TOWARD UNDERSTANDING THE B[e] PHENOMENON. IV. MODELING OF IRAS 00470+6429

For the problem at hand, hdust must be provided with a description of the properties of the central star (effective temperature, T_eff, and radius, R) and the physical properties of the CS envelope (density and velocity distribution of the gas and the characteristics of the CS dust, such as composition and grain size distribution). In addition, two geometrical parameters must be specified: the distance to the star, d, and the inclination angle, i, which is the angle between the observer's LOS and the symmetry axis of the system. Finally, the interstellar reddening, E(B − V)_IS, must also be specified. Even though it is not an intrinsic parameter of the system, we include it as a model parameter because of the uncertainty in its determination, mainly owing to the difficulty to separate the interstellar from the CS reddening.

A long standing issue regarding the stars with the B[e] phenomenon is that very little is known about the processes that lead to the formation of their CS environment (see Zickgraf 2006 for a review). Also, in spite of the effort of numerous observers and theoreticians, the most fundamental properties of the CS envelope remain quite uncertain. In the case of supergiants B[e] (sgB[e]), a subclass of the stars with the B[e] phenomenon (Lamers et al. 1998), it is generally believed that their CS envelope is bimodal, composed of a slowly outflowing, possibly rotating disk and a fast polar wind (Zickgraf et al. 1985). Two scenarios invoked to explain this bimodality are the rotationally induced bi-stability (Lamers & Pauldrach 1991) and rotationally supported Keplerian outflows (Porter 2003).

CS envelopes of FS CMa objects are even less understood, since very little theoretical work has been done with those objects so far. For this initial theoretical effort, we adopt a model for IRAS 00470+6429 that has two key assumptions. First, we assume that the star has a luminosity similar to that of a main-sequence star with the same spectral type, i.e., the stellar luminosity is in the range of a few thousand solar luminosities. We believe this assumption is well supported by the available data (Paper III). Second, we assume that some sort of bimodal, two-component wind is present, similar in structure to the CS envelopes of sgB[e]. The results shown in Section 3 largely corroborate the above assumptions, but possible shortcomings are discussed in Section 4.

We adopt the following parametric form for the CS density and velocity structure. We assume that the underlying physical reason for the bimodal envelope is that the mass loss is, somehow, enhanced around the equator. We express the stellar mass-loss rate per unit solid angle as

$$\begin{equation} \frac{d\dot{M}(\theta)}{d\Omega } = \frac{d\dot{M}(0)}{d\Omega } \left[ 1 + A_1\sin ^m(\theta) \right], \end{equation} \tag{ 1 }$$

where θ is the colatitude measured from the pole. A₁ is a parameter that controls the ratio between the equatorial and polar mass-loss rates

$$\begin{equation} A_1 = \frac{d\dot{M}(90\mbox{$^\circ $})/d\Omega }{d\dot{M}(0)/d\Omega }-1. \end{equation} \tag{ 2 }$$

Since the mass-loss rate is enhanced at the equator, A₁ > 0. The parameter m controls how fast the mass-loss rate drops from the equator to the pole. Defining the disk opening angle, Δθ_op, as the latitude for which $d\dot{M}(\theta)/d\Omega$ has dropped to half of its equatorial value, we have

$$\begin{equation} \Delta \theta _{\rm op} = \sin ^{-1} \left[ \left(\frac{A_1-1}{2A_1} \right)^{1/m} \right]. \end{equation} \tag{ 3 }$$

For the velocity structure, we assume that the gas motion is purely radial, with a latitude-dependent radial velocity given by

$$\begin{equation} v_r(r,\theta) = v_0 + [v_\infty (\theta)-v_0](1-R/r)^{\beta (\theta)}, \end{equation} \tag{ 4 }$$

where r is the distance to the center of the star. For a given θ, Equation (4) is a standard β-law for radiatively driven winds (Lamers & Cassinelli 1999). We assume that both the wind terminal velocity, v_∞, and the acceleration parameter, β, are functions of θ, and have the same latitudinal dependence as the mass loss. Thus,

$$\begin{equation} v_\infty (\theta) = v_\infty (0) [ 1+A_2\sin ^m(\theta)], \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \beta (\theta) = \beta (0) [ 1+ A_3 \sin ^m(\theta) ]. \end{equation} \tag{ 6 }$$

