THE Hα PROFILES OF Be SHELL STARS

J. Silaj; C. E. Jones; T. A. A. Sigut; C. Tycner

doi:10.1088/0004-637X/795/1/82

1. INTRODUCTION

Be shell stars are a subset of (classical) Be stars—rapidly rotating, non-supergiant B-type stars that have shown Balmer emission at some epoch, indicating the presence of ionized circumstellar gas (see Rivinius et al. 2013 for a recent review). Spectroscopic and polarimetric measurements have long provided indirect evidence that the gas surrounding Be stars is arranged in a thin, equatorial disk-like structure, and more recently, interferometric observations have provided direct evidence of this geometry (see, e.g., Dougherty & Taylor 1992; Quirrenbach et al. 1993 and subsequent interferometric studies such as Stee et al. 1995; Quirrenbach et al. 1997; Tycner et al. 2005; Stee 2011.)

The emission lines that characterize all Be stars arise from radiative processes in the disk, such as recombination and free–free emission, and shell stars are distinguished observationally by the superposition of narrow absorption cores on their highly rotationally broadened emission lines. They are commonly understood as ordinary Be stars seen nearly edge-on. In such a model, the narrow absorption cores can be explained simply and naturally as arising from re-absorption and scattering out of the line of sight toward the observer. Porter (1996) tested the hypothesis that shell stars are Be stars seen edge on, and concluded that this is the case.

While all Be stars (ordinary and shell) show one or more emission lines in the Balmer series (in addition to various metal emission lines), Hα emission is typically the strongest feature, and modeling it reveals insights into the average properties of the disk since it is formed over a large region in the disk. In Silaj et al. (2010, hereafter referred to as Paper I), the authors created theoretical Hα profiles by systematically varying model parameters and illustrated the importance of fully understanding the physical conditions (i.e., temperature and density structure) within the disk, as these conditions significantly affect the appearance of the resultant profiles. In the aforementioned paper, however, profiles were computed under the simplifying assumption that a spectrum computed at i = 0° would remain valid for other inclinations and could be Doppler shifted by the appropriate amount for a given i value to simulate the effect of viewing the system at other orientations. This technique is a reasonable approximation at low i values, but is not sufficient to describe Be stars at high inclination angles (such as those expected in shell stars) because it does not account for the re-absorption that occurs when viewing a dense disk nearly edge-on.

Beray is a new code that solves the transfer equation along a series of rays (≈10⁵) through the star+disk system (Sigut 2011). It therefore accounts for the re-absorption along the line of sight and can adequately describe the resultant emission in cases where the star+disk system is at a high inclination angle. The objective of this work then is two-fold: to re-compute Hα profiles in this new regime to compare those results with the previous results of Paper I, and to model the shell spectra of Paper I that were previously left un-analyzed.

The paper is organized as follows. The observations are discussed in the next section, followed by a review of the basic theory and construction of the models (Section 3). In Section 4, we compare our new results to our previous work. Section 5 presents the detailed modeling for eight well-known shell stars, discussing each star in its own subsection. Finally, in Section 6, we give a recap of the main results of both the comparison with previous work and those obtained from the individual modeling and state our conclusions.

2. OBSERVATIONS

The observations modeled in Section 4 of this paper are a subset of the observations originally presented in Paper I. It was our intention to model the 11 spectra designated "sh" in Table 3 of that work: 28 Tau (Pleione), epsilon Aur, ν Gem, 4 Her, 51 Oph, 88 Her, 4 Aql, 1 Del, 60 Cyg, epsilon Cap, and MCW 166. However, since the time of publication, new papers on three of the stars— epsilon Aur, 51 Oph, and MCW 166—have been published that strongly indicate that they are not classical Be stars. epsilon Aur is an eclipsing binary of Algol type (detached); it is believed to have an F-type primary, and recent VEGA/CHARA visibility measurements indicate the formation of Hα emission wings arise in a stellar wind (see Mourard et al. 2012). 51 Oph and MCW 166 both appear to be young stars still surrounded by their nascent disks (see Thi et al. 2013 for a recent study of 51 Oph, and Alecian et al. 2013 for a discussion of MCW 166.) They were therefore removed from the study. The remaining eight stars are all well known classical Be (shell) stars, most of which have been observed and studied by various other groups of authors (see, e.g., Jaschek et al. 1980; Slettebak 1982 and, more recently, Rivinius et al. 2006; Saad et al. 2006).

As detailed in Paper I, all of the observations presented in our catalog were obtained between 2005 and 2008 with the fiber-fed échelle spectrograph attached to the 42 inch John S. Hall telescope at the Lowell Observatory, located near Flagstaff, AZ. The spectra were processed using standard routines developed specifically for the instrument and have a resolving power of 10,000 in the Hα region. In this work, we model one representative Hα spectrum for each of our eight targets that shows a clear shell signature: emission peaks with a narrow absorption core that extends below the stellar flux continuum. We note, however, that for many of our stars, we have several observations dating back to 2003. We consult these archival data to aid in determining the variability of our targets; in some cases, a clear trend of disk growth or disk loss can be detected.

3. THEORY AND MODELS

As discussed in Paper I, the radiative transfer code Bedisk computes the thermal structure and atomic level populations for a star of given mass, radius, effective temperature, and gravity. The reader is referred to Paper I and to Sigut & Jones (2007) for a detailed discussion of the theoretical background of Bedisk, but we recall here some of the key assumptions of the code.

First, the disk is assumed to be axisymmetric about the star's rotation axis and symmetric about the midplane of the disk. We use R to denote the radial distance from the star's rotation axis, and Z for the height above the equatorial plane. The radial density structure of the disk is determined by choosing a base, or initial, density (ρ₀) of the disk at the star's surface (i.e., R = 1 R_*, Z = 0), and assuming it decreases with increasing distance in the equatorial plane according to a simple power law index n. The parameters ρ₀ and n are essentially free parameters that are specified by the user, but previous studies suggest that ρ₀ is typically 1 × 10⁻¹² g cm⁻³ to 1 × 10⁻¹⁰ g cm⁻³, and n ranges from ≈2 to 4 (see, e.g., Jones et al. 2008) for most Be stars. To compute the vertical density structure, it is assumed that the gas is in isothermal hydrostatic equilibrium. Thus, the density of the disk at any position is given by

$\begin{equation} \rho (R,Z) = \rho _0 \left(\frac{R_*}{R}\right)^{n} e^{-(Z/H)^2}, \end{equation} \tag{ 1 }$

where ρ₀ is the initial density in the equatorial plane, n is the index of the radial power law, and H is the scale height in the Z-direction and is given by

$\begin{equation} H = \sqrt{\frac{2\,R^3}{\alpha _0}} \end{equation} \tag{ 2 }$

where the parameter α₀ is of the form

$\begin{equation} \alpha _0 = GM_* \frac{\mu _0 m_H}{k T_0}. \end{equation} \tag{ 3 }$

In these expressions, M_* and R_* are the stellar mass and radius, respectively, T₀ is an assumed, isothermal temperature used solely to fix the vertical structure of the disk, and μ₀ is the mean molecular weight of the gas. The remaining variables have their usual meanings: G is the gravitational constant, k is the Boltzmann constant, and m_H is the mass of a hydrogen atom.

Finally, the central star (which is assumed to be the sole source of photoionizing radiation for the disk) is also assumed to be spherically symmetric, and for all models we have set the star's rotation to be 0.8 v_crit, where v_crit is given by³

$\begin{equation} v_{\rm crit} = \sqrt{\frac{2\,GM_*}{3\,R_*}}. \end{equation} \tