Axi-symmetric Models of B[e] Supergiants: I. The Effective Temperature and Mass-loss Dependence of the Hydrogen and Helium Ionization Structure

Abstract

We calculate the hydrogen and helium ionization in B[e] envelopes and explore their dependence on mass-loss and effective temperature. We also present simulated observations of the Halpha emission line and the C IV 1550 doublet, and study their behavior. This paper reports our first results in an ongoing study of B[e] supergiants, and provides a glimpse on the ionization of the most important elements in self-consistent numerical simulations. Our newly developed 2D stellar atmosphere code, ASTAROTH, was used for the numerical simulations. The code self-consistently solves for the continuum radiation, non-LTE level populations, and electron temperature in axi-symmetric stellar envelopes. Observed profiles were calculated by an auxiliary program developed separately from ASTAROTH. In all but one of our models, H remained fully ionized. Due to ionizations from excited states it is much more difficult to get a H neutral disk than indicated by previous analytical calculations. Near the poles, the ionization is high in all models, while helium recombined in the equatorial regions for all but our lowest mass-loss rate. Although the model parameters were not adjusted to provide fits to any particular star, the theoretical profiles show some features seen in the profiles of R126. These include the partially resolved double peaked profile of Halpha, and the weak emission associated with the UV C IV resonance line. Comment: Accepted for publication in A&A

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arXiv:0712.0870v1 [astro-ph] 6 Dec 2007
Astronomy & Astrophysics manuscript no. 8293ms
February 2, 2008
c ? ESO 2008
Axi-symmetric Models of B[e] Supergiants:
I. The Effective Temperature and Mass-loss Dependence of the
Hydrogen and Helium Ionization Structure
J. Zsarg´ o1, D. J. Hillier1, and L. N. Georgiev2
1Dept. of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA
2Instituto de Astronomia, Universidad Nacional Autonoma de Mexico (UNAM), CD. Universitaria, Apartado Postal
70-264, 04510, M´ exico DF, M´ exico
Received DD MM 200Y/ Accepted DD MM 200Y
ABSTRACT
Aims. We calculate the hydrogen and helium ionization in B[e] envelopes and explore their dependence on mass-loss and
effective temperature. We also present simulated observations of the Hα emission line and the CIV λλ1550 doublet,
and study their behavior. This paper reports our first results in an ongoing study of B[e] supergiants, and provides a
glimpse on the ionization of the most important elements in self-consistent numerical simulations.
Methods. Our newly developed 2D stellar atmosphere code, ASTAROTH, was used for the numerical simulations.
The code self-consistently solves for the continuum radiation, non-LTE level populations, and electron temperature in
axi-symmetric stellar envelopes. Observed profiles were calculated by an auxiliary program developed separately from
ASTAROTH.
Results. In all but one of our models, H remained fully ionized — only for˙M>10−5M⊙ yr−1and Teff ≤18,000 K did
we obtain a neutral H disk, and then only for radii beyond 3R∗. Due to ionizations from excited states it is much more
difficult to get a H neutral disk than indicated by previous analytical calculations. Near the poles, the ionization is high
in all models, while helium recombined in the equatorial regions for all but our lowest mass-loss rate (10−6M⊙ yr−1).
Although the model parameters were not adjusted to provide fits to any particular star, the theoretical profiles show
some features seen in the profiles of R126. These include the partially resolved double peaked profile of Hα, and the
weak emission associated with the UV Civ resonance line.
Key words. Physical data and processes: Radiative transfer – Stars: early-type – Stars: atmospheres – Stars: mass-loss
1. Introduction
Massive stars are very important constituents of the
Universe despite their low share of galactic masses. They
are the primary sources of elements heavier than Li and
power the internal evolution of galaxies by their radiation,
winds, and explosions. Many of these stars, and their im-
mediate environment, cannot be fully understood in the
context of plane-parallel or spherical models, and their mul-
tidimensional nature has to be taken into account if their
evolution or physical state is to be adequately described.
The origin of their asphericity is either fast rotation, the
presence of a dynamically important magnetic field, or their
interaction with a companion.
