A&A 366, 263-275 (2001)
DOI: 10.1051/0004-6361:20000011
J. C. B. Papaloizou - R. P. Nelson - F. Masset
Astronomy Unit, Queen Mary & Westfield College, Mile End Rd, London E1 4NS, UK
Received 8 June 2000 / Accepted 15 November 2000
Abstract
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 consider disc
models that are thought to be typical of protostellar discs during the
planet forming epoch, with characteristic surface densities similar to
standard minimum mass solar nebula models. We consider
companions, ranging in mass between 1 and 30 Jupiter masses
that are initially
embedded within the discs on circular orbits about a central solar mass.
We find that a transition in orbital behaviour
occurs at a mass in the range 10-20
For low mass
planetary companions,
we find that the orbit remains essentially
circular. However, for companion masses
we find that non
steady behaviour of the orbit occurs,
characterised by a growth
in eccentricity to values of
.
Analysis of the disc response to the presence of a perturbing companion
indicates that for the higher masses, the inner parts of the disc
that lie exterior to the companion orbit
become eccentric through an instability driven by the
coupling of an initially
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,
leading to a growth of the disc eccentricity.
The interaction of the companion with this eccentric disc, and
the driving produced by direct resonant wave excitation at the 1:3 resonance,
can lead to the growth of orbital eccentricity, with the driving
provided by the eccentric disc being the stronger. Eccentricity growth
occurs when the tidally induced gap width is such that eccentricity damping
caused by corotating Lindblad resonances is inoperative.
These simulations indicate that for standard disc models,
gaps become wide enough for the 1:3 resonance to dominate,
such that the transition from circular orbits can occur, 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 for these masses.
Another possibility is that an eccentric disc is produced
by an alternative mechanism, such as viscous
overstability resulting in a slowly
precessing non axisymmetric mass distribution.
A large eccentricity
in a planet orbit contained within an inner cavity
might then be produced.
Key words: accretion, accretion disks - methods: numerical - stars: planetary systems
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, , that are comparable to that of Jupiter ( ), have orbital semi-major axes in the range , and orbital eccentricities in the range (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 45 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 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 20 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 20 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.
We work with flat 2-dimensional disc models. In a cylindrical coordinate system centred on the central star, the disc rotation axis and the central star-companion object orbital angular momentum vectors are in the z direction. The equations of motion that describe the disc are the vertically integrated Navier-Stokes equations. The disc evolves in the combined gravitational field of the star and companion, and as a result of pressure and viscous forces. The star-companion orbit evolves under their mutual gravitational interaction and the gravitational field of the disc. For the calculations reported here, the companion gravitational potential was taken to be that of a softened point mass with softening parameter equal to being the Roche lobe radius. As with similar cases dealt with in NPMK, the companion was not permitted to accrete mass but could maintain a static atmosphere.
We use a locally isothermal equation of state, and prescribe the local sound speed to be such that the disc aspect ratio H/r=0.05 throughout the disc, being the Keplerian velocity. Thus the disc Mach number is 20 everywhere. We employ a constant value of the kinematic viscosity in dimensionless units (described below). The assumption implicit within this formalism is that the process that causes angular momentum transport in astrophysical discs may be modeled simply using an anomalous viscosity coefficient in the Navier-Stokes equations, even though it probably arises through complicated processes such as MHD turbulence generated by the Balbus-Hawley instability (Balbus & Hawley 1991, 1998).
For computational convenience we adopt dimensionless
units. The unit of mass is taken to be the sum
of the mass of the central star (M_{*}) and companion ().
The unit of length
is taken to be the initial orbital radius of the companion, .
The gravitational constant G=1, so that the natural unit of time becomes
The disc models used in all simulations had uniform surface density, with an imposed taper near the disc edge, initially. The value of was chosen such that when there exists the equivalent of 2 Jupiter masses in the disc interior to the initial orbital radius of the companion. Then in our dimensionless units.
Different mass ratios between the companion and the central star , were considered, such that . For the lower end of this range corresponds to a Jupiter mass protoplanet and the upper end to a Brown Dwarf of mass The inner radius of the disc was located at r=0.4 and the outer radius at r=6.
The calculations presented here were performed with the three dimensional MHD code NIRVANA (here adapted to 2D) that has been described in depth elsewhere (Ziegler & Yorke 1997). Viscous forces have been added as described by Kley (1998).
