Transport in Astrophysics: I. Diffusion of Solar and Galactic Cosmic Rays

Some solutions for the diffusion phenomenon as a function of time and space are reviewed. Two new solutions of the homogeneous diffusion equation in 1D and 2D are derived in the presence of an existing fixed number of particles. The initial conditions which allow deriving a power law behavior for the energy of the cosmic rays (CR) are derived. The superposition of transient diffusive phenomena on an existing power law distribution for the energy of CR allows simulating the knee, the second knee, and the ankle.


Introduction
We will briefly review the meaning of typical features of Cosmic Rays (CR). After their discovery in 1912 by Hess [1], this term started to appear in [2] [3] [4] [5]. The term "solar modulation" is connected with the solar activity and its 11-year cycle; this term was introduced by [6] followed by [7] [8] and an example is visible in Figure 6 in [9]. The solar modulation of low energy CR, as an example 0.32 GeV, can decrease the intensity by a factor of ≈20, see Figure 5 in [10]. The term "knee" in the CR energy distribution was introduced for the first time in 1961 by [11] and subsequently widely used, see as an example [12] [13] [14] [15]. The term "ankle" refers to a change in the energy distribution of CR at ≈10 7 TeV; this term was introduced in 1996 by [16] followed by [17] [18]. The term "galactic cosmic rays" was introduced by Compton [19] in order to predict asymmetries in the intensities of CR due to the motion of our sun with a velocity of 300 km/sec, which now is taken to be 230 km/s; this trend continued with [20], who introduced the scattering due to the stars and with [21] [22] where the radio-emission of our galaxy radiation was supposed to be due to the gyration of P k , [38], the modulation of CR by an interplanetary shock wave [39], time profiles in the intensity of solar CR at various distances from the source as a function of the ratio of the mean free path to the focusing length of the interplanetary field [40], solution of the diffusion equations adopting realistic models of the galactic field and using diffusion coefficients appropriate for strong turbulence [41], the computation of the perpendicular diffusion coefficients and mean free paths of particles for an anisotropic Alfvénic turbulence spectrum, [42], asymmetric diffusion in the presence of high-amplitude magneto-hydrodynamic turbulence [43], a two-component model for the evolution of fluctuations of solar wind plasma [44] and anomalous transport phenomena associated with galactic CR propagating through interstellar space [45]. The present paper reviews the existing situation of the solutions of the diffusion equation and derives two new solutions in 1D and 2D, see Section 2. The astrophysical applications to CR allow building an energy spectrum similar to the observed one, and simulate the knee, the second knee and the ankle, see Section 3.

Transient Diffusion
This section reviews the definition of the diffusion coefficient, Fick's second law for diffusion, the impulsive 1D, 2D and 3D solutions, introduces two new solutions (2D and 3D) for the impulsive case in the presence of an existing fixed profile and reviews the diffusion internal to a ball when the density on the boundary is constant over time.

The Diffusion Coefficient
The dependence for the mean square displacement, where d is the number of spatial dimensions. From Equation (1), the diffusion coefficient is derived in the continuum: Using discrete time steps, the average square radius after N steps, Equation (12.5) in [46], is ( ) from which the diffusion coefficient is derived: when the step length of the walker or mean free path between successive collisions is λ and the transport velocity is tr v .

Fick's Second Law
Fick's second law in 3D states that a change in concentration, N, in any part of the system is due to an inflow and an outflow of material into and out of that part of the system ( ) where D is the diffusion coefficient, t is the time and 2 ∇ is the Laplacian operator.

1D Case, Impulsive Injection
In 1D, Fick's second law is and a first solution is of Gaussian type A second solution is given by and ( ) erf x is the error function [47]. Figure 2 reports a comparison of the two solutions here analysed.

2D Case, Impulsive Injection
In 2D, Fick's second law in polar coordinates is A solution in the impulsive case is   (13) where 0 N is the number of particles injected in the time dt. The concentration as a function of r has a maximum at

3D Case, Impulsive Injection
In 3D, Fick's second law in spherical coordinates is which has a solution ( ) ( )

1D Case, Existing Profile
We now solve Fick's second law in 1D as given by Equation (7) where s is an adjustable parameter and 0 N the number of particles at 0 x = .
The boundary conditions are assumed to be and the solution is

2D Case, Existing Profile
We now solve Fick's second law in 2D, which is assuming the following profile of density at where L is the side of the square. The boundary conditions are assumed to be An example is reported in Figure 4.

3D Case, in the Ball
The propagation of the temperature in a ball has been investigated, see https://www.math.hmc.edu/ajb/PCMI/lecture_schedule.html, and due to the analogy between the heat equation and the diffusion equation, we adopt the same , . Figure 5 reports an example of the time necessary to fill the ball.

