 │ 273.18 │59,800 │ 16.1 │ −100.08│ └──────────┴────────────────┴──────────┴────────┴───────┴───────┴────────┴─────────┴──────────┴─────────┴────────┘

Moons

┌──────────────┬────────────────┬──────────┬──────────┬───────┬───────┬────────┬─────────┬─────────┬───────┐ │ Moon │ Mass (kg) │ Radius │ a │ e │ inc │ Ω │ ω │ Period │ M │ │ │ │ (km) │ (km) │ │ (°) │ (°) │ (°) │ (days) │ (°) │ ├──────────────┼────────────────┼──────────┼──────────┼───────┼───────┼────────┼─────────┼─────────┼───────┤ │ Moon (Earth) │ 7.35×10²² │ 1,738 │ 384,400 │ 0.055 │ 5.16 │125.08 │ 318.15 │ 27.322 │135.27 │ │ Io │ 8.93×10²³ │ 1,822 │ 421,800 │ 0.004 │ 0.00 │ 0.0 │ 49.1 │ 1.763 │330.9 │ │ Europa │ 4.80×10²³ │ 1,561 │ 671,100 │ 0.009 │ 0.50 │184.0 │ 45.0 │ 3.525 │345.4 │ │ Ganymede │ 1.48×10²⁴ │ 2,631 │1,070,400 │ 0.001 │ 0.20 │ 58.5 │ 198.3 │ 7.156 │324.8 │ │ Callisto │ 1.08×10²⁴ │ 2,410 │1,882,700 │ 0.007 │ 0.30 │309.1 │ 43.8 │ 16.690 │ 87.4 │ │ Titan │ 1.35×10²⁴ │ 2,575 │1,221,900 │ 0.029 │ 0.30 │ 78.6 │ 78.3 │ 15.945 │ 11.7 │ └──────────────┴────────────────┴──────────┴──────────┴───────┴───────┴────────┴─────────┴─────────┴───────┘

Reference Frames

Source: https://ssd.jpl.nasa.gov/orbits.html

Planets: All orbital elements are referenced to the mean ecliptic and equinox of J2000.
Moons: The source data for moons is referenced to the Laplace plane (Jupiter and Saturn's moons) or the ecliptic (Earth's Moon). The Laplace plane is a hybrid reference plane between a planet's equator and its orbital plane around the Sun.
Important: Moon inclination and node values are not referenced to the same plane as the planets. Converting to a common frame is required before combining into a single simulation.

Moon Frame Transformation Plan

Source data provides for each moon: Tilt (angle between planet's equator and Laplace plane), R.A. and Dec. (Laplace plane pole position in ICRF).

Transformation approach using in-engine primitives:

Build a rotation matrix from Laplace plane to equatorial plane using the Tilt angle and pole position (R.A., Dec.)
Apply the rotation to the moon's position/velocity vectors via Mat3 × Vec3
Reconstruct orbital elements from the rotated Cartesian state using cartesian_to_orbital_elements()

This leverages the existing mat3_rotation_x, mat3_rotation_z, and mat3_multiply functions to compose the frame-rotation matrix, then uses the engine's built-in cartesian_to_orbital_elements() to extract the new (i, Ω, ω) values in the equatorial frame.

J2000 Starting Positions

Source: Table 1 from https://ssd.jpl.nasa.gov/orbits.html (valid 1800–2050 AD, no perturbation terms needed).

Mean anomaly at J2000: M = L − ϖ, where L is mean longitude and ϖ is longitude of perihelion.

To get the true anomaly ν (which the TOML orbit.true_anomaly expects), solve Kepler's equation:

M = E − e·sin(E)   →   solve for eccentric anomaly E
tan(ν/2) = √((1+e)/(1−e)) · tan(E/2)

Once ν is computed for each body, set it as true_anomaly in the config. The engine will then propagate from the J2000 snapshot forward.

Laplace Plane Data Limitations

Reliable, authoritative data for the Laplace plane parameters (pole R.A./Dec. and tilt relative to each planet's equator) is difficult to find in standard planetary data sources. JPL Horizons and the JPL orbits page provide moon orbital elements relative to the Laplace plane but do not publish the Laplace plane's own orientation in ICRF.

Decision: Use the planet's equatorial frame for moon orbital elements instead of converting from the Laplace plane. The Laplace plane is very close to the equatorial plane — tilted by only ~1° for Jupiter and ~0.3° for Saturn — so the resulting errors are negligible:

Inclination offset: ~0.3–1°
Node and periapsis offset: similar small amounts
Angular position error in space: ~0.3–1°

This error is smaller than the uncertainties from using mean orbital elements (which ignore perturbations and resonances) and has no practical impact for simulation purposes.

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