2 Brief description of the method of calculations

We have assumed a spherical shape of the reflecting atmosphere to simplify the computational problems of radiative transfer. The geometry of the model is shown in Fig. 1.

  

Figure 1: Schematic diagram of the binary components with incident radiation from the surface of the secondary. O and $O^\prime$ are the centres of gravity of the two components

Let O and $O^\prime$ be the centres of the primary and the secondary respectively. The atmosphere of the primary is assumed to be spherical and divided into several discrete shells. We calculate the source functions of the radiation field emerging from the companion whose centre is at $O^\prime$ (see Fig. 1) and incident on the atmosphere of the component whose centre is at O. We consider the set of rays such as STP, $E \tau P$, $O^\prime T_{1} P$,$E^\prime \tau^\prime P$, WT₂P etc., emerging from the surface SW of the companion and meeting at a point P in the atmospheres of the component. These rays lie within the quadrilateral such as $PSO^\prime W$ and enter the boundary of the atmosphere of the component at points $T, \ \tau, \ T_{1}, \ \tau^\prime, \ T_{2}$ etc. The surfaces of the companion such as SW will be different for different points P in the atmosphere. The radiation field at P is estimated by calculating the source function whose contribution comes from self radiation of the primary and that due to the incident radiation from the surface SW of the secondary facing the primary. We need to estimate geometrical length of the ray segments such as $P\tau$,$P\tau^\prime$ etc, inside the atmosphere so that the transfer of radiation along these segments is estimated and its contribution to the source function at the point P due to the incident radiation at $\tau, \ \tau^\prime$ etc. The length of the segments such as $P\tau$ in $SPO^\prime$ are given in the Appendix A (see Peraiah 1983 and Peraiah & Rao 1983).

For a given density distribution we need to calculate the optical depth along the segments $P\tau$, $P\tau^\prime$ etc. The source function at points such as P due to the irradiation are calculated using the one-dimensional transfer (Wing 1962; Sobolev 1963; Grant 1968). We will describe this procedure briefly below.

  

Figure 2: Schematic diagram of the rod model

We consider a segment AB (see Fig 2) which has two rays oppositely directed to each other. The optical depth $\tau$ is given by,

$$\tau = \tau (x) = -\int^x_{\rm L} \sigma (x^\prime) {\rm d}x^\prime \ \ ; \ \ \tau (0)=T$$ (1)

where $\sigma (x^\prime)$ is the extinction coefficient and T is the total optical depth. The optical depth is measured in the direction opposite to that of the geometrical segment.

We assume a steady state, monochromatic condition with local source function $B^+(\tau)$ in the direction of increasing $\tau$ and $B^- (\tau)$ in the reverse direction. $U^+(\tau)$ and $U^-(\tau)$ are the specific intensities in the $\tau$ increasing and decreasing directions respectively.

The two equations of transfer for $U^+(\tau)$ and $U^-(\tau)$ are

$$\frac {{\rm d}U^+} {{\rm d}\tau} + U^+ = S_1^+,$$ (2)

$$\frac {{\rm d}U^-} {{\rm d}\tau} + U^- = S_1^-,$$ (3)

where

$$S_1^+ = B^+ (\tau) + \omega (\tau) \bigl [p (\tau) U^+(\tau) + (1 - p (\tau) U^- (\tau)\bigr],$$ (4)

$$S_1^- = B^- (\tau) + \omega(\tau) \bigl [(1 - p (\tau)) U^+ (\tau) + p(\tau) U^- (\tau)\bigr],$$ (5)

are the source functions and $\omega(\tau)$ is the albedo for single scattering and the phase function $p=\frac {1} {2}$. The boundary conditions at $\tau = 0$ and $\tau = T$ are given by

$$U^+ (\tau = 0) = U_1,$$ (6)

$$U^- (\tau = T) = U_2,$$ (7)

where U₁ and U₂ will be specified later. The total source function $S_{\rm d}$at any point $\tau$ is the combination of the scattered part of the local intensities $U^{\pm}(\tau)$ in either directions and the diffuse radiation generated by the incident radiation at the boundaries $\tau = 0$ and $\tau = T$(the quantities U₁ and U₂ in Eqs. (6) and (7) respectively). This added to the local sources would give the total source function given by,

$$\begin{eqnarray} S^+_{\rm d} (\tau){=}S_1^+\! (\tau)\! +\! \omega (\tau)\! \big... ...u} \!\!+\! p (\tau) U_{2} {\rm e}^{-(T - \tau)}\bigg]\nonumber\\ \end{eqnarray}$$ (8)

In this case, the boundary conditions are,

U⁺ (0) = U^- (T) = 0.

