Modulation of Galactic Cosmic Ray Intensity in the Approximation of Small Anisotropy

Abstract—

The propagation of cosmic rays in the interplanetary medium based on the transport equation is considered. The solution of the cosmic ray transport equation is obtained for the known energy distribution of high-energy charged particles at the heliospheric boundary. The spectrum of galactic cosmic rays in the local interstellar medium is taken on the basis of the data from the Voyager 1 and 2 spacecraft. The flux of galactic cosmic rays in different periods of solar activity is calculated. Cosmic ray intensity gradients are estimated, and these calculations are compared to the data from space missions. The anisotropy of the angular distribution of cosmic rays is calculated. It is shown that the flux of galactic cosmic rays in the Earth’s orbit has an azimuthal direction, and the value of the anisotropy of protons with energies from 1 MeV to 1 Gev is of the order of 0.5%.

INTRODUCTION

The intensity of galactic cosmic rays (GCRs) is modulated by the interaction of high-energy charged particles with the electromagnetic fields of the solar wind. Numerous long-term measurements of the intensity of cosmic rays by the ground-based neutron monitor network showed that there is an 11-year cycle of GCR intensity (which anticorrelates with solar activity) as well as cosmic ray intensity (CR) variations with a 22-year period [10, 12, 12, 16]. In order to elucidate the mechanisms of GCR modulation in the interplanetary medium, it is necessary to know the structure of the heliosphere and the spatial distribution of the interplanetary magnetic fields. The data obtained by the Pioneer and Voyager spacecraft (SC), which explored remote regions of the heliosphere, are of great importance. In particular, the data obtained by the Voyager-1 SC, which left the heliosphere in 2012, made it possible to determine the CR energy spectrum in the local interstellar medium to MeV energies inclusive [21, 23]. Information about the unmodulated energy spectra of GCRs is necessary in calculations of the spatial–energy distribution of particles within the heliosphere. The Ulysses space mission explored the polar regions of the heliosphere and obtained important data on the structure of the heliosphere and on the distribution of GCRs in the interplanetary medium [10–12, 16, 17]. The trajectory of the Ulysses spacecraft provides a unique opportunity to study the spatial modulation of cosmic rays under conditions of solar minimum [10, 11, 16, 17]. The simultaneous detection of CR intensity by the Ulysses spacecraft (at high heliolatitudes) and satellites IMP-8, AMS, and PAMELA (at near-Earth orbits) made it possible to obtain unique data regarding the spatial distribution of GCRs during different periods of solar activity [10–12, 16, 17].

The theoretical study of the CR intensity modulation process is based on solutions of the CR transport equation with appropriate conditions at the heliospheric boundary. It is known that the observed anisotropy of the angular distribution of GCRs is a small value for the particles of wide energy range. This experimental fact allows us to apply the approximation of small anisotropy to the theoretical investigation of charged particle propagation in magnetic heliospheric fields. In [9, 22], a method for the approximate solution of the transport equation taking into account the small GCR anisotropy was proposed. The radial flux density of particles with a given momentum value is assumed to be zero. The method consists in solving the corresponding partial derivative equation provided that the CR energy spectrum at the boundary of the modulation region is known [9, 22]. This approach was elaborated in [4], where an iterative procedure for taking into account a small parameter, the anisotropy of the angular distribution of GCRs, was proposed. The approximation of small anisotropy was used, e.g., in the works on the CR propagation in heliospheric magnetic fields [5, 7, 8, 13–15, 17]. In this work, the CR radial flux density and the particle flux along the heliolatitude based on the solution of the CR transport equation were calculated. The gradients of the GCR intensity and anisotropy of the angular distribution of the particles were estimated.

