Tidal truncation of the viscous disk

Artymowicz & Lubow (1994) investigated the tidal/resonant truncation of the accretion disks in eccentric binary systems and found that the disk size becomes smaller in a system with larger eccentricity. Following their formulation of the companion-disk interaction, we study below the tidal truncation of the Be-star disk in 4U0115+634.

In order to evaluate the tidal/resonant torque on the Be-star disk exerted by the neutron star, we consider the binary potential in a coordinate system $(r, \theta, z)$ in which the origin is attached to the Be star primary:

$$\Phi (r, \theta, z) = -{{GM_1} \over r} -{{GM_2} \over{[r^2+r_2^2-2rr_2 \cos(\theta-f)]^{1/2}}}$$

$$+{{GM_2r} \over r_2^2} \cos(\theta-f),$$ (1)

where $M_1$ and $M_2$ are masses of the Be and neutron stars, respectively, $r_2$ is the distance of the neutron star from the primary, and $f$ is the true anomaly of the neutron star. The third term in the right hand side of Eq.(1) is the indirect potential arised because the coordinate origin is at the primary.

We expand the potential by a double series as

$$\Phi (r, \theta, z) = \sum_{m,l} \phi_{ml} \exp[i(m \theta - l \Omega_B t)],$$ (2)

where $m$ and $l$ are the azimuthal and time-harmonic numbers, respectively, and $\Omega_B = [G(M_1+M_2)/a^3]^{1/2}$ is the mean motion of the binary with semimajor axis $a$. The pattern speed of each potential component is given by $\Omega_p=(l/m)\Omega_B$.

Inverting Eq.(2) and denoting the angle $\theta-f$ as $\varphi$, we have

$$\phi_{ml} = -{{GM_2} \over a} \left\{{2 \over \pi^2} \int_0^\pi d(\Omega_B t) {a \over r_2} \cos(mf - l\Omega_B t) \right.$$

$$\times \int_0^\pi d\varphi {{\cos m\varphi} \over (1+\beta^2-2\beta \cos \varphi)^{1/2}}$$

$$\left. -{{\delta_{m1}} \over \pi} \int_0^\pi d(\Omega_B t) {a \over r_2} \beta \cos(mf - l\Omega_B t) \right\},$$ (3)

where $\beta = r/r_2$ and $\delta_{m1}$ is the Kronecker delta function.

For each potential component $\phi_{ml}$, there can be outer and inner Lindblad resonances at radii where $\Omega_p=\Omega \pm \kappa/m$ and a corotation resonance at the radius where $\Omega_p=\Omega$. Here, $\kappa$ is the epicyclic frequency, and here and hereafter the upper and lower signs correspond to the outer Lindblad resonance (OLR) and inner Lindblad resonance (ILR), respectively. The radii of these resonances are given by

$$r_{\rm LR} = \left( {{m \pm 1} \over l} \right)^{2/3} (1+q)^{-1/3} a$$ (4)

and

$$r_{\rm CR} = \left( {m \over l} \right)^{2/3} (1+q)^{-1/3} a,$$ (5)

where $q=M_2/M_1$.

For near Keplerian disks, where $\Omega \sim \kappa \sim (GM_1/r^3)^{1/2}$, the Goldreich & Tremaine (1979, 1980)'s standard formula for torques $T_{ml}$ at the outer Lindblad resonance (OLR) and inner Lindblad resonance (ILR) is reduced to

$$T_{ml} = \pm {{m(m \pm 1)\pi^2 \sigma (\lambda \mp 2m)^2 \phi_{ml}^2} \over {3l^2 \Omega_B^2}},$$ (6)

where $\sigma$ is the surface density of the disk at the resonance radius and $\lambda = (d \ln \phi_{ml}/d \ln r)_{\rm LR}$. Similarly, torques at the corotation resonance (CR) is written as

$$T_{ml} = {{2m^3 \pi^2 \sigma \phi_{ml}^2} \over {3l^2 \Omega_B^2}}.$$ (7)

Note that angular mommentum is removed from the disk at the ILRs, whereas it is added to the disk at the OLRs and CRs.

In near Keplerian disks, the viscous torque formula derived by Lin & Papaloizou (1986) is written as

$$T_{\rm vis} = 3 \pi \alpha GM_1 \sigma r \left( {H \over r} \right)^2$$ (8)

[see also Artymowicz & Lubow (1994)], where $\alpha$ is the Shakura-Sunyaev viscosity parameter and $H$ is the vertical scale-height of the disk given by

$${H \over r} = {c_{\rm s} \over {V_{\rm K}(R_*)}} \left({r \over R_*} \right)^{1/2}$$ (9)

for the isothermal disk. Here, $c_{\rm s}$ is the sound speed and $V_{\rm K}(R_*)$ is the Keplerian velocity at the stellar surface. In the systems we will discuss later, $c_{\rm s}/V_{\rm K}(R_*)$ ranges $3.1-4.1 \cdot 10^{-2}(T_{\rm d}/T_{\rm eff})^{1/2}$, where $T_{\rm d}$ and $T_{\rm eff}$ are the disk and stellar temperatures, respectively. In what follows, we adopt $T_{\rm d}=0.8\,T_{\rm eff}$.

The disk is truncated if the viscous torque is smaller than the resonant torque. The criterion for the disk truncation at a given resonance radius is, in general, written as

$$T_{\rm vis} + \sum_{ml}(T_{ml})_{\rm ILR} +\sum_{ml}(T_{ml})_{\rm OLR}+\sum_{ml}(T_{ml})_{\rm CR} \le 0,$$ (10)

where the summation is taken over all combination of $(m,l)$ which gives the same resonance radius. Actually, criterion (10) is determined only by the viscous torque and the torques from the ILRs of several lowest-order potential components, because the torques from the ILRs dominate those from the OLR and CR in circumstellar disks and high-order potential components little contribute to the total torque, even if the eccentricity of the orbit $e$ is not small (Goldreich & Tremaine 1980; Artymowicz & Lubow 1994).

Note that the criterian (10) at a given resonance is met for $\alpha$ smaller than a critical value $\alpha_{\rm crit}$.