The Effect of Solar Gravitational Potential On GPS Clocks

1
The Effect of Solar Gravitational PotentialOn
GPS Clocks

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• Tom Van Flandern Thomas B. Bahder
• Army Research Laboratory
• PAWG, Colorado Springs
• 1998 August 19
• (last two slides updated 2002 March 30)

2
What affects clock rates?

• SR motion slows clocks by the factor (t
  ypically lt 10-8)
• GR gravitational potential slows clocks by the
  factor (typically ? 10-8)
• N.B. neglected (?10-16)

3
GPS satellite clock rates (ns/day)

• Orbital speed slowing 7,200
• Gravitational potential slowing at GPS orbital
  altitude 14,400
• Gravitational potential slowing on Earths
  geoid 60,300
• Net GPS satellites run faster than clocks on
  geoid by 38,700

4
Question Why is solar potential effect not
included in GPS?

• Predicted effect 12-ns amplitude, 12-hour period
• GPS raw pseudo-ranges show unexplained 12-hour
  periods correlated with Sun direction
• Hoffman (1963) Clocks in freefall should show
  no effect. But counter-examples exist.

5
What does not affect clock rates?

• Cyclotron experiments with meson decay times show
  that acceleration as such does not affect clock
  rates even at 1019 g.
• F m a and equivalence principle If
  acceleration does not affect clocks,
  gravitational force cannot either.

6
Inside a uniform spherical shell, force is zero,
potential is constant, clocks slow
Alley experiment speed altitude affect clocks,
but not acceleration (e.g. when plane circles)
7
Fixed and accelerating clocks with same speed and
height tick at same rates
Suspension/free fall doesnt affect clock rates
8
Compared to pole, clocks on equator slow due to
higher rotation speed, but speed up due to weaker
gravitational potential. These effects
cancel.
Result All clocks at sea level tick at the same
rate!
9
Satellite clocks tick slower than clocks at
infinity because speed and potential effects
accumulate.
Also, as satellite falls toward perigee,
potential gets stronger and speed increases.
Again, the two clock-slowing effects accumulate.
10
As Earth and satellite fall toward perihelion,
all their clocks slow in unison relative to
distant pulsar clocks. The two slowing effects
accumulate.
11
But as the satellite orbits, do the two
clock-slowing effects from gravitational
potential and velocity in the Suns field cancel
or accumulate?
predicted amplitude of solar potential effect 12
ns (3.6m)
To Sun
12
Or if Earth is not present but the satellite
follows the same path as before using rocket
propulsion, do the two solar clock-slowing
effects cancel or accumulate?
predicted amplitude of solar potential effect 12
ns (3.6m)
To Sun
13
Alley experiment Clocks flown over south pole to
New Zealand South pole in winter was always
farther from Sun than Earth center
No effect of Suns potential was seen.
14
Solar clock-slowing computation

• Suns potential at Earths surface is 14 x
  stronger than Earths potential.
• Mathematica applied to GR metric equation to
  compute satellite proper time in Suns field for
  any orbit.
• Lorentz transformation applied to convert from
  SSB coordinate time to ECI coordinate time.
• Result potential and velocity effects from Sun
  on Earth-satellite cancel!

15
A
Explanation As Earth orbits Sun (center path),
aphelion satellite positions must travel at
higher speed to get from A to B, and perihelion
satellite positions must travel at slower speed
to get from C to D.
C
D
B
This centripetal potential is speed difference
added by Suns presence.
16
Summary of clock-slowing effects
?Motion free ?Motion forced
17
Conclusions

• Solar potential effect does not exist in GPS
  data motion is forced
• Unexplained 12-hour periods correlated with Sun
  direction must have some other explanation
• No unresolved relativity issues remain at the
  1-meter level for GPS