6 Applications of anomalous UV FeII lines in $\eta$ Carinae

From the qualitative standpoint, the main result of this computer consideration of Eqs. (43), (46), and (47), which are presented in Fig. 12, is as follows: the change of the attenuation coefficients of two lines is inversely proportional to the resonance scattering coefficients (see the asymptotes for $\kappa_{\rm {sc}}R\ll1$ in Fig. 12). This asymptotic behaviour can be approximated by the simple expression giving the intensity ratio between the blob lines A and b on the surface of the blob:

$$\left( \frac{I_{\rm A}}{I_{\rm b}}\right) \sim\left( \frac{\kappa_{\rm {sc}}^{\rm b}}% {\kappa_{\rm {sc}}^{\rm A}}\right)\cdot$$ (53)

Using a more detailed notation for the anomalous spectral lines and the corresponding quantum transitions and energy levels (Fig. 13), one can estimate the ratio between the resonance scattering coefficients:

$$\frac{\kappa_{\rm {sc}}^{\rm A}}{\kappa_{\rm {sc}}^{\rm b}} \frac{\tau_{\rm {1b-2a}}}{\tau_{\rm {1a-2a}}} =\frac{N_{\rm {1a}}}{N_{\rm {1b}}}\frac{\sigma_{\rm {1a-2a}}} {\sigma_{\rm {1b-2a}}}$$ =

$$\left( \frac{N_{\rm {1b}}}{N_{\rm {1a}}}\right) \left( \frac{A_{\... ...\right) \left( \frac{g_{\rm {1a}}/g_{\rm 2a}}{g_{\rm {1b}}/g_{\rm {2a}}}\right)$$ =

$$\left( \frac{T_{\rm {1a}}}{T_{\rm {1b}}}\right) \left( \frac{A_{\... ...}}{A_{\rm {2a-1b}}}\right) \left( \frac{g_{\rm {1a}}}{g_{\rm {1b}}}\right)\cdot$$ = (54)

Using the asymptotic ratio (53), we can obtain the following expression for the blob and laboratory line intensity ratios:

$$\left( \left. \frac{I_{\rm A}}{I_{\rm b}}\right) ^{\rm {blob}... .../ \left( \frac{I_{\rm A}}{I_{\rm b}}\right) ^{\rm {Lab.}}\cdot$$ (55)

A similar expression can also be obtained for the (C,d) pair of UV FeII lines:

$$\left(\left. \frac{I_{\rm C}}{I_{\rm d}}\right) ^{\rm {blob}}... .../ \left( \frac{I_{\rm C}}{I_{\rm d}}\right) ^{\rm {Lab.}}\cdot$$ (56)

The model considered above for the difference in attenuation between spectral lines in resonance-scattering and nonresonance-absorbing media requires that the difference between the lifetimes $T_{\rm {1a}}$ and $T_{\rm {1b}}$ of the low-lying levels (1a) and (1b) (or $\rm c^{4}F_{9/2}$ and $\rm c^{4}F_{7/2}$ according to the present identification of FeII levels) should be substantial:

$$T_{\rm {1a}} \gg T_{\rm {1b}}.$$ (57)

Figure 9: Spectral radiation density at the edge of the nebula, $\bar{J} (r=R,\omega)$ , as a function of the optical density $\tau _{12} =\kappa _{\rm {sc}}^{} R$ in the absence of nonresonance absorption ($\beta =0$), normalized to the spectral density value at the edge of the nebula in the absence of scattering.

Figure 10: Spectral radiation density at the edge of the nebula, $\bar{J} (r=R,\omega)$, as a function of the optical density $\tau_{12} =\kappa_{\rm {sc}}R=0.5\ldots200$ at various nonresonance absorption cross sections, normalized to the spectral density value at the center of the nebula in the absence of scattering: a) $\beta =0.001;$ b) $\beta =0.005;$ c) $\beta =0.01;$ d) $\beta =0.1$.

Figure 11: Total photon density at the center of the nebula as a function of the optical density $\tau _{12}$ at various continuous absorption parameter values, normalized to the total photon density value in the absence of scattering.

Figure 12: Total photon density at the edge of the nebula as a function of the optical density $\tau _{12}$ at various continuous absorption parameter values, normalized to the total photon density value in the absence of scattering.

Figure 13: Detailed notation for the anomalous UV FeII spectral lines and the corresponding energy levels.

It is only in this case that the weak (in lab.) lines are becoming more strongly attenuated in the blob than their strong counterparts. This requirement contradicts the existing data on the lifetimes of the lower levels: $T_{\rm {1a}}=1.35$ ms, $T_{\rm {1b}}=1.6$ ms (Kurucz 1988).

