Spacecraft Orbit State

SMA contains information on the type and size of an orbit. If SMA > 0 the orbit is elliptic. If SMA <0 the orbit is hyperbolic. SMA is infinite for parabolic orbits.

ECC contains information on the shape of an orbit. If ECC = 0, then the orbit is circular. If 0 < ECC < 1, the orbit is elliptical. If , ECC = 1 the orbit is parabolic. If ECC > 1 then the orbit is hyperbolic.

INC is the angle between the orbit angular momentum vector and the z-axis. If INC < 90 deg., then the orbit is prograde. If INC > 90 deg, then the orbit is retrograde

RAAN is defined as the angle between x-axis and the node vector measured counterclockwise. The node vector is defined as the cross product of the z-axis and orbit angular momentum vector. RAAN is undefined for equatorial orbits.

AOP is the angle between a vector pointing at periapsis and a vector pointing in the direction of the line of nodes. AOP is undefined for circular orbits.

TA is defined as the angle between a vector pointing at periapsis and a vector pointing at the spacecraft. TA is undefined for circular orbits.

SMA is infinite and cannot be used to define the size of the orbit. GMAT requires ECC < 0.9999999 or ECC > 1.0000001 when setting ECC or when performing conversions. For transformations performed near these limits, loss of precision may occur.

AOP is undefined. If ECC <= 1e-11, GMAT sets AOP to zero in the conversion from Cartesian to Keplerian/ModKeplerian and includes all orbital-plane angular displacement in the true anomaly.

Results in a singular conic section. GMAT requires |SMA| > 1 meter when inputting SMA.

SMA is infinite and another parameter is required to capture the size of the orbit. Keplerian elements are not supported.

RAAN is undefined. If INC < 6e-10, GMAT sets RAAN to 0 in the conversion from Cartesian to Keplerian/ModKeplerian. Then, if ECC < 1e-11, AOP is set to 0 and GMAT includes all angular displacement between the x-axis and the spacecraft in the true anomaly. If ECC ≥ 1e-11, then AOP is computed as the angle between the eccentricity vector and the x-axis.

RAAN is undefined. If INC > (180 - 6e-10), GMAT sets RAAN to 0 in the conversion from Cartesian to Keplerian/ModKeplerian. Then, if ECC < 1e-11, AOP is set to 0 and GMAT includes all angular displacement between the x-axis and the spacecraft in the true anomaly. If ECC ≥ 1e-11, then AOP is computed as the angle between the eccentricity vector and the x-axis.

Singular conic section. GMAT requires RadPer > 1 meter in state conversions.

Singular conic section. GMAT requires abs(RadApo) > 1 meter in state conversions.

The mean anomaly. It is related to uniform angular motion on a circle of radius SMA.

See “Keplerian State” section, AOP

See “Keplerian State” section, RAAN

Related to the two-body orbital energy. DelaunayL = sqrt(mu*SMA)

Magnitude of the orbital angular momentum vector. DelaunayG = DelaunayL*sqrt(1-ECC^2)

The K component of the orbital angular momentum. DelaunayH = DelaunayG * cos(INC)

DelaunayL is not real for hyperbolic orbits by its definition.

BrouwerLongAOP

BrouwerShortAOP

Brouwer-Lyddane long-term averaged (short-term averaged) mean argument of periapsis.

BrouwerLongMA

BrouwerShortMA

Brouwer-Lyddane long-term averaged (short-term averaged) mean MA (mean anomaly).

BrouwerLongECC

BrouwerShortECC

Brouwer-Lyddane long-term averaged (short-term averaged) mean eccentricity.

BrouwerLongINC

BrouwerShortINC

Brouwer-Lyddane long-term averaged (short-term averaged) mean inclination.

BrouwerLongRAAN

BrouwerShortRAAN

Brouwer-Lyddane long-term averaged (short-term averaged) mean RAAN (right ascension of the ascending node).

BrouwerLongSMA

BrouwerShortSMA

Long-term averaged (short-term averaged) mean semi-major axis.

Because Brouwer’s formulation based on Earth’s zonal harmonics, BrouwerMeanShort and BrouwerMeanLong cannot address orbits with mean perigee distance is smaller than Earth’s radius, 3000 km because of numerical instability.

If given BrouwerLongINC (long-term averaged INC only) is close to i_c= 63 deg. or 117 deg., the algorithm is unstable because of singular terms (non-zero imaginary components). Thus, GMAT cannot calculate osculating elements.

If BrouwerECC is larger than 0.9, or BrouwerECC is smaller than 1E-7, it has been reported that Cartesian to BrouwerMeanLong state does not converge statistically. For these cases, GMAT gives a warning message with the current conversion error.

The magnitude of the position vector.

The right ascension which is the angle between the projection of the position vector into the xy-plane and the x-axis measured counterclockwise.

The declination which is the angle between tjhe position vector and the xy-plane.

The magnitude of the velocity vector.

The vertical flight path angle. The angle measured from a plane normal to the postion vector to the velocity vector , measured in the plane formed by position vector and velocity vector.

The flight path azimuth. The angle measured from the vector perpendicular to the position vector and pointing north, to the projection of the velocity vector, into a plane normal to the position vector.

The right ascension of velocity. The angle between the projection of the velocity vector into the xy-plane and the x-axis measured counterclockwise.

The flight path azimuth. The angle between the velocity vector and the xy-plane.