Three groups of such stars are particularly relevant for
this paper. These are the classical Be stars, the Luminous
Blue Variables (LBV), and the B[e] supergiants (sgB[e]),
all of which share many similar characteristics. The 2D
or 3D nature of these objects was recognized through
both observations and theoretical calculations. For ex-
ample, the presence of disks around classical Be stars
was inferred from line modeling and polarimetric stud-
ies (Poeckert & Marlborough 1978a,b), and has been con-
firmed by interferometric observations (Stee et al. 1995;
Quirrenbach et al. 1997). Similarly, a growing body of
evidence suggests that the LBV phenomenon includes
2D or 3D processes. For example, η Car is a binary
(Damineli et al. 2000) and the Homunculus nebula around
it is the most cited example of a bipolar outflow. The as-
pherical envelope of η Car is not unique among LBVs;
Groh et al. (2006) found that AG Car is a fast rotator, and
hence may have a latitude dependent wind. The relevance
of the binarity and fast rotation in the LBV phenomenon
in general is not yet clear and is the subject of vigorous
research (e.g., Martin et al. 2006; Nielsen et al. 2007).
B[e] stars, or rather the stars that show B[e] phe-
nomenon (Lamers et al. 1998), are characterized by strong
broad H Balmer emission lines (as are the classical Be
stars) and by the presence of narrow permitted and forbid-
den emission lines from low-ionization species, like FeII,
[FeII], or [OI]. B[e] stars also show strong near/mid-IR ex-
cess that is attributed to hot cirumstellar dust. Interested
readers should refer to Zickgraf et al. (1985, 1986, 1992);
Miroshnichenko et al. (2005) and references therein for
further information on the B[e] phenomenon. A particu-
larly interesting subclass of the B[e] stars are the sgB[e]-
s (Lamers et al. 1998). These are single B supergiants

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2 Zsarg´ o et al.: Ionization in the wind of sgB[e] stars
1984). Their location on the HR diagram may suggest a
link to the more massive LBVs or, alternatively, they may
represent a second evolutionary path between Of and Wolf-
Rayet (W-R) stars (Schulte-Ladbeck 1998; Zickgraf 1998).
The envelope of sgB[e] stars is thought to be lat-
itude dependent, with a normal B supergiant wind
at the pole and a dense and slowly moving equato-
rial flow (Zickgraf et al. 1985, 1986). Polarimetric ob-
servations by Magalhaes (1992); Oudmaijer et al. (1998);
Melgarejo et al. (2001) revealed that sgB[e]-s have strong
intrinsic polarization consistent with this picture. The most
widely accepted theory for the bi-modal nature of the enve-
lope is the rotationally induced bi-stability. Around Teff=
20,000 – 25,000 K the characteristics of the line-driven
wind abruptly change due to the recombination of FeIV to
FeIII (Pauldrach & Puls 1990; Lamers & Pauldrach 1991).
If the sgB[e] stars are fast rotators, as suggested, the en-
suing gravity-darkening can place the equatorial and po-
lar regions on the opposite sides of the bistability, hence
providing a mechanism to produce the bi-modal envelope.
However, the bi-stability model is not without its weak-
nesses. For example, gravity darkening reduces the flux
at the equator, potentially inhibiting any equatorial en-
hanced mass-flux. It is also difficult to maintain multi-
ple scattering in the equator when the photons can eas-
ily escape in the polar direction. (see discussion in §7).
Consequently, other sgB[e] models have been proposed, like
the presence of a Keplerian disk similar to those around
classical Be stars. Further, disk winds as proposed by
Oudmaijer et al. (1998), the traditional wind compressed
disk model of Bjorkman & Cassinelli (1993), and even the
magnetically confined disk models (ud-Doula & Owocki
2002; Owocki & ud-Doula 2004) cannot be excluded. So
far, theoretical efforts have primarily focused on under-
standing the hydrodynamic structure in the context of
the bi-stability model, and trying to incorporate the bi-
stability jump into the Castor et al. (1975) formalism (e.g.,
Pelupessy et al. 2000; Cur´ e et al. 2005).
There have only been a few efforts to understand the
ionization structure of the wind and to perform spectral
analysis of B[e] stars. Semi-analytical calculations of hydro-
gen and helium ionization were done by Kraus & Lamers
(2003) and Kraus (2006), and they found the equatorial
disk essentially neutral in these species all the way down
to the stellar surface. Porter (2003) attempted to repro-
duce the optical to infra-red continuum of R 126 (prototype
sgB[e] in the LMC) by using bi-stability and Keplerian vis-
cous disk models. Both of them reproduced the optical and
near-IR emission, but they failed to account for the dust
emission by an order of magnitude. Recently, Kraus et al.