COSMIC RAY TRANSPORT EQUATION

Let us write down the equation of cosmic ray transport in the interplanetary medium as follows [1, 2, 20]

$$\frac{{\partial N}}{{\partial t}} + {\text{div}}\vec {j} + \frac{1}{{{{p}^{2}}}}\frac{\partial }{{\partial p}}{{p}^{2}}{{j}_{p}} = 0,$$

(1)

where

$${{j}_{\alpha }} = - {{\kappa }_{{\alpha \beta }}}\frac{{\partial N}}{{\partial {{r}_{\beta }}}} - \frac{{{{u}_{\alpha }}p}}{3}\frac{{\partial N}}{{\partial p}}$$

(2)

is the particle flux density with concentration N(r, p, t), κ_αβ is the CR diffusion tensor, and u is the solar wind speed. The value of

$${{j}_{p}} = \frac{{up}}{3}\frac{{\partial N}}{{\partial r}}$$

(3)

is the flux density of particles in the space of absolute momentum values.

The CR diffusion tensor has the following form [6]

$${{\kappa }_{{\alpha \beta }}} = {{\kappa }_{ \bot }}{{\delta }_{{\alpha \beta }}} + ({{\kappa }_{{11}}} - {{\kappa }_{ \bot }}){{h}_{\alpha }}{{h}_{\beta }} + {{\kappa }_{A}}{{\varepsilon }_{{\alpha \beta \gamma }}}{{h}_{\gamma }},$$

(4)

where δ_αβ is the Kronecker symbol, ε_αβγ is the unit antisymmetric tensor, h = H/H is the unit vector in the direction of large-scale interplanetary magnetic field H, κ₁₁ is the particle diffusion coefficient in the direction of the mean magnetic field, κ_⊥ is the particle diffusion coefficient in the direction perpendicular to the magnetic field, and the value κ_A defines antisymmetric components of the CR diffusion tensor.

Components of the CR flux density are as follows

$${{j}_{r}} = - {{\kappa }_{{rr}}}\frac{{\partial N}}{{\partial r}} - \frac{{up}}{3}\frac{{\partial N}}{{\partial p}} - {{\kappa }_{A}}{{h}_{\varphi }}\frac{1}{r}\frac{{\partial N}}{{\partial \theta }},$$

(5)

$${{j}_{\theta }} = - {{\kappa }_{ \bot }}\frac{1}{r}\frac{{\partial N}}{{\partial \theta }} + {{\kappa }_{A}}{{h}_{\varphi }}\frac{{\partial N}}{{\partial r}}.$$

(6)

Consider the spatial distribution of particles averaged over the solar rotation period so that the concentration of particles does not depend on the azimuthal angle φ. Let us define the dimensionless coordinate

$$\rho = \frac{r}{{{{r}_{0}}}}$$

(7)

and the dimensionless momentum of the particle

$$\varsigma = \frac{p}{{mc}},$$

(8)

where m is the rest mass of the particle and r₀ is the distance to the boundary of the CR modulation region. We assume that the energy spectrum of GCRs is given at the boundary of the heliosphere. The transport path of the particle increases as the energy of the particle increases. Assume that all components of the CR diffusion tensor change in proportion to the momentum of the particle. In this case

$${{\kappa }_{{rr}}} = {{\kappa }_{0}}\varsigma ;\,\,\,\,{{\kappa }_{ \bot }} = {{\delta }_{ \bot }}{{\kappa }_{0}}\varsigma ;\,\,\,\,{{\kappa }_{A}} = {{\delta }_{A}}{{\kappa }_{0}}\varsigma .$$

(9)

In dimensionless variables, the components of particle flux density take the following form.

$${{j}_{r}} = - \frac{{u\varsigma }}{\mu }\left\{ {\frac{{\partial N}}{{\partial \rho }} + \frac{\mu }{3}\frac{{\partial N}}{{\partial \varsigma }} + {{\delta }_{A}}{{h}_{\varphi }}\frac{1}{\rho }\frac{{\partial N}}{{\partial \theta }}} \right\},$$

(10)

$${{j}_{\theta }} = - \frac{{u\varsigma }}{\mu }\left\{ {{{\delta }_{A}}{{h}_{\varphi }}\frac{{\partial N}}{{\partial \rho }} - \frac{{{{\delta }_{ \bot }}}}{\rho }\frac{{\partial N}}{{\partial \theta }}} \right\},$$

(11)

where

$$\mu = \frac{{u{{r}_{0}}}}{{{{\kappa }_{0}}}}$$

(12)

is the CR modulation parameter.