We can write the Eqs. (2) and (3) as

$$M \frac {{\rm d}U} {{\rm d}\tau} + U = S_1,$$ (10)

where

$${M}= \left[ \begin{array} {ll} 1 & 0 \\ 0 & -1 \end{array... ... \left[ \begin{array} {l} S_1^+ \\ S_1^- \end{array}\right],$$ (11)

and Eqs. (4) and (5) and (8) and (9) will be written as,

$$S_1 (\tau) = B (\tau) + \omega (\tau) P (\tau) U (\tau),$$ (12)

$$S_{\rm d} (\tau) = S_1 (\tau) + \omega (\tau) P (\tau) U_{\rm b} (\tau),$$ (13)

where

$${B}= \left[ \begin{array} {l} B^+ \\ B^- \end{array}\righ... ...ft[ \begin{array} {ll} p & 1-p \\ 1-p & p\end{array}\right],$$ (14)

$${U_{\rm b}(\tau)}= \left[ \begin{array} {ll} U_1 & {\rm e}^{-\tau} \\ U_2 & {\rm e}^{-(T-\tau)} \end{array}\right].$$ (15)

The mathematical aspects of the solution of the Eqs. (2) and (3) are discussed in Sobolev (1963) and Grant (1968). We shall merely quote the relevant results and these are given in Appendix B.

Using equations (B1) and (B2) one can calculate the source function at points such as P, due to the incident radiation from the secondary component.

Now, we need to estimate the source function due to self radiation of the component. This can be done by solving the line transfer equation for a Non-LTE two-level atom in the comoving frame in spherical symmetry. This is given by (see Peraiah 1984; Peraiah et al. 1987)

$$\begin{eqnarray} & & \mu \frac {\partial I(x, \mu, r)}{\partial r} + \frac {(1 -... ...} {{\rm d}r}\bigg] \frac {\partial I (x, \mu, r)} {\partial x}, \end{eqnarray}$$ (16)

and

$$\begin{eqnarray} & & -\mu \frac {\partial I (x, -\mu, r)} {\partial r} - \frac {... ...} {{\rm d}r} \bigg] \frac {\partial I (x, -\mu, r)} {\partial x}, \end{eqnarray}$$ (17)

where $I(x, \pm\mu, r)$ is the specific intensity of the ray at an angle $\cos^{-1} \mu \ \ [\mu \epsilon (0, 1)]$ with the radius vector at the radial point r with frequency $x(=(\nu - \nu_0)/\Delta\nu_{\rm D}$ where $\nu_0$ and $\nu$ are the frequency points at the line centre and at any point in the line and $\Delta\nu_{\rm D}$ is the standard frequency interval such as Doppler width) V(r) is the velocity of the gas at r in units of mean thermal units (mtu) and K(x, r) and $K_{\rm c}(r)$ are the absorption coefficients per unit frequency interval in the line and the continuum respectively. The quantities $S_{\rm L}$ and $S_{\rm c}$ are the line and continuum source functions given by,

$$S_{\rm L}(r) = (1 - \epsilon) \int_{-\infty}^{+\infty} J(x, r) \phi (x) {\rm d}x + \epsilon B(x, T(r)),$$ (18)

$$S_{\rm c}(r) = \rho(r) B(x, T(r)),$$ (19)

$$K(x, r) = K_{\rm L}(r) \phi(x),$$ (20)

where $K_{\rm L}(r)$ is the line-centre absorption coefficient and $\phi(x)$ is the normalized line profile and $\rho(r)$ is an arbitrary factor less than one and B(x, T(r)) is the Planck function with frequency x, and temperature T at the radial point r. J(x, r) is the mean intensity given by,

$$J(x, r) = \frac {1} {2} \int^{+1}_{-1} I(x,\, r,\, \mu) {\rm d}\mu.$$ (21)

The quantity $\epsilon$ is the probability per scattering that a photon is thermalised by collisional de-excitation of the excited states, and this is given by,

$$\epsilon=C_{21} \bigg [C_{21}+A_{21} \bigl [1-\exp (-h \nu_{0}/ kT)\bigr ] \bigg ]^{-1}$$ (22)

where C₂₁ is the collisional transition rate from level 2 to 1 and A₂₁ is the Einstein spontaneous emission probability for transition from level 2 to 1. The quantities h and k are the Planck constant and Boltzmann constant respectively. From Eqs. (17) to (22), we obtain the source function due to self radiation, given by

$$S_{\rm s}(x, r) = \frac {\phi(x)} {\beta + \phi(x)} S_{\rm L}(r) + \frac {\beta} {\beta + \phi(x)} S_{\rm c} (r).$$ (23)

where $\beta$ is the ratio of absorption coefficients in the continuum and line centre.

