The artificial satellite is rotating. How many artificial satellites orbit the earth?
Have you ever wondered how many satellites orbit the Earth?
The first artificial satellite was launched into earth orbit on October 4, 1957. Over the years of space exploration, several thousand flying objects have accumulated in near-Earth space.
Flies over our heads 16 800 artificial objects, among them 6000 satellites, the rest are considered space debris- these are accelerating blocks and debris. There are fewer actively functioning devices - about 850 .
AMSAT OSCAR-7, launched into orbit on November 15, 1974, is considered the longest-lived satellite. This small device (its weight is 28.8 kilograms) is intended for amateur radio communications. The largest object in orbit is the International Space Station (ISS). Its weight is about 450 tons.
Satellites providing communications mobile operators(“Beeline”, MTS and “Megafon”) are placed in two types of orbits: low and geostationary.
At a low altitude, 780 kilometers from Earth, there is a used mobile operators global system Iridium communications. The idea of its creation was proposed in the 1980s by Motorola. The system owes its name to the chemical element iridium: it was supposed to contain 77 devices, which is equal to the atomic number of iridium. Iridium currently has 66 satellites.
Geostationary orbit is located at an altitude of 35,786 kilometers above the equator. It is more profitable to place communication satellites on it, since you do not need to constantly point the antenna - the devices rotate with the Earth and are always located above one point. The geostationary station has 178 satellites. The largest group in Russia belongs to the Federal State Unitary Enterprise "Space Communications": 9 satellites of the "Express" series provide television and radio broadcasting, mobile, as well as government and presidential communications, and the Internet. Meteorological and observation satellites are also located in geostationary orbit. Meteorological satellites record changes in the atmosphere, “observers” determine the degree of ripening of grain, the degree of drought, etc.
A body taken outside the Earth's atmosphere is affected, like any celestial body, only by gravitational forces from the Earth, the Sun and other celestial bodies. Depending on the initial speed imparted to the body when it takes off from the surface of the Earth, the further fate of the body can be different: at a low initial speed, the body falls back to the Earth; at higher speeds, the body can turn into an artificial satellite and begin to rotate around the Earth, like its natural satellite- Moon; at an even higher speed, the body can move so far from the Earth that the force of gravity will have virtually no effect on its movement and it will turn into an artificial planet, that is, it will begin to revolve around the Sun; finally, at an even higher speed, the body can leave the solar system forever into outer space.
We will consider only the case when a body turns into an artificial Earth satellite. When studying its motion relative to the Earth, we will take into account only the force of its attraction by the Earth. We will see that a body can become a satellite of the Earth only if its speed lies within relatively narrow limits: from 7.91 to 11.19 km/s. At a speed less than 7.91 km/s, the body will fall back to Earth; at a speed greater than 11.19 km/s, the body will leave the Earth irrevocably.
For start artificial satellites they use special rockets that lift the satellite to a given height and accelerate it to the required speed; After this, the satellite is separated from the launch vehicle and continues its movement under the influence of only gravitational forces. Rocket engines must do work against the forces of gravity and against the forces of air resistance, and also impart greater speed to the satellite. To do this, rocket engines must develop enormous power (millions of kilowatts).
If the distance from the satellite to the surface of the Earth changes insignificantly compared to the distance to the center of the Earth, then the force of attraction of the satellite by the Earth can (for rough calculations) be considered constant in magnitude, as we did in § 113 when studying the flight of a body thrown at an angle to the horizon . But the direction of gravity can no longer be considered constant, as for short trajectories of bullets and shells; Now we must take into account that the force of gravity is directed at any point along the radius towards the center of the Earth.
We will consider only the movement of artificial satellites in circular orbits. The force of gravity of the Earth creates a centripetal acceleration of the satellite equal to , where is the radius of the orbit, and is the currently unknown speed of the satellite. Let us assume that the orbit passes near the surface of the Earth, so that it is practically equal to the radius of the Earth. Then, if we neglect atmospheric resistance, the satellite will move with acceleration directed towards the center of the Earth. Hence,
Radius of the Earth. From here we find that the speed of a satellite describing a circular orbit near the Earth’s surface should be equal to
Substituting and , we find
This speed is called first escape velocity. Moving at such a speed, the satellite would circle the Earth in .
