3.1. Coordinate-systems and time. Seeber 2.1.
Z
NON INERTIAL
SYSTEM
Mean-rotationaxis
1900.
Gravity-centre
Y- Rotates with
the Earth
Greenwich
X
CTS:
Conventional
Terrestrial System
1
CIS
• Zero-meridian for Bureau Internationale de l’
Heure (BHI) determined so that star-catalogues
agree in the mean with observations from
astronomical observatories.
• The connection to an Inertial System is
determined using knowledge of the Z-axís (Polar
motion), rotational velocity and the movement of
the Earth Center.
• We obtain an Quasi-Inertial system, CIS.
• More correct to use the Sun or the centre of our
galaxe !
2
Kap. 3
POLAR MOTION
• Approximatively circular
• Period 430 days (Chandler period)
• Main reason: Axis of Inertia does not coinside with axis of rotation.
• Rigid Earth: 305 days: Euler-period.
3
Ch. 3
POLBEVÆGELSEN
• .
4
Kap. 3
POLAR MOVEMENT
• Coordinates for the Polen and Rotational velocity
• IERS (http://www.iers.org)
• International Earth Rotation and Reference
System service (IAG + IAU)
• http://aiuws.unibe.ch/code/erp_pp.gif
• Metods:
VLBI (Radio astronomi)
LLR (Laser ranging to the Moon)
SLR (Satellite Laser ranging)
GPS, DORIS
5
Kap. 3
• Polbevægelse, 1994-1997, Fuld linie : middel pol bevægelse, 19001996
6
Kap. 3.
International Terrestrial Reference System (ITRS)
• Defined, realised and controlled by IERS
ITRS Center.
http://www.iers.org/iers/products/itrs/
• Geocentric, mass-centre from total Earth
inclusive oceans and atmosphere.
• IERS Reference Pole (IRP) and Reference
Meridian (IRM) konsist with BIH directions
within +/- 0.005".
7
Kap. 3, ITRS.
• Time-wise change of the orientations
secured through 0-rotation-condition taking
into account horizontal tectonic movements
for the whole Earth.
• ITRS realised from estimate of coordinates
for set of station with observations of VLBI,
LLR, GPS, SLR, and DORIS. See:
ftp://lareg.ensg.ign.fr/pub/itrf/old/itrf92.ssc
8
Kap. 3
•
•
•
•
•
•
•
Paris, 1 July 2003
Bulletin C 26
INFORMATION ON UTC - TAI
NO positive leap second will be introduced at the end of December
2003.
The difference between UTC and the International Atomic Time TAI
is :
from 1999 January 1, 0h UTC, until further notice : UTC-TAI = -32 s
Leap seconds can be introduced in UTC at the end of the months of
December or June, depending on the evolution of UT1-TAI. Bulletin
C is mailed every six months, either to announce a time step in UTC,
or to confirm that there will be no time step at the next possible date.
http://www.iers.org/iers/products/eop/leap_second.html
9
Kap. 3
•
10
Kap. 3 Variationer jord-rotationen.
11
Kap. 3
12
Ch. 3, Transformation CIS - CTS
•
•
•
•
Precession
Nutation
Rotation+
Polar movement
Sun+Moon

rCTS  SNPrCIS
13
Ch. 3,
Precession.
P  R3 ( z ) R2 ( ) R3 ( )
z, ,  given by 3' rd order polynomials in T .
T  (t - t 0 ) in Julian centuries,36525days.
t 0 : J2000,2000- 01- 01.
• Example: t-t0=0.01 (2001-01-01)
• .
14
Ch. 3,
Nutation – primarily related to the Moon.
• Movement takes place in Ecliptica
  obliquity of theeclipt ic,  nutationof ob.
  nutationin longitude(in eclipt ic)
N  R1 (   ) R3 ( ) R1 ( )
  230 26'21"  23.0 43916667
15
Ch. 3,
Nutation:
  mean eclipticlongitudeof thelunar ascendingnode
D  mean elongationof theMoon from thesun
  17.1996" sin   1.3187" sin(2 F  2 D  2)
- 0.2274"sin(2F- 2)
  9.2025"cos  0.5736cos(2F - 2D - 2)
 0.0977"cos(2F- 2)
Example: D  0,   0, T  0
• .
16
Ch. 3,
Earth rotation and polar motion (ERP).
GAST  Greenwich apparantsiderial time
x p , y p pole - coordinates (fromIERS)
S  R 2 ( x p ) R1 ( y p ) R3 (GAST)
small angles : sinv -  v, cosv -  1 :
 1

