Cosmic ray transport near the Sun

In the absence of magnetic turbulence the magnetic moment of particles are 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 $B_{\mathrm{Earth}}$ with a pitch-angle $\alpha$, and where $\mu=\cos\alpha$, will not be able to penetrate a region of magnetic field strength $B$, if

We can now calculate this critical angle $\mu_{c}$ for the case of GCRs, starting out isotropically from Earth, and propagating towards the Sun along $\vec{B}$. In Fig. 1 we show the mirror ratio $B_{\mathrm{Earth}}/B$, 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 become negligible beyond Earth’s orbit as the mirror ratio approaches unity. Without any additional process that can drive anisotropic behaviour, GCR 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 $\mu_{c}$ to estimate the associated loss-cone distributions at different distances (these distances are indicated by vertical arrows on the upper panels). These loss-cone distribution illustrate how difficult it is for GCR to reach the inner heliosphere; an effect also studied by e.g. Hutchinson et al. (2021).

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, $f(z,\mu,t)$, is given by the so-called focused transport equation,

where $\mu$ is the cosine of the particle pitch-angle, $v$ is the particle speed so that $v_{||}=\mu v$ is the parallel (with respect to the magnetic field) projection of the particle speed. Particle mirroring/focused is included via the focusing/mirroring length

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

Once $f$ is obtained, an omni-directional distribution function can be calculated as

with an associated first-order anisotropy, given by

and varying between $-3$ (all particles propagating towards 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

which is related to the parallel diffusion coefficient through

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

Following Litvinenko & Schlickeiser (2013), we can derive an approximate transport equation for $F(z,t)$ from Eq. 2 by expanding $f$ in terms of $F$ and a small correction factor $F_{1}(z,\mu,t)$,

where

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 $z$, $\mu$, 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 pseudo-diffusion equation

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

Note that the $\kappa_{||}$ introduced here differs from $\kappa^{0}_{||}$ as defined in Eq. 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

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

In the case of vanishing focusing/mirroring, $L\rightarrow\infty$, we have the evolution of the isotropic diffusion equation

which follows directly from Eq. 10.

We start with solving Eq. 13 by assuming a steady state solution, $\partial F/\partial t=0$, from which we directly obtain

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

A steady-state assumption for Eq. 10 directly leads to a gradient of

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

By using Eq. 12 for the streaming flux, the anisotropy follows as

Note that this expression diverges for strong focusing/mirroring, $A\rightarrow-\infty$ as $L\rightarrow 0$. This is due to the assumption of weak focusing/mirroring used when driving Eq. 10, i.e. $\lambda_{||}\ll L$. This strict condition is not satisfied for all parameters assumed in this work.

For the simulations presented here we assume a parametrized version of $D_{\mu\mu}$ from quasi-linear theory (Jokipii, 1996),

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 $D_{0}$ from first principles (as is done by e.g. Strauss et al., 2017), but rather specify $\lambda_{||}$ from which $D_{0}$ can be calculated from Eq. 6. Eq. 2 is solved with the open-source numerical model presented in van den Berg et al. (2020) and available from https://github.com/RDStrauss/SEP_propagator.

The resulting particle intensities, for different choices of $\lambda_{||}$ are shown in Fig. 2. For all simulations, protons with a kinetic energy of 100 MeV is assumed. However, with this specific model set-up (including an energy independent $\lambda_{||}$, no energy losses, and the assumption of a steady-state solution) the results are independent of the particle energy considered.

The left panels of Fig. 2 corresponds to the case of $\lambda_{||}=0.1\text{au}$, while the right panels correspond to $\lambda_{||}=1\text{au}$. 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 $\sim 0.1-1$ au for protons in the rigidity range of $\sim 10-10^{4}$ MV. The top panels show the computed omni-directional intensity, the calculated gradient, and the resulting anisotropy as a function of magnetic distance, $z$. Results from the numerical model, solving Eq. 2, are shown as red symbols, the case of weak scattering, i.e. solving Eq. 10, as the solid blues line, and the case of the isotropic diffusion equation, Eq. 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 compares relatively well. However, the analytical approach cannot capture the radial dependence of the anisotropy as $\lambda_{||}\ll L$ is generally violated for all assumed parameters. The diffusion equation, assuming no focusing/mirroring, gives a smaller gradient as compared to the other approaches.

The bottom panels of Fig. 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 form the numerical model. In these graphs, $\alpha(\mu)=0^{\circ}(1)$ indicates particles streaming away from the Sun and $\alpha(\mu)=180^{\circ}(-1)$ labels particles moving towards the Sun. The black shaded regions show the pitch-angle regions close to the model’s boundaries where information about the the particle distribution is not available. For the assumed values of $\lambda_{||}$, strong anisotropies, and significant anisotropic behaviour, is only observed very close to the inner boundary (e.g. inside 0.1 au).