As before, A₂ and A₃ control the ratio between the values of each quantity at the equator and at the pole,

$$\begin{equation} A_2 = \frac{v_\infty (90\mbox{$^\circ $})}{v_\infty (0)}-1, \end{equation} \tag{ 7 }$$

and

$$\begin{equation} A_3 = \frac{\beta (90\mbox{$^\circ $})}{\beta (0)}-1. \end{equation} \tag{ 8 }$$

Since the terminal velocity at the equator is smaller than at the pole, A₂ < 0.

Assuming radial outflow, the mass continuity equation is,

$$\begin{equation} \frac{d\dot{M}(\theta)}{d\Omega } = r^2 \rho (r,\theta) v(r,\theta), \end{equation} \tag{ 9 }$$

so we can write the CS gas mass density distribution as

$$\begin{equation} \rho (r,\theta) = \frac{d\dot{M}(\theta)/d\Omega }{r^2 v(r,\theta) }. \end{equation} \tag{ 10 }$$

Having chosen a prescription for the CS gas, we must now provide a site for dust formation and describe the properties of the dust grains that are formed. In an initial study of the continuum emission of sgB[e], Porter (2003) used two criteria to determine the dust formation site. In his models, dust was formed where the gas temperature, which he assumed to be an ad hoc function of the distance from the star, was lower than a given value (≈1500 K) and the gas density was larger than a critical value.

In our models, the coolest part of the envelope is in the denser equatorial regions, an expected result based on previous works (e.g., Zsargó et al. 2008; Carciofi & Bjorkman 2006; see Figure 5 below). However, even in the midplane the gas temperatures never fall below about 4000 K. This temperature is much larger than the temperature needed for dust grain condensation to occur, which is thought to be below about 1500 K (Porter 2003). It is likely that the gas temperatures predicted by hdust are overestimated, because no cooling by helium or other metals is included. However, Zsargó et al. (2008), in a study of the properties of the CS envelope of sgB[e], found that the temperature never falls below 2000–3000 K, even though they used more realistic cooling terms. These authors argue that a possible mechanism for further cooling the gas is adiabatic cooling, or an even more complex chemical composition.

Because of the above, it is clear that the gas temperature cannot be used in our models to trace the dust formation site and we use, for this purpose, a different approach than the one used by Porter (2003). We define the dust formation site using two complementary criteria.

If dust is to be present at a given location of the CS envelope,

• 1.  
  the equilibrium temperature of the dust grains (not the gas temperature) must be smaller than a given grain destruction temperature, T_destruction;
• 2.  
  furthermore, the gas density must be larger than a critical value, such that grain growth can occur.

The first criterion was implemented in hdust using a very simple procedure. For a given model, we start from a guess of the distance from the star beyond which the dust grains begin to condensate. During the iterative process used for determining the temperatures, whenever the dust temperature in a given cell becomes larger than T_destruction, the code removes the dust from that particular cell. Conversely, if in a given dust-free cell the temperature of the dust, if present, would have been smaller than the destruction temperature, then that cell becomes a dusty cell. This simple procedure ensures a self-consistent determination of the location of the dust destruction radius as a function of latitude. Note that the dust destruction radius is highly model-dependent: it is mainly controlled by the gas opacity in the inner part of the envelope and the dust model used.

The second criterion for defining the dust formation site follows a simple argument outlined by Gail & Sedlmayr (1988) and adopted by Porter (2003). The idea is that a critical gas number density for the dust formation can be derived by comparing two timescales, the timescale for chemical reactions, τ_ch, and the timescale for the gas expansion, τ_exp. Grain formation and growth occurs if τ_ch < τ_exp. Gail & Sedlmayr (1988) found that this occurs if the number density of the growth species (silicon, in this case) is larger than the following critical value

$$\begin{equation} n_{\rm Si}(r,\theta) = \frac{v_r(r,\theta)}{10^{-16} r v_{\rm rel}(r,\theta) } \;, \end{equation} \tag{ 11 }$$

