6 Discussion

In this paper we have investigated the driving of orbital eccentricity of giant protoplanets and brown dwarfs through disc-companion tidal interactions by means of two dimensional numerical simulations. Disc models thought to be typical of protostellar discs during the planet forming epoch, with characteristic surface densities similar to standard minimum mass solar nebula models were considered. We examined the evolution of companions ranging in mass between 1 and 30 $M_{\rm J}$ initially embedded within the discs on circular orbits. For low mass companions, typical of giant planets, we found, in agreement with our previous work (NPMK) that the orbit remains essentially circular. For companion masses that are greater than $\sim$20 $M_{\rm J}$, however, we found that a transition occurs, leading to a non steady behaviour of the companion orbit. This is characterised by a growth in eccentricity such that $0.1 \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;e \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;0.25$.

The inner parts of the disc that lie exterior to the companion orbit become eccentric through an instability driven through the coupling of the non circular motions associated with a small disc eccentricity to the companion's tidal potential. This coupling leads to the excitation of an m=2 spiral wave at the 1:3 outer eccentric Lindblad resonance, which transports angular momentum outwards. A similar picture of disc eccentricity driving for inner discs has been discussed by Lubow (1991) in the context of Cataclysmic Variables. As the disc eccentricity corresponds to a negative angular momentum mode, this angular momentum loss leads to a growth of the disc eccentricity. In addition to the effects of resonant wave excitation at the 1:3 resonance produced by the direct forcing of the companion in its eccentric orbit, the gravitational interaction of the companion with this eccentric disc leads to the growth of eccentricity of the companion orbit, where this latter effect is found to be moderately larger.

For a companion orbiting within a disc, the effects of the 1:3 resonance lead to growth of the eccentricity while the effects of corotation and coorbital Lindblad resonances lead to its damping (e.g. Artymowicz 1993; Ward & Hahn 2000). For a very wide gap or isolation of the companion from the disc material, the effects of the 1:3 resonance win and the eccentricity grows (Lin & Papaloizou 1993a).

However, our simulations indicate that for standard disc models, sufficient clearance due to the companion tides occurs only for masses in the brown dwarf range. However, the transition mass might be reduced into the range for extrasolar planets if the disc viscosity is significantly lower enabling wider gaps to occur. One can estimate the viscosity required by noting that the gap must extend out to the 1:2 resonance. For $\nu=10^{-5}$ as adopted here the gap half width is $\sim$0.2 for $1~M_{\rm J}.$To reach the 1:2 resonance this has to be three times larger. From Lin & Papaloizou (1993a) and Bryden et al. (1999), the tidal torque, which varies as the inverse cube of the gap width, is then reduced by a factor of 27. To prevent the gap filling the viscosity would have to be reduced by at least the same factor requiring $\nu \le 3\; 10^{-7}.$ This corresponds to the Shakura & Sunyaev (1973) viscosity parameter, $\alpha \le 1.5\; 10^{-4}.$ Note that this is significantly smaller than values normally adopted for protostellar discs (e.g. Papaloizou & Terquem 1999).

We also found that when the angular momentum content of the disc material within a scale characteristic of the inner edge radius is comparable to that of the companion in a circular orbit, the eccentricity of the disc and companion are coupled. This behaviour occurs because the gravitational potential produced by the disc is similar to that of another companion in eccentric orbit. A coupling is then expected from standard secular perturbation theory. When the companion and disc masses are disparate, orbital eccentricity would be expected only for the smaller of the two.

Although the extrasolar planet mass range is too small (Marcy & Butler 2000) for eccentricity driving due to the 1:3 resonance assuming standard disc parameters, it is possible that it could be produced if the protoplanet orbits in a cavity with an eccentric external disc. This would require disc m=1 modes to be excited by some other mechanism e.g. viscous overstability (Kato 1978; Papaloizou & Lin 1988). A slowly precessing non axisymmetric mass distribution would then be produced. A configuration like the one described above might be produced during the phase of disc clearance. There is observational evidence that this occurs on a 10⁵ yr time scale working from the inside out (Shu et al. 1993). The precession period of the nonaxisymmetric mass distribution would be time variable and could potentially equal that of the inner protoplanet orbit at some stage. Because the average disc and protoplanet orbits would then maintain a fixed orientation, a large eccentricity in the protoplanet orbit can be produced by gravitational torques on the precession time scale. Notably the effect need not be correlated with the protoplanet mass. A mechanism operating on the same principle has been proposed by Ward et al. (1976) as a mechanism for producing the eccentricity of Mercury. The precession frequency induced by the disc (assumed to have constant surface density) in the protoplanet orbit is $\omega_{\rm p} = \Omega(r_{\rm p}) (r_{\rm p}/r)^3 (M_{\rm d}/M_*).$ For $r_{\rm p} =1$ au, r =10 au, $M_{\rm d}/M_* \sim 0.01,$ this gives a precession period of $\sim$10⁵ yr comparable to estimated disc dispersal times (Shu et al. 1993).

Finally, the presence of the brown dwarf Gliese 229B with mass $\sim$45 $M_{\rm J}$ in a binary system indicates the potential existence of a separate population, at the one percent incidence level, of brown dwarfs (Oppenheimer et al. 2000) which could form from discs. The simulations presented here indicate eccentricity excitation due to the effects of the 1:3 resonance plays a role for these masses. The different type of disc behaviour could result in a distinct orbital and mass distribution for these objects as compared to extrasolar planets.

Acknowledgements

This work was supported by PPARC grant number PPARC GR/L 39094. It was also supported in part (F.M.) by the European Commission under contract number ERBFMRX-CT98-0195 (TMR network "Accretion onto black holes, compact stars and protostars''). We thank Udo Ziegler for making a FORTRAN Version of his code NIRVANA publicly available. The calculations reported here were carried out using GRAND, a high performance computing facility funded by PPARC.