2. Orbital Properties

The effective gravitational potential in a corotating frame is the Roche potential. This has the form

$$\Phi_{R}^{} \bf r {GM_{\rm c}\over\left\vert{\bf r} - {\bf r}_{\rm c}\right\vert} {GM_{\rm x}\over\left\vert{\bf r} - {\bf r}_{\rm x}\right\vert} {1\over 2} \left(\vphantom{{\bf\omega}\times{\bf r}}\right. \bf\omega \bf r \left.\vphantom{{\bf\omega}\times{\bf r}}\right)^{2}_{}$$ (4)

where $\bf r_{\rm c}^{}$, $\bf r_{\rm x}^{}$, M_c, and M_x are the position vectors and masses of the companion and accreting stars, respectively (Frank et al. (1985)). The orbital angular velocity of the binary is $\omega$. This form for the potential assumes that the companion is rotating synchronously. If the companion's rotation is not synchronized with $\omega$ the effective potential will be the generalized Roche potential (Avni and Bahcall (1975) and references therein):

$$\Phi \left[\vphantom{{1\over\left\vert{\bf r} - {\bf r}_{\rm c}\right\... ...{\bf r}_{\rm x}\right\vert} -{x\over D} +{1+q\over2}\Omega^2(x^2+y^2)}\right. {1\over\left\vert{\bf r} - {\bf r}_{\rm c}\right\vert} {q\over\left\vert{\bf r} - {\bf r}_{\rm x}\right\vert} {x\over D} {1+q\over2} \Omega^{2}_{} \left.\vphantom{{1\over\left\vert{\bf r} - {\bf r}_{\rm c}\right\... ...{\bf r}_{\rm x}\right\vert} -{x\over D} +{1+q\over2}\Omega^2(x^2+y^2)}\right]$$ (5)

where q = M_x/M_c, D is the separation of the stars, and $\Omega$ = $\omega_{c}^{}$/$\omega$ is the ratio of the angular speed of the companion's rotation to the orbital angular speed. The x and y-axes are in the orbital plane as shown in Figure 2.2. In either case the qualitative features of the potential are the same. There is a critical surface consisting of two Roche lobes, one surrounding each star and connected by a saddle point (the first Lagrange or L₁ point). Matter may pass most easily from the companion to the primary through the L₁ point. Figure 2.2 shows the critical Roche surface of an X-ray binary similar to Cen X-3.

Figure 3: Roche Lobe geometry of an X-ray binary. The orbital separation D = a_x + a_c and the mass ratio q $\equiv$ M_x/M_c have been chosen to be similar to those of Cen X-3. The high-mass companion is close to filling its Roche lobe and is tidally distorted. The X- and Y-axes lie in the orbital plane and the accreting neutron star is at the origin.