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Neil Ashby
Dept. of Physics, University of Colorado
National Institute of Standards and Technology Affiliate
Email: [email protected]
1. Navigation—why you need a clock
2. Brief history of relativity in the GPS
3. What the GPS is
4. Relativistic effects:
Relativity of synchronization;
Time dilation;
Gravitational frequency shifts;
Sagnac effect;
5. Observations: testing relativity
TOPEX;
Frequency jumps;
Unmodeled effects;
6. Applications
Relativity in Global Satellite
Navigation Systems
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Latitude
3
w
4
Moon, Jupiter & Satellites
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GPS RELATIVITY MEETINGS
• 1979 SAMSO Relativity Seminar (Boulder)
• 1985 JASON Study
• 1986 Air Force Studies Board
• 1988-98 Various Working Group Meetings
• 1995 ARL-Chapel Hill
• 1997 ICD-200 Relativity Review (Boulder)
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GPS RELATIVITY MEETINGS
• 1979 SAMSO Relativity Seminar
• 1985 JASON Study
• 1986 Air Force Studies Board
• 1988-98 Various Working Group Meetings
• 1995 ARL-Chapel Hill
• 1997 ICD-200 Boulder
Erroneous Reports
• 1977-83 Moses, Cohen, Rosenblum
• 1992 Deines
• 1996 Fliegel & DiEsposti (Aerospace Corp)
• 2000-2006 Hatch
• 2008 Beisner
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GPS Constellation
• 24 Satellites (now>30)
• 6 orbital planes, 55o inclination
• Period: half a sidereal day
• Several atomic clocks/satellite
• Several spare satellites
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Control Segment
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GPS IIR Satellite
IIF
III
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Block III satellite
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Block III GPS satellite
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Other GNSS Satellites
Beidou GALILEO
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Constellation Status-GPS
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GPS Constellation Status
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GALILEO Constellation Status 11/21/2016
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Planned Beidou Constellation
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Beidou Constellation Status
• As of November 2016: 20 operational satellites of 35 planned
– 6 satellites in geostationary orbits;
– 8 in 55-degree inclined geosynchronous orbits;
– 6 in medium earth orbits at altitude 21,500 km
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• Constancy of the speed of light– The speed of light, c, is a constant independent of the
motion of the source (or of the observer);
• Principle of Equivalence (“weak form”)– Over a small region of space and time, the fictitious
gravitational field induced by acceleration cannot be
distinguished from a real gravitational field due to a
mass.
Fundamental Principles
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Constancy of c
,1,2,3,4
j jc t t
j
r - r
Synchronization
Is the key!
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Reciprocity
j jc t t r -r
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Exploded Block IIR Satellite View
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Clock Improvement Since 1000 A.D.
*
1 ns/day=
10-14
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Why are atomic clocks needed?
To reduce the effect of clock error to < 2 meters,
the clock error must be less than 2/c = 6.7 x 10-9 sec.
Half a day = 43200 seconds, so the fractional clock
error must be less than:
(2 m)/(43200 s x c) = 1.5 x 10-13.
Only atomic clocks can achieve such stability.
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25
Constancy of the speed of light
implies time dilation
(These clocks are at
rest in the moving frame.)
(This clock at rest in “rest”
frame, coincides with upper
clock in moving frame.)
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Einstein’s Light Clock
2 2c v
2 2
2 2
2 2
// ;
1 /
' / 1 /
L ct L c v
v c
t L c v c t
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How Big is Time Dilation?
22 2
2
211
2
11 / 1 ;
2
4000m/s;
18.9 10
2
vv c
c
v
v
c
(about 8 microseconds per day)
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Accounting For Relativistic Effects
Example: Time Dilation:
2 2
1/ 22 2
2
2
2
2
path
1 / ;
1 /
11 .
