Cosmic-Ray Transport near the Sun

In the absence of magnetic turbulence, the magnetic moment of particles is conserved during propagation and, along with the conservation of kinetic energy, any mirroring/focusing force is accompanied by an interchange between parallel and perpendicular energy: As a particle moves into a region of larger magnetic field strength, its perpendicular speed increases, with the effect that its parallel speed decreases. Ultimately, this causes the particle's motion to be reversed and the particle is mirrored. However, not all particles entering a magnetic bottle will be mirrored. A particle starting out in a region with field strength B_Earth with a pitch angle α, and where $\mu =\cos \alpha$, will not be able to penetrate a region of magnetic field strength B, if

$\begin{eqnarray}&&| \mu | \gt {\mu }_{c}=\sqrt{1-\displaystyle \frac{{B}_{\mathrm{Earth}}}{B}}.\end{eqnarray} \tag{ 1 }$

We can now calculate this critical angle μ_c for the case of GCRs, starting out isotropically from Earth, and propagating toward the Sun along B . In Figure 1 we show the mirror ratio B_Earth/B and the resulting ${\alpha }_{c}={\cos }^{-1}{\mu }_{c}$, as a function of radial distance. The effectiveness of the focusing/mirroring process decreases quickly away from the Sun and becomes negligible beyond Earth's orbit as the mirror ratio approaches unity. Without any additional process that can drive anisotropic behavior, GCRs are generally observed to be nearly isotropic after propagating (diffusively) from the heliopause to Earth (see, e.g., Gil et al. 2021). In the bottom panels of the figure, we use the calculated μ_c to estimate the associated loss-cone distributions at different distances (these distances are indicated by vertical arrows on the upper panels). These loss-cone distributions illustrate how difficult it is for GCRs to reach the inner heliosphere, an effect also studied by, e.g., Hutchinson et al. (2022).

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While these results are useful for quantifying the effect of mirroring in the Parker (1958) HMF, they do not capture the full picture of particle transport: The HMF is turbulent on all scales, and these turbulent fluctuations will lead to pitch-angle scattering that tends to isotropize the GCR distribution and also disrupt the general particle drift picture (e.g., van den Berg et al. 2020, 2021). The interplay between scattering (leading to isotropic distributions) and focusing/mirroring (leading to anisotropic distributions) is therefore essential to understand. The additional effect of pitch-angle scattering is considered in the next sections.

The evolution of a nearly gyrotropic distribution function, f(z, μ, t), is given by the so-called focused transport equation,

$\begin{eqnarray}&&\displaystyle \frac{\partial f}{\partial t}+\mu v\displaystyle \frac{\partial f}{\partial z}+\displaystyle \frac{v}{2L}\left(1-{\mu }^{2}\right)\displaystyle \frac{\partial f}{\partial \mu }=\displaystyle \frac{\partial }{\partial \mu }\left({D}_{\mu \mu }\displaystyle \frac{\partial f}{\partial \mu }\right),\end{eqnarray} \tag{ 2 }$

where μ is the cosine of the particle pitch angle, and v is the particle speed so that v_∣∣ = μ v is the parallel (with respect to the magnetic field) projection of the particle speed. Particle mirroring/focused is included via the focusing/mirroring length

$\begin{eqnarray}&&\displaystyle \frac{1}{L}=-\displaystyle \frac{1}{B}\displaystyle \frac{{dB}}{{dz}}.\end{eqnarray} \tag{ 3 }$

and pitch-angle diffusion is included via the pitch-angle diffusion coefficient D_{μ μ} . Equation (2) is also referred to as the Roelof (1969) equation.

Once f is obtained, an omnidirectional distribution function can be calculated as

$\begin{eqnarray}&&F(z,t)=\displaystyle \frac{{\int }_{-1}^{+1}f(z,\mu ,t)d\mu }{{\int }_{-1}^{+1}d\mu }=\displaystyle \frac{1}{2}{\int }_{-1}^{+1}f(z,\mu ,t)d\mu ,\end{eqnarray} \tag{ 4 }$

with an associated first-order anisotropy, given by

$\begin{eqnarray}&&\begin{array}{l}A(z,t)=3\displaystyle \frac{{\int }_{-1}^{+1}\mu f(z,\mu ,t)d\mu }{{\int }_{-1}^{+1}f(z,\mu ,t)d\mu }\\ \quad =\displaystyle \frac{3}{2F(z,t)}{\int }_{-1}^{+1}\mu f(z,\mu ,t)d\mu ,\end{array}\end{eqnarray} \tag{ 5 }$

and varying between −3 (all particles propagating toward the Sun) and +3 (all particles propagating away from the Sun).

