#### Transcript The Effect of Solar Gravitational Potential On GPS Clocks

```The Effect of Solar
Gravitational Potential
On GPS Clocks
Tom Van Flandern & Thomas B. Bahder
Army Research Laboratory
1998 August 19
(last two slides updated 2002 March 30)
What affects clock rates?
SR: motion slows clocks by the
factor  1 v 2 
1 

-8)
2 
(typically
<
10
 2c 
GR: gravitational potential slows
clocks by the factor
 GM    
1  2   1  2  (typically  10-8)
 rc   c 
4
2
v

• N.B. 4 , 4
c c
neglected (10-16)
GPS satellite clock rates (ns/day)
Orbital speed slowing:
7,200
Gravitational potential slowing
at GPS orbital altitude:
14,400
Gravitational potential slowing
on Earth’s geoid:
60,300
Net: GPS satellites run faster
than clocks on geoid by:
38,700
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.
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.
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)
Fixed and
accelerating
clocks with
same speed
and height tick
at same rates
Suspension/free fall doesn’t affect clock rates
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!
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.
As Earth and
satellite fall
toward
perihelion, all
their clocks
slow in unison
relative to
distant pulsar
clocks. The two
slowing effects
accumulate.
But as the satellite
orbits, do the two
clock-slowing
predicted amplitude effects from
gravitational
of solar potential
effect: 12 ns (3.6m) potential and
velocity in the
Sun’s field cancel
or accumulate?
To Sun
Or if Earth is not
present but the
satellite follows
predicted amplitude the same path as
before using rocket
of solar potential
effect: 12 ns (3.6m) propulsion, do the
two solar clockslowing effects
cancel or
accumulate?
To Sun
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 Sun’s potential was seen.
Solar clock-slowing computation
Sun’s potential at Earth’s surface is
14 x stronger than Earth’s potential.
Mathematica applied to GR metric
equation to compute satellite proper
time in Sun’s 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!
A
B
D
Explanation: As Earth
orbits Sun (center path),
aphelion satellite
C
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.
This “centripetal potential” is speed
Summary of clock-slowing effects
Case
Geoid
Earth
satellite
Sun satellite
Geodesic Accumulate
(free fall)? Or Cancel?
No
Cancel
Yes
Yes
Earth sat.
Yes
in Sun field
Rocket in
No
Sun field
Motion free
Accum.
Accum.


Cancel
Cancel
Motion
forced
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
```