Jet propulsion. Motion of a body with variable mass

Astronautics regularly achieves stunning successes. Artificial satellites are constantly being used in more and more diverse ways. An astronaut's stay in low-Earth orbit has become commonplace. This would have been impossible without the main formula of astronautics - the Tsiolkovsky equation.

In our time, the study of both planets and other bodies of our solar system (Venus, Mars, Jupiter, Uranus, Earth, etc.) and distant objects (asteroids, other systems and galaxies) continues. Conclusions about the characteristics of the cosmic motion of Tsiolkovsky bodies laid the foundation theoretical foundations astronautics, which led to the invention of dozens of models of electric jet engines and extremely interesting mechanisms, for example, a solar sail.

The main problems of space exploration

Three areas of research and development in science and technology are clearly identified as problems of space exploration:

1. Flights near the Earth or construction of artificial satellites.
2. Lunar flights.
3. Planetary flights and flights to solar system objects.

Tsiolkovsky's equation for jet propulsion has contributed to the fact that humanity has achieved amazing results in each of these areas. And also many new applied sciences have appeared: space medicine and biology, life support systems on a spacecraft, space communications, etc.

Most people today have heard about the major achievements: the first moon landing (USA), the first satellite (USSR) and the like. In addition to the most famous achievements that everyone knows, there are many others. In particular, the USSR belongs to:

• first orbital station;
• the first flyby of the Moon and photographs of the far side;
• first automated landing on the Moon;
• the first flights of vehicles to other planets;
• first landing on Venus and Mars, etc.

Many people don’t even realize how enormous the achievements of the USSR were in the field of astronautics. In any case, they were much more than just the first satellite.

But the United States made no less contribution to the development of astronautics. In the USA they carried out:

• All major achievements in the use of low-Earth orbit (satellites and satellite communications) for scientific purposes and solving applied problems.
• Many expeditions to the Moon, exploration of Mars, Jupiter, Venus and Mercury from flyby distances.
• Many scientific and medical experiments conducted in zero gravity.

And although at the moment the achievements of other countries pale in comparison to the USSR and the USA, China, India and Japan actively joined the study of space in the period after 2000.

However, the achievements of astronautics are not limited to top layers planets and high scientific theories. On simple life she also provided big influence. As a result of space exploration, the following things have come into our lives: lightning, Velcro, Teflon, satellite communications, mechanical manipulators, wireless instruments, solar panels, artificial heart and much more. And it was Tsiolkovsky’s speed formula, which helped overcome gravitational attraction and contributed to the emergence of space practice in science, that helped achieve all this.

The term "cosmodynamics"

The Tsiolkovsky equation formed the basis of cosmodynamics. However, this term should be understood in more detail. Especially in the matter of concepts close to it in meaning: cosmonautics, celestial mechanics, astronomy, etc. Cosmonautics is translated from Greek as “swimming in the Universe.” In the usual case, this term refers to the mass of all technical capabilities and scientific achievements that make it possible to study cosmic space and celestial bodies.

Space flight is something that humanity has dreamed about for centuries. And these dreams turned into reality, from theory into science, and all thanks to Tsiolkovsky’s formula for rocket speed. From the works of this great scientist we know that the theory of astronautics stands on three pillars:

1. Theory describing the movement of spacecraft.
2. Electric rocket engines and their production.
3. Astronomical knowledge and research of the Universe.

As previously noted, in the space age many other scientific and technical disciplines appeared, such as: spacecraft control systems, communication and data transmission systems in space, navigation in outer space, space medicine and much more. It is worth noting that at the time of the birth of the foundations of astronautics there was not even a radio as such. The study of electromagnetic waves and the transmission of information over long distances with their help was just beginning. Therefore, the founders of the theory seriously considered light signals reflected towards the Earth as a method of data transmission Sun rays. Today it is impossible to imagine astronautics without all the related applied sciences. In those distant times, the imagination of a number of scientists was truly amazing. In addition to communication methods, they also touched upon such topics as the Tsiolkovsky formula for a multi-stage rocket.

Is it possible to single out any discipline as the main one among all the diversity? It is the theory of the movement of cosmic bodies. It is she who serves as the main link, without which astronautics is impossible. This area of ​​science is usually called cosmodynamics. Although it has many identical names: celestial or cosmic ballistics, mechanics of flight in space, applied celestial mechanics, the science of the movement of artificial celestial bodies, etc. They all denote the same field of study. Formally, cosmodynamics is part of celestial mechanics and uses its methods, but there is an extremely important difference. Celestial mechanics only studies orbits; it has no choice, but cosmodynamics is designed to determine the optimal trajectories for reaching certain celestial bodies by spacecraft. And the Tsiolkovsky equation for jet propulsion allows ships to determine exactly how they can influence the flight path.

Cosmodynamics as a science

Since K. E. Tsiolkovsky derived the formula, the science of the movement of celestial bodies has firmly taken shape as cosmodynamics. It allows spacecraft to use methods to find the optimal transition between different orbits, which is called orbital maneuvering, and is the basis of the theory of movement in space, just as the basis for flight in the atmosphere is aerodynamics. However, she is not the only scientist dealing with this issue. In addition to it, there is also rocket dynamics. Both of these sciences form a solid basis for modern space technology and both are included in the section of celestial mechanics.

Cosmodynamics consists of two main sections:

1. The theory about the movement of the center of inertia (mass) of an object in space, or the theory about trajectories.
2. The theory of the motion of a cosmic body relative to its center of inertia, or the theory of rotation.

To understand what the Tsiolkovsky equation is, you need to have a good understanding of mechanics, i.e. Newton's laws.

Newton's first law

Any body moves uniformly and rectilinearly or is at rest until external forces applied to it force it to change this state. In other words, the speed vector of such movement remains constant. This behavior of bodies is also called inertial motion.

Any other case in which there is any change in the velocity vector means that the body has acceleration. An interesting example V in this case is the movement of a material point in a circle or of any satellite in orbit. In this case it happens uniform motion, but not rectilinear, because the velocity vector constantly changes direction, which means the acceleration is not zero. This change in speed can be calculated using the formula v 2 / r, where v is a constant speed value, and r is the orbital radius. Acceleration in this example will be directed towards the center of the circle at any point in the trajectory of the body.

Based on the definition of the law, the cause of a change in the direction of a material point can only be force. Its role (in the case of a satellite) is the gravity of the planet. The attraction of planets and stars, as you can easily guess, has great importance in cosmodynamics in general and when using the Tsiolkovsky equation in particular.

