Re: The physics of music.

Hey Potter....

Ref: http://math.ucr.edu/home/baez/RelWWW/wrong.html#gps

In a paper remarkable chiefly for the extraordinary number of obvious
errors it contained (see above), Tom Van Flandern, ("The speed of
gravity-- what the experiments say" Phys.Lett.A 250 (1998) 1-11, also
available here), stated:

    "the Global Positioning System (GPS) showed the remarkable fact that
all atomic clocks on board orbiting satellites moving at high speeds in
different directions could be simultaneously and continuously
synchronized with each other and with all ground clocks. No "relativity
of simultaneity" corrections, as required by SR, were needed. This too
seemed initially to falsify SR. But on further inspection, continually
changing synchronization corrections for each clock exist such that the
predictions of SR are fulfilled for any local co-moving frame. To avoid
the embarrassment of that complexity, GPS analysis is now done
exclusively in the Earth-centered inertial frame (the local gravity
field). And the pre-launch adjustment of clock rates to compensate for
relativistic effects then hides the fact that all orbiting satellite
clocks would be seen to tick slower than ground clocks if not
rate-compensated for their orbital motion, and that no reciprocity
would exist when satellites view ground clocks".

At first glance, Van Flandern here appears to be claiming that the fact
that the GPS continues to operate with great accuracy has in fact
disproven the predictions of str concerning moving clocks (Van Flandern
doesn't mention the gtr effects, but they are also significant). On
careful reading, in this paper he actually appears to be saying in
effect that anything that can be explained using str can be explained
just as well using the Lorentz ether theory (let), a theory which he
has never specified but which is usually taken to be mathematically
equivalent to str, but with a different interpretation of Lorentz
transformations, one which most physicists since Lorentz's day have
found implausible. However, more recently, in postings to
sci.physics.relativity, Van Flandern has clearly stated that he
believes that changes in electrostatic and gravititostatic potentials
are transmitted instantly (literally!), just as if electromagnetism and
gravity were truly governed by the Poisson equation, a viewpoint which
is mathematically utterly inconsistent with both str and gtr, contrary
to his claims in an earlier (and also wildly erroneous) paper,
"Possible new properties of gravity", Astrophysics and Space Science
244 (1996), also available here.

Since Van Flandern also claims special expertise in the GPS system, by
virtue of having worked as a "consultant" in its design, some
nonphysicists might take his claims seriously. However, his name does
not appear anywhere in the official GPS bibliography kept by NOAA, and
according to the article by Farrell, Van Flandern "left the U.S. Naval
Observatory under something of a cloud."

(On the other hand, Neil Ashby (Physics, University of Colorado) has
written extensively in journals such as GPS World, IEEE Spectrum, and
has written some of the official documentation for the GPS system; at
this point, some readers may want to skip directly to General
relativity in the global positioning system, a short paper by Ashby
which quickly debunks Van Flandern's claims.)

Before we can understand why, contrary to Van Flandern's assertions,
relativity theory is actually working just fine in the GPS, we need to
understand the basic principles behind its design and daily operation,
so I'll begin by explaining (in an oversimplified way) how the GPS
works, what the most important non-relativistic sources errors are, and
how they are overcome. Once this is out of the way, I'll discuss the
relativistic sources of error, and how they are overcome.

The GPS has revolutionized the transportion industry, as well as
offering unprecedented position and chronometer accuracy to field
researchers involved in biology, botany, ecology, geology, and
petroleum exploration, among others. While the system was not designed
as a test of gtr, it turns out that in addition to numerous extremely
complex Newtonian physical issues which must be taken account of, there
are also about a dozen distinct str and gtr effects which must be taken
into account in the design and operation of the system. This takes
quite a bit of explaining, since the actual system is quite complex,
and I certainly won't attempt to explain all the engineering details
here (although I'll provide links to sites where you can obtain more
detailed information).

The purpose of the GPS is, of course, to allow users with a GPS
receiver to determine his/her location on the Earth, including
altitude, latitude, longitude, and to inform the user of the precise
Universal Coordinated Time (UTC) maintained by a reference atomic clock
at the U.S. Naval Observatory in Bethesda, MD, as well as velocity and
heading (if the user is in a moving vehicle, aircraft, or vessel),
using coded signals transmitted by a constellation of 24 Earth orbiting
satellites.

