Conventional wisdom
There are special bodies in the Solar System - comets.
A comet is a small body several kilometers in size. Unlike an ordinary asteroid, a comet contains various ices: water, carbon dioxide, methane and others. When a comet enters the orbit of Jupiter, these ices begin to quickly evaporate, leave the surface of the comet along with dust and form the so-called coma - a cloud of gas and dust surrounding the solid core. This cloud extends hundreds of thousands of kilometers from the core. Thanks to reflected sunlight, the comet (not itself, but only the cloud) becomes visible. And thanks to light pressure, part of the cloud is pulled into the so-called tail, which stretches from the comet for many millions of kilometers (see photo 2). Due to very weak gravity, all the matter in the coma and tail is irretrievably lost. Therefore, flying close to the Sun, a comet can lose several percent of its mass, and sometimes more. Her life time is insignificant by astronomical standards.
Where do new comets come from?
According to traditional cosmogony, they arrive from the so-called Oort cloud. It is generally accepted that at a distance of one hundred thousand astronomical units from the Sun (half the distance to the nearest star) there is a huge reservoir of comets. Nearby stars periodically disturb this reservoir, and then the orbits of some comets change so that their perihelion is near the Sun, gases on its surface begin to evaporate, forming a huge coma and tail, and the comet becomes visible through a telescope, and sometimes with the naked eye. Pictured is the famous Great Comet Hale-Bopp, in 1997.
How did the Oort cloud form? The generally accepted answer is this. At the very beginning of formation solar system In the region of the giant planets, many icy bodies with a diameter of ten kilometers or more formed. Some of them became part of the giant planets and their satellites, and some were thrown to the periphery of the Solar system. Jupiter played a major role in this process, but Saturn, Uranus and Neptune also contributed their gravitational fields to it. In the most general outline this process looked like this: a comet flies near the powerful gravitational field of Jupiter, and it changes its speed so that it ends up on the periphery of the Solar system.
True, this is not enough. If the comet's perihelion is inside the orbit of Jupiter, and its aphelion is somewhere on the periphery, then its period, as is easy to calculate, will be several million years. During the existence of the Solar System, such a comet will have time to approach the Sun almost a thousand times and all its gas, which can evaporate, will evaporate. Therefore, it is assumed that when the comet is on the periphery, disturbances from nearby stars will change its orbit so much that the perihelion will also be very far from the Sun.
So it's a four-step process. 1. Jupiter throws a piece of ice onto the periphery of the solar system. 2. The nearest star changes its orbit so that the perihelion of the orbit is also far from the Sun. 3. In such an orbit, a piece of ice remains intact for almost several billion years. 4. Another star passing nearby again disturbs its orbit so that the perihelion is close to the Sun. As a result, a piece of ice flies to us. And we see it like a new comet.
To modern cosmogonists all this seems quite plausible. But is it? Let's take a closer look at all four steps.
GRAVITY MANEUVER
First meeting
I first became acquainted with the gravitational maneuver in the 9th grade at the regional physics Olympiad. The task was this.
A rocket starts from the Earth with a speed V (sufficient to fly out of the gravitational field). The rocket has an engine with thrust F, which can operate for a time t. At what point in time should the engine be turned on so that the final speed of the rocket is maximum? Neglect air resistance.
At first it seemed to me that it didn’t matter when to turn on the engine. Indeed, due to the law of conservation of energy, the final speed of the rocket should be the same in any case. It remains to calculate the final speed of the rocket in two cases: 1. we turn on the engine at the beginning, 2. we turn on the engine after leaving the Earth’s gravitational field. Then compare the results and make sure that the final speed of the rocket is the same in both cases. But then I remembered that power is equal to: traction force times speed. Therefore, the power of the rocket engine will be maximum if you turn on the engine immediately at the start, when the rocket speed is maximum. So, the correct answer is: we turn on the engine immediately, then the final speed of the rocket will be maximum.
And although I solved the problem correctly, the problem remained. The final speed, and therefore the energy of the rocket, DEPENDS on at what point in time the engine is turned on. It seems to be a clear violation of the law of conservation of energy. Or not? What's the matter? Energy must be conserved! I tried to answer all these questions after the Olympics
The thrust of a rocket DEPENDS on its speed. This important point, and it's worth discussing.
Let us have a rocket of mass M with an engine that creates thrust of force F. Let's place this rocket in empty space (far from stars and planets) and turn on the engine. At what acceleration will the rocket move? We know the answer from Newton's Second Law: acceleration A is equal to:
A = F/M
Now let's move to another inertial reference frame, in which the rocket is moving at high speed, say, 100 km/sec. What is the acceleration of the rocket in this reference frame?
Acceleration DOES NOT DEPEND on the choice of inertial reference frame, so it will be the SAME:
A = F/M
The mass of the rocket also does not change (100 km/sec is not yet a relativistic case), therefore the thrust force F will be the SAME.
