﻿ Orbit Element Types

Orbit Element Types

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The orbital elements are six independent parameters that describe the shape and orientation of a Spacecraft's orbit. The orbital element systems available in FreeFlyer are described below. The ElementConvert function can be used to convert orbital elements between the different systems.

The diagram below illustrates the orbital element systems that are available for each central body in FreeFlyer. Note that the Spherical (Latitude, Longitude), Brouwer-Lyddane Mean, J2 Brouwer-Lyddane, and SGP4 elements are intended for use only when the Spacecraft's central body is Earth. Cartesian

The Cartesian element set defines the position and velocity of a Spacecraft with respect to the origin of an inertial rectangular coordinate system. Positions and velocities are expressed by their components along the three principal axes of the MJ2000 Earth Equator reference frame.

 Element Description X Position along the X-axis Y Position along the Y-axis Z Position along the Z-axis VX Velocity along the X-axis VY Velocity along the Y-axis VZ Velocity along the Y-axis

 // Assign values to the Cartesian orbital elements with respect to the MJ2000 Earth Equator frame Spacecraft1.X = -3410.673; Spacecraft1.Y = 5950.957; Spacecraft1.Z = -1788.627; Spacecraft1.VX = 1.893; Spacecraft1.VY = -1.071; Spacecraft1.VZ = -7.176;

You can also set and get a Spacecraft's or CelestialObject's Cartesian state with respect to other reference frames using the methods:

Keplerian

The Keplerian element set defines six parameters that describe the shape, size, and orientation of a Spacecraft's orbit, as well as the Spacecraft's current location in the orbit (represented by the True Anomaly). The Keplerian element set assumes perfect Keplerian orbital motion, so all gravitational forces are treated as point masses and no perturbing forces are considered when calculating these parameters.

 Element Description A Semi-major axis E Eccentricity I Inclination RAAN Right Ascension of the Ascending Node W Argument of Periapsis TA True Anomaly

 // Assign values to the Keplerian orbital elements with respect to the MJ2000 Earth Equator frame Spacecraft1.A = 7088.42; Spacecraft1.E = 5.17E-08; Spacecraft1.I = 98.22; Spacecraft1.RAAN = 301.97; Spacecraft1.W = 310.24; Spacecraft1.TA = 244.52;

You can also set and get a Spacecraft's or CelestialObject's Keplerian state with respect to other reference frames using the methods:

Spherical

The spherical element set defines the position and velocity of a Spacecraft with respect to the MJ2000 Earth Equator reference frame in spherical coordinates.

 Element Description Radius Magnitude of the position vector RA Right Ascension DEC Declination Vi Magnitude of the velocity vector Azimuth Velocity azimuth angle VFPA Vertical Flight Path Angle

 // Assign values to the Spherical orbital elements Spacecraft1.SphericalRadius = 7088.42; Spacecraft1.RA = 119.81; Spacecraft1.DEC = -14.61; Spacecraft1.Vi = 7.498; Spacecraft1.SphericalAzimuth = 188.50; Spacecraft1.VerticalFPA = 90.00;

Spherical (Latitude, Longitude)

The spherical element set defines the position and velocity of a Spacecraft with respect to the Earth Fixed in spherical coordinates, using the Earth's latitude and longitude in place of right ascension and declination. This element set is intended for use only when the Spacecraft's central body is Earth.

 Element Description Radius Magnitude of the position vector Latitude Geodetic latitude of the spacecraft's subsatellite point Longitude Longitude of the spacecraft's subsatellite point Vi Magnitude of the velocity vector Azimuth Velocity azimuth angle HFPA Horizontal Flight Path Angle

 // Assign values to the Spherical Latitude/Longitude orbital elements Spacecraft1.LatLongRadius = 7088.42; Spacecraft1.Latitude = -14.72; Spacecraft1.Longitude = 64.13; Spacecraft1.LatLongVi = 7.58; Spacecraft1.LatLongAzimuth = 192.29; Spacecraft1.HorizontalFPA = 0;

Nonsingular Keplerian

A representation of the Keplerian element set which removes the mathematical singularities at E = 0 and I = 0 degrees.

 Element Description A Semi-major axis e1 E cos(RAAN + W) e2 E sin(RAAN + W) e3 sin(I/2) sin(RAAN) e4 sin(I/2) cos(RAAN) e5 RAAN + W + MA

 // Assign values to the Nonsingular Keplerian orbital elements Spacecraft1.NonSingularA = 7070.81; Spacecraft1.NonSingularE1 = 0.00181; Spacecraft1.NonSingularE2 = -0.00170; Spacecraft1.NonSingularE3 = -0.641; Spacecraft1.NonSingularE4 = 0.400; Spacecraft1.NonSingularE5 = 136.72;

Equinoctial

A representation of the Keplerian element set which removes the mathematical singularities at E = 0 and I = 0 and 90 degrees. This element set has a singularity at an inclination of 180 degrees. Direct Equinoctial Reference Frame

Note: The equinoctial reference frame in FreeFlyer is the direct (prograde) equinoctial reference frame.

