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What did the Beidou satellite do for you?

On June 23, 2020, a Changsan B carrier rocket successfully ignited and took off at the Xichang Satellite Launch Center.

After nearly 26 minutes of flight and attitude adjustment, the navigation satellite located in the upper instrument module was sent to the predetermined orbit, and the deployment of the Beidou global navigation satellite system constellation was successfully completed.

This is the latest Beidou-3 global network satellite. On July 29, Beidou officially announced that the satellite had completed orbit testing and officially entered the network.

What exactly does the Beidou system mean? How difficult is precise positioning to cover the world?

The technology that uses satellites to achieve navigation and positioning is called GNSS (Global Navigation Satellite System). Currently, there are mainly GPS in the United States, GLONASS in Russia, Galileo in the EU, and BeiDou in China.

It is not easy to build a satellite navigation system that can cover the world.

Since 1994, China has built the Beidou satellite navigation system in three steps.

Two satellites were launched in 2000, Beidou-1 was created to provide services to Chinese users; 14 satellites were launched in 2012, Beidou-2 was completed, and the scope of the service expanded to the Asia-Pacific region; 30 satellites were launched and networked in 2020, Completion of Beidou-3, which covers the entire world.

In fact, Beidou has launched a total of 59 satellites in 26 years, excluding dismantling, failure, and testing, 45 satellites are in orbit normally, more than the other three systems.

Furthermore, the structure of the Beidou constellation is also different.

The GPS, GLONASS and Galileo satellites are basically in the mid-circle Earth orbit with an altitude of approximately 20,000 kilometers. This undulatory path is projected onto the earth: satellites cover different areas in periods of time and multiple satellites surround the world.

In addition to the 27 satellites in the middle circular Earth orbit, Beidou orbits around the world, there are also 10 satellites in inclined geosynchronous orbits. The trajectory in figure eight can improve signal coverage in the Asia-Pacific region. There are also 8 satellites in the geostationary orbit, which can The time interval serves the Asia-Pacific region.

More satellites and types of orbits have improved Beidou’s global coverage, especially the eastern hemisphere.

These two images are the trajectories of the Beidou and GPS satellites in service.

And this is an image of the number of satellites that can be seen anywhere in the world: the darker the color, the more satellites can be seen locally. Beidou can see 14-16 satellites or more in most regions of the eastern hemisphere, and at least 6 satellites can be seen in other regions.

So how do these satellites achieve global positioning?

When you are at a certain location on earth, holding a device that can receive satellite signals, such as a mobile phone, and the satellites are flying in the sky, you want to request their x, y, and z coordinates in space. The key is to measure the satellite and the distance S between you.

First, the x1, y1, and z1 coordinates of the satellite can be calculated from the satellite’s ephemeris data, which are known.

You just need to find the diagonal length of the cube according to the Pythagorean theorem, and you can use this formula to express the distance S between the satellite and you.

Second, the electromagnetic wave signal sent by the satellite can also measure distance. Knowing the speed of propagation of electromagnetic waves, the speed of light c is almost 300,000 kilometers per second (299,792,458 m / s), multiplied by the time tB-tA that it takes for the electromagnetic wave to be sent from the satellite to mobile phone reception is distance.

The calculation of this time and distance actually depends on the range code.

Both the satellite and the mobile phone will continue to generate a range code according to the same rules at the same time, and the satellite will send the range code to the mobile phone via electromagnetic waves.

However, the sending process takes time, so when the mobile phone receives the range code sent by the satellite, it will find an offset from the self-generated range code, which is the propagation time of the electromagnetic wave.

Taking Beidou’s B1C signal as an example, the range code rate is 1,023 Mbps, and the width of a single chip is the reciprocal of the rate, which is 0.977517 μs. When the displacement is 120,000 chips, the propagation time of the electromagnetic wave is approximately 0.117 seconds, and the calculated distance is approximately 35,166 kilometers.

Knowing the distance, you can get this equation, where the three unknowns x, y, and z still can’t be solved. But as long as there are 3 satellites, 3 equations can be numbered to form a system of equations, and then their x, y, z coordinates can be calculated.

It seems easy and simple, however it is not exact to obtain such coordinates.

Because even the smallest error of any parameter in it will cause the positioning to shift considerably, such as the propagation time of electromagnetic waves, as long as the error of 0.000001 seconds, the calculated distance changes by 300 meters.

To reduce this error, we must first consider the accuracy of the clock on the satellite.

Today, the atomic clock carried by the Beidou satellite is accurate to 3 million years old and is only a second away, yet you will encounter relativity effects when operating in space.

Simply put, according to the special theory of relativity, if the satellite moves rapidly relative to the ground, the time to observe the satellite from the ground will slow down.

According to the general theory of relativity, satellites are further from the center of the earth than the ground, have a smaller absolute value of gravitational potential energy, and time will be faster than the ground.

The combined effect of the two on the satellite clock is faster than the clock on the ground, and the amount of change can be expressed with this formula.

So how can we eliminate this amount of change?

The first half of the formula can be calculated by the gravitational constant μ and the speed of light c. Assuming that the satellite is operating in a circular orbit 36,000 kilometers from the center of the Earth, the time on the satellite will be 0.00000000051 seconds faster than the ground per second.

To eliminate this part of the variation, you can reduce the frequency of the atomic clock on the satellite by a specified multiple before the satellite is launched, slowing it down.

However, the actual orbit of the satellite is actually an ellipse, and the changes caused by the relativity effect will have periodic changes, mainly in the second half of the formula.

Of course, this can also be corrected in real time by calculating parameters such as orbital eccentricity, semi-major axis, and angle outside the anchor.

But this is not enough. The atomic clock itself still has errors when it runs without maintenance. There are also errors in calculating the satellite coordinates. The speed of the electromagnetic waves affected by the ionosphere and the troposphere will change when they propagate in the atmosphere.

But don’t worry, there are several mathematical models that can calculate these errors.

However, there is another troublesome error, which is the ground receiver time error. Devices like mobile phones have errors due to various reasons, and it is difficult to calculate and correct the error directly.

Don’t panic!

We can also set this error as an unknown quantity, enter the fourth satellite and obtain four equations to obtain the four unknown quantities of the time error and the coordinates of the receiver, and then determine its position with greater precision.

Of course, Beidou’s technology to achieve precise positioning is much more than that. Today, Beidou can achieve positioning accuracy better than 10 meters in the world and better than 5 meters in the Asia-Pacific region.

And through methods such as ground-based enhancement and satellite-based enhancement based on a large number of ground reference stations, highly accurate positioning can be achieved at the decimeter level, centimeter level, and even level of a millimeter after processing.

The application of this technological force has been seen everywhere in contemporary life. From dam monitoring, power communications, precision agriculture, to buses, bike sharing, and mobile phones, you can see Beidou.

And in this process of using satellites to achieve global positioning, countless scientists and engineers must compare errors.

So when you look at the sky again, don’t forget that in addition to the Seven Stars, Beidou also has 45 satellites floating in the sky over tens of thousands of kilometers, allowing you to know where you are and where you are going.

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