2 A model of the origin of the extremely bright UV FeII lines

According to observational data (Davidson & Humphreys 1997), the Weigelt blobs (Weigelt & Ebersberger 1986) close to $\eta$ Car (for example, blob B) are compact gas condensations (low-intensity compact ejecta with a high hydrogen concentration $N_{\rm H} \simeq 10^6$ cm^-3 and a Fe concentration of $N_{\rm Fe} \simeq 10^2$ cm^-3).

The blob diameter is $D \leq 10^{15}$ cm, and the distance from the central star is $R_{\rm b} \simeq 3\times 10^{16}$ cm. The diameter of the photosphere of the central star is $2r_{\rm s} = d_{\rm s} \simeq 3\times 10^{13}$ cm, and its effective temperature is $T_{\rm s} = (20{-}30) \times 10^3$ K. These parameter values are preliminary. For example, according to recent HST/STIS observations (T. Gull, private communication, April 2001), the distance $R_{\rm b}$ between the blob B and the central star is around $3 \times 10^{15}$ cm, i.e. an order of magnitude smaller than derived from previous data. The fact that the blob is located close enough to the central star is of importance for our model. The dilution factor of the radiation from $\eta$ Car, $\Omega \cong \frac {1}{4}(r_{\rm s}/R_{\rm b})^2$, is around 10^-6 - 10^-5, which is many orders of magnitude greater than that for a typical planetary nebula. For illustration, we present in Fig. 1 a schematic picture of a gas condensation having a spherical shape with an average diameter D (as judged from density) and an average volume concentration $N_{\rm H}$ of hydrogen atoms and ions and being located at the distance $R_{\rm b} \gg D$ from the star.

Figure 1: Model geometry of the compact gas condensation (blob) in the vicinity of the central star of $\eta$ Carinae.

The most intense part of the stellar radiation with h $\nu > {\rm h}\nu_{\rm c} = 13.6$ eV is absorbed almost completely by neutral hydrogen having a concentration of $N_{\rm HI}$( $N_{\rm {HI}} \ll N_{\rm {HII}} \approx N_{\rm H}$), provided that the optical density $\tau (\nu_{\rm c})$ of the Lyman-continuum is high enough, i.e.,

$$\sigma_{\rm {ph}} (\nu_{\rm c}) N_{\rm {HI}} D \gg 1.$$ (1)

Condition (1) is satisfied if the hydrogen density $N_{\rm H}$ exceeds some critical value $N_{\rm H}^{\rm {cr}}$:

$$N_{\rm H}^{\rm {cr}} = \left(\frac{I_{{\rm ph}}}{\alpha D} \r... ...c},T_{\rm s})\Delta \nu_{\rm {ph}}}{\alpha D} \right )^{1/2} ,$$ (2)

where $\alpha$ is the rate coefficient of the HII $\rightarrow$HI recombination process, and $I_{\rm {ph}}$ is the effective intensity of the radiation incident on the blob surface and ionizing hydrogen. As an estimate we use, as in our previous paper (Johansson & Letokhov 2001b), average values of the blackbody radiation intensity $P(\nu_{\rm c})$ and photoionization cross section $\sigma_{\rm {ph}}(\nu)$ in the effective frequency interval $\Delta \nu_{\rm {ph}} \simeq 3$ eV above $\nu_{\rm c}$, where $P(\nu)$ and $\sigma_{\rm {ph}}(\nu)$ are sufficiently large. Condition (1) means that the local Strömgren boundary separating the HII and HI regions is inside the gas condensation being considered, as shown in Fig. 1. For blob B near $\eta$ Carinae at $D \leq 10^{15}$ cm and $T_{\rm s} = 30\,000$ K, the critical density $N_{\rm H}^{\rm {cr}} \geq 10^8$ cm^-3.

The physical depth $l_{\rm {ph}}$ at which the photoionization state of hydrogen is sustained at $N_{\rm H} >N_{\rm H} ^{\rm {cr}}$ is defined by the expression

$$\frac {l_{\rm {ph}}}{D} \simeq \left ( \frac {N_H^{\rm cr}}{N... ...2 = \frac {1}{N^2_{\rm H}} \frac{I_{\rm {ph}}}{\alpha D}\cdot$$ (3)

At $N_{\rm H} \gg N_{\rm H} ^{\rm {cr}}$, the depth $l_{\rm {ph}} \ll D$, i.e. there is a thin front layer of the blob facing the central star (Fig. 1).