(2007) proposed that the dust and optical/near-IR contin-
uum emission can be reconciled if the free-free and bound-
free emission originates from the polar wind rather than
the equatorial disk.
A self-consistent spectral analysis of a stellar envelope
needs to solve the coupled equations for the radiation field
and level populations. Obviously, this is a very difficult
task in a non-LTE 2D or 3D envelope and compromises
and simplifications are inevitable. Kraus & Lamers (2003)
and Kraus (2006), for example, used a simplified radiative
transfer, limited their analysis to optically thin or optically
thick cases, and neglected ionization from excited levels; all
gained by simple calculations, one needs numerical simula-
tions for a fully self-consistent spectral analysis.
To provide a tool for such studies of Be/B[e] stars, bi-
naries, and LBVs, we have been developing a code for axi-
symmetric models. So far, only basic tests and code verifi-
cation were performed (Georgiev et al. 2006; Zsarg´ o et al.
2006), but we have now reached the point where scientif-
ically meaningful simulations can be performed. The ion-
ization structure of hydrogen and helium in axi-symmetric
sgB[e] envelopes offers such an opportunity. The details of
the program are discussed in Georgiev et al. (2006) and
Zsarg´ o et al. (2006) so we only briefly describe it in §2. The
hydrodynamic structures and the atomic models that are
used in our simulations are discussed in §3; computational
issues are discussed in §4; and our results are presented in
§5. Simulated observations are shown in §6, and we draw
our conclusions in §7 and §8.
2. The Code
Our C++ code, ASTAROTH, was developed for simula-
tions of stellar envelopes. However, it is flexible enough to
be applied to any hot axi-symmetrical object with velocity
gradients (including extragalactic objects, like AGN-s).
The non-LTE level populations, the radiation field, and
the electron temperature are calculated by simultaneously
solving the equations of statistical equilibrium, radiative
transfer, and energy conservation. The short-characteristic
method (see e.g., Mihalas et al. 1978; Kunasz & Auer 1988;
Busche & Hillier 2000) is used to treat the continuum ra-
diation transfer while bound-bound transitions are treated,
for simplicity, by the Sobolev approximation. The simul-
taneous solution of the equations is found by an approx-
imate lambda iteration (Rybicki & Hummer 1991, 1992).
The code has been tested by solving 2D pure scattering
problems with grey opacity, as well as reproducing spheri-
cal symmetric models of CMFGEN (Hillier & Miller 1998),
a well-established code in stellar studies. In these tests, the
new code reproduced the reference results within a few per-
cent (see, Zsarg´ o et al. 2006; Georgiev et al. 2006).
3. Models
To simulate the hydrodynamical structure of a sgB[e] atmo-
sphere we followed the approach of Kraus & Lamers (2003),
but relaxed two of their simplifications. We used a β-law
(Castor et al. 1975) to describe the radial velocity and al-
lowed for a varying electron temperature. Our velocity field
was still simplified; only a latitude dependent radial velocity
was included and the azimuthal and latitudinal velocities
were set to zero. To simulate the bi-modal wind, we de-
scribed the radial velocity and the mass-loss per unit solid
angle by
Vr(R,θ) = V∞(θ = 0) · 10bvsinsθ·
?
1 −R∗
R
?β
(1)
and
∂2M
∂t∂Ω(θ) =∂2M
respectively. The parameters R and θ are the traditional
polar coordinates, and s controls the thickness of the equa-
∂t∂Ω(θ = 0) · 10bmsinsθ,
(2)

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Zsarg´ o et al.: Ionization in the wind of sgB[e] stars3
0 306090
θ (deg)
105
106
107
Hydrogen density (cm-3)
Fig.1. The density as a function of latitude at R= 100 R∗
in our models A and B (see Table 1). Note that the density
scale is logarithmic. The density distribution for models
C and D is similar in shape, but the values are 10 times
greater.