Newton's second law

Acceleration is directly proportional to force and inversely proportional to body mass. Or in mathematical form: a = F / m, or more commonly - F = ma, where m is the coefficient of proportionality, which is a measure of the inertia of the body.

Since any rocket is represented as the movement of a body with a variable mass, the Tsiolkovsky equation will change every unit of time. In the above example about a satellite moving around a planet, knowing its mass m, you can easily find out the force under which it rotates in its orbit, namely: F = mv 2 /r. Obviously, this force will be directed towards the center of the planet.

The question arises: why does the satellite not fall on the planet? It does not fall, since its trajectory does not intersect with the surface of the planet, because nature does not force it to move along the action of the force, since only the acceleration vector is codirected with it, not the velocity.

It should also be noted that under conditions where the force acting on the body and its mass are known, the acceleration of the body can be determined. And using it, mathematical methods are used to determine the path along which this body moves. Here we come to two main problems that cosmodynamics deals with:

1. Identifying the forces that can be used to manipulate the motion of a spacecraft.
2. Determination of the motion of this ship if the forces acting on it are known.

The second problem is a classic question for celestial mechanics, while the first shows the exceptional role of cosmodynamics. Therefore, in this area of ​​physics, in addition to the Tsiolkovsky formula for reactive motion, it is extremely important to understand Newtonian mechanics.

Newton's third law

The cause of a force acting on any body is always another body. But the opposite is also true. This is the essence of Newton's third law, which states that for every action there is an action equal in magnitude but opposite in direction, called a reaction. In other words, if body A acts with a force F on body B, then body B acts on body A with a force -F.

In the example of a satellite and a planet, Newton’s third law leads us to understand that with the same force the planet attracts the satellite, exactly the same force with which the satellite attracts the planet. This attractive force is responsible for imparting acceleration to the satellite. But it also gives acceleration to the planet, but its mass is so great that this change in speed is negligible for it.

Tsiolkovsky's formula for jet propulsion is entirely based on an understanding of Newton's last law. After all, it is precisely due to the ejected mass of gases that the main body of the rocket acquires acceleration, which allows it to move in the desired direction.

A little about reference systems

Considering any physical phenomena, it is difficult not to touch upon such a topic as the frame of reference. The movement of a spacecraft, like any other body in space, can be recorded in different coordinates. There are no wrong frames of reference, only more convenient ones and less convenient ones. For example, the movement of bodies in solar system best described in heliocentric system reference, that is, in coordinates associated with the Sun, also called the Copernican system. However, the movement of the Moon in this system is less convenient to consider, so it is studied in geocentric coordinates - the countdown is relative to the Earth, this is called the Ptolemaic system. But if the question is whether an asteroid flying nearby will hit the Moon, it will be more convenient to use heliocentric coordinates again. It is important to be able to use all coordinate systems and be able to look at a problem from a different points vision.

Rocket movement

The main and only way to travel in outer space is a rocket. This principle was first expressed, according to the Habr website, by Tsiolkovsky’s formula in 1903. Since then, astronautics engineers have invented dozens of types of rocket engines using the most various types energy, but they are all united by one operating principle: the ejection of part of the mass from the reserves of the working fluid to obtain acceleration. The force that is generated as a result of this process is usually called the traction force. Let us present some conclusions that will allow us to arrive at the Tsiolkovsky equation and the derivation of its basic form.

It is obvious that the traction force will increase depending on the volume of mass ejected from the rocket per unit time and the speed that this mass can impart. Thus, the relation F = w * q is obtained, where F is the traction force, w is the speed of the thrown mass (m/s) and q is the mass expended per unit time (kg/s). It is worth separately noting the importance of the reference system associated specifically with the rocket itself. Otherwise, it is impossible to characterize the thrust of a rocket engine if everything is measured relative to the Earth or other bodies.

Research and experiments have shown that the relation F = w * q remains valid only for cases when the ejected mass is liquid or solid. But rockets use a jet of hot gas. Therefore, a number of corrections need to be introduced into the relation, and then we obtain an additional term of the relation S * (p r - p a), which is summed with the original w * q. Here p r is the pressure exerted by the gas at the nozzle exit; p a - Atmosphere pressure and S is the nozzle area. Thus, the refined formula will look like this:

F = w * q + Sp r - Sp a.

This shows that as the rocket gains altitude, the atmospheric pressure will become less and the thrust will increase. However, physicists love convenient formulas. Therefore, a formula similar to its original form is often used: F = w e * q, where w e is the effective mass outflow rate. It is determined experimentally during testing of the propulsion system and is numerically equal to the expression w + (Sp r - Sp a) / q.

Let's consider the concept identical to w e - specific thrust impulse. Specific means relating to something. In this case it refers to the gravity of the Earth. To do this, in the formula described above, the right side is multiplied and divided by g (9.81 m/s 2):

F = w e * q = (w e / g) * q * g or F = I beat * q * g

This value of I beat is measured in N*s/kg or, which is the same, m/s. In other words, the specific impulse of thrust is measured in units of speed.

Tsiolkovsky formula

As you can easily guess, in addition to the engine thrust, many other forces act on the rocket: the gravity of the Earth, the gravity of other objects in the solar system, atmospheric resistance, light pressure, etc. Each of these forces imparts its own acceleration to the rocket, and the total effect affects the final acceleration. Therefore, it is convenient to introduce the concept of reactive acceleration or a r = F t / M, where M is the mass of the rocket in a certain period of time. Jet acceleration is the acceleration with which the rocket would move in the absence of external forces acting on it. Obviously, as mass is consumed, acceleration will increase. Therefore, there is another convenient characteristic - the initial reactive acceleration a r0 = F t * M 0, where M 0 is the mass of the rocket at the moment it begins to move.

It would be logical to ask what speed a rocket can develop in such an empty space after it has consumed a certain amount of working fluid mass. Let the mass of the rocket change from m 0 to m 1. Then the speed of the rocket after uniform consumption of mass to a value of m 1 kg will be determined by the formula:

V = w * ln(m 0 / m 1)

This is nothing more than the formula for the motion of bodies with variable mass or the Tsiolkovsky equation. She characterizes energy resource rockets. And the speed obtained by this formula is called ideal. You can write this formula in another identical version:

V = I beat * ln(m 0 / m 1)

It is worth noting the use of the Tsiolkovsky Formula for calculating fuel. More precisely, the mass of the launch vehicle that will be required to launch a certain weight into Earth orbit.

