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tokens: respect to the geometric centre, with a sinusoidal function was derived by Williams et al. ( 2016 ). Fig. 5 shows the resulting expansion velocity of the forward shock as a function of the position angle for Model B at t = T age = 450 yr; the detected velocities of the proper motion from the X-ray and the radio in Williams et al. ( 2016 ) for a distance of 2.3 kpc are also indicated. The velocities from the simulation for Model B at different position angles of the forward shock range from 3000 to 4000 km s −1 . For the shock velocities from the simulation and the ones derived from X-ray/radio observations according to the position angle, a significant deviation exists between 50° and 100°. In particular, the detected velocity at θ = 80° is about 1000 km s −1 smaller than the simulation value. Moreover, the velocities from θ = 200° to 250° are about 500 km s −1 smaller than the detected ones. Apart from this variation, at θ = 15° and 30°, the resulting velocities are consistent with those detected in the radio. From 120° to 150°, coincident with the protrusion to the south-east, the detected values of velocities and the decreasing trend are reproduced. The increasing trend, which is indicated in both the X-ray and the radio observations, is also consistent with the simulation of the velocities for the position angle from 150° to 180°. Moreover, except for 276°, the simulated velocities from 250° to 350° are about 3500 km s −1 , which is in agreement with the averaged values of the detected ones in the radio and the X-ray band. Figure 5. The distribution of the speed of the forward shock in relation to the position angle in the plane x = 0 for Model B at t = T age = 450 yr. The detected velocities at different positions on the periphery of Tycho's SNR for a distance of 2.3 kpc based on the radio and the X-ray observations are also indicated for comparison (Williams et al. 2016 ). 4 DISCUSSION AND CONCLUSIONS A series of 3D MHD simulations have been performed both to investigate the formation mechanism of the peculiar morphology of Tycho's SNR and to derive the ambient material distribution prior to the supernova. These simulations are based on the assumption that the progenitor is imbedded in a bubble that is blown by the wind, which has a latitude-dependent density distribution. Moreover, the inclination angle between the pole axis of the wind and the north direction, and the spatial motion of the progenitor are included in order to reproduce the asymmetry of the profile of the SNR shock. With a pole velocity of v p = 100 km s − 1 , ξ = 20 and a duration of T w ∼ 10 5 yr, an inclination angle of ∼50° between the pole of the wind and the north direction and a spatial velocity of v s ∼ 0.3 v p are derived based on the simulation to reproduce the detected profile of the outer border of the remnant. Furthermore, because a wind with a velocity greater than 100 km s −1 can be produced as a result of accretion onto a WD, an SD scenario for the