cET 318 Fundamental Knowledge Quick Overview Kelperian 3 Laws Fourth Lecture First Law: ellipse, Sun is a focus 4. GPS Satellite Orbits Second law: the same area in same time Third Law: Ti-a Book: p. 39-70 Dr Guoqing Zhou 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 nit sphere is termed the nodes, where the ascending node defines the northward crossing of the equator 4.1 Introduction satellite Why do We study Orbit? The applications of GPs depend substantiall Orbit Information and SA Technology knowing the satellite orbits How to obtain Orbital Information: For single receiver positioning, an orbital error is highly correlated with the positional error either transmitted by the satellite as part of 2. In relative positioning, relative orbital errors approximately equal to can be obtained(typically some days after the sources relative baseline errors
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
Orbit Inf and SA: The activation of sa in the block li satellites may lead to a degradation of the broadcast orbit up to 50-100 m 4.2 Orbit Description Civil Community Since some users need more precise ephemerides, the civil community must generate its own precise satellite ephemerid 4.2.1 Keplerian Motion Artificial Earth Satellite: Orbital parameters Mass: negligible The movement of mass m2 relative to mI is defined by the homogeneous 2d order differential equation u=GMa=3986005105m3s2 G(m2+m2) The analytical solution of differential equation leads the well-known Keplerian motion define The orbital parameters correspond to the six integration econd-order vector equation. 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 2丌 Argument of perigee For GPS orbits, a=26560 km, so, an orbital pe of 12 sidereal hours. The ground track of Numerical eccentricity of ellipse satellites ery sidereal day To Epoch of perigee passag
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 m2 relative to m1 is defined by the homogeneous 2nd order differential equation 0 ( ) 3 1 2 = + + r r G m m r r m1 m2 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 8 3 2 3986005 10 − u = GM = ⋅ m s G Par. Notation Ω 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 P a n π µ = =
Orbit Representation p 42 In orbital plane, the position vector r and the velocity The transformation of and r into the equatoria vector i=dl(with eccentric true anomaly) system x' is performed by a rotation matrix X3 -esine P =R satellite p=Rr e"lco 3D rotation R, e3=0 vernal equinox X2 R=R3{-9}R1{-}R3{-}=le1g2gl] Differential Relations P 45-46 Eq4.1P.45 The derivatives of p and e with respect to the six Keplerian parameters are required in one of the In order to rotate the syster ubsequent sections an additional rotation Greenwich sidereal transformation matrix, therefore become The vectors r and i depend only on the parameters a,e, To, whereas the matrix is only a function of the R=R3{60R3{-9R1{-i}R3{-m} emaining parameters a i, 12 The differential relations The meani Orbital Plane Space-fixed Sys ->Terrest 中=面+且+如+回a+ ,盐m 中=R如+Bd+R如+d+ 4.2.2 Perturbed Motion Keplerian Motion vs Perturbed Motion The Keplerian orbit is a theoretical orbit and does not include actual perturbations The parameters p; are constant. based on an inhomogeneous Thus, for the position and velocity vector of the perturbed For GPS satellites, the acceleration B is at least 10 times p{,pP2(m)} due to th attractive force A p=p{t,P(1)} Analytical solution Au=l u=? (p.47-50) =?