At $N_{\rm H} >N_{\rm H} ^{\rm {cr}}$ the neutral component HI absorbs almost all of the power ${\cal P}$ (in photons/s) of the Lyman-continuum radiation coming from the star:

$${\cal P}_{\rm {abs}} ( \nu > \nu_c) \simeq \Omega_0 S_{\rm {abs}} \int_{\nu_{\rm c}}^{\infty}{ P(\nu, T_{\rm s}){\rm {d\nu}}}$$

$$\simeq \Omega \pi^2 D^2 \overline{P(\nu_{\rm c}, T_{\rm s})}\Delta \nu_{\rm {ph}} ,$$ (4)

where $\Omega_0 = 4\pi \Omega$ is the solid angle and $S_{\rm {abs}}=\frac{\pi}{4}D^2$ is the disc area of the blob absorbing the incident radiation from the central star. Most of the absorbed energy is reemitted due to radiative recombination of the hydrogen ions formed. The major proportion ( $\eta \approx 0.7$) of the absorbed energy is emitted in the resonance line HI Ly$\alpha$. The maximum value of the optical density $\tau_0$(Ly$\alpha$) with reference to the center of Ly$\alpha$ in the HII region is

$$\tau_0^{\rm m} \simeq \frac {\sigma_0}{\sigma_{\rm {ph}} (\nu_{\rm c})} \simeq 5 \times 10^3 ,$$ (5)

where we have set $\sigma_0 \simeq 1.4 \times 10^{-14}$ cm² and $\sigma_{\rm {ph}} (\nu_{\rm c}) \simeq 3 \times 10^{-18}$ cm² (average value) in the frequency interval ( $\nu_{\rm c}, \nu_{\rm c} + \Delta \nu_{\rm {ph}})$. For illustration, we show in Fig. 2 the qualitative distribution of the average concentrations of HII and HI inside the blob, as well as the evolution of the Ly$\alpha$ radiation spectrum in various zones of the blob.

Figure 2: Radial profiles of the ionized ( $N_{\rm HII}$) and neutral hydrogen ( $N_{\rm HI}$) concentrations and a qualitative evolution of the spectral shape of the Ly$\alpha$ radiation in the active HII region, the Strömgren boundary region, and the passive fluorescence region relative to the FeII absorption profile.

The Ly$\alpha$ radiation proves to be trapped in the HII region, but the diffusion confinement time is limited by the fact that the number of scattering events is limited because of the Doppler frequency redistribution of the scattered Ly$\alpha$ photons. This redistribution makes them leave the trapping region relatively fast via the wings of the Doppler profile (Osterbrock 1967). Since the optical density $\tau_0$ is limited to $\tau_0^{\rm m} = (\sigma_0/\sigma_{\rm {ph}})$, and the damping factor for Ly$\alpha$ is $a = \delta \nu_{\rm {rad}}/\delta \nu_{\rm D} \simeq 4 \times 10^{-4}$, the optical density in the Lorentz wings is $a \tau_0^{\rm m} \simeq 1$, where we assume the Doppler spectral width to be $\delta \nu_{\rm D} \simeq 6$ cm^-1 and the natural spectral width, $\delta \nu_{\rm {rad}} \simeq 2.5 \times 10^{-3}$ cm^-1. Therefore, the Ly$\alpha$ photons leave the HII region by diffusion while suffering an increase in the Doppler width $\Delta \nu_{\rm D}$ by a factor of $(\ln \tau_0^{\rm m})^{1/2} \approx 3$, so that the total power emitted by the surface of the blob into Ly$\alpha$ is

$${\cal P}_{\rm {em}}({\rm {Ly}} \alpha) \simeq 4 \pi S_{\rm {e... ...rm {br}})(\Delta\nu_{\rm D} \sqrt {{\rm {ln}}\tau_0^{\rm m}}),$$ (6)

where $S_{\rm {em}}= \pi D^2$ is the area of the emitting surface of the spherical blob and $P(\nu _{\rm {Ly\alpha}}, T_{\rm {br}})$ is the average spectral intensity of Ly$\alpha$ on the blob surface, given by the Planck distribution at a frequency of $\nu _{\rm {Ly\alpha}}$ and a spectral brightness temperature of $T_{\rm {br}}$, account being taken of the spectral line broadening due to trapping. In the steady-state case where the absorption of the trapped Ly$\alpha$ in the HII region of the blob is negligible, we have

$$\eta {\cal P}_{\rm {abs}} (\nu > \nu_{\rm c}) = {\cal P}_{\rm {em}}(\rm {Ly\alpha}).$$ (7)