Finally, something should be said about such a great scientist as Meshchersky. Together with Tsiolkovsky, they are the forefathers of astronautics. Meshchersky made a huge contribution to the creation of the theory of motion of objects of variable mass. In particular, the formula of Meshchersky and Tsiolkovsky is as follows:

m * (dv / dt) + u * (dm / dt) = 0,

where v is the speed of the material point, u is the speed of the thrown mass relative to the rocket. This relationship is also called the Meshchersky differential equation, then the Tsiolkovsky formula is obtained from it as a particular solution for a material point.

In this section we will consider the movement of bodies of variable mass. This type of movement is often found in nature and in technical systems. As examples, we can mention:

Fall of an evaporating drop;

The movement of a melting iceberg on the surface of the ocean;

Movement of a squid or jellyfish;

Rocket flight.

Below we will derive a simple differential equation that describes the motion of a body of variable mass, considering the flight of a rocket.

Differential equation of jet propulsion

Jet propulsion is based on Newton's third law , according to which “the action force is equal in magnitude and opposite in direction to the reaction force.” Hot gases escaping from the rocket nozzle create an action force. A reaction force acting in the opposite direction is called traction force. This force is what ensures the acceleration of the rocket.

Let the initial mass of the rocket be \(m,\) and its starting speed is \(v.\) After some time \(dt\) the mass of the rocket will decrease by the amount \(dm\) as a result of fuel combustion. This will increase the rocket speed by \(dv.\) Apply law of conservation of momentum to the "rocket + gas flow" system. At the initial moment of time, the momentum of the system is \(mv.\) After a short time \(dt\), the momentum of the rocket will be \[(p_1) = \left((m - dm) \right)\left((v + dv) \right),\] and the momentum associated with the exhaust gases in the coordinate system relative to the Earth will be equal to \[(p_2) = dm\left((v - u) \right),\] where \(u\) − gas flow rate relative to the Earth. Here we took into account that the speed of gas outflow is directed in the direction opposite to the speed of the rocket (Figure \(1\)). Therefore, there is a minus sign in front of \(u\).

In accordance with the law of conservation of total momentum of the system, we can write: \[ (p = (p_1) + (p_2),)\;\; (\Rightarrow mv = \left((m - dm) \right)\left((v + dv) \right) + dm\left((v - u) \right).) \]

Fig.1

Transforming given equation, we get: \[\require(cancel) \cancel(\color(blue)(mv)) = \cancel(\color(blue)(mv)) - \cancel(\color(red)(vdm)) + mdv - dmdv + \cancel(\color(red)(vdm)) - udm. \] In the last equation, the term \(dmdv,\) can be neglected when considering small changes in these quantities. As a result, the equation will be written in the form \ Divide both sides by \(dt,\) to transform the equation into the form Newton's second law :\ This equation is called differential equation of jet motion . Right part equation represents traction force\(T:\) \ From the resulting formula it is clear that the traction force is proportional gas flow rates And fuel combustion rate . Of course, this differential equation describes the ideal case. It doesn't take into account gravity And aerodynamic force . Taking them into account leads to a significant complication of the differential equation.

Tsiolkovsky formula

If we integrate the differential equation derived above, we obtain the dependence of the rocket speed on the mass of the burned fuel. The resulting formula is called ideal jet propulsion equation or Tsiolkovsky formula , who brought it out in \(1897\) year.

To obtain the indicated formula, it is convenient to rewrite the differential equation in the following form: \ Separating the variables and integrating, we find: \[ (dv = u\frac((dm))(m),)\;\; (\Rightarrow \int\limits_((v_0))^((v_1)) (dv) = \int\limits_((m_0))^((m_1)) (u\frac((dm))(m)) .) \] Note that \(dm\) denotes a decrease in mass. Therefore, let's take the increment \(dm\) with negative sign. As a result, the equation takes the form: \[ (\left. v \right|_((v_0))^((v_1)) = - u\left. (\left((\ln m) \right)) \right |_((m_0))^((m_1)),)\;\; (\Rightarrow (v_1) - (v_0) = u\ln \frac(((m_0)))(((m_1))).) \] where \((v_0)\) and \((v_1)\) are the initial and final speed of the rocket, and \((m_0)\) and \((m_1)\) are the initial and final mass of the rocket, respectively.

Assuming \((v_0) = 0,\) we obtain the formula derived by Tsiolkovsky: \ This formula determines the speed of the rocket depending on the change in its mass as the fuel burns. Using this formula, you can roughly estimate the amount of fuel required to accelerate a rocket to a certain speed.

Jet propulsion is based on the recoil principle. In a rocket, when fuel burns, gases heated to high temperature, are ejected from the nozzle at high speed U relative to the rocket. Let us denote the mass of ejected gases by m, and the mass of the rocket after the outflow of gases by M. Then for the closed system “rocket + gases” we can write based on the law of conservation of momentum (by analogy with the problem of firing a gun):, V = - where V - the speed of the rocket after the exhaust gases.

Here it was assumed that the initial speed of the rocket was zero.

The resulting formula for the speed of the rocket is valid only under the condition that the entire mass of burnt fuel is ejected from the rocket at the same time. In fact, the outflow occurs gradually throughout the entire period of accelerated motion of the rocket. Each subsequent portion of gas is ejected from the rocket, which has already acquired a certain speed.

To obtain an accurate formula, the process of gas outflow from a rocket nozzle needs to be considered in more detail. Let the rocket at time t have mass M and move with speed V. During a short period of time Dt, a certain portion of gas will be ejected from the rocket with a relative speed U. The rocket at moment t + Dt will have a speed and its mass will be equal to M + DM , where DM< 0 (рис. 1.17.3 (2)). Масса выброшенных газов будет, очевидно, равна -ДM >0. The speed of gases in the inertial frame OX will be equal to V+U. Let's apply the law of conservation of momentum. At the moment of time t + Дt, the momentum of the rocket is equal to ()(M + ДМ) and the momentum of the emitted gases is equal to At the moment of time t, the momentum of the entire system was equal to MV. Assuming the “rocket + gases” system is closed, we can write:

The value can be neglected, since |DM|<< M. Разделив обе части последнего соотношения на Дt и перейдя к пределу при Дt >0, we get

