tokens: width of the horseshoe region in front and at the rear of the planet in order to minimize the effect of an asymmetry, we prefer not to introduce a source of asymmetry of the horseshoe region. 4 The planet mass is introduced progressively over two orbital periods, and the width of the horseshoe region is measured 20 orbits after the introduction of the planet. We perform a systematic study for three values of the aspect ratio: h = 0.03, h = 0.04 and h = 0.05. For each of these values, we perform 30 calculations with a planetary mass varying in a geometric sequence from 0.05 h 3 to 4 h 3 . The width of the horseshoe region is determined automatically by dichotomy, averaging the results in φ = 1 rad and φ = −1 rad. We find that more accurate results are obtained if the disc is inviscid when q 0.4 h 3 , and viscous for larger masses. For lower mass indeed, a viscous drift of the disc can distort severely the horseshoe region (Masset 2002 ), whereas for high masses, the planet can trigger vortices on the edges of its horseshoe region if the disc is inviscid, which precludes an accurate determination of the width of the horseshoe region. For low masses, we have the scaling (Lega et al. 2015 ; Masset Benítez-Llambay 2016 ): whereas for large masses we recover the scaling found in 2D simulation with softened potential (Masset et al. 2006 ): Using the width normalized to the pressure length scale X s = x s /( hr p ) and denoting with Q the planetary mass normalized to the thermal mass : we can rewrite equations ( 24 ) and ( 25 ) as and The lack of an explicit reference to h in these expressions suggests that the normalized width of the horseshoe region has a universal dependence on the mass expressed in thermal masses. Fig. 6 confirms this expectation. We find that a function that gives the asymptotic behaviours of equation ( 27 ) and ( 28 ) and that matches satisfactorily the width behaviour for intermediate masses is a blend of the asymptotic values at low- and large mass with weights respectively ε and 1 − ε, where ε = 1/(1 + 2 Q 2 ). This yields or Figure 6. Normalized half width of the horseshoe region as a function of the mass in thermal masses. The data from the three different aspect ratios coincide nearly exactly. The solid curve corresponds to equation ( 29 ), the dotted curve to equation ( 27 ) and the dashed curve to equation ( 28 ). This figure should be compared to fig. 7 of Lega et al. ( 2015 ). From now on we will use equation ( 30 ) to evaluate the width of the horseshoe region. As a side result, this leads us to a reasonable definition of an intermediate mass protoplanet as a protoplanet that has a horseshoe width intermediate between the widths given by the low-mass and high-mass scaling (equations 24 and 25 ). Examination of Fig. 6 shows that this broadly corresponds to 0.2 h 3 ≲ q ≲ 2 h 3 . We assume that the corotation torque tends towards its linear value when dissipation (either viscous or thermal) is large, regardless of the
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