In each case the equations are solved using a finite difference scheme on a discretised computational domain containing grid cells, where the grid spacing in the coordinates is uniform.
For the calculations presented here and listed in Table 1, no accretion onto the companion was allowed and three different resolutions have been used. The run with q=10^{-3}, listed as N1 in Table 1 used and . The other lower resolution runs used and One higher resolution run used and . The numerical method is based on the monotonic transport algorithm of Van Leer (1977), leading to the global conservation of mass and angular momentum. The evolution of the companion orbit was computed using a standard leapfrog integrator.
Run | Resolution | q | e |
growth? | |||
N1 | 10^{-3} | No | |
N2 | 0.01 | Yes | |
N3 | 0.02 | Yes | |
N4 | 0.025 | Yes | |
N5 | 0.03 | Yes | |
N6 | 0.03 | Yes |
NIRVANA has been applied to a number of different problems including that of an accreting protoplanet embedded in a protostellar disc (NPMK). It was found to give results that are very similar to those obtained with other finite difference codes including FARGO (described in Masset 2000) and RH2D (described in Kley 1999).
In order to establish the reliability of the numerical results, simulations were also performed using an alternative code based on the FARGO fast advection method (see Masset 2000, NPMK). In this scheme, which is able to operate with longer time steps than NIRVANA, a fourth-order Runge Kutta scheme was used as orbit integrator. Results obtained with the two codes were very similar.
Figure 1: This figure shows the evolution of the eccentricity versus time (measured in units of P_{0}) for the calculations, N2, N3, N4, N5 listed in Table 1. The lines corresponding to each companion of different mass are indicated on the figure in units of | |
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Recent simulations of protoplanets with 0.001 < q < 0.01,interacting with discs with physical parameters similar to those adopted here, being thought to be appropriate to protoplanetary discs, have been carried out by Bryden et al. (1999), Kley (1999), and Lubow et al. (1999). Fixed circular orbits were assumed. It was found that a gap was formed that deepened and widened with increasing q. The disc interior to the orbit accreted onto the central star forming an inner cavity. Subsequent work by NPMK which allowed the orbit to evolve found that it remained essentially circular while migrating towards the central star.
We remark that Artymowicz (1992) indicated that orbital eccentricity growth might occur for sufficiently large while Lin & Papaloizou (1993a) argued that for sufficiently wide gaps, eccentricity growth could be induced by interaction at the 1:3 resonance in the outer disc and the 3:1 resonance in the inner disc. As the latter is absent in the work presented here, only the 1:3 resonance in the outer disc will concern us.
The aim of the work presented here is to examine for what value of the companion-star mass ratio, if any, eccentricity growth of the companion orbit is induced by its interaction with the disc model that we assume. Should it occur, we wish to understand the dominant mechanism by which growth occurs and in particular whether the 1:3 resonance is involved.
The numerical calculations are summarised in Table 1. The case of q=10^{-3} was considered in some detail in NPMK. The planet/companion in this case was found to migrate in towards the central star on a time scale given by the viscous evolution time at the initial position of the planet (also see Lin & Papaloizou 1986) remaining on an essentially circular orbit. We will not give further discussion here on this case, since eccentricity growth was not observed.
Figure 2: This figure shows the evolution of the surface density profile of the disc in calculation N5 described in Table 1. The time, in units of P_{0}, is shown at the top right hand corner of each panel. The initial clearing of a gap is shown in the first and second panel, and the growth of eccentricity of the disc interior to the 1:3 resonance (located at ) may be observed in the third panel. The fourth panel shows the disc at a later time when the companion has developed an eccentricity of , and indicates that the behaviour of the disc becomes more complicated and unsteady during these later times | |
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The evolution of the eccentricity for the remaining cases listed in Table 1 is shown in Fig. 1. It is apparent from this figure that strong eccentricity growth occurs for mass ratios in the range , with the strongest eccentricity growth being observed at the higher end of this range. From here onwards we will concentrate on describing the evolution of the q=0.03 case.
