True anomaly
In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit.It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits).The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π rad).The true anomaly f is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as: where: For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis.Instead the argument of latitude u is used: where: For circular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes.One uses the true longitude instead: where: The relation between the true anomaly ν and the eccentric anomalyis: or using the sine[1] and tangent: or equivalently: so Alternatively, a form of this equation was derived by [2] that avoids numerical issues when the arguments are nearare always in the same quadrant, there will not be any sign problems.via a Fourier expansion:[3] with Bessel functionsOmitting all terms of order), it can be written as[3][4][5] Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricityis known as the equation of the center, where more details about the expansion are given.The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula where a is the orbit's semi-major axis.In celestial mechanics, Projective anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit.It is the angle between the direction of periapsis and the current position of the body in the projective space.The projective anomaly is usually denoted by theand is usually restricted to the range 0 - 360 degree (0 - 2is one of four angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly, true anomaly and the mean anomaly.In the projective geometry, circle, ellipse, parabolla, hyperbolla are treated as a same kind of quadratic curves.An orbit type is classified by two project parametersis semi major axis,Position and heliocentric distance of the planetcan be calculated as functions of the projective anomaly− β + α cos θ1 + α β cos θ1 + α β cos θα − β cos θ1 + α β cos θcan be calculated from the eccentric anomaly
Orbital mechanicsOrbital elementsArgument of periapsisEccentricityInclinationMean anomalyOrbital nodesSemi-major axistwo-body orbitsCircular orbitElliptic orbitTransfer orbitHohmann transfer orbitBi-elliptic transfer orbitParabolic orbitHyperbolic orbitRadial orbitDecaying orbitDynamical frictionEscape velocityKepler's equationKepler's laws of planetary motionOrbital periodOrbital velocitySurface gravitySpecific orbital energyVis-viva equationCelestial mechanicsBarycenterHill spherePerturbationsSphere of influenceN-body orbitsLagrangian pointsHalo orbitsLissajous orbitsLyapunov orbitsEngineering and efficiencyMass ratioPayload fractionPropellant mass fractionTsiolkovsky rocket equationGravity assistOberth effectOrbital maneuverOrbit insertionparameterKeplerian orbitperiapsisellipseGreek lettersLatin lettereccentric anomalyorbital state vectorsorbital velocity vectoreccentricity vectororbital position vectorcircular orbitsargument of latitudetrue longitudetangentFourier expansionBessel functionsequation of the centerperihelion distanceaphelion distanceTwo body problemprojective geometryHyperbolaBibcodeorbitsCaptureCircularEllipticalHighly ellipticalEscapeHorseshoeHyperbolic trajectoryInclinedNon-inclinedKeplerLagrange pointOsculatingParabolic trajectoryParkingPrograde / RetrogradeSynchronousGeocentricGeosynchronousGeostationaryGeostationary transferGraveyardHigh EarthLow EarthMedium EarthMolniyaNear-equatorialOrbit of the MoonSun-synchronousTransatmosphericTundraVery low EarthAreocentricAreosynchronousAreostationaryDistant retrogradeLissajousLibrationHeliocentricEarth's orbitMars cyclerHeliosynchronousLunar cyclerParametersSemi-minor axisApsidesLongitude of the ascending nodeLongitude of the periapsisMean longitudeMean motionOrbital speedManeuversBi-elliptic transferCollision avoidance (spacecraft)Delta-vDelta-v budget