The efficiency of pitch-angle scattering is usually quantified by the parallel mean free path, calculated here following Hasselmann & Wibberenz (1968) as

$\begin{eqnarray}&&{\lambda }_{\parallel }^{0}=\displaystyle \frac{3}{8}v{\int }_{-1}^{1}\displaystyle \frac{{(1-{\mu }^{2})}^{2}}{{D}_{\mu \mu }(\mu )}{\rm{d}}\mu ,\end{eqnarray} \tag{ 6 }$

which is related to the parallel diffusion coefficient through

$\begin{eqnarray}&&{\kappa }_{| | }^{0}=\displaystyle \frac{v}{3}{\lambda }_{| | }^{0}.\end{eqnarray} \tag{ 7 }$

For this work, we assume ${\lambda }_{\parallel }^{0}=\lambda$ to be a constant.

Following Litvinenko & Schlickeiser (2013), we can derive an approximate transport equation for F(z, t) from Equation (2) by expanding f in terms of F and a small correction factor F₁(z, μ, t),

$\begin{eqnarray}&&f(z,\mu ,t)=F(z,t)+{F}_{1}(z,\mu ,t),\end{eqnarray} \tag{ 8 }$

where

$\begin{eqnarray}&&\int {F}_{1}d\mu =0,\end{eqnarray} \tag{ 9 }$

and $\left|{F}_{1}\right|\ll f$. This corresponds to the case of a nearly isotropic distribution, which is possible when ${\lambda }_{| | }^{0}\ll L$ so that the transport is diffusion dominated in this weak-focusing/mirroring limit. Note that, from here, the explicit dependence on the quantities z, μ, and t are not included. By furthermore assuming that ${\lambda }_{| | }^{0}$ and L are constant, Litvinenko & Schlickeiser (2013) show that the evolution of F is governed by the following pseudodiffusion equation:

$\begin{eqnarray}&&\displaystyle \frac{\partial F}{\partial t}+\displaystyle \frac{{\kappa }_{| | }}{L}\displaystyle \frac{\partial F}{\partial z}={\kappa }_{| | }\displaystyle \frac{{\partial }^{2}F}{\partial {z}^{2}},\end{eqnarray} \tag{ 10 }$

which is in the form of a diffusion equation but with the addition of a coherent advection speed

$\begin{eqnarray}&&{u}_{| | }=\displaystyle \frac{{\kappa }_{| | }}{L}.\end{eqnarray} \tag{ 11 }$

Note that the κ_∣∣ introduced here differs from ${\kappa }_{| | }^{0}$ as defined in Equation (7) and includes a correction to account for focusing/mirroring effects. However, in the weak-focusing/mirroring case considered here, we can approximate ${\kappa }_{| | }\approx {\kappa }_{| | }^{0}$ and use the two quantities interchangeably.

The diffusive streaming flux for this scenario is given by

$\begin{eqnarray}&&S=\displaystyle \frac{v}{2}\int \mu {F}_{1}d\mu \mathop{=}\limits^{{\lambda }_{| | }\ll L}-{\kappa }_{| | }\displaystyle \frac{\partial F}{\partial z}.\end{eqnarray} \tag{ 12 }$

In deriving the results presented here, Litvinenko & Schlickeiser (2013) assume a constant L, which is definitely not the case for a Parker (1958) HMF; see the top panel of Figure 1. However, these authors also state that the analytical approximation should still be valid as long as L is constant on the scale of λ_∣∣. This is a reasonable approximation for constant (i.e., independent of radial distance) λ_∣∣, with values between 0.1 and 1 au, assumed in this work. However, modeling results of mostly solar energetic particles have shown that λ_∣∣ can vary significantly from event to event (e.g., Dröge 2000), while theoretical calculations indicate that λ_∣∣ can have a complicated radial dependence inside 1 au (e.g., Strauss & le Roux 2019).

In the case of vanishing focusing/mirroring, L → ∞, we have the evolution of the isotropic diffusion equation,

$\begin{eqnarray}&&\displaystyle \frac{\partial F}{\partial t}={\kappa }_{| | }\displaystyle \frac{{\partial }^{2}F}{\partial {z}^{2}},\end{eqnarray} \tag{ 13 }$

which follows directly from Equation (10).