where v_rel is the relative velocity between the dust grain and an atom or molecule, which is of the order of the sound speed and, thus, position-dependent. Since n_Si of Equation (11) was defined from simple physical arguments, and, therefore, is probably just a rough estimate of the critical density, we introduce a scaling factor, η, so that the condition for dust formation becomes

$$\begin{equation} \frac{\epsilon \rho (r,\theta)}{\mu m_{\rm H}} \le \eta n_{\rm Si}(r,\theta)\;, \end{equation} \tag{ 12 }$$

where m_H is the mass of the H atom and μ is the gas molecular weight, which is typically about 0.6 for an ionized gas with solar abundance. is the relative abundance in number of Si to H, for which we adopt the solar value 3.6 × 10⁻⁵. The IR excess is a measure of the amount of dust that is formed in the CS envelope. Hence, the value of η can be found by matching the IR excess. The second criterion for the spatial distribution of the CS dust effectively confines the dust grains to the dense and slow equatorial regions.

Finally, given a site for the dust in the CS envelope, we need to describe the properties of the dust grains, which requires several additional parameters. We assume that at each location of the envelope where dust can form, a certain fraction of the gas mass is converted into dust. This fraction, known as the gas-to-dust ratio, was assumed to be 200. We further assume that grains are spherical and their composition is oxygen-based, i.e., the dust is mainly comprised of silicates. This assumption is supported by evolutionary arguments, since carbon is depleted due to the CNO processing cycle in massive stars, therefore ruling out carbon-based grains, such as graphite, for the object in question. Furthermore, IR spectra of nearly 20 FS CMa objects obtained with the Spitzer Space Telescope (IRAS00470+6429 was not observed due to the presence of a bright nearby IR source) show silicate emission features (Miroshnichenko et al. 2008).

For this initial analysis, we adopt an MRN law for the distribution of grain sizes (Mathis et al. 1977), according to which

$$\begin{equation} f(a) = C a^n\;, \end{equation} \tag{ 13 }$$

where f is the fractional number of dust grains with radius a and C is a normalization constant. In the standard MRN law, n = −3.5 and the distribution is confined between a_min = 0.05 μm and a_max = 0.25 μm. In our models, we let both a_min and a_max to be free parameters in order to investigate the possible existence of large dust grains (Paper III). Given a dust mass density and f(a), we further need the average density of the grain material, ρ_dust, in order to calculate the number density of dust grains. Since we do not know how fluffy the grains are, we let ρ_dust be a free parameter. We note, however, that ρ_dust and the gas-to-dust ratio are not independent parameters, since both are contained in the dust optical depth, the one quantity that effectively controls the radiative transfer in the CS dust shell.

A few words are needed about the grid structure we adopted for the current problem. In hdust, the CS envelope is divided into N_r radial cells, N_μ colatitude cells and N_ϕ azimuth cells, and it is assumed that the state variables of the CS material (temperature and level populations) are constant in each cell.

The density law of Equation (4) can present difficulties for a radiative transfer solver, because it has a large radial gradient close to the star and it may have a sharp enhancement at the equator, depending on the adopted value of m. For the radial spacing we adopt cells sizes such that the radial electron scattering optical depth is the same for all cells, which results in very small cells close to the star and large cells in the outer envelope. This ensures that the inner regions of the envelope are sampled adequately. For the colatitude bins, we define the cell sizes is such a way that from the equator to the pole, along a ray with constant radius, the density at the center of consecutive cells falls in steps of 2/N_μ. Therefore, when m is large, which corresponds to a disk with small opening angle, the cells are small at the equator and large at the pole.

Two competing issues must be taken into account when choosing the values of N_r and N_μ. Since hdust employs a statistical method for sampling cell-dependent quantities such as radiative heating (see Carciofi & Bjorkman 2006), the larger the number of cells the larger the computational effort necessary to reach a given sampling error in the rates. On the other hand, the grid structure must be fine enough to correctly describe the adopted density and to resolve possible regions of ionization changes, where a transition from ionized to neutral hydrogen occurs. The minimum number of cells necessary for a given problem can be estimated by running models with an increasing number of cells. If the cell spacing is properly defined, there will be a certain number of cells beyond which no modification of the solution is found by further refining the grid. We found that N_r = 100 (60 for the gas shell and 40 for the dusty shell) and N_μ = 40 are a good compromise between speed and accuracy. For the current problem the number of azimuth cells was set to 1 because axisymmetry was assumed.

