Some known analytical solutions for a cascade shower were used for determining the intensity of excess electrons' radio-emission. It is shown that the energy spectrum has a sharp decline in the region of high frequencies. At a usual root-mean-square radius of the disk within the Moon's regolith ~ 0.06m the maximum of the spectrum is in the region ~ 0.5-0.6 GHz while in the range 2-2.5 GHz the pulse's amplitude drops by 5-6 orders. This fact is a consequence of loss of coherence due to finite dimensions of the cascade disk and demands conducting a significant correction of "radio-frequency window" for measuring cosmic rays' flux with a radio telescope.
It is known that motion of a charge with super-light velocity in a dielectric is accompanied by Cherenkov radiation. In an idealized model of Cherenkov radiation at motion of a point charge with constant velocity through a homogeneous matter the duration of emitted radio pulse is indefinitely small and the radiation spectrum is practically unlimited if the angle of observation is ) n / 1 arccos( = q . A real system of charges has finite dimensions, a limited region of motion and an observation angle d i f f e r e n t t h a n t h e C h e r e n k o v o n e . I n t h i s p a p e r , t h e s e e m i s s i o n p a r a m e t e r s a r e investigated for a real object, which appears during the formation of a cascade shower of ultrahigh-energy particles ( eV 10 10 22 16 ¸) in a gaseous or solid media. As a result of its interaction with the environment a clot of electrons and positrons, having in the maximum stage of the shower values quantities of the order of 13 7 10 10 ¸, is being formed. This cluster has the shape of a disk and moves faster than the speed of light in this environment. The thickness of the disk is thousands of times smaller than its radius.
The highest charge density is achieved in a typical disk at its center, and one may approximately assume that the particle distribution has axial symmetry. Due to various processes associated with interaction of atoms of the medium with the particles of the shower, an excess of electrons in the disk is created whose energy corresponds to the Lorentz factor g » 1 [1]. Therefore, the motion of particles of the disk, even in such a rarefied medium as the Earth’s atmosphere, is accompanied by a coherent Cherenkov radiation in radio frequencies, because in standard atmosphere, with the rate 100 50 ~ģ its velocity is greater than the velocity of electromagnetic wave [2].
Knowledge of the characteristics of the radiation field of the cascade disk is of interest in connection with the attempts to use coherent radiation for the detection of cosmic rays of ultrahigh energies. In this regard, since 1965, experiments were conducted studying the radio emission produced by extensive air showers (see, for example, the survey [3]). In [4] for the first time appeared the idea of the possibility of recording a radio pulse, caused by ultrahigh-energy particles on the Moon’s surface using a radio telescope. Currently there are about a dozen experimental studies, but none of them have reliably reported signs of radio pulses from lunar origin [5 -12].
A detailed study of the characteristics of the radiation with the help of software systems has been carried out in [13], whose results so far are essential for conducting experimental work. Parametrization of the field at the Cherenkov angle of observation is presented in this paper by the expression
where GHz 5 . 0 0 = n . The energy power spectrum of this radio pulse in relative units is shown in Fig. 5 (curve 1).
Subsequently, these results were slightly modernized, but the main feature has remained virtually unchanged. It’s the increase in the energy power spectrum
r up to ~10-100 GHz [7,12] without a significant decrease. And consequently in later experiments the frequency range for the registration of radio pulses was chosen to be in the region of several GHz or even higher [10].
On the pulse duration.
Model of a shower in the form of a disk propagating in a homogeneous medium leads to some conclusions about the duration of the radio pulse. On Fig. 1 АВ is a disk with an excess of charged particles moving in the direction of v r . AD and BC -direction of radio waves incident on the antenna CD. . This duration differs significantly from the results of calculations [13]. This refers to the following. Expression (1) allows building a radio pulse as a function of time using the Fourier transform. The phase required for this is presented in Fig. 16 [13]. According to [13] in the frequency range 0 -10 GHz the phase varies by no more than 40 degrees. It means that the phase dependence on frequency can be almost neglected, and we can write the inverse Fourier transform in the form of
The result of this recovery is shown in Fig. 2 (curve 1). . This fact could explain the growth of the spectrum to such high frequencies (~ 10 GHz). However, above this frequency, the phase is unknown and so this explanation is not convincing. However, the expression (2) for the upper limit GHz 10 1 = w
shows that the time of the jump is 0.08 ns. This data, together with the expected duration ns 55 . 0 s 10 5 . 5 10 = = t i s v ery al arm i ng an d stim u l at e to review the expression (1) and modernize it, as it’s performed in several works. For example, in the experiment GLUE [7] they used the parametrization of the spectral energy in the form of a radio pulse (Fig. 3, curve 1)
where GHz 5 . 2 0 = n . In the NuMoon project [11] they used parametrization (Fig. 3,
where GHz 5 . 2 0 = n and in the RESUN project (12) they calculated the intensi
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