Astrophysical Applications
In the following, the space is expressed in pc and the time in years.

The Relativistic Gyroradius
The relativistic gyroradius or Larmor radius is where m is the mass of the particle, c is the speed of light, is the Lorentz factor, v is the velocity of the particle, q is the charge of the particle, B is the magnetic field and v ⊥ is the velocity perpendicular to the magnetic field, see formula (1.54) in [48] or formula (7.3) in [49]. In the case of CR we express the Larmor radius in pc 6 GeV 15 6 6 1.081 10 1.081 pc , where Z is the atomic number, GeV E is the energy expressed in GeV, 15 E is the energy expressed in 10 15 eV, and 6 B − is the magnetic field expressed in 10 −6 gauss, see [50]. On assuming that the CR diffuse with a mean free path equal to the relativistic gyroradius, the transport velocity is equal to the speed of light and 3 d = , the diffusion coefficient according to Equation (5), is pc . year year

Cosmic Rays
The observed differential spectrum of CR according to [51] [52] [53] [54] is reported in Figure 6 in the H case ( H I ).
Flux of H versus energy per nucleus in Gev: experimental data (empty stars) and theoretical power law (full line), see Figure 6.

The Spectral Index
In the 3D impulsive case, see solution (16), the ratio between the two values of concentration characterized by min D and max D is

Modulation of CR at Low Energies
The spectrum of CR without the influence of the heliosphere was measured in the summer of 2012 by the Voyager 1 spacecraft, see [55] [56] [57], and is reported in Figure 9 where the range [2.06 × 10 −3 GeV -0. 58  3.992 10 a − = × , see Figure 10.
In the framework of the 3D impulsive case, see solution (16), we now report in Figure 11 the number of CR at 1 au, imp N , as a function of the energy in GeV. The resulting total number of CR, tot N , is Figure 12 reports the total number of CR as evaluated in Equation (34).

Variable Number of Injected Particles
We now analyse the case where the number of injected particles at 0 t = varies as a function of the coefficient of diffusion, analysing the following 3D solution     , , , , L. Zaninetti International Journal of Astronomy and Astrophysics The distribution in energy of this formula is reported in Figure 14 for the case of the Voyager 1 spacecraft and in Figure 15 for the whole spectrum of CR [54].

The Knee and the Second Knee for CR
The Kascade experiment [59] has measured the CR with energies in the range International Journal of Astronomy and Astrophysics   , where a f is the concentration at ( ) 0, a and min E the minimum energy considered. We now assume that for a preexisting spectrum of energy, .
The preexisting number of CR is assumed to be  Table 1. Figure 17 reports tot N as well the data of the Kascade experiment.

The ankle for CR
The ankle in the distribution of high-energy CR is at ≈10 7 TeV and characterizes the transition from galactic to extra-galactic CR [62]; Figure 18 reports the experimental energy distribution according to Figure 1 in [63].
We now test solution (40), which represents an impulsive 3D solution in the presence of a variable number of injected particles situated at 20 pc r = , which ( ) tot N E as function of energy (full line) with data as in Table 1 and data of the Kascade experiment (empty stars). Figure 18. CR energy in TeV: experimental data (empty stars) according to [63].
is the distance of the expanding surface of the local bubble. Figure 19 reports the existing number of CR, exi N , evaluated as in Equation (40). We now add to an International Journal of Astronomy and Astrophysics existing power law distribution an impulsive phenomena; Figure 20 reports the number of CR, imp N , in a burst at 20 pc r = .
The total number of CR as evaluated with formula (34) is reported in Figure   21, which thus makes visible a theoretical explanation for the ankle. Figure 19. Flux of H versus energy per nucleus in Gev: experimental data (empty green stars) according to [63] and theoretical existing distribution (red full line) when

PDE & Boundary Conditions
Two new solutions for the diffusion equation have been derived: the first one was derived for the case of a 1D diffusion and an exponential profile for the number of particles, see Equation (19); the second one gives the 2D diffusion for the case of a power law profile for the number of particles, see Equation (22).

CR and Diffusion
In order to obtain a distribution in energy for cosmic rays (CR) in 3D which can be compared to the observed one, the number of injected particles should be a function of the diffusion coefficient, see Equation (35), or its energy equivalent, see Equation (37). The theoretical distribution has a maximum at the value of the diffusion coefficient given by Equation (36) or at the value of energy represented by Equation (38).

Knee and Ankle
In the framework of summing an existing distribution in energy for the CR originating from the local bubble with that originating from other sources, see Equation (42), it is possible to simulate the knee and the second knee, see Figure   17. The ankle is simulated in Figure 19 through the superposition of two events of solar origin assuming a time independent solution for the energy as given by Equation (40).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.