3. Mass Transfer

The mass transfer can take place by capture of the companion's stellar wind (see Figure 2.3) or by Roche lobe overflow as shown in Figure 2.3. These two mechanisms represent the extreme cases. However the difference between these two cases is not distinct for companions that come close to filling their Roche Lobes. Because of the presence of the accreting object, the stellar wind will be enhanced along the line of centers, forming a tidal stream that flows through the first Lagrange (L₁) point (Blondin et al. (1991); Petterson (1978)). Such a system may still be said to be wind-fed but most of the accreted matter flows through the L₁ point.

Figure 4: Mass-transfer in an X-ray binary by wind accretion.

Figure 5: Mass-transfer in an X-ray binary by Roche lobe overflow.

Whether or not an accretion disk forms depends on the mass transfer mechanism. For Roche lobe overflow an accretion disk is required to lower the specific angular momentum of the transferred matter before it can accrete onto the compact object. In the case of wind accretion an accretion disk may not be required depending on the orbital angular speed of the accreting star.

Following Frank et al. (1985) we can verify that an accretion disk should form in the case of Roche lobe overflow. The specific angular momentum of the material spilling through the L₁ point is essentially b²$\omega$, where b is the distance from the accreting object to the L₁ point and $\omega$ is the orbital angular speed. In order to accrete onto the compact object this matter must shed angular momentum. The stream can lose energy through shocks but it is difficult for it to lose angular momentum. Thus the accreting matter will initially settle into the lowest energy orbit consistent with its specific angular momentum. Assuming the matter stream has lost no angular momentum since passing through the L₁ point this will be a circular orbit with radius R_circ = b²$\omega$/v_$\phi$ where

$$\phi \sqrt{G M_{\rm x} M_{\odot} \over R_{\rm circ}}$$ (6)

is the tangential velocity. Viscous interactions will cause the plasma to spread out and form an accretion disk. The circularization radius R_circ may be expressed in terms of the orbital separation a and the ratio of the masses of the accreting star and the companion q $\equiv$ M_x/M_c as

$$\left(\vphantom{{b\over a}}\right. {b\over a} \left.\vphantom{{b\over a}}\right)^{4}_{}$$ R_circ = (7)

$$\approx$$ a(1 + q)(0.500 - 0.227log₁₀q)⁴

$$\approx \odot$$

where P_day is the orbital period in days. Here the approximate formula

$$\left[\vphantom{ 0.500 - 0.227 \log_{10} q}\right. \left.\vphantom{ 0.500 - 0.227 \log_{10} q}\right]$$ (8)

has been used (Frank et al. (1985)). If the accreting star is a white dwarf, a neutron star, or a black hole, R_circ will be larger than the stellar radius, allowing an accretion disk to form. R_circ will also be smaller than the primary's Roche lobe radius which is

$${0.49\over 0.6 + q^{2/3}\ln(1+q^{-1/3})}$$ (9)

using the approximation of Eggelton (1983).