Finally we calculate the total source function by adding $S_{\rm d}(\tau)$ in Eq. (14) and $S_{\rm s}(x, r)$ in Eq. (24) and obtain,

$$S = S_{\rm s} + S_{\rm d}.$$ (24)

We calculate the set of source functions at the points of intersection of the ray parallel to the line of sight and the shell boundaries. These source functions are used to calculate the emergent specific intensities at infinity (or at the observer's point), by using the formula (see Peraiah & Srinivasa Rao 1983)

$$I_{n+1} (r) = I_0(n) {\rm e}^{-\tau} + \int_o^\tau S(t) {\rm e}^{-[-(\tau - t)]}{\rm d}t,$$ (25)

where I_n(r) corresponds to the specific intensity of the ray passing through shell numbers n and n+1 and corresponding to the perpendicular to the axis $OO^\prime$ at different radii. I₀(n) corresponds to the incident intensity at the boundary of the shell and $\tau$ is the optical depth in the sector along the ray path. The source function S(t) is calculated by linear interpolation between S(t_n) and S(t_n+1). The specific intensity at the boundary of each shell is calculated by using Eq. (26).

The atmosphere in question is divided in to n shells (see Fig. 1) where n = 1 corresponds $\tau = T$ and n = 100 corresponding to $\tau = 0$, and $\tau$ is the optical depth at any point and T is the total optical depth. The total optical depth is set in advance. The incident radiation at Q, the bottom of the atmosphere, (see Fig. 1) is given as

$$I_{\rm s} (\tau = T, \ \mu_j) = 1.$$ (26)

The incident radiation from the secondary is given in terms of $I_{\rm s}$ in the ratio I, where I is given by

$$I = \frac {U_1} {I_{\rm s}}.$$ (27)

The velocities of expansion of the gas are expressed in terms of mean thermal units V_T (mtu) given by,

$$V_{T} = \bigg [{\frac {2kT} {m_i} } \bigg ] ^ {\frac {1} {2} }$$ (28)

where k is the Boltzmann constant and T is the temperature and m_i is the mass of the ion. The velocity at n=1 or $\tau = T$ is V_A and V_B is the velocity at n = 100 or $\tau = 0$ are given in terms of mtu or V_T as

$$V_{A}=\frac {v_{A}} {V_{T}}$$ (29)

$$V_{B} = \frac {v_{B}} {V_{T}}$$ (30)

where v_A and v_B are the velocities at the inner radius A, outer radius B at radial point r respectively in units of mtu of gas. We assume uniform expansion of the gases for the sake of simplicity. The proximity of the component is measured in terms of separation parameter r₁/R, where r₁ is the radius of the primary and R is the separation of centres of gravity of the two components. The ratio of the outer to the inner radii (B/A) of the atmosphere is always taken to be 2. The actual thickness in the components could be much larger than what we have considered here. As the number of parameters is large, we restricted our calculations to this modest thickness of the atmosphere.

The variation of the source functions are shown against the shell numbers (n = 1 to 100) for different parameters are shown. The line profile fluxes $(F_Q/F_{\rm c})$ are plotted against the normalized frequency Q, where

$$Q = x/x_{\rm max},$$ (31)

F_Q = F(x_Q),

$$F_C = F(x_{\rm max}),$$ (33)

and

$$x = (\nu - \nu_0)/\Delta\nu_D,$$ (34)

$$x_{\rm max} = \mid x \mid + V_B,$$ (35)

$$\Delta\nu_D = \nu_0 \frac {v_T} {c},$$ (36)

x lies between $\pm$ 5 units. The equivalent widths are calculated by the relation,

$$EQ. W = \int_{x_{\rm min}}^{x_{\rm max}} (1 - F_{Q}/F_{\rm c}){\rm d}x$$ (37)

where

$$x_{\rm min} = - (\mid x \mid + V_{B}).$$ (38)