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.

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