cET 318 The ThIrd Lecture 1 Introduction 3. Reference Systems Book: p 25-38 Dr Guoqing Zhou 1.1 Overview 1.2 A Group of Basic Ce GPS Basic Positioning Principle 1. Sun and Earth Spring Two reference systems: Time and Coordinate 2. Earth 3. Celestial body .lz1 4. Ecliptic Plane 5. Earth equator Autumn n-known 6. Celestial equator known 3 7. Greenwich Meridian Coordinat 8. Vernal equinox 9. Node 2. Space-fixed Coordinate System 0. Perigee 13. Earth-fixed Coordinate System I 1.Apogee 14. Artificial Earth Satellite 15. Orbital Plane 16. Orbital plane Coordinate 巴:条
1 Dr. Guoqing Zhou 3. Reference Systems CET 318 Book: p. 25-38 1. Introduction • GPS Basic Positioning Principle • Two reference systems: Time and Coordinate Syst a 2 3 known known known 1 Un-known a a GPS Satellite GPS Receiver Time System Coordinate System S=vt 1.1 Overview 1.2 A Group of Basic Concepts 1. Sun and Earth 2. Earth 3. Celestial body 4. Ecliptic Plane 5. Earth Equator 6. Celestial equator 7. Greenwich Meridian 8. Vernal Equinox Sun Spring Winter Autumn Summer 9. Node 10.Perigee 11.Apogee 12. Space-fixed Coordinate System 13. Earth-fixed Coordinate System 14. Artificial Earth Satellite 15. Orbital Plane 16. Orbital plane Coordinate System
1. A uniform coordinate syster 2. A three-dimensional Cartesian system The angle o between the two systems is called The X Greenwich sidereal time system points towards the vernal X,=X The X axis being ortho o both the x.-axis X,-axis of the and the x,-axis completes a right-handed earth-fixed system coordinate frame defined by the ntersection line of the equatorial plane vith the plane represented by the meridian 2.1 A Group of Terms Precession(Sun-Earth-Moon, Fixed! ) Vernal Equinox moves towards west 50.26second. The difference between equinox calendar year and sidereal year 0.014day 2. Coordinate Systems Nutation: The oscillation withrespect to the inertial space (Sun-Earth-Moon, Change! )(p. 27) nutation: Chandler period: The period of the free motion amounts to about 430 days and is known as the Chandler period 1. Conventional Celestial Reference System 2. Conventional Terrestrial Reference System l. Definition of CCRS: p Definition: Terrestrial Reference Frame(TRF)(p. 28) CRF: Since this system is defined conventionally and TRF: is defined by a set of terrestrial control stations the practical realization does not necessarily coincide serving as reference points. Most of the reference stations with the theoretical system, it is called (conventional) are equipped with Satellite Laser Ranging(SLR)or Celestial Reference Frame(CRE Very Long Baseline Interferometry (VLBI) pabilities gorously inertial because of the accelerated motion of the earth around the sun o World Geodetic System 1984(WGS-84 ), 1500 o the International Earth Rotation Service (IERS) called IcRF o ITRF-94 established by the IERS, 180 points
2 1. A uniform coordinate system, 2. A three-dimensional Cartesian system -axis of the earth-fixed system is defined by the intersection line of the equatorial plane with the plane represented by the Greenwich meridian. X1 The -axis for the space-fixed system points towards the vernal equinox. 0 X 1 The angle Θ0 between the two systems is called Greenwich sidereal time. The -axis being orthogonal to both the -axis and the -axis completes a right-handed coordinate frame. X1 0 X 1 0 X 1 2. Coordinate Systems Precession (Sun-Earth-Moon, Fixed!!!): Vernal Equinox moves towards west 50.26second, The difference between equinox calendar year and sidereal year 0.014day Nutation: The oscillation with respect to the inertial space is called nutation (Sun-Earth-Moon, Change!!!) (p. 27). 1. Secular precession: 2. Periodic nutation: Polar motion: The oscillation with respect to the terrestrial system is named polar motion. Chandler period: The period of the free motion amounts to about 430 days and is known as the Chandler period. ω 2.1 A Group of Terms 1. Definition of CCRS: p.28. CRF: Since this system is defined conventionally and the practical realization does not necessarily coincide with the theoretical system, it is called (conventional) Celestial Reference Frame (CRF). o "quasi-inertial" means a geocentric system is not rigorously inertial because of the accelerated motion of the earth around the sun. o the International Earth Rotation Service (IERS), called ICRF 1. Conventional Celestial Reference System (p. 28) 2. Conventional Terrestrial Reference System (p. 28) Definition: Terrestrial Reference Frame (TRF) (p. 28) TRF: is defined by a set of terrestrial control stations serving as reference points. Most of the reference stations are equipped with Satellite Laser Ranging (SLR) or Very Long Baseline Interferometry (VLBI) capabilities. X1 o World Geodetic System 1984 (WGS-84), 1500 sites/points o ITRF-94 established by the IERS, 180 points