And, therefore, the power of a rocket DEPENDS on its speed. After all, power is equal to force multiplied by speed. It turns out that if a rocket is moving at a speed of 100 km/sec, then the power of its engine is 100 times more powerful than the EXACT same engine on a rocket moving at a speed of 1 km/sec.
At first glance, this may seem strange and even paradoxical. Where does the huge extra power come from? Energy must be conserved!
Let's look into this issue.
A rocket always moves on jet propulsion: it throws various gases into space at high speed. For definiteness, we assume that the gas emission speed is 10 km/sec. If a rocket moves at a speed of 1 km/sec, then its engine accelerates mainly not the rocket, but the rocket fuel. Therefore, the engine power to accelerate the rocket is not high. But if the rocket moves at a speed of 10 km/sec, then the ejected fuel will be REST relative to the external observer, that is, all the engine power will be spent on accelerating the rocket. What if the rocket moves at a speed of 100 km/sec? In this case, the ejected fuel will move at a speed of 90 km/sec. That is, the fuel speed will DECREASE from 100 to 90 km/sec. And ALL the difference in the kinetic energy of the fuel, due to the law of conservation of energy, will be transferred to the rocket. Therefore, the power of the rocket engine at such speeds will increase significantly.
Simply put, for a fast-moving rocket, its fuel has enormous kinetic energy. And from this energy additional power is drawn to accelerate the rocket.
Now it remains to figure out how this property of the rocket can be used in practice.
An attempt at practical application
Suppose in the near future you are planning to fly a rocket to the Saturn system to Titan (see photos 1-3) to study anaerobic life forms. We flew to the orbit of Jupiter and it turned out that the rocket’s speed had dropped to almost zero. The flight path was not calculated properly or the fuel turned out to be counterfeit :) . Or maybe a meteorite hit the fuel compartment, and almost all the fuel was lost. What to do?
The rocket has an engine and a small amount of fuel left. But the maximum that the engine is capable of is to increase the speed of the rocket by 1 km/sec. This is clearly not enough to reach Saturn. And so the pilot offers this option.
“We enter the gravitational field of Jupiter and fall onto it. As a result, Jupiter accelerates the rocket to enormous speed - approximately 60 km/sec. When the rocket accelerates to this speed, turn on the engine. Engine power at this speed will increase many times over. Then we fly out of Jupiter's gravitational field. As a result of such a gravitational maneuver, the speed of the rocket increases not by 1 km/sec, but significantly more. And we can fly to Saturn."
But someone objects.
“Yes, the power of the rocket near Jupiter will increase. The rocket will receive additional energy. But, flying out of Jupiter’s gravitational field, we will lose all this additional energy. The energy must remain in the potential well of Jupiter, otherwise there will be something like a perpetual motion machine, and this is impossible. Therefore, there will be no benefit from the gravity maneuver. We’ll just waste our time.”
So, the rocket is not far from Jupiter and is almost motionless relative to it. The rocket has an engine with enough fuel to increase the rocket's speed by only 1 km/sec. To increase the efficiency of the engine, it is proposed to perform a gravity maneuver: “drop” the rocket on Jupiter. It will move in its field of attraction along a parabola (see photo). And at the lowest point of the trajectory (marked with a red cross in the photo) it will turn on b engine. The rocket's speed near Jupiter will be 60 km/sec. After the engine accelerates it further, the rocket's speed will increase to 61 km/sec. What speed will the rocket have when it leaves Jupiter's gravitational field?
This task is within the capabilities of a high school student, if, of course, he knows physics well. First you need to write a formula for the sum of potential and kinetic energies. Then remember the formula for potential energy in the gravitational field of a ball. Look in the reference book to find out what the gravitational constant is, as well as the mass of Jupiter and its radius. Using the law of conservation of energy and performing algebraic transformations, obtain the general final formula. And finally, by substituting all the numbers into the formula and doing the calculations, you get the answer. I understand that no one (almost no one) wants to delve into any formulas, so I will try, without bothering you with any equations, to explain the solution to this problem “on your fingers”. I hope it works! :) .
If the rocket is stationary, its kinetic energy is zero. And if a rocket moves at a speed of 1 km/sec, then we will assume that its energy is 1 unit. Accordingly, if a rocket moves at a speed of 2 km/sec, then its energy is 4 units, if 10 km/sec, then 100 units, etc. It's clear. We have already solved half of the problem.
At the point marked with a cross (see photo), the rocket speed is 60 km/sec and the energy is 3600 units. 3600 units is enough to fly out of Jupiter's gravitational field. After the rocket accelerated, its speed became 61 km/sec, and the energy, accordingly, was 61 squared (take a calculator) 3721 units. When a rocket leaves Jupiter's gravitational field, it only spends 3600 units. 121 units remain. This corresponds to a speed (take the square root) of 11 km/sec. The problem is solved. This is not an approximate answer, but an EXACT answer.
