Growth of the Eccentric Mode

We study the evolution of the eccentricity of the Be disc by decomposing the surface density distribution into Fourier components which vary as $\exp i(k \phi - \ell \Omega_{\rm B} t)$, where $k$ and $\ell$ are the azimuthal and time-harmonic numbers, respectively, and $\Omega_{\rm B} = [G(M_{*}+M_x)/a^3]^{1/2}$ is the frequency of mean binary rotation with semimajor axis $a$.

As in Lubow (1991), we define the mode strength

$$\begin{eqnarray*} S_{f, g, k, l} &=& \frac{2}{\pi M_{\rm d} (1+\delta_{k, 0}) ... ...nonumber\\ && \times \Sigma(r, \phi, t) f(k \phi) g(\ell t'), \end{eqnarray*}$$

where $f$ and $g$ are either $\sin$ or $\cos$ functions, $M_{\rm d}$ is the disc mass, and $\Sigma$ is the surface density of the disc. For reducing computing time, we compute the surface density by summing up $\delta$ functioins at particle positions. Then, the total strength of the mode $(k, \ell)$ is defined by

$$S_{k, \ell}(t) = (S_{\cos, \cos, k, \ell}^2 + S_{\cos, \si... ..._{\sin, \cos, k, \ell}^2 + S_{\sin, \sin, k, \ell}^2)^{1/2}.$$

Fig. 4 shows the strengths of several modes. The growth rate of the (1,0) mode (i.e., eccentric mode) averag ed over $1 \le t \le 5$ is $5.58\cdot 10^{-2}\Omega_{\rm B}^{-1}$.

Figure 4: Strengths of several modes. The solid, the dashed, the dash-dotted, and the dotted lines denote the strengths of the (1,0) mode, the (2,3) mode, the (1,1) mode, and the (2,2) mode, respectively.