3 Normal modes

We have calculated the lowest order global normal modes for two different configurations involving protoplanets orbiting interior to the disc in an inner cavity. The first, approximating the Upsilon Andromedae system involves two protoplanets with mass ratios 0.00383and 0.00196 orbiting at $0.6R_{\rm in}$ and $0.194R_{\rm in}$respectively. However, the disc normal modes do not change much in character if the protoplanet orbital radii are scaled to somewhat smaller fractions of $R_{\rm in}.$ But the ratio of protoplanet to disc eccentricity in the joint modes changes more significantly. The second configuration we consider is a single protoplanet with mass ratio 0.002 orbiting at $0.6R_{\rm in}.$Finally we consider a disc with no interior orbiting protoplanets. In all of these cases we consider both model A and model B discs.

The pattern speeds for the highest frequency normal modes and the protoplanet orbital eccentricities occurring jointly with the normal modes are given in Table 1. For the results presented here, the modes are normalized such that the disc eccentricity at the inner boundary is 0.1.

Figure 2: This figure shows the first two modes and the associated equilibrium eccentricities as a function of radius r in units of $R_{\rm in}$for the two protoplanet case with disc model A. The two modes (full curves) are normalized so that e=0.1 at r=1. The lower curve corresponds to the mode of highest frequency given in Table 1. The dotted curves give the associated equilibrium eccentricities calculated for a low mass protoplanet orbiting within the disc, the lowermost corresponding to the highest frequency mode. The dashed curve gives the equilibrium eccentricity for a low mass protoplanet orbiting within the disc calculated for the highest frequency mode but neglecting the non axisymmetric component of the disc potential. The dot dashed curve gives the corresponding plot for the other mode. The latter curves indicate the presence of secular resonances at $r \sim 1.1$ associated with both modes.

Many of the properties of these modes can be understood with reference to the local dispersion relation for density waves in the low frequency limit (Lin & Shu 1969) in the form

$$2\Omega(\omega_{\rm p} -\Omega_{\rm p}) = - 2\pi G \Sigma \vert k\vert +c^2k^2 ,$$ (22)

where k is the radial wavenumber. The above indicates that when self-gravity, governed by the first term on the right hand side, is unimportant as occurs for either low mass discs or large |k|, $\Omega_{\rm p}$ is negative. On the other hand positive $\Omega_{\rm p}$ can occur for low |k|or massive discs. Thus in that case the lowest order modes (in terms of numbers of nodes) should be prograde with more of these existing for more massive discs, a trend we find in our results. There can also be very low frequency (in magnitude) modes for which the self-gravity and pressure terms approximately cancel. This occurs when $\vert k\vert = 2\pi G \Sigma/c^2 \sim 2/(QH).$For $Q \sim r/H$ as in the models here, this is comparable to the radius making a very global mode.

Figure 3: As in Fig. 2 but for disc model B. In this case there is a secular resonance asociated with the highest frequency mode at $r \sim 1.1$. For the other mode this occurs at $r \sim 1.4.$

As we calculate normal modes jointly involving protoplanets and disc, the protoplanets have associated eccentricities. In general the ${\cal N}$ highest frequency modes predominantly involve the protoplanets while the others predominantly involve the disc, ${\cal N}$ being the number of protoplanets. The two highest frequency modes for the two protoplanet case are plotted in Fig. 2 for disc model A and in Fig. 3 for disc model B. For the second highest frequency mode calculated for disc model A, the protoplanet eccentricity ratio is 1.38, while for model B it is 1.88. The latter result is similar to that given by Chiang et al. (2001) who considered the current Upsilon Andromedae system with no disc and concluded it was predominantly in this mode.

The results found here suggest the disc and protoplanet orbits were antialigned. With the outermost protoplanet at $0.6R_{\rm in}$ the disc inner boundary and outer protoplanet eccentricities are comparable for model A while for model B the relative disc eccentricity is 56 percent smaller. However, when the outer protoplanet orbits at  $0.5R_{\rm in}$ the disc inner boundary eccentricity is only 1/3 that of the protoplanet for model A and 19 percent for model B. Apart from the eccentricity scaling relative to the interior protoplanets, the spatial form of the eigenfunction in the disc remains almost identical.

The four highest frequency modes for the one protoplanet case with disc model A are plotted in Figs. 4 and 5 for model B. Pure disc modes with no protoplanet are plotted in Fig. 6 for model A and Fig. 7 for model B. The modes develop increasing numbers of nodes as their frequency decreases but the mode with lowest frequency in absolute magnitude can be very global with only a few nodes out to $r=R_{\rm out}.$We comment that the modes are rather non compressive requiring eccentricities of order unity to provide Lagrangian changes in surface density of order unity. As the motion in the modes in linear theory is assumed epicyclic, this is suggestive that the analysis should be valid as long as the epicyclic approximation is.

Figure 4: This figure shows the four highest frequency modes as a function of radius r in units of $R_{\rm in}$for the one planet model with disc A. The eccentricities are all normalized so that e = 0.1 at r=1.The modes can be identified by noting that the number of nodes increases with decreasing eigenfrequency.

Figure 5: As in Fig. 4 but for the one planet model with disc model B.

Apart fom the two modes with highest frequency, other normal modes are essentially pure disc modes with only small associated protoplanet eccentricities.

Figure 6: As in Fig. 4 but for the disc model A with no protoplanets.

Figure 7: As in Fig. 6 but with disc model B.

Having calculated these modes that may be largely driven by protoplanets in eccentric orbits or possibly self-excited or long lived structures we now go on to consider the migration of low mass protoplanets embedded in such eccentric discs.

  

Table 1: This table gives the protoplanet orbital eccentricities where appropriate together with the pattern speeds for the normal modes calculated. The disc model used is indicated in the first column, the protoplanet mass ratios to the central star in the second and third columns, their orbital eccentricities in the fourth and fifth column and the mode frequency or pattern speed in the sixth column. The modes are normalized such that the disc eccentricity at the inner boundary is 0.1. A negative protoplanet eccentricity then indicates the orbit is antialigned with the disc.

$$\begin{tabular}{lllllr@{$~\times~$}l} \hline \hline Disc &$m... ... B& - & -& - & - &$-5.77$&$10^{-4}$ \\ \hline \end{tabular}$$