tokens: We perform this analysis for the outer and the inner (if present) spirals. On the one hand, we observe that all the outer spirals have similar pitch angles (|φ| ∼ 16°), but the simulation with T s = 20 kyr (|φ| ∼ 20°). On the other hand, we find that the inner spirals for the moving shadows (|φ| ∼ 23°) are bit larger than the planetary ones (|φ| ∼ 14°). Again, we see that the pitch angle increases with T s (see Table 1 ). In Fig. 3 , we show a synthetic image of the emission of the disc at a wavelength equal to 1 μm, where the intensity has been normalized and scaled by r 2 . This calculation is done by means of the radmc-3d code (Dullemond et al. 2012 ), modifying it to mimic the shadows by blocking the photons sent from the star in the shadowed regions. It is based on the density field for the simulation with prograde shadows with T s = 10 kyr, assuming a dust-to-gas ratio of 0.01. Figure 3. Radiative transfer calculation at 1 μm for the simulation of prograde shadows with T s = 10 kyr, shown in the third column of Fig. 2 . We assume a dust-to-gas ratio equal to 0.01. 4 DISCUSSION Moving shadows dramatically changes the disc structure and evolution. For retrograde rotations, we do not observe any spirals, despite the fact that there are two shadows cast in the disc. Because the shadows are always confronting the gas motion, there is not enough time for it to cool down in order to create a gradient pressure to act as a symmetry breaking. This could be different if a more efficient cooling mechanism is taken into account (e.g. β cooling prescription, Gammie 2001 ). Realistically, it is expected that upper layers of the gas cool down faster, resulting in a colder shadowed region able to produce a gradient pressure, promoting sharper spiral features for this mode. For static shadows, we obtain similar results as in Montesinos et al. ( 2016 ). However, in the prograde rotation case, a gas portion under the shadow moving at the same shadow's speed will always be in the shade. This corotating position can be obtained by equating the azimuthal velocity with the shadow speed v s = (2π/ T s ) r . For a Keplerian disc, where , we obtain R co = ( GM ⋆ ( T s /2π) 2 ) 1/3 . In Fig. 4 , we plot R co as a function of the shadow's period and the azimuthal velocity along with the shadow speed for the different periods T s . The intersection between both curves gives the corotating location for each model (red dots). These points are in excellent agreement with the location of the local perturbations observed in the density fields shown in Fig. 2 (see also Table 1 ). We also found that the perturbation development is independent of the stellar luminosity or the gas density. Figure 4. Left-hand panel: Shadow rotation speed ( v s = 2π/ T s r ) for different periods T s (colored dotted lines) along with the azimuthal velocity (solid curve). The intersections (red dots) give the corotating locations, in remarkable agreement with the location of the perturbations
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