012345678
0
1
9.0
9.0
10.0
11.0
Fig.2. A meridional snapshot of the logarithmic hydrogen
density near the star for models A and B. The pole is to-
ward the top of the page and the equator is horizontal. The
envelope is axi-symmetric around the pole and top-bottom
symmetric around the equator. The units are in stellar ra-
dius and the thick circle at the center represents the surface
of the star. A similar snapshot for models C and D would
look identical except with larger density values. Note, that
the open contour lines near the stellar surface are artifacts
of omitting the region R<1.1 R∗from the plot which was
done for clarity.
where 10 and 100 were used. In these initial calculations
we avoided the higher values because of the need to have a
much finer spatial grid which would have substantially in-
creased the computational effort. In a future, detailed anal-
ysis of sgB[e]s, the effects of varying disk thickness will need
to be explored.
Using Eq. 2 and assuming top-bottom symmetry, the
total mass-loss rate can be calculated by
˙M = 4π ·
?
π
2
0
∂2M
∂t∂Ω(θ) · sinθdθ .
(3)
Also, because all non-radial velocities are zero, the gas den-
sity in the wind, ρ(r,θ) is given by
ρ(R,θ) =∂2M
∂t∂Ω(θ) ·
1
R2Vr(R,θ).
(4)
In our models we used bv= −2 and bm= 1 which resulted
in a 3 order of magnitude density enhancement and a 2
order of magnitude velocity decrease between the pole and
the disk (see §7 for discussion on the validity of Sobolev
approximation in the disk). The 2D density structure is
displayed in Fig 1 and Fig 2. The bi-modal structure of
Table 1. Description of the Models
Model
M∗
R∗
Teff
∂2M
∂t∂Ω(θ=0)
ABCD
30 M⊙
82 R⊙
22500 K
8×10−8
18000 K
M⊙
yr str
22500 K
8×10−7
18000 K
M⊙
yr str
∂2M
∂t∂Ω(θ=π
2)8×10−7
M⊙
yr str
8×10−6
M⊙
yr str
˙ M
V∞(θ=0)
V∞(θ=
βa
a– Power for the standard β velocity law (Castor et al. 1975).
3.3×10−6M⊙ yr−1
3.3×10−5M⊙ yr−1
2000 km s−1
20 km s−1
0.8
π
2)
There is a significant difference between our approach
and that of Kraus & Lamers (2003). They did not deal with
the photosphere, and used Kurucz (1979) spectra to sim-
ulate the stellar radiation field. ASTAROTH, similarly to
CMFGEN, solves for the unknown level populations and
ionization structure, and calculates spectra for both the
photosphere and the wind. It uses the same methods as
CMFGEN (Hillier & Miller 1998) to create an approximate
hydrostatic photosphere. In brief, the wind follows a simple
beta-like velocity law (Castor et al. 1975), and for this work
starts at a radius Rphotdefined by Vr(Rphot,θ)= 1 km s−1.
Below this radius the density follows an exponential law
and the velocity is set by the continuity equation for the
given mass-loss rate. The value of Rphotis latitude depen-
dent in the models presented here, but it was always nearly
R∗(<1-2% difference).
For the stellar parameters, listed in Table 1, we chose
values that are broadly representative of the most lumi-
nous sgB[e]-s (see, Zickgraf et al. 1985, 1986) and some
less luminous LBVs (e.g., P Cygni, Pauldrach & Puls 1990;
Drew 1985). The mass-loss range covers the upper end
of the range used in Kraus & Lamers (2003) and in-
cludes their baseline models (models A, C, and F in
Kraus & Lamers 2003). The suspected mass-loss rate of
R 126 (∼4×10−5M⊙yr−1, Bjorkman 1998) is also within
our mass-loss range. We vary the effective temperature be-
tween Teff∼22,500 K (L= 1.5×106L⊙), that of R 126, to
Teff∼ 18,000 K (L=6×105L⊙) which approximately cov-
ers the sgB[e] temperature range in the Magellanic Clouds
(see, Zickgraf et al. 1985, 1986).
Finally, the atomic model is presented in Table 2. We
used essentially H/He atmospheres with a fairly large num-
ber of levels included. A few levels and ionization states of
carbon were also included to achieve more realistic heating
and cooling terms. The fractional (number) abundances of
He and C was 0.1 and 9.82×10−4, respectively. The later
number represents the net abundance of CNO elements.
