© Copyright 1997, 2002

Chapter 4, A Tale of Two Brothers

It is the best of times, it is the worst of times, for brothers Albert and Henry, identical twin brothers, aged 20.

It is the best of times because brother Albert is going to have a inter-star voyage for the first time in human history. That star system Albert about to explore is so similar to our Solar System, there are planets around the star. Most recent research indicates that one of the planets in the star system is very suitable for life. It is even possible that Albert will be able to find some kind of highly developed intelligent creatures on it.

However, the star's distance from our Solar System is so huge, measured almost 30 light years. A light year is the distance that light travels in a year, which is

186,000miles/second×86,400seconds/day×365days/year=5,865,696,000,000 miles or

300,000km/second×86,400seconds/day×365days/year=9,460,800,000,000 kilometers

Even if Albert's spaceship can travel approaching the speed limit of the Universe, the speed of light, it is going to take several decades for the round trip and back to the earth. The theory of Relativity dictates that nothing travels faster than the speed of light.

Unfortunately, Henry can not go with his twin brother Albert since he has other important things to do here at the earth home base. They all realize that by the time Albert comes back home, Albert's clock on the spaceship will indicate that only few years past during the trip, several decades will have past for Henry here on Earth.

It is the worst of times because this means that they may never see each other again. Even if Henry can live long enough, he will be more than 80 years old, meeting with his twin brother Albert coming back from the epic space odyssey, still in his early twenties. This is going to be very hard for them emotionally.

No Kidding

Believe it or not, this is not a "Back to the Future" kind of science fiction nor movie. Remember, everything in this book is about TRUE GPS.

To Be Continued……

It is absolutely nothing to do with the human metabolism either. It is something to do with how the times run differently on the spaceship with a tremendous speed and on the earth.

Einstein's special Relativity theory states that time as well as distance is affected by relative motion. The clock on the spaceship runs slower than an identical clock here on the Earth and the distance between a star and our solar system will be shortened a great deal for the spaceship traveling at a speed comparable to the speed of light. The shrinking of the distance because of the relative motion is called the Lorentz contraction after Hendrik Lorentz (1853-1928), a Dutch physicist who formulated it.

From Albert's point of view in the spacecraft, his clock is as usual as when he is on earth. However, the distance between earth and the star system is much shorter (Lorentz contracted) than the same distance measure on earth. So, only less than few years is needed to reach the distant star system. Same for the trip back to the family reunion, the distance is again Lorentz contracted and less than few years is needed to come back to meet his brother Henry. However, for brother Henry always stays on the earth, the distance between the star system and our solar system is always 30 light years apart, no wonder that Henry sees his brother Albert's trip will take several decades for the adventure. From Henry's point of view, he can only find Albert's clock or time run much slower than his clock or time on earth.

This concept of modern physics' relative time and distance with its consequences is an absolute breathtaking feat for our perception and logic of classical Physics' absolute time and space.

There is another conclusion deducted from the theory of Relativity you may already be familiar with, is the exchangeability of energy and mass i.e. the formula

E = m c2where c is the constant speed of light

c = 300,000,000 meters/second. The value of c2will be almost

9×10 to the 16th, this means that a minute amount of mass can be converted to a huge amount of energy. This is how nuclear energy and endless sun's energy source arise.

Relativity of GPS

GPS is one of the first operational systems human created, other than the particle accelerators, that has significant relativistic effects which must be considered. One reason is that the GPS satellites may have a large speed relative to the earth. We learned that the satellites have speed relative to the earth since they are not in the geo-stationary orbit in Chapter 2. Even though GPS satellites' speed around the earth is not comparable to the speed of Albert's spaceship, it does cause a minor synchronization problem which needs to be adjusted by the system operation and the user equipment (GPS receivers). The clocks on the satellites would run a little slower than the identical clocks down on earth if it is not detuned somehow because of the relativistic effects. These effects by themselves may not be significant because they are so minor. Nevertheless, considering that this system carries precision atomic frequency standard, that measurements can be made to accuracy in the nanosecond range, and that positioning precision can be of the less-than-a-inch level, relativistic effects can be significant and need to be taken into account. These effects must be corrected by your receiver. Your receiver needs to first figure out the relative speed between a satellite and itself from the almanac, orbital data of the satellites, or Doppler measurements and then combine the relativistic correction with other clock correction factors broadcast by the satellite message data to get the TRUE GPS time needed to solve for position.

Please note that only the velocity component along the line of sight counts. If a satellite is directly ahead of the receiver, the satellite's velocity is perpendicular to the line of sight and there is no relative velocity between them at that moment. See Figure 4-1 and Figure 2-6.

Figure 4-1. The satellites’ velocity are represented by the arrows, which is perpendicular to the lines drawn from the center of the Earth since the orbits are nearly circular. The relative speeds between you and the satellites are the projection of the arrows on the Lines of Sight (LOS).SatA is moving towards you, SatB is moving away.SatC is right above you and there is no relative speed.

If a GPS system design is not considering the Relativistic effect, it may introduce an error similar to the Selective Availability SA effect, a deliberate downgrade of the accuracy by DoD for the concern of exploitation by the terrorist or hostile forces against U.S. and its allies. SA effect causes the position to drift randomly in a limited region.

Also needed to be accounted for is the rotation of the Earth during the time of transit of the GPS signal from satellite to receiver since the speed of light is a finite number. The satellite's position is computed at the time of transmission of the signal, however, the user's receiver is computing position at a slightly later time. During this transit time the earth has rotated a little and this rotation must considered also. Note that these times of transit for different satellites are not equal because the ranges or distances from each of them are not equal.

Thanks to the complicated, smart algorithms of the software program running on the microcomputer in your receiver takes care of all that again.

This gives you some feeling of how many details, no matter how small they may be, the receiver system must get into in order to compute a precise position for you.

Copyright 1997, 2002. All rights reserved.
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