A satellite orbiting the Earth near the Earth's surface has an acceleration directed toward the center of the Earth, i.e., the same gravitational acceleration as a body freely flying along a parabolic trajectory or falling vertically near the Earth's surface. This means that the motion of the satellite is simply a free fall, similar to the motion of bullets and shells or ballistic missiles. The only difference is that the speed of the satellite is so high that the radius of curvature of its trajectory is equal to the radius of the Earth: the fall (i.e., movement with acceleration directed towards the center of the Earth) is reduced to circling the globe.
Rice. 203. Drawing from the works of Newton: the trajectories of a body thrown from the top of a high mountain with different horizontal speeds. Newton also understood that in order to launch a body into orbit around the Earth, the body must have a sufficiently high speed. - points at which trajectories end as speed increases
From formula (125.1) it is clear that if the speed of the body is less than the first cosmic speed, then the force of gravity will force it to move along a trajectory with a radius of curvature less than the radius of the Earth. This means that at this speed the body will fall to the ground. At higher speeds, the radius of curvature of the trajectory will be larger and the body will describe an elliptical trajectory (Fig. 203).
In reality, a satellite cannot be launched into a radius orbit due to the enormous air resistance near the Earth's surface. Let us find what should be the speed of movement in a circular orbit of any radius greater than . To do this, we will use a formula similar to (125.2), taking into account that the acceleration of gravity decreases with distance from the center of the Earth in a ratio inverse to the ratio of the squares of the distances from the center. We will find the acceleration at a distance from the center of the Earth using the formula. The speed of the satellite in a circular orbit of radius is obtained from the equality
,
(125.3)
Thus, as the orbital radius increases, the speed of the artificial satellite decreases.
This does not mean, however, that in order to launch a satellite into an orbit of a larger radius, the rocket engines must do less work. Only the fraction of work required to impart kinetic energy to the satellite decreases. But at the same time, the satellite must be raised to a greater height above the Earth; This means that it will be necessary to do a lot of work against the force of gravity, i.e., to impart greater potential energy to the satellite. As a result, it turns out that as the orbital radius increases, the total work required to launch a satellite increases.
In fact, let us calculate how, depending on the radius of the orbit, the work required to lift the satellite from the earth's surface to orbit and to impart to it the speed necessary to move in orbit varies. According to formula (125.3), the kinetic energy of a satellite of mass , moving in an orbit of radius , is equal to
,
where is the first escape velocity. Substituting instead the value determined by formula (125.2), the expression for kinetic energy can be given the form
Let us consider the flight of a satellite of mass in an orbit of radius and in an orbit of radius, where is a positive increment of the radius, much less than the radius itself. According to (125.4), the kinetic energy of the satellite when flying in these orbits is, respectively,
.
where is the increment in the kinetic energy of the satellite during the transition from the first orbit to the second. This increment is equal to
In accordance with the fact that when moving from the first orbit to the second, the speed of the satellite decreases, it turned out to be negative.
On the other hand, the work against gravity when moving from the first orbit to the second is equal to the force of gravity acting on the satellite multiplied by . Since it is small, the change in gravity during the transition can be neglected and considered equal to . Consequently, the work against gravity during the transition from the first orbit to the second
This work is spent on increasing the potential energy of the satellite during the transition from the first orbit to the second. Thus,
. (125.6)
Comparison of expressions (125.5) and (125.6) shows that the increase in potential energy is twice as large as the decrease in the kinetic energy of the satellite:
Let us imagine the transition of a satellite from an orbit of radius to an orbit of radius , which is very different from as a series of successive transitions, with each of which the orbital radius increases by a small amount. At each such transition, relation (125.7) is satisfied. Consequently, this relationship also holds when moving from an orbit of radius to an orbit of radius:
(see formula (125.4)). The resulting equality will be satisfied if we put the distance from the center of the Earth equal
, (125.8)
where is an arbitrary constant. Let us recall that potential energy can always be determined up to an arbitrary additive constant, the value of which depends on the choice of the position of the body in which its potential energy is assumed to be zero.
In this case, at . At any finite distance from the center of the Earth, potential energy is negative. Expression (125.9) can be given a different form by replacing, according to (124.3), with:
. (125.10)
We have obtained expression (125.8) for a satellite moving in an orbit of radius . However, it does not contain velocity and, therefore, is valid for any body of mass, regardless of whether this body is moving or at rest.
If we take it equal to zero when the body is on the Earth’s surface (i.e.), then , and the expression for potential energy takes the form
.
