Orbital elements

Definition

Fix a reference plane through the central body, a reference direction in it, and a reference time $t_0$. Let a body move on an elliptic Kepler orbit whose plane is not the reference plane. Its Keplerian orbital elements are the six numbers:

1. the semi-major axis $a > 0$ of the ellipse;
2. the eccentricity $e \in [0, 1)$;
3. the inclination $i \in [0, \pi]$, the angle between the orbital plane and the reference plane;
4. the longitude of the ascending node $\Omega \in [0, 2\pi)$, the angle in the reference plane from the reference direction to the ascending node;
5. the argument of periapsis $\omega \in [0, 2\pi)$, the angle in the orbital plane from the ascending node to the periapsis;
6. the mean anomaly $M_0 \in [0, 2\pi)$ at time $t_0$.

The ascending node is the point where the body crosses the reference plane going upwards. The mean anomaly at time $t$ is $n (t - \tau)$, where $n = 2\pi / T$ is the mean motion, $T$ the period and $\tau$ a time of periapsis passage.