3 Boundary conditions

The atmosphere in question is divided into n shells 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. We assume that no radiation is incident from outside the spherical shell at r=B and $\tau = 0$, while radiation of unit intensity is incident at r=A, $\tau=\tau_{\rm max}=T$, in the case of purely scattering medium. Thus we have,

$${\bf U}^-(\tau=T, \mu_j)=1$$

$${\bf U}^+(\tau=0, \mu_j)=0, \quad (\epsilon=0, \beta=0)$$ (4)

and in the case of the boundary condition of the frequency derivative, we have:

$$% \frac {\partial {\bf U}} {\partial X}\quad ({\rm at} X=\mid X_{\rm max}\mid)=0.$$ (5)

We have set the velocities at r=A as V_A, and at r=B as V_Bwhich is set equal to 0 and 50 mtu. If the spherical shell expands with constant velocity, we have;

V_A=V_B (6)

if there are velocity gradients, then we have;

$$% V_A=0 \quad V_B > 0.$$ (7)

Finally we calculate total source function S by adding the source function (1) due to reflected radiation $S_{\rm d}$ and due to the self radiation $S_{\rm s}$ (See Paper I)

$$% S = S_{\rm s} + S_{\rm d}.$$ (8)

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)

$$% I_{n+1} (r) = I_0(n) {\rm e}^{-\tau} + \int_o^\tau S(t) {\rm e}^{-[-(\tau - t)]}\ {\rm d}t,$$ (9)

where I_n(r) corresponds to the specific intensity of the ray passing through the shell bounded by r_n and r_n+1 corresponding to perpendicular to the axis OO$^\prime$ (See Fig. 1 of Paper I) 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. (9).

Figure: The source functions S are shown with respect to the shell numbers in a scattering medium for different dust optical depths, velocities, with $\frac{r_1}{R}=\frac {1}{2}, \frac{1}{5}$ and I=5

The incident radiation at Q, the bottom of the atmosphere, (See Fig. 1 of Paper I) is given as

$$% I_{\rm s} (\tau = T, \ \mu_j) = 1.$$ (10)

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

$$% I = \frac {U_1} {I_{\rm s}}\cdot$$ (11)

The velocities of expansion of the gas are expressed in terms of mean thermal units $V_{\rm T}$ (mtu) given by,

$$% V_{\rm T} = \bigg [{\frac {2kT_1} {m_i} } \bigg ] ^ {\frac {1} {2} }$$ (12)

where k is the Boltzmann constant and T₁ 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 $V_{\rm T}$ as

$$% V_{A}=\frac {v_{A}} {V_{\rm T}}$$ (13)

$$% V_{B} = \frac {v_{B}} {V_{\rm T}}$$ (14)

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 the 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 $\frac {r_1} {R}$, where r₁ is the radius of the primary and R is the separation of centre of gravity of the two components. The ratio of the outer to the inner radii $\frac {B}{A }$ 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 line profile fluxes $\frac {F_Q}{F_c}$ are plotted against the normalized frequency Q, where

$$% Q = \frac {x}{x_{\rm max}},$$ (15)

F_Q = F(x_Q), (16)

$$% F_C = F(x_{\rm max}),$$ (17)

and

$$% x = (\nu - \nu_0)/\Delta\nu_{\rm D},$$ (18)

$$% x_{\rm max} = \mid x \mid + V_B,$$ (19)

$$% \Delta\nu_{\rm D} = \nu_0 \frac {v_{\rm T}} {C},$$ (20)

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

$$% EQ. W = \int_{x_{\rm min}}^{x_{\rm max}} \Bigl(1 - \frac {F_Q}{F_{c}}\Bigr) {\rm d}x$$ (21)

where

$$% x_{\rm min} = - (\mid x \mid + V_{B}).$$ (22)