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Satellite Engineering Research
Corporation
Practical Relativistic
Timing Effects
in GPS and Galileo
Robert A.
Nelson
Satellite Engineering Research
Corporation
Bethesda,
MD
301-657-9641
CGSIC Timing Subcommittee
Meeting
Thursday, March 20, 2003
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Special and General Theories of
Relativity
Special relativity
Created in 1905
Concerns kinematics, mechanics, and
electromagnetism
General relativity
Completed in 1916
Concerns gravitation
Not a separate theory: includes special
relativity
Today the general theory of relativity is not
simply a subject of theoretical scientific speculation, but rather it has
entered the realm of practical engineering necessity.
Relativistic effects must be considered in the
transport of atomic clocks and the propagation of electromagnetic
signals.
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Proper Time vs. Coordinate
Time
Proper time
The time provided by an ideal clock in its own rest frame
Different for clocks in different states of
motion and in different gravitational potentials
“Hardware” proper time
The time provided by a real clock in its own
rest frame corrupted by noise and environmental
effects
Coordinate time
The time coordinate in the chosen space-time coordinate system
A global coordinate
Has same value everywhere for a given event
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Relativistic
Effects
Three effects contribute to the net relativistic
effect on a transported clock
Velocity (time dilation)
Makes transported clock run slow relative to a clock on the geoid
Function of speed only
Gravitational potential (red shift)
Makes transported clock run fast relative to a clock on the geoid
Function of altitude
only
Sagnac effect
Makes transported clock run fast or slow relative to a clock on the geoid
Depends on direction and path
traveled
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Time dilation of muon
lifetime
B. Rossi and D.B. Hall (1941); D.H. Frisch and J.H.
Smith (1963)
Muons observed in 1 h at top of Mt. Washington (elev. 1910 m) and at sea level.
Number observed at elev. 1910 m is 568.
Number observed at sea level is 412.
Exponential law of decay with mean proper
lifetime = 2.2 s
Muons selected with velocity 0.9952
c
Time of flight in laboratory frame = 6.4
s
Time of flight in muon rest frame = 0.63 s
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Around the world atomic clock
experiment
(J.C. Hafele and R.E. Keating (1971)
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Around the world atomic clock
experiment
(Flying clock – Reference clock)
predicted effect direction
East West
Gravitational potential
(redshift) + 144
ns
+ 179 ns
Velocity (time
dilation) 51
ns
47
ns
Sagnac
effect 133
ns
+ 143 ns
Total
40
23
ns + 275 21 ns
Measured 59 10 ns + 273 7 ns
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Gravitational redshift of an
atomic clock
C.O. Alley, et al. (1975)
Gravitational redshift 52.8 ns
Time dilation 5.7 ns
Net effect 47.1 ns
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TWTT Flight
Tests
Tests conducted by Timing Solutions Corp., Zeta
Associates, and AFRL
Flight clock data collected on a C-135E aircraft
to demonstrate TWTT in
background of an active communications channel
6 flights in November 2002 from
WPAFB
L-Band Antenna
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Relativistic
Effects
Relativity effects on flight clock computed based
on the position record over the flight interval
Gravitational (redshift)
effect, velocity (time dilation) effect and Sagnac effect combine to a predicted
net change in flight clock phase of 15 ns
Relativistic Effects (Reference Clock – Flying Clock)
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Processed TWTT
Data
Averaging instantaneous data results in a
sub-nanosecond, continuous record of the clock difference over the flight
interval
Collected data agree well with predicted clock
differences based on relativity calculations
TWTT Data (60 s
average)
Approach/Landing
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Sagnac effect (TWSTT)
NIST to
USNO via Telstar 5 at 97 WL
Uplink 24.1
ns
Downlink 57.7
ns
Total Sagnac correction 81.1 ns
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GPS
Gravitational redshift (blueshift)
Orbital altitude 20,183 km
Clock runs fast by 45.7 s per
day
Time dilation
Satellite velocity 3.874 km/s
Clock runs slow by 7.1 s per day
Net secular effect (satellite clock runs fast)
Clock runs fast by 38.6 s per
day
Residual periodic effect
Orbital eccentricity 0.02
Amplitude of periodic effect 46
ns
Sagnac effect
Maximum value 133 ns for a stationary receiver
on the geoid
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GPS (Summary)
Net secular relativistic effect is 38.6 s per day
Nominal clock rate is 10.23 MHz
Satellite clocks are offset by – 4.464733 parts in 1010 to compensate effect
Resulting (proper) frequency in orbit is 10229999.9954326 Hz
Observed average rate of satellite clock is
same as clock on the geoid
Residual periodic effect
Maximum amplitude 46 ns
Correction applied in
receiver
Sagnac effect
Maximum value 133 ns
Correction applied in
receiver
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Galileo
Gravitational redshift (blueshift)
Orbital altitude 23,616 km
Clock runs fast by 47.3 s per
day
Time dilation
Satellite velocity 3.645 km/s
Clock runs slow by 6.3 s per day
Net secular effect (satellite clock runs fast)
Clock runs fast by 47.3 s per
day
Residual periodic effect
Orbital eccentricity 0.02
Amplitude of periodic effect 49
ns
Sagnac effect
Maximum value 153 ns for a stationary receiver
on the geoid
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Molniya satellite
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Molniya orbit ground
trace
Period = 11.967 h Apogee altitude = 39,362 km Perigee altitude = 1006 km
Eccentricity = 0.722 Inclination = 63.4 Argument of perigee = 250
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Eccentricity correction for Molniya orbit
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GPS ICD-200
Must also consider effect of moving receiver on
signal propagation time.
Paragraph on “Geometric Range” in GPS ICD-200
revised in 1998.
In the past, the ICD assumed the receiver was at rest on the rotating Earth. Paragraph is now completely general.
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Measurement of
pseudorange
(Coordinate time)
(“Hardware” proper time)
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Additional relativistic
effects
Contribution to gravitational redshift due to Earth oblateness
Amplitude of periodic effect for GPS is 24
ps
Tidal potentials of the Moon and Sun
Amplitude of periodic effect is on the order of
1 ps
Effect of gravitational potential on time of signal propagation
On the order of 3 ps
Intersatellite links (GPS III and beyond)
Eccentricity correction on the order of tens of nanoseconds
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Conclusion
Relativity has become an important practical
engineering consideration for modern precise timekeeping
systems.
Far from being simply a textbook problem or
merely of theoretical scientific interest, the analysis of relativistic effects
is an essential practical engineering consideration.
These relativistic effects are well understood
and have been applied successfully in the GPS.
Similar corrections will need to need to be
applied in Galileo.
Common geodetic and time scale references will be needed for possible interoperability between GPS and Galileo.
Terrestrial reference system (WGS-84 and ITRF-2000)
Time (realization of common coordinate time by
satellite clocks)
Of these two considerations, the measurement of time will be the most important.