tokens: et al. ( 2006 ) and Kuwabara et al. ( 2009 ) in a numerical model best fitted to the measurements taken by the muon network at Earth (rigidities 50–100 GV) and was utilized to determine the orientation of the interplanetary FR. The diffusion–expansion approach was also implemented by Subramanian et al. ( 2009 ) and Arunbabu et al. ( 2013 ) in an analytical model and was compared to the GRAPES-3 muon telescope (rigidities 12–42 GV). These authors estimated the radial perpendicular diffusion coefficient to be of the order of 10 21 . However, it should be noted that their studies are adjusted to large events associated with a shock and observed at high particle rigidities. At rigidities of about 1 GV, Cane et al. ( 1995 ) estimated a diffusion coefficient of about 10 19 by utilizing spacecraft measurements and less energetic ejecta. Both are roughly in agreement with values obtained from the typical empirical expression used in numerical models (see Equations (23) and (24) in Potgieter 2013 ). Therefore, these studies outline a rough constraint on the diffusion coefficient used to model FR-associated FDs. Since both diffusion and expansion are processes that presumably start close to the Sun, FRs, that is, CME initial properties, need to be considered in the model. This ejecta-only FD model primarily describes FDs associated with ICMEs without a shock/sheath region, because a shock/sheath region presumably introduces an additional decrease via different physical mechanisms, but could also be applied to explain the ejecta part of an FD in two-step FDs. The aim of this study is to take these considerations into account in the diffusion–expansion model to improve our understanding on the cause, formation, and evolution of FDs. 2. Basic Diffusion Model First we consider an analytical FD model based on the perpendicular diffusion of GCRs into the FR, similar to Cane et al. ( 1995 ). In this basic diffusion model, the FR is regarded as a closed magnetic structure, rooted at the Sun, that is initially empty of GCRs (Figure 1 (a)). The FR is of cylindrical form (hereafter approximated by a long cylinder), it moves with constant velocity, and as it moves, it does not vary in shape or size (Figure 1 (b)). We note that the assumption of constant velocity is in general not valid, as it was shown that CMEs slower than the solar wind accelerate, whereas CMEs faster than the solar wind decelerate (e.g., Sheeley et al. 1999 ; Gopalswamy et al. 2000 ), which is attributed to the magnetohydrodynamical drag (see, e.g., Vršnak et al. 2013 ; Hess Zhang 2014 ; Sachdeva et al. 2015 , and references therein). However, since the acceleration/deceleration depends on the difference between the CME and solar wind speed, for a substantial subset of CMEs (and especially the slower ones that do not drive shocks), a constant velocity can be taken as a relatively fair approximation. We also note that the assumption of constant shape and size might not hold true, as it was
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