A good way to start thinking about the general principle behind GPS is
as follows. Suppose you know your precise range r1 to an object S1 with
precisely known position x1 (a point in E^3, ordinary Euclidean space).
Then you know you are located somewhere on a sphere of radius r1 and
center x1. Next, suppose you also know your precise range r2 to a
second object S2, with precisely known position x2. Then you know you
are located somewhere on the circle which is the intersection of the
first sphere with the sphere of radius r2 and center x2. Now suppose
you also know your precise range r3 to a third object S3, with
precisely known position x3. If you know all three things at the same
time, then you know you are located on one of the two points in which
three circles intersect! This process is called trilateration by
geographers.

The basic idea behind GPS is to adapt this idea by providing users with
a "constellation" of satellites as "orbiting landmarks", which always
know their precise position with respect to the Earth's surface, as
well as the precise UTC at their location, and which continually
transmit this dual information at regular intervals. The current (Block
2) GPS constellation consists of (at least) 24 Earth orbiting
satellites, called SV's, in circular orbits about 11,000 nm (20,200 km)
above the Earth's surface (that is, 26,750 km above the center of the
Earth), traveling at 4 km/sec, giving an orbital period of precisely
twelve sidereal hours; that is, the satellites rotate once every twelve
hours with respect to the fixed stars, not with respect to the Earth,
which is of course itself rotating underneath the satellites. (The
actual number of satellites in orbit varies from time to time because
new ones are launched, with a Delta 2 rocket, as old ones begin to wear
out. The design life of each satellite is 7.5 years.)

This means that each satellite comes over the same location along the
same track over a fixed location on the surface of the Earth every 24
hours, or rather, four minutes earlier each day; the four minute
discrepancy is due to the Earth's advance in its own orbit around the
Sun. Incidentally, a common question is: why are the satellites not in
geostationary orbits, like many communication satellites? The answer is
that geostationary orbits are only possible over the equator, and as
we've seen, trilateration won't work unless you can measure your
distance to satellites in more than one plane. Since the designers of
GPS were also looking ahead to future space-based applications, e.g.
actively steering spacecraft in perfect formation (despite buffeting
from the solar wind) by keeping their position using GPS, arranging the
satellites so that their orbital period is precisely 12 sidereal hours
turns out to be the simplest choice.

Each satellite is visible above the horizon of a stationary user for
about five hours. The satellites each orbit in one of six equally
spaced planes (sixty degrees apart), each inclined at about 55 degrees
to the equatorial plane of the Earth, and with (at least) four SV's in
each plane. This configuration ensures that at any given at time and
any given place on the Earth, five to eight satellites are in a direct
line-of-sight from the user.

Now, if one can provide a way for a ground receiver to synchronize its
clock with the satellite clock, a simple measurement of the time delay
in the signal received from a satellite in view of the Earthbound user,
should result in a precise range to that satellite. Imagine for example
a ground receiver which finds and locks onto one visible satellite,
synchronizes its clock with the satellite clock, measures the time
delay, computes the range, stores this number along with the reported
position of the satellite, then locks onto a second visible satellite,
and repeats this process until ranges and positions from three
satellites have been obtained. This isn't how GPS works, but it's
getting close!

Before delving into a more accurate explanation of how GPS works, let's
examine some nonrelativistic sources of error in the simple procedure I
have just outlined. As I said, in principle, determining the range to
each satellite is a matter of a simple computation: the time delay of
each signal from the satellite, multiplied by the speed of light, gives
the range! However, the signals are not in fact transmitted in along
the geometric line of sight, because they are diffracted at the upper
boundary of the ionosphere (at about 1000 km above the Earth's surface)
and then again at the boundary between the ionosphere and the
troposphere (at about 70 km above the Earth's surface). These boundary
layers are move up and down, depending upon the time of day, the
latitude, and other factors. Moreover, the lower 8-13 kilometers to the
atomosphere are active participants in local weather conditions, and
the speed of light in air is different from the speed of light in vacuo
(and the temperature of the air matters!). In addition, satellites
which are closer to the horizon will transmit signals which are subject
to more atmospheric disturbance than those near the zenith. In sum,
ionospheric, tropospheric, and local weather conditions can result in
errors of ten meters or more. Furthemore, it is possible for the same
signal to arrive at the receiver at different times, having taken
multiple paths to get there; this can account for another half meter or
so of inaccuracy. All these sources of error must be taken account of
and corrected by the GPS system.

