5 Result and discussion

Results are presented in Figs. 1 to 7 for different parameters. The figures are self explanatory regarding the parameters used. The atmosphere of the primary is divided into 100 shells. The separation of the components $\frac {r_1} {R}$ where r₁ is the radius of the primary component and R is the separation of the centers of gravity of the components. We have considered two cases of separation $\frac {r_1}{R}=\frac {1}{2}$ and $\frac {1}{5}$, and the atmosphere extension $\frac {B}{A}=2$. The total radial optical depth T is taken to be 10⁴. The velocities of expansion are measured in terms of mean thermal units and uniform expansion velocity law is assumed. If V_Aand V_B are the velocities at A and B respectively, then the velocity at any shell boundary $V_n=V_A+\bigl[ \frac {V_B-V_A}{N}\times n \bigr]$. At $A(\tau=\tau_{\rm max} =T)$ the velocity is V_A and at $B(\tau=0)$ the velocity is V_B. The parameter $\epsilon$< 1 for non-LTE line formation and this is set to equal to 0. The quantity $\beta$ is set equal to 0 in our calculations. The dust optical depth is taken be $\tau_{\rm d}=0, 1, 2, 5$. The expansion velocities are taken V_B=0, 5, 25, 50. But the results are presented for static medium with V_A=0, V_B=0 and expanding medium with V_A=0, V_B=50. The irradiation from the secondary component is taken to be I=1, 5, 10 times the self radiation of the star. However the results are presented for I=5 only.

Figure 4: Comparison of line profiles for different irradiation (I=1, 5, 10) when $\frac {r_1}{R}=\frac {1}{2}$ and $\frac {1}{5}$ for different dust optical depths

Figure 5: Same as Fig. 4 but VB=50

In Fig. 1a, the source function S are plotted for a static dusty medium, with $\frac {r_1}{R}=\frac {1}{2}$ and I=5 when $\tau_{\rm d}=0$ (dust free medium), the source function falls rapidly in magnitude from the boundary $\tau = T$ towards the boundary $\tau = 0$. It reaches a minimum at around n=20 at which point it starts to raise because of the diffuse radiation incident from the secondary. It reaches a maximum at about n=80. This is due to the diffuse reflection of the incident light from the secondary by the gaseous medium immediately inside the atmosphere at the boundary at $\tau = 0$. When dust is introduced the source functions have been reduced proportionately to the amount of dust present in it but nature of variation of the source function remain the same. The reduction in the source functions in dusty medium is due to the physical effect that dust removes photons from the radiation field and that these are not replaced by emission. In Fig. 1b we introduced expansion with V_B=50 mtu. The source functions fall more rapidly near the boundary $\tau = T$(n=1) but otherwise the variation is similar to those shown in the case of static medium (Fig. 1a). Figures  1c and 1d are the same as those given in Figs. 1a and 1b respectively but with $\frac {r_1}{R}=\frac {1}{2}$ replaced by $\frac {1}{5}$. In both the situations there is no change in variation of the source functions but the values of the source functions are much smaller because of the fact that the distance between the components is increased and therefore the incident radiation is reduced by an approximate factor of $\frac {1}{(R-r_1)^2}$.

Figures 2a,b gives the source function $S_{\rm s}$ and S given in Eq. (8) for various parameters shown in the figure, across the atmosphere for n=1 to n=100. Figure 2a represents a static and scattering medium with V_A=V_B=0and $\epsilon=\beta=0$ and the incidence radiation factor I=5 (see Eq. (10)). The source function $S_{\rm s}$decreases slowly from the point $\tau = T$ to the point $\tau = 0$ in the scattering medium, with T=10⁴. When the reflected radiation is included, the source function for $\frac {r_1}{R}=\frac {1}{2}$ and $\frac {1}{5}$ are considerably enhanced as these source functions include the incident radiation from the companion. Similar trend is observed in Fig. 2b when the expansion velocity is V_B=50.

In Figs. 3a-d the line fluxes are plotted against the normalized frequency points Q (see Eq. (15)) along the line of sight. Each one of the graphs in Figs. 3a-d gives the line profiles for different $\tau_{\rm d}$'s. Figures 3a and 3b represent the static atmosphere but with $\frac {r_1}{R}=\frac {1}{2}$ and $\frac {1}{5}$ respectively. As the fluxes are normalized, there do not seem to be any change in the symmetric profiles. However, the effect of dust is significant in that dust scatters more photons into line core thereby increasing the emission in the core of the line, and more emission in the cores of the lines formed in the former case (i.e.) $\frac {r_1}{R}=\frac {1}{2}$ appears and this is because more light falls on the component in the former case than in the latter case. Figures  3c and 3d give the flux profiles for the same parameters as those in Figs. 3a and 3b respectively except that a velocity of expansion V_B=50 mtu is introduced into the medium. The profiles show asymmetry with red emission and blue absorption and a blue shift of the centre of the line - a P Cygni type profile. The part played by the dust is similar to that shown in Figs. 3a and 3b. It scatters more photons into the line centre and at high dust optical depth, the line may even disappear altogether.

In Figs. 4a-f line profiles without reflection are compared with those formed with the incident radiation from the secondary component i.e., I=1, 5, 10. The line profiles are plotted with normalized frequency points Q, and the ratios of the flux $\frac {F_Q}{F_C}$. The solid curves (scale given on the right side of the figure) are for the case without irradiation while the dashed are the ones formed with irradiation from the secondary. As the medium is static, the profiles are symmetric with central absorption. Figures  4a-f give a comparison of profiles formed in the condition of no irradiation and those with irradiation from the secondary component, and each of the six figures drawn for different $\tau_{\rm d}$'s are shown in the respective figures. Each figure contains profiles for I=0, 1, 5 and 10 where 0 means no incident radiation. It appears that the higher values of I give higher line fluxes and in particular, more emission is seen in cores of the lines.

Figures 5a-f contain the flux profiles for the same parameters as those in Figs. 4a-f but with V_B=50 mtu. The P Cygni type profiles are obtained similar to those shown in Fig. 3.

Figure 6: Equivalent widths are plotted against the expansion velocity with reflection and without reflection

Figures  6a-c give the variation of equivalent widths against the expansion velocity V_B for the parameters shown in the figure. We can see that when there is no radiation incident from the companion, the equivalent widths are much larger as the absorption core is deeper. In the presence of dust $\tau_{\rm d}=1, 2, 5$ the equivalent widths reduce considerably for the reason given above that more photons are scattered into the core, thus reducing absorption and increasing core emission.

Figure 7: The ratios of the heights of the emission to the depth of the absorption in the lines for both the case of reflected radiation and non reflected radiation are shown against the velocity expansion VB

Figure 7 gives the variation of the ratio of height of emission $(H_{\rm e})$ to that of depth absorption $(H_{\rm a})$. There is no change between the reflection case and the non-reflection case. However it is noteworthy that this ratio reaches a maximum at about V_B=4 to 10 mtu and then falls slowly as the expansion velocity increases. Similar kind of trend is observed in the presence of dust. It is different for $\tau_{\rm d}=5$ in that case the ratio increases slowly as the expansion velocity increases. This can be understood again from the argument that more dust scatters more photons into the line core enhancing emission of the core of the line.