The mathematical principles behind Asteroid3D's real-time 3D Keplerian orbital propagation engine.
1. The 6 Classical Keplerian Orbital Elements
1. **$a$ (Semi-major Axis)**: Determines orbit scale in Astronomical Units (AU).
2. **$e$ (Eccentricity)**: Elliptical elongation ($0 \le e < 1$).
3. **$i$ (Inclination)**: Orbital plane tilt relative to the ecliptic.
4. **$\Omega$ (Longitude of Ascending Node)**: Orientation of nodal crossing.
5. **$\omega$ (Argument of Perihelion)**: Orientation of the ellipse within its plane.
6. **$M$ (Mean Anomaly)**: Epoch fraction representing time since perihelion.
2. Solving Kepler's Transcendental Equation
$$M = E - e \sin E$$
Solved numerically via Newton-Raphson iteration:
$$E_{n+1} = E_n - \frac{E_n - e \sin E_n - M}{1 - e \cos E_n}$$
3. Euler Rotation to 3D Heliocentric Cartesian Coordinates
Applying successive orthogonal transformation matrices $R_z(\Omega) R_x(i) R_z(\omega)$ maps planar orbital coordinates into 3D Cartesian space $(X, Y, Z)$.