A plot of the surface density evolution of run N5 described in Table 1 is shown in Fig. 2. The times corresponding to each panel are shown in the top right hand corner in units of P_{0}. It can be seen that the disc interior to is cleared out by the action of the companion tides on a relatively short time scale, and remains tidally truncated. Further evolution of the system leads to the formation of an eccentric outer disc, as may be observed in the third panel of Fig. 2. Figure 4 also shows the formation of an eccentric disc for a companion on a fixed circular orbit, and is described below. As the eccentricity of the companion increases beyond , we observe that the disc response becomes non steady as the companion orbits between apocentre and pericentre. The evolution of the orbital radius of the companion and its eccentricity evolution (on both linear and log scales) are presented in Fig. 3. This figure shows that the eccentricity undergoes a period of rapid growth after a time of P_{0}. The logarithmic plot of e versus t shown in the second panel of this figure shows that the growth rate is initially constant (approximately), but then decreases as the system evolves beyond P_{0}, indicating exponential growth during these early stages. Analysis of the torques being exerted on the companion between the times P_{0} shows that the angular momentum is removed from the companion orbit and transferred to the disc as it approaches apocentre, leading to further growth of the orbital eccentricity. This, however, does not explain why the disc itself becomes eccentric during its early evolution. At a later time P_{0}, the eccentricity reaches a maximum value 0.25. It subsequently enters into a sequence of cyclic variations, decreasing to small values at P_{0}, before increasing again. After the initial saturation the disc is always very eccentric and in contact with the companion at some orbital phases. This interaction of the companion with material with higher specific angular momentum causes the early net inward orbital migration to reverse such that the final semi-major axis at P_{0} exceeds the initial one. But note that the details of this evolution depend on the manner in which the disc interacts with the companion and will be discussed by one of us in a future publication. Clearly the disc-companion interaction is very different in the brown dwarf regime from the planetary one.
Figure 3: This figure shows the evolution of the companion-star orbital separation as a function of time in the first panel, and the orbital eccentricity plotted as a log function (dotted line) and linear function (solid line) in the second panel. It is apparent that the eccentricity undergoes a period of rapid growth after P_{0}. Between and 100 P_{0}, the growth is approximately exponential, as indicated by the plot of in the second panel which is close to being a straight line during these times. At later times, the growth rate decreases and the plots show a change of behaviour at P_{0} when . This change in the growth of e is accompanied by a change in the behaviour of the disc surface density response, which becomes very unsteady. This latter point was discussed in the caption of Fig. 2. On average there is net inward orbital migration until P_{0}. After this time the eccentricity starts to decrease before increasing again in a cyclic variation. During these later phases the companion has contact with material with higher specific angular momentum from the eccentric disc. The interaction causes the inward migration to reverse | |
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Figure 4: This figure shows the eccentricity of the inner parts of the outer disc for a run in which the companion was maintained on a fixed circular orbit, with . The times of each panel are shown in the top right corner in units of the orbital period. The figure is plotted in a frame that is corotating with the binary orbit, and the dotted line indicates the position of the 1:2 outer Lindblad resonance | |
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Figure 5: This figure shows the evolution of the m=2 component of the Fourier transform of the surface density, in the inertial frame, for the case of a companion on a fixed circular orbit. The time elapsed when moving between the panels is shown in the top right hand corner, in units of P_{0}, where we have arbitrarily denoted the time of the first panel to be t=0.0. From the discussion in the text, it is expected that an m=2 spiral density wave will be excited at the 1:3 outer eccentric Lindblad resonance, located at (shown by the dashed line), if the eccentricity growth of the disc arises because of a period doubling instability. This wave should have a pattern speed . For an m=2 feature, the observed pattern should repeat every P_{0}, but not every P_{0}/2 (as would be the case if the wave had a frequency ). Comparing the first and fifth panels, it is apparent that the m=2 pattern does indeed repeat after every P_{0}. Comparing the first and third panels, it is obvious that the m=2 feature does not repeat every P_{0}/2 | |
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In order to explore why the disc itself becomes eccentric, an identical simulation to N5 was performed, except that the companion was maintained on a fixed circular orbit. According to the discussion given below, non linear coupling between an eccentric disc mode (with small e) corresponding to an essentially time independent m=1 pattern and the m=1component of the tidal potential and its disc response propagating with pattern speed equal to the orbital frequency, is expected to give rise to the excitation of an m=2 spiral density wave emitted at the 1:3 outer eccentric Lindblad resonance. The pattern speed of this wave is half the orbital frequency. This wave can in turn couple back through the tidal potential to produce a time independent wave and associated potential with m=1. The removal of angular momentum from the disc through such a potential will cause the disc eccentricity to increase. This is because no energy is removed along with the angular momentum, from the fluid orbits, so they cannot remain circular. If a wave with pattern speed equal to half the orbital frequency is present and launched from the 1:3 resonance, in a simulation with a companion on a fixed circular orbit, then it accounts naturally for the growth of disc eccentricity. Wave excitation is also expected to cause a growth in orbital eccentricity when that is present (see Lin & Papaloizou 1993a, and below). The excited wave is expected to have a pattern speed equal to Because the disc configuration changes to one in which the pattern rotation period has doubled, the instability produces a period doubling and so resembles a parametric instability.
