# Navigation in space, how could that work

## Navigating by satellite: GPS

**Just a few years ago it was expensive additional equipment in luxury limousines, today anyone can afford it: a Global Positioning System, or GPS for short. Not only in the car, but also as a portable device on a bike or on foot, GPS reliably shows the position, accurate to about 10 meters. But how does this location system work? How does it know that I have to “please turn left at the next intersection”. And why would it lead us astray without corrections to Einstein's theory of relativity?**

The basis is the determination of the location via the exact time span that a signal needs from one place to another, a so-called transit time measurement. Everyone knows the following experiment: In a thunderstorm lightning flashes, we count "21, 22, 23", then the thunder cracks - and we are reassured. Because this lightning struck about a kilometer away. The speed of sound in air is about 330 meters per second. However, the light from the flash travels much faster, namely at the speed of light of around 300,000 kilometers per second (exact value *c* = 299,792.458 km / s) to the observer. After the three seconds of counting, the sound has traveled three times 330 meters. On the other hand, the light only needed three millionths of a second to cover this kilometer, so for us without any noticeable loss of time. By determining the transit time of a signal, you can calculate the distance between the source and receiver if you know the propagation speed of the signal.

### Structure of the GPS system

Transporting a satellite into space

This principle is now being used in the Global Positioning System (GPS). The GPS is currently available to everyone and has been fully operational since 1995. The US system consists of 24 satellites. The European Galileo system is also likely to start operating in the coming years. Each GPS satellite is labeled with a number from 1 to 24. All orbit the earth at the same height of around 20,200 kilometers on six orbits offset by 60 degrees from one another and inclined by 55 degrees from the equatorial plane.

In fact, there are currently 32 satellites in total, but some of them are "sleeping" in reserve and some are "under repair" because their average lifespan is only about ten years. Or the position of the orbit is corrected by the earth-based control stations and the transmitter is switched off in the process. For the GPS to function, it is important that a receiver is in contact with at least four satellites at the same time anywhere on earth in the sky - even if you are walking to the North Pole. From the consideration that centrifugal force and gravity are balanced in orbit, it can be calculated that the orbital time of a satellite is almost exactly twelve hours. This results in a circulation speed of 3.9 kilometers per second.

For an exact measurement of the signal transit times between the satellite and the earth's surface, every satellite in its orbit even has several atomic clocks on board, and can indicate its time to 15 places after the decimal point. A network of five monitor stations controls the satellites. The stations are distributed around the world at geographically precisely known fixed points on the ground near the equator and transmit the information to the satellites where they are currently located. They also ensure that all atomic clocks on board all satellites and in the ground stations are synchronized and tick exactly the same - so there is a constant clock comparison.

### Positioning with the GPS

Now every satellite sends electromagnetic signal waves at the speed of light on a carrier frequency used by civilians (L1 = 1575.42 megahertz (MHz)) - this corresponds to a wavelength of a good 19 centimeters. The signal with the modulation frequency of 1023 MHz carries the so-called C / A code ("Coarse / Acquisition"), which the GPS receiver receives on earth. It essentially contains three pieces of information:

GPS satellite

- Identification: "I am the satellite S
_{x}.“ - Position statement: “My current position is
*y*Degrees north and*z*Degree west. " - Signal transmission time: “According to my synchronized on-board atomic clock, I have this information exactly at the time
*t*_{S.}Posted."

To simplify matters, it is assumed that the satellite being targeted is stationary and that the receiver with its GPS device does not move either. If the receiver in the GPS is equipped with a good quartz radio clock that shows the time with absolute accuracy to about a thousandth of a second, then the absolute time information on the satellite atomic clock and the receiver clock match each other to the nearest thousandth of a second. Assume that the receiver receives the satellite signal with position and identification number 0.067 seconds after it was sent. Since the signal was transmitted at the speed of light, the distance can be calculated from the transit time, as in the lightning example, and a distance of 20,100 kilometers between the transmitter satellite and the GPS receiver is obtained.

But this result contains even more information: Since the satellite transmitter sends in all directions of the room, it forms a “signal ball” around itself. At the time when the receiver receives the signal, the receiver is somewhere on a sphere with a radius of 20,100 kilometers around the satellite, and thereby also transmits the position of the satellite.

In order to be able to determine the exact location on the earth's surface, one needs the data from at least three satellites. Again using time-of-flight measurements, a total of three radius curves are obtained, which intersect at a certain point on the earth. This is exactly the position at which the GPS receiver is currently located.

### Positioning accuracy

But how accurate is this position determination with three satellite signals? The answer to this lies in the accuracy of the receiver clock. In the interval of a thousandth of a second - that is the reliable time indication of the clock in the receiver - the signal, regardless of which satellite, has traveled exactly 300 kilometers. With this method, our location information is at best correct for this distance. If this were the state of affairs, the GPS would be completely unusable. However, modern GPS devices can determine the location with an accuracy of at least 30 meters, i.e. ± 15 meters around the position. How does this work?