 0
 x
• . p
0
1
yp
x p  cos

 y p  sin 
1  0
sin 
cos
0
0

0
1
17
Ch. 3,
Example for point on Equator.
• Suppose θ=0, xp=yp =1” (30 m)
1
0
1 / 200000  0 





0
1
1 / 200000  0 

 1 / 200000 1 / 200000
 6371km
1
• .
18
Ch. 3,
Exercise.
2 May 1994:
x”=0.1843”=0.000000893,
y”=0.3309”=0.0000014651
(x,y,z)=(3513648.63m,778953.56m,5248202.81m)
Compute changes to coordinates.
19
Ch. 3, Time requirement
• 1 cm at Equator is 2*10-5 s in rotation
• 1 cm in satellite movement is 10-6 s
• 1 cm in distance measurement is 3*10-11 s
• We must measure better than these
quantities.
• Not absolute, but time-differences.
20
Ch. 3,
Siderial time and UT. (see fig. 2.13).
• Siderial time: Hour-angle of vernal equinox
in relationship to the observing instrument
• LAST: Local apparent siderial time: true
hour angle
xp
• GAST: LAST for Greenwich
• LMST: Local hour angle of mean equinox
• GMST: LMST for Greenwich
• GMST-GAST=Δψcosε
• LMST-GMST=LAST-GAST=Λ
21
Ch. 3,
UT
• UT= 12 hours + Greenwich hourangle for
the mean sun. Follows siderial time.
• 1 mean siderial day = 1 mean solar day 3m55.909s.
• UT0B is time at observation point B, must
be referred to conventional pole
• UT1= UT0B + ΔΛP
22
Ch. 3,
UT1, GMST and MJD
GMST at 0 h UT 1 6 h 41m50.s 54841
8640184.81
2866s Tu 
0.093104 Tu2  ......
Tu is timefrom J2000  2000- 01- 01: 12h UT1,
countedin Julian centuries
MJD  JD - 2400000.5,for 2005:
MJD  53371 DOY
• .
23
Ch. 3,
Dynamic time
• ET: Ephemeis time (1952) to make equatins
of motion OK.
• TDB= Barycentric time – refers to the Sun
• TDT=Terrestrial time
• From general relativity: clock at the earth moving around
the sun varies 0.0016 s due to change in potential of sun
(Earth does not move with constant velocity).
• TDB=ET on 1984-01-01
24
Ch. 3,
•
•
•
•
GPS Time
GPS time = UTC 1980-01-05
Determined from Clocks in GPS satellites
GPS time – UTC = n * s-C0,
C0 about 300 ns
25
Ch. 3,
Clocks and frequency standards.
• With GPS we count cycles. Expect the fequency
to be constant.
1
Ideal ( I) clock : TI 
fi
If we count N I cyclesin interval
NI
(t - t 0 )  N I  TI 
, but
fi
f i (t )  f I  f i  fi (t  t0 )  ....... timeerror
ti  ti  t  Bias  Drift (t  t0 )  aging......
Must be determinedby measurements!
26
Ch. 3,
Praxis, see Seeber, Fig. 2.15.
• Precision quarts crystal: temperature
dependent, aging
• Rubidium: good stability, long term
• Cesium: stable both on short term and long
term – transportable, commercially
available.
• Hydrogen masers: 10-15 stability in periods
of 102 to 105 s.
• Pulsars: period e.g. 1.6 ms.
27