4 The motion of a low mass protoplanet in an eccentric disc

The evolution of the coplanar orbit of an interior protoplanet in the earth mass range embedded in the disc due to the gravitational interaction with the slowly precessing disc can be found using secular perturbation theory. We regard the embedded protoplanet as a test particle evolving in a prescribed gravitational potential and neglect the influence of the protoplanet on the disc mode. This should be reasonable when as here, the angular momentum of the protoplanet is significantly less than that of the material supporting the normal mode (e.g. Papaloizou et al. 2001).

The orbit evolves under the perturbing potential per unit mass,

$${\cal{ V}} \equiv \Phi_{\rm D} + \Phi_{\rm ext} = \Phi_0(r) +\Phi_1(r)\cos(\varphi -\Omega_{\rm p} t).$$ (23)

Here $\Phi_0(r)$ is the axisymmetric component of the potential but excluding that due to the central mass and $\Phi_1(r)$ is the radial amplitude of the m=1 component arising from the normal mode.

On performing a time average over the protoplanet orbit one obtains the Hamiltonian system:

$${{\rm d}h\over {\rm d}t} = -{\partial {\cal{H}}\over \partial... ...a \over {\rm d}t} ={\partial {\cal{H}} \over \partial h }\cdot$$ (24)

Here $\alpha= \varpi -\Omega_{\rm p} t,$ with $\varpi$ being the longtitude of periapse and ${\cal{ H}} ={\cal{V}} -\Omega_{\rm p} h .$The specific angular momentum is $h= \sqrt{GM_* a_{\rm p}(1-e_{\rm p}^2)},$ $a_{\rm p}$ is the constant semi-major axis, and $e_{\rm p}$ is the eccentricity. To leading powers of $e_{\rm p}$ we have

$${\cal{H}} = A(a_{\rm p})+ B(a_{\rm p}) e_{\rm p}^2 -\Omega_{\... ... +C(a_{\rm p})e_{\rm p}^4 + e_{\rm p}D(a_{\rm p})\cos\alpha .$$ (25)

Hamilton's equations then give to leading powers of $e_{\rm p}$

$${{\rm d}\alpha \over {\rm d}t} = -{2(B+e_{\rm p}^2(2C-B/2))\o... ...a_{\rm p})\cos\alpha\over \sqrt{GM_*a_{\rm p}} e_{\rm p}}\cdot$$ (26)

$${{\rm d} e_{\rm p}\over {\rm d}t} = {\sqrt{1-e_{\rm p}^2}\ove... ...ha} = -{D(a_{\rm p}) \over \sqrt{GM_* a_{\rm p}}}\sin\alpha .$$ (27)

Here the precession rate of the apsidal line induced by the axisymmetric potential, to first order in the perturbing potential, expressed as an expansion in powers of $e_{\rm p}$ is given by the first term on the right hand of Eq. (26) as

$$\omega_{\rm pg} =-{2\over \sqrt{GM_*a_{\rm p}}} \left(B+e_{\rm p}^2(2C-B/2)\right) .$$ (28)

This may also be written in terms of the axisymmetric component of the potential directly as

$$2\omega_{\rm pg}\sqrt{GM_*a_{\rm p}^5} =(1+e_{\rm p}^2){{\rm ... ...{\rm p}^2\over 8a_{\rm p}^2}{{\rm d}^4\Phi_0\over {\rm d}u^4},$$ (29)

where u=1/r, and r is taken to be equal to $a_{\rm p}.$

Equilibrium or steady state solutions of (26) and (27) with $e_{\rm p}$ and $\alpha$ constant occur when $\alpha=0$ or $\alpha=\pi.$Adopting the convention that $e_{\rm p}$ is positive suffices to select one of the latter possibilities as (26) gives an expression from which $e_{\rm p}$can be determined in the form

$$e_{\rm p} = -{{D(a_{\rm p})\cos\alpha}\over{{2(B+e_{\rm p}^2(2C-B/2))} +\Omega_{\rm p} \sqrt{GM_*a_{\rm p}} }}$$ (30)

or equivalently

$$e_{\rm p} = {{D(a_{\rm p})\cos\alpha}\over{(\omega_{\rm pg} -\Omega_{\rm p})\sqrt{GM_*a_{\rm p}} }} \cdot$$ (31)

Alternatively one may fix $\alpha$ to be 0 or $\pi$and allow $e_{\rm p}$ to change sign as we have done for the normal modes.