4. Computational Issues
Previous works on stellar winds and sgB[e] stars (e.g.,
Kraus & Lamers 2003) have found that ionization changes
can occur suddenly, in the form of ionization fronts. One
of our main concerns was, therefore, whether our spatial
grid would adequately resolve a potential 2D ionization

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4 Zsarg´ o et al.: Ionization in the wind of sgB[e] stars
Table 2. Atomic Model
Specie
HI
HII
HeI
HeII
HeIII
CII
CIII
CIV
CV
Number of Levels
20
1
11
20
1
9
10
5
1
semi-irregularly with the limitation that all latitude must
have the same radial grid. The only problem we encountered
was the sharp hydrogen recombination front that occurred
in model D, and the spatial grid had to be adjusted by hand
to better resolve the front (adaptive gridding capability is
not yet included in ASTAROTH). Otherwise the standard
grid was adequate for our models. Simulations on a denser
grid with 120 depth and 20 latitude points revealed no qual-
itative changes in the populations and temperatures.
The convergence was also sometimes slowed by oscil-
lations in the low radiation field/high density equatorial
disk. As the statistical equilibrium and energy conserva-
tion equations are highly nonlinear in the populations,
radiation field, and electron temperature, solution tech-
niques can provide oscillating solutions. Often the oscil-
lations arose at depths, and in populations, where changes
would have a negligible influence on the emergent spectrum.
Unfortunately, since we currently use a stopping criterion
based on the maximum population change, such oscilla-
tions can greatly increase the total computational effort. To
dampen the oscillations we linearized the statistical equi-
librium equations with populations averaged over several
previous iteration cycles. Switching to linear interpolations
of quantities, instead of the standard cubic or parabolic
approximations, also improved the convergence. The lin-
ear interpolation is an extremely well-behaved and stable
approximation and it would be the preferred method if it
provided the necessary accuracy. Unfortunately, this is not
the case for spatial grids that are practical for ASTAROTH
simulations; therefore, linear interpolation was limited to
grid-points immediately around the trouble spot.
5. Ionization and Temperature Structure
Figs. 3, 4, and 5 show the electron temperature, hydrogen,
and helium ionization structures, respectively, in our model
envelopes.
5.1. Electron Temperature
The electron temperature varies between ∼100,000 K to
∼20,000 K in the inner hydrostatic atmosphere (not shown
in Fig. 3), and it was a few 1,000 K in the outer envelope.
The behavior of Tein the wind is similar, as expected, for
all models — the equator is cooler than the pole at the
same radius. The effect of the lower luminosity in models B
and D is a shift of the overall structure closer to the stellar
surface (see Fig 3), while increasing the mass-loss results
in a thicker cool disk. It is immediately obvious from Fig 3
The temperature in our models is too hot for dust for-
mation. Nowhere in our models the temperature falls be-
low ∼1,500 K which is the upper limit for dust formation
(Porter 2003). We cannot conclude, however, that dust is
formed beyond R > 100 − 200R∗ (the outer boundary of
our models) because of the relative simplicity of our models
and the neglect of adiabatic cooling in our simulations. We
will address this question in follow-up studies with models
constrained by the observations of individual stars.
5.2. Hydrogen Ionization
Fig. 4 shows the ionized to neutral H ratio in logarithmic
scale where neutral H means the total population of all
HI levels. It was very difficult to display the wide range of
ionization levels occurring in our models. Since there were
orders of magnitude differences even within a single model,
we opted for using a separate set of contour levels for each
model.
The hydrogen ionization shows latitudinal variations
similar to those of the temperature. The equatorial region
is more neutral than the pole in all models, and the lower
the effective temperature or higher the mass-loss the more
neutral the inner envelope. In only one model, model D,
does a neutral H disk form, and then only beyond ∼3R∗.
This contradicts the result of Kraus & Lamers (2003) who
found predominantly neutral hydrogen disks, even nearly
at the surface, for very similar models. In our models A, B,
and C the neutral hydrogen is negligible compared to HII
(inside 10R∗), although there is a tendency for the neutral
hydrogen fraction to rise at larger radii. The high level of
ionization prevails even if the effective temperature is low-
ered (model B) or the mass-loss rate is increased (model C).