The value is fuel consumption per unit time. The quantity is called the reactive thrust force F p The reactive thrust force acts on the rocket from the side of the outflowing gases, it is directed in the direction opposite to the relative speed. Ratio

expresses Newton's second law for a body of variable mass. If gases are ejected from the rocket nozzle strictly backward (Fig. 1.17.3), then in scalar form this relationship takes the form:

where u is the relative velocity module. Using the mathematical operation of integration, from this relationship we can obtain a formula for the final speed x of the rocket:

where is the ratio of the initial and final masses of the rocket. This formula is called the Tsiolkovsky formula. It follows from it that the final speed of the rocket can exceed the relative speed of the outflow of gases. Consequently, the rocket can be accelerated to the high speeds required for space flights. But this can only be achieved by consuming a significant mass of fuel, constituting a large proportion of the initial mass of the rocket. For example, to achieve the first cosmic velocity x = x 1 = 7.9 10 3 m/s at u = 3 10 3 m/s (gas outflow velocities during fuel combustion are on the order of 2-4 km/s), the starting mass of a single-stage The rocket should be approximately 14 times its final mass. To achieve the final speed x = 4u the ratio must be = 50.

A significant reduction in the launch mass of a rocket can be achieved when using multi-stage rockets, when the rocket stages are separated as the fuel burns out. The masses of containers that contained fuel, spent engines, control systems, etc. are excluded from the process of subsequent rocket acceleration. It is along the path of creating economical multi-stage rockets that modern rocket science is developing.

Material from Wikipedia - the free encyclopedia

However, the first to solve the equation of motion of a body with a variable mass were the English researchers W. Moore, as well as P. G. Tate and W. J. Steele from the University of Cambridge, respectively, in 1810-1811. and in 1856.

The Tsiolkovsky formula can be obtained by integrating the Meshchersky differential equation for a material point of variable mass:

$m\; \backslash cdot\; \backslash frac\; (d\backslash vec(V))(dt)+\; \backslash vec(u)\; \backslash cdot\; \backslash frac\; (dm)(dt)=0$, in which $m$ - point mass; $V$ - point speed; $u$ - the relative speed with which the part of its mass moving away from the point moves. For a rocket engine, this value is its specific impulse $I$ $\backslash Delta\; v\_(g)\backslash \; =\; \backslash int\backslash limits\_(0)^(t)\; g(t)\backslash cdot\; \backslash cos(\backslash gamma\; (t))\backslash ,dt$,

Where $g(t)$ And $\backslash gamma\; (t)$- local gravitational acceleration and the angle between the engine thrust vector and the local gravity vector, respectively, which are functions of time according to the flight program. As can be seen from Table 1, the largest part of these losses occurs in the first stage flight segment. This is explained by the fact that in this section the trajectory deviates from the vertical to a lesser extent than in sections of subsequent steps, and the value $\backslash cos(\backslash gamma\; (t))$ close to the maximum value - 1.

Aerodynamic losses are caused by the resistance of the air environment when the rocket moves in it and are calculated using the formula:

$\backslash Delta\; v\_(a)\backslash \; =\; \backslash int\backslash limits\_(0)^(t)\; \backslash frac\; (A(t))(m(t))\; \backslash ,dt$,

Where $A(t)$ is the force of frontal aerodynamic drag, and $m(t)$- current mass of the rocket. The main losses from air resistance also occur in the operating section of the 1st stage of the rocket, since this section takes place in the lower, densest layers of the atmosphere.

The ship must be launched into orbit with strictly defined parameters; for this, the control system during the active phase of the flight deploys the rocket according to a certain program, while the direction of engine thrust deviates from the current direction of the rocket's movement, and this entails speed losses for control, which are calculated by formula:

$\backslash Delta\; v\_(u)\backslash \; =\; \backslash int\backslash limits\_(0)^(t)\; \backslash frac\; (F(t))(m(t))\; \backslash cdot(1\; -\; \backslash cos(\backslash alpha\; (t)))\; \backslash ,\; dt$,

Where $F(t)$- current engine thrust, $m(t)$ is the current mass of the rocket, and $\backslash alpha\; (t)$- the angle between the thrust and velocity vectors of the rocket. The largest part of the rocket control losses occurs in the 2nd stage flight section, since it is in this section that the transition from vertical to horizontal flight occurs, and the engine thrust vector deviates to the greatest extent from the rocket velocity vector.

Using the Tsiolkovsky formula in rocket design

Developed at the end of the 19th century, Tsiolkovsky’s formula still forms an important part of the mathematical apparatus used in the design of rockets, in particular, in determining their main mass characteristics.

By simple transformations of the formula we obtain the following equation:

$\backslash frac\; (M\_(1))\; (M\_(2))\; =\; e^(V/I)$ (1)

This equation gives the ratio of the rocket's initial mass to its final mass for given values ​​of the rocket's final velocity and specific impulse. Let us introduce the following notation:

$M\_(0)$ - payload mass; $M\_(k)$ - mass of the rocket structure; $M\_(t)$ - mass of fuel.

The mass of the rocket structure in a wide range of values ​​depends on the mass of the fuel almost linearly: the larger the fuel supply, the larger the size and mass of the tanks for storing it, the larger the mass of the load-bearing structural elements, the more powerful (and therefore more massive) the propulsion system. Let us express this dependence in the form:

$M\_(k)=\backslash frac\; (M\_(t))\; (k)$, (2) where $k$- coefficient showing how much fuel is per unit mass of the structure. With rational design, this coefficient primarily depends on the characteristics (density and strength) of the structural materials used in the production of the rocket. The stronger and lighter the materials used, the higher the coefficient value. $k$. This coefficient also depends on the average density of the fuel (less dense fuel requires containers of larger size and weight, which leads to a decrease in the value $k$).

Equation (1) can be written as:

$\backslash frac\; (M\_(0)+\; M\_(t)+M\_(t)/k)\; (M\_(0)+M\_(t)/k)=e^(V/I)$,

which, through elementary transformations, is reduced to the form:

$M\_(t)=\backslash frac\; (M\_(0)\; \backslash cdot\; k\; \backslash cdot\; (e^(V/I)-1))(k+1-\; e^(V/I))$ (3)

This form of the Tsiolkovsky equation allows one to calculate the mass of fuel required for a single-stage rocket to achieve a given characteristic speed, given the payload mass, specific impulse value, and coefficient value $k$.

