IZhO 2010, Теория, английски — Задача 3. The relic radiation and cosmic rays

VI International Zhautykov Olympiad. Theoretical Competition · 13 януари 2010 г. · 10 т.

Условие

According to modern cosmology, our universe is filled with electromagnetic radiation, remained after the Big Bang and called the cosmic relict radiation, or cosmic microwave background (shortly, CMB). With good accuracy, the CMB is homogeneous and isotropic in the reference system associated with our Galaxy which we take as the laboratory reference system. The frequency distribution of the CMB coincides with the spectrum of a blackbody at temperature $T = 2.7\ \mathrm{K}$. Discovery and study of properties of the CMB were awarded Nobel Prizes in Physics in 1978 and 2006.

Relict photons are extremely numerous and, therefore, may affect Galaxy radiation of another nature called cosmic rays (shortly, CR). It is believed that CR are formed in stellar explosions. CR consist mainly of protons whose energy can exceed by many orders the energy of terrestrial accelerators. The mechanism of generation of ultrahigh energy CR is not entirely clear, but the experimentally observed energy distribution of CR is limited above by the energy $E_p^{max} = 10^{21}\ \mathrm{eV}$. In this problem it is assumed that this limitation is caused by energy loss in the process of interaction between protons and the CMB. Protons can participate in the Compton scattering

and cause the reaction

where $p$ is a proton, $\gamma$ is a relict photon, $\Delta$ is the lightest, as compared to the nucleon, baryon with the rest mass $m_\Delta = 1.232 \times 10^6\ \mathrm{eV}/c^2$, which quickly decays into pi-meson $\pi^0$ and proton $p$. ($\Delta$-particle decays also to $\pi^+$-meson and neutron. Neutron quickly turns into proton by $\beta$-decay, so the consideration of this reaction channel is not essential in this Problem). Due to the birth of $\Delta$-particles, the probability of interaction of protons with $\gamma$-quanta increases dramatically.

The purpose of this Problem is to determine the upper limit of the observed energy spectrum of CR, supposing that reaction (2) is the main mechanism of energy loss of CR, and compare the loss of energy protons in the reactions (1) and (2).

In the following questions, you have to measure the energy in electron-volts $\mathrm{eV}$, and momenta in $\mathrm{eV}/c$, where $c$ is the speed of light in vacuum. The rest mass of the proton is $m_p = 938 \times 10^6\mathrm{eV}/c^2$, the rest mass of pi-meson is $m_\pi = 140 \times 10^6\mathrm{eV}/c^2$. Boltzmann constant is $k = 1.38 \times 10^{-23}\text{Дж} \cdot K^{-1}$.

1. Estimate most probable energy $E_\gamma$ and the corresponding momentum $p_\gamma$ of relict photons, given that they correspond to the blackbody radiation at temperature $T = 2.7\ \mathrm{K}$.

It is assumed in the following questions that the initial photon in reactions (1) and (2) has the energy and the momentum found Question 1.

2. The rest mass $m$ of a particle is related to the total relativistic energy $E$ and momentum $\vec{p}$ of this particle in an arbitrary inertial reference system as $E^2/c^2 - \vec{p}^{\,2} = m^2c^2$. Here, the quantity $mc^2$ does not depend on the reference system and is a complete internal energy of the particle. Write a corresponding expression for the total internal energy of a system of two non-interacting particles (i.e. the total energy in the reference frame in which the total momentum of physical system is equal to zero) having the total energies $E_1$, $E_2$ and momenta $\vec{p}_1$, $\vec{p}_2$.

For further analysis you may need the relativistic law of transformation of momentum and energy of the particle. In the transition from the inertial reference system $S$ to an inertial system $S'$, moving along the positive direction of the axis $Z$ ($OZ \uparrow\uparrow OZ'$) with the speed $V_0$ in the reference system, energy and momentum transform like the coordinates of space-time point $(x, y, z, t) \to (x', y', z', t')$:

Here,

3. Determine the lowest possible energy of the proton at which reaction (2) turns possible:

a) in the center of mass of proton and photon $p + \gamma$ (i.e., the reference system in which the total momentum of the proton and photon is zero);

b) in the Galaxy system of reference.

Starting from the obtained results, determine the maximum energy of protons in cosmic rays.

4. Assuming that in the Galaxy system a proton has the maximum energy $E_p^{max}$ and commits a head-to-head collision with a relict photon, determine the momentum of pi-meson $\pi^0$ in reaction (2) in this frame of reference for the case when the emitted particles move along or against the direction of the initial momentum of the proton. What is the change of the momentum of the proton in this case?

5. What is the value of momentum of the outgoing photon in the Compton scattering (1) under the same conditions as in the Question 4?

6. Is reaction (2) with the cosmic relict radiation possible, if the initial momenta of the proton and the photon are parallel in the Galaxy system? If the reaction is possible, what is the minimum value of the photon momentum in the Galaxy system?

Решение

Оригинал в Архива: IZhO-2010-Theory_eng.pdf · официални решения: IZhO-2010-Theory_eng_sol.pdf