1 Dr. Guoqing Zhou 4. GPS Satellite Orbits CET 318 Book: p. 39-70 Fundamental Knowledge Quick Overview Kelperian 3 Laws First Law: ellipse, Sun is a focus Second Law: the same area in same time Third Law: 3 2 3 1 2 2 2 1 a a T T = Sun Earth Perigee and Apogee: The point of closest approach of the satellite with respect to the earth's center of mass is called perigee and the most distant position is the apogee. Nodes: The intersection between the equatorial and the orbital plane with the unit sphere is termed the nodes, where the ascending node defines the northward crossing of the equator. 4.1 Introduction Why do We Study Orbit? The applications of GPS depend substantially on knowing the satellite orbits. 1. For single receiver positioning, an orbital error is highly correlated with the positional error. 2. In relative positioning, relative orbital errors are considered to be approximately equal to relative baseline errors. Orbit Information and SA Technology How to obtain Orbital Information: either transmitted by the satellite as part of the broadcast message, or can be obtained (typically some days after the observation) from several sources. 2 Orbit Inf. and SA: The activation of SA in the Block II satellites may lead to a degradation of the broadcast orbit up to 50-100 m. Civil Community: Since some users need more precise ephemerides, the civil community must generate its own precise satellite ephemerides. 4.2 Orbit Description 4.2.1 Keplerian Motion Orbital Parameters The movement of mass m 2 relative to m 1 is defined by the homogeneous 2 nd order differential equation 0 )( 3 21 = + + r r mmG rG6G6 r m 1 m 2 t=? The analytical solution of differential equation leads to the well-known Keplerian motion defined by six orbital parameters The orbital parameters correspond to the six integration constants of the second-order vector equation. Artificial Earth Satellite: – Points: – Mass: negligible 238 103986005 ? ?== smGMu G Par. Notation ? Right ascension of ascending node i Inclination of orbital plane ω Argument of perigee a Semi-major axis of orbital ellipse e Numerical eccentricity of ellipse To Epoch of perigee passage Six orbital parameters The mean angular satellite velocity n (also known as the mean motion) with revolution period P follows from Kepler's Third Law given For GPS orbits, a = 26560 km, so, an orbital period of 12 sidereal hours. The ground track of the satellites repeats every sidereal day. 3 2 aP n μπ == 3 In orbital plane, the position vector and the velocity vector (with eccentric + true anomaly): Orbit Representation ? ? ? ? ? ? + ? ? = ev v ea u r cos sin )1( 2 D ) 12 ( ar ur ?=D r dt rd r =D ? ? ? ? ? ? = ? ? ? ? ? ? ? ? = v v r Ee eE ar sin cos sin1 cos 2 ve ea Eear cos1 )1( )cos1( 2 + ? =?= p.42 The transformation of and into the equatorial system is performed by a rotation matrix rRp = rRp DD = r rD 0 i X 3D rotation R, e3 = 0 In order to rotate the system into the terrestrial system , an additional rotation through the angle Θ 0 , the Greenwich sidereal time, is required. The transformation matrix, therefore, becomes }{}{}{}{' 31303 ω????Θ= RiRRRR 0 i X i X Orbital Plane Space-fixed Sys. Terrestrial Sys. ][}{}{}{ 321313 eeeRiRRR =????= ω Eq. 4.11, P. 45 ? ? Differential Relations The derivatives of and with respect to the six Keplerian parameters are required in one of the subsequent sections. The vectors and depend only on the parameters a, e, To, whereas the matrix is only a function of the remaining parameters ω, i, ?. p pD r rD P.45 ~ 46 The differential relations The meaning? P. 46 ? ?? ? + ? ? + ? ? + ? ? + ? ? + ? ? = dr R dir i R dr R dm m r Rde e r Rda a r Rdp ω ω ? ?? ? + ? ? + ? ? + ? ? + ? ? + ? ? = dr R dir i R dr R dm m r Rde e r Rda a r Rpd DDD DDD D ω ω 4.2.2 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 pdp p u p DDDD =+ 3 For GPS satellites, the acceleration is at least 10 4 times larger than the disturbing accelerations due to the central attractive force. pCC Analytical solution (p. 47-50) luA = ?=A ?=u ?=l Keplerian Motion vs. Perturbed Motion The parameters p i are constant. They are time dependent. Thus, for the position and velocity vector of the perturbed motion, we have )}(,{ tpt i ρρ = )}(,{ tpt i ρρ G6G6 = 4 4.2.3 Disturbing Accelerations In reality, many disturbing accelerations act on a satellite and are responsible for the temporal variations of the Keplerian elements. Solar radiation pressure (direct and indirect ) Air drag Relativistic effects Others (solar wind, magnetic field forces, etc. ) Non- gravitational Nonsphericity of the Earth Tidal attraction (Direct and Indirect ) Gravitational They can be divided into: Disturbing For GPS satellites, altitude is about 20200 km, the indirect effect of solar radiation pressure and air drag may be neglected. The shape of the satellites is irregular which renders the modeling of direct solar radiation pressure more difficult. Different Satellites are different radiation pressures The variety of materials used for the satellites has a different heat-absorption which results in additional and complicated perturbing accelerations. Accelerations may arise from gas leaks in the container of the gas-propellant. Example: P. 53 1. Nonsphericity of the Earth: Example: P. 51 for GPS The numerical values 5·10 -2 ms -2 