GPS and relativity

According to an article in the November 2004 edition of
Scientific American, the GPS satellites' clocks have a
7 microsec per day effect due to relativistic motion
and a 45 microsec per day effect due to the lower
gravity in orbit. The two effects partially cancel, so
the net adjustment is 38 microseconds.

It is left as an exercise for the student to determine
the effect of an uncompensated time value on position
values.

Well, I'm doing really well if I can keep the ILS needle off the end of
the scale...

(Everett M. Greene) wrote in
:

According to an article in the November 2004 edition of
Scientific American, the GPS satellites' clocks have a
7 microsec per day effect due to relativistic motion
and a 45 microsec per day effect due to the lower
gravity in orbit. The two effects partially cancel, so
the net adjustment is 38 microseconds.

It is left as an exercise for the student to determine
the effect of an uncompensated time value on position
values.

It's many miles. Nanosecond accuracy is required for the current ~10
meter accuracy. Linear extrapolation would lead to ~10 kilometers.

--
Regards,

Stan

"They that can give up essential liberty to obtain a little temporary
safety deserve neither liberty nor safety." B. Franklin

It's many miles. Nanosecond accuracy is required for the current ~10
meter accuracy. Linear extrapolation would lead to ~10 kilometers.

The speed of light is about 30 cm per nanosecond (a foot, give or take
a bit) so 10 meters calls for a resolution of about 30 nsec. Not
really a big deal these days (I design systems that must resolve to
better than 0.5 nsec). But the extrapolation to 10 km (which is indeed
how far light travels in 38 microseconds) doesn't work.

GPS receivers don't rely on an internal, independent clock. They
synchronize to the satellite - which is a sloppy way of putting it
anyway. The real issue is the difference between travel times of
signals from different satellites, not the absolute travel time. Thus
what matters here (to a first approximation, anyway) is that the
satellites are synchronized to each other, not to any earthbound clock.
To a second approximation, it is important that the almanac be right.
In other words, the satellite needs to be where it is expected to be at
the time it transmits. However, now the errors measured in
microseconds are much smaller - the key parameter is not how far light
travels in those microseconds (about 10 km as you noted), but how far
the satellite travels in those microseconds (more properly measured in
centimeters rather than kilometers).

Of course the error, if not corrected, is cumulative. After a few
weeks it would be quite significant.

Michael

Stan Gosnell wrote:

(Everett M. Greene) wrote in
:

According to an article in the November 2004 edition of
Scientific American, the GPS satellites' clocks have a
7 microsec per day effect due to relativistic motion
and a 45 microsec per day effect due to the lower
gravity in orbit. The two effects partially cancel, so
the net adjustment is 38 microseconds.

It is left as an exercise for the student to determine
the effect of an uncompensated time value on position
values.

It's many miles. Nanosecond accuracy is required for the current ~10
meter accuracy. Linear extrapolation would lead to ~10 kilometers.


Is it as great as that? The relativistic effects would be the same for all
satellites, so, while a clock on Earth may disagree with a clock on the
satellite, all satellites would disagree by the same amount. Therefore,
while the uncompensated effect may well be several kilometres, wouldn't it
always be the _same_ kilometres?

--
Nick

Nick wrote in
:

Is it as great as that? The relativistic effects would be the same
for all satellites, so, while a clock on Earth may disagree with a
clock on the satellite, all satellites would disagree by the same
amount. Therefore, while the uncompensated effect may well be several
kilometres, wouldn't it always be the _same_ kilometres?

No. They originally tried it without any corrections, because many of
the design engineers didn't think it would matter. Turned out it did
matter. The theoretical error is large, although I can't remember the
exact numbers. BTW, it was September 2004, not November.

The GPS receiver doesn't really know the time, it just synchronizes with
the time reported by the satellites. If it thinks the time is different
than what it is, then it thinks it's in the wrong position, because it
calculates position based on the difference in time it takes the signals
to travel from different satellites. The SciAm article has a fuller
explanation, and you can also find several explanations on the net. You
can start at http://www.gpsinformation.net.

--
Regards,

Stan

"They that can give up essential liberty to obtain a little temporary
safety deserve neither liberty nor safety." B. Franklin

In article ,
Stan Gosnell wrote:
(Everett M. Greene) wrote in
:

According to an article in the November 2004 edition of
Scientific American, the GPS satellites' clocks have a
7 microsec per day effect due to relativistic motion
and a 45 microsec per day effect due to the lower
gravity in orbit. The two effects partially cancel, so
the net adjustment is 38 microseconds.

It is left as an exercise for the student to determine
the effect of an uncompensated time value on position
values.

It's many miles. Nanosecond accuracy is required for the current ~10
meter accuracy. Linear extrapolation would lead to ~10 kilometers.

An easy back-of-the-envelope way to look at this is that light travels
roughly a foot per nanosecond. 38 microseconds = 38000 nanoseconds.
Thus, in 38 microseconds, light will travel about 38000 feet, or about
7 miles.

-- Dane

Others have covered this amply already it's really not about hor far
light goes in 38us. However, I do want to plug one of my favorite
internet tools: The Google Calculator.

If you google on "38 microseconds * c in miles" it'll print out 7.079
for you. It does dimensional analysis and has a half decent library of
constants, too. By using 'in' at the end of the calculation you get to
specify what form you'd like to see your result.

-- dave j