We see that gravity assist can be used to generate additional energy. Instead of accelerating a rocket to 1 km/sec, it can be accelerated to 11 km/sec (121 times more energy, 12 thousand percent efficiency!) if there is some massive body like Jupiter nearby.
How did we get a HUGE energy gain? Due to the fact that they left the spent fuel not in empty space near the rocket, but in a deep potential hole created by Jupiter. The spent fuel received greater potential energy with a MINUS sign. Therefore, the rocket received greater kinetic energy with a PLUS sign.
Rotate a vector
Suppose we are flying a rocket near Jupiter and want to increase its speed. But we have NO fuel. Let's just say we have some fuel to correct our course. But it is clearly not enough to significantly accelerate the rocket. Can we significantly increase the speed of a rocket using gravity assist?
In the very general view this task looks like this. We fly into the gravitational field of Jupiter at some speed. Then we fly out of the field. Will our speed change? And how much can it change?
Let's solve this problem.
From the point of view of an observer who is on Jupiter (or rather, motionless relative to its center of mass), our maneuver looks like this. First the rocket is at long distance from Jupiter and moves towards it with speed V. Then, approaching Jupiter, it accelerates. In this case, the trajectory of the rocket is curved and, as is known, in its most general form it is a hyperbola. The maximum speed of the rocket will be at minimum approach. The main thing here is not to crash into Jupiter, but to fly next to it. After minimal approach, the rocket will begin to move away from Jupiter, and its speed will decrease. Finally, the rocket will fly out of Jupiter's gravitational field. What speed will it have? Exactly the same as it was upon arrival. The rocket flew into the gravitational field of Jupiter with a speed V and flew out of it with exactly the same speed V. Has anything changed? No it has changed. The DIRECTION of speed has changed. It is important. Thanks to this, we can perform a gravity maneuver.
Indeed, what is important for us is not the speed of the rocket relative to Jupiter, but its speed relative to the Sun. This is the so-called heliocentric speed. At this speed the rocket moves through the solar system. Jupiter also moves through the solar system. The rocket's heliocentric velocity vector can be decomposed into the sum of two vectors: orbital speed Jupiter (approximately 13 km/sec) and the speed of the rocket RELATIVE to Jupiter. There is nothing complicated here! This is a common triangle rule for vector addition taught in 7th grade. And this rule is ENOUGH to understand the essence of the gravity maneuver. 
We have four speeds. U(1) is the speed of our rocket relative to the Sun BEFORE the gravity maneuver. V(1) is the rocket's speed relative to Jupiter BEFORE the gravity maneuver. V(2) is the rocket's speed relative to Jupiter AFTER the gravity maneuver. In magnitude V(1) and V(2) are EQUAL, but in direction they are DIFFERENT. U(2) is the rocket's speed relative to the Sun AFTER the gravity maneuver. To see how all these four speeds are related to each other, let's look at the figure.
The green arrow AO is the speed of Jupiter's movement in its orbit. The red arrow AB is U(1): the speed of our rocket relative to the Sun BEFORE the gravitational maneuver. The yellow arrow OB is the speed of our rocket relative to Jupiter BEFORE the gravity maneuver. The yellow arrow OS is the speed of the rocket relative to Jupiter AFTER the gravity maneuver. This speed MUST lie somewhere on the yellow circle of radius OB. Because in its coordinate system, Jupiter CANNOT change the value of the rocket’s speed, but can only rotate it by a certain angle (alpha). And finally, AC is what we need: the rocket's speed U(2) AFTER the gravity maneuver.
Look how simple it is. The speed of the rocket AFTER the gravity maneuver AC is equal to the speed of the rocket BEFORE the gravity maneuver AB plus vector BC. And the vector BC is a CHANGE in the speed of the rocket in the Jupiter reference frame. Because OS - OV = OS + VO = VO + OS = BC. The more the rocket’s velocity vector rotates relative to Jupiter, the more effective the gravitational maneuver will be.
So, a rocket WITHOUT fuel flies into the gravitational field of Jupiter (or another planet). The value of its speed BEFORE and AFTER the maneuver relative to Jupiter DOES NOT CHANGE. But due to the rotation of the velocity vector relative to Jupiter, the speed of the rocket relative to Jupiter still changes. And the vector of this change is simply added to the velocity vector of the rocket BEFORE the maneuver. I hope I explained everything clearly.
To better understand the essence of the gravitational maneuver, let's look at it using the example of Voyager 2, which flew near Jupiter on July 9, 1979. As can be seen from the graph (see photo), it approached Jupiter at a speed of 10 km/sec, and flew out of its gravitational field at a speed of 20 km/sec. Only two numbers: 10 and 20.
