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Subsections

   
2 The physical model

We solve the time dependent evolution equations for a system composed of a primary star, small mass companion, and viscous disc. The equations of motion and numerical procedures adopted are described in NPMK. For the sake of brevity we refer the reader to that paper for details.

We work with flat 2-dimensional disc models. In a cylindrical coordinate system $(r, \varphi, z)$ 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 $0.4 r_{\rm L},$ $r_{\rm L}$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 $c_{\rm s}=(H/r) v_{\rm K}$ to be such that the disc aspect ratio H/r=0.05 throughout the disc, $v_{\rm K}=r \Omega_{\rm K}$ being the Keplerian velocity. Thus the disc Mach number is 20 everywhere. We employ a constant value of the kinematic viscosity $\nu=10^{-5}$ 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 ($M_{\rm p}$). The unit of length is taken to be the initial orbital radius of the companion, $r_{\rm p}$. The gravitational constant G=1, so that the natural unit of time becomes

\begin{displaymath}P_0/(2\pi) = \omega^{-1} = \sqrt{\frac{r_{\rm p}^3}{G(M_*+M_{\rm p})}},
\end{displaymath}

with P0 being the initial orbital period of the binary, which in all cases was initiated on a circular orbit.

   
2.1 Initial conditions

The disc models used in all simulations had uniform surface density, $\Sigma,$ with an imposed taper near the disc edge, initially. The value of $\Sigma$was chosen such that when $M_* = 1~M_{\odot},$ there exists the equivalent of 2 Jupiter masses $(M_{\rm J})$ in the disc interior to the initial orbital radius of the companion. Then $\Sigma=6 \; 10^{-4}$ in our dimensionless units.

Different mass ratios between the companion and the central star $q=M_{\rm p}/M_*$, were considered, such that $10^{-3} \le q \le 0.03$. For $M_* = 1~M_{\odot},$ the lower end of this range corresponds to a Jupiter mass protoplanet and the upper end to a Brown Dwarf of mass $30~M_{\rm J}.$The inner radius of the disc was located at r=0.4 and the outer radius at r=6.

2.2 Boundary conditions

Since material in a viscous accretion disc will naturally flow onto the central star, an outflow condition was applied at the inner boundary. The outer boundary condition was the same as that described in NPMK.


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