Of course, this formula makes sense only when the value obtained by substituting the original data is positive. Since the exponent for a positive argument is always greater than 1, the numerator of the formula is always positive, therefore, the denominator of this formula must also be positive: $k+1-\; e^(V/I)>0$, in other words, $k>e^(V/I)-1$ (4)

This inequality is achievability criterion single-stage rocket of a given speed $V$ at given specific impulse values $I$ and coefficient $k$. If the inequality is not satisfied, the given speed cannot be achieved at any cost of fuel: with an increase in the amount of fuel, the mass of the rocket structure will increase and the ratio of the initial mass of the rocket to the final mass will never reach the value required by the Tsiolkovsky formula to achieve the given speed.

Example of rocket mass calculation

It is required to launch an artificial Earth satellite with a mass of $M\_(0)=10$ T into a circular orbit at an altitude of 250 km. The located engine has a specific impulse $I=2900$ m/c. Coefficient $k=9$- this means that the mass of the structure is 10% of the mass of the fueled rocket (stage). Let us determine the mass of the launch vehicle.

The first escape velocity for the chosen orbit is 7759.4 m/s, to which are added the estimated losses from gravity of 600 m/s (this, as can be seen, is less than the losses given in Table 1, but the orbit to be reached is - twice as low), the characteristic speed will thus be $V=8359.4$ m/c(the remaining losses can be neglected as a first approximation). With such parameters, the value $e^(V/I)=17.86$. Inequality (4) obviously does not hold; therefore, with a single-stage rocket under these conditions it is impossible to achieve the goal.

Calculation for a two-stage rocket. Let's divide the characteristic speed in half, which will be the characteristic speed for each of the stages of a two-stage rocket. $V=4179.7$ m/c. This time $e^(V/I)=4.23$, which satisfies the achievability criterion (4), and, substituting values ​​into formulas (3) and (2), for 2nd stage we get: $M\_(t2)=\backslash frac\; (10\; \backslash cdot\; 9\; \backslash cdot\; (4.23-1))(9+1-\; 4.23)=50.3$ T; $M\_(k2)=\backslash frac\; (50.3)\; (9)=5.6$ T; the total mass of the 2nd stage is $55,9$ T. For 1st stage The total mass of the 2nd stage is added to the payload mass, and after appropriate substitution we obtain: $M\_(t1)=\backslash frac\; ((10+55.9)\; \backslash cdot\; 9\; \backslash cdot\; (4.23-1))(9+1-\; 4.23)=331.3$ T; $M\_(k1)=\backslash frac\; (331.3)\; (9)=36.8$ T; the total mass of the first stage is $368,1$ T; the total mass of the two-stage rocket with payload will be $10+55,9+368,1=434$ T. Calculations are performed in a similar way for a larger number of stages. As a result, we get: -The launch mass of a three-stage rocket will be $323,1$ T. -Four-speed - $294,2$ T. -Five-speed - $281$ T.

This example shows how it is justified multi-stage in rocket science - at the same final speed, a rocket with a larger number of stages has less mass.

It should be noted that these results were obtained under the assumption that the coefficient of design perfection of the rocket $k$ remains constant, regardless of the number of steps. A closer look reveals that this is a gross oversimplification. The steps are connected to each other by special sections - adapters- load-bearing structures, each of which must withstand the total weight of all subsequent stages, multiplied by the maximum overload value that the rocket experiences during all flight segments in which the adapter is part of the rocket. As the number of stages increases, their total mass decreases, while the number and total mass of adapters increases, which leads to a decrease in the coefficient $k$, and, along with it, a positive effect multi-stage. In modern rocket science practice, more than four stages, as a rule, are not made.

This kind of calculations are carried out not only at the first stage of design - when choosing a rocket layout option, but also at subsequent stages of design, as the design is detailed, the Tsiolkovsky formula is constantly used when verification calculations, when characteristic speeds are recalculated, taking into account the ratios of the initial and final mass of the rocket (stage) formed from specific details, specific characteristics of the propulsion system, clarification of speed losses after calculating the flight program in the active section, etc., in order to control the achievement of the given rocket by the rocket speed.

Generalized Tsiolkovsky formula

For a rocket flying at a speed close to the speed of light, the generalized Tsiolkovsky formula is valid:

$\backslash frac(M\_(2))(M\_(1))=\backslash left\; (\backslash frac(1-\backslash frac(V)(c))(1+\backslash frac(V)(c))\; \backslash right)^(\backslash \; frac(c)(2I))$,

Where $c$- speed of light. For a photon rocket $I=c$ and the formula looks like:

$\backslash frac(M\_(1))(M\_(2))=\backslash sqrt\; (\backslash frac(1+\backslash frac(V)(c))(1-\backslash frac(V)(c)))$,

The speed of a photon rocket is calculated by the formula:

$\backslash frac(V)(c)\; =\; \backslash frac(1-\; \backslash left(\backslash frac(M\_(2))(M\_(1))\; \backslash right)^(2))(1+\; \backslash left(\backslash frac(M\_(\; 2))(M\_(1))\; \backslash right)^(2))$,

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Notes

Literature

• Levantovsky V.I. Mechanics of space flight in an elementary presentation. - M.: Nauka, 1980. - 512 p.

Laws and tasks Celestial sphere Orbit parameters Movement celestial bodies Astrodynamics Space flight Spacecraft orbits