You'll be surprised how much information you can extract from these numbers:
1. We will calculate what speed Voyager 2 had when it left the Earth's gravitational field.
2. Let's find the angle at which the device approached the orbit of Jupiter.
3. Let's calculate the minimum distance at which Voyager 2 flew up to Jupiter.
4. Let's find out what its trajectory looked like relative to an observer located on Jupiter.
5. Let's find the angle at which the spacecraft deviated after meeting Jupiter.
We will not use complex formulas, but will do calculations, as usual, “on our fingers,” sometimes using simple drawings. However, the answers we receive will be accurate. Let's just say they probably won't be accurate because the numbers 10 and 20 are probably not accurate. They are taken from the graph and rounded. In addition, other numbers we will use will also be rounded. After all, it is important for us to understand the gravity maneuver. Therefore, we will take the numbers 10 and 20 as exact, so that we have something to build on.
Let's solve the 1st problem.
Let's agree that the energy of Voyager 2, moving at a speed of 1 km/sec, is 1 unit. The minimum speed of departure from the solar system from the orbit of Jupiter is 18 km/sec. The graph of this speed is in the photo, and it is located like this. You need to multiply the orbital speed of Jupiter (about 13 km/sec) by the root of two. If Voyager 2, when approaching Jupiter, had a speed of 18 km/sec (energy 324 units), then its total energy (the sum of kinetic and potential) in the gravitational field of the Sun would be EXACTLY zero. But the speed of Voyager 2 was only 10 km/sec, and the energy was 100 units. That is, less by the amount:
324-100 = 224 units.
This lack of energy CONSISTS as Voyager 2 moves from Earth to Jupiter.
The minimum speed of departure from the solar system from Earth's orbit is approximately 42 km/sec (slightly more). To find it, you need to multiply the Earth's orbital speed (about 30 km/sec) by the root of two. If Voyager 2 was moving from Earth at a speed of 42 km/sec, its kinetic energy would be 1764 units (42 squared), and its total kinetic energy would be ZERO. As we have already found out, the energy of Voyager 2 was 224 units less, that is, 1764 - 224 = 1540 units. We take the root of this number and find the speed with which Voyager 2 flew out of the Earth’s gravitational field: 39.3 km/sec.
When a spacecraft is launched from Earth outer part Solar system, then they launch it, as a rule, along the orbital speed of the Earth. In this case, the speed of the Earth's movement is ADDED to the speed of the apparatus, which leads to a huge gain in energy.
How is the issue with the DIRECTION of speed resolved? Very simple. They wait until the Earth reaches the desired part of its orbit so that the direction of its speed is the one needed. Let's say, when launching a rocket to Mars, there is a small “window” in time during which it is very convenient to launch. If, for some reason, the launch failed, then the next attempt, you can be sure, will not be earlier than in two years.
When at the end of the 70s of the last century the giant planets lined up in a certain order, many scientists - specialists in celestial mechanics suggested taking advantage of the happy accident in the location of these planets. A project was proposed as minimal costs carry out the Grand Tour - a trip to ALL the giant planets at once. Which was done successfully.
If we had unlimited resources and fuel supplies, we could fly wherever we wanted, whenever we wanted. But since energy has to be saved, scientists carry out only energy-efficient flights. You can be sure that Voyager 2 was launched along the direction of the Earth's motion.
As we calculated earlier, its speed relative to the Sun was 39.3 km/sec. When Voyager 2 reached Jupiter, its speed dropped to 10 km/sec. Where was she headed?
The projection of this velocity onto the orbital velocity of Jupiter can be found from the law of conservation of angular momentum. The radius of Jupiter's orbit is 5.2 times greater than the Earth's orbit. This means that you need to divide 39.3 km/sec by 5.2. We get 7.5 km/sec. That is, the cosine of the angle we need is equal to 7.5 km/sec (projection of Voyager’s speed) divided by 10 km/sec (Voyager’s speed), we get 0.75. And the angle itself is 41 degrees. At this angle, Voyager 2 approached the orbit of Jupiter. 
Knowing the speed of Voyager 2 and the direction of its movement, we can draw a geometric diagram of the gravitational maneuver. It's done like this. We select point A and draw from it the vector of Jupiter’s orbital velocity (13 km/sec on the selected scale). The end of this vector (green arrow) is designated by the letter O (see photo 1). Then from point A we draw the velocity vector of Voyager 2 (10 km/sec on the selected scale) at an angle of 41 degrees. The end of this vector (red arrow) is designated by the letter B.
Now we build a circle ( yellow) with center at point O and radius |OB| (see photo 2). The end of the velocity vector both before and after the gravity maneuver can only lie on this circle. Now we draw a circle with a radius of 20 km/sec (on the chosen scale) with a center at point A. This is the speed of Voyager after the gravitational maneuver. It intersects with the yellow circle at some point C.