Only the combined effect of the two (model D) produced
a neutral hydrogen disk. We will further discuss the dif-
ferences between our results and those of Kraus & Lamers
(2003) in §7.
5.3. Helium Ionization
Fig. 5 shows the logarithm of the ionized to neutral He ra-
tio. Ionized He means the total population of all HeII and
HeIII levels. The ionization level of helium is markedly
lower than that of hydrogen, a result to be expected given
the low effective temperatures which were adopted. As op-
posed to H, He forms a neutral disk in all but one of our
models — in model A, only an enhancement of neutral he-
lium occurs in the equatorial region. The neutral disks show
characteristics similar to those of the temperature distribu-
tion or hydrogen ionization; e.g, lowering Teffresults in an
inward shift of the neutral regions as seen in the last two
panels of Fig. 5. Higher mass-loss causes the neutral disk
to become thicker. It also appears that the thicker the disk,
the sharper its ionization boundaries. The neutral He disk
reaches down deeper than the hydrogen disk in model D,
but it is still truncated at ∼2 R∗.
6. Observed Profiles
Observed profiles for a converged ASTAROTH model are
calculated independently by an auxiliary routine (described

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Zsarg´ o et al.: Ionization in the wind of sgB[e] stars5
012345678
0
1
1.2
1.2
1.6
1.6
012345678
0
1
0.8
0.8
1.2
1.2
1.6
012345678
0
1
0.8
0.8
1.2
1.2
1.6
1.6
012345678
0
1
0.8
0.8
1.2
1.6
Fig.3. The temperature structure near the star in models
A, B, C, and D (top to bottom). The layout of the plots
is the same as that of Fig. 2. The contour levels are from
6,000 K to 16,000 K in 2,000 K increments and shown in
10,000 K units.
the results of the auxiliary routine. Below we discuss the
properties of the observed Hα profile, which arises from the
disk, and the CIV λλ1550 doublet profile, which primarily
arises in the polar wind. We also discuss similarities and
differences between the computed lines, and those seen for
our benchmark B[e] supergiant R 126, although we stress
that this work is not a spectral analysis of this star. For
the calculations presented in the following sections we gen-
erally adopted 10kms−1for the Doppler parameter of the
intrinsic line absorption/emission profile. Because of the
low velocities in the disk, the adopted Doppler parameter
can have a substantial influence on the line profile shape,
and model profiles calculated with Sobolev approximation
are significantly different. Furthermore, as the azimuthal
velocities are zero in our models, the profiles are not rota-
tionally broadened.
6.1. H Line Profiles
The shape of the Hα lines in Fig. 6 are very sensitive to the
observer’s viewing inclination (i). Depending on the model
either a single or double emission peak is seen; sometimes
the emission peaks are well resolved, while at other times
they blend into a continuous emission feature. For i = 0,
the half-widths at the line base (HWLB) are 30–50 km s−1
which is slightly larger than V∞ in the equatorial plane
(20 km s−1). Very weak wings, arising from emission in
the wind outside of the equatorial latitudes, is also present.
At higher inclinations the profiles are broader, and gen-
012345678
0
1
5.6
7.0
7.0
012345678
0
1
4.2
5.6
5.6
012345678
0
1
3.2
4.8
4.8
6.4
6.4
012345678
0
1
-0.80
-0.60
-0.40
0
0
2
2
4
4
Fig.4. The log
D (top to bottom). The layout of the figures is the same as
that of Fig. 2. The thin solid lines represent ratios above 0
(ionized H) and the thick dotted lines show neutral regions.
Note, that the lower the ratio the more neutral the hydro-
gen is! The spacing between levels is different from figure
to figure and between neutral and ionized regions.
?
NH II
NH I
?
contours for models A, B, C, and
012345678
0
1
1.2
1.2
2.4
2.4
3.6
3.6
4.8
012345678
0
1
-1.8
-1.2
-0.6
-0.6
0.0
0.0
1.6
1.6
3.2
3.2
012345678
0
1
-1.2
-0.4
-0.4
0
0
2
2
4
4
012345
?
678
0
1
-6
-6
-4
-4
-2
-2
0.0
0.0
1.6
1.6
3.2
3.2
Fig.5. The log
B, C, and D (top to bottom). The layout of the plots and
?
NHe III+NHe II
NHe I
contours for models A,