Global Positioning System

In space are three satellites, A, B and C. Each satellite broadcasts its position to receivers D and E. These then compute the distances AB, BC and CA and the base of a three-sided pyramid is known by both D and E. The satellites also transmits their own time, and the distances AD, BD and CD are the emission speed of the signal relative to the satellite, 299,792.4562 kilometers per second, plus the velocity of the satellite in the direction of the receiver (negligible) multiplied by the time it takes for the signal, Dt. The value of Dt is simply t_D-t_Awhere t_D is the time of arrival and t_A the time of emission. The time t_D is not known exactly because the receiver does not have an atomic clock, but can be and is computed. This computation causes the following effect:

t_D is synchronized to t_A, t_B and t_C. Any error e in t_A is also in t_D. Thus the distance AD is found by:

where c = 299,792.4562 kilometers per 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom, also known as a second.

Each receiver manufacturer will have his own algorithm for the computation, with a better approximation using a fourth satellite.

For example, there will be a point in space which is within the triangle ABC, and the time t =t_A=t_B=t_C can be computed from a planar triangle (t being the same value anywhere by Sir Isaac Newton's proclamation, the distance along the white lines being computable with the angles between them being 120^o

All GPS calculations are carried out by the receiver and will vary according to the manufacturer. Create your own if you wish.

Regardless of the method employed in synchronizing it is nevertheless merely a matter of offset, setting all clocks to some nominal value such as 12:00:00.0000000 simultaneously, wherever they may be. If one hour later or 25 hours later one clock should display 1:00:00 and another 1:01:00 then they no longer synchronise. We would say the second clock is running faster than the first or "gaining", or the first is running slower than the second, or "losing". Which is master we cannot say without reference to some other standard.

The confusion that specimens of homo neanderthalensis have is one between gain and offset.

It is gain that is the subject of Einstein's relativity, not offset, they are entirely different issues and clearly gain doesn't exist within GPS, all satellites remain synchronized with each other and with clocks on the ground within the limits of engineering tolerances.

Suppose I have a GPS receiver and I'm located at

40°26'29.03"N, 79°59'41.41"W, which happens to be the USX (formerly US Steel) building and the tallest building in Pittsburgh, which anyone can visit. There is a restaurant at the top. My GPS receiver isn't perfect and it says I'm located at 40°26'30"N, 79°59'0"W, an error of 0.97" N, 1.41" W, which is 2948 feet away in the Monongahela River. You happen to be somewhere in the vicinity, but your GPS receiver has the same error. Because I know exactly where I am I can contact you by ordinary cell phone and tell you to adjust your position by adding 0.97" N, 1.41" W which will tell you exactly where you are. Now of course this can be fully automated, there is no requirement for human intervention. All that is needed is a continual transmission of the offset (which will change as the satellites move and from ionosphere effect) from the USX building to your GPS receiver over the normal phone system. I have deliberately exaggerated the offset for this example of the simplicity of DGPS.

Note to Einstein worshippers: Any artificial offset k in t_A(= t_B=t_C) is also an offset in t_D+k and is irrelevant to the triangulation of the receiver's position, it is Dt = (t_D+k) - (t_A+k) that is needed. Hence any artefact introduced by a "GR correction" is automatically eliminated, although it will cause the receiver to wrongly show satellite time and not local ground time. This is of little consequence, 38 ms fast or slow matters not one iota to the user. Corrections to satellite time are regularly made from ground stations to keep them synchronized with each other and UTC. Arguments that GPS requires General Relativity are simply stupid.

Every satellite orbit is perturbed by the presence of the Moon and Sun, the two bodies responsible for Earth's tides.

It is a strange fact, always overlooked by the brainless relativist, that satellites have no initial idea where they are and hence cannot inform a receiver until they find out where they are.

Anyone with common sense would realize that a GPS satellite could (and does) find it's position from precisely located ground stations specifically designed to inform the satellite where it is at any time. To extrapolate to the next position prior to transmission a strange device called a "computer" is carried by each satellite, and the computer has a "clock" which works by "counting" ticks from an oscillator. The basic oscillator is a caesium atom (Satellites carry caesium clocks) which emits seconds for 9,192,631,770 "ticks". See: http://physics.nist.gov/cuu/Units/second.html

Ticks are used by the computer to perform "digital" calculations.

Hence GR advocates are saying gravity causes a caesium atom to emit fewer ticks (=smaller seconds) at sea level than a caesium atom at higher altitude.

The Master Control facility of the GPS system is located at Schriever Air Force Base (formerly Falcon AFB) in Colorado,
70 miles from downtown Mile High City (formerly and informally known as Denver), where caesium atoms are given special military training in how to breathe in phase with caesium atoms at sea level.

To do this a pipe

is connected from Mile High City to the Johnson Space Centre in Houston Texas, 833.43 miles away and one mile lower. The atom then breathes in and out down the pipe whilst an instructor atom at the other end of the pipe breathes out and in. In this way the pressure in the pipe has to remain constant. Atoms which fail this training are not selected for space flight. The next stage of their training is to learn the mathematics of General Relativity and to breathe at a rate more suitable in airless conditions at altitude.
Only when this training is completed does the caesium atom go to the employment office in Houston (they slide down the pipe) for a full medical check-up and astroatom training in a satellite simulator. After that they are shipped to Cape Canaveral-by-sea and launched, piloting their own satellite.
Because the British United Long-Lived Caesium Atomic Clock Association (BULLCACA) is asking for biohazard money where one atom has to breathe the expelled air of another, the astroatoms are "working to rule" (a funny British term for go-slow) and are breathing at the standard rate of 9,192,631,770 breaths per second. Some of them failed their GRE (General Relativity Education) tests anyway, caesium atoms are only slightly more intelligent than relativists.