Further "Newtonian" sources of error arise in determining the precise
position of each satellite. In principle, a Keplerian orbit about the
Earth is determined by six orbital parameters, usually taken to be the
following:

    * the eccentricity of the elliptical orbit,
    * semi-major axis of the orbit,
    * the inclination of the orbital plane to the equatorial plane of the Earth,
    * the right ascension of the ascending node
    * the angular location of the perigee (point of closest approach),
    * the time at which the satellite passes the perigee

However, there are numerous influences which perturb the orbits of the
satellites in an extremely complicated way. First, the Earth is not a
perfect oblate spheroid, but has numerous bumps and valleys and
mountains, and the mountains, being closer to the satellite, "tug" a
bit more firmly. In addition, the Earth has an inhomogeneous mass
density, so that even if it were a perfect sphere, its "gravitational
tug" on the satellite would still vary from place to place in the
orbit. For this reason, spherical harmonics up to order eight are used
to model the gravitational potential at each point on the orbit. This
requires a detailed gravimetric (density inhomogeneity) and geodetic
(shape inhomogeneity) surveys of the Earth! Just to make things more
complicated, the shape of the Earth is dynamic, principally because the
lunar tides pull the surface of the Earth closer to the satellite twice
a day (and the Moon itself also tugs on the satellite more when it is
closer). Furthermore, the varying radiation pressure from sunlight
striking the satellite turns out to be a significant source of error.
This effect is extremely complex because the different parts of the
satellite have different albedos, diffusivities, etc., and of course
present different aspects to the Sun as the satellite and the Sun and
Earth move around. This radiation pressure effect turns out to be the
most difficult kind of error to eliminate in the GPS system. And of
course, orbital injection is never perfect, and none of the GPS
satellites are in perfect circular orbits, or aligned precisely as
designed. There is also slight frictional drag due to the (extremely
diffuse) atmosphere at the location of the satellite orbits, which
causes a steady decay of each orbit, but this effect is largely
negligible. (If you are curious about the details, see this file for
the names and orbital parameters as determined by NORAD of the
currently operational GPS satellites.)

Now, as I said, I have not yet described how GPS really works. There
are two fundamental issues I have not addressed:

    * How does the ground receiver synchronize its (cheap, inaccurate)
clock with the (highly accurate and expensive) atomic clocks carried
aboard a given SV?

    * How do the SV's synchronize their atomic clocks with each other,
and how is this collaborative timekeeping (called GPS time) converted
by the receiver to the correct UTC for a given location on the ground?

There is also a further component to the GPS system which I have not
yet discussed. In addition to the satellites themselves, the space
segment of the system, and the millions of hand held GPS receivers
operated by people around the world, the user segment, there are also
four unmanned ground stations (one located in Hawaii, a second on
Kwajalein, an atoll in the Pacific Ocean; a third on Diego Garcia, an
island in the Indian Ocean, and a fourth on Ascension Island in the
Atlantic Ocean), which are devoted to carefully tracking the position
of each GPS satellite, taking readings twice every three seconds. Each
station automatically uses local weather and ionospheric conditions to
average the tracking data, and every quarter hour, reports its best
estimate of the position of each visible satellite to the master ground
control station, which is located at Schriever AFB, Colorado Springs,
CO. Here, the orbits and on-board clocks of each satellite can be
adjusted if neccessary, using more sophisticated computer models (and
bigger computers!) than can be carried on the SV's. This is the third
component of GPS, the control segment.

