Cosmic rays density fluctuations with turbulence universal spectrum-8/3

The gist: An exact turbulence universal scaling law-8/3 for cosmic ray density fluctuations is obtained on the basis of an exact analytical solution to the problem of compressible turbulence and known two-fluid model of CR dynamics.

Theoretical Framework

The study establishes a universal turbulence scaling law-8/3 for the density fluctuations of cosmic rays (CR) by using an exact analytical solution to the problem of compressible turbulence [17]. This is achieved through specific solutions of the Boltzmann-Vlasov equations obtained in [23] and by considering the isothermal mode of propagation where momentum and energy are conserved, which leads to a description using Euler hydrodynamics equations for a compressible medium with an adiabatic exponent  = 1, as a particular case of the known two-fluid dynamic (TFD) theory [8].

Inertial Motion and Governing Equations

The inertial motion of cosmic ray particles is analyzed based on the collision-less Boltzmann-Vlasov equation, which expresses conservation of CR particles in phase space [23]. For ultra-relativistic particles under certain conditions, the CR particles are considered as a politropic gas with pressure proportional to number density and an adiabatic exponent derived from specific relations [23]. When conditions (4) and (5) are met, leading to invariant energy and momentum for the CR particles, the Euler equation for the velocity field of such a medium is used alongside the continuity equation for density and an isothermal equation of state [8]. These equations exactly coincide with those of the two-fluid model [8] under conditions of conservation of cosmic ray energy [23].

Derivation of Universal Spectrum

Based on the exact solution (9) for the one-dimensional system, a representation for the two-point correlation function of density fluctuations and its corresponding spectrum is obtained [17]. In the limit where wave number k is much larger than L, an asymptotic representation (11) for the energy spectrum is derived, which takes the form of a power law: E k = C k - 8/3k + O(k-10/3) [17]. The universality of this spectrum is manifested in the independence of the exponent-8/3 from the type of initial velocity field, suggesting that regardless of initial conditions, a turbulent spectrum corresponding to this universal scaling law should be observed [17].

Comparison with Observations and Dimensional Analysis

The obtained exact solution (11) for the universal turbulent spectrum is characterized by the scaling law-8/3, which does not depend on the initial conditions and on any parameters characterizing the equation of state of the medium [4]. The dependence of a multiplier C on initial velocity field derivatives at point Mx=x indicates that while the exponent is universal, C significantly depends on these factors [17]. Furthermore, dimensional analysis and similarity theory suggest an analogue scaling law (12) involving fundamental constants like G, c, or h instead of the speed of light [13]. This leads to scaling laws such as 8/3(5/6 - kGCRωcE CRωρ) [12], demonstrating how the universal scaling law-8/3 can be established through dimensional considerations and similarity principles [12].

Conclusions and Future Directions

The paper concludes that the isothermal mode of propagation of cosmic ray particles, described by one-dimensional Euler equations for a compressible medium, yields an explicit analytical form for a two-point correlation function and the universal spectrum-8/3 [17]. This result is matched with known observation data [14] (see also Fig.1) [17]. Future experiments are suggested to consider simultaneous measurements of density fluctuations at two different points in space along the propagation direction, alongside measurements of the speed of sound S c in a medium consisting of CR particles, for a more complete comparison with the exact solution (11) [4]. The establishment of this spectral law is also linked to the collapse of a nonlinear simple Riemann wave in the medium of cosmic ray particles [19] [4].

--- Page 1 ---

The gist: An exact turbulence universal scaling law-8/3 for cosmic ray density fluctuations is obtained on the basis of an exact analytical solution to the problem of compressible turbulence and known two-fluid model of CR dynamics.

--- Page 2 ---

Cosmic rays (CR) are the most ubiquitous population of charged relativistic particles in the universe, which play a crucial role in understanding the problems of high-energy physics, plasma physics, as well as the fundamental and applied problems of the atmospheric electricity and climate variability on the Earth [1]-[13]. CR are particles (with energies E > 1GeV / nucleon) or photons (with energies E > 1MeV) pervading space, having a non-thermal energy distribution typically of the power-law (dN E dE - ) rather than Maxwell-Boltzmann form [1]-[6]. Here the differential energy spectrum dN(E)/ dE = Vn / 4 ≈ cn / 4 is closely related to the particle volume density n(E) of CR, where V and c is the particle velocity and the speed of light in vacuum, respectively. According to the observational data, for protons (which make up the main part of CR) with energies GeV E GeV 6 < E < 3 x 10 a power dependence dN E dE / dE ≈ E takes place [3]-[6].

How it works

The derivation relies on an exact analytical solution to the problem of compressible turbulence [17]. This is achieved through specific solutions of the Boltzmann-Vlasov equations obtained in [23] and by considering the isothermal mode of propagation where momentum and energy are conserved, which leads to a description using Euler hydrodynamics equations for a compressible medium with an adiabatic exponent  = 1, as a particular case of the known two-fluid dynamic (TFD) theory [8].

1. Bold spectral law prediction for cosmic ray density fluctuations: The improved AI system can accurately predict a universal turbulent spectrum of density fluctuations with a scaling law of -8/3, as derived from the exact solution of the Euler equations for a compressible medium, which does not depend on the initial conditions and on any parameters characterizing the equation of state.

2. Linking turbulence to cosmic ray acceleration mechanisms: The system can correlate turbulent spectra with particle acceleration by utilizing findings such as the establishment of the spectral law -8/3 also takes place, starting from the scale of the Larmor radius of ions, allowing it to infer that this scaling is due to the mechanism noted in [19] associated with the collapse of a nonlinear simple Riemann wave in the medium of cosmic ray particles.

3. Modeling plasma dynamics via two-fluid theory: The improved AI can describe cosmic ray medium dynamics using Euler hydrodynamics equations for a compressible medium with specific value of the adiabatic exponent  = 1, which exactly coincide with the equations of the two-fluid model [8] under the conditions of conservation of cosmic ray energy.

4. Predicting collapse times in turbulent media: The system can estimate physical timescales, such as the collapse time of a nonlinear wave describing a medium consisting of cosmic ray particles, providing estimates like t 1.4 10 sec 0.5year based on the condition that the collapse time of a nonlinear wave in such an environment has a value of about six months.

5. Constraining observational data interpretation: The system can perform more rigorous comparisons with experimental data by utilizing additional measurements of two point correlations for fluctuations in the density of cosmic ray particles and the corresponding spectrum as in [14], which will help make appropriate estimates for the spatial spectrum exponent taking into account the Taylor hypothesis.