2 Basic equations and their linearized form for large scale global modes with $\vec{m} = \mathsf{1}$

We work in a non rotating cylindrical coordinate system $(r,\varphi)$ which may initially be considered to be centred on the primary star. The basic equations of motion in a two dimensional approximation which should be appropriate for a large scale description of the disc are taken to be

$${\partial v_{\rm r} \over\partial t}+ v_{\rm r}{\partial v_{\... ...partial \Pi \over\partial r}- {\partial \Phi \over\partial r},$$ (1)

$${\partial v_{\varphi} \over\partial t}+ v_{\rm r}{\partial v_... ...arphi}- {1\over r} {\partial \Phi \over\partial \varphi }\cdot$$ (2)

In addition we have the two dimensional form of the continuity equation

$${\partial (r\Sigma) \over\partial t}+ {\partial \left(r\Sigma... ...tial \left(\Sigma v_{\varphi} \right)\over\partial \varphi}=0.$$ (3)

Here the velocity is ${\vec v} = ( v_{\rm r} , v_{\varphi}),$ $\Pi=\int^{\infty}_{-\infty}P{\rm d}z$represents a vertically integrated pressure which we assume to be a function of the surface density $\Sigma$ defined by

$$\Sigma =\int^{\infty}_{-\infty} \rho {\rm d}z.$$ (4)

The sound speed is then given by $c = \sqrt{{\rm d}\Pi /{\rm d} \Sigma}.$The gravitational potential $\Phi= -GM_*/r +\Phi_{\rm D} + \Phi_{\rm ext},$with G being the gravitational constant, has a point mass contribution arising from the central mass M_*, a contribution, $\Phi_{\rm D},$ due to the disc and a contribution, $\Phi_{\rm ext},$ due to external protoplanets.

The unperturbed disk is axisymmetric with no radial motion such that the velocity ${\vec v}=(0,r\Omega)$ with $\Omega > 0.$In equilibrium we then have from Eq. (1)

$${\Omega^2 r} = {1\over \Sigma}{\partial \Pi \over\partial r}+ {\partial \Phi \over\partial r}$$ (5)

and the gravitational potential due to the disc is given by

$$\Phi_{\rm D} = -G\int \Sigma K(r,r')r'{\rm d}r',$$ (6)

with

$$K_0(r,r') = \int^{2\pi}_0 {1 \over \sqrt{(r^2+r'^2 -2rr'\cos(\varphi))}} {\rm d}\varphi.$$ (7)

We allow for the effects of orbiting external protoplanets which provide the external potential $\Phi_{\rm ext}.$ In this paper we are primarily interested in phenomena which vary on a time scale long compared to a local orbital period. Accordingly we use the time averaged or secular protoplanet perturbing potential which is derived in Appendix 1. For a single protoplanet of mass $m_{\rm p}$ and orbital radius  $r_{\rm p},$ the contribution to the external potential is given by

$$\Phi_{\rm ext} = -G { m_{\rm p} \over 2\pi } K_0(r,r_{\rm p}).$$ (8)

In a thin disc of the kind considered here, the contributions due to pressure and self-gravity are comparable and small leading to nearly Keplerian rotation which has $\Omega \propto r^{-3/2}.$When $c \propto r^{-1/2},$the disc then has a nearly constant putative aspect ratio $H/r = c/(r\Omega)$, H being the putative semi thickness.

2.1 Linearization

Here we are interested in global m=1 modes with slowly varying pattern when viewed in the adopted reference frame. To study these, we linearize the basic equations about the equilibrium state denoting perturbations to quantities with a prime. As usual we assume that the dependence of all perturbations (in cylindrical coordinates) on $\varphi$ and tis through a factor $\exp i\left( m\left (\varphi -\Omega_{\rm p} t\right) \right).$For the slowly varying modes we consider $\Omega_{\rm p} << \Omega.$ The linearized forms of Eqs. (1)-(3) are then

$$im(\Omega-\Omega_{\rm p}) v'_{\rm r} - 2\Omega v'_{\varphi} = -{\partial W \over\partial r},$$ (9)

$$im(\Omega-\Omega_{\rm p}) v'_{\varphi} + {\kappa^2\over 2\Omega} v'_{\rm r} = -{im W\over r},$$ (10)

and

$${im (\Omega-\Omega_{\rm p})\Sigma ( W-\phi') \over c^2}=- {1\... ...right)\over \partial r} -{im \Sigma v'_{\varphi} \over r}\cdot$$ (11)

Here $W = \Pi'/\Sigma +\Phi' = \Sigma'c^2/\Sigma+\Phi'$, and $\kappa^2 =(2\Omega/r)({\rm d}(r^2\Omega)/{\rm d}r)$ denotes the square of the epicyclic frequency.

For the gravitational potential perturbation we have $\Phi' =\Phi_{\rm D}'+\Phi'_{\rm ext}.$Here we shall allow for the contribution of the secular effects of other sources such as protoplanets to the modes through the gravitational potential perturbation $\Phi'_{\rm ext}.$

2.2 Slowly varying modes with m = 1

We consider linear perturbations of an axisymmetric disc that correspond to normal modes with m=1that make it eccentric. The modes we consider are such that $\Omega_{\rm p} << \Omega$ and the dominant motion is epicyclic. In this case, to lowest order, we may neglect the pressure and self-gravity term Was well as $\Omega_{\rm p}$ and assume $\Omega$is keplerian, to obtain from Eq. (10)

$$i v'_{\varphi} =- {1\over 2} v'_{\rm r}.$$ (12)

Using the above and introducing the radial Lagrangian displacement $\xi _{\rm r} = -i v'_{\rm r}/(\Omega - \Omega_{\rm p}),$ Eq. (11) gives the surface density perturbation in the low frequency limit in the form

$$\Sigma'= -r{{\rm d} ( \Sigma e(r) )\over {\rm d}r},$$ (13)

with $e(r) \equiv \xi_{\rm r}/r$ being the disc eccentricity.