The Comparison of WGS-84 and ITRF 2.2 Transformation of Coordinate System General remarks bservations from the tranSi transformation between the celestial Reference while itrf is based on nd VlBI me(CRF)and the Terrestrial Reference Frame (TRF) by accuracy of th reference stations is XERRRRx estimated to be in of 1 to 2 meters while Where the accuracy of the erence stations is at the RM rotation matrix for polar motion RS rotation matrix for sidereal time Why do we need RN rotation matrix for nutation learn the The comparison of parameters of wGS-84 and ITRF RP rotation matrix for precession transformation of reveals remarkable differences. WHY? coordinate systems? 1. Precession T represents the time-span expressed in Julian centuries of 365.25 mean solar days between the standard epoch The position of the mean vernal equinox at the standard J2000.0 and the epoch of observation. epoch to is denoted by Eo and the position at the observation epoch t is denoted by E. The precession matrix x(to R3{-=R2{-)R;{ related to time? mean equator(t) The precession parameters are computed from this time mean equator(to) series 5=23062181T+0.3018872+001799873 z=23062181T+1”.0946872+0″01820373 Xi(to) =2004.3109T-04266572-0”41833T3 The definition of Precession 2. Nutation The mean obliquity of the ecliptic e has been determined denoted by E and the true equinox by Er. The nutation E=23°2621:"448-46″81501-0.″0005912+0.″001813T matrix" is composed of three successive rotation matrice where T is the same time factor RN=R-(e+△e)R3-△o)}Rle} The nutation parameters△rand△4 are computed from the harmonic series. nutation△in longitude△e P35,Eq.3.15 mean equator and the nutation be treated differential true equator 方=12 quantities
3 The Comparison of WGS-84 and ITRF 1. The WGS-84 was established through Doppler observations from the TRANSIT satellite system while ITRF is based on SLR and VLBI observations. 2. The accuracy of the WGS-84 reference stations is estimated to be in the range of 1 to 2 meters while the accuracy of the ITRF reference stations is at the centimeter level. The comparison of parameters of WGS-84 and ITRF reveals remarkable differences. WHY? 2.2 Transformation of Coordinate System General remarks The transformation between the Celestial Reference Frame (CRF) and the Terrestrial Reference Frame (TRF) by Where: RM rotation matrix for polar motion RS rotation matrix for sidereal time RN rotation matrix for nutation RP rotation matrix for precession CRF M S N P TRF x = R R R R x Why do we need learn the transformation of coordinate systems? 1. Precession The position of the mean vernal equinox at the standard epoch t0 is denoted by E0 and the position at the observation epoch t is denoted by E. The precession matrix R = R3{−z}R2{−ϑ}R3{−ς} P The precession parameters are computed from this time series 2 3 ς = 2306′′.2181T + 0′′.30188T + 0′′.017998T 2 3 ϑ = 2004′′.3109T − 0′′.42665T − 0′′.41833T 2 3 z = 2306′′.2181T +1′′.09468T + 0′′.018203T The definition of Precession T represents the time-span expressed in Julian centuries of 365.25 mean solar days between the standard epoch J2000.0 and the epoch of observation. Why does the precession is related to time? 2. Nutation The mean vernal equinox at the observation epoch is denoted by E and the true equinox by Et . The nutation matrix is composed of three successive rotation matrices N R R R1{ (ε ε )}R3{ φ)}R1{ε} N = − + ∆ −∆ where both the nutation in longitude ; and the nutation in obliquity can be treated as differential quantities. ∆ε ∆φ The mean obliquity of the ecliptic has been determined as ε ∆ε ∆φ ε P35., Eq. 3.15 where T is the same time factor. = 23°26'21.″448 - 46.″8150T – 0.″00059T2 + 0.″001813T3 The nutation parameters and are computed from the harmonic series:
3. Sidereal time 4. Polar motion The rotation matrix for sidereal time rs is The previous computations yield the instantaneous CEP. R=R36 The cep must still be rotated into the cio The computation of the apparent Greenwich sidereal This is achieved by means of the pole coordinates xp, yp time eo which define the position of the CEp with respect to the CIO. The pole coordinates are determined by the IErS is shown in the section on time systems. and are available upon request. The rotation matrix for R=R2(-XPiRi(-yp)=0 y R=RR 3. Time Systems The rotation matrices R andr are often combined to a single matrix for earth rotation What is Time? 3. 1 A Group of Time System Definition of time criterion must be motion object Several time systems are in current use. They are based on various periodic processes such as earth rotation Periodie process Earth rotation Greenwich Sidereal Time(00) 1. Earth rotation by itself( Sidereal time, Solar time Earth revolution Terrestrial Dynamic Time(DT) 2. Earth rotation around Sun(Calendar time Barycentric Dynamic Time(BDT) Atomic oscillations nternational 3. Oscillation frequency of atomie motion(AtomIc UT Coordinated (UTC)
