Trajectory of a spacecraft (red arrows) through the model of the expanding toroidal flux rope, corresponding to the fit from Fig. 9b. The trajectory is shown in the intrinsic Cartesian system x, y, and z of the toroid, Eqs. (19)–(21) (the coordinates are different from the GSE coordinates), in projection to the a) yz-plane, b) xz-plane, and d)xy-plane; panel c displays the trajectory in a cross section of the toroid, where coordinates r and θ serve as polar coordinates. The blue lines are the boundaries of the toroid at the spacecraft entry, the green lines are the boundaries at the exit. According to the model assumptions, the minor radius of the toroid increases during expansion, while the major radius remains constant.