APPROXIMATE SOLUTION OF THE TRANSPORT EQUATION

It is known that the angular distribution of GCRs in the interplanetary space is close to isotropic so that the CR flux density turns out to be a small value [9, 5, 18, 22]. By equating expression (11) for the heliolatitude particle flux density to zero, we can express the derivative $\frac{{\partial N}}{{\partial \theta }}$ through the CR concentration derivative ρ. Let us use the obtained relation in radial particle flux expression (10) and equate the value j_r (10) to zero. We obtain the following equation

$$\left( {1 + \frac{{\delta _{A}^{2}}}{{{{\delta }_{ \bot }}}}h_{\varphi }^{2}} \right)\frac{{\partial N}}{{\partial \rho }} + \frac{\mu }{3}\frac{{\partial N}}{{\partial \varsigma }} = 0.$$

(13)

Consider that the particle concentration at the boundary of the heliosphere (at the point ρ = 1) is a given function of the particle momentum:

$$N(1,\varsigma ) = {{N}_{0}}(\varsigma ).$$

(14)

The solution to Eq. (13) satisfying boundary condition (14) is as follows:

$$N(\rho ,\theta ,\varsigma ) = {{N}_{0}}(\xi ),$$

(15)

where

$$\xi = \varsigma + \frac{\mu }{3}\int\limits_\rho ^1 {\frac{{d\rho }}{{1 + \frac{{\delta _{A}^{2}}}{{{{\delta }_{ \bot }}}}h_{\varphi }^{2}}}} .$$

(16)

In the model of the interplanetary magnetic field proposed by Parker [19], the heliolatitude component of the magnetic field is zero (H_θ = 0). The radial and azimuthal components of the magnetic field strength are defined by the following relations [12, 19]

$${{H}_{r}} = {{H}_{{0r}}}{{\left( {\frac{{{{r}_{s}}}}{r}} \right)}^{2}},$$

(17)

$${{H}_{\varphi }} = - {{H}_{{0r}}}\frac{{r_{s}^{2}\Omega }}{{ru}}\sin \theta ,$$

(18)

where r_s is the solar radius, Ω is the angular velocity of the solar rotation, and H_0r is the radial component of the interplanetary magnetic field strength near the solar surface.

The azimuthal component of the unit magnetic field vector is as follows:

$${{h}_{\varphi }} = - \frac{{\rho \omega \sin \theta }}{{\sqrt {1 + {{\rho }^{2}}{{\omega }^{2}}{{{\sin }}^{2}}\theta } }}\left[ {1 - 2\Theta \left( {\theta - \frac{\pi }{2}} \right)} \right],$$

(19)

where

$$\omega = \frac{{{{r}_{0}}\Omega }}{u},$$

(20)

and Θ(x) is the unit Heaviside function. Note that relation (19) corresponds to the A⁺ epoch. During this epoch, the radial component of the interplanetary magnetic field strength in the northern hemisphere of the heliosphere has a positive sign (H_0r > 0).

By integrating with formula (16), we obtain the relation

$$\xi = \varsigma + \frac{{\mu {{\delta }_{ \bot }}}}{{3({{\delta }_{ \bot }} + \delta _{A}^{2})}}\left\{ {1 - \rho + \frac{{\delta _{A}^{2}}}{{\sqrt {{{\delta }_{ \bot }}({{\delta }_{ \bot }} + \delta _{A}^{2})} }}} \right.\left[ {\arctan \frac{{\sqrt {{{\delta }_{ \bot }} + \delta _{A}^{2}} }}{{\sqrt {{{\delta }_{ \bot }}} }}} \right.\omega \sin \theta - \left. {\left. {\arctan \frac{{\sqrt {{{\delta }_{ \bot }} + \delta _{A}^{2}} }}{{\sqrt {{{\delta }_{ \bot }}} }}\rho \omega \sin \theta } \right]} \right\}.$$

(21)

Therefore, the CR concentration with the momentum ζ at a point with coordinates ρ, θ inside the heliosphere is determined by the energy distribution of particles specified at the boundary of the modulation region according to relations (15) and (21).

