12. 540 Principles of the Global Positioning System Lecture 04 Prof. Thomas Herring 02/803 12540Lec04 Review So far we have looked at measuring coordinates with conventional methods and using gravity field Examine definitions of coordinates Relationships between geometric coordinates Time systems Start looking at satellite orbits
02/18/03 12.540 Lec 04 1 12.540 Principles of the Global Positioning System Lecture 04 Prof. Thomas Herring 02/18/03 12.540 Lec 04 2 Review – Examine definitions of coordinates – – Time systems – Start looking at satellite orbits • So far we have looked at measuring coordinates with conventional methods and using gravity field • Today lecture: Relationships between geometric coordinates 1
Coordinate types Potential field based coordinates Astronomical latitude and longitude Orthometric heights(heights measured about an equipotential surface, nominally mean-sea-level (MSL) Geometric coordinate systems Cartesian xYZ Geodetic latitude, longitude and height 02/803 12540Lec04 Astronomical coordinates Astronomical coordinates give the direction of the normal to the equipotential surface Measurements Latitude: Elevation angle to North Pole(center of star rotation field Longitude: Time difference between event at Greenwich and locally
02/18/03 12.540 Lec 04 3 Coordinate types – – Orthometric heights (heights measured about an (MSL) – Cartesian XYZ – Geodetic latitude, longitude and height • Potential field based coordinates: Astronomical latitude and longitude equipotential surface, nominally mean-sea-level • Geometric coordinate systems 02/18/03 12.540 Lec 04 4 Astronomical coordinates – Latitude: Elevation angle to North Pole (center of star rotation field) – Longitude: Time difference between event at Greenwich and locally • Astronomical coordinates give the direction of the normal to the equipotential surface • Measurements: 2
Astronomical Latitude Normal to equipotential defined by local gravity vector Direction to North pole defined by position of rotation axis. however rotation axis moves with t to crust of earth Motion monitored by International Earth RotationseRviceIerShttp://www.iers.org/ Astronomical latitude pa=z66人9a δ To Celestial body
02/18/03 12.540 Lec 04 5 Astronomical Latitude http://www.iers.org/ • Normal to equipotential defined by local gravity vector • Direction to North pole defined by position of rotation axis. However rotation axis moves with respect to crust of Earth! • Motion monitored by International Earth Rotation Service IERS 02/18/03 12.540 Lec 04 6 Astronomical Latitude d fa zd To Celestial body Rotation Axis fa = Zd-d declination Zenith distance= 90-elevation Geiod 3
Astronomical Latitude By measuring the zenith distance when star is at minimum, yields latitude Problems Rotation axis moves in space, precession nutation Given by International Astronomical Union (AU) precession nutation theory Rotation moves relative to crust 02/803 12540Lec04 Rotation axis movement Precession Nutation computed from Fourier Series of motions Largest term 9"with 18.6 year period Over 900 terms in series currently(see http://bowie.mitedu/tah/mhb2000/jb000165onlinepdf Declinations of stars given in catalogs Some almanacs give positions of"date meaning precession accounted for
02/18/03 12.540 Lec 04 7 Astronomical Latitude – Rotation axis moves in space, precession nutation. Given by International Astronomical Union (IAU) precession nutation theory – Rotation moves relative to crust 02/18/03 12.540 Lec 04 8 Rotation axis movement http://bowie.mit.edu/~tah/mhb2000/JB000165_online.pdf) • By measuring the zenith distance when star is at minimum, yields latitude • Problems: • Precession Nutation computed from Fourier Series of motions • Largest term 9” with 18.6 year period • Over 900 terms in series currently (see • Declinations of stars given in catalogs • Some almanacs give positions of “date” meaning precession accounted for 4
Rotation axis movement Movement with respect crust called"polar motion". Largest terms are Chandler wobble (natural resonance period of ellipsoidal body) and annual due to weathe Non-predictable: Must be measured and monitored 12540Lec04 Evolution(IERS CO1) "8 E0.00 1920.0 19600 1980.0
