3 The 80-level collision radiative model with fluorescence

A CR model with 80 levels for Fe VI is employed to calculate level populations, N_i, relative to the ground level. The emitted flux per ion for transition $j\rightarrow i$, or the emissivity $\epsilon_{ji}$ (ergs cm^-3 s^-1), is given by,

$$% \epsilon_{ji}=N_jA_{ji}h\nu _{ij}.$$ (10)

The line intensities ratios, for example, $\epsilon _{ji}/\epsilon _{kl}$for transitions j-i and k-l are then calculated. A computationally efficient code to set up CR matrix in CR model is employed to solve the coupled linear equations for all levels involved (Cai & Pradhan 1993).

We can write the rate coupled equations of statistical equilibrium in CR matrix form:

$$% {\bf C=NeQ+A}$$ (11)

where Ne is the electron density, and

$$% C_{ii}\! =\! 0\\ \ C_{ij}\! =\! q_{ij}Ne ~~~(j>i)\\ \ C_{ij}\! =\! q_{ij}Ne+A_{ij} ~~~(j<i).\\$$ (12)

In the above model, we have assumed the optically thin case. FLE by continua pumping and other pumping mechanisms are not considered in this mode. Also, the line emission photons from the tramsitions within the ions are assumed to escape directly without absorption (Case A).

In a thermal continuum radiation field of optically thick gaseous nebulae, the ion can be excited by photon pumping or de-excited by induced emission. With the FLE mechanism for excitation or de-excitation in the CR model, Eqs. (10, 11) should be replaced by

C_ij = q_ijNe+J_ijB_ij    (j>i)

C_ij = q_ijNe+A_ij+J_ijB_ij    (j<i) (13)

where J_ij is the mean intensity of the continuum at the frequency for transition $i\rightarrow j$. B_ij is the Einstein absorption coefficient or induced emission coefficients. For a blackbody radiation with effective temperature $T_{\rm eff}$, J_ij can be expressed as

$$% J_{ij}=WF_{\nu}=W\frac{8\pi h\nu ^3}{c^2}\frac 1{{\rm e}^{h\nu/kT_{\rm eff}}-1}$$ (14)

where $\pi F_{\nu}$ is the monochromatic flux at the photosphere with an effective temperture $T_{\rm eff}$. $W=\frac 14(\frac Rr)^2=1.27\,\, 10^{-16}(\frac {R_{\ast}/R_{\odot}}{r/{\rm pc}})^2$is the geometrical dilution factor, R and rbeing the radius of the photosphere and the distance between the star and the nebula, respectively. More accurate specific luminosity density function versus frequency, or the mean intensity, may be employed in ohter applications, e.g. in the synchrotron continuum pumping in the Crab nebula (Davidson & Fesen 1985; Lucy 1995).

With the notations used and explanations given above, the equation of statistical equlibrium for the k-th level has the form

$$% \sum_{j\neq k}(N_jC_{jk}-N_kC_{kj})=0.$$ (15)

The attenuation effect in continuum intensity J_ij has been neglected in the above rate equations, i.e. there are no optical depths along the line of observation in the nebula to the continuum radiation source. This approximation may be responsible for part of the difference between the calculated and observed line ratios as shown in Table 6 below. With this approximation, the rate coefficients C_ij are independent of level population N_k; the rate equations are therefore linear and can be solved directly.