We plotted the gravity maneuver that Voyager 2 performed on July 9, 1979. AO is the vector of Jupiter's orbital velocity. AB is the velocity vector with which Voyager 2 approached Jupiter. Angle OAB is 41 degrees. AC is the velocity vector of Voyager 2 AFTER the gravity maneuver. From the drawing it can be seen that the angle OAC is approximately 20 degrees (half the angle OAB). If desired, this angle can be calculated accurately, since all the triangles in the drawing are given.
OB is the velocity vector with which Voyager 2 approached Jupiter, FROM THE POINT OF VIEW of an observer on Jupiter. OS is the Voyager velocity vector after the maneuver relative to the observer on Jupiter.
If Jupiter did not rotate, and you were in the subsolar side (the Sun is at its zenith), then you would see Voyager 2 moving from West to East. First it appeared in the western part of the sky, then, approaching, it reached the Zenith, flying next to the Sun, and then disappeared behind the horizon in the East. Its velocity vector turned, as can be seen from the drawing, by approximately 90 degrees (angle alpha).
Pulses along the axis of motion affect the shape and orientation* of the orbit and do not change its inclination.
Gravitational maneuver as a natural phenomenon was first discovered by astronomers of the past, who realized that significant changes in the orbits of comets, their period (and therefore their orbital speed) occur under the gravitational influence of the planets. Thus, after the transition of short-period comets from the Kuiper belt to the inner part of the Solar system, a significant transformation of their orbits occurs precisely under the gravitational influence massive planets, when exchanging angular momentum with them, without any energy costs.
Samu the idea to use gravity maneuver for spaceflight purposes was developed by Michael Minovich in the 60s, when, as a student, he interned at JPL*. The idea was quickly picked up and implemented in many space missions. But at first glance, the possibility of significantly accelerating the movement of the device without spending energy seems strange and requires explanation.
We often hear about the “capture” of asteroids and comets by the field of planets. Strictly speaking, capture without loss of energy is impossible: if some body approaches a massive planet, its velocity module first increases as it approaches, and then decreases by the same amount as it moves away. But the body can still move into the orbit of the planet’s satellite if it is decelerated at the same time (for example, there is deceleration in upper layers atmosphere, if the approach is close enough; or if significant tidal energy dissipation occurs; or, finally, if the body is destroyed within the Roche limit with different velocity vectors acquired by the fragments). At the stage of formation of the Solar System, an important factor was also the deceleration of the body in the gas-dust nebula. As for spacecraft, then only in the case of launching a satellite into orbit, braking in the upper layers of the atmosphere (aerobraking) is used. In a “pure” gravitational maneuver, the rule of equality of the velocity module before and after approaching the planet is strictly preserved (as intuition suggested: what you came with is what you left with). What's the gain?
The gain becomes obvious if we move from planetocentric to heliocentric coordinates.
The most beneficial maneuvers are near the giant planets, and they significantly reduce the flight duration. Maneuvers are also used Earth and Venus, but this significantly increases the duration of space travel. All data given in the table refers to a passive maneuver. But in some cases, at the pericenter of the flyby hyperbola, the device, with the help of its propulsion system, is given a small reactive impulse, which gives a significant additional gain.
In flight, the device often requires deceleration rather than acceleration. It is easy to choose such a rendezvous geometry when the speed of the vehicle in heliocentric coordinates drops. This depends on the position of the velocity vectors during the exchange of angular momentum. Simplifying the problem, we can say that the approach of the device to the planet with inside its orbit leads to the fact that the device gives up part of its angular momentum to the planet and slows down; and vice versa, getting closer to outside orbit leads to an increase in the moment and speed of the apparatus. It is interesting that it is impossible to register changes in the speed of the vehicle during maneuvers with any accelerometers on board - they constantly record the state of weightlessness.
Advantages of gravity maneuver compared to Homan flight to the giant planets are so large that the payload of the device can be doubled. As already mentioned, the time to reach the target during a gravitational maneuver for massive giant planets is reduced very significantly. The development of the principles of the maneuver showed that it is possible to use less massive bodies (Earth, Venus and, in special cases, even the Moon). Only the mass, in a sense, is exchanged for the flight time, which forces researchers to wait 2-3 extra years. However, the desire to reduce costs for expensive space program makes you come to terms with such a waste of time. Now the choice of flight route is, as a rule, multi-purpose, covering several planets. In 1986, a gravitational maneuver near Venus allowed the Soviet spacecraft VEGA-1 and VEGA-2 to meet Halley's Comet.
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Books
- Things of the 20th century in drawings and photographs. Forward to space! Discoveries and achievements. Set of 2 books, . "Forward, into space! Discoveries and achievements" Since ancient times, man has dreamed of getting off the ground and conquering the sky, and then space. More than a hundred years ago, inventors were already thinking about creating...
