3. Tidal Truncation of Viscous Disks around Be Stars

From the direct comparison of the viscous torque and the tidal torque applied to the gas at a given resonance radius, we have the criterion for marginal tidal truncation as

$$3 \pi \alpha \left( {H \over r} \right) ^2 = {T_{\rm tidal} \over {\Sigma \Omega^2 r^4}}$$ (1)

(Artymowicz & Lubow 1994), where H is the scale-height of the disk, $T_{\rm tidal}$ is the tidal torque by the compact star, $\Sigma$ is the surface density of the disk, and $\Omega$ is the angular frequency of disk rotation.

We evaluate this criterion for two Be/X-ray binary systems, A0535+262 and 4U0115+634. The A0535+262 system consists of a neutron star orbiting the O9.7IIIe star HD245770 (=V725 Tau). The orbit is wide ( $P_{\rm orb}=110.3\,{\rm d}$) and eccentric (e = 0.47). The 4U0115+634 system consists of a neutron star orbiting the B0.2Ve star V635 Cas in a relatively close ( $P_{\rm orb}=24.3\,{\rm d}$) and eccentric (e=0.34) orbit. These systems are chosen because the system parameters are observationally well determined. Table 1 lists some resonance radii and the critical values of viscosity parameter, $\alpha_{\rm crit}$, for which the criterion (1) for marginal tidal truncation is met. The disk is truncated at a given resonance if $\alpha < \alpha_{\rm crit}$.

 

Table 1: Criterion for marginal tidal truncation.

resonance	A0535+262		4U0115+634
	r/a	$\alpha_{\bf crit}$	r/a	$\alpha_{\bf crit}$
3:1	0.47	$4.1\cdot 10^{-1}$	0.47	$7.4\cdot 10^{-1}$
4:1	0.39	$8.0\cdot 10^{-2}$	0.39	$1.6\cdot 10^{-1}$
5:1	0.34	$2.8\cdot 10^{-2}$	0.33	$3.3\cdot 10^{-2}$

In principle, we can constrain the value of $\alpha$ in decretion disks around Be stars in Be/X-ray binaries by evaluating the criterion (1) at the observed disk sizes. Since no direct measurement of disk sizes has been done for Be/X-ray binaries, we take the size of the H$\alpha$-emitting region, which can be estimated from the separation of double peaks of the line profile, as an approximate value to the disk size.

Clark et al. (1998) studied the long-term variability of A0535+262 and found that the H$\alpha$ line profiles taken from 1994 September to 1996 Feburuary were relatively stable. The peak separation of the H$\alpha$ line during this period varied around 5.2Å. Taking the stellar parameters from Vacca et al. (1996) and the inclination angle of 46$^\circ$ (Negueruela, private communication), we have the disk size $r_{\rm disk}/a \sim 0.57$ for A0535+262, which can be marginally consistent with being the 3:1 resonance radius.

For 4U0115+634, Negueruela et al. (1997) reported that the peak separation of the H$\alpha$ line was about 440kms^-1 and did not change noticeably in 1992 and 1993. Using the stellar parameters and the inclination angle of Negueruela et al. (1998b), we obtain $r_{\rm disk}/a \sim 0.35$ for 4U0115+634. The Be-star disk of 4U0115+634 can be truncated at $\sim$4:1 resonance radius. Note that $\alpha \sim 0.1$ is compatible with the observed disk sizes for these systems.