An excerpt characterizing the Tsiolkovsky Formula

“Well, they should have slept,” said the Cossack.
“No, I’m used to it,” answered Petya. - What, you don’t have flints in your pistols? I brought it with me. Isn't it necessary? You take it.
The Cossack leaned out from under the truck to take a closer look at Petya.
“Because I’m used to doing everything carefully,” said Petya. “Some people just don’t get ready, and then they regret it.” I don't like it that way.
“That’s for sure,” said the Cossack.
“And one more thing, please, my dear, sharpen my saber; dull it... (but Petya was afraid to lie) it was never sharpened. Can this be done?
- Why, it’s possible.
Likhachev stood up, rummaged through his packs, and Petya soon heard the warlike sound of steel on a block. He climbed onto the truck and sat on the edge of it. The Cossack was sharpening his saber under the truck.
- Well, are the fellows sleeping? - said Petya.
- Some are sleeping, and some are like this.
- Well, what about the boy?
- Is it spring? He collapsed there in the entryway. He sleeps with fear. I was really glad.
For a long time after this, Petya was silent, listening to the sounds. Footsteps were heard in the darkness and a black figure appeared.
- What are you sharpening? – the man asked, approaching the truck.
- But sharpen the master’s saber.
“Good job,” said the man who seemed to Petya to be a hussar. - Do you still have a cup?
- And over there by the wheel.
The hussar took the cup.
“It’ll probably be light soon,” he said, yawning, and walked off somewhere.
Petya should have known that he was in the forest, in Denisov’s party, a mile from the road, that he was sitting on a wagon captured from the French, around which the horses were tied, that the Cossack Likhachev was sitting under him and sharpening his saber, that there was a big black spot to the right is a guardhouse, and a bright red spot below to the left is a dying fire, that the man who came for a cup is a hussar who was thirsty; but he knew nothing and did not want to know it. He was in a magical kingdom in which there was nothing like reality. A large black spot, perhaps there was definitely a guardhouse, or perhaps there was a cave that led into the very depths of the earth. The red spot might have been fire, or maybe the eye of a huge monster. Maybe he’s definitely sitting on a wagon now, but it’s very possible that he’s not sitting on a wagon, but on a terrible high tower, from which if you fall, you would fly to the ground for a whole day, a whole month - you’d keep flying and never reach it. It may be that just a Cossack Likhachev is sitting under the truck, but it may very well be that this is the kindest, bravest, most wonderful, most excellent person in the world, whom no one knows. Maybe it was just a hussar passing for water and going into the ravine, or maybe he just disappeared from sight and completely disappeared, and he was not there.
Whatever Petya saw now, nothing would surprise him. He was in a magical kingdom where everything was possible.
He looked at the sky. And the sky was as magical as the earth. The sky was clearing, and clouds were moving quickly over the tops of the trees, as if revealing the stars. Sometimes it seemed that the sky cleared and a black, clear sky appeared. Sometimes it seemed that these black spots were clouds. Sometimes it seemed as if the sky was rising high, high above your head; sometimes the sky dropped completely, so that you could reach it with your hand.
Petya began to close his eyes and sway.
Drops were dripping. There was a quiet conversation. The horses neighed and fought. Someone was snoring.
“Ozhig, zhig, zhig, zhig...” the saber being sharpened whistled. And suddenly Petya heard a harmonious choir of music playing some unknown, solemnly sweet hymn. Petya was musical, just like Natasha, and more than Nikolai, but he had never studied music, did not think about music, and therefore the motives that unexpectedly came to his mind were especially new and attractive to him. The music played louder and louder. The melody grew, moving from one instrument to another. What was called a fugue was happening, although Petya had no the slightest idea about what a fugue is. Each instrument, sometimes similar to a violin, sometimes like trumpets - but better and cleaner than violins and trumpets - each instrument played its own and, not yet finishing the tune, merged with another, which started almost the same, and with the third, and with the fourth , and they all merged into one and scattered again, and again merged, now into the solemn church, now into the brightly brilliant and victorious.
“Oh, yes, it’s me in a dream,” Petya said to himself, swaying forward. - It's in my ears. Or maybe it's my music. Well, again. Go ahead my music! Well!.."
He closed his eyes. And from different sides, as if from afar, sounds began to tremble, began to harmonize, scatter, merge, and again everything united into the same sweet and solemn hymn. “Oh, what a delight this is! As much as I want and how I want,” Petya said to himself. He tried to lead this huge choir of instruments.
“Well, hush, hush, freeze now. – And the sounds obeyed him. - Well, now it’s fuller, more fun. More, even more joyful. – And from an unknown depth arose intensifying, solemn sounds. “Well, voices, pester!” - Petya ordered. And first, male voices were heard from afar, then female voices. The voices grew, grew in uniform, solemn effort. Petya was scared and joyful to listen to their extraordinary beauty.
The song merged with the solemn victory march, and drops fell, and burn, burn, burn... the saber whistled, and again the horses fought and neighed, not breaking the choir, but entering into it.
Petya didn’t know how long this lasted: he enjoyed himself, was constantly surprised by his pleasure and regretted that there was no one to tell it to. He was awakened by Likhachev's gentle voice.
- Ready, your honor, you will split the guard in two.
Petya woke up.
- It’s already dawn, really, it’s dawning! - he screamed.
The previously invisible horses became visible up to their tails, and a watery light was visible through the bare branches. Petya shook himself, jumped up, took a ruble from his pocket and gave it to Likhachev, waved, tried the saber and put it in the sheath. The Cossacks untied the horses and tightened the girths.
“Here is the commander,” said Likhachev. Denisov came out of the guardhouse and, calling out to Petya, ordered them to get ready.

