12.215 Modern NavigationSummary of last classSatellite OrbitsDynamics of satellite orbitsSimple dynamicsSolution for central force modelKeplerain elements: Orbit planeKeplerain elements in planeSatellite motionTrue anomalyEccentric anomalyVector to satelliteFinal conversion to Earth Fixed XYZPerturbed motionsPerturbation from Flattening J2J2 PerturbationsGravitational perturbation stylesOther perturbation on orbits and approximate sizeGPS OrbitsBasic ConstellationBroadcast EphemerisDistribution of EphemeridesRINEX and SP3 standardAccuracy of orbitsSummary of Today’s class12.215 Modern NavigationThomas Herring ([email protected]), MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/~tah/12.21512/07/2009 12.215 Lec 21 2Summary of last class–Ionospheric delay effects in GPS•Look at theoretical development from Maxwell’s equations•Refractive index of a low-density plasma such as the Earth’s ionosphere.•Most important part of today’s class: Dual frequency ionospheric delay correction formula using measurements at two different frequencies–Effects of ionospheric delay are large on GPS (10’s of meters in point positioning); 1-10ppm for differential positioning–Largely eliminated with a dual frequency correction (most important thing to remember from this class) at the expense of additional noise (and multipath)–Residual errors due to neglected terms are small but can reach a few centimeters when ionospheric delay is large.12/07/2009 12.215 Lec 21 3Satellite Orbits•Treat the basic description and dynamics of satellite orbits•Major perturbations on GPS satellite orbits•Sources of orbit information:–SP3 format from the International GPS service–Broadcast ephemeris message•Accuracy of orbits and health of satellites12/07/2009 12.215 Lec 21 4Dynamics of satellite orbits•Basic dynamics is described by F=Ma where the force, F, is composed of gravitational forces, radiation pressure (drag is negligible for GPS), and thruster firings (not directly modeled).•Basic orbit behavior is given by€ ˙ ˙ r = −GMer3r12/07/2009 12.215 Lec 21 5Simple dynamics•GMe =  = 3986006x108 m3s-2•The analytical solution to the central force model is a Keplerian orbit. For GPS these are elliptical orbits.•Mean motion, n, in terms of period P is given by •For GPS semimajor axis a ~ 26400km€ n =2πP=μa312/07/2009 12.215 Lec 21 6Solution for central force model•This class of force model generates orbits that are conic sections. We will deal only with closed elliptical orbits.•The orbit plane stays fixed in space•One of the foci of the ellipse is the center of mass of the body•These orbits are described Keplerian elements12/07/2009 12.215 Lec 21 7Keplerain elements: Orbit planeNodeiωΩνZθ0GreenwichVernalequinoxSatelliteperigeeequatori InclinationΩ Right Ascension of ascending nodeω Argument of perigeeν True anomaly12/07/2009 12.215 Lec 21 8Keplerain elements in planeaFocusCenter of MassaeSatellitePerigeeApogeebEνra semimajor axisb semiminor axise eccentricityν True anomalyE Eccentric anomalyM Mean anomaly12/07/2009 12.215 Lec 21 9Satellite motion•The motion of the satellite in its orbit is given by•To is time of perigee€ M (t) = n(t −T0)E(t) = M (t)+ esin E(t)ν(t) = tan−11− e2sin E(t)/(1− ecos E(t))(cos E(t) − e)/(1− ecos E(t)) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥12/07/2009 12.215 Lec 21 10True anomaly0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 104-0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25Difference between true anomaly and Mean anomaly for e 0.001-0.10012/07/2009 12.215 Lec 21 11Eccentric anomaly0 0.5 1 1.5 2 2.5 3 3.5 4x 104-0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25Difference between eccentric anomaly and Mean anomaly for e 0.001-0.10012/07/2009 12.215 Lec 21 12Vector to satellite•At a specific time past perigee; compute Mean anomaly; solve Kepler’s equation to get Eccentric anomaly and then compute true anomaly. •Vector r in orbit frame is€ r = acos E − e1− e2sin E ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= rcosνsinν ⎡ ⎣ ⎢ ⎤ ⎦ ⎥r = a(1− ecos E) =a(1− e2)1+ ecosν12/07/2009 12.215 Lec 21 13Final conversion to Earth Fixed XYZ•Vector r is in satellite orbit frame•To bring to inertial space coordinates or Earth fixed coordinates, use•This basically the method used to compute positions from the broadcast ephemeris•(see Lecture 9 for discussion of rotation matrices)€ ri= R3(−Ω)R1(−i)R3(−ω)rre= R3(−Ω +θ )R1(−i)R3(−ω)r12/07/2009 12.215 Lec 21 14Perturbed motions•The central force is the main force acting on the GPS satellites, but there are other significant perturbations. •Historically, there was a great deal of work on analytic expressions for these perturbations e.g. Lagrange planetary equations which gave expressions for rates of change of orbital elements as function of disturbing potential•Today: Orbits are numerically integrated although some analytic work on form of disturbing forces.12/07/2009 12.215 Lec 21 15Perturbation from Flattening J2•The J2 perturbation can be computed from the Lagrange planetary equations€ ˙ Ω = −32nae2cosia2(1− e2)2J2˙ ω =34nae25 cos2i −1a2(1− e2)2J2˙ M = n +34nae23cos2i −1a2(1− e2)3J212/07/2009 12.215 Lec 21 16J2 Perturbations•Notice that only   and n are effected and so this perturbation results in a secular perturbation•The node of the orbit precesses, the argument of perigee rotates around the orbit plane, and the satellite moves with a slightly different mean motion•For the Earth, J2 = 1.08284x10-312/07/2009 12.215 Lec 21 17Gravitational perturbation stylesParameter Secular Long period Short perioda No No Yese No Yes Yesi No Yes YesYes Yes YesYes Yes YesM Yes Yes Yes12/07/2009 12.215 Lec 21 18Other perturbation on orbits and approximate sizeTerm Acceleration (m/sec2)Central 0.6J25x10-5Other gravity 3x10-7Third body 5x10-6Earth tides 10-9Ocean tides 10-10Drag ~0Solar radiation 10-7Albedo radiation 10-912/07/2009 12.215 Lec 21 19GPS Orbits•Orbit characteristics are–Semimajor axis 26400 km (12 sidereal hour period)–Inclination 55.5 degrees–Eccentricity near 0 (largest 0.02)–6 orbital planes with 4-5 satellites per plan•Design lifetime is 6 years, average lifetime 10 years•Generations: Block II/IIA 9729 kg, Block IIR 11000 kg12/07/2009 12.215 Lec 21 20Basic ConstellationOrbits shown in inertial space and size relative to Earth is correct12/07/2009 12.215 Lec 21 21Broadcast Ephemeris•Satellites transmit as part of their data