The adopted model described above has many parameters but for this initial study we chose to let just part of them be free parameters, while keeping the rest fixed. We chose as fixed parameters the ones which can either be sufficiently well constrained by a model-independent analysis, or are associated with the basic model assumptions defined in Section 2.1. Examples of fixed parameters are the stellar characteristics and the dust composition.

Free parameters, on the other hand, are the ones about which no information can be readily obtained from the observations and, therefore, require a detailed modeling. The main free parameters in our study describe the density and velocity of the CS gas and the latitude-dependent stellar mass-loss rate. We believe that this approach is appropriate for this initial study, since it allows us to focus on the least known properties of the CS envelope. A list of model parameters for IRAS 00470+6429 is given in Table 1.

Table 1. Model Parameters for IRAS 00470+6429

Parameter	Value	Type
Stellar Parameters
Spectral type	B2V	Fixed
R	6 R☉	Fixed
Teff	20, 000 K	Fixed
L	5100 L☉	Fixed
Geometrical Parameters
i	84°	Free
d	1100 ± 100 pc	Free
Wind Parameters
$d\dot{M}(90\mbox{$^\circ $})/d\Omega$	(1.4–1.6) × 10−8 M☉ yr−1 sr−1	Calculated
$d\dot{M}(0)/d\Omega$	<3 × 10−9 M☉ yr−1 sr−1	Free
A1	>19	Free
v∞(90°)	24 km s−1	Calculated
v∞(0)	1200 km s−1	Fixed
A2	−0.98	Free
β(0)	4	Free
A3	0	Free
m	92	Free
Rw	≳105 R☉	Fixed
Common Dust Parameters
Composition	Amorphous silicate a	Fixed
n	−3.5	Fixed
Gas-to-dust ratio	200	Fixed
Dust Model 1
amin–amax	0.05–0.25 μm	Fixed
ρdust	1.5 g cm−3	Free
Tdestruction	1500 K	Free
Rdestruction	540 R	Calculated
η	0.02	Free
E(B − V)IS	1.1	Free
Dust Model 2
amin–amax	0.05–0.10 μm	Fixed
ρdust	0.15 g cm−3	Free
Tdestruction	1500 K	Free
Rdestruction	540 R	Calculated
η	0.02	Free
E(B − V)IS	1.1	Free
Dust Model 3
amin–amax	1–50 μm	Fixed
ρdust	0.1 g cm−3	Free
Tdestruction	1600 K	Free
Rdestruction	79 R	Calculated
η	0.02	Free
E(B − V)IS	1.2	Free

Note. ^aThe optical constants of Ossenkopt et al. (1992) were used.

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The stellar parameters are all fixed in this study. As mentioned above, one of our key assumptions is that IRAS 00470+6429 has a typical luminosity of a normal main-sequence B star. Following the analysis in Paper III, we chose a spectral type of B2V for the central star. This sets the radius to ≈6 R_☉, the effective temperature to ≈20, 000 K and the luminosity to ≈5100 L_☉ (Harmanec 1988). We investigated the effects of changing T_eff and L, and we found that varying T_eff by ≈10% (and thus L by ≈40%) has only small effects on the results. For this initial study we considered a static, non-rotating central star. This is not a limitation of hdust (see, e.g., Carciofi et al. 2008) but, instead, reflects the fact that the observed line profile is controlled mostly by the CS gas (Section 3.1).

For the stellar spectrum, we used the model atmosphere of Kurucz (1994) for T_eff = 20,000 K and log g = 4.⁴ To describe the photospheric specific intensity as a function of direction we adopted the limb darkening models of Claret (2000).

The geometrical parameters (i, d, and interstellar reddening) are all free parameters. Nothing is known about i, but our previous analysis allows us to impose some limits on d and E(B − V)_IS. Following Paper III, we adopt d in the range of 1000–2000 pc and E(B − V)_IS in the range 0.8–1.2 mag. Given a model for the CS gas and dust, d and E(B − V)_IS are obtained by fitting the observed flux level and shape of the visible SED, respectively.

