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Appendix A: Flux weighting

Appendix A.1: Upstream conservation of flux

The flux of particles through a given cross-sectional area of the flux tube towards the shock is given as (A.1)which satisfies the steady state requirement of It is noteworthy that although the mean direction of particles is in the negative x-axis direction, we defined the total net flux across the shock as being positive. We considered the shock to be an absorbing boundary, that is, the distributions and fluxes represent particles that have yet to encounter the shock. The particle flux, which is constant regardless of position x₁, is easily evaluated far from the shock, where only the scattering process affects the angular distribution of the particles. The distribution is assumed to be isotropic in the local plasma frame. Thus, sufficiently far upstream, the flux is given by the form (A.2)where n_∞ is the particle density and f_∞ the particle distribution at infinity.

At velocities v < u₁, all particles travel towards the shock, and thus, information of the shock at x₁ = 0 cannot propagate into the upstream. The distribution remains isotropic in the local plasma frame, and the differential flux impacting the shock is simply (A.3)Considering particles with speeds v > u₁, the picture becomes more complicated. Information of the shock can propagate against the flow of plasma, because μv − u₁ can be positive. This means that the angular distribution of particles that have not yet interacted with the shock becomes anisotropic, as the distribution f(x₁ = 0,v,μ) = 0 for μv > u₁, due to the absorbing boundary at the shock. However, at a distance of x₁ = 2λ, for instance, we can still assume isotropy and take f(2λ,v,μ) ≈ f_∞(v), so that (A.4)is valid for all velocities including v > u₁. In this flux, particles with μv > u₁ cause a negative contribution to the net flux, because they propagate outwards across the flux tube cross-section surface at x₁ = 2λ.

For the purpose of constructing a semi-analytical model of particle injection at high particle speeds, we simplified the anisotropies of the particle distribution at the shock. We assigned the modified differential flux of particles with v > u₁ at the shock as (A.5)where is a scaled distribution function yielding the correct total net flux, that is, for v < u₁, but for v > u₁, Thus, for the particles that have not yet interacted with the shock, we find (A.6)valid for particles of speed v > u₁. This accounts for the absorbing boundary at the shock whilst maintaining the conservation of total flux at a given particle speed v.

For Monte Carlo simulations, the same consideration of flux conservation must be made. The flux, that is, the total amount of particles encountered by the shock within time dt, is formulated as (A.7)For particles of speeds v < u₁, the whole population is advected towards the shock, meaning that no information of the approaching shock can reach the particle distribution before impact. Thus, for these particles, the differential flux, extended to encompass all pitch-angles, can be written as (A.8)Thus, the probability of a particle with speed v exhibiting pitch-angle μ when impacting the shock is given as (A.9)This can be integrated to find the cumulative distribution function for a value of μ as (A.10)From this, the Monte Carlo randomisation formula for μ can be solved as (A.11)where μ receives values from the range −1 < μ ≤  + 1 and is a uniformly distributed random number in the range [0,1).

For particles with speeds v > u₁, information of the propagating shock can extend into the upstream, affecting the incident particle pitch-angle distribution. Thus, we initialised the particle distribution in the upstream of the shock at a distance of x₁ = 2λ, and allowed particles to convect towards the shock. This resulted in a realistic pitch-angle distribution at x₁ = 0₊, without having to resort to flux modification (Eq. (A.6)).

This pre-propagation is limited to the region x₁ ∈  [0,2λ] with particles initialised isotropically at values μ < u₁/v. As the total population is advected towards the shock, all particles that escape to x₁ > 2λ will eventually return to the initialisation boundary of x₁ = 2λ, isotropised in the fluid frame. Thus, particles escaping to the upstream can be simply re-initialised at that position.

The distribution of pitch-angles μ for a given speed v, limiting the valid pitch-angle range to values and normalising the total probability to 1, is (A.12)This will result in the shock-incident flux (A.13)Using Eq. (A.12), the Monte Carlo randomisation formula for μ can be solved (similar to Eq. (A.10)) as (A.14)where μ receives values from the range .