3 In orbital plane, the position vector and the velocity vector (with eccentric + true anomaly): Orbit Representation + − − = v e v a e u r cos sin (1 ) 2 D ) 2 1 ( r a rD = u − r dt d r rD = = − − = v v r e E E e r a sin cos 1 sin cos 2 e v a e r a e E 1 cos (1 ) (1 cos ) 2 + − = − = p.42 The transformation of and into the equatorial system is performed by a rotation matrix p = Rr pD = RrD r rD 0 X i 3D rotation R, e3 = 0 In order to rotate the system into the terrestrial system , an additional rotation through the angle Θ0, the Greenwich sidereal time, is required. The transformation matrix, therefore, becomes ' { } 3{ } 1{ } 3{ } R = R3 Θ0 R −Ω R −i R −ω 0 X i X i Orbital Plane Space-fixed Sys. Terrestrial Sys. { } { } { } [ ] 3 1 3 1 2 3 R = R −Ω R −i R −ω = e e e 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, Ω. p pD r rD P.45 ~ 46 The differential relations The meaning? P. 46 Ω ∂Ω ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = rd R rdi i R rd R dm m r de R e r da R a r dp R ω ω Ω ∂Ω ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = rd R rdi i R rd R dm m r de R e r da R a r dp R D D D D D D 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 p dp p u DpD + = DD 3 For GPS satellites, the acceleration is at least 104 times larger than the disturbing accelerations due to the central attractive force. CpC Analytical solution (p. 47-50) Au = l A = ? u = ? l = ? Keplerian Motion vs. Perturbed Motion • The parameters pi are constant. • They are time dependent. Thus, for the position and velocity vector of the perturbed motion, we have {t, p (t)} ρ = ρ i {t, p (t)} ρ ρ i =
4.2.3 Disturbing Accelerations Oa、 Refletion Mor ty, many disturbing accelerations act on a satellite responsible for the temporal var ian elements They can be divided int Gravitational ricity of the Earth Tidal attraction(Direct and Indirect Non- Solar radiation pressure(direct and indirect Solar radiation g Gravity I Others(solar wind, magnetic field forces Tidal Earth Disturbing The variety of materials used for the satellites has a satellites. altitude is about 20200 km the different heat-absorption which results in addi ffect of solar radiation pressure and air and complicated perturbing accelerations be neglected. The shape of the satellites is irregular which renders Accelerations may arise from gas leaks in the the modeling of direct solar radiation pressure more container of the gas-propellant. difficult. Different Satellites are different radiation xample: P53 2. Tidal Effects A cel I Nonsphericity of the Earth Example: P 51 for GPS Among all the celestial bodies in the solar system, only the sun The numerical values 5.10-2 ms-2 and the moon must be considered because the effects of the when the three bodies are situated in a straight line the soli unt The model for the indirect effect due to the oceanic tides is
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. ) Nongravitational • 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
3. Solar Radiation Pressure: The perturbing acceleration due to the direct solar radiati -The first component is in the order of 10-7ms'2 pressure has two components nt 1. The principal component is directed away from the sun. believed to be caused by a combination of 2. The smaller component acts along the satellites y-axis misalignments of the solar panels and thermal This is an axis orthogonal to both the vector In and the antenna which is nominally directed towards the center of the earth The solar radiation pressure which is reflected back from the earth's surface causes an effec alled albedo. For GPS, the associated perturbing accelerations are smaller than the y-bias and can Earth 4. Relativistic Effect: he relativistic effect on the satellite orbit is caused b ity field of the earth and gives rise to a This effect is smaller than the indirect effects by one 4.3 Orbit determination order of magnitude The numerical values of perturbing acceleration results Orbit Determination: orbital parameters and satellite Position vector is a function of ranges, whereas the In principle, the problem is inverse to the navigational or velocity vector is determined by range rates reeving goal At present, the observations for the orbit determination are Fundamental performed at terrestrial sites, such as TOPEX/Poseidon. P=p=pa"velocity The Gps data could also be obtained from orbiting The position vector and the velocity vector of the The position vector of the observing site is assumed to be known in a geocentric system
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 p = p − p R R S R S p p p p p pD D − − = 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
Clues of orbital determination: 4.3.1 Keplerian Orbi The actual orbit determination is performed in two It is assumed that both the position and the velocity vector of the satellite have been derived from observations 1. A Kepler ellipse is fitted to the observations Now, the question arises of how to use these data for the (theoretically) (1)Initial value problem (2)Boundary value problem The position and velocity vector given at the same epoch i define an initial value problem, and two position vectors taking into account perturbing accelerations problem In principle, a second and a third boundary value Add all perturbation parameters into Kepler Orbi problem could also be defined; however, these probler (2)Numerical solution are not of practical importance in the context of GPS and e not treated here Initial value Problem The derivation of the Keplerian parameters from Boundary value Problem ion and velocity vectors, given at the same h and expressed in an equatorial system, is ar It is assumed that two position vectors S(t,)and initial value problem for solving the differential 2. Note that position vectors are preferred for orbit etermination since they are more accurate tha Recall that the two given vectors cont omponents(six Keplerian parameters both vectors are given at the same epoch 中 5. the solutio d aa Co basic second-order differer Orbit Improvement 4.3.2 Perturbed Orbit If there are redundant observations, the parameters of ar instantaneous Kepler ellipse can be improved because each observed range gives rise to an equation. Definition: p. 58 neters. Thus. it actually contains the differential Earth) potential is expressed as a function increments for the six orbital parameters. of the Keplerian parameters. Eq. 4.57, p. 59 (Eq4.57,P.59) course of GPS data processing when, in addition to terrestrial position vectors, the increments were determined has a harmonic repre en failed for small us the tidal perturbations can be analyticall lly three degrees of freedom were assigned to the orbit (p. 58)