Results in a singular conic section: declination and flight path angle are undefined. GMAT will not allow transformations if RMAG < 1e-10. For RMAG values greater than, but near 1e-10, loss of precision may occur in transformations.

Results in a singular conic section: velocity declination and flight path angle are undefined. GMAT will not allow transformations if VMAG < 1e-10.For VMAG values greater than, but near 1e-10, loss of precision may occur in transformations.

Magnitude of the orbital radius vector.

Planetodetic longitude.

Planetodetic latitude, using the Flattening of the central body.

Magnitude of the orbital velocity vector in the fixed frame.

Orbital velocity azimuth in the fixed frame.

Horizontal flight path angle. HFPA = 90 - VFPA

Results in a singular conic section: declination and flight path angle are undefined. GMAT will not allow transformations if PlanetodeticRMAG < 1e-10. For PlanetodeticRMAG values greater than, but near 1e-10, loss of precision may occur in transformations.

Results in a singular conic section: velocity declination and flight path angle are undefined. GMAT will not allow transformations if PlanetodeticVMAG < 1e-10. For PlanetodeticVMAG values greater than, but near 1e-10, loss of precision may occur in transformations.

See Keplerian section.

A measure of the orbital eccentricity and argument of periapsis. EquinoctialH and EquinoctialK together govern how elliptical an orbit is and where the periapsis is located. EquinotialH = ECC * sin(AOP).

A measure of the orbital eccentricity and argument of periapsis. EquinoctialH and EquinoctialK together govern how eliptical an orbit is and where the periapsis is located. EquinotialK = ECC * cos(AOP)

A measure of the orientation of the orbit. EquinoctialP and EquinoctialQ together govern how an orbit is oriented. EquinotialP = tan(INC/2)*sin(RAAN).

A measure of the orientation of the orbit. EquinoctialP and EquinoctialQ together govern how an orbit is oriented. EquinotialQ = tan(INC/2)*cos(RAAN).

A measure of the mean location of the spacecraft in its orbit. MLONG = AOP + RAAN + MA.

RAAN is undefined. If INC > 180 - 1.0e-11, GMAT sets RAAN to 0 degrees. GMAT does not support Equinoctial elements for true retrograde orbits.

Equinoctial elements are not defined for parabolic or hyperbolic orbits.

See Keplerian section.

See Equinoctial section.

See Equinoctial section.

A measure of the orientation of the orbit. AltEquinoctialP and AltEquinoctialQ together govern how an orbit is oriented. AltEquinotialP = sin(INC/2)*sin(RAAN).

A measure of the orientation of the orbit. AltEquinoctialP and AltEquinoctialQ together govern how an orbit is oriented. AltEquinotialP = sin(INC/2)*cos(RAAN).

See Equinoctial section.

Magnitude of the position vector when at true anomaly of 90 deg SemilatusRectum = SMA*(1-ECC^2)

Components of eccentricity vector (with ModEquinoctialG). Projection of eccentricity vector onto x. ModEquinoctialF = ECC * cos (AOP+RAAN)

Components of eccentricity vector (with ModEquinoctialF). Projection of eccentricity vector onto y. ModEquinoctialG = ECC * sin (AOP+RAAN)

Identical to EquinoctialQ.

Idential to EquinoctialP.

A measure of the true location of the spacecraft in its orbit. TLONG = AOP + RAAN + TA.

Similar to Equinoctial elements, there is singularity at INC = 180 deg. GMAT does not support ModifiedEquinoctial elements for retrograde equatorial orbits.

IncomingRadPer

OutgoingRadPer

The orbital radius of periapsis. The radius of periapsis is the minimum distance (osculating) between the spacecraft and celestial body at the origin of coordinate system. IncomingRadPer/OutgoingRadPer differ from RadPer only in that they are associated with the IncomingAsymptote and OutgoingAsymptote state representations, respectively.

IncomingC3Energy

OutgoingC3Energy

C3 energy. C3Energy = -mu/SMA. IncomingC3Energy/OutgoingC3Energy differ only in that they are associated with the IncomingAsymptote and OutgoingAsymptote state representations, respectively.

IncomingRHA

OutgoingRHA

IncomingRHA/OutgoingRHA is the right ascension of the incoming/outgoing asymptote. If C3Energy < 0 the apsides vector is substituted for the incoming/outgoing asymptote.

IncomingDHA

OutgoingDHA

IncomingDHA/OutgoingDHA is the declination of the incoming/outgoing asymptote. If C3Energy < 0 the apsides vector is substituted for the incoming/outgoing asymptote..

IncomingBVAZI

OutgoingBVAZI

IncomingBVAZI/OutgoingBVAZI is the B-vector azimuth at infinity of the incoming/outgoing asymptote measured counter-clockwise from south. If C3Energy < 0 the apsides vector is substituted for the outgoing/incoming asymptote.

See Keplerian.

If IncomingC3Energy/OutgoingC3Energy = 0 the spacecraft has a parabolic orbit. Hyperbolic asymptote states do not support parabolic orbits. It must be avoided that -1E-7 ≤ IncomingC3Energy/OutgoingC3Energy ≤ 1E-7 by choosing a proper set of elements.

For the case of circular orbits, TA is undefined. It must be avoided that ECC ≤ 1E-7 by choosing a proper set of elements. GMAT does not support hyperbolic asymptote representation for true circular orbits.

If the asymptote vector is parallel or antiparallel to coordinate system’s z-direction, then the B-plane is undefined. It must be avoided by choosing either a proper coordinate system or set of elements.