In contrast to previous studies, the MFP expressions employed here are defined to include independent rigidity and nonlinear radial dependences. Furthermore, this approach does not make the simplifying assumption that λ
⊥ is proportional to λ
∥ (see, e.g., Ferreira et al. 2001). In order to study the radial and rigidity dependences of MFPs required to fit observations, as well as the values of these quantities at 1 au so as to compare these with theoretical predictions of the same, this study still implements simplified parametric expressions for these MFPs, motivated by the fact that the transport code here employed is written to perform on graphic processing units GPUs in CUDA, which allows for the vast performance increase required of such a study, but limits the number of definable variables due to memory constraints (see Dunzlaff et al. 2015, for additional information). Subsequently, λ
∥ and λ
⊥ are defined as
5



λ∥=λ∥,0PP0αrreβ,



6



λ⊥=λ⊥,0PP0γrreδ,

where r

e
 = 1 au, and P
0 = 1 GV, in an approach similar to that employed by Engelbrecht & Di Felice (2020). Here, parameters α and β specify the respective rigidity and radial dependences of the parallel MFP, which assumes a value of λ
0,∥ (in astronomical units) at Earth, with γ, δ, and λ
0,⊥ playing the same roles with respect to the perpendicular MFP. In the parameter studies employed here, these quantities are varied over a broad range of values, chosen so as to include, as well as extend beyond, various theoretical predictions for these quantities. In terms of MFP values at 1 au, the ranges λ
0,∥ ∈ [0.05, 1.0]  au and λ
0,⊥ ∈ [0.001, 0.5]  au are chosen, so as to include values reported in previous such studies for the parallel MFP (e.g., Palmer 1982; Bieber et al. 1994; Dröge et al. 2014) as well as for the perpendicular MFP (e.g., Ferrando et al. 1993; Giacalone 1998; Burger et al. 2000; Zhang et al. 2007; Vogt et al. 2020), while taking into account various theoretical predictions and expressions previously used in modeling studies for these MFPs (e.g., Ferreira et al. 2003; Engelbrecht & Burger 2013b; Nndanganeni & Potgieter 2018; Engelbrecht 2019). For the rigidity dependences, it is assumed that α ∈ [−0.1, 0.67] and γ ∈ [−0.1, 0.33], incorporating the flat rigidity dependence expected of λ
∥ for low-energy electrons (e.g., Dröge 1994; Potgieter 1996; Evenson 1998) within a range that includes dependences expected at higher rigidities as well as from different scattering theories (e.g., Dröge 2000; Teufel & Schlickeiser 2002; Shalchi & Schlickeiser 2004; Burger et al. 2008; Engelbrecht & Burger 2010). The range of rigidity dependences for λ
⊥ was chosen following similar reasoning, with the aim of incorporating a flat rigidity dependence within a range that incorporates moderate rigidity dependences expected from various theories (e.g., Shalchi et al. 2004a; Gammon & Shalchi 2017; Dempers & Engelbrecht 2020) as well as prior particle transport studies (e.g., Zhang et al. 2007; Dröge & Kartavykh 2009) and numerical test particle simulations of diffusion coefficients (e.g., Minnie et al. 2007a; Dundovic et al. 2020). Lastly, radial dependences were varied such that β ∈ [0.5, 1.5] and δ ∈ [0.5, 1.5], taking into account various possible radial dependences for these MFPs reported by studies employing turbulence quantities yielded by various turbulence transport models as inputs for several theoretical expressions (see, e.g., Engelbrecht & Burger 2013b; Wiengarten et al. 2016; Chhiber et al. 2017; Zhao et al. 2017; Adhikari et al. 2021). In order to study the various dependencies in Equations (5) and (6), four initial electron energies are chosen, motivated by the fact that for these energies spacecraft data is available for periods of good and bad magnetic connection to the Jovian source as well as at the Jovian source itself. The three higher energies furthermore correspond to the three lowest-energy channels of the ISEE 3 data by Moses (1987). As the results of the parameter studies have to be compared to spectral fits (given by the dotted lines in Figure 1), this choice allows for the most transparent analysis of the deviation of the simulation results from in situ data. The six parameters governing the MFP expressions used in this study, along with the relevant initial energies, are
7



E=[0.0025,0.006,0.0085,0.0125]GeVλ∥,0=[0.05,0.106,0.224,0.473,1.0]auλ⊥,0=[0.001,0.00473,0.02236,0.1057,0.5]auα=[−0.1,−0.01,0.01,0.04,0.165,0.67]γ=[−0.1,−0.01,0.01,0.03,0.1,0.33]β=[0.5,0.658,0.866,1.397,1.5]δ=[0.5,0.658,0.866,1.397,1.5],

which form a parameter space of consisting of 90,000 grid points, resulting in 180,000 simulations for comparison with observational points both well and badly connected to the Jovian source as shown in Figure 1.
Citation: Zhang et al. 2007
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