A plot of the disc surface density for the run with the companion on a fixed circular orbit is shown in Fig. 4, in a frame corotating with the orbit. The panels are separated by one third of the orbital period. The 1:2 outer Lindblad resonance is shown by the dotted line, and the eccentricity of the inner parts of the disc that lie just exterior to this resonance may be observed.
The disc surface density that resulted from the simulation in which the companion remained on a fixed circular orbit was Fourier transformed in azimuth, and the m=2 component was examined. In line with our expectations, it was observed that once the inner disc was cleared of material, the presence of the m=2 wave being launched from the 1:3 outer eccentric Lindblad resonance became apparent, travelling with a pattern speed . This m=2 component of is plotted in Fig. 5 at different times during an orbit. Comparison between the first, fifth, and ninth panels, which are separated by an orbital period of the star-companion system, show that the pattern does indeed repeat after this time interval. Comparison between panels separated by half an orbital period, such as the first and third, shows that the pattern does not repeat after half an orbital period. An m=2 wave travelling with the orbital frequency would repeat after every half an orbital period. The fact that the m=2 wave only repeats after every orbital period indicates that it is being excited at the 1:3 resonance (indicated by the dashed line in Fig. 5) with a pattern speed . We conclude that a parametric instability is operating to excite this wave, giving rise to an eccentric disc. In calculation N5 where the star-companion orbit is able to evolve, the interaction between the companion and the eccentric disc leads to the growth of eccentricity of the companion orbit.
As described above, previous work on orbital eccentricity growth through disc-companion interactions indicated that direct resonant wave excitation at inner and outer eccentric Lindblad resonances should drive eccentricity, whereas interaction with material at corotating Lindblad resonances should cause it to damp (Artymowicz 1993; Lin & Papaloizou 1993a). In the simulations presented here, only the outer 1:3 resonance is important, and it is of interest to ask whether direct wave excitation at this resonance, or interaction with the disc eccentricity, is primarily responsible for the eccentricity growth of the companion orbit. To address this question, we performed simulations similar to run N5, but in which the surface density of the disc was Fourier analysed in azimuth. In these calculations the gravitational force of the disc acting on the star-companion orbit only included a contribution from an individual Fourier component (m=1, 2, 3, or 4), though the full potential of the star-companion system acting on the disc was included. We expect that if the interaction of the companion with the disc eccentricity is dominant in driving the companion eccentricity then the calculation including only the m=1 component of the disc gravitational potential will show more rapid growth of orbital eccentricity. Conversely, if resonant wave excitation of an m=2 wave at the 1:3 resonance was primarily responsible, then the calculation including only the m=2 component of the disc potential will show more rapid eccentricity growth. In fact, we find that the m=1 run showed a moderately larger eccentricity growth than the m=2 run, suggesting that the orbital interaction with the eccentric disc produces the stronger growth. This conclusion is further suggested by the estimates of the eccentricity growth rates presented in Sect. 5.10.
The runs including only the m=3 and m=4 terms showed negligible eccentricity growth.
In order to help understand some of the simulation results described above, we consider a simplified model system of primary, orbiting companion and disc. We suppose the companion orbits inside a disc with truncated inner edge. We make the additional simplification that the central star remains fixed at the origin of the coordinate system. The disc itself is taken to be inviscid with a simple relation between vertically integrated pressure, Pand surface density, Much of the complex nonlinear dynamics involving the balance between tides and viscosity determining the structure of the inner disc edge is thus omitted. However, enough content remains in order to display an instability leading to eccentricity growth in both disc and companion orbit, as well as the excitation of an outward propagating density wave with half the pattern speed. The existence of this has been described above.
