2016A&A...589A.133F

Context. Planetesimals and planets embedded in a circumstellar disk are dynamically perturbed by the disk gravity. It causes an apsidal line precession at a rate that depends on the disk density profile and on the distance of the massive body from the star.
Aims. Different analytical models are exploited to compute the precession rate of the perihelion p. We compare them to verify their equivalence, in particular after analytical manipulations performed to derive handy formulas, and test their predictions against numerical models in some selected cases.
Methods. The theoretical precession rates were computed with analytical algorithms found in the literature using the Mathematica symbolic code, while the numerical simulations were performed with the hydrodynamical code FARGO.
Results. For low-mass bodies (planetesimals) the analytical approaches described in Binney & Tremaine (2008, Galactic Dynamics, p. 96), Ward (1981, Icarus, 47, 234), and Silsbee & Rafikov (2015ApJ...798...71S) are equivalent under the same initial conditions for the disk in terms of mass, density profile, and inner and outer borders. They also match the numerical values computed with FARGO away from the outer border of the disk reasonably well. On the other hand, the predictions of the classical Mestel disk (Mestel 1963MNRAS.126..553M) for disks with p=1 significantly depart from the numerical solution for radial distances beyond one-third of the disk extension because of the underlying assumption of the Mestel disk is that the outer disk border is equal to infinity. For massive bodies such as terrestrial and giant planets, the agreement of the analytical approaches is progressively poorer because of the changes in the disk structure that are induced by the planet gravity. For giant planets the precession rate changes sign and is higher than the modulus of the theoretical value by a factor ranging from 1.5 to 1.8. In this case, the correction of the formula proposed by Ward (1981) to account for the presence of a gap is a better approximation, at least in predicting a positive precession rate.
Conclusions. Analytical modeling of the precession rate of massive bodies embedded in a circumstellar disk is accurate for planetesimals and terrestrial planets, but it becomes inaccurate when Jupiter-sized planets are considered. The changes they induce in the disk reverse the precession rate and increase it by more than 50%. This discrepancy can be partly solved by including the effects of a gap in the disk density profile in the analytical equations, as done in Ward (1981).