Further, it is necessary to set the energy spectrum of the GCRs at the boundary of the heliosphere. Let us use the results obtained by the Voyager-1 spacecraft in the local interstellar medium after it left the heliosphere in August 2012 [21, 23]. It is known that the modulation of particles with energies greater than 30 GeV is negligibly small. Therefore, in the high energy region, we can use, e.g., the data from the PAMELA SC and AMS-02 module [21, 23]. In the region of ultrarelativistic energies, the CR intensity is proportional to the particle kinetic energy E_k raised to the power of –2.78 [21, 23]:

$$I({{E}_{k}}) \propto E_{k}^{{ - 2.78}}.$$

According to the Voyager-1 data in the nonrelativistic energy region [21, 23],

$$I({{E}_{k}}) \propto E_{k}^{{0.12}}.$$

From the known dependence of the CR intensity on the kinetic energy of the particle, we can determine the CR concentration dependence on the momentum [18, 23]. For nonrelativistic protons (ζ $\ll$ 1), we will have

${{N}_{0}}(\varsigma ) \propto {{\varsigma }^{{ - \beta }}}$, where β = 1.76.

In the high energy region (ζ $\gg$ 1),

${{N}_{0}}(\varsigma ) \propto {{\varsigma }^{{ - \gamma }}}$, where γ = 4.78.

Let us give a formula for the proton energy distribution at the heliospheric boundary [7, 8]:

$${{N}_{0}}(\varsigma ) = {{q}_{0}}{{\varsigma }^{{ - \beta }}}{{(1 + {{\varsigma }^{2}})}^{{\frac{{\beta - \gamma }}{2}}}},$$

(22)

where q₀ is the constant that can be determined, e.g., on the basis of the CR energy density value in the interstellar medium. According to formula (22), the spectrum of ultrarelativistic particles (ζ $\gg$ 1) turns out to be a power-law with exponent γ. For nonrelativistic particles (ζ $\ll$ 1), a power dependence of CR concentration on momentum with exponent β follows from formula (22). For the chosen values of the parameters β, γ particle spectrum (22) agrees with the energy distribution of CRs, which was recorded by Voyager-1 after it left the heliosphere [21, 23].

Figure 1 shows the dependence of the CR concentration (15), (21), (22) on the heliocentric distance. The concentration N of particles is normalized to the corresponding value N₀ at the boundary of the modulation region (22).

Fig. 1.

Dependence of the cosmic ray concentration on the heliocentric distance.

The following parameters are chosen: μ = 1.5, δ_⊥ = 0.02, δ_A = 0.01. The curves correspond to particle kinetic energies of 5 GeV, 1 GeV, 500 MeV, and 100 MeV. The value of kinetic energy is given near the corresponding curves. It can be seen that the CR modulation level decreases as the kinetic energy of the particles increases (Fig. 1). The CR concentration monotonically increases with increasing heliocentric distance. Note that the dependence of CR concentration on heliolatitude is insignificant for these parameter values, and it cannot be seen in Fig. 1.

COSMIC RAY FLUX DENSITY

Let us write CR radial flux (10) as the following sum

$${{j}_{r}} = j_{r}^{s} - \frac{{u\varsigma }}{\mu }{{\delta }_{A}}{{h}_{\varphi }}\frac{1}{\rho }\frac{{\partial N}}{{\partial \theta }},$$

(23)

where

$$j_{r}^{s} = - \frac{{u\varsigma }}{\mu }\left\{ {\frac{{\partial N}}{{\partial \rho }} + \frac{\mu }{3}\frac{{\partial N}}{{\partial \varsigma }}} \right\}.$$

(24)

Cosmic ray transport Eq. (1) takes the following form:

$$\frac{1}{{{{\rho }^{2}}}}\frac{\partial }{{\partial \rho }}{{\rho }^{2}}j_{\rho }^{s} + {{w}_{r}}\frac{{\partial N}}{{\partial \rho }} + {{w}_{\theta }}\frac{1}{\rho }\frac{{\partial N}}{{\partial \theta }} - \frac{{u\varsigma }}{\mu }\frac{{{{\delta }_{ \bot }}}}{{\rho \sin \theta }}\frac{\partial }{{\partial \theta }}\sin \theta \frac{{\partial N}}{{\partial \theta }} + \frac{u}{{3{{\varsigma }^{2}}}}\frac{\partial }{{\partial \varsigma }}{{\varsigma }^{3}}\frac{{\partial N}}{{\partial \rho }} = 0,$$

