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ABOUT PHOTONS AND PHOTON ROCKETS

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In the focus of the ideal parabolic mirror is the source of photons, resulting from the annihilation of matter and antimatter. After reflection from the mirror, photons fly in a parallel beam. Find the speed of the spaceship if its mass before the start of the movement is equal to m, and after acceleration - m ₀ . What part of the initial mass can be accelerated to a speed of 0.999s?

In connection with the publications in the newspaper "Physics", concerning the methods of teaching the theory of relativity, in particular in connection with the question of the mass, depending on the speed, does it make sense to solve this problem? Maybe this problem can be solved without using relativistic laws? "

The answer may be of interest to many teachers of physics. Therefore, we give the solution of this problem with the comments.

1. In analyzing the possibilities of a hypothetical photon rocket, we must separate the question of how the photon engine works, from the question of how the missile accelerates with such an engine. In both cases, for the analysis it is necessary to use the laws of the particular theory of relativity (WHAT). Solving the proposed problem with the help of nonrelativistic laws of conservation of energy and momentum is in principle wrong.

2. The essence of the photon engine design is the use of the annihilation reaction of matter and antimatter, in which photons are formed.

An example of such a reaction can be the annihilation of a proton-antiproton pair with the formation of two g-quanta. In general, any particle, annihilating with its antiparticle, can become a pair of photons. This reaction is theoretically the most advantageous for creating a jet engine, because In it particles (photons) are formed, flying at the maximum possible speed c. As Tsiolkovsky established, the efficiency of a jet engine is directly proportional to the jet flow rate (in this case, the flux of photons). Secondly, in the annihilation reaction there is theoretically the most effective "combustion" of fuel, because The rest energy of particles and antiparticles is completely transformed into photon energy.

The often used expression "when a photon engine is running, the mass is converted into energy" - it is unsuccessful. It is correct to speak about the transition of energy from one form (rest energy of matter and antimatter) to another (photon energy).

We do not discuss the question of whether it is possible to actually create a photon engine.

3. The questions posed in the problem are connected with the stage of dispersal of the photon rocket. The solution should use the relativistic energy-momentum conservation laws and the Einstein relation that relates the energy, momentum and mass of particles. We recall these relations. Let E, p, v and m be the energy, momentum, velocity and mass of a particle, respectively. Then:

E ² = p ² c ² + m ² c ⁴ ; (1)

P = Ev / c ² . (2)

Let each photon reflected from a mirror have a momentum p _g . Since the photon mass is zero, by virtue of the relation (1), the energy of each photon is equal to E _g = cp _g , where p _g is the modulus of the photon momentum. Since all photons fly parallel to each other, the total momentum carried away by photons during the acceleration of the rocket, p = ep _g . Hence the total energy of the emitted photons is E = cp.

By virtue of the law of conservation of momentum, the total momentum of the system "rocket and radiated photons" is zero, i.e. The rocket itself will receive at the end of the acceleration the momentum p ₀ = -p. Modulo these pulses are equal, therefore, the total energy of the emitted photons can be expressed through the modulus of the rocket's momentum:

E = cp ₀ . (3)

Let us write down the law of conservation of energy:

Mc ² = E ₀ + E, (4)

Where the initial energy is equal to the rest energy of the rocket before the acceleration, E0 is the energy of the rocket after the acceleration, and E is the energy of the photons emitted during the acceleration time. Taking into account relation (3), formula (4) can be written in the form:

Mc ² = E ₀ + cp ₀ . (5)

In addition, it follows from the general relation (1) that after acceleration

E ₀² - (cp ₀ ) ² = m ₀² c ⁴ . (6)

Equations (5) and (6) allow us to find the relationship between the initial and final masses of the rocket and the speed v obtained as a result of the acceleration. Indeed, by virtue of the relation (2), cp ₀ = E ₀ v / c = bE ₀ , where b = v / c. Substituting this relation into formulas (5) and (6), we obtain:

Mc ² = (1 + b) E ₀ (7)

M ₀² c ⁴ = (1 - b ² ) E ₀² . (8)

Dividing expression (8) by the square of expression (7), we find:

(9)

This is the answer to the second question of the problem.

Expressing b through the mass ratio, we get the answer to the first question of the problem:

(10)

At v = 0.999c the ratio is m ₀ / m »0.02.
We note that this problem was taken from the book by II Vorob'ev, "The Theory of Relativity in Problems" (Moscow: Nauka, 1989).

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PS The material is protected.
Date of publication 01.10.2003гг

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