3 Results and discussion

The equations of line transfer given in the Eqs. (17) and (18) are solved following the procedure described in Peraiah et al. (1987). The optical depths along these segments $P\tau$ $P\tau^\prime$ are calculated using the Eq. (1). We set $\sigma$ as the electron scattering coefficient equal to (Thomson cross section) to $6.6525 \ 10^{-25}$ cm². The lengths of the segments change between 0 and 2r where r is the radius of the component. We have set an electron density of 10¹⁴ cm^-3. The maximum optical depth is 97.5 while optical depth of the segment along the x-axis $OO^\prime$ is 66.525 where the radius of the star is taken to be 10¹² cm and the thickness of the atmosphere as 10¹² cm. The parameters that are used in the calculations are listed below.

B/A = ratio of the outer to the inner radii of the atmosphere of the primary component and whose reflection effect is being studied(=2). n = number of shells into which the atmosphere of the component is divided,

r₁/R = ratio of the radius of the component to that of the line joining the centres of gravity of the two components $r_{1}=2 \ 10^{12}$ cm.

V_A = initial velocity of expansion in units of mtu at n = 1 (see Eqs. (29) and (30)).

V_B = final velocity in units of mtu at n = 100 (see Eq. (31)).

S = total source function (see Eq. (25)),

$S_{\rm s}$ = source function due to self radiation (see Eq. (24)),

I = ratio of incident radiation to that of self radiation of the star (see Eq. (28)),

$\epsilon$ = probability per scatter that a photon is thermalised by collisional de-excitation (see Eq. (23)),

$\beta$ = ratio of absorption coefficient in continuum to that in the line,

T = total optical depth,

Q = $x/x_{\rm max}$ (see Eq. (32)),

${F_{Q}}/{F_{\rm c}}$ = ratio of the line flux at the normalized frequency Q to that in the continuum or at $x_{\rm max}$ (see Eqs. (33) and (34)),

(R; N.R) = with reflection and with no reflection,

${H_{\rm e}}/{H_{\rm a}}$ = height of the emission to the depth of absorption in the line,

$T_{\rm e}$ = temperature in the atmosphere.

Few results are presented in Figs. 3, 4, and 5 for different parameters. The figures are self explanatory as far as the parameters are concerned. The atmosphere of the primary whose centre is at O is divided into 100 shells (see Fig. 1). The separation of the components as r₁/R where r₁ is the radius of the primary and $R (=OO^\prime$) is the separation of the centres of gravity of the components. We have considered two cases of separation r₁/R= 1/ 2 and 1/5 and the atmospheric extension is set to equal to stellar radius or B/A=2. The total radial optical depth T is taken to be 10³, 10⁴ and 10⁵. The velocities of expansion are measured in terms of mean thermal units and uniform expansion velocity law is assumed. If V_A and V_B are the velocities at A and B respectively, then the velocity at any shell boundary $V_n = \left[V_{A}+ \frac {V_B - V_A} {N}\right]$. At $A(\tau=\tau_{\rm max}=T)$the velocity is V_A and at $B (\tau=0)$ the velocity is V_B. The parameters $\epsilon$ which is defined in Eq. (23) is the probability that a photon is destroyed by collisional de-excitation and it is < 1 for non-LTE line formation and this is set to equal to 0 and 10^-4. The quantity $\beta$ which is the ratio of absorption in the continuum to that in the line centre is set to 0 and 10^-4 in our calculations.

  

Figure 3: a) The source functions S and $S_{\rm s}$ are shown with respect to the shell numbers in a scattering medium for velocity VA=0;VB=0. b) Line profiles with reflection and without reflection from the secondary components for a static and scattering medium with total optical depth T=104 and VA=0;VB=0. c) The source functions S and $S_{\rm s}$ are shown with respect to the shell numbers in a scattering medium for velocity VA=0;VB=50. d) Line profiles with reflection and without reflection from the secondary components for a static and scattering medium with total optical depth T=104 and VA=0;VB=50. e) Equivalent widths of the lines are plotted against the expansion velocity VB. f) The ratios of the height 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 of expansion VB