The distance to satellite i from the receiver is

Ri = c (Ti - t)

Where

** Ri = This distance
** Ti = Time reading from this satellite
** t = Time reading from the receiver plus offset

We really don't care about the receiver time t because with the
fourth satellite it can be calculated. However, all satellites should
have their time synchronized with each other. The distance from the
satellite to the receiver is also the following.

Ri^2 = (x - Xi - Vxi (Ti - t))^2 + (y - Yi - Vxi (Ti - t))^2 +
(z - Zi - Vxi (Ti - t))^2

Where

** x, y, z = Coordinate of the receiver
** Vxi, Vyi, Vzi = Velocity of the satellite at time = Ti

The velocity of the satellite can be embedded in the almanac
information sent down from the satellite.

So, we have

c^2 (Ti - t)^2 = (x - Xi - Vxi (Ti - t))^2 + (y - Yi - Vxi (Ti
- t))^2 + (z - Zi - Vxi (Ti - t))^2

Since these satellites are very close to ground, we have

** abs(x - Xi) >> Vxi (Ti - t)
** abs(y - Yi) >> Vyi (Ti - t)
** abs(z - Zi) >> Vzi (Ti - t)

With the above simplification, we have a cheaper algorithm as follows.

c^2 (Ti - t)^2 = (x - Xi)^2 + (y - Yi)^2 + (z - Zi)^2

Collecting the information from four satellites at the same time, we
have the following four equations.

** c^2 (T1 - t)^2 = (x - X1)^2 + (y - Y1)^2 + (z - Z1)^2
** c^2 (T2 - t)^2 = (x - X2)^2 + (y - Y2)^2 + (z - Z2)^2
** c^2 (T3 - t)^2 = (x - X3)^2 + (y - Y3)^2 + (z - Z3)^2
** c^2 (T4 - t)^2 = (x - X4)^2 + (y - Y4)^2 + (z - Z4)^2

Or you can use polar coordinate. Anyhow, solve for the four unknowns
(t, x, y, z). You get your position. Again, the time of all
satellites have to be synchronized.

** No GR
** No SR
** Sagnac lack of effect

Androcles is correct after all. This should make [Van de] moortel extremely
happy.

It's wrong at a very fundamental level -- it requires a minimum of FOUR
GPS satellites to determine one's position AND TIME. The GPS receivers
have no clock (or at least no clock that is usefully accurate).

Humpty carries four wristwatches, but doesn't understand triangulation.

Also the velocity of the satellite in the direction of the receiver is
not "negligible".

And an offset in time of all satellites would destroy differential GPS,
which is used in the most critical applications (e.g. busy harbors).

Humpty doesn't understand Dt.

And the website omits several other important corrections, among the
most prominent is the Sagnac effect.

On Tue, 02 Jan 2007 20:01:13 GMT, Humpty Roberts wrote:

"the basic equations of SR are only APPROXIMATELY valid."

and he should know, he wrote a lot in the FAQ's.

The basic equations of NM are what the basic equations of SR approximate to.

"Tom Roberts" <tjroberts137@sbcglobal.net> wrote in message
news:WpbYg.14819$6S3.6439@newssvr25.news.prodigy.net...
| I see there are two different systems that have been known as
| "differential GPS".
|
| What I described as differential GPS is an earthbound clock and
| transmitter that belong to the GPS constellation and behave just like a
| satellite with an orbital period of 1 sidereal day and an altitude of a
| few meters.
| It is useful only to receivers within line-of-sight of its
| antenna.
| This was described in a paper I heard at MILCOM '88, and I was
| struck by the simplicity of giving an earthbound satellite such orbital
| parameters. Because it broadcasts on the GPS satellite frequencies, no
| modification to GPS receivers is required.
|
| [It is possible this had some other name, and my memory morphed
| it into "differential GPS". Unfortunately my personal copy of
| the proceedings of the conference was destroyed long ago, so I
| cannot check this.]
|
| That is obviously an expensive and complicated method, because the
| ground system must have several atomic clocks, must be part of the GPS
| constellation, must be managed by the GPS control segment, etc. The
| latter is difficult as the control segment must have a local receiver to
| monitor the ground system. I now suspect this was never actually
| implemented, except possibly for some early studies of continental
drift....
|
| The current implementation of differential GPS involves a receiver at a
| known location, and a second communication channel completely separate
| from the GPS radio channel (usually another radio channel, but some now
| use the Internet(!)). This is a MUCH cheaper and less complicated ground
| system, but is more complicated to implement in the receiver; with
| modern electronics it is not outrageously so.
|
| Sorry for the confusion.

If the satellites gained an hour a day the receiver would gain it also. GR "correction" is hocus pocus.

If you want to know the time ask a policeman
The proper rhythmic time, ask a policeman