As always, the devil is in the details, and it is the facts reported in
the previous two paragraphs which have permitted Van Flandern to make a
plausible sounding, but nonetheless completely incorrect, allegation.
Let's take a closer look at how the above two issues are met in the
design of the GPS.

Each GPS satellite (SV) has a mass of 930 kg and a length of 5.1 m, and
carries four atomic clocks (two Cesium and two Rubidium oscillitors).
These clocks are highly stable and accurate to within about 3
nanoseconds per day. They are occasionally reset from the ground, as
required, in order to maintain sufficient synchrony with GPS time (for
the moment, we can consider this to be the same as UTC time). Each
satellite transmits two types of signals:

    * The L1 signal has a carrier wave frequency of 1575.42 MHz
(wavelenth 19 cm) and is used in the Standard Positioning Service
(SPS), which is available to civilian users. Superimposed on this
carrier frequency is a coded modulation called the coarse/acquistition
code (C/A code) which is repeated over and over. Instead of thinking of
the C/A modulation in terms of time, it is useful to think of the SPS
signal as a light ray, in fact as linear yardstick marked into 300 m
intervals. IOW, the SPS user can determine his range to each visible
satellite to within 300 m.

    * The L2 signal has a carrier wave frequency of 1227.60 MHz
(wavelength 24 cm) and is used in the Precise Positioning Service
(PPS), which is available to military users. The satellites recieve
signals from ground control stations at a frequency of 1783.74 MHz.
Superimposed on this carrier frequency is a coded modulation called the
Precise code (P code), which is also repeated over and over (with a
period of about one week). The P modulation corresponds to a distance
of approximately 181,440,000,000 km! However, I lied a bit in the
previous paragraph: in fact, the P code modulation is also performed on
the L1 frequency signal, but in a clever way which ensures that it does
not interfere with SPS users. The reason for this complication is to
encrypt the military grade data in order to deny it to civilian users
(or hostile military forces). Without going into the details (which are
irrelevant to our purposes), suffice it to say that military GPS
receivers are able to decode the data transmited by each satellite, so
that for purposes of the PPS, the PPS signals can be thought of as
linear yardsticks marked at 30 m intervals. IOW, the PPS user can
determine his range to each visible satellite to within 30 m.

(This description is oversimplified; in fact, there are three
modulations superimposed on the two carrier frequencies, as shown in
this figure, taken from Peter Dana's website. Incidentally, one reason
for using two frequencies in the GPS system is that the ionospheric
disturbances are frequency dependent, so PPS users can use both
frequencies to partially compensate for this source of error, using a
mathematical model of the ionosphere. [SPS users are stuck with the
errors!] The tropospheric disturbance is partially compensated for by
estimating it using another mathematical model. However, all these
ramifications are irrelevant for our purposes.)

In addition to transmitting its on-board clock time and estimated
orbital parameters and its own estimated current location (the basis
ephemeris information), its assessment of its own "health", its best
estimate of the current location of the other GPS satellites (the basic
almanac information), each satellite must also transmit identifying
data so that the ground receivers can distinguish between signals
coming from different visible satellites, and lock onto four or more in
turn. In addition, the system is designed so that it takes only 30
seconds to obtain a first, rough position, and 630 seconds to obtain a
full precision position. Moreover, it is desirable to build in some
degree of error protection, so that if the signal is slightly
disrupted, the receiver can compensate for some lost bits (or if
neccessary wait for the next transmission).

To meet these requirements, the signals transmitted by the satellites
are neccessarily rather complex. Each signal consists of nine packets,
with an overall transmission rate of 50 bits per second (including
parity check digits). Each packet consists of about 200 Bytes and is
divided in turn into five 300 bit subpackets, with one packet
transmitted every 30 seconds. The first subpacket contains the clock
corrections; the next two contain the precise location (ephemeris data)
of the transmitting satellite; the next contains the UTC time and some
ionospheric correction data, and the last contains partial almanac data
for all 24 satellites. (The complete almanac and ionospheric data is
built up over the 7.5 minutes required to transmit nine packets, making
up one complete signal.) There is some error-detection/correction built
into the coding: the subpackets consist of 30 bit words, of which 24
bits contain data and the remaining 6 bits allow parity checks.