We start with solving Equation (13) by assuming a steady-state solution, ∂F/∂t = 0, from which we directly obtain

$\begin{eqnarray}&&\displaystyle \frac{\partial F}{\partial z}=\mathrm{constant}\,\Rightarrow \,F=\mathrm{constant}\,\cdot \,z,\end{eqnarray} \tag{ 15 }$

with a gradient of g(z) = 1/z and, of course, an anisotropy value of A = 0.

A steady-state assumption for Equation (10) directly leads to a gradient of

$\begin{eqnarray}&&\displaystyle \frac{1}{F}\displaystyle \frac{\partial F}{\partial z}=\displaystyle \frac{1}{L}=g(z),\end{eqnarray} \tag{ 16 }$

while F(z) can be obtained by substituting the definition of L(z), given by Equation (3), and solving for F(z) in terms of the magnetic field B(z), to obtain $d\mathrm{ln}F=-d\mathrm{ln}B$, which in turn leads to

$\begin{eqnarray}&&F(z)\cdot B(z)=\mathrm{constant}.\end{eqnarray} \tag{ 17 }$

By using Equation (12) for the streaming flux, the anisotropy follows as

$\begin{eqnarray}&&A(z)=\displaystyle \frac{3S}{{vF}}=-\displaystyle \frac{{\lambda }_{| | }}{L(z)}.\end{eqnarray} \tag{ 18 }$

Note that this expression diverges for strong focusing/mirroring, A → −∞ as L → 0. This is due to the assumption of weak focusing/mirroring used when driving Equation (10), i.e., λ_∣∣ ≪ L. This strict condition is not satisfied for all parameters assumed in this work.

For the simulations presented here we assume a parameterized version of D_{μ μ} from quasi-linear theory (Jokipii 1966),

$\begin{eqnarray}&&{D}_{\mu \mu }={D}_{0}(1-{\mu }^{2})\left(| \mu {| }^{q-1}+H\right),\end{eqnarray} \tag{ 19 }$

with q = 5/3 representing the inertial range turbulence spectral index and H = 0.05 accounting for possible dynamic effects in an ad hoc fashion (e.g., Beeck & Wibberenz 1986). We do not, however, calculate D₀ from first principles (as is done by, e.g., Strauss et al. 2017), but rather specify the λ_∣∣ from which D₀ can be calculated from Equation (6). Equation (2) is solved with the open-source numerical model presented in van den Berg et al. (2020) and available online. ⁵

The resulting particle intensities, for different choices of λ_∣∣, are shown in Figure 2. For all simulations, protons with a kinetic energy of 100 MeV are assumed. However, with this specific model setup (including an energy-independent λ_∣∣, no energy losses, and the assumption of a steady-state solution), the results are independent of the particle energy considered.

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The left panels of Figure 2 correspond to the case of λ_∣∣ = 0.1 au, while the right panels correspond to λ_∣∣ = 1 au. Although the mean free path is treated, in this work, as a free parameter, it is chosen to correspond to consensus values at Earth from, e.g., Bieber et al. (1994), who reported values ranging from ∼ 0.1 to 1 au for protons in the rigidity range of ∼10–10⁴ MV. The top panels show the computed omnidirectional intensity, the calculated gradient, and the resulting anisotropy as a function of magnetic distance, z. Results from the numerical model, solving Equation (2), are shown as red symbols, the case of weak scattering, i.e., solving Equation (10), as the solid blue line, and the case of the isotropic diffusion equation, Equation (13), as the gray dashed lines. The intensities, and hence also the gradients, as calculated by the numerical model and the analytical approach assuming weak focusing/mirroring compare relatively well. However, the analytical approach cannot capture the radial dependence of the anisotropy as λ_∣∣ ≪ L is generally violated for all assumed parameters. The diffusion equation, assuming no focusing/mirroring, gives a smaller gradient compared to the other approaches.

The bottom panels of Figure 2 show the pitch-angle distribution of the particles at different radial positions (these positions are indicated by the vertical arrows in the upper panels) as calculated from the numerical model. In these graphs, α(μ) = 0°(1) indicates particles streaming away from the Sun and α(μ) = 180° (−1) labels particles moving toward the Sun. The black shaded regions show the pitch-angle regions close to the model's boundaries where information about the particle distribution is not available. For the assumed values of λ_∣∣, strong anisotropies, and significant anisotropic behavior, are only observed very close to the inner boundary (e.g., inside 0.1 au).