(25)

where

$${{w}_{r}} = {{\delta }_{A}}\frac{{u\varsigma }}{\mu }\frac{1}{{\rho \sin \theta }}\frac{\partial }{{\partial \theta }}({{h}_{\varphi }}\sin \theta ),$$

(26)

$${{w}_{\theta }} = - {{\delta }_{A}}\frac{{u\varsigma }}{\mu }\frac{1}{\rho }\frac{\partial }{{\partial \rho }}(\rho {{h}_{\varphi }}).$$

(27)

Note that CR transport equation (25) corresponds to the stationary particle distribution. From Eq. (25), we have the following relation for $j_{r}^{s}$ (24):

$$j_{r}^{s} = - \frac{1}{{{{\rho }^{2}}}}\int\limits_0^\rho {d\rho {{\rho }^{2}}} {{w}_{r}}\frac{{\partial N}}{{\partial \rho }} - \frac{1}{{{{\rho }^{2}}}}\int\limits_0^\rho {d\rho \rho } {{w}_{\theta }}\frac{{\partial N}}{{\partial \theta }} + \frac{{u\varsigma }}{\mu }\frac{\delta }{{\rho \sin \theta }}\frac{\partial }{{\partial \theta }}\int\limits_0^\rho {d\rho } \frac{{\partial N}}{{\partial \theta }} - \frac{u}{{3{{\varsigma }^{2}}{{\rho }^{2}}}}\frac{\partial }{{\partial \varsigma }}{{\varsigma }^{3}}\int\limits_0^\rho {d\rho {{\rho }^{2}}} \frac{{\partial N}}{{\partial \rho }}.$$

(28)

Taking into account the relation between the radial velocity w_r (26) and the azimuthal component of the magnetic field strength unit vector h_φ (19), let us write the first term of formula (28) as follows

$${{j}_{1}} = \frac{{u\varsigma }}{\mu }\frac{{{{\delta }_{A}}\omega }}{{{{\rho }^{2}}}}\int\limits_0^\rho {d\rho {{\rho }^{2}}} \frac{{\partial N}}{{\partial \rho }}\left\{ {\frac{{2 + {{\rho }^{2}}{{\omega }^{2}}{{{\sin }}^{2}}\theta }}{{1 + {{\rho }^{2}}{{\omega }^{2}}{{{\sin }}^{2}}\theta }}\cos \theta \left[ {1 + 2\Theta \left( {\theta - \frac{\pi }{2}} \right)} \right] - \frac{2}{{\sqrt {1 + {{\rho }^{2}}{{\omega }^{2}}} }}\delta \left( {\theta - \frac{\pi }{2}} \right)} \right\},$$

(29)

where δ(x) is the Dirac delta function. Note that relation (29) corresponds to the A⁺ epoch when the radial component of the interplanetary magnetic field strength is positive in the northern part of the heliosphere (H_0r > 0). For the A^– epoch, j₁ has the opposite sign. For the A⁺ epoch, j₁ is positive at all heliolatitudes except for the helioequator (θ ≠ π/2). In the helioequator plane (θ = π/2), the particle flux j₁ (29) is negative, i.e., it is directed toward the Sun. During the time A^–, the direction of flux (29) is opposite. Under the condition θ ≠ π/2, the particle flux is directed toward the Sun (j₁ < 0), while particle flux (29) is directed out of the heliosphere in the helioequator plane.

