,
and orbital
eccentricities in the range
Orbital migration originally considered as a mechanism operating in protoplanetary discs by Goldreich & Tremaine (1980) has been suggested as an explanation for the existence of giant planets close to their central star (e.g. Lin et al. 1996). Protoplanetary cores are thought to form at several astronomical units and then migrate inwards either before accumulating a gaseous envelope and while in the earth mass range (type I migration) or in the form of a giant planet (type II migration) (see for example Lin & Papaloizou 1993; Ward 1997). In the former case the interaction with the disc is treated in a linearized approximation (e.g. Goldreich & Tremaine 1980) while in the latter nonlinearity is important leading to gap formation (e.g. Bryden et al. 1999; Kley 1999). Both types of migration have been found to occur on a timescale more than one order of magnitude shorter than that required to form giant planets or the expected lifetime of the disc and accordingly the survival of embryo protoplanets is in question (e.g. Ward 1997; Nelson et al. 2000). Accordingly mechanisms that slow or halt migration such as the entry into a magnetospheric cavity close to the star (Lin et al. 1996) have been suggested. However, the existence of giant planets over a range of semi-major axes suggests that a more general mechanism for halting migration should be sought.
In this paper we shall consider effects produced when the large scale motion of the disc gas deviates from that of pure circular motion about the central mass as could be produced when the disc becomes eccentric as happens when it supports a global m = 1 mode. Such modes could be produced by disc protoplanet interactions (e.g. Papaloizou et al. 2001). However, for standard parameters, it was found that an instability leading to an eccentric disc together with an eccentric protoplanet orbit occurred only for large masses exceeding about fifteen Jupiter masses.
In this context note these calculations were done for laminar model discs with anomalously large viscosity coefficient and results may be somewhat different for turbulent discs. These simulations as well as resonant torque calculations appropriate to embedded cores by Papaloizou & Larwood (2000) have indicated possible reversal of orbital migration associated with eccentric orbits with modest eccentricity comparable to the disc aspect ratio.
This leads us to study further the dynamics of protoplanetary cores, massive enough for tidal interactions with the disc to be more important than effects due to gas drag, in an eccentric protoplanetary disc. In this paper we consider large scale m = 1 modes in gaseous protoplanetary discs that correspond to making them eccentric. These modes have a low frequency branch for which the disc gas follows trajectories differing from Keplerian eccentric orbits by small corrections depending on forces due to disc self-gravity and pressure. These modes can be global in that they may vary on a length scale comparable to that of the whole disc even though it might have a large dynamic range. We shall consider a ratio of outer to inner radius of one hundred. These modes are of interest because, even though no definitive excitation mechanism of general applicability has yet been identified, their large scale implies a long life time comparable to the viscous time of the disc making them of potential interest in Astrophysics (e.g. Ogilvie 2001). They have also been considered as a potential source of angular momentum transport by Lee & Goodman (1999) in a tight winding approximation. Furthermore a disc composed of many stars on near Keplerian orbits has been postulated to occur in such objects as the nucleus of M 31 (Tremaine 1995).
Here we consider global m = 1 modes for various disc models
neglecting viscous processes which are presumed to act over
a longer time scale than that appropriate to the
phenomena of interest. For global disturbances
in protoplanetary discs of the type we consider, inclusion of self-gravity
is important. Even though the discs are gravitationally stable,
pressure and self-gravity can be equally important on scales
comparable to the current radius, r, when the Toomre stability
parameter
,
with H being the disc semi-thickness.
This condition is satisfied for typical protostellar disc
models (e.g. Papaloizou & Terquem 1999). In addition to this,
disc self-gravity has to be considered in the description of the motion
of embedded protoplanets and it generally causes them to move in
eccentric orbits, with eccentricity comparable to or exceeding that
in the disc. We find that circularization due to tidal interaction
with the disc may play only a minor role if the local test particle
precession frequency relative to the disc is large compared to
the circularization rate. We also find that for modest
disc eccentricities comparable to the aspect ratio H/r,disc tidal interactions may differ significantly from those found
in axisymmetric discs. This in turn may have important consequences
for estimates of orbital migration rates
for protoplanetary cores in the earth mass range.
In Sect. 2 we give the Basic equations and the linearized form we use to calculate global m=1 modes with low pattern speed corresponding to introducing a finite disc eccentricity. We go on to describe the disc models used which may contain protoplanets orbiting in an inner cavity. In Sect. 3 we present the results of normal mode calculations. We go on to discuss the motion of a protoplanet in the earth mass range in an eccentric disc in Sect. 4, determining the equilibrium (non precessing) orbits which maintain apsidal alignment with the disc gas orbits. We then formulate the calculation of the tidal response of an eccentric disc to a low mass protoplanet in Sect. 5 determining the time rate of change of the eccentricity and orbital migration rate. We find that aligned orbits with very similar eccentricity to that of the gas disc may suffer no eccentricity change while undergoing inward migration in general. However, when the non precessing aligned orbit has a significantly higher eccentricity than the disc, as can occur for modes with very small pattern speed, orbital migration may be significantly reduced or reverse from inwards to outwards for the disc models we consider. This finding is supported by a local dynamical friction calculation applicable when the protoplanet eccentricity is much larger than the disc aspect ratio. Finally we go on to discuss our results in Sect. 7.
Copyright ESO 2002