Some information about the gas dynamics can be readily obtained by analyzing the emission line profiles. For instance, the broad blue absorption profile of the H i lines suggests that the absorbing material has large velocities in the LOS and the blue edge of the absorption profile indicates the terminal velocity of the fast wind. For v_∞(0), we adopted a value of 1200 km s⁻¹ and set this as a fixed parameter in our modeling. We note that this value of the terminal velocity is consistent with what is expected for main-sequence B stars (Grady et al. 1987).

In addition, the relatively narrow peaks of the Balmer lines suggest that the bulk of the emitting material has small velocity along the LOS. Supposing that most of the H i line emission comes from the equatorial region (an assumption corroborated by our detailed calculations), this suggests that the equatorial outflow velocities are very small compared to the velocities in the polar region.

For the present work, hundreds of models were run covering a wide range of model parameters. Before presenting our solution, it is useful to discuss some trends that became clear once a large range in the parameter space was covered.

From our results we can safely rule out both large (Δθ_op ≳ 15° or m ≲ 15) and small (Δθ_op ≲ 5° or m ≳ 180) opening angles for the outflowing disk around IRAS 00470+6429. The opening angle, together with the equatorial mass-loss rate, controls the strength of the H i line emission. For large opening angles, relatively low mass-loss rates are necessary to reproduce the observed emission strength, i.e., the larger the opening angle, the lower the equatorial densities for a given emission strength. Also, the lower the density, the farther away from the star is the dust destruction surface and, therefore, the lower the density in the dust condensation zone. The two effects combined have the consequence that models with Δθ_op ≳ 15° have densities below the critical density necessary to form dust (Equation (12)).

Models with small opening angles require high equatorial densities to reproduce the line emission and, therefore, can form dust. However, as we shall see in Section 3.2, in our models dust is typically confined within a wedge about one opening angle away from the equator. This is because the density drops quickly toward the pole and soon falls below the critical density for grain formation. If the opening angle is too small (Δθ_op ≲ 5°), the dust will encompass too small a solid angle for the radiation that comes from the inner envelope (both stellar radiation and reprocessed radiation by the CS gas) and, as a consequence, will reprocess little radiation. This is a geometrical effect and is true even for very large dust opacities. Therefore, small opening angles can be ruled out on the grounds that they produce too small IR excess.

Another trend that became evident is that $d\dot{M}(0)/d\Omega$ cannot be larger than about 3 × 10⁻⁹ M_☉ yr⁻¹ sr⁻¹. A large density in the poles implies that the polar regions would contribute significantly to the overall line emission. Since the polar material has a large terminal speed, the polar emission would appear as a broad red emission wing, which is not observed. The polar emission begins to alter the emission profile significantly if mass-loss rate exceeds the quoted value.

This upper limit for $d\dot{M}(0)/d\Omega$ has an important implication on the possible range of inclination angles. Since the polar regions must be relatively optically thin, they cannot account for the broad and deep absorption component observed in the Balmer lines (see, for instance, Figure 3 of Paper III). Models with 0° < i ≲ 75° do not produce the necessary absorption and, therefore, this range of inclination angles can be excluded. Besides this interval, our models can also exclude inclination angles too close to edge-on (i ≈ 90°). In all our models, the radial optical depth along the equator is very large (τ ≈ 10⁶ in the Lyman continuum and ≈10² in the Balmer continuum). As discussed in Paper III, some features of the stellar photosphere are observed, which, therefore, excludes edge-on viewing.

Our best-fit SED, Hα, Hβ, Hγ, and Hδ line profiles for the parameters listed in Table 1 are compared to the observations in Figures 1–3. The CS gas density and velocity profiles along several latitudes are shown in Figure 4.