It follows from (7) that the brightness temperature $T_{\rm {br}}$ of the Ly$\alpha$ radiation on the open surface of the blob can be estimated by the expression

$$P(\nu _{\rm {Ly\alpha}}, T_{\rm {br}}) \simeq \frac {\eta}{4}... ...qrt{\ln \tau_0^{\rm m}}}\overline{ P(\nu_{\rm c}, T_{\rm s}}).$$ (8)

It can be seen that the dilution factor $\Omega$ can largely be compensated for by the effect of the spectral compression of the absorbed energy into the relatively narrow Ly$\alpha$ radiation. Note that the intensity of the Ly$\alpha$ radiation inside the HII region of the blob is higher than that on the blob surface, $P(\nu _{\rm {Ly\alpha}}, T_{\rm {br}})$, by a factor of ${\rm {ln}}\tau_0^{\rm m}$ due to diffusion confinement. Therefore, to determine the brightness temperature $\tilde T_{\rm {br}}$ of the radiation incident on the boundary between the HII and HI regions inside the blob (the Strömgren boundary in Fig. 1), the right-hand side of expression (7) should be increased by ${\rm {ln}}\tau_0^{\rm m} \simeq 7$ times, which is equivalent to the same increase in $\Omega$:

$$P_{\rm {St.b}}(\nu _{\rm {Ly\alpha}}, \tilde T_{\rm {br}}) \s... ...m}}}{\Delta \nu_{\rm D}}\overline{ P(\nu_{\rm c}, T_{\rm s}}).$$ (9)

The temperature $\tilde T_{\rm {br}}$ is fairly close to the temperature of the blackbody radiation of the central star's photosphere (for $\Omega \simeq 10^{-6} {-} 10^{-5}$, the temperature $\tilde T_{\rm {br}}$ is $(10{-}20)\times 10^3$ K), but the spectral width of the line radiation leaving the active HII region is still insufficient to compensate for the difference in wavelength between the Ly$\alpha$ emission and the FeII absorption lines (see the top part of Fig. 2). This radiation suffers further spectral diffusion both in the HII/HI transition region (the Strömgren boundary), which is optically denser for Ly$\alpha$, and mainly in the passive, weakly ionized HI region.

The physical depth $\delta l_{\rm {ph}}$ of the HII/HI transition region is shallow:

$$\delta l_{\rm {ph}} \simeq \frac{1}{\sigma_{\rm {ph}}N_{\rm H}} \ll l_{\rm {ph}},$$ (10)

in comparison with the physical length $l_{\rm {ph}}$ of the photoionization depth (the physical depth of the active HII part of blob B):

$$l_{\rm {ph}} \simeq \delta l_{\rm {ph}} \frac{N_{\rm {HII}}}{N_{\rm {HI}}} \leq D,$$ (11)

defined by expression (3). Note that, for the reader's convenience, we present here these well-known expressions (Mihalas 1978) in order to give explanations to the designations used. The optical thickness of the HII/HI transition layer with physical width $\delta l_{\rm {ph}}$ can be estimated as:

$$\tau_0^{\rm {tr}} \simeq N_{\rm {HI}}^{\rm {tr}}\delta l_{\rm... ...}{2} \frac {\sigma_0}{\sigma_{\rm {ph}}} \simeq 2 \times 10^3,$$ (12)

where $N_{\rm {HI}}^{\rm {tr}} \simeq \frac{1}{2}N_{\rm H}$ is the average concentration of HI in the transition layer, and $\delta l_{\rm {ph}}$ is determined by Eq. (10). The high density of HI compensates the small width of this layer and as a result the optical thickness of the transient layer $\tau_0^{\rm {tr}}$ is almost equal to the optical thickness $\tau_0^{\rm m}$ of the whole active zone. Nevertheless, these values of $\tau_0$ are much smaller than the magnitude required for the Ly$\alpha$ excitation of FeII:

$$\tau_0^{\rm {exc}} = \sigma_0 N_{\rm H} D = \frac {\sigma_0}{... ...delta \nu_{\rm {rad}}\delta \nu_{\rm D}} \simeq 7 \times 10^6.$$ (13)

However, the remaining, dissipating (passive) volume of the blob is also large enough to provide for $\tau_0^{\rm {exc}} \simeq 10^7$ and the appropriate broadening of the Ly$\alpha$ line as a result of the Doppler frequency diffusion of the radiation on the Lorentz wing of the Ly$\alpha$ resonance line.