Appendix B: Analytical injection thresholds

Appendix B.1: Reflection threshold

In attempting to determine particle injection, we can solve certain seed particle speed thresholds. A particle is reflected (see Eq. (9)), and thus, injected, if (B.1)The right-hand side (RHS) of the equation is constant. Through roots of derivatives of the left-hand side (LHS), we can find LHS maxima at if , or μ = 1, if . Thus, if , the LHS maximum is given as (B.2)This results in no possibility of reflection, if u₁/r_B < v < v_R1, where (B.3)If , the LHS has a maximum value of 2vu₁ − v². This results in no possibility of reflection, if v < v_R2, where (B.4)The LHS minimum is found at μ = −1. With these shock and solar wind parameters, there exists no valid speed v for which this value of the LHS would be positive. Under these circumstances, no seed particle speed v results in certain reflection, regardless of pitch-angle μ.

Appendix B.2: Threshold for return from the downstream

To split the transmitted particle population into portions with either possible or impossible injection, we examined Eqs. (10)and (11)to find (B.5)We found the maxima of downstream speed v′ for a given v, because this can result in an injection velocity threshold. Maxima for v′ can be found at μ = −1 and μ =  + 1 or by solving the roots of the derivative of v′(μ). Assigning (B.6)and finding the first derivative for v^′2 gives Solving the roots provides an extremum, if μ ∈ (−1, + 1) and if If holds true, only the solution with the minus sign is of interest. These, in addition to the possible extrema at μ = −1 and μ =  + 1, result in several possible threshold velocities.

Appendix B.2.1: Extrema at μ =-1 and μ =  + 1

The possible downstream velocity extrema at μ = −1 and μ =  + 1 can be solved by refining Eq. (B.5). Formulating this as the threshold for no injection results in (B.11)where the term inside the square root is positive for all transmitted particles. Thus, a transmitted particle can return to the upstream only if For particles with v > u₁, injection is always possible. For particles with v < u₁, we found the additional requirements of For our parameters, only v_Tμ− provides a valid condition for injection.

Appendix B.2.2: Extrema at μ ≠  ± 1

Solving the thresholds speeds for the one or two extrema given in Eq. (B.8)in an analytical fashion does not result in easily applicable equations. However, solving the velocity thresholds for these extrema in a numerical fashion revealed valid extrema only at shock-normal angles θ_Bn ≤ 4°. At these shock-normal angles, the existing extrema at μ =  ± 1 already allow theoretical return of particles regardless of their speed. Thus, these extrema do not affect the found particle return speed thresholds. However, in a more general case, they cannot be ignored.

Appendix C: Statistical handling of uninjected particles

In Sect. 7, we presented a method for estimating the capability of a shock to inject particles using Monte Carlo simulations. In our method, we propagated the particle in the downstream until it was either injected into the upstream, or the cumulative probability of return at the downstream boundary fell below 10^-6 and it was removed from the simulation as an uninjected particle. For some shock parameters, however, this method will result in very low statistics for the injected particles. To improve the statistical accuracy, we used the fact that successes and failures – injections and non-injections – are distributed according to a negative binomial distribution. When the numbers of successes, ℛ, and failures, , are known, the probability for success can be evaluated using the minimum-variance unbiased estimator (see, e.g., Lehmann & Casella 1998), which gives (C.1)It should be noted, however, that the actual probability of injection for a particle is (C.2)

where P_s is the injection probability associated with the last encountered success.

To evaluate the unbiased estimator, we randomised particles in groups. For each particle within the group, the newly randomised values of v and μ were used to test the particle for reflection, as explained in Sect. 5.1. Reflected particles were considered successes and have P_s = 1. Non-reflected particles are transmitted to the downstream, with v′ and μ′ calculated according to Eqs. (10)and (11). They were then followed in the downstream, as described in Sect. 7, until they were either injected, incrementing ℛ, or considered uninjected, incrementing . This was continued until ℛ = 5. The last particle of the group was then injected with the probability P_inj.

As a precaution against excessively low success probabilities, the group size was limited to . If this limit was reached and at least two successes were encountered, the values of ℛ_i and associated with the last encountered success were used to calculate . If the tests resulted in only 0 or 1 success, the group was considered to result in no injection.

With these methods, the injected weight of the Monte Carlo particle was found to be W_inj = W_seedP_inj, where W_seed is the representative weight of upstream seed particles assigned to this group, and was found based on the plasma density. The total injected weight was then used to calculate the particle flux.

© ESO, 2013