The signal travels this distance of 30 meters at the speed of light in ten billionths of a second. But the quartz watch in the receiver does not achieve this accuracy. The GPS developers use an elegant solution here: the information from a fourth satellite. In order to achieve the simultaneous initialization of on-board and receiver clocks in the range of billionths of a second, the following fact is used: Due to the accuracy problems with the time specification of the receiver clock, the same error always occurs in all distance measurements to the satellites. For example, if the receiver clock goes ahead 0.002 seconds, the distance to all satellites is measured 600 kilometers (\ (s = 300,000 \, \ textrm {km / s} \ cdot 0.002 \, \ rm s \)) too large. However, if a fourth satellite is used for observation, this error can be eliminated. Because the computer of the GPS device has to solve a system of four equations with four unknowns with the values of four satellites. This system of equations has a unique solution, which leads to the synchronization of the receiver clock with the atomic clocks of the satellites and thus allows the position to be determined more precisely. The only requirement: the receiver must see at least 4 satellites at the same time.

### Relativistic corrections according to Einstein

For position information with accuracies of ± 15 meters, time measurements with accuracies in the range of ± 0.000.000.010 seconds are necessary. In this range of accuracy, relativistic corrections according to Einstein must also be taken into account. Einstein's special theory of relativity is based on his considerations and experimentally confirmed fact that the speed of light *c* is constant. Nothing is faster than light. From this basic statement follows the so-called relativistic time dilation, which says: "A moving clock goes slower than a stationary clock". This means that when determining a time interval it makes a difference whether you measure with a clock on the solid earth or in a moving system, e.g. with an atomic clock on a satellite. From Einstein's theories it follows that the clock on the satellite is slower than on earth. How much slower the satellite clock goes depends on the speed of the satellite. If the GPS satellite moves relative to the receiver at a speed of 3.9 kilometers per second, the satellite clock runs a tiny fraction of \ (0.83 \ cdot 10 ^ {- 8} \) percent too slowly.

At first glance, this effect appears to be negligibly small for the GPS. But for a fixed measurement time we get an error in the running distance from this error in the time determination. And as time goes on, this error gets bigger and bigger, it accumulates. So after only twelve hours the determined GPS position is a good kilometer away from the actual position.

### Influence of gravity

More serious, in the truest sense of the word, is the influence of gravity for the accuracy of the GPS. This is where Einstein's theory comes into play for the second time, now with the general theory of relativity, which says: “The greater the force of gravity, the slower the time passes!” Since the force of gravity in the satellite (0.56 Newton) is only about six percent of the force of gravity Earth is (9.81 Newtons), according to Einstein's theory, the clock in the satellite goes faster. This is exactly the opposite of the behavior caused by the rapid satellite movement and the time deviation according to Einstein's special theory of relativity. But the two effects do not exactly cancel each other out. The influence of gravity means that a time interval measured on earth is always smaller than in a satellite. But this effect must be precisely determined. A more precise calculation shows that, according to the general theory of relativity, the time in the satellite runs too fast by a proportion of \ (5.28 \ cdot 10 ^ {- 8} \) percent. Taken together, the two relativistic deviations result in the following overall error: the atomic clock in a GPS satellite goes a total of 0.445 billionths of a second too fast. In practice this means that after only 24 hours the GPS will miss the actual position by about 11.5 kilometers. So without considering Einstein's theories, GPS was misleading.

### Corrections according to Einstein

However, since the error is linear in time, i.e. it always increases in the same way over time, it is easy to remedy the situation. The modulation frequencies for the GPS system have been slightly corrected: They are reset from 1023 MHz to 1022,999,999,545 MHz. This corrects the relativistic errors for the satellite clock. With this simple trick, you no longer have to deal with the theory of relativity when determining your position with the GPS. Einstein would certainly regret that. On the other hand, the GPS also proves the correctness of his considerations. That, in turn, would certainly have pleased Einstein.

### Egging satellites: Further fixes for more accurate positions

Modern GPS systems are becoming even more accurate. This requires a lot of further corrections and data and expensive electronics not only in the receivers. With the help of what is known as “differential technology” (DGPS), in which a second stationary receiver is used for reference, accuracies of less than ± 1 meter can be achieved. If you want to be precise to the centimeter, further errors must be corrected.

Periodic errors result, for example, from the fact that the orbits of the satellites are not completely round but elliptical, and / or the satellite does not run straight on its orbit but "ejaculates". Disturbances in signal propagation result from scattering and refraction of the electromagnetic signal waves on the charged molecules of the ionosphere. The theory of relativity also requires further corrections, e.g. through the so-called Sagnac effect, the additional consideration of the earth's rotation when determining the transit time. The fact that the earth is not a sphere also makes small corrections. The European Galileo system, which is scheduled to start in 2010, will cover the entire globe with its 30 satellites, deliver further improvements and become more precise than GPS. A merger of the two satellite navigation systems GPS and Galileo is under discussion and would revolutionize the accuracy of location determination. Positioning to the millimeter is not a dream of the future, but can be achieved - even within buildings. A satellite navigation system can then even be used to determine whether a pool table is crooked or not.

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