4 The effect of different resonance scattering lengths in optically dense media in the presence of nonresonant absorption

An essential result of the discussion in (Johansson & Letokhov 2001b) is the possible evidence for a (in the case of bright UV lines) significant optical density, $\tau_{12}>1$, for UV transitions terminating on long-lived states. A second important feature is the physical difference between the strong and weak pairs of UV lines, namely, that they terminate on different fine-structure levels (!), which we have denoted as levels 1. Under laboratory conditions (optically thin, non-absorbing media), the intensity ratio between, for example, transition A (2a $\rightarrow$1b) and transition b (2a $\rightarrow$1a) is:

$$\left( I_{\rm A}/I_{\rm b}\right) ^{\rm {Lab.}}=R_{\rm {A/b}}... ...\frac{A_{\rm { 2a\rightarrow 1b}}}{A_{\rm {2a\rightarrow1a}}}%$$ (20)

i.e., equal to the ratio between the Einstein coefficients A for the corresponding transitions.

Under blob conditions, we should take into account the possible difference in optical pathway for photons associated with different transitions as a result of a possible difference in the mean resonance scattering lengths in optically thick media ( $\tau_{12}>1$). The resulting difference in attenuation, $K_{\rm {att}}$, in the presence of weak nonresonant (continuous) absorption

$$K_{\rm {att}}^{\rm {A,b}} \simeq\exp\left( -\kappa_{\rm {abs}} <L_{\rm {A,b}}>\right) ,$$ (21)

where $\kappa_{\rm {abs}}$ is the nonresonance absorption coefficient (cm^-1), $L_{\rm A}$ and $L_{\rm b}$ are the optical pathways of photons with different resonance scattering coefficients, $\kappa_{\rm {sc}}^{\rm A}$ and $\kappa_{\rm {sc}}^{\rm b}$, or different resonance scattering with mean free paths $\left\langle \ell _{\rm {sc}}^{\rm A}\right\rangle$ and $\left\langle \ell_{\rm {sc}}^{\rm b}\right\rangle$. In the diffusion approximation $\left( <\ell_{\rm {sc}}^{\rm {A,b}}>\,\ll D\right)$

$$\left\langle L_{\rm {A,b}}\right\rangle \simeq\frac{D^{2}}{\l... ...{\rm {A,b}}\right\rangle }=\kappa_{\rm {sc}}^{\rm {A,b}}D^{2},$$ (22)

where D is the distance from the emitting source to the boundary of the medium (in our case, it is the size of the diameter of the blob), the resonance scattering coefficients are determined by the populations of the lower states ( $N_{2}\ll N_{1}$), and the corresponding cross-sections or optical densities $\tau _{12}$ of the transitions are given by

$$\kappa_{\rm {sc}}^{\rm A}=N_{\rm {1b}}\sigma_{\rm {1b-2a}} =N... ...}}\frac{g_{\rm {2a}}}{g_{\rm {1b}}}% \propto\tau_{\rm {1b-2a}}$$ (23)

and

$$\kappa_{\rm {sc}}^{\rm b}=N_{\rm {1a}}\sigma_{\rm {1a-2a}}= N... ...}\frac{g_{\rm {2a}}}{g_{\rm {1a}}}% \propto\tau_{\rm {1a-2a}}.$$ (24)

According to Eqs. (21) and (22), the intensity ratios of the blob lines should be

$$\left( \frac{I_{\rm A}}{I_{\rm b}}\right) ^{\rm {blob}}= \lef... ...}}D^{2}(\kappa_{\rm {sc}}^{\rm A}-\kappa_{\rm {sc}}^{\rm b})}.$$ (25)

Thus, in the presence of nonresonant absorption ( $\kappa_{\rm {abs}} \neq0$) the difference between the resonance scattering lengths $1/\kappa_{\rm {rs}}^{\rm A}$ and $1/\kappa_{\rm {sc}}^{\rm b}$ for different spectral lines will lead to a difference between the blob and laboratory intensity ratios. In Fig. 5 we have plotted the intensities of lines A and b (normalized to laboratory values) as subject to radiative transfer from the center of the blob (r=0) to its surface (r=D/2). (Note that the label on the vertical axis in a similar figure in our previous work (Johansson & Letokhov 2001a) is incorrect.)

The effect of nonresonant absorption of the flux of emission lines having different optical path lengths in a resonance-scattering medium of large optical thickness was considered long ago (Hummer 1968). We have, in essence, rediscovered this effect when trying to find an explanation for the anomalous behavior of the intensities of two pairs of close spectral lines. The qualitative picture above does not take into account the Doppler frequency shift caused by resonance scattering by moving FeII ions. Actually the frequency-shifted photons can escape from the blob at the depth $r_{\rm {esc}} \ll D$. This effect will limit the optical pathways $\left\langle L_{\rm {A,b,C,d}}\right\rangle$ for trapped spectral lines (Adams 1972; Harrington 1973; Hummer & Kunasz 1980). Hummer (1968) investigated in detail the special functions arising in radiative transfer due to nonresonance scattering and the numerical solution of the transport equation for the case of scattering with complete frequency redistribution. Subject to consideration in Hummer's work (1968) was radiative transfer in a plane-parallel layer with radiation sources both in a line spectrum and in a continuous spectrum. In addition, the source of the line radiation in Hummer's work contained some part due to the absorption of photons as a result of electron collisions with the excited atom. The radiative transfer process in a medium with non-resonant absorption in the case of partial frequency redistribution was investigated both numerically and asymptotically in (Hummer & Kunasz 1980; Frisch 1980), where effective scaling laws were found for the case of small absorption coefficients and large optical thicknesses of the layer.

In the present work, the transport equations have a somewhat different form, because the only source of radiation in the UV lines is the spontaneous emission of the FeII ions excited by the wide-band Ly$\alpha$ radiation at different wavelength. Furthermore, the optical thickness $\tau_{12}^{0}$ of the $1\rightarrow2$ transition, which is responsible for the intense UV radiation, is substantial $(\tau_{12}^{0}\gg1)$ (Johansson & Letokhov 2001b), but not so great that it is necessary to take into account the scattering in the Lorentzian wings, i.e. the parameter $a\tau_{12}^{0}<1$, where the damping parameter $a=\Delta\nu _{\rm {rad}}/\Delta\nu_{\rm {Dopp}}$ for the FeII lines of interest. And finally, the geometry of the passive region of blob B exposed to the Ly$\alpha$radiation is more spherical than plane, even though it is a convenient assumption to simplify the calculations. Thus, for the present problem with the anomalous Fe II lines it would be desirable to calculate the radiative transfer in an optically dense medium featuring nonresonant absorption in a spherical geometry.