4 3. Sidereal time The rotation matrix for sidereal time RS is The computation of the apparent Greenwich sidereal time is shown in the section on time systems. { } R = R3 Θ0 S Θ0 The previous computations yield the instantaneous CEP. The CEP must still be rotated into the CIO. This is achieved by means of the pole coordinates xp, yp which define the position of the CEP with respect to the CIO. The pole coordinates are determined by the IERS and are available upon request. The rotation matrix for polar motion is given by 4. Polar motion − = − − = − 1 0 1 1 0 2{ } 1{ } P P P P P P S x y y x R R x R y The rotation matrices and are often combined to a single matrix for earth rotation: R M S R = R R S R M R 3. Time Systems What is Time? Definition of time criterion must be motion object • Periodic • Stable • Observable 1. Earth rotation by itself (Sidereal time, Solar time 2. Earth rotation around Sun (Calendar time 3. Oscillation frequency of atomic motion (Atomic time) 3.1 A Group of Time System International Atomic Time (IAT) UT Coordinated (UTC) Atomic oscillations Terrestrial Dynamic Time (TDT) Barycentric Dynamic Time (BDT) Earth revolution Universal Time (UT) Greenwich Sidereal Time (00) Earth rotation Periodic Process Time System Several time systems are in current use. They are based on various periodic processes such as earth rotation
1. Solar and sidereal times Universal Time (UT): is defined by the greenwich Notice Ir angle augmented by 12 hours of a fictitious Both solar and sidereal time are not uniform since mly orbiting in the equatorial plane(p. 35) the angular velocity we is not constant. Changes in the polar moment of inertia exerted Sidereal Time(ST: is defined by the hor by tidal deformation the vernal Taking the mean equinox as the reference leads to Oscillations of the earth's rotational axis itself 2. Dynamic Times Dynamic Times: The time systems derived from planetary onions in the solar system are called dynamic times In 1991. the International Astronomical Unior yeentrio Dynamic Time(BDT): is an inertial time introduced the term Terrestrial Time(r)to stem 1 Newtonian sense and pr ovides the time TDT. Furthermore, the terminology of coordinate variable in the equations of motion. according to the theory of general relativity was merly called ephemeris time and serves for the integration of the differential equations for the orbital motion of satellites around the earth 3. Atomic Times 3.2 Conversions Dynamic Time System is achieved by the use of atomic time UTC S c second. but to k The unit of the system is the keep the system clos e=1.0027379093Um1+uo+△cos UTI and integer leap seconds are inserted at distinct epochs 1. The first term. the different scales of solar and sidereal GPS Time is also related to the atomic time system 2. The quantity Do represents the actual sidereal time at Naval Observatory (USNO) 3. The third term: the projection of Ao onto the equator and 2. GPS time system nominally has a constant offset of considers the effect of nutation seconds with iat 3. GPS Time was coincident with Utc at the Gp standard epoch 1980, January 6. 0
5 1. Solar and Sidereal Times Universal Time (UT): is defined by the Greenwich hour angle augmented by 12 hours of a fictitious sun uniformly orbiting in the equatorial plane (p. 35). Sidereal Time(ST): is defined by the hour angle of the vernal equinox. Taking the mean equinox as the reference leads to mean sidereal time and using the true equinox as a reference yields true or apparent sidereal time (p. 35). Both solar and sidereal time are not uniform since the angular velocity wE is not constant. • Changes in the polar moment of inertia exerted by tidal deformation • Other mass transports • Oscillations of the earth's rotational axis itself Notice 2. Dynamic Times Dynamic Times: The time systems derived from planetary motions in the solar system are called dynamic times. Barycentrio Dynamic Time (BDT): is an inertial time system in the Newtonian sense and provides the time variable in the