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As said above, the opening angle and the equatorial mass-loss rate are the primary parameters that control the strength of the line emission for a given i. Our best-fit value for the opening angle is 7°, which corresponds to m = 92. For the equatorial mass-loss rate our best-fit value is in the range $d\dot{M}(90\mbox{$^\circ $})/d\Omega =(1.4$–1.6) × 10⁻⁸ M_☉ yr⁻¹ sr⁻¹. The lower limit corresponds to the best-fit value for the line emission observed on 2006 December 13, whereas the upper limit corresponds to the 2006 December 27 observations, which have substantially stronger emission (see Figure 2). This result indicates that IRAS 00470+6429 presents significant changes in the mass-loss rate in very short timescales (two weeks).

Both m and $d\dot{M}(90\mbox{$^\circ $})/d\Omega$ are well-constrained parameters in our models, but we note that reasonable fits of the emission strength could also be obtained with slightly larger opening angles (say, by 1° or 2°) and slightly smaller (a few tens of per cent) $d\dot{M}(90\mbox{$^\circ $})/d\Omega$. The opposite is also true, i.e., similar fits can be obtained by slightly smaller opening angles and slightly larger mass-loss rates.

The opening angle and the mass-loss rate also control the SED shape. Their effects on the SED are complex and depends significantly on the adopted i. Dust also plays a major role in defining the shape of the SED, not only in the IR but also on the visible spectral region provided that the LOS of the observer crosses dusty material. For this reason, we postpone the discussion of the model SED and its comparison to observations to Section 3.2.

Contrary to the equatorial mass-loss rate, the polar mass-loss rate is rather poorly constrained. As explained above, an upper limit of about $d\dot{M}(0)/d\Omega =3\times 10^{-9}\,M_\odot \rm \;yr^{-1}\;sr^{-1}$ could be set on the basis of the absence of a broad emission component. Given the best-fit value of $d\dot{M}(90\mbox{$^\circ $})/d\Omega$, above, this corresponds to a lower limit of A₁ = 19, which means that the ratio between the equatorial and polar mass-loss rates is at least as high as 20, but our models do not exclude much larger values of A₁. The reason why our models do not constrain well the polar densities is that the H i recombination lines are not suitable probes of low density regions. UV resonance lines of high ionization species are the ideal probes of those regions, but unfortunately no UV data are available for this object.

It should be noted that even though the lower limit for the ratio between the equatorial and polar mass-loss rates is 20, the ratio of the actual densities is much larger. As shown in Figure 4 this ratio is about 200 or larger, depending on the adopted value for A₁.

The shape of the absorption component of the H i lines is highly variable, with the blue absorption edge moving from ≈400 to ≈800 km s⁻¹ on very short timescales (see Figure 3 of Paper III). For this reason, it is difficult to reproduce the detailed shape of the absorption profile. The high variability of the observed line profiles suggests variable mass loss on short timescales. For instance, our models reproduce very well the Hβ absorption profile of 2006 December 27, but not the profile that was observed two weeks earlier. This may indicate that not only the mass loss varies but also the kinematic properties of the CS envelope change. The depth of the absorption component, contrary to its shape, is remarkably constant. It depends on two competing processes: the absorption of resonant radiation by H i atoms in a given excitation state and the emission filling-in, i.e., the generation of new photons both by line emission and bound–free and free–free continuous emission that partially fills in the absorption component. The filling-in by continuous emission is particularly important because of the very large densities close to the star (see Figure 4). As shown in Figure 3, our models also fit reasonably well the depth of the absorption component for other lines of the Balmer series, which is not a trivial result.

Another important parameter is the value of β that tells how fast the wind accelerates. We found that the best models for the line profiles are the ones with large values of β for all latitudes. Low values of β (say, β ≲ 2) would result in a too broad emission component, due to the large velocities in the inner part of the wind, and also a sharp absorption edge at high velocities. Both these features are not observed. Even though in our models it was possible to have different values of β for different latitudes, our best-fit value was for β = 4 and A₃ = 0, i.e., a constant value of β. Again, this is a somewhat uncertain result. For instance, a model with A₃ = 1/3, for which β(0) = 3, is also a good model and cannot be ruled out from the present analysis.