Quickly in the semi-darkness they dismantled the horses, tightened the girths and sorted out the teams. Denisov stood at the guardhouse, giving the last orders. The party's infantry, slapping a hundred feet, marched forward along the road and quickly disappeared between the trees in the predawn fog. Esaul ordered something to the Cossacks. Petya held his horse on the reins, impatiently awaiting the order to mount. Washed with cold water, his face, especially his eyes, burned with fire, a chill ran down his back, and something in his whole body trembled quickly and evenly.
- Well, is everything ready for you? - Denisov said. - Give us the horses.
The horses were brought in. Denisov became angry with the Cossack because the girths were weak, and, scolding him, sat down. Petya took hold of the stirrup. The horse, out of habit, wanted to bite his leg, but Petya, not feeling his weight, quickly jumped into the saddle and, looking back at the hussars who were moving behind in the darkness, rode up to Denisov.
- Vasily Fedorovich, will you entrust me with something? Please... for God's sake... - he said. Denisov seemed to have forgotten about Petya’s existence. He looked back at him.
“I ask you about one thing,” he said sternly, “to obey me and not to interfere anywhere.”
During the entire journey, Denisov did not speak a word to Petya and rode in silence. When we arrived at the edge of the forest, the field was noticeably getting lighter. Denisov spoke in a whisper with the esaul, and the Cossacks began to drive past Petya and Denisov. When they had all passed, Denisov started his horse and rode downhill. Sitting on their hindquarters and sliding, the horses descended with their riders into the ravine. Petya rode next to Denisov. The trembling throughout his body intensified. It became lighter and lighter, only the fog hid distant objects. Moving down and looking back, Denisov nodded his head to the Cossack standing next to him.
- Signal! - he said.
The Cossack raised his hand and a shot rang out. And at the same instant, the tramp of galloping horses was heard in front, screams from different sides and more shots.
At the same instant as the first sounds of stomping and screaming were heard, Petya, hitting his horse and releasing the reins, not listening to Denisov, who was shouting at him, galloped forward. It seemed to Petya that it suddenly dawned as brightly as the middle of the day at that moment when the shot was heard. He galloped towards the bridge. Cossacks galloped along the road ahead. On the bridge he encountered a lagging Cossack and rode on. Some people ahead - they must have been French - were running from the right side of the road to the left. One fell into the mud under the feet of Petya's horse.
Cossacks crowded around one hut, doing something. A terrible scream was heard from the middle of the crowd. Petya galloped up to this crowd, and the first thing he saw was the pale face of a Frenchman with a shaking lower jaw, holding onto the shaft of a lance pointed at him.
“Hurray!.. Guys... ours...” Petya shouted and, giving the reins to the overheated horse, galloped forward down the street.
Shots were heard ahead. Cossacks, hussars and ragged Russian prisoners, running from both sides of the road, were all shouting something loudly and awkwardly. A handsome Frenchman, without a hat, with a red, frowning face, in a blue overcoat, fought off the hussars with a bayonet. When Petya galloped up, the Frenchman had already fallen. I was late again, Petya flashed in his head, and he galloped to where frequent shots were heard. Shots rang out in the courtyard of the manor house where he was with Dolokhov last night. The French sat down there behind a fence in a dense garden overgrown with bushes and fired at the Cossacks crowded at the gate. Approaching the gate, Petya, in the powder smoke, saw Dolokhov with a pale, greenish face, shouting something to the people. “Take a detour! Wait for the infantry!” - he shouted, while Petya drove up to him.
“Wait?.. Hurray!..” Petya shouted and, without hesitating a single minute, galloped to the place from where the shots were heard and where the powder smoke was thicker. A volley was heard, empty bullets squealed and hit something. The Cossacks and Dolokhov galloped after Petya through the gates of the house. The French, in the swaying thick smoke, some threw down their weapons and ran out of the bushes to meet the Cossacks, others ran downhill to the pond. Petya galloped on his horse along the manor's yard and, instead of holding the reins, strangely and quickly waved both arms and fell further and further out of the saddle to one side. The horse, running into the fire smoldering in the morning light, rested, and Petya fell heavily onto the wet ground. The Cossacks saw how quickly his arms and legs twitched, despite the fact that his head did not move. The bullet pierced his head.
After talking with the senior French officer, who came out to him from behind the house with a scarf on his sword and announced that they were surrendering, Dolokhov got off his horse and approached Petya, who was lying motionless, with his arms outstretched.
“Ready,” he said, frowning, and went through the gate to meet Denisov, who was coming towards him.
- Killed?! - Denisov cried out, seeing from afar the familiar, undoubtedly lifeless position in which Petya’s body lay.
“Ready,” Dolokhov repeated, as if pronouncing this word gave him pleasure, and quickly went to the prisoners, who were surrounded by dismounted Cossacks. - We won’t take it! – he shouted to Denisov.
Denisov did not answer; he rode up to Petya, got off his horse and with trembling hands turned Petya’s already pale face, stained with blood and dirt, towards him.
“I’m used to something sweet. Excellent raisins, take them all,” he remembered. And the Cossacks looked back in surprise at the sounds similar to the barking of a dog, with which Denisov quickly turned away, walked up to the fence and grabbed it.
Among the Russian prisoners recaptured by Denisov and Dolokhov was Pierre Bezukhov.

There was no new order from the French authorities about the party of prisoners in which Pierre was, during his entire movement from Moscow. This party on October 22 was no longer with the same troops and convoys with which it left Moscow. Half of the convoy with breadcrumbs, which followed them during the first marches, was repulsed by the Cossacks, the other half went ahead; there were no more foot cavalrymen who walked in front; they all disappeared. The artillery, which had been visible ahead during the first marches, was now replaced by a huge convoy of Marshal Junot, escorted by the Westphalians. Behind the prisoners was a convoy of cavalry equipment.
From Vyazma, the French troops, previously marching in three columns, now marched in one heap. Those signs of disorder that Pierre noticed at the first stop from Moscow have now reached the last degree.
The road along which they walked was littered with dead horses on both sides; ragged people lagging behind different teams, constantly changing, then joined, then again lagged behind the marching column.
Several times during the campaign there were false alarms, and the soldiers of the convoy raised their guns, shot and ran headlong, crushing each other, but then they gathered again and scolded each other for their vain fear.
These three gatherings, marching together - the cavalry depot, the prisoner depot and Junot's train - still formed something separate and integral, although both of them, and the third, were quickly melting away.
The depot, which had initially contained one hundred and twenty carts, now had no more than sixty left; the rest were repulsed or abandoned. Several carts from Junot's convoy were also abandoned and recaptured. Three carts were plundered by the backward soldiers from Davout's corps who came running. From conversations of the Germans, Pierre heard that this convoy was put on guard more than the prisoners, and that one of their comrades, a German soldier, was shot on the orders of the marshal himself because a silver spoon that belonged to the marshal was found on the soldier.

Physics

The laws of nature around us can only be called cruel in a figurative sense. We have created machines that can free us from the bonds that hold all of humanity in the gravitational well, but some aspects of them remain beyond our control. If we want to begin our journey through the solar system, then these restrictions will have to be circumvented somehow.

Modern rockets eject some of their own mass as gas from their engine nozzles, allowing them to move in the opposite direction. This is possible thanks to Newton's third law, which was formulated in 1687. We owe all of our rocket propulsion to Tsiolkovsky’s 1903 formula.

There are only four variables in the formula (from left to right): the final speed of the aircraft, the specific impulse of the rocket engine (the ratio of engine thrust to the second mass consumption of fuel), the initial mass of the aircraft (payload, structure and fuel) and its final mass (payload and design).

How can you change one of the variables if the other three are already set? This is simply impossible; no form of desire, desire or request will help here.

It is the losses due to gravity that define the limits of human space exploration, and we are forced to take them into account when we choose where we want to go. Today there are not so many such places. WITH earth's surface we can find ourselves in Earth's orbit, from Earth's orbit we can go to the surface of the Moon, or to the surface of Mars, or to the space between the Moon and the Earth. Various combinations are possible, but with current technology developments these are the most likely destinations.

The values ​​presented below do not take into account any losses due to, for example, atmospheric resistance, but the values ​​are close enough to illustrate what should be taken for granted. This is in some way the cost of the flight.

As you can see, the path from Earth to orbit, these measly 400 kilometers, is the most expensive part of the flight. This is a whole half the “cost” of a flight to Mars; even getting to the Moon “costs” less. All this is connected with the gravitational attraction of our cosmic home.