Let , where be a very small value compared to. Then expression (125.11) is simplified as follows:
.
We have arrived at a well-known expression for the potential energy of a body raised above the Earth to a height of .
Let us recall that potential energy determines the work done by gravitational forces on a body when it moves from a position with energy to a position in which the potential energy is zero. Consequently, expression (125.11) determines the work done by gravitational forces when moving from a point located at a distance from the center of the Earth to a point on the Earth’s surface. From formula (125.11) it follows that when a body of mass moves from infinity to the surface of the Earth, gravitational forces perform work on the body equal to . Accordingly, the work that needs to be done against the forces of gravity in order to remove a body from the surface of the Earth to infinity is also equal to . This work is finite, despite the fact that the path along which it is accomplished is infinitely large. This is explained by the fact that gravitational forces decrease rapidly with increasing distance from the Earth - inversely proportional to the square of the distance.
Using expressions for kinetic and potential energies, one can determine the work that needs to be done to launch a satellite of mass into an orbit of radius . Before launch, the satellite's total energy (kinetic plus potential) is zero. Moving in orbit, the satellite has kinetic energy, determined by expression (125.4), and potential energy, determined by expression (125.11). The work we are interested in is equal to the total energy of the satellite moving in orbit:
This expression does not take into account the work that must be done when launching a satellite against atmospheric drag forces. From (125.12) it is clear that as the orbital radius increases, the work that needs to be spent to put the satellite into orbit increases.
Work (125.13) is performed due to the reserve of kinetic energy, which is imparted to the satellite during launch. Minimum speed, from which the satellite must be launched so that it moves away to infinity, is determined by the condition
This speed is called the second escape velocity. Comparison with (125.2) shows that the second escape velocity is times greater than the first:
When a body is launched at a speed greater than the second escape velocity, it will also not return to Earth, but in this case, as the body moves away from the Earth, its speed will not tend to zero.
125.1. At what speed must a body be thrown vertically upward so that it reaches a height above the Earth's surface equal to the radius of the Earth? When calculating, neglect air resistance, but take into account the change in gravity.
125.2. At what distance from the center of the Earth will the orbital period of an artificial satellite be equal to 24 hours, so that the satellite can occupy a constant position relative to the rotating Earth (“synchronous satellites”)?
Before we begin discussing the Baron’s project, let’s remember what the first and second escape velocities are. In order to anticipate in advance all possible bewilderments of students, we will present our reasoning in the form of an imaginary dialogue between the Author and the Reader.
Reader: I think no. In the end, everything that falls will fall.
Reader: Satellites? But they rotate around the Earth, and do not fall on it!
Reader: Then I don’t understand how satellites manage to stay in orbit...
Instead of the tension force of the thread, the satellite is acted upon by the force of gravity (Fig. 5.2).
To make it clearer, let's conduct the following thought experiment. Let's go up to Very a high tower - about a hundred kilometers high (at this height the force of air resistance is practically absent) - and we will throw pebbles from the tower, as shown in Fig. 5.3. The faster we throw a pebble, the farther it will fall from the base of the tower. Finally, at a certain speed, it will not fall to the ground at all, but will return to us from the opposite side.
Caution is required here: given that the speed of such a pebble should be 10 times more speed artillery shell, then the consequences may be... you understand. And the speed of such a pebble is called the first cosmic speed. Let's formulate this more clearly.
The first cosmic velocity is the speed that must be imparted to a body so that it becomes a satellite of the Earth and moves in a circular orbit at a low altitude compared to the radius of the Earth.
Let's immediately calculate the first escape velocity X I. Since the body is at a low altitude h R, then the acceleration of gravity will be considered equal to g= 9.8 m/s 2. The only force that acts on a body moving in a circular orbit around the Earth is gravity. It is she who imparts centripetal acceleration to the body, where R- radius of the Earth.
According to Newton's second law:
Let's substitute the numerical values ( R= 6.400·10 6 m, g= 9.8 m/s 2), we get:
Let's remember: first escape velocity X I = 7.9 km/s.
Note that using formula (5.1) we can calculate the first escape velocity not only for the Earth, but also for any other planet.
Reader: And if the pebble in Fig. 5.3 report speed X> 7.9 km/s?
