1 Introduction

Disc-companion interactions are important in a variety of astrophysical contexts ranging from orbital evolution, tidal interaction, and accretion in close binary systems (e.g. Lin & Papaloizou 1979; Artymowicz & Lubow 1994; Papaloizou & Terquem 1995; Larwood & Papaloizou 1997) to the evolution of black hole binary pairs in active galaxies (Begelman et al. 1980; Ivanov et al. 1999).

The recent discovery of a number of extrasolar giant planets orbiting around nearby solar-type stars (Marcy & Butler 1998, 2000) has stimulated renewed interest in the theory of planet formation and disc-companion interaction. These planetary objects have masses, $m_{\rm p}$, that are comparable to that of Jupiter ( $0.25 \; M_{\rm J} \; \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\... ...} \ \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\; 11 \; M_{\rm J}$), have orbital semi-major axes in the range $0.03 \; {\rm AU} \; \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\; a \; \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\; 2.5 \; {\rm AU}$, and orbital eccentricities in the range $0.0 \; \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\; e \; \; \raisebox{-.8ex}{$\buildrel{\textstyle<}\over\sim$ }\;\; 0.67$ (Marcy & Butler 2000). Explaining these data is one of the major challenges faced by planet formation theory.

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 also form from discs, possibly through a different mechanism to that for extrasolar planets.

Recent simulations of protoplanets in the observed mass range (Kley 1999; Bryden et al. 1999; Lubow et al. 1999) interacting with a disc with parameters thought to be typical of protoplanetary discs, indicate gap formation and upper mass limit consistent with the observations. In addition simulations by Nelson et al. (2000) (hereafter NPMK) which allowed the protoplanet orbit to change found inward migration on near circular orbits leaving the observed eccentricities of extrasolar planets unexplained.

However, previous discussions of this problem (Artymowicz 1992; Lin & Papaloizou 1993a) have indicated that orbital eccentricity might be driven by disc-companion interactions. This might be expected from the general theory of tidal interaction. Jeffreys (1961) showed that a body in circular orbit around a rapidly rotating central object could have an instability driving orbital eccentricity. A similar instability might be expected for a body orbiting inside a rotating disc. In this case resonant wave excitation at the inner and outer eccentric Lindblad resonances leads to the excitation of eccentricity. However, a sufficiently wide gap is required to exclude coorbital and near coorbital disc material which would damp the eccentricity through coorbital Lindblad torques (Artymowicz 1993). Thus an eccentric instability would only be expected for a mass sufficiently large to clear a wide gap. The mass above which eccentric orbits might be excited is expected to be a function of the disc parameters.

Determination of this mass is important for the theory of disc-companion interactions and its implications for binary systems and the formation theory of planets and brown dwarfs. We find that the transition to an eccentric orbit can be linked to the driving of an eccentric disc, and it appears that the interaction between the companion and this disc eccentricity produces eccentricity driving that moderately exceeds that due to direct resonant wave excitation at the 1:3 outer eccentric Lindblad resonance discussed above. As the behaviour of the disc then also has a transition, different predictions for the mass spectrum and orbital distribution of more massive objects above the transition mass may result.

In this paper we investigate the driving of orbital eccentricity of giant protoplanets and brown dwarfs through disc-companion tidal interactions by means of two dimensional numerical simulations. We examined the evolution of companions ranging in mass between 1 and 30 $M_{\rm J}$orbiting within protoplanetary discs about a central solar mass. For standard parameters, we find the transition to eccentric orbits occurs for companion masses that are greater than $\sim$20  $M_{\rm J}.$ In these cases the inner disc has accreted onto the central star. The inner parts of the disc that lie exterior to the companion orbit were found to become eccentric. This latter process is associated with the excitation of a m=2 spiral wave at the 1:3 outer eccentric Lindblad resonance. This effect has a counterpart in the modelling of discs in Cataclysmic binaries. Here, in the lower companion mass range, an eccentric circumprimary disc has been seen (Whitehurst 1988; Hirose & Osaki 1990; Lubow 1991). This in turn is related to the excitation of a m=2 spiral wave at the 3:1 inner eccentric Lindblad resonance, as described by Lubow (1991).

Our present results indicate eccentric orbits only for masses in the brown dwarf range. However, the transition mass might be reduced into the range for extrasolar planets if wider gaps or more extensive disc clearance occurs while allowing an outer eccentric disc to still exist.

The plan of the paper is as follows. In Sect. 2 we describe the physical model of the disc-companion system used. In Sect. 3 we describe the numerical methods. In Sect. 4 we present the numerical results which indicate the transition from circular companion orbit to eccentric companion orbit together with an eccentric outer disc for companion masses exceeding $\sim$20 $M_{\rm J}.$In Sect. 5 we give a simple analytic model illustrating how the eccentricity can be driven both in the companion orbit and exterior disc by an instability operating through density wave excitation at the outer 1:3 Lindblad resonance leading to a pattern rotating with 1/2 the companion orbital frequency as seen in the numerical calculations. A similar analysis for the driving of eccentricity in circumstellar discs, but with fixed circular companion orbit, is discussed in Lubow (1991). Finally in Sect. 6 we discuss our results and speculate on how eccentric discs might produce eccentric orbits for masses in the planetary range.