Natural Science Forum / Physics / General Physics / July 2008

http://www.aticourses.com/global_positioning_system.htm

RELATIVITY

The precision of GPS measurements is so great that it requires the
application of Albert Einstein’s special and general theories of
relativity for the reduction of its measurements. Professor Carroll
Alley of the University of Maryland once articulated the significance
of this fact at a scientific conference devoted to time measurement in
1979. He said, “I think it is appropriate ... to realize that the
first practical application of Einstein’s ideas in actual engineering
situations are with us in the fact that clocks are now so stable that
one must take these small effects into account in a variety of systems
that are now undergoing development or are actually in use in
comparing time worldwide. It is no longer a matter of scientific
interest and scientific application, but it has moved into the realm
of engineering necessity.”

According to relativity theory, a moving clock appears to run slow
with respect to a similar clock that is at rest. This effect is called
“time dilation.” In addition, a clock in a weaker gravitational
potential appears to run fast in comparison to one that is in a
stronger gravitational potential. This gravitational effect is known
in general as the “red shift” (only in this case it is actually a
“blue shift”).

GPS satellites revolve around the earth with a velocity of 3.874 km/s
at an altitude of 20,184 km. Thus on account of the its velocity, a
satellite clock appears to run slow by 7 microseconds per day when
compared to a clock on the earth’s surface. But on account of the
difference in gravitational potential, the satellite clock appears to
run fast by 45 microseconds per day. The net effect is that the clock
appears to run fast by 38 microseconds per day. This is an enormous
rate difference for an atomic clock with a precision of a few
nanoseconds. Thus to compensate for this large secular rate, the
clocks are given a rate offset prior to satellite launch of - 4.465
parts in 1010 from their nominal frequency of 10.23 MHz so that on
average they appear to run at the same rate as a clock on the ground.
The actual frequency of the satellite clocks before launch is thus
10.22999999543 MHz.

Although the GPS satellite orbits are nominally circular, there is
always some residual eccentricity. The eccentricity causes the orbit
to be slightly elliptical, and the velocity and altitude vary over one
revolution. Thus, although the principal velocity and gravitational
effects have been compensated by a rate offset, there remains a slight
residual variation that is proportional to the eccentricity. For
example, with an orbital eccentricity of 0.02 there is a relativistic
sinusoidal variation in the apparent clock time having an amplitude of
46 nanoseconds. This correction must be calculated and taken into
account in the GPS receiver.

The displacement of a receiver on the surface of the earth due to the
earth’s rotation in inertial space during the time of flight of the
signal must also be taken into account. This is a third relativistic
effect that is due to the universality of the speed of light. The
maximum correction occurs when the receiver is on the equator and the
satellite is on the horizon. The time of flight of a GPS signal from
the satellite to a receiver on the earth is then 86 milliseconds and
the correction to the range measurement resulting from the receiver
displacement is 133 nanoseconds. An analogous correction must be
applied by a receiver on a moving platform, such as an aircraft or
another satellite. This effect, as interpreted by an observer in the
rotating frame of reference of the earth, is called the Sagnac effect.
It is also the basis for a laser ring gyro in an inertial navigation
system.