Orbit Element Types

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.

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 ICRF 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 ICRF 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:

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 ICRF 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:

The spherical element set defines the position and velocity of a Spacecraft with respect to the ICRF 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;

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;

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;

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.

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;

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. The Brouwer-Lyddane Mean element type, also referred to as Brouwer-Lyddane Long Transformation (BLLT), includes averaging the osculating elements using both short (days/weeks) and long-term periods (months/years). 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;

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. The J2 Brouwer-Lyddane element type, also referred to as Brouwer-Lyddane Short Transformation (BLST), includes averaging the osculating elements using short-period terms (days/weeks). 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;

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;

Orbit Element Types

Brouwer-Lyddane Mean

See Also