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

Satellite Engineering Research Corporation 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

Satellite Engineering Research Corporation Around the world atomic clock experiment

(J.C. Hafele and R.E. Keating (1971)

Satellite Engineering Research Corporation Around the world atomic clock experiment

(Flying clock – Reference clock)

predicted effect

East

directio

n West Gravitational potential (redshift) + 144 ns + 179 ns Velocity (time dilation) Sagnac effect  51 ns  47 ns  133 ns + 143 ns Total Measured  40  23 ns + 275  21 ns  59  10 ns + 273  7 ns

Satellite Engineering Research Corporation Gravitational redshift of an atomic clock

C.O. Alley, et al. (1975)

Gravitational redshift Time dilation Net effect 52.8 ns 5.7 ns 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 Ku Band Satellite Terminal

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) 10 5 0 -5 -10 -15 -20 -25 8:19:41 9:19:41 10:19:41 11:19:41

Zulu

12:19:41 13:19:41 14:19:41 Gravity Velocity Sagnac Net

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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) -60 -65 -70

Approach/Landing

-75 -80 -85 8:19:41 9:19:41 10:19:41 11:19:41

Zulu

12:19:41 13:19:41 14:19:41 Predicted Measured

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NIST to USNO via Telstar 5 at 97



WL

Uplink Downlink Total Sagnac correction 24.1 ns 57.7 ns 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 10 10 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 Eccentricity = 0.722

Apogee altitude = 39,362 km Inclination = 63.4

 Perigee altitude = 1006 km 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.