The physical depth $l_{\rm {br}}$ of the passive HI region, at which the diffusion spectrum broadening necessary for the excitation of FeII is reached, is defined by the expression

$$l_{\rm {br}} \simeq \frac {1}{\Delta \nu N_{\rm H}} \simeq \frac {\tau_0^{\rm {exc}}}{\sigma_0 N_{\rm H}}\cdot$$ (14)

The depth $l_{\rm {br}}$ (see Figs. 1 and 2) can be compared with the depth $l_{\rm {ph}}$ of the photoionization region. Their ratio is given by

$$\frac {l_{\rm {br}}}{l_{\rm {ph}}} \simeq \frac {\tau_0^{\rm ... ...m H} D} \left (\frac {N_{\rm H}}{N_{\rm {cr}}}\right )^2\cdot$$ (15)

At $N_{\rm H} = N_{\rm {cr}}$ the depth $l_{\rm {br}} \simeq$ (0.01 to 0.1) $l_{\rm {ph}}$, so that the transition region fully provides for the necessary diffusion broadening of Ly$\alpha$. At $N_{\rm H} = 10 N_{\rm {cr}}$, the depth $l_{\rm {br}}$ increases to (0.1 to 1) $l_{\rm {ph}}$. Finally, $l_{\rm {br}} \simeq l_{\rm {ph}}$ at a hydrogen concentration in the gas condensation defined as

$$N_{\rm H} \simeq \frac{\sigma_0 I_{\rm {ph}}}{\alpha \tau_0^{\rm {exc}}}\cdot$$ (16)

For example, for blob B in the vicinity of $\eta$ Car, this regime is reached at $N_{\rm H} \simeq 10^8 {-} 10^9$ cm^-3.

Figure 3: Energy levels and quantum transitions in FeII relevant to the anomalous UV FeII spectral lines.

The following two well-known processes take place in the passive HI region where the Ly$\alpha$ radiation is transferred in a medium with $\tau_0^{\rm {exc}} \gg \tau_0^{\rm {m}}$. At first, a more effective increase of the radiation spectral width (in proportion to $\sqrt {\tau_0}$) occurs as a result of the Doppler frequency redistribution in the Lorentz wings, which compensates, according to (12), for the frequency difference  $\Delta \nu$ between the Ly$\alpha$ and the FeII absorption lines. The diffusion broadening of the Ly$\alpha$ radiation spectrum in the HI region of the blob is limited exactly by the photoselective absorption of this radiation by FeII at $\lambda = 1218$ Å and intensity transformation in the blue wing of Ly$\alpha$ into an intense UV fluorescence of FeII (see the bottom right-hand part of Fig. 2). Secondly, a more effective diffusive confinement of the Ly$\alpha$ radiation takes place, resulting in a substantial increase, by at least $(\Delta \nu /\Delta \nu_{\rm D}) \simeq$ 50 times, of its spectrum-integrated intensity. However, to carry out a quantitative analysis of the radiation transfer in the HI-to-HII boundary region, in which the HI and HII concentrations change sharply over a short distance of $l_{\rm {tr}} \simeq (\sigma_{\rm {tr}}N_{\rm H})^{-1} \simeq 10^9 {-} 10^{10} ~\rm {cm} \ll D$ requires a special computer modeling, and this will be the subject of a special publication. So, in the case of blob B near $\eta$ Car, the extreme spectral brightness of the Ly$\alpha$ radiation causes a very strong excitation of FeII ions contained in the blob matter as a result of the Ly$\alpha$ quasi-coincidence ( $\Delta \lambda \simeq 3$ Å) with the absorption line of FeII in a low-lying metastable state. The subsequent extremely bright fluorescence on UV FeII lines ensures an effective radiative cooling of the blob.

To conclude this brief consideration of the evolution of the spectrum and intensity of the Ly$\alpha$ radiation, let us emphasize that Ly$\alpha$ has a very high optical density, so that one should take into account the diffusive frequency redistribution far in the Lorentz wings. At the same time the transfer of the UV FeII radiation (considered in Sect. 5 below) takes place at a more moderate optical density $\tau({\rm {UV Fe II}}) \leq 2 \times 10^2$, so that the Lorentz wings take no part in the evolution of the intensity and spectrum of UV FeII. This remark is specially made to avoid confusion between similar designations for the two different spectral lines, because in both cases (for the Ly$\alpha$ and the UV FeII line) the transfer of radiation in an optically dense medium plays an important part.