At a speed greater than the first cosmic speed, the trajectory of the pebble (or space station) will turn from a circle into an ellipse, which will become more and more elongated as the speed increases (Fig. 5.4). Finally, at speed X= 11.2 km/s, which is called second space, the trajectory of the body from an ellipse will turn into a parabola and the body will forever leave the limits of gravity.
The idea of a high-speed satellite
Now about the idea of the baron. The speed with which its satellite rotates around the Earth - 30 km/s - is significantly greater than the first cosmic speed, which, as we found out, is only 7.9 km/s! But the baron's companion, as can be seen from the picture on the poster, has engine, which ejects a jet stream in the direction from the center of the orbit! This engine creates an additional force, which now, together with the force of gravity, imparts centripetal acceleration to the satellite. In other words, the centripetal force increased by the amount of traction force jet engine, and, therefore, the centripetal acceleration also increased. Now Newton's second law for the satellite will look like:
Where f- Reactive force, X- satellite speed, R- orbital radius, m is the mass of the satellite, and g- free fall acceleration (Fig. 5.5).
From formula (5.2) it is clear that by increasing the reactive force f, we can increase the satellite rotation speed X. In theory no one is stopping us from making the reactive force as large as we like, which means the satellite’s rotation speed can be in theory increase indefinitely up to the speed of light. The problems begin where we move from theory to practice.
First, let's answer the Professor's objection. He fears that since the speed of the satellite exceeds not only the first, but also the second cosmic speed, our satellite will move away from the Earth to an infinite distance. The professor simply forgot that this is only true for celestial body- that is, a satellite that does not have any engines. Having an engine changes everything fundamentally. With an engine you can fly away from Earth with any, even at a very low speed (if you don’t mind the fuel), but you can don't fly far from her, moving very fast!
So we do not accept the Professor’s objection.
Now let's look at the Engineer's objection: why doesn't the speed increase if the engine is running? That is, why is it not increasing? speed, if the satellite is affected by force?
A counter-question is appropriate here: why doesn’t the speed of the satellite, which moves around the Earth in a circular orbit at the first escape velocity, increase (see Fig. 5.2)? After all, the force of gravity also acts on it. Why doesn’t the speed of the ball that we spin on a rope increase (see Fig. 5.1)? After all, the tension force of the thread also acts on it!
The fact is that all these forces are directed perpendicular to the direction of speed, so they do not make mechanical work: the angle that each of these forces makes with the small displacement vector is equal to 90°, so the work done by all these forces is equal to: A = F·Δ s cos90° = 0. And all these forces are “engaged” not in increasing the magnitude of the body’s velocity, but in changing directions speed.
One may ask: what is it spent on then? energy fuel, it can’t disappear, can it? Alas, it is spent quite wastefully - on increasing the internal energy of fuel combustion products.
The most unpleasant question for the baron was asked by the Businessman: “How much fuel will you need?”
Let's not upset the baron: Very a lot, it’s better not to even count so as not to get upset. The engine must operate at full power all the time, but fuel still needs to be delivered into orbit! True, the baron did not say anything about the design of his engine. Maybe he has already learned to draw energy “from the physical vacuum,” as some modern inventors suggest? Then it's a different matter!
We'd better make a different assessment. Let's calculate what overload the baron will experience if he finds himself inside his own satellite. That is, let’s calculate how many times the baron’s weight in the satellite will be greater than his weight on Earth.
Note that there is no weightlessness in the Baron's satellite - which, of course, is good if the weight is not too large, but very bad if the weight becomes too large!
So, let our baron have a mass of 100 kg and move in his satellite in an orbit with a radius of 6400 km, that is, in low-Earth orbit. Then the acceleration of free fall is equal to g= 9.8 m/s 2 (Fig. 5.6). Satellite speed v=30 km/s
The baron is acted upon by two forces: the reaction force from the floor and the force of gravity. Let's write Newton's second law in projection onto the direction of the normal:
It is clear that, according to Newton’s third law, the baron will press on the floor with exactly the same force:
R = N= 1300 kgf.
At the same time, on Earth the weight of a baron with a mass of 100 kg is 100 kgf. Thus, the weight of the baron in the satellite will increase by 13 times!
In the history of astronautics, there have been cases when cosmonauts withstood similar overloads for several seconds and still remained alive. But our baron is a man of exceptional physical strength, so perhaps he can withstand several minutes of such a flight. Although, to be honest, it would be better to slow down to at least 20 kilometers per second: ambition is ambition, but life is still more expensive!