- Let's go to space! Discoveries and achievements, Klimentov Vyacheslav Lvovich, Sigorskaya Yulia Aleksandrovna. Since ancient times, man has dreamed of breaking away from the earth and conquering the sky, and then space. More than a hundred years ago, inventors were already thinking about creating spaceships, but the beginning of space...
It is difficult to imagine how much fuel gravity maneuvers saved spacecraft. They help to reach the vicinity of the giant planets and even go beyond the solar system forever. Even for the study of comets and asteroids relatively close to us, it is possible to calculate the most economical trajectory using gravitational maneuvers. When did the idea of the “space sling” come about? And when was it first implemented?
Gravitational maneuver as a natural phenomenon was first discovered by astronomers of the past, who realized that significant changes in the orbits of comets, their periods (and, consequently, their orbital speed) occur under the gravitational influence of the planets. Thus, after the transition of short-period comets from the Kuiper belt to the inner part of the Solar system, a significant transformation of their orbits occurs precisely under the gravitational influence of massive planets, when exchanging angular momentum with them, without any energy costs.
The idea of using gravity maneuvers to achieve the goal of space flight was developed by Michael Minovich in the 60s, when, as a student, he interned at NASA's Jet Propulsion Laboratory. The idea of a gravitational maneuver was first implemented in the flight path of the automatic interplanetary station Mariner 10, when the gravitational field of Venus was used to reach Mercury.
In a “pure” gravitational maneuver, the rule of equality of the velocity modulus before and after approaching a celestial body is strictly preserved. The gain becomes obvious if we move from planetocentric coordinates to heliocentric ones. This is clearly visible in the diagram shown here, adapted from the book “Mechanics of Space Flight” by V.I. Levantovsky. The trajectory of the apparatus is shown on the left, as seen by an observer on planet P. The speed v in at “local infinity” is equal in absolute value to v out. All that the observer will notice is a change in the direction of movement of the apparatus. However, an observer located in heliocentric coordinates will see a significant change in the speed of the vehicle. Since only the module of the vehicle’s velocity relative to the planet is preserved, and it is comparable to the module of the orbital velocity of the planet itself, the resulting vector sum of velocities can become either greater or less than the velocity of the vehicle before approaching. The vector diagram of this exchange of angular momentum is shown on the right. The equal velocities of entry and exit of the vehicle relative to the planet are denoted by v in and v out, and by V close, V removed and V pl - the speed of approach and removal of the device and the orbital speed of the planet in heliocentric coordinates. The increment ΔV is the velocity impulse that the planet imparted to the apparatus. Of course, the moment that the apparatus itself transmits to the planet is negligible.
Thus, by appropriately choosing the rendezvous route, you can not only change the direction, but also significantly increase the speed of the vehicle without any expenditure of its energy sources.
This diagram does not show that at first the speed increases sharply and then drops to a final value. Ballisticians usually do not care about this; they perceive the exchange of angular momentum as a “gravitational blow” from the planet, the duration of which is negligible compared to the total duration of the flight.
The critical factors in a gravitational maneuver are the mass of the planet M, the target range d and the speed vin. It is interesting that the velocity increment ΔV is maximum when vin is equal to the circular velocity at the surface of the planet.
Thus, maneuvers near the giant planets are most advantageous, and they significantly reduce the flight duration. Maneuvers near Earth and Venus are also used, but this significantly increases the duration of space travel.
After the success of Mariner 10, gravity assist maneuvers were used on many space missions. For example, the mission of the Voyager spacecraft was extremely successful, with the help of which studies of the giant planets and their satellites were carried out. The devices were launched in the United States in the fall of 1977 and reached the mission's first target, the planet Jupiter, in 1979. After completing the research program at Jupiter and studying its moons, the vehicles performed a gravity maneuver (using Jupiter's gravitational field), which allowed them to be sent along slightly different trajectories to Saturn, which they reached in 1980 and 1981, respectively. Voyager 1 then performed a complex maneuver to pass within just 5,000 km of Saturn's moon Titan, and then found itself on a trajectory out of the solar system.
Voyager 2 also performed another gravity maneuver and, despite some technical problems, was sent to the seventh planet, Uranus, which was encountered in early 1986. After approaching Uranus, another gravity maneuver was performed in its field, and Voyager 2 headed towards Neptune. Here, the gravitational maneuver allowed the device to get quite close to Neptune’s satellite Triton.
In 1986, a gravitational maneuver near Venus enabled the Soviet spacecraft VEGA-1 and VEGA-2 to encounter Halley's comet.
At the very end of 1995, a new apparatus, Galileo, reached Jupiter, the flight path of which was chosen as a chain of gravitational maneuvers in the gravitational fields of the Earth and Venus. This allowed the device to visit the asteroid belt twice in 6 years and get close to the rather large bodies Gaspra and Ida, and even return to Earth twice. After launching in the USA in the fall of 1989, the device was sent to Venus, with which it approached in February 1990, and then returned to Earth in December 1990. The gravity maneuver was performed again, and the device went to the inner part of the asteroid belt. To reach Jupiter, Galileo returned to Earth again in December 1992 and finally set its flight path to Jupiter.