equations of motion. Quasi-inertial Terrestrial Dynamic Time (TDT): was formerly called ephemeris time and serves for the integration of the differential equations for the orbital motion of satellites around the earth. In 1991, the International Astronomical Union (IAU) introduced the term Terrestrial Time (TT) to replace TDT. Furthermore, the terminology of coordinate times according to the theory of general relativity was introduced 3. Atomic Times Dynamic Time System is achieved by the use of atomic time scales. UTC System is a compromise. The unit of the system is the atomic second, but to keep the system close to UT1 and approximate civil time, integer leap seconds are inserted at distinct epochs. GPS Time is also related to the atomic time system. 1. GPS time is referenced to UTC as maintained by the U.S. Naval Observatory (USNO). 2. GPS time system nominally has a constant offset of 19 seconds with IAT. 3. GPS Time was coincident with UTC at the GPS standard epoch 1980, January 6.d0. 3.2 Conversions The conversion between the times is achieved by the formula Θ0 = 1.0027379093 UT1 + υ0 + ∆φcosε 1. The first term: the different scales of solar and sidereal time, 2. The quantity υ0 represents the actual sidereal time at Greenwich midnight (i.e., 0h UT). 3. The third term: the projection of ∆φ onto the equator and considers the effect of nutation
Thus, the mean solar time is corrected for polar motion Uo=24110.54841+8640184581286610+ 09093104T20-62.106T UTI and the apparent sidereal time eo where To represents the timespan between the standard epoch J2000.0 (Julian centuries of 365.25 mean solar The mean sidereal time by neglecting the nutation term days) and the day of observation at oh UT. is a part of the navigation message broadcast by the UT1 UTC GPS satellites UTl=UTC+duTI When the absolute value of dUTI becomes larger than 0. 9, a leap second is inserted into the UTC system 3.3 Calendar Dynamic Time and Atomic Time System (GPS) The Julian Date (D) defines the number of mean solar days elapsed since the epoch 4713 B. C, January 1.5 IAT=GPS+19000 constant offset IAT= TDT-32 184 constant offset (3. 22) Modified . ulian I MuD) is obtained by subtracting 00.5 days from his convention saves digits and IAT=UTC +1.000 n variable offset as leap seconds are MJD commences at civil midnight instead of noon. ubstituted for example, the calculation of the parameter T for the dard epoch. Subtracting the respective Julian dates ing by 36525 (i.e, the number of days in a Julian entury)yields T=-0. 1998767967 Assignment 3 Summary 1. What are space-fixed and earth-fixed coordinate systems? Give the examples What have we learnt? 2. What are ecliptic plane, vernal equinox, node, Which parts are important? perigee, apogee, orbital plane? 3. What is orbital plane? 4. How many time system do you know? What is sidereal time and solar time 6
6 Thus, the mean solar time is corrected for polar motion UT1 and the apparent sidereal time Θ0. The mean sidereal time by neglecting the nutation term is a part of the navigation message broadcast by the GPS satellites. Time Series for υ0 υ0 = 24110s .54841 + 8640 184.s 812866 T0+ 0.s 093104 T2 0 -6.s 2 .10-6 T3 0 where T0 represents the timespan between the standard epoch J2000.0 (Julian centuries of 365.25 mean solar days). and the day of observation at oh UT. UT1 and UTC UT1 = UTC + dUT1 When the absolute value of dUTl becomes larger than 0.s 9, a leap second is inserted into the UTC system. Dynamic Time and Atomic Time System (GPS) IAT = GPS + 19.s 000 constant offset IAT = TDT -32.s 184 constant offset (3.22) IAT = UTC + l.s 000 n variable offset as leap seconds are substituted. 3.3 Calendar Definitions The Julian Date (JD) defines the number of mean solar days elapsed since the epoch 4713 B.C., January 1.d5. The Modified Julian Date (MJD) is obtained by subtracting 2400000.5 days from JD. This convention saves digits and MJD commences at civil midnight instead of noon. the Julian date for two standard epochs is given. This table enables, for example, the calculation of the parameter T for the GPS standard epoch. Subtracting the respective Julian dates and dividing by 36525 (i.e., the number of days in a Julian century) yields T = -0.1998767967. Summary What have we learnt? Which parts are important? Assignment 3 1. What are space-fixed and earth-fixed coordinate systems? Give the examples. 2. What are ecliptic plane,vernal equinox, node, perigee, apogee,orbital plane? 3. What is orbital plane? 4. How many time system do you know? 5. What is sidereal time and solar time? 6. What is GPS time?
