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Gravitational Potential and Coordinate Time Increment on Earth's Oblate Geoid:
Gravitational Potential:
Grav. Calculation:
Earth radius
height ( h )
in air = 0
Grav. (1)
Quadrupole (2)
Centripetal (3)
Earth radius (pole)
degrees
min
at latitude
above sea level
in rock = 1
-GM/r
(GMa^2J/2r^3)*(3cos^2
q
-1)
(-
W
^2r^2sin^2
q/2
)
Total: (1+2+3)
F
(r): (1+2)
-GM/r^2
F
(r)/r
Total/r
latitude 1
Earth radius average
latitude 2
ratio:
inv. ratio:
Earth radius (equator)
Kinematic Calculation:
Clock Velocity E (m/s)
trip duration (hrs)
Rotv (m/s)
ECIv (m/s)
ECIv - Rotv (ns)
Earth
W
average
latitude 1
latitude 2
Earth Velocity r
W
Coordinate Time Increment at Latitude and height (h) (dt):
cos(lat)
sin(lat)
p
r2 =A
Grav.
Quadrupole
Centripetal
(-GM/r)/c^2
GMa^2J(3Cos^2
q
-1)/(2c^2r^3)
(W
*r*sin
q)
^
2
/2c^2
Total:
a*a*cos(lat)
b*b*sin(lat)
Earth Mass
a*cos(lat)
b*sin(lat)
ns in one day
Coordinate Time Increment at Latitude and sea level (dt):
cos(lat)
sin(lat)
G
Grav.
Quadrupole
Centripetal
(-GM/r)/c^2
GMa^2J(3Cos^2
q
-1)/(2c^2r^3)
(W
*r*sin
q)
^
2
/2c^2
Total:
a*a*cos(lat)
b*b*sin(lat)
GM
a*cos(lat)
b*sin(lat)
quadrupole coeff. J
Hafele and Keating Experiment Calculation (n/s):
c^2
Clock
Velocity
D
t
Grav.
Quadrupole
Centripetal
Total Grav.
D
t
Total in ns
diff. (m)
from ideal Geoid:
latitude 1
m
latitude 2
m
diff. in ns
Note: In the Hafale and Keating Simulation, the time differences shown for each moving clock (Lat.1 and Lat 2) are compared to a clock at sea level at latitude1
The sign of the velocities needs to be corrected for Latitude 2 when the ECIv exceeds the Rotv.
Gravitational potential = mgh. Thus the ratios of the radius's = the inverse ratio of the
g's at the different latitudes. This makes earth at sea level an equipotential surface.
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