And we will have to fly on a rocket with chemical engines; Although there are promising developments, the traditional engines that have been used for more than 60 years in manned space flight remain realistic. Chemical fuels place a limit on the amount of energy that can be extracted from them and therefore put into a rocket, and we use the most efficient reactions known to mankind. And again we will have to come to terms with some value of the variable that we cannot change.

Below are some types of rocket fuel that have been used at least once to propel vehicles with a person on board or are planned for use, as well as their specific impulses. Methane-oxygen is under consideration for future missions to the Moon and Mars. Self-igniting two-propellant liquid rocket propellant was used for the Apollo lunar module landing because of its simplicity.

The most effective pair remains oxygen-hydrogen, and chemistry cannot give us more. At the end of the 70s of the last century, nuclear rocket engine with hydrogen as a working fluid, which was accelerated by the heat of a controlled nuclear reaction, produced 8.3 km/s.

So, the only thing we can now change in Tsiolkovsky’s formula is the ratio of the masses of the aircraft. The rocket must be built in such a way that this ratio has some given value, otherwise it simply will not reach its goal. Something can be done by adding a few ingenious solutions to the design, but in general this will have little effect on the result - the chemistry of the fuel and the gravity of celestial bodies cannot be changed.

So what do we have? Here percentage fuel from the total mass of the rocket, necessary for the rocket to enter Earth's orbit.

The obtained figures do not take into account various losses of atmospheric resistance, incomplete combustion and other negative factors, so the real ratio is slightly closer to 100%. Excellent engineering solutions such as staging, multiple fuel types (for example, kerosene or solid fuel for the first stage, hydrogen for the rest) are very helpful in a situation where only about 10% of the mass of the device remains for the rocket itself. The mass of the payload is sometimes literally worth its weight in gold.

The characteristics of real rockets do not differ much from these ideal values ​​obtained without taking into account many factors. The largest rocket in the history of mankind, the Saturn 5, had 85% of its entire mass fueled on the launch pad. It had three stages: the first ran on kerosene and oxygen, the second and third – on hydrogen and oxygen. The Shuttle has the same indicator. The Soyuz uses kerosene on all its stages, so its fuel mass makes up 91% of the total mass of the rocket. The use of a hydrogen-oxygen pair is associated with big amount technical difficulties, but this combination is more effective; Kerosene paired with oxygen provides the opportunity to use simpler and more reliable solutions.

15% of the rocket's mass is much less than it seems. The rocket must have tanks, pipes leading to the engines, a body that must be able to withstand both supersonic flight in the atmosphere after the inhuman heat of the launch pad, and the cold of airless space. The rocket must be guided and controlled using supersonic rudders and maneuvering engines. The fragile bodies of people in a spaceship need to be provided with oxygen, as well as carbon dioxide removed, they need to be protected from heat and cold, and given the opportunity to safely return to the surface of their home planet. Finally, people are not the only load on the rocket: we do not launch people just for fun, or rather, we can launch a person for the sake of the fact, but only once. People also carry a variety of equipment into space to conduct experiments, since space flights are aimed at scientific research.

The actual mass of the rocket payload is much less than this 10%-15%. Saturn V, the only rocket that helped put man on the Moon, delivered only 4% of its total mass into Earth orbit, with a total of 120 tons delivered into orbit. The Shuttles could deliver about the same amount (100 tons), but the actual payload was about 20 tons, 1% of the total mass.

Let's compare rockets with the vehicles we are used to. (Of course, the rocket has tanks with oxidizers, and earthly transport uses oxygen from the air for this.)

It's easy to see the difference in materials and design vehicle depending on the relative mass of the fuel. Transport with fuel weighing less than 10% of its total mass is usually made of steel, and there is no need to think much about its design: attach this part to that and strengthen the body where intuition requires. A ten-ton truck can be heavily overloaded, but it will continue to move, albeit slowly.

Air transport requires a more serious approach and lightweight structures made of aluminum, magnesium, titanium, and composite materials. You can’t just change anything here, and you need to think twice about any small detail. Machines of this nature cannot operate so far beyond their load limits. 60%-70% of the mass of these devices is the actual weight of the vehicle with payload, and with the use of some engineering solutions, comfortable, safe and profitable operation is possible.

And rockets, where 85% is fuel, are at the limit of our engineering capabilities. We can barely produce them, they require constant improvement to be able to use them. Externally, small changes require a huge amount of various analysis and testing of prototypes in wind tunnels, vibration stands, and for a test launch, personnel should be removed to a bunker a couple of kilometers from the launch pad - even after all these checks, accidents are possible. Very often it is impossible to exceed loads by more than 10% of those specified by technical requirements. This is similar to the situation when, after accelerating to 44 kilometers per hour, a bicycle will fall apart into tiny screws simply because the maximum speed is 40 km/h.

A marvel of mass production, the aluminum beer can is approximately 94% of its contents, with only 6% being the body, but somehow this figure is better than the Shuttle's external tank, despite the fact that it does not contain a drink slightly colder than room temperature. temperatures, and highly active liquids at about 20 degrees above absolute zero, compressed to terrible pressure. At the same time, this fuel tank can withstand an overload of 3 g, maintaining the flow of oxidizer and fuel at a level of 1.5 tons per second.

Don Pettit describes details of the STS-126 November 2008 mission. The shuttle's engines were supposed to shut down when the speed reached 7824 m/s, but if this happened at 7806 m/s, the spacecraft would become a satellite of the Earth, but would not reach the target orbit. To put it simply, Endeavor would not have reached the ISS. Is this a big difference? This is roughly the same as paying $10 and only missing two cents (0.2%). Well, in this case it would be possible to use some of the fuel for orbital maneuvers. If the speed had been only 3% lower, then these reserves would not have been enough, and the shuttle would have had to be landed somewhere in Spain. This 3% could have been lost if the main engine had turned off just 8 seconds earlier.

Let's imagine the best combination of circumstances: a tank for the Shuttle (we will discard the mass of the engines) and hydrogen-oxygen fuel. If we substitute the values ​​into Tsiolkovsky’s formula, it will become clear that with the radius of our planet one and a half times greater than its current one, we would never have reached space only due to the technology of chemical rocket engines.

And all this is the consequences of Tsiolkovsky’s formula. If we want to break free from its brutal dominance, we will have to create working versions of fundamentally new engines. Perhaps then rockets will become as safe, familiar and reliable as jet passenger aircraft.