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