 Element Description A Keplerian semi-major axis h E sin(RAAN + W) k E cos(RAAN + W) p tan(I/2) sin(RAAN) q tan(I/2) cos(RAAN) Longitude Mean Longitude: RAAN + W + MA

 // Assign values to the Equinoctial orbital elements Spacecraft1.EquinoctialA = 7070.81; Spacecraft1.EquinoctialH = 0.00237; Spacecraft1.EquinoctialK = 0.000731; Spacecraft1.EquinoctialP = -0.735; Spacecraft1.EquinoctialQ = 0.458; Spacecraft1.EquinoctialLongitude = 4.412;

Modified Equinoctial

A representation of the Keplerian element set which removes the mathematical singularities at E = 0 and I = 0 and 90 degrees. This element set has a singularity at an inclination of 180 degrees.

 Element Description p A(1 - E2) f E cos(RAAN + W) g E sin(RAAN + W) h tan(I/2) cos(RAAN) k tan(I/2) sin(RAAN) L True Longitude: RAAN + W + TA

 // Assign values to the Modified Equinoctial orbital elements Spacecraft1.ModifiedEquinoctialP = 7070.766; Spacecraft1.ModifiedEquinoctialF = 0.00180; Spacecraft1.ModifiedEquinoctialG = -0.00170; Spacecraft1.ModifiedEquinoctialH = 0.610; Spacecraft1.ModifiedEquinoctialK = -0.980; Spacecraft1.ModifiedEquinoctialL = 136.64;

Brouwer-Lyddane Mean

The Brouwer-Lyddane Mean element contains similar parameters to the traditional Keplerian elements, but represent values that are averaged over time rather than instantaneous values. These elements account for gravitational perturbations due to the J2-J5 oblateness terms. This element set is intended for use only when the Spacecraft's central body is Earth. Mean elements are particularly useful when designing orbits that are sensitive to gravitational perturbations.

 Element Description A Mean semi-major axis E Mean eccentricity I Mean inclination RAAN Mean Right Ascension of the Ascending Node W Mean Argument of Periapsis MA Mean Anomaly

 // Assign values to the Brouwer-Lyddane Mean orbital elements Spacecraft1.BL_A = 7088.42; Spacecraft1.BL_E = 5.17E-08; Spacecraft1.BL_I = 98.22; Spacecraft1.BL_RAAN = 301.97; Spacecraft1.BL_W = 310.24; Spacecraft1.BL_MA = 244.52;

J2 Brouwer-Lyddane

This element set is a mean element set similar to the Brouwer-Lyddane Mean element set, but accounts for only the J2 gravitational perturbation term. This element set is intended for use only when the Spacecraft's central body is Earth. Mean elements are particularly useful when designing orbits that are sensitive to gravitational perturbations.

 Element Description A J2 Mean semi-major axis E J2 Mean eccentricity I J2 Mean inclination RAAN J2 Mean Right Ascension of the Ascending Node W J2 Mean Argument of Periapsis MA J2 Mean Anomaly

 // Assign values to the J2 Brouwer-Lyddane Mean orbital elements Spacecraft1.BLJ2A = 7088.42; Spacecraft1.BLJ2E = 5.17E-08; Spacecraft1.BLJ2I = 98.22; Spacecraft1.BLJ2RAAN = 301.97; Spacecraft1.BLJ2W = 310.24; Spacecraft1.BLJ2MA = 244.52;

SGP4

The SGP4 derived elements are associated with the Two-Line Element (TLE) definitions. This element set is intended for use only when the Spacecraft's central body is Earth, and should be used with the SGP4 propagator.

 Element Description I Mean inclination RAAN Mean Right Ascension of the Ascending Node E Mean Eccentricity W Mean Argument of Perigee MA Mean Anomaly N Mean Motion

 // Assign values to the SGP4 derived elements Spacecraft1.SGP4.I = 98.25; Spacecraft1.SGP4.RAAN = 302.04; Spacecraft1.SGP4.E = 0.00264; Spacecraft1.SGP4.W = 357.18; Spacecraft1.SGP4.MA = 197.55; Spacecraft1.SGP4.MeanMotion = 14.61;