02/20/02 12.540 Lec 05 1 12.540 Principles of the Global Positioning System Lecture 05 Prof. Thomas Herring 02/20/02 12.540 Lec 05 2 Satellite 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 satellites ? Logistics: Who can attend lecture on Fridays at 11-12:30? 02/20/02 12.540 Lec 05 3 Dynamics 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 Y Y r =? GM e r 3 r 02/20/02 12.540 Lec 05 4 Simple dynamics ?GM e = μ = 3986006x10 8 m 3 s -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 = 2π P = μ a 3 02/20/02 12.540 Lec 05 5 Solution 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 elements 02/20/02 12.540 Lec 05 6 Keplerain elements: Orbit plane Node i ω ? ν Z θ 0 Greenwich Vernal equinox Satellite perigee equator i Inclination ? Right Ascension of ascending node ω Argument of perigee ν True anomaly 02/20/02 12.540 Lec 05 7 Keplerain elements in plane a Focus Center of Mass ae Satellite Perigee Apogee b E ν r a semimajor axis b semiminor axis e eccentricity ν True anomaly E Eccentric anomaly M Mean anomaly 02/20/02 12.540 Lec 05 8 Satellite motion ? The motion of the satellite in its orbit is given by ?T o is time of perigee M (t)= n(t ?T 0 ) E(t)= M (t)+esin E(t) ν(t)= tan ?1 1?e 2 sin E(t)/(1?ecos E(t)) (cos E(t)?e)/(1?ecos E(t)) ? ? ? ? ? ? 02/20/02 12.540 Lec 05 9 True anomaly 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Difference between true anomaly and Mean anomaly for e 0.001-0.100 02/20/02 12.540 Lec 05 10 Eccentric anomaly 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Difference between eccentric anomaly and Mean anomaly for e 0.001-0.100 02/20/02 12.540 Lec 05 11 Vector to satellite ? At a specific time past perigee; compute Mean anomaly; solve Kepler’s equation to get Eccentric anomaly and then compute true anomaly. See Matlab/truea.m ?Vector r in orbit frame is r = a cos E ?e 1?e 2 sin E ? ? ? ? ? ? = r cosν sinν ? ? ? ? ? ? r = a(1?ecos E)= a(1?e 2 ) 1+ecosν 02/20/02 12.540 Lec 05 12 Final 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 r i = R 3 (??)R 1 (?i)R 3 (?ω)r r e = R 3 (??+θ)R 1 (?i)R 3 (?ω)r 02/20/02 12.540 Lec 05 13 Perturbed 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. 02/20/02 12.540 Lec 05 14 Perturbation from Flattening J 2 ? The J 2 perturbation can be computed from the Lagrange planetary equations Y ? =? 3 2 na e 2 cosi a 2 (1?e 2 ) 2 J 2 Y ω = 3 4 na e 2 5 cos 2 i?1 a 2 (1?e 2 ) 2 J 2 Y M = n+ 3 4 na e 2 3cos 2 i?1 a 2 (1?e 2 ) 3 J 2 02/20/02 12.540 Lec 05 15 J 2 Perturbations ? Notice that only ?ωand 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, J 2 = 1.08284x10 -3 02/20/02 12.540 Lec 05 16 Gravitational perturbation styles Parameter Secular Long period Short period aNoNoYes eNYes iNo Yes ? Yes Yes Yes ω Yes Yes Yes M Yes Yes Yes 02/20/02 12.540 Lec 05 17 Other perturbation on orbits and approximate size Term Acceleration (m/sec 2 ) Central 0.6 J 2 5x10 -5 Other gravity 3x10 -7 Third body 5x10 -6 Earth tides 10 -9 Ocean tides 10 -10 Drag ~0 Solar radiation 10 -7 Albedo radiation 10 -9 02/20/02 12.540 Lec 05 18 GPS 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 kg 02/20/02 12.540 Lec 05 19 Basic Constellation Orbits shown in inertial space and size relative to Earth is correct 02/20/02 12.540 Lec 05 20 Broadcast Ephemeris ? Satellites transmit as part of their data message the elements of the orbit ? These are Keplerian elements with periodic terms added to account for solar radiation and gravity perturbations ? Periodic terms are added for argument of perigee, geocentric distance and inclination ? The message and its use are described in the ICD- GPS-200 icd200c123.pdf (page 105 in PDF) 02/20/02 12.540 Lec 05 21 Distribution of Ephemerides ? The broadcast ephemeris is decoded by all GPS receivers and for geodetic receivers the software that converts the receiver binary to an exchange format outputs an ASCII version ? The exchange format: Receiver Independent Exchange format (RINEX) has a standard for the broadcast ephemeris. ? Form [4-char][Day of year][Session].[yy]n e.g. brdc0120.02n 02/20/02 12.540 Lec 05 22 RINEX standard ? Description of RINEX standard can be found at ftp://igscb.jpl.nasa.gov/igscb/data/format/rinex2.txt ? Homework number 1 also contains description of navigation file message (other types of RINEX files will be discussed later) ? 12.540.HW1.PDF is first homework: Due Fri March 8.