Figure 3a gives the source functions $S_{\rm s}$ and S given in Eqs. (24) and (25) for various parameters shown in the figure, across the atmosphere from n=1 to n=100. These results represent a static and scattering medium with V_A=V_B=0 and $\epsilon = \beta = 0$ and the incidence radiation factor I=1 (see Eq. (28)). The incident radiation at A is given according to Eq. (27). The source function $S_{\rm s}$ (which does not contain the reflected radiation) 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 functions for r₁/R = 1/ 2 and 1/5 are considerably enhanced as these source functions include the incident radiation from the companion along the axis $OO^\prime$. Figure 3b gives the line profile in the direction of the line of sight, corresponding to the source functions given in Fig. 3a. The line fluxes are plotted against the normalized frequency points Q (see Eq. (32)). As the medium is static, the profile are symmetric with central absorption. More photons are removed from the centre when the line central optical depth is 10⁴, and the centre becomes almost black. When the incident radiation from the component is added, there is more emission in the central portion of the line. The shapes of the lines in all these cases remain symmetric about the centre of the line. Figure 3c gives the variation of the same quantities as those given in Fig. 3a except that velocity V_B = 50 mean thermal units. There is a marked difference in the variation of the source functions in the two cases when V_B=0 in Fig. 3a and V_B=50 mtu in Fig. 3c. There is a sudden fall in the source functions near $\tau = T$ and these remain almost constant throughout the rest of the atmosphere. Figure 3d presents the line profiles along the line of sight corresponding to the source functions presented in Fig. 3c. These are similar to P-Cygni type profiles formed in an expanding media with blue shifted absorption. However the emission although small, confines more or less to the centre of the line formed in static medium. The reason for this is that the absorption core is formed in the portion of the atmosphere which is directly in between the star and the observer. As it is moving towards the observer there will be a Doppler shift of the frequencies of the line photons towards the blue side of the centre of the line. The photons that are emitted in the side lobes of the atmosphere are merely scattered and the Doppler effect due to the velocities in the farther part and nearer part (with respect to the observer's point) will nearly counter each other, maintaining an approximate symmetric emission about the centre. Therefore the asymmetry caused by the Doppler shifts is minimal in the emission part of the line. Figure 3e gives the variation of equivalent widths against the expanding velocities V_B for the parameters shown in Fig. 3. We can see that when no radiation is incident from the companion the equivalent widths are much larger than when there is incident light falling on the component from the companion. This can be understood from the fact that more photons are emitted through the line when external radiation is falling on the atmosphere from out side, which is also clear from a comparison of profiles given in Figs. 3b and 3d. However in both the cases of reflection and no reflection the equivalent widths increase with the increasing velocities of expansion. Figure 3f 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 in the reflection and non reflection cases. 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 velocities increase.

  

Figure 4: a) Same as those given in Fig. 3a but with $\epsilon = 1 \ 10^{-4}$ b) Same as those given in Fig. 3b but with $\epsilon = 1 \ 10^{-4}$ c) Same as those given in Fig. 3c but with $\epsilon = 1 \ 10^{-4}$ d) Same as those given in Fig. 3d but with $\epsilon = 1 \ 10^{-4}$ e) Same as those given in Fig. 3e but with $\epsilon = 1 \ 10^{-4}$

  

Figure 5: a) Same as those given in Fig. 3a but with $\epsilon = 1 \ 10^{-4}$ b) Same as those given in Fig. 3b but with $\epsilon = 1 \ 10^{-4}$ c) Same as those given in Fig. 3c but with $\epsilon = 1 \ 10^{-4}$ d) Same as those given in Fig. 3d but with $\epsilon = 1 \ 10^{-4}$ e) Same as those given in Fig. 3e but with $\epsilon = 1 \ 10^{-4}$

Figures 4a to 4e gives the results for the set $\epsilon=10^{-4}$ and $\beta={0}$. The Planckian B(T(r)) is set to equal to 1 uniformly throughout the medium. The source functions described in Fig. 4a represent the internal emission of photons. Although we have given a uniform emission B(T(r))=1 throughout more of the radiation is scattered towards the boundary $\tau = 0$ ($r=r_{\rm max}=B)$ this is the effect of sphericity or curvature scattering. The incident radiation increases the source function only marginally. Corresponding line profiles (along the line of sight) are given in Fig. 4b. these profiles show emission with absorption at line centre and these are symmetric with respect to the centre of the line as the medium treated here is static. Figure 4c gives the source functions of a medium which is expanding with V_B=50 mtu. These are different from those given in Fig. 4a for a static medium. The maximum is spread over a larger spatial extent which is the effect of scattering in an expanding gas. The corresponding line profiles are given in Fig. 4d. The emission is spread through the line except for a small absorption around the central position of the line. The equivalent widths of those lines are plotted in Fig. 4e with respect to expansion velocities $V_{B}^\prime$s. It is interesting to note that when there is no reflection, the equivalent widths of the emission lines increase with velocities of expansion while the incident radiation from outside reduces the emission line equivalent widths considerably.

In Figs. 5a to 5e, the results of the cases $\epsilon=\beta=10^{-4}$are plotted. These results show similar characteristics those given in Figs. 4a to 4e.

We have performed several calculations for different parameters to study the effects of irradiation on line formation in the expanding atmosphere of the component of a close binary system. We have studied the variation of source functions, with different velocity gradients and also various values of irradiation from the secondary component. The line profiles computed with reflection are compared with those computed without reflection and for several cases of the proximity of the two components. We obtained P-Cygni type profiles. These results are described in six sets. The figures are self explanatory as far as the parameters that are used, are concerned.