Appendix A: Adjustment of the effective interaction

Our results (Eq. (11)) were derived by using the screened interaction URð Þr . As already mentioned, the use and the modeling of such an effective two-body interaction are a major but indispensable approximation for a BC treatment where the full ion-target interaction is replaced by an accumulation of isolated ion-electron collisions. The replacement of the complicated real non-spherically symmetric potential, like the wake fields as shown and discussed in Ref. [60], with a spherically symmetric one is, however, well motivated by earlier studies on a BC treatment at vanishing magnetic field (see Refs. [53–55]). It was shown by comparison with 3D self-consistent PIC simulations that the drag force from the real nonsymmetric potential induced by the moving ion can be well approximated by an BC treatment employing a symmetric Debye-like potential with an effective velocity-dependent screening length λð Þ vi . In these studies also a recipe was given how to derive the explicit form of λð Þ vi , which turned out to be not too much different from a dynamic screening length of the simple form λð Þ¼ vi λ<sup>D</sup> 1 þ ð Þ vi=vth <sup>2</sup> h i<sup>1</sup>=<sup>2</sup> . Here, λ<sup>D</sup> ¼ vth=ω<sup>p</sup> is the Debye screening length at vi ¼ 0, ω<sup>p</sup> is the electron plasma frequency, and vth is a thermal velocity of electrons. Although no systematic studies about the use of such an effective interaction with a screening length λð Þ vi have been made for ion stopping in a magnetized electron plasma, the replacement of the real interaction by a velocity-dependent spherical one should be a reasonable approximation also in this case. The introduced dynamical screening length λð Þ vi also implies the assumption of a weak perturbation of the electrons by the ion and linear screening where the screening length is independent of the ion charge Ze, which coincide with the regimes of perturbative BC (see, e.g., Ref. [54]). Therefore, we do not consider here possible nonlinear screening effects.

Next, we specify the parameter ƛ which is a measure of the softening of the interaction potential at short distances. As we discussed in the preceding sections, the regularization of the potential (Eq. (10)) guarantees the existence of the s integrations, but there remains the problem of treating accurately hard collisions. For a perturbative treatment, the change in relative velocity of the particles must be small compared to vr, and this condition is increasingly difficult to fulfill in the regime vr ! 0. This suggests to enhance the softening of the potential near the origin of the smaller vr. Within the present perturbative treatment, we employ a dynamical regularization parameter <sup>ƛ</sup> ð Þ vi [44, 45], where <sup>ƛ</sup><sup>2</sup> ð Þ¼ vi Cb<sup>2</sup> <sup>0</sup>ð Þþ vi <sup>ƛ</sup><sup>2</sup> <sup>0</sup> and <sup>b</sup>0ð Þ¼ vi <sup>∣</sup>Z∣=e<sup>2</sup>=m v<sup>2</sup> <sup>i</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> th . Here, <sup>b</sup><sup>0</sup> is the averaged distance of the closest approach of two charged particles in the absence of a magnetic field, and ƛ<sup>0</sup> is some free parameter. In addition we also introduced C≃ 0:292 in ƛð Þ vi . In Refs. [44, 45], this parameter is deduced from the comparison of the second-order scattering cross sections with an exact asymptotic expression derived in Ref. [61] for the Yukawa type (i.e., with ƛ ! 0) interaction potential. As we have shown in Refs. [44, 45] employing the dynamical parameter ð Þ vi , the second-order cross sections for electron-electron and electron-ion collisions excellently agree with CTMC simulations at high velocities. Also, the free parameter ƛ<sup>0</sup> is chosen such that ƛ<sup>0</sup> ≪ b0ð Þ0 , where b0ð Þ0 is the distance b0ð Þ vi at vi ¼ 0. From the definition of ð Þ vi , it can be directly inferred that ƛ<sup>0</sup> does not play any role at low velocities, while it somewhat affects the size of the stopping force at high velocities when b0ð Þ vi ≲ ƛ0. More details on the parameter ƛ<sup>0</sup> and its influence on the cooling force are discussed in Ref. [47].

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