So, how can the ground receiver synchronize its clock with the
satellite clock, in order to carry out the basic time delay computation
outlined above? The answer is that it doesn't--- instead of
sequentially locking onto and synchronizing clocks with four different
satellites, as suggested above, the simplest (and cheapest) GPS
receivers first lock onto the signal of one satellite, i.e. compare
their internally generated C/A signal with the satellite signal until
they get their own (cheap, inaccurate) clock in "roughly in synch"
(modulo a still undetermined offset) with GPS time, and then records a
reception time (by its own clock). It then locks onto the next signal,
and repeats this process until four reception times (modulo
undetermined offsets) have been recorded. Only then does it combine
this data to determine the common offset of its clock from the
satellite clocks, in effect resynchronizing its clock with three
on-board atomic clocks. It then uses trilateralization and the declared
position of each satellite as described above to determine the its own
position. One way to think about this is that by locking onto four
satellite signals, the receiver can determine its position and one more
number, the offset of its clock from the highly stable, accurate, and
mutually synchronized clocks carried by the satellites.

The grand result is that SPS users can obtain their position accurate
to within 100 meters (latitude and longitude), their altitude to within
150 meters, and the universal time to within 350 nanoseconds. PPS users
can obtain their position accurate to within 20 meters, their altitude
to within 30 meters, and the universal time to within 200 nanoseconds.
Actually, even this is merely the tip of the iceberg--- more
sophisticated users can use two or more sequential ground receivers, or
a parallel receiver capable of locking onto several satellite signals
at once, and can use various least squares estimations to collate the
data collected in order to considerably improve on the stated
accuracies, typically to a few meters in the position accuracy.
Furthermore, stationary receivers (e.g. used by geologists to study the
motion of tectonic plates, or volcanic terrain) currently achieve
position accurracy on the order of one mm!

So far I've said quite a lot about GPS, and have yet to mention
relativity. It is when we begin to discuss how GPS time, a sort of
"imaginary time" stitched together from the onboard clocks of the 24
satellites, as correlated by the master control station, is converted
to UTC time, that we can see some explicit relativistic computations
entering into the daily operation of the GPS; see for example this
figure, also from Peter Dana's site. I won't try to explain how GPS
time is constructed, other than to say that one basic idea is to
compare time signals from a given satellite as received by two ground
stations, using a ground link between the two stations, with suitable
time delays for the transmission times. Because the sphere in which the
SV orbits reside is quite large (more than 40 million meters), the
light time travel delays (which are of course the key to making the
system work in the first place) must be very carefully compensated for
in correlating the onboard clocks.

The first 10 GPS satellites, comprising Block I, were used for testing
and for military geolocation, and were launched beginning in 1978. The
next 24 satellites, comprising Block II, were launched between 1989 and
1994; these are the SV's used in the operational GPS system. The way in
which Van Flandern's claims quoted above are misleading is now easily
summarized:

    * It is true that the current (Block II) satellites carry clocks
which are occasionally adjusted from the master ground control station.

    * It is completely false that the GPS somehow defies the
predictions of relativity theory. Indeed, when the first atomic clock
was sent into orbit in June 1977 (aboard a satellite which was a
testbed for the Block I GPS), and I quote from Ashby's paper:

	  there were some who doubted that relativistic effects were
real. A frequency synthesizer was built into the satellite clock system
so that after launch, if in fact the rate of the clock in its final
orbit was that predicted by GR, then the synthesizer could be turned on
bringing the clock to the coordinate rate necessary for operation. The
atomic clock was first operated for about 20 days to measure its clock
rate before turning on the synthesizer. The frequency measured during
that interval was +442.5 parts in 10^12 faster than clocks on the
ground; if left uncorrected this would have resulted in timing errors
of about 38,000 nanoseconds per day. The difference between predicted
and measured values of the frequency shift was only 3.97 parts in
10^12, well within the accuracy capabilities of the orbiting clock.
This then gave about a 1% validation of the combined motional and
gravitational shifts for a clock at 4.2 earth radii [the radius of the
satellite's orbit].