2
11
2
d v c dt
dt v c d
vd
c
vt d
c
Elapsed Coordinate time:
Observed Proper Time
Note: 2 2 22 2 2 2 2(1 / )cd cdt v c cdt dx dy dz
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Fundamental line element
2 2 2 2 2
2 2 22 2 2
2 2
2 2 2 2
2 2 2 2 2
2 2
0 ( )
1( ) ( ) 1
( )
2 21 ( ) 1
0.path
ds cdt dx dy dz
dx dy dzds cd cdt
c dt
cdt dx dy dz
ds cdt dx dy dzc c
ds
For light:
Time dilation:
With gravity:
Motion of
Planets:
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Coordinate Time
• In special relativity:
To each real clock, corrections are applied such
that at each instant, the clock would read the same
as a hypothetical clock at rest at the same point
in the underlying inertial frame.
• When gravitational fields are present:
Additional corrections compensate for gravitational
frequency shifts relative to a reference on earth’s geoid.
• GPS time is an example of coordinate time, in which
the reference is on the earth’s rotating geoid.
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Sagnac Effect
2 2 2 2 2 2
2 22 2 2 2 2 2
2
( ) ( ( ))
1 ( ) 2
ds c dt dr r d t dz
rc dt r d dt dr r d dz
c
w
ww
2 2 2 2 2 2 2( ) ( ) ( )ECIcd ds cdt dr r d dz
For light: solving for dt to first order in
the ddt term gives rise to the
Sagnac effect.
,w
This is the Langevin metric.
In a rotating coordinate system such as one fixed to the earth, let the
axis representing the zero for the angle rotate with constant
angular speed:
2
2
d r ddt
c c
w
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Sagnac Effect on Synchronization
in a Rotating System
w
Sagnac Correction =2
2zA
c
w
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Over a small region of space and time,
a fictitious gravity field induced by
acceleration cannot be distinguished
From a gravity field produced by mass.
Equivalence principle and
gravitational frequency shifts
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Gravitational Frequency Shift
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Gravitational Frequency Shift
2 2
/ ;
;
.
t L c
gLv gt
c
f v gL
f c c c
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Relativity of Simultaneity
To an observer on the ground, let two lightning
strokes at the front and back of the train
be simultaneous.
The “moving” observer at the train’s midpoint finds
the event at front occurs first.
2 2'
vxt t t
c c
V r
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Principle of Equivalence
V
V+V
Acceleration
Local Inertial
Frame
r
2
2 2
2
2
2
2
Induced potential difference/c ;
Gravitational potential difference/c ;
Net potential difference/c = 0;
c Tc
c
c
A r V r
r
A r r
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Fundamental Scalar Invariant
2 2 2 2 2
2 2
2 2
2 1
2 2
2 2( ) 1 ( ) 1 ( )
3 11
2
c d c dt dx dy dzc c
J aGM z
r r r
For a clock near earth,
2
2 2
path
11
2
vt d
c c
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Earth-fixed Clock
0
2
path
2 2
0 2 1
2 2 2 2
1 1
10
10
1 ;
2 2
6.95348 .00376 .01203 10
6.96927 10
t dc
GMJ aGM
c c a c a c
w
This is the fractional frequency shift of an atomic clock
fixed on earth, relative to an atomic clock at infinity.
Note about centripetal term
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Clocks on earth’s geoid beat at equal rates
Clocks at rest on geoid
beat at equal rates, defining
International Atomic Time.
They are synchronized in
the underlying inertial frame.
Centripetal potential,
monopole potential,
quadrupole, and higher
potential terms conspire
to give an equipotential
in the rotating frame.
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Earth-based Time Scale
SI Second:
0
2
2 2 2 2 20
2 2
1
2( ) 2( ) 1 ( ) 1 ( )
t tc
cd cdt dx dy dzc c
(Basis for International Atomic Time, Universal Coordinated Time.)