An equilibrium solution so determined corresponds to the situation when the protoplanet orbit precesses at the same rate, $\Omega_{\rm p},$ as the mass distribution that produces the gravitational potential while maintaining a constant eccentricity. Assuming that $\omega_{\rm pg}$ and $\Omega_{\rm p}$ are of comparable magnitude, in general for modest eccentricity,

$$e_{\rm p} \sim D(a_{\rm p})/(\Omega_{\rm p} \sqrt{GM_*a_{\rm p}}) \equiv e_0$$

and is proportional to the magnitude of the nonaxisymmetric potential. An exception occurs when

$${2B +\Omega_{\rm p} \sqrt{GM_*a_{\rm p}} }=0.$$

In this case, the precession frequency of a free orbit with small eccentricity, $\omega_{\rm pg},$ matches that of the nonaxisymmetric mass distribution $\Omega_{\rm p}$ corresponding to a secular resonance. When this occurs Eq. (30) indicates that $e_{\rm p} \sim (e_0)^{1/3}.$Accordingly significantly larger equilibrium eccentricities are expected close to a secular resonance. We comment that for fixed $\alpha$$e_{\rm p}$ changes sign as a secular resonance is passed through corresponding to an alignment change through a rotation of the axis of the ellipse through $\pi$.

One may also investigate the effect of orbital circularization by adding a term $-e_{\rm p}/\vert t_{\rm e}\vert$ to the right hand side of (27), where $\vert t_{\rm e}\vert$ is the circularization time. In this case one can still find an equilibrium but with orbital apsidal line rotated. Restricting consideration to the situation away from secular resonance, one finds that the equilibrium eccentricity is reduced by a factor $\sqrt{1+1/((\omega_{\rm pg} -\Omega_{\rm p})^2t_{\rm e}^2)}$and $\vert\sin \alpha\vert = 1/\sqrt{1+(\omega_{\rm pg} -\Omega_{\rm p})^2t_{\rm e}^2}.$Thus when $\vert(\omega_{\rm pg} -\Omega_{\rm p})t_{\rm e}\vert$ is large, the effect of the circularization term is to produce a small rotation of the apsidal line of the orbit.

As indicated above equilibrium solutions correspond to the situation where the apsidal precession of the protoplanet orbit is locked to that of the underlying nonaxisymmetric disc.

One can find solutions undergoing small librations in the neighbourhood of equilibrium solutions (e.g. Brouwer & Clemence 1961) and when dissipative forces are added these may decay making the attainment of equilibrium solutions natural. As we indicate below, tidal interaction of a protoplanet with the disc may produce such dissipative forces. Accordingly we shall focus on equilibrium solutions in what follows below.

4.1 Determination of equilibrium eccentricities corresponding to normal modes

It is a simple matter to determine equilibrium eccentricities for protoplanets moving under the gravitational potential appropriate to a normal mode corresponding to an eccentric disc, with pattern speed $\Omega_{\rm p},$ of the type calculated above. We recall the normal mode Eq. (18) which can be regarded as determining the equilibrium disc eccentricity in the form

$$2\left(\Omega_{\rm p} - \omega_{\rm p} \right) \Omega r^3 e(r... ...ight) - {{\rm d}\left( r^2\Phi'\right) \over {\rm d} r} \cdot$$ (32)

If the pressure forces are neglected by dropping the terms involving c², this is exactly equivalent to Eq. (31) for determining the protoplanet eccentricity with e(r) corresponding to $e_{\rm p}$ and $\omega_{\rm p}$ replaced by $\omega_{\rm pg}.$ Thus a local protoplanet equilibrium eccentricity appropriate to a disc mode or response can be determined by using the disc normal mode Eq. (18) retaining the term resulting from the disc self-gravity but omitting pressure contributions.

Figure 8: This figure shows the form of the equilibrium eccentricity as a function of radius r in units of $R_{\rm in}$for the modes shown in Fig. 4, the latter being normalized so that e=0.1 for r=1.Curves are associated with normal modes according to increasing number of nodes. Thus the curve with the most nodes is associated with the normal mode with the most nodes. Note the secular resonances for r <5.

Figure 9: As in Fig. 8 but for the one planet model with disc model B. Equilibrium curves are given for the two highest frequency modes only. That associated with the highest frequency decays to small values at about r=5.The other curve shows a strong secular resonance at $r \sim 20.$

Figure 10: As in Fig. 8 but for the disc model A with no protoplanets.

Figure 11: As in Fig. 10 but for the disc model B with no protoplanets. In this case the longest wavelength equilibrium eccentricity indicates antialignment with the disc.

This procedure should be applicable provided the eccentricities are not too large. Equilibrium eccentricities are plotted for some of the normal modes we calculated in Figs. 2 and 3 and also 8-11. In Figs. 2 and 3 we also plot equilibrium eccentricities but calculated with the nonaxisymmetric contribution to the gravitational potential due to the disc removed which is equivalent to assuming that it remains circular. We see that the assumption that disc remains circular results in a significantly smaller equilibrium eccentricity for disc model A, particularly for the mode with second highest frequency. For the lower mass disc model B, differences arising fom assuming the disc remains circular are less pronounced. We also coment that the occurence of secular resonances in the inner parts of the disc where $r/R_{\rm in} < 10$is common.

One expects that dissipative torques produced by protoplanet-disc tidal interaction may result in the approach to such equilibrium solutions from general initial conditions. This we now discuss.