Relations (15) and (21) make it possible to calculate partial derivatives of the particle concentrations $\frac{{\partial N}}{{\partial \rho }}$ and $\frac{{\partial N}}{{\partial \theta }}$ that determine particle flux density (28). The particle flux density j_θ is determined by relations (11), (15), and (21). Figure 2 shows the dependence of the flux density j_θ on the angle θ at the point ρ = 0.01 for the A⁺ epoch (solid curve) and for the A^– epoch (dashed curve). It can be seen that the flux density j_θ is negative in the northern hemisphere (θ < π/2) during the A⁺ periods of solar activity. If θ > π/2, the flux density j_θ is positive. Therefore, the particle flux is directed from the helioequator toward high latitudes. At the point θ = π/2, there is a jump in the j_θ value. This effect is due to the choice of the interplanetary magnetic field model (17), (18), in which the azimuthal component of the magnetic field strength unit vector looks like (19). Calculations were performed for particle energies E_k = 1 GeV for the following parameter values: μ = 1.5, δ_⊥ = 0.02, δ_A = 0.01. In the A^–-epoch, on the contrary, the particle flux j_θ (28) is directed toward the helioequator plane (the dashed curve in Fig. 2).

Fig. 2.

Dependence of the heliolatitude cosmic ray flux density on the angle θ. The solid curve represents the A⁺ epoch, and the dashed curve represents the A^–.

The CR radial flux density is determined by relations (23), (28). Figure 3 shows the dependence of the radial particle flux density on the angle θ at the point ρ = 0.01. The kinetic energy of the particles E_k = 1 GeV, the values of other parameters are the same as in Fig. 2. The solid curve corresponds to the A⁺ epoch, while the dashed curve corresponds to the A^– epoch. Note that the radial particle flux in the A⁺ epoch is directed toward the Sun in the helioequator plane, while the radial particle flux has the opposite direction (from the heliosphere) in the A^– epoch. During the A⁺ epoch, the radial particle flux (outside the equatorial plane) is positive and directed away from the Sun, except for the high polar regions, where the flux has the opposite direction (Fig. 3). In the A^– epoch, the radial particle flux is directed toward the Sun (j_r < 0), except for a small region near the helioequator (Fig. 3).

Fig. 3.

Dependence of the CR radial flux density on the angle θ. The solid curve denotes the A⁺ epoch, and dashed curve represents the A^– epoch.

COSMIC RAY CONCENTRATION

Let us write down the cosmic ray concentration $\tilde {N}(\rho ,\theta ,\varsigma )$ in the following form [4, 7, 8, 14, 15]

$$\tilde {N}(\rho ,\theta ,\varsigma ) = {{N}_{0}}(\xi ) + \delta N(\rho ,\theta ,\varsigma ),$$

(30)

where δN is small (compared to N₀) and the variable ξ is defined by formula (21). Here is an equation that satisfies the value δN. Based on relations (23), (24), the following equation for the value δN can be obtained

$$\frac{{\partial \delta N}}{{\partial \rho }} + \frac{\mu }{3}\frac{{\partial \delta N}}{{\partial \xi }} = \Phi (\rho ,\theta ,\varsigma ),$$

(31)

where

$$\Phi (\rho ,\theta ,\varsigma ) = - \frac{{\partial {{N}_{0}}(\xi )}}{{\partial \rho }} - \frac{\mu }{3}\frac{{\partial {{N}_{0}}(\xi )}}{{\partial \xi }} - \frac{\mu }{{u\varsigma }}j_{r}^{s},$$

(32)

and the value j_r^s is determined by relation (28). Since the CR concentration N₀(ξ) satisfies boundary condition (14), the function δN should be zero at the boundary of the modulation region (at the point ρ = 1).

The solution to Eqs. (31), (32) that satisfies the zero-boundary condition at the heliopause is as follows

$$\delta N(\rho ,\theta ,\varsigma ) = \int\limits_1^\rho {d{{\rho }_{1}}\Phi ({{\rho }_{1}},\theta ,{{\xi }_{1}})} ,$$

(33)

where

$${{\xi }_{1}} = \varsigma + \frac{\mu }{3}\left( {{{\rho }_{1}} - \rho } \right).$$

(34)

Figure 4 shows the dependence of the CR $\tilde {N}(\rho ,\theta ,\varsigma )$ (30) concentration at the point ρ = 0.01 on the angle θ. The kinetic energy of the particles is E_k = 100 MeV. The parameter values are the same as in the previous figures. The dotted curve in Fig. 4 corresponds to the particle concentration N₀ (15), (21). Note that this value is almost independent of the angle θ. The solid curve in Fig. 4 corresponds to the A⁺ epoch and the dashed one to the A^– epoch. According to the above calculations, the GCR intensity in the A⁺ epoch of solar activity increases with increasing heliolatitude (Fig. 4). In the A^– periods of solar activity, the opposite effect is observed: the CR intensity is maximum in the helioequator region.