(This number is now a defined quantity.)100
26.969290134 10 .GL
c
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Atomic Clock in a Satellite
122 2
0 0
2 2 2 2
0SV 2 2 2
path
10
path
2
2( ) ( )1 ; 1
2
3 2 1 11
2
[1 4.4647 10 ]
2sin (sec)
meter
v vd dt dt d
c c c c
GM GMt d
ac c c a r
d
GM ae E
c
21; a = semimajor axis
2 2
GMv
a
38.6 s/day
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Factory Frequency Offsets
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Frequency shifts cancel at this radius
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Three Important Effects (GPS)
#1: Scale correction to satellite clock:
10.23000000000 MHz 10.22999999543 MHz
#2: Receiver must implement the eccentricity correction:
104.4428 10 sin (sec)meter
ae E
#3: User must account for time required for signal propagation
(Sagnac effect) if relevant.
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SV#13 eccentricity effect
(TOPEX receiver)
0.013e
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GALILEO Satellites in unintended orbits
Normal radius of a GALILEO satellite: 29599.8 km
Eccentricity: 0
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Quasi-Zenith Satellite System (Japan)
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Frequency “Breaks”
Due to Orbit Adjustments
0
2 2 2
3 2 1 1.
2
f GM GM
f c a c c a r a
If eccentricity is small,
2 2
-13
-13
3;
2
FOR SV#43:
Measured: -1.85 10
Predicted: -1.77 10
Now implemented.
f GM a
f c a
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Unmodeled Relativistic Effect: Oblateness
Effect of Earth’s oblateness on satellite orbit:
2
2 2
2
3 2
3;
2J
GMJ z r
r r
Change in monopole potential:2 2
2 1
3
sincos2( ) ...
4
GM GMJ a If
r a w
Change in kinetic energy:
2 2 2
2 1
3
sincos2( ) ...
2 2
v GMJ a If
a w
Change in frequency:2 2
2 1
3 2
sincos2( ) ...
f GMJ a If
f a c w
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A Coincidence?
There are many terms in the perturbations arising
from Earth’s oblateness with coefficient
231 (sin )
2I
For GPS, this is nearly zero. ( 55 )I
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Shapiro delay SV to earth surface
2 1 1 2 2 1
3
1 2 2 1
logG e
gravity
r r L r r r rGMt
c c r r r r
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Spectrum of lunar tidal potential
The coefficients are functions of the eccentricities and inclinations of the SV and the
moon with respect to the equator. The phases are functions of the altitudes of perigee
and the angles of the lines of nodes.
There are significant contributions from many frequencies
in the neighborhood of 6 hours. (These correspond to )
The short-period terms are sufficiently close together that they can beat against
each other, reinforcing and cancelling. They can combine and have amplitudes
that are estimated to be greater than about
Detailed calculation of the lunar tidal potential gives perturbations in terms of
cos( )i i sat i moon i
i
A n t m tw w
6,... 8; 7,... 8i in m
152 10 .
2.in
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Unmodeled Relativistic Effects:
Lunar and Solar Tides
Lunar and solar tidal perturbations are estimated to affect the
fractional frequency shifts of GPS SV clocks in a predictable way
by about153.7 10
The principal periods with which this occurs are near 6 hours
but there are many nearly equal frequencies.
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Surface Plate Velocities
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Control of Monster Machines
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Autonomous Operation
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Precision Agriculture
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Surveying
Finding boundary markers
lost for a century.
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Animal
Tracking
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GNSS-other satellite navigation
systems
GLONASS-Russia
GALILEO-ESA
BEIDOU--China
IRNSS--INDIA
QZSS--JAPAN
AUGMENTATION SYSTEMS:
WAAS
EGNOS
QZSS
All use the same fundamental relativity concepts.
The GALILEO specs state “all relativity corrections are
the responsibility of the user.” ????
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Clock Coefficient a0 of
GPS Satellite clocks, 1992-2014
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References:
http://relativity.livingreviews.org/Articles/lrr-2003-1/index.html
“100 Years of Relativity,” World Scientific,
A. Ashtekar, ed., (2005), Chapter 10
“Handbook of Spacetime,” Springer,
Ashtekar and Petkov, eds, (2014), Chapter 24
“General Relativity: The Most Beautiful of Theories,”
Rovelli, ed., de Gruyter, (2015) pp 165-188
END
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GPS DEVELOPMENT KIT
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