2. Tidal Effects Among all the celestial bodies in the solar system, only the sun and the moon must be considered because the effects of the planets are negligible. –The maximum of the perturbing acceleration is reached when the three bodies are situated in a straight line. – Apart from the direct effect of the tide generating bodies, indirect effects due to the tidal deformation of the solid earth and the oceanic tides must be taken into account. – The model for the indirect effect due to the oceanic tides is more complicated. Sat. Cel. Ear. 5 3. Solar Radiation Pressure: The perturbing acceleration due to the direct solar radiation pressure has two components: 1. The principal component is directed away from the sun. 2. The smaller component acts along the satellite's y-axis. This is an axis orthogonal to both the vector pointing to the sun and the antenna which is nominally directed towards the center of the earth. x y z Sun Earth – The first component is in the order of 10 -7 ms -2 – The second component is called y-bias, and is believed to be caused by a combination of misalignments of the solar panels and thermal radiation along the y-axis. The solar radiation pressure which is reflected back from the earth's surface causes an effect called albedo. For GPS, the associated perturbing accelerations are smaller than the y-bias and can be neglected. 4. Relativistic Effect: The relativistic effect on the satellite orbit is caused by the gravity field of the earth and gives rise to a perturbing acceleration. This effect is smaller than the indirect effects by one order of magnitude. The numerical values of perturbing acceleration results in an order of 3·10 -10 ms -2 4.3 Orbit Determination Orbit Determination: orbital parameters and satellite clock biases. (p. 54) In principle, the problem is inverse to the navigational or surveying goal. The position vector and the velocity vector of the satellite are considered unknown. The position vector of the observing site is assumed to be known in a geocentric system. R S ppp ?= R R S R S p pp pp p DD ? ? = Fundamental equation Position Velocity Position vector is a function of ranges, whereas the velocity vector is determined by range rates. At present, the observations for the orbit determination are performed at terrestrial sites, such as TOPEX/Poseidon. The GPS data could also be obtained from orbiting receivers. 6 The actual orbit determination is performed in two steps. 1. A Kepler ellipse is fitted to the observations (theoretically). (1) Initial value problem (2) Boundary value problem 2. This ellipse serves as reference, then is improved by taking into account perturbing accelerations. Add all perturbation parameters into Kepler Orbit (1) Analytical solution (2) Numerical solution Clues of Orbital Determination: 4.3.1 Keplerian Orbit Clues: – It is assumed that both the position and the velocity vector of the satellite have been derived from observations. Now, the question arises of how to use these data for the derivation of the Keplerian parameters. – The position and velocity vector given at the same epoch i define an initial value problem, and two position vectors at different epochs t l and t 2 define a (first) boundary value problem. In principle, a second and a third boundary value problem could also be defined; however, these problems are not of practical importance in the context of GPS and are not treated here. 0 )( 3 21 = + + r r mmG rG6G6 Initial Value Problem The derivation of the Keplerian parameters from position and velocity vectors, given at the same epoch and expressed in an equatorial system, is an initial value problem for solving the differential. Recall that the two given vectors contain six components (six Keplerian parameters). Since both vectors are given at the same epoch, the time parameter is omitted. 0 )( 3 21 = + + r r mmG r>> Boundary Value Problem 1. It is assumed that two position vectors S(t 1 ) and S(t 2 ) at epochs t 1 and t 2 are available. 2. Note that position vectors are preferred for orbit determination since they are more accurate than velocity vectors. 3. The given data correspond to boundary values in the solution of the basic second-order differential equation. Orbit Improvement If there are redundant observations, the parameters of an instantaneous Kepler ellipse can be improved because each observed range gives rise to an equation. The vector can be expressed as a function of the Keplerian parameters. Thus, it actually contains the differential increments for the six orbital parameters. In the past, orbit improvement was often performed in the course of GPS data processing when, in addition to terrestrial position vectors, the increments were determined. The procedure became unstable or even failed for small networks. In the case of orbit relaxation, only three degrees of freedom were assigned to the orbit (p. 58). 4.3.2 Perturbed Orbit In order to be suitable for Lagrange's equations, the disturbing (Earth) potential is expressed as a function of the Keplerian parameters. Eq. 4.57, p. 59. R= ****** (Eq. 4.57, P. 59) The tidal potential also has a harmonic representation, and thus the tidal perturbations can be analytically modeled. Analytical Solution Definition: p. 58 7 Numerical Solution With initial values of the position and velocity vectors at a reference epoch t 0 , a numerical integration of the following Eq. can be performed. Definition: p. 60 ? ?? ? + ? ? + ? ? + ? ? + ? ? + ? ? = dr R dir i R dr R da m r Rde e r Rda a r Rdp ω ω ? ?? ? + ? ? + ? ? + ? ? + ? ? + ? ? = dr R dir i R dr R da m r Rde e r Rda a r Rpd DDD DDD D ω ω This simple concept can be improved by the introduction of a Kepler ellipse as a reference. 