Fig. 4.

Dependence of the CR concentration on the angle θ at the point ρ = 0.01. The solid curve indicates the A⁺ epoch, and dashed curve represents the A^– epoch.

These calculations agree with the observed data. Measurements of the Ulysses SC far from the ecliptic plane and on satellites in the near-Earth space in 1994 (near the solar minimum, A⁺ phase of solar activity), showed that there were heliolatitudinal gradients of GCR intensity. The latitudinal gradients (toward the polar regions) of all types of nuclei were positive in both the northern and southern hemispheres [10–12, 16, 17]. The measured heliolatitudinal gradients were relatively small. The most significant heliolatitudinal dependence of the CR intensity was observed for particles with a rigidity of 1 GV. The value of the GCR heliolatitudinal gradient decreased both in the direction of high and low particle energies [10–12, 16, 17].

Studies conducted by the Ulysses, PAMELA, and other missions in 2006–2007 made it possible to determine radial and latitudinal gradients of CR intensity during the minimum of solar cycle 23 (A^– epoch). It was shown that the GCR intensity is maximum in the equatorial region in the A^– epoch of solar activity [10, 21, 23].

The presented calculations make it possible to estimate the anisotropy of the angular distribution of GCRs. The CR anisotropy is proportional to the particle flux density and is defined by the following relation:

$$\xi = \frac{{3j}}{{vN}}.$$

(35)

Let us estimate the CR anisotropy at the Earth’s orbit. The CR flux densities in radial (10) and latitudinal (11) directions appear to be small in absolute value. A suitable value of the anisotropy of the angular distribution of particles at the heliocentric distance of 1 au is negligibly small. At the Earth’s orbit, the flux line of the interplanetary magnetic field forms an angle close to π/4 with the radial direction [3, 19]. Therefore, the CR diffusive flux in the azimuthal direction is close to the diffusive component of the radial flux. The radial diffusive flux is approximately equal to the convection flux [3, 9, 18, 22]. Therefore, we obtain the following ratio of the CR azimuthal anisotropy at a heliocentric distance of 1 au:

$$\xi _{\varphi }^{{}} = \frac{{u\varsigma }}{{vN}}\frac{{\partial N}}{{\partial \varsigma }}.$$

(36)

Thus, the GCR flux in the Earth’s orbit is directed in the azimuthal direction, which agrees with the observed diurnal variation of CR intensity [3]. According to formula (36), we obtain the CR anisotropy ξ_φ ≈ −5 × 10⁻³ for particles with energies from 1 MeV to 1 GeV.

CONCLUSIONS

The propagation of cosmic rays in the interplanetary medium under the approximation of small anisotropy of the angular distribution of particles was considered. Under the condition of the known energy distribution of particles set at the boundary of the heliosphere, the solution of the cosmic ray transport equation was obtained. It was shown that the GCR heliolatitudinal flux is directed from the helioequator to high latitudes in the A⁺ epoch. In contrast, during A^– epochs of solar activity, the particle flux j_θ is directed toward the helioequator. In the helioequator region, the radial particle flux in the A⁺ epoch is directed toward the Sun, while the radial flux has the opposite direction (from the heliosphere) in the A^– epoch. Outside the helioequator plane, the radial particle flux is directed away from the Sun during the A⁺ epoch and toward the Sun during the A^– epoch.

We have shown that the GCR intensity in the A⁺ epoch of solar activity increases with increasing heliolatitude. During A^– epochs of solar activity, the opposite effect is observed: the CR intensity is maximum in the helioequator region. These calculations agree with the data from the Voyager 1, 2, Ulysses, PAMELA, IMP-8, and AMS spacecraft. The anisotropy of the angular distribution of CRs was estimated. We showed that the CR flux at the Earth’s orbit has an azimuthal direction, and the value of anisotropy of particles with energies from 1 MeV to 1 GeV is almost independent of the energy, approximately 0.5%.