In October 1997, also in the United States, the Cassini spacecraft was launched towards Saturn. Its flight program provides for 4 gravity maneuvers: two at Venus and one each at Earth and Jupiter. After the first maneuver in approach to Venus (in April 1998), the device went to the orbit of Mars and again (without the participation of Mars) returned to Venus. The second maneuver at Venus (June 1999) returned Cassini to Earth, where it also performed a gravity assist (August 1999). Thus, the device gained sufficient speed for a quick flight to Jupiter, where at the end of December 2000 its last maneuver on the way to Saturn will be performed. The device should reach its goal in July 2004.
L. V. Ksanfomality, Doctor of Physics and Mathematics. Sciences, head of the laboratory of the Institute of Space Research.
, Earth, Mars and even the Moon.
Physical essence of the process
Let's consider the trajectory of a spacecraft flying near some large celestial body, for example, Jupiter. In the initial approximation, we can neglect the effect of gravitational forces from other celestial bodies on the spacecraft.
A complex combination of gravitational maneuvers was used by the Cassini spacecraft (for acceleration, the device used gravitational field of three planets - Venus (twice), Earth and Jupiter) and "Rosetta" (four gravitational maneuvers near Earth and Mars).
In art
An artistic description of such a maneuver can be found in the science fiction novel “2010: Odyssey 2” by A. Clark.
In the science fiction film Interstellar, the Endurance orbital station does not have enough fuel to reach the third planet, located next to the black hole Gargantua (named after the literary giant glutton). Main character Cooper takes a risky step: Endurance must pass close to Gargantua's event horizon, thereby giving the station acceleration due to the attraction of the black hole.
In the science fiction novel “The Martian” and the film of the same name, using a gravitational maneuver around the Earth, the team accelerates the Hermes ship for a second flight to Mars.
see also
Write a review about the article "Gravity maneuver"
Notes
Links
- // crydee.sai.msu.ru
- (navigation calculations for space simulator"Orbiter", allows you to calculate, including gravity maneuvers)
- // novosti-kosmonavtiki.ru
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An excerpt characterizing the Gravity Maneuver
- Oh my God!- Why are you pushing, is the fire about you alone, or what? See... it fell apart.
From behind the established silence, the snoring of some who had fallen asleep was heard; the rest turned and warmed themselves, occasionally talking to each other. A friendly, cheerful laugh was heard from the distant fire, about a hundred paces away.
“Look, they’re roaring in the fifth company,” said one soldier. – And what a passion for the people!
One soldier got up and went to the fifth company.
“It’s laughter,” he said, returning. - Two guards have arrived. One is completely frozen, and the other is so courageous, dammit! Songs are playing.
- Oh oh? go have a look... - Several soldiers headed towards the fifth company.
The fifth company stood near the forest itself. A huge fire burned brightly in the middle of the snow, illuminating the tree branches weighed down with frost.
In the middle of the night, soldiers of the fifth company heard footsteps in the snow and the crunching of branches in the forest.
“Guys, it’s a witch,” said one soldier. Everyone raised their heads, listened, and out of the forest, into bright light fire, two strangely dressed human figures appeared, holding each other.
These were two Frenchmen hiding in the forest. Hoarsely saying something in a language incomprehensible to the soldiers, they approached the fire. One was taller, wearing an officer's hat, and seemed completely weakened. Approaching the fire, he wanted to sit down, but fell to the ground. The other, small, stocky soldier with a scarf tied around his cheeks, was stronger. He raised his comrade and, pointing to his mouth, said something. The soldiers surrounded the French, laid out an overcoat for the sick man, and brought porridge and vodka to both of them.
The weakened French officer was Rambal; tied with a scarf was his orderly Morel.
When Morel drank vodka and finished a pot of porridge, he suddenly became painfully cheerful and began to continuously say something to the soldiers who did not understand him. Rambal refused to eat and silently lay on his elbow by the fire, looking at the Russian soldiers with meaningless red eyes. Occasionally he would let out a long groan and then fall silent again. Morel, pointing to his shoulders, convinced the soldiers that it was an officer and that he needed to be warmed up. The Russian officer, who approached the fire, sent to ask the colonel if he would take the French officer to warm him up; and when they returned and said that the colonel had ordered an officer to be brought, Rambal was told to go. He stood up and wanted to walk, but he staggered and would have fallen if the soldier standing next to him had not supported him.
- What? You will not? – one soldier said with a mocking wink, turning to Rambal.