4. Orbit Dissemination 4.1 Tracking Networks 1. Objectives and Strategies The official orbit determination for GPS satellites is based on observations at the five monitor stations of the control segment. 1.The broadcast ephemeredes for Block I satellites: ~5 m. 2.For the Block II satellites: up to 50-100 m by SA. An orbital accuracy of about 20 cm is required for specific missions such as TOPEX/Poseidon or for investigations which require an accuracy at the level of 10 -9 . Minimum Number of Sites: in a global network is six, if a configuration is desired where at least two satellites can be tracked simultaneously any time from two sites. Global Network and Regional Network: Global Network result in higher accuracy and reliability compared to regional networks. Orbit System Tie: The tie of the orbital system to terrestrial reference frames is achieved by the collocation of GPS receivers with VLBI and SLR trackers. GPS Site Distribution: The distribution of the GPS sites is essential to achieve the highest accuracy. A Comparison of Two Distribution of GPS Sites – The sites are regularly distributed around the globe; – Each network site is surrounded by a cluster of additional points to facilitate ambiguity resolution Examples for Global Networks Several networks have been established for orbit determination. Regional Continental size (the Australian GPS) Global networks 1. Global Orbit Tracking Experiment (GOTEX): p. 63. 2. The Cooperative International GPS Network (CIGNET): p.64. 3. In 1990, IAG installed an International GPS Service for Geodynamics (IGS): p. 64. 8 4.2 Ephemerides Three sets of data are available to determine position and velocity vectors of the satellites in a terrestrial reference frame at any instant: – Almanac data, – Broadcast ephemerides, and – Precise ephemerides 1. Almanac Data Purpose: provide the user with less precise data to facilitate receiver satellite search or for planning tasks e.g., the computation of visibility charts. The almanac data are updated at least every six days and are broadcast as part of the satellite message. The almanac message essentially contains parameters for the orbit and satellite clock correction terms for all satellites. All angles are expressed in semicircles. Table 4.6: Almanac Data, P. 68 2. Broadcast Ephemerides Purposes: to compute a reference orbit for the satellites 1. The broadcast ephemerides are based on observations at the five monitor stations. 2. Additional tracking data are entered into a Kalman filter and the improved orbits are used for extrapolation. 3. The orbital data could be accurate to approximately 5 m based on three uploads per day; with a single daily update one might expect an accuracy of 10 m. 4. The Master Control Station is responsible for the computation of the ephemerides and the upload to the satellites. The ephemerides are broadcast (mostly) every hour and should only be used during the prescribed period of approximately four hours to which they refer . Table 4.7: Broadcast Ephemerides, P. 67 3. Precise Ephemerides 1. The official precise orbits are produced by the NSWC together with the DMA and are based on observed data in the (extended) tracking network. 2. The post-mission orbits are available upon request about four to eight weeks after the observations. 3. The most accurate orbital information is provided by the IGS with a delay of about two weeks. 4. Less accurate information is available about two days after the observations. 5. Currently, IGS data and products are free of charge for all users. The precise ephemerides – satellite positions and – velocities at equidistant epochs. Since 1985, NGS began to distribute precise GPS orbital data. Formats: – the specific ASCII formats SP1 and SP2 –their binary counterparts ECF1 and ECF2. – Later, ECF2 was modified to EF13 format. Typical spacing of the data is 15 minutes. 9 Each NGS format consists of a header containing general information (epoch interval, orbit type, etc.) followed by the data section for successive epochs. – The position: kilometer – The velocity: kilometer/second NGS formats are described in Remondi (1989, 1991b ). NGS provides software to translate orbital files from one format to another. NGS Format Summary What have we learnt? Which parts are important? Assignment 4 1. Illustrate 6 Keplerian orbit parameters 2. Use eccentricity, true anomaly to represent Keplerian orbit. 3. Represent perturbed motion. 4. Please list the sources of disturbing accelerations, and lists their characteristics. 5. Why do we neglect the GPS solar radiation pressure and air drag? 6. How to determine the GPS orbit? 7. What is the Initial value problem? What is boundary value problem? 8. What is analytical solution and numerical solution of perturbed orbit? 9. Please describe in detail the Almanac data, Broadcast emphemerides, and Precision emphemerides. 10. Please describe the NGS GPS data format