Classical Orbital Elements

Six numbers that describe the orbit (and where the satellite is on it), as an alternative to the Cartesian state vector. See also ENAE 441: Orbits and ENAE 441: State Information.

Orbital Motion So Far

• Summary of what the conservation laws give us:
  • The orbit is in a plane (constant angular momentum)
    • , semimajor axis: size
    • , eccentricity: shape
    • these are orbital elements
    • true anomaly a function of time ???
  • It is a conic section
• Next steps (start of “Orbital state and elements”):
  • Go from the 2D orbit in space to
    • 3D
    • the orbit in time
  • Orbital elements and the state vector: the state of a system

Conserved Quantities

• Conserved quantities:
  • Energy (1)
  • Angular momentum (3)
  • Eccentricity vector (3)
• These are constant, so the true anomaly can’t be extracted from them
• 7 conserved, yet ?
  • for one of these, only two components are required
  • Energy is redundant: it can be calculated from the angular momentum and eccentricity
• So the conserved quantities supply 5 elements; the sixth must be a time-dependent one

Orbit in Three Dimensions

• Embed the orbital plane in 3D space:
  • three orbital elements from the plane (physics): , , and a time element
  • three orbital elements from the 3D embedding (geometry): , ,
• Reference: the geocentric equatorial, Earth-centered inertial (ECI) frame
  • toward the vernal equinox, toward the north pole
  • “inertial” “non-rotating”, i.e., not rotating with the central body
  • more in Coordinate Systems
• The orientation is a set of 3-1-3 Euler angles:
  • , inclination: the angle from the equatorial to the orbital plane (from to )
  • , right ascension of the ascending node (RAAN): angle between and the ascending node , which is the intersection of the orbital and equatorial planes on the side where the satellite goes from the southern to the northern hemisphere
  • , argument of perigee: angle from the ascending node to the eccentricity vector
• Sketches (drawn after screen sharing failed): an inclined orbit seen edge-on, crossing the equator at angle ; and a view down the pole of an orbit circling the rotating Earth

Classical Element Set

• Special cases: (circular), (equatorial)
  • here some angles are undefined: a “singular orbital element set” (fixes in Equinoctial Elements)
• Classical or Kepler orbital element set:
  3. [time dependent]: e.g. the true anomaly ; a time element, or position angle
• Description of the orbital state: the orbital state vector in Cartesian space
• Conversions both ways: Orbital Elements from State Vector and State Vector from Orbital Elements

Variants of Orbital Elements

1. Handy variations of classical elements, e.g. , instead of ,
2. Changing the time element / position angle (e.g. mean anomaly instead of )
3. Nonsingular elements: circular elements, equinoctial
4. Canonical elements

• Comparison of the element sets:

Set	Position variable	Singularities	Nature
Classical	or	, ,	Geometric
Equinoctial			Nonsingular
Modified Equinoctial	(true longitude)	None ( handled via factor)	Universal
Delaunay	(mean anomaly)	, ,	Canonical
Poincaré	(mean longitude)	None	Canonical / Nonsingular

Argument of Latitude

• For circular orbits there is no perigee, so measure the satellite’s position from the ascending node instead
• Argument of latitude : the angle from the ascending node to the satellite
• If the orbit is not circular:
• Sketch: line of nodes, perigee at from the node, satellite at from perigee; spans both