We end this section discussing the temperature and ionization structure of the CS gas. Figure 5 shows the midplane temperature and the fractional number of H atoms in the ground level as a function of the distance from the star. Clearly, the gas is highly non-isothermal, a result already obtained by Drew (1985) and, more recently, by of Carciofi & Bjorkman (2006) and Zsargó et al. (2008). The temperature reaches a minimum of about 5000 K at a distance of around 5 R and then steadily rises to an equilibrium temperature of about 12,000 K for r ≳ 100 R. This rise in the temperature is probably a result of our unrealistic chemical composition. The inclusion of the cooling terms by other elements will result in a different temperature profile (see, for instance, Drew 1985).

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The gas is essentially ionized everywhere in the envelope. The maximum value for the relative fraction of H i and H ii is never larger than 10%. This result agrees with the recent work by Zsargó et al. (2008), who studied the envelopes of sgB[e], and found that H is generally ionized, although this may depend on both the effective temperature of the central star and on the density scale of the wind.

Many processes are involved in defining the shape of the SED, and both gas and dust can play an important role depending on the wavelength range considered. In the visible region, the dominant processes are continuum absorption by both gas and dust, and emission by continuum processes in the gas (bound–free and free–free emission). The SED is controlled by both gas and dust emission at wavelengths of ≲2 μm in the near-IR, but the dust becomes the major player at longer wavelengths.

For this initial study of the dusty content of IRAS 00470+6429, three different models for the CS dust were investigated. For all models the grain size distribution has the form of Equation (13) with n = −3.5, and each model explores different values for the minimum and maximum sizes of the distribution. The first model studied corresponds to the standard MRN model for interstellar dust grains (Mathis et al. 1977), for which a_min = 0.05 and a_max = 0.25 μm. The second model employs a much larger upper limit for the distribution (a_max = 10 μm), whereas the third model has very large grains only (a_min = 1 and a_max = 50 μm). The three dust models are listed in Table 1.

At this point, it is useful to consider what controls the shape and level of the IR excess arising from dust thermal emission. The shape of the emitted spectrum is controlled mainly by the dust absorption coefficient in the IR domain, which is a function of chemical composition and size of the grain. Other factors controlling the shape are the temperature and density structure. The level of the IR excess depends on the fraction of the radiation coming from the central regions that is reprocessed by the dust grains. Since in our model the dust is confined to the dense equatorial outflow, the amount of reprocessing will be a function of the disk opening angle, Δθ_op, and the dust optical depth, which is set by the gas-do-dust ratio and ρ_dust (the bulk density of the grain material).

The role of η in defining the dust condensation site deserves a more careful examination. Recall that the dust condensation site is defined by two conditions: the density of the growing material (Si) must satisfy Equation (12), in which η is a scaling factor, and the dust equilibrium temperature must be smaller than a given value that corresponds to the temperature above which the dust grains are destroyed (Section 2.2). In Figure 6, we plot the equilibrium temperature of the smallest grains considered by us (a = 0.05 μm). As we see below, the smallest grains are always hotter than the larger grains, and, therefore, are used as the tracers of the dust formation site. The plot represents a cut through the envelope such that the direction of the symmetry axis is vertical and the direction of the midplane is horizontal. The latitude-dependent inner dust radius represents our solution for the dust destruction surface. Near the midplane, where the densities are larger and the material is more shielded from the central radiation, the dust destruction surface is closer to the star, and this surface moves outward for higher latitudes.

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The solid white lines in Figure 6 mark the points of the wind for which the right side of Equation (12) is equal to the left side, and it encloses the region around the equator which have densities larger than the critical density for grain formation (to draw this line we used η = 0.02, see below). According to our two assumptions, the site for dust formation is the region of the wind both beyond the destruction surface and inside the region enclosed by the solid white lines. The dust condensation site has an angular size of 2Δθ_dust, the value of which is controlled by the parameter η: the larger η the smaller Δθ_dust.

It is important to note that the site for dust formation does not necessarily coincide with the site where dust actually is, because the dust formed in the innermost region of the condensation site will be carried outward along the wind streamlines. Therefore, dust that was formed at high latitudes will be dragged to regions where the density is much too low to form dust. This mechanism allows dust to exist at high latitudes in the outer envelope, where the density is lower than the critical value. Therefore, in our models dust is confined within a wedge with a (half) opening angle of Δθ_dust. Δθ_dust is a function of both Δθ_op and η.