- Eh, fool! Why are you lying awkwardly! “It’s a man, really, a man,” they heard from different sides reproaches to the soldier who joked. They surrounded Rambal, lifted him into his arms, grabbed him, and carried him to the hut. Rambal hugged the necks of the soldiers and, when they carried him, spoke plaintively:
- Oh, nies braves, oh, mes bons, mes bons amis! Voila des hommes! oh, mes braves, mes bons amis! [Oh well done! O my good, good friends! Here are the people! O my good friends!] - and, like a child, he leaned his head on the shoulder of one soldier.
Meanwhile Morel sat on best place surrounded by soldiers.
Morel, a small, stocky Frenchman, with bloodshot, watery eyes, tied with a woman's scarf over his cap, was dressed in a woman's fur coat. He, apparently drunk, put his arm around the soldier sitting next to him and sang a French song in a hoarse, intermittent voice. The soldiers held their sides, looking at him.
- Come on, come on, teach me how? I'll take over quickly. How?.. - said the joker songwriter, who was hugged by Morel.
Vive Henri Quatre,
Vive ce roi vaillanti –
[Long live Henry the Fourth!
Long live this brave king!
etc. (French song)]
sang Morel, winking his eye.
Se diable a quatre…
- Vivarika! Vif seruvaru! sit-down... - the soldier repeated, waving his hand and really catching the tune.
- Look, clever! Go go go go!.. - rough, joyful laughter rose from different sides. Morel, wincing, laughed too.
- Well, go ahead, go ahead!
Qui eut le triple talent,
De boire, de batre,
Et d'etre un vert galant...
[Having triple talent,
drink, fight
and be kind...]
– But it’s also complicated. Well, well, Zaletaev!..
“Kyu...” Zaletaev said with effort. “Kyu yu yu...” he drawled, carefully protruding his lips, “letriptala, de bu de ba and detravagala,” he sang.
- Hey, it’s important! That's it, guardian! oh... go go go! - Well, do you want to eat more?
- Give him some porridge; After all, it won’t be long before he gets enough of hunger.
Again they gave him porridge; and Morel, chuckling, began to work on the third pot. Joyful smiles were on all the faces of the young soldiers looking at Morel. The old soldiers, who considered it indecent to engage in such trifles, lay on the other side of the fire, but occasionally, raising themselves on their elbows, they looked at Morel with a smile.
“People too,” said one of them, dodging into his overcoat. - And wormwood grows on its root.
- Ooh! Lord, Lord! How stellar, passion! Towards the frost... - And everything fell silent.
The stars, as if knowing that now no one would see them, played out in the black sky. Now flaring up, now extinguishing, now shuddering, they busily whispered among themselves about something joyful, but mysterious.
X
The French troops gradually melted away in a mathematically correct progression. And that crossing of the Berezina, about which so much has been written, was only one of the intermediate stages in the destruction of the French army, and not at all a decisive episode of the campaign. If so much has been and is being written about the Berezina, then on the part of the French this happened only because on the broken Berezina Bridge, the disasters that the French army had previously suffered evenly here suddenly grouped together at one moment and into one tragic spectacle that remained in everyone’s memory. On the Russian side, they talked and wrote so much about the Berezina only because, far from the theater of war, in St. Petersburg, a plan was drawn up (by Pfuel) to capture Napoleon in a strategic trap on the Berezina River. Everyone was convinced that everything would actually happen exactly as planned, and therefore insisted that it was the Berezina crossing that destroyed the French. In essence, the results of the Berezinsky crossing were much less disastrous for the French in terms of the loss of guns and prisoners than Krasnoye, as the numbers show.
The only significance of the Berezina crossing is that this crossing obviously and undoubtedly proved the falsity of all plans for cutting off and the justice of the only possible course of action demanded by both Kutuzov and all the troops (mass) - only following the enemy. The crowd of Frenchmen fled with an ever-increasing force of speed, with all their energy directed towards achieving their goal. She ran like a wounded animal, and she could not get in the way. This was proven not so much by the construction of the crossing as by the traffic on the bridges. When the bridges were broken, unarmed soldiers, Moscow residents, women and children who were in the French convoy - all, under the influence of the force of inertia, did not give up, but ran forward into the boats, into the frozen water.
This aspiration was reasonable. The situation of both those fleeing and those pursuing was equally bad. Remaining with his own, each in distress hoped for the help of a comrade, for a certain place he occupied among his own. Having given himself over to the Russians, he was in the same position of distress, but he was on a lower level in terms of satisfying the needs of life. The French did not need to have correct information that half of the prisoners, with whom they did not know what to do, despite all the Russians’ desire to save them, died from cold and hunger; they felt that it could not be otherwise. The most compassionate Russian commanders and hunters of the French, the French in Russian service could not do anything for the prisoners. The French were destroyed by the disaster in which the Russian army was located. It was impossible to take away bread and clothing from hungry, necessary soldiers in order to give it to the French who were not harmful, not hated, not guilty, but simply unnecessary. Some did; but this was only an exception.
