Orbital Elements from State Vector

Computing the classical orbital elements from the state vector, position and velocity (Curtis section 4.4, algorithm 4.2).

The Map

Vectors First

1. Angular momentum vector:
2. Semilatus rectum:
3. Eccentricity vector:
4. Node direction (toward the ascending node):

• Then the size and shape:

RAAN and Inclination

• RAAN, quadrant-safe with the two-argument arctangent:
• or, from the cosine with a quadrant check:
• Inclination:
  • no quadrant check needed: already covers to

Argument of Perigee

• Angle between the ascending node and :
• If , then the perigee is below the equatorial plane:

True Anomaly

• Angle between and , or from the orbit equation:
• If , then the true anomaly is below the apse line (major axis), so
  • (radial rate): negative radial velocity means the satellite is heading back toward perigee
• These give us the orbital elements from the orbital state vector, position and velocity
• The reverse direction: State Vector from Orbital Elements

Singular cases

• is undefined for () and for : then , , are undefined; use Equinoctial Elements