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 tl and t2 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 1 2 = + + r r G m m r 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 1 2 = + + r r G m m > r> Boundary Value Problem 1. It is assumed that two position vectors S(t1) and S(t2) at epochs t1 and t2 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
Numerical Solution Definition: p 60 With initial values of the position and velocity vectors at a reference epoch to, a numerical integration of the following Eq can be performed 4. Orbit Dissemination 中=B山+B+B+*Bda 中=且山+由+B些如+d+坚边+强a This simple concept can be improved by introduction of a Kepler ellipse as a reference Minimum Number of Sites: in a global network is 4.1 Tracking Networks six, if a configuration is desired where at least two satellites can be tracked simultaneously any time from 1. Objectives and Strategies two sites The official orbit determination for gPs satellites is Network and Regional Network: Global based on observations at the five monitor stations of the k result er accuracy and reliability control segment. 1. The broadcast ephemeredes for Block I satellites:5 Orbit Syste The tie of the orbital system to 2. For the Block Il satellites: up to 50-100 m by $A frames is achieved by the collocation An orbital accuracy of about 20 cm is required for receivers with VLBi and SLR specific missions such as TOPEX/Poseidon or for investigations which require an accuracy at the level of GPS Site distribution: the distribution of the gps 10 sites is essential to achieve the highest accuracy. A Comparison of Two Distribution of GPS Sites Examples for Global Networks The sites are regularly distributed around the globe Several networks have been established for orbit Each network unded by a cluster of Continental size(the Australian GPS) dditional poir cilitate ambi 1. Global Orbit Tracking Experiment (GOTEX): D. perative In 3. In 1990. IAG installed an International GPS Geodynamics(IGS): p 64
7 Numerical Solution With initial values of the position and velocity vectors at a reference epoch t0, a numerical integration of the following Eq. can be performed. Definition: p. 60 Ω ∂Ω ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = rd R rdi i R rd R da m r de R e r da R a r dp R ω ω Ω ∂Ω ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = rd R rdi i R rd R da m r de R e r da R a r dp R D D D D D D 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
4.2 Ephemerides 1. Almanac data Purpose: provide the user with less precise data to facilitat Three sets of data are available to determine position satellite search or for planning tasks e.g., th and velocity vectors of the satellites in a terrestrial computation of visibility charts. reference frame at any instant The almanac data are updated at least every six days and are broadcast as part of the satellite Almanac data message. Broadcast ephemerides, and The almanac message essentially contains parameters for the orbit and satellite clock orrection terms for all satellites All angles are expressed in semIcircles Table 4.6: P.68 2. Broadcast Ephemerides Purposes: to compute a reference orbit for the satellites des are broadcast(mostly) every hour 1. The broadcast ephemerides are based on observations d should only be used during the prescribed period at the five monitor stations of approximately four hours to which they refer 2. Additional tracking data are entered into a Kalman filter and the improved orbits are used for Table 4.7: Broadcast Ephemerides 3. The orbital data could be accurate to approximately 5 P.67 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 che satiates of the ephemerides and the upload t 3. Precise Ephemerides 1. The official precise orbits are produced by the NSwC The precise ephemerides together with the dMa and are based on observed atellite positions and data in the(extended )tracking network. velocities at equidistant epochs 2. The post-mission orbits are available upon request Since 1985, NGs began to distribute precise GPS about four to eight weeks after the observations. 3. The most accurate orbital information is provided by ecific AsCll formats spl and SP2 the IGs with a delay of about two weeks. binary counterparts ECFI and ECF 4. Less accurate information is available about two days ECF2 was modified to EF13 format after the observations 5. Currently, IGs data and products are free of charge Typical spacing of the data is 15 minutes. for all users
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
NGS Format Each NGS format consists of a header containing Summary general information(epoch interval, orbit type, etc. )followed by the data section for successive What have we learnt? epochs Which parts are The velocity: kilometer/second NGS formats are described in Remondi (1989, 1991b) NGS provides software to translate orbital files from one format to another Illustrate 6 Keplerian orbit parameters 2. Use eccentricity, true anomaly to represent Keplerian orbit 4. Please list the sources of disturbing accelerations, and lists 5. Why do we neglect the GPS solar radiation pressure and air 6. How to determine the GPs orbit? 7. What is the Initial value problem? what is boundary value perturbed orbit? 9. Please describe in detail the Almanac data, Broadcast ephemerides, and Precision ephemerides 10. Please describe the Ngs gps data format
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