02/18/03 12.540 Lec 04 9 Rotation axis movement monitored • Movement with respect crust called “polar motion”. Largest terms are Chandler wobble (natural resonance period of ellipsoidal body) and annual term due to weather • Non-predictable: Must be measured and 02/18/03 12.540 Lec 04 10 Evolution (IERS C01) -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 1900.0 1920.0 1940.0 1960.0 1980.0 2000.0 X Pole (") Pole Position (") (0.5"=15m) Year Y Pole (") CIO 1900-1905 5
Volution oT uncertainty co1900-190 0.14 o XPole ( g Pole 50.10 1900.0 1940.0 2000.0 02/83 12540Lec04 Behavior 1993-2001 0.0
6 02/18/03 12.540 Lec 04 11 Evolution of uncertainty 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 1900.0 1920.0 1940.0 1960.0 1980.0 2000.0 s X Pole (") s Y Pole (") s Pole Position (") (0.02"=0.6m) Year CIO 1900-1905 02/18/03 12.540 Lec 04 12 Behavior 1993-2001 -10.0 -5.0 0.0 5.0 10.0 20.0 15.0 10.0 5.0 0.0 Pole Position X Pole (m) Y Pole (m) 1993 2001
Astronomical Longitude Based on time difference between event in Greenwich and local occurrence Greenwich sidereal time(GsT) gives time relative to fixed stars GST=1.002737909371+0b+△ cost =24110.54841+8640184.812866T+ ulian Centuries 0.093104T2-6.2×10-673 Universal time UT1: Time given by rotation of Earth. Noon is mean"sun crossing meridian at Greenwich UTC: UT Coordinated. Atomic time but with leap seconds to keep aligned with UT1 UT1-UTC must be measured
† 02/18/03 12.540 Lec 04 13 Astronomical Longitude GST =1.0027379093UT1+ J0 GMST { y e Precession 34 J0 = 24110.54841+ 8640184.812866 T Julian Centuries { + 0.093104T 2 - 6.2 ¥10-6 T 3 • Based on time difference between event in Greenwich and local occurrence • Greenwich sidereal time (GST) gives time relative to fixed stars + D 1 2 4 cos 02/18/03 12.540 Lec 04 14 Universal Time • UT1: Time given by rotation of Earth. Noon is “mean” sun crossing meridian at Greenwich • UTC: UT Coordinated. Atomic time but with leap seconds to keep aligned with UT1 • UT1-UTC must be measured 7
ength of day (LOD) LOD Difference of day from 8640 1850.0 2000.0 12540Lec04 Recent LOD LOd= Difference of day from 86400 seconds 置 19920 1994.0 20020
8 02/18/03 12.540 Lec 04 15 Length of day (LOD) -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 1800.0 1850.0 1900.0 1950.0 2000.0 LOD = Difference of day from 86400. seconds LOD (ms) LOD (ms) LOD (ms) Year 02/18/03 12.540 Lec 04 16 Recent LOD -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1992.0 1994.0 1996.0 1998.0 2000.0 2002.0 LOD = Difference of day from 86400. seconds LOD (ms) LOD (ms) LOD (ms) Year
LOD to UT1 Integral of LOD is UT1 (or visa-versa If average LOD is 2 ms, then 1 second difference between ut1 and atomic time develops in 500 days Leap second added to utc at those times 02/803 12540Lec04 UT1-UTC UTl-UTC(discontinuities are leap-seconds 0.40 1993.01994.01995.01996.01997.019980199902000.0
9 02/18/03 12.540 Lec 04 17 LOD to UT1 • Integral of LOD is UT1 (or visa-versa) • If average LOD is 2 ms, then 1 second difference between UT1 and atomic time develops in 500 days • Leap second added to UTC at those times. 02/18/03 12.540 Lec 04 18 UT1-UTC -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1993.0 1994.0 1995.0 1996.0 1997.0 1998.0 1999.0 2000.0 2001.0 UT1-UTC (discontinuities are leap-seconds) UT1-UTC (s) UT1-UTC (s) Year
Transformation from Inertial space to Terrestrial Frame To account for the variations in earth rotation parameters, as standard matrix rotation is made w x, Inertial Precession Nutation Spin Polar Motion Terrestrial 02/803 12540Lec04 Geodetic coordinates Easiest global system is Cartesian XYZ but not common outside scientific use Conversion to geodetic Lat, Long and Height X=(N+h)cosφcosλ Y=(N+h)cosφsinλ N /a?cos +basin?p
† † 02/18/03 12.540 Lec 04 19 Transformation from Inertial Space to Terrestrial Frame made xi Inertial { = P Precession { N Nutation { S Spin { W Polar Motion { xt Terrestrial { • To account for the variations in Earth rotation parameters, as standard matrix rotation is 02/18/03 12.540 Lec 04 20 Geodetic coordinates X = (N + h) f l Y = (N + h) f sinl Z = ( b2 a 2 N + h)sinf N = a 2 a 2 2 f +b2 sin2 f • Easiest global system is Cartesian XYZ but not common outside scientific use • Conversion to geodetic Lat, Long and Height cos cos cos cos 10