The gravitational potential perturbation induced by the disc is

$$\Phi'_{\rm D} = -G\int \Sigma' K_1(r,r')r'{\rm d}r',$$ (14)

with

$$K_1(r,r') = \int^{2\pi}_0 {\cos(\varphi) \over \sqrt{(r^2+r'^2 -2rr'\cos(\varphi))}} {\rm d}\varphi -{\pi r\over r'^2},$$ (15)

where the second term corresponds to the indirect term which results from the acceleration of the coordinate system produced by the disc material.

The existence of a mode with m=1 in the disc causes the orbits of the protoplanets interior to the disc to become eccentric. Noting the different form of indirect term used, the secular perturbing potential derived in Appendix 1 then gives the contribution of an orbiting protoplanet to the external potential perturbation to be

$$\Phi'_{\rm p}(r) = - {G m_{\rm p} \xi_{\rm r}(r_{\rm p}) \ove... ...partial r_{\rm p}}\left( K_1(r,r_{\rm p}) r_{\rm p}^2 \right).$$ (16)

Here the eccentricity of the companion orbit is related to the displacement by $e_{\rm p} = \xi_{\rm r}(r_{\rm p})/r_{\rm p}.$The total external potential perturbation is then found by summing over the perturbing protoplanets:

$$\Phi'_{\rm ext}(r) = \sum_{p} \Phi'_{\rm p}(r).$$ (17)

Eliminating $v_{\varphi}'$ from Eqs. (9) and (10), using Eq. (13) for $\Sigma'$ and expanding to first order in the small frequencies $\Omega_{\rm p}$ and $\omega_{\rm p} =\Omega - \kappa,$the fluid element orbital precession frequency in the absence of surface density or gravitational potential perturbations, one obtains a normal mode equation relating $e(r), e_{\rm p}$ and $\Omega_{\rm p}$in the form (see also Papaloizou et al. 2001)

$$2\left(\Omega_{\rm p} - \omega_{\rm p} \right) \Omega r^3 e(r... ...}r}\right) - {{\rm d}\left( r^2\Phi'\right) \over {\rm d} r} ,$$ (18)

where, c, is the local sound speed.

In addition we have an equation for each protoplanet of the form

$$2\left(\Omega_{\rm p} - \omega_{\rm p} \right) \Omega(r_{\rm ... ...\left( r^2\Phi'\right) \over {\rm d} r}\right]_{r=r_{\rm p}} ,$$ (19)

where of course for a particular protoplanet there is no self-interaction term in the sum for $\Phi'_{\rm ext}.$ The inclusion of self-gravity in the eigenvalue problem is essential if modes with prograde precession frequency are to be obtained. For typical protoplanetary disc models, self-gravity can be strong enough to induce prograde precession for the long wavelength m=1 modes considered here.

Normal modes were calculated by discretizing (18) and (19) and formulating a matrix eigenvalue problem on an unequally spaced grid with 200 grid points with intervals increasing in geometric progression (see Terquem & Papaloizou 2000 for consideration of a related problem).

2.3 Disc models

We present here results obtained for a disc model with equation of state given by a polytrope of index n=1.5.The sound speed is given by

$$c^2 ={GM_* H^2\over R_{\rm in}^3} \left(1-\left(R_{\rm in}/r ... ... \left(1-\left( r/R_{\rm out}\right)^{10} \right) R_{\rm in}/r.$$ (20)

Here the first two factors provide sharp edges at the disc inner and outer boundaries and H/r= 0.05 is the disc aspect ratio away from these boundaries.

The surface density is given by

$$\Sigma = \Sigma_0 (c^2)^n .$$ (21)

The boundary regions are chosen to be of order the disc scale height in width and away from these $\Sigma \propto r^{-3/2}.$In the above $R_{\rm in}$ is the inner boundary radius which is taken to be the unit of length. The unit of mass is the central mass M_*and the unit of time is $R_{\rm in}^{3/2}/\sqrt{GM_*}.$The arbitrary scaling factor $\Sigma_0$ was chosen such that in model A the total disc mass was $4.0\times 10^{-2}M_*.$ In model B the total disc mass was $4.0 \times 10^{-3}.$ $R_{\rm out}$ is the outer boundary radius, here, in order to study large scale modes, taken as  $100R_{\rm in}.$

The importance of self gravity is measured by the Toomre parameter $Q=\Omega c/(\pi G \Sigma).$For model A this has a minimum value of 5.2 while for the lower mass disc model of model B this minimum value is 52. Another quantity of interest is the local precession frequency of a test particle orbit $\omega_{\rm pg}$ in the axisymmetric component of the total gravitational potential (see Eq. (29) below).

Figure 1: This figure shows the local test particle orbital precession frequency in dimensionless units as a function of radius, r in units of $R_{\rm in}$in the range 1.1 <r< 100for the two disc models A (lower curve) and B with no interior orbiting protoplanets. Apart from small regions near to the boundaries, the frequency is negative corresponding to retrograde precession.

This is plotted for the two disc models in Fig. 1. It is small and negative in dimensionless units over much of the discs, corresponding to retrograde precession, scaling with the disc mass as it is determined by the disc self-gravity. As we shall see this behaviour provides scope for secular resonance associated with low frequency modes.