This occurs because the slowly precessing nonaxisymmetric pattern in the disc makes it behave similarly to a planet of mass From standard secular perturbation theory (Brouwer & Clemence 1961) eccentricity is freely exchanged between this planet and the companion when their orbital angular momenta are comparable, a situation that occurs here when the masses are comparable.
(9) |
Although Eq. (10) applies to the joint disc and companion system,
we shall separate the contributions to the potential from
the companion and disc, writing
(11) |
(13) |
(14) |
with
with the coupling coefficient given by
We comment that although the gravity of the disc acting on both itself and the companion has been included in the above formalism, the self-interaction only affects the precise determination of the and u_{j}.The above discussion still applies if that is neglected, as long as the interaction with the companion is retained. This is the case for the simulations presented here.
The effective angular momentum content in the joint mode can be
evaluated using standard methods (e.g. Goodman & Ryu 1992;
Lin & Papaloizou 1993b).
For the low frequency modes considered here, we obtain after
a straightforward calculation
The modes discussed above, in the time averaging approximation,
do not lose angular momentum. However,
only the time averaged potential due to the companion has been included.
When the effects of the full potential are included,
additional perturbation of the disc occurs. This can lead to
angular momentum loss through resonant torques
(Goldreich & Tremaine 1978).
In that case, the mode eccentricities grow and instability occurs.
The full potential due to the companion located
at
can be written
(22) |
The angular momentum is transported outwards via a wave with pattern speed It has to be extracted from the m=1 joint mode and the companion orbit which have combined together to give the resonant forcing.
Extraction from an eccentric companion orbit can occur directly through the excited m=2 wave. Extraction from the m=1 mode can occur through a recoupling of the m=2 wave with the tidal potential with pattern speed to produce a time independent forcing with m=1. Similarly extraction from the orbit can occur through recoupling of the m=2wave to the m=1 joint mode to produce an m=1 disturbance with the circular orbit pattern speed
If the resonantly excited m=2 wave carries an angular momentum , the associated energy is If this energy is supplied by the circular orbit tide, then the angular momentum supplied along with it will be leaving to be extracted from the joint m=1 mode.
The above discussion suggests the growth rate of this mode can be obtained by assuming half the resonantly induced angular momentum loss is extracted from it.
The growth rate of the combined mode is then estimated as
(26) |
Note that here we neglect any change in the companion semi-major axis due to the time dependent terms in the disc potential. To set the magnitudes of typical growth rates, we first set the terms involving the disc eccentricity to zero. We find from (28) that for the 1:3 resonance with with P_{0} being the orbital period. Thus for here we allow a factor of two surface density enhancement above the initial value at the 1:3 resonance, we find a growth time 120 companion orbital periods. This is comparable, but a factor of 3 smaller, than the eccentricity growth rate obtained for calculation N5, and is consistent with the idea that the orbital eccentricity driving arises through effects due to both the orbital eccentricity itself and the eccentricity induced in the disc, the latter actually giving stronger effects. We comment that for these parameters the disc and companion orbit eccentricities are likely to be coupled. This is because typically Then the precession frequency induced by the companion in externally orbiting disc matter is given by Similarly the precession frequency induced by the disc (assumed to have constant surface density) in the companion orbit is These are comparable leading to the likely setting up of a joint mode. We comment that this condition is the same as requiring that the companion orbital angular momentum and the orbital angular momentum contained in assumed in circular orbit be comparable.
Under the conditions of the simulations considered here, Then it is also likely that the radial action in the disc is comparable to that of the companion and so modifies the growth rate. However, as long as they are comparable our estimate is probably reasonable. Note that as long as is comparable to or less than If the companion mass dominates, so fixing it in a circular orbit, then e=0, in (28) and attaining a limiting value independent of the disc mass. Our simulation with fixed companion orbit indicates that the growth rate does not change very much as is reduced significantly below
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 as adopted here the gap half width is 0.2 for 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 This corresponds to the Shakura & Sunyaev (1973) viscosity parameter, 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 10^{5} 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 For au, r =10 au, this gives a precession period of 10^{5} yr comparable to estimated disc dispersal times (Shu et al. 1993).
Finally, the presence of the brown dwarf Gliese 229B with mass 45 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.