5 Results of calculations

The six panels of Fig. 5 show the bird-eye view of density isosurfaces in the vicinity of the Be star at the level $\rho =5\times 10^{-5}\cdot \rho (L_1^{\rm rot})$. These isosurfaces are shown for six moments of time: $t=0.514P_{\rm orb}$, $0.545P_{\rm orb}$, $0.586P_{\rm orb}$, $0.608P_{\rm orb}$, $0.639P_{\rm orb}$, and $0.660P_{\rm orb}$, respectively. Projections of the envelope into XY plane (density distribution and velocity vectors in equatorial plane) and XZ plane (density distribution in the frontal plane) are also depicted in those six panels. An analysis of the results shows that the outflow of matter from the Be star in the vicinity of $L_1^{\rm rot}$ point results in the formation of an envelope with a fast retrograde apsidal motion. The mean angular velocity of apsidal motion can be derived from data of Fig. 6 where the time evolution of two angles, the angle between X axis and the direction to the centre of mass of the envelope (dashed line), and the angle between X axis and the direction to the centre of mass of the outer part of the envelope (solid line; values of density $\rho \in [10^{-3}, 10^{-2}]$ are shown). It follows from the data of Fig. 6 that $\Omega_{\rm aps}=-6\Omega$, and $P_{\rm aps}=\hbox{$~^{1}\!/_{6}$ }P_{\rm orb}$, respectively. Moreover, it is clearly seen that the velocity of the apsidal motion is variable; the motion gets slower in the interval $\alpha\in[0.4\pi,0.6\pi]$. This region corresponds to the zone of interaction between the envelope and the outflowing stream of matter from $L_1^{\rm rot}$. It seems that this interaction causes a standstill of the apsidal motion. The analysis of Fig. 6 also shows that the centre of mass of the whole envelope oscillates within the interval $\alpha\in[0.35\pi,0.65\pi]$ while the centre of mass of the outer layers makes a full revolution within the interval $\alpha\in[-\pi,\pi]$. This finding seems to indicate the presence of a strong differential rotation of the envelope. It is interesting to note that the elongated shape of the envelope implies varying velocity of rotation, the velocity being larger than the Keplerian one in one part of the orbit and lower than Keplerian in the rest of the orbit. Figure 7 illustrates this phenomenon and depicts the ratio of angular velocity to the Keplerian one along the isolines $\rho =9\times 10^{-5}$ and $\rho =5\times 10^{-3}$ for the time $t=0.514P_{\rm orb}$ (see also Fig. 5a). The length of each bar characterizes the deviation of angular velocity from the Keplerian one $\nu =\vert V-V_{\rm Kep}\vert/V_{\rm Kep}$ (the bar in upper right corner corresponds to $\nu =1$) and its direction - inward or outward - denotes the sub-Keplerian or super-Keplerian flow regime, respectively.

Figure 6: Time variation of the angle between X axis and the direction to the centre of mass of the envelope (dashed line), and the angle between X axis and the direction to the centre of mass of the outer part of the envelope (with values of density $\rho \in [10^{-3}, 10^{-2}]$ - solid line). Six asterisks correspond to six moments of time presented in Fig. 5.

Figure 7: The ratio of angular velocity to Keplerian one along the isolines $\rho =9\times 10^{-5}$ and $\rho =5\times 10^{-3}$ for the time $t=0.514P_{\rm orb}$. The length of each bar characterizes how much the angular velocity deviates from the Keplerian one $\nu =\vert V-V_{\rm Kep}\vert/V_{\rm Kep}$ (the bar in the upper right corner corresponds to $\nu =1$) and its direction - inward or outward - characterizes the sub-Keplerian or super-Keplerian flow regime, respectively. The dashed circle represents the surface of star 1. Coordinates x and y are expressed in $R_\odot$.

The drawings of the envelope in Fig. 5 are given for six moments of time that cover the entire $P_{\rm aps}$. Our analysis provides, therefore, a complete picture of the flow and gives an estimate of the main parameters of the envelope as well. It is seen that for some time ( $t=0.514P_{\rm orb}$, $t=0.545P_{\rm orb}$, and $t=0.586P_{\rm orb}$ (which corresponds to $\alpha_{\rm out}\in[\pi,0.3\pi]$ for the vector pointed to the centre of mass of the outer layers of the envelope), the envelope has a torus-like shape, the thickness of the envelope being $h\sim R_1$, i.e. exceeding the polar radius which equals $\sim$0.65R₁. In the rest of the time ( $t=0.608P_{\rm orb}$, $t=0.639P_{\rm orb}$, and $t=0.660P_{\rm orb}$ (corresponding to $\alpha_{\rm out}\in[0.3\pi,-\pi]$), one can see the elongation of the envelope. Its shape becomes nearly disk-like with a characteristic thickness $h\sim\hbox{$~^{1}\!/_{2}$ }R_1$. It is obvious that the changes in the envelope shape result from both the presence of the binary companion and the interaction of the gas in the envelope (during its apsidal motion) with the stream of gas leaving $L_1^{\rm rot}$. The characteristic linear sizes of the envelope in the equatorial plane are the following: for times when it has a torus-like shape it is $\sim$3R₁ (on the level $\rho=5\times10^{-5}\times\rho(L_1^{\rm rot})$), while for the time when it has disk-like shape, its size increases to $\sim$4.5R₁.

A test calculation for a binary composed from the same two stars but with an orbital period of 15 days also led to rapid retrograde revolution of the outflowing disk, though with a less regular pattern.