8 Summary

This paper examines the torque exerted on a protoplanet held on a fixed circular orbit by a uniform surface density and uniform aspect ratio viscous protoplanetary disk, by means of numerical simulations. A number of runs have been performed, varying the disk aspect ratio, the planet mass and the disk viscosity, which enables one to disentangle the differential Lindblad torque/corotation torque additional term from the co-orbital corotation torque main term. The differential torque functional dependence upon the width of the librating fluid elements region and upon viscosity is in good agreement with previous expectations. The behavior of the total torque as a function of viscosity and planet mass is presented. It is found that the curves of the normalized total torque as a function of viscosity are lift up as the planet mass increases, eventually leading to a positive torque peak value in the thinnest disks. This behavior is explained as due to the non-linear Lindblad torque cut-off and to the corotation torque additional term, which both conspire in lifting the normalized torque as the planet mass increases. Smoothing issues are discussed. The smoothing length is shown to have a different impact on the Lindblad torque and on the corotation torque, and the corotation torque is found to depend dramatically on the smoothing length value, which shows that there is no such thing as a "magic value'' for the smoothing length which would give unconditionally correct results (i.e. compatible with fully three dimensional calculations). A method is given to evaluate the value of the smoothing length which gives correct results for the corotation torque (i.e. identical to what one expects in a 3D situation). If the motion in the disk in the co-orbital region is purely horizontal, this method should give in principle an exact result. Note however that the functional dependence upon viscosity of the 3D corotation torque and the 2D one with the appropriate smoothing length may be different, and that the method exposed here is exact only for the case of a fully unsaturated torque, i.e. at large viscosity (although before the cut-off). High resolution runs are presented in which the smoothing length is chosen conservatively, and these runs show that Neptune-sized planets undergo a positive torque in thin, viscous disks with a shallow surface density profile, although only a lower limit of the torque value can be inferred from the runs.

Acknowledgements

I wish to thank Prof. M. Tagger for fruitful discussions, as well as the anonymous referee for comments and a careful reading of the manuscript. The early stages of this work were supported by the research network "Accretion onto Black Holes, Compact Stars, and Protostars'' funded by the European Commission under contract ERBFMRX CT 98-0195. Computational resources were available at the Rechenzentrum Garching and at the CGCV Grenoble, and are gratefully acknowledged. The last stages of this work were carried out at the Institute of Astronomy, UNAM, Mexico, and I wish to thank Drs. J. Cantó, A. Raga and A. Santillán for hospitality.