To fulfill the requirement, we should consider the possibility of the photoionization depopulation of the lower levels by the Ly$\alpha$radiation (Johansson & Letokhov 2001b). The rate of this process is too low to provide for any population inversion, but is quite sufficient to provide for the depopulation of the lower levels at a rate much higher than the radiative decay rate 1/T₁. According to Johansson & Letokhov (2001b), for the effective temperature of Ly $\alpha,T_{\rm {eff}}({\rm Ly}\alpha)\simeq10\,000$ K, the resonance enhancement of the photoionization cross-section can provide for the depopulation of the lower states at a rate of $W_{\rm {ph}}^{\rm {1i}}\gg1/T_{1}$. It is most important that the narrow resonances of the photoionization cross-sections of Fe II (Nahar & Pradhan 1994) and the energy difference between the levels $\rm c^{4}F_{7/2}$ and $\rm c^{4}F_{9/2}$ (Fig. 4, $\Delta E=30$ cm^-1) must provide for the preferable depopulation of the $\rm c^{4}F_{7/2}$ state (1b) and the resulting weaker absorption of the A and C ultraviolet lines. According to (38), it is necessary that the $\sigma_{\rm {ph}}^{\rm {1i}}$ ratio should be

$$\frac{\sigma_{\rm {ph}}^{\rm {1b,i}}}{\sigma_{\rm {ph}}^{\rm ... ...\left( \frac{I_{\rm A}}{I_{\rm b}}\right) ^{\rm {lab}}\simeq30$$ (58)

which can be proved by precise calculations similar to those reported in (Nahar & Pradhan 1994) in the range slightly above the ionization limit of FeII. The requirement in Eq. (58) is rahter unusual for photoionization of levels belonging to the same LS term of an electron configuration. Perhaps, this contradiction can be overcome in the frame of plausible photodepletion of the c⁴F levels by relatively narrow autoionization levels (Johansson & Letokhov 2001b, Fig. 9). However, this is subject to more detailed and elaborative atomic structure calculations and/or experiments, which are out of the scope of the present paper.

The estimates in Eq. (58) require the following value of $\kappa_{\rm {abs}}$(UV):

$$\kappa_{\rm {abs}}^{\rm {(UV)}}\simeq(10^{-2}{-}10^{-1})\kappa_{\rm {sc}}^{\rm {(UV)}}% .$$ (59)

This requirement must be compatible with the need for the Ly$\alpha$photons to be scattered many times, $\tau_{0}\simeq10^{7}$, prior to their decay (Johansson & Letokhov 2001b) to provide for a substantial diffusive broadening of the Ly$\alpha$ spectrum. The most suitable non-resonance absorption mechanism is the photoionization absorption of He I to long-lived metastable states that are known to accumulate in the passive HI region during the HeI/HeII/He*I photoionization/recombination cycle under the effect of the shorter-wavelength EUV radiation from the central star (Ambartzumian 1939). Let us assume that the continuous absorption coefficients $\kappa_{\rm {abs}}$ are of the same order of magnitude for Ly$\alpha$ and UV FeII photons, i.e. $\kappa_{\rm {abs}}({\rm {Ly}}\alpha)$. It follows from these crude estimates that the ratio between the resonance scattering coefficients for these two wavelengths is

$$\frac{\kappa_{\rm {sc}}({\rm {Ly}}\alpha)}{\kappa_{\rm {sc}}(... ...}\frac{\sigma_0({\rm {Ly}}\alpha)}{\sigma_{12}({\rm {FeII}})},$$ (60)

where $N_{\rm HI}$ and $N_{\rm {Fe}}$ are the hydrogen and iron concentrations in the blob, respectively. The factor $N^1_{\rm {FeII}}$ is the concentration of FeII ions in the long-lived state 1 (c⁴F), which governs the optical density of the $1\rightarrow2$ transition and the value is, according to the estimates presented in Johansson & Letokhov (2001b), $N^1_{\rm {FeII}} \simeq 3\times10^{-3}$ $N^m_{\rm {FeII}}$, where m is the metastable state. If we take the standard estimate for $N_{\rm {HI}}/N_{\rm {FeII}} \simeq10^{4}$, we get from relation (60)

$$\kappa_{\rm {sc}}({\rm {Ly}}_{\alpha})\simeq3\times10^{6}\kappa_{\rm {sc}}({\rm {UV}}),$$ (61)

and so requirement (59) corresponds to

$$\kappa_{\rm {abs}}({\rm {Ly}}_{\alpha})\simeq\kappa_{\rm {abs... ...=3~(10^{-9}{-}10^{-8})\kappa _{\rm {sc}}({\rm {Ly}}_{\alpha}).$$ (62)

Such a weak absorption of the Ly$\alpha$ photons with their short scattering length is known to provide for the necessary number of their scattering events, $\tau_{0}\simeq10^{7}$.