Figure 1 compares the best-fit SED obtained for each of the three dust models tested. We did not attempt to fit the IRAS fluxes, but focus instead on the Berkeley Automated Supernova Search (BASS) and MSX data, since the large aperture of IRAS has probably lead to some degree of contamination by the interstellar cirrus. Also, we recall, from the data description in Paper III, that the NIRIS data have large errors of up to 20% in the photometric calibration.

The MRN model (model 1, top panel of Figure 1) does not fit the data well. The integrated IR flux is in rough agreement with the observations, which means that the model does reprocess the correct fraction of the bolometric stellar luminosity. It reproduces the SED reasonably well in the 2–3 μm range, but fails to reproduce the slope of the SED for larger wavelengths. The model with a large upper limit for the grain size (model 2, middle panel of Figure 1) fits both the level and shape of the SED in the 2–7 μm range. The reason why model 2 fits better the shape of the near-IR SED than model 1 resides in an interesting physical effect: differently sized grains have different equilibrium temperatures (Carciofi et al. 2004). Figure 7 shows the temperature of the smallest and largest grains of dust models 1 and 2. For model 1, the span in grain sizes is not very large and the difference between the coolest and hottest grains is, at most, 300 K. For model 2, however, we find that the very large grains are much cooler then the small grains. This explains the difference in the slope of the SED: model 2, having on average a lower dust equilibrium temperature, has proportionally larger emission at longer wavelengths.

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The 10 μm region of the spectrum presents a big challenge for models 1 and 2, since for both the 9.7 μm silicate emission feature is quite strong, in sharp contrast to the observations. This situation is much improved for the model with only large grains (model 3, bottom panel of Figure 1). This model reproduces reasonably well the shape of the thermal IR emission in the entire 2–13 μm range. As discussed above, the shape of the IR thermal continuum is controlled mainly by the dust absorption opacity, which is a function of grain size and composition. Under our assumption that the grains in IRAS 00470+6429 are oxygen-based, the observed IR excess seems to indicate that large grains (a ≳ 1 μm) are the dominant constituents of the CS dust.

The large difference between the equilibrium temperatures of small and large grains have another consequence for the models. Recall that one of the conditions for determining the dust condensation site is that the temperature of the hottest grains cannot exceed T_destruction. For models 1 and 2 this condition is satisfied only for r ≳ 540 R, but for model 3 the dust destruction surface is much closer to the star r ≳ 79 R.

Another important difference between models with small and large grains lies in the bulk density of the grain material (ρ_dust). At a given location of the envelope, a constant fraction of the gas is converted into dust. Since the mass of the grain increases as a³ and its cross section as ∼ a², we have to decrease the bulk density of the material so that the dust optical depth remains approximately constant. Therefore, our modeling for the IR excess of IRAS 00470+6429 suggests that the dust grains are both large (a ≳ 1 μm) and fluffy (ρ_dust ∼ 0.1 g cm⁻³).

Finally, the three models also reproduce well the visible SED (Figure 1), but require different amounts of interstellar reddening: for the models with small grains, E(B − V)_IS = 1.1, and for the model with large grains, E(B − V)_IS = 1.2. Recall that E(B − V)_IS is a free parameter in the modeling and was set by fitting the visible SED. The different values of E(B − V)_IS can be easily understood by considering that the CS extinction of model 3 is essentially gray in the visible and for this reason its intrinsic CS reddening is smaller than the models with small grains.

The extent to which the CS disk modifies the stellar radiation can be assessed by comparing the model SED with the unattenuated stellar radiation (dashed line in Figure 1). Since the star is viewed close to edge-on, this causes the extinction of UV and visible radiation that is reprocessed into longer wavelengths. Most of the reprocessed radiation escapes in the polar direction, due to the fact that the vertical optical depth is much smaller than the radial optical depth, but part of it escapes radially. The U-band flux is particularly important for the SED fitting. Our models predict that when the object is viewed pole-on, the large bound–free excess produced by the inner envelope completely fills in the photospheric Balmer jump, thus producing a power-law SED up to the UV region. The fact that the Balmer jump is present is another evidence in support of larger inclination angles (Section 3.1).