    * It is true that GPS is not used as a test of gtr, because it is
simply not designed for that purpose. In particular, the orbiting
clocks are occasionally reset from the ground to maintain the best
possible synchrony of the orbiting clocks with one another and with UTC
time.

    * It is completely false that the design of the GPS system ignores
relativity theory. Relativistic effects in the GPS system are vitally
important. The total difference in the rate of atomic clocks on board a
GPS satellite and the reference clock at the USNO amounts to some
38,600 nanoseconds per day. (This is mostly due to a combination of the
Sagnac effect for a clock which is moving wrt the GPS receiver, and the
relative gravitational time dilation between a stationary clock on the
Earth's surface and a stationary clock 20,200 km above the surface, as
mentioned in the above quoted paragraph from Ashby's paper; frequency
shifts in clocks on the ground wrt UTC due to inhomogeneties in the
shape of the Earth also play a role.) In contrast, in order to maintain
the accuracies listed above, the GPS system must maintain a timekeeping
synchrony within 10 nanoseconds variation per day, indefinitely! The
major way in which the 38,600 nanosecond per day discrepancy due to
relativistic effects is accounted for is by building into the GPS
software used to keep the satellite clocks in synch with each other and
to synchronize GPS time with UTC an effective downward frequency shift
of 446.47 parts per trillion in the orbiting atomic clocks. In addition
to this basic conversion factor, GPS receivers are programmed to take
account for the fact that slight eccentricities in the satellite orbits
result in tiny periodic changes in the frequency of the orbiting
clocks.

At this point, I can do no better than send readers who have not
already been there to Neil Ashby's paper for a detailed accounting of
str and gtr effects which are significant in the GPS system. For a
slightly longer version of this paper, look at this issue of Matters of
Gravity, a newsletter in the field of gravitation physics. See also
this paper by Thomas Bahder and references therein. You can also try
this paper by Charles W. Misner (Physics, University of Maryland), and
this one by Clifford Will (Physics, Washington University). You can
find additional references in the posts by Tom Roberts included in this
collection.

By the way, there is a Russian system, GLOSNASS, which has fewer
satellites but is similar to GPS. I have seen the claim that this
system does not exhibit the relativistic effects mentioned in Ashby's
paper. This is of course completely false.

.

Relevant Pages

  • Re: The Speed of Gravity =?windows-1252?Q?=96_What_the_Exp?= =?windows-1252?Q?eriments_Say?=
    ... continually changing synchronization corrections for each clock exist such that the predictions of SR are fulfilled for any local co-moving frame. ... To avoid the embarrassment of that complexity, GPS analysis is now done exclusively in the Earth-centered inertial frame. ... And the pre-launch adjustment of clock rates to compensate for relativistic effects then hides the fact that all orbiting satellite clocks would be seen to tick slower than ground clocks if not rate-compensated for their orbital motion, and that no reciprocity would exist when satellites view ground clocks. ... Now, if one can provide a way for a ground receiver to synchronize its clock with the satellite clock, a simple measurement of the time delay in the signal received from a satellite in view of the Earthbound user, should result in a precise range to that satellite. ...
    (sci.physics)
  • Re: NEW GPS with the best sensitivity of antenna!!!
    ... > I have head that under tree cover after rain GPS doesn't work. ... the geometry of the satellites being used with respect to receiver ... diffraction, ans scattering of the satellite signal by trees, utility ... Satellite clock--Errors in the transmitted clock, ...
    (sci.geo.satellite-nav)
  • Re: to sam
    ... Einstein's Relativity and Everyday Life ... But GPS is an exception. ... twice per day, much faster than clocks on the surface of the Earth, ... The net result is that time on a GPS satellite clock advances ...
    (sci.physics)
  • Re: Einstein (1905) Absurdities
    ... According to Hatch GPS is rather Lorentzian relativity ... than Einsteinian SRT. ... The main argument is, the satellite clock ...
    (sci.math)
  • Re: Einstein (1905) Absurdities
    ... According to Hatch GPS is rather Lorentzian relativity ... than Einsteinian SRT. ... The main argument is, the satellite clock ...
    (sci.physics)
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