Keplerain elements: Orbit plane Satellite equinox Node A greenwich Inclination Q2 Right Ascension of ascending node oo Argument of perigee V True anomaly Principles of the Global Positioning System 2005-3-11(11 Keplerain elements in plane Apogee Perigee FocuS Center of Mass a semimajor axIs V True anomaly b semiminor axis E Eccentric anomaly e eccentricity M Mean anomaly Principles of the Global Positioning System 20053-11(12
6 Principles of the Global Positioning System 2005-3-11 11 Keplerain elements: Orbit plane Node i ω Ω ν Z θ0 Greenwich Vernal equinox Satellite perigee equator i Inclination Ω Right Ascension of ascending node ω Argument of perigee ν True anomaly Principles of the Global Positioning System 2005-3-11 12 Keplerain elements in plane a Focus Center of Mass ae Satellite Apogee Perigee b E ν r a semimajor axis b semiminor axis e eccentricity ν True anomaly E Eccentric anomaly M Mean anomaly
The mean angular satellite velocity n an angular satellite velocity n(also known the mean motion) with revolution period P follows from Kepler's Third Law given 2 For gPS orbits, a=26560 kn, so, an orbital period 12 sidereal hours. The ground track of the satellites repeats every sidereal day 電 Principles of the Global Positioning System 2005-3-11(13 Orbit Representation n orbital plane, the position vector r and the velocit tv vector i=drdt (with eccentric true anomaly cos e-e coSy X3=X3 Principles of the Global Positioning System 2005-3-11(14
7 Principles of the Global Positioning System 2005-3-11 13 The mean angular satellite velocity n Principles of the Global Positioning System 2005-3-11 14 Orbit Representation
Orbit Representation The transformation of r and r into the equatorial system x o is performed by a rotation matrix X satellite perigee R 3D rotation R. e3=0 LOT orbit &e Principles of the Global Positioning System 2005-3-11(15 Orbit Representation R=R3{-91{-i}R3{-m}=[eg2e3] In order to rotate the system x into the terrestrial system X, an additional rotation through the angle Oo, the transformation matrix, therefore, becomes quired The R=R3O。}3{-2R1{-i}R3{ Orbital Plane Space-fixed Sys. ->Terrestrial Sys. 電 Principles of the Global Positioning System 2005-3-11(16
8 Principles of the Global Positioning System 2005-3-11 15 Orbit Representation Principles of the Global Positioning System 2005-3-11 16 Orbit Representation
Differentia/ Relations The derivatives of p and p with respect to the six Keplerian parameters are required in one of the The vectors r and i depend only on the parameters a.e. To. whereas the matrix is only a function of the haining parameters a i, s2 The differential relations The meaning? aR 中=Rdn+Rsd+Rsdm R 電 Principles of the Global Positioning System 20053117 Perturbed Motion The Keplerian orbit is a theoretical orbit and does not include actual perturbations The perturbed motion is based on an inhomogeneous differential equation of second order p+“P= For GPS satellites, the acceleration p is at least 10+times attractive force A=? Analytical solution Au =l Principles of the Global Positioning System 2005-3-11
9 Principles of the Global Positioning System 2005-3-11 17 Differential Relations Principles of the Global Positioning System 2005-3-11 18 Perturbed Motion