Cartesian Orbital Elements in Satellite Initial State

Cartesian Orbital Elements

• The Cartesian orbital elements are used to describe the initial state of a satellite in ECI (Earth-Centered Inertial) coordinates.
• The Cartesian orbital elements are represented by the vector ce containing the initial position and velocity components: r1, r2, r3, r˙1, r˙2, r˙3.

Classical Orbital Elements

• The classical orbital elements are a, e, i, Ω, ω, M0.
• a is the orbit semimajor axis.
• e is the orbit eccentricity, ranging from 0 to 1.
• i is the orbital plane inclination, ranging from 0 to 180 degrees.
• Ω is the right ascension of the ascending node (RAAN), ranging from 0 to 360 degrees.
• ω is the anomaly of the periapsis, ranging from 0 to 360 degrees.
• M0 is the mean anomaly at epoch, ranging from 0 to 360 degrees.

Perifocal Coordinate System (PCS)

• The PCS is inertially fixed only in the absence of orbital perturbations.
• The PCS has three base vectors: p̂1, p̂2, p̂3.
• p̂1 is in the direction of the periapsis (equivalent to the eccentricity vector).
• p̂2 defines the orbital plane together with ê1.
• p̂3 is perpendicular to the orbital plane and positive in the direction of the orbital angular momentum.

Hill Cartesian Coordinate System (Hill CCS)

• The Hill CCS is centered at the spacecraft center of mass.
• The Hill CCS has three base vectors: ĥ1, ĥ2, ĥ3.
• ĥ1 is in the direction of the instantaneous satellite position (radius vector).
• ĥ2 is in the transversal direction, defining the orbital plane together with ĥ1.
• ĥ3 is perpendicular to the orbital plane and positive in the direction of the orbital angular momentum.

Circular Orbit

• A circular orbit has an eccentricity e equal to 0.
• The orbital trajectory is a circle with a radius R coincident with the semilatus rectum.
• The satellite flies at a constant orbital speed, equivalent to its transversal velocity, while the radial velocity stays zero.
• The angular momentum h is related to the orbital radius R by the equation h = µR.

This quiz covers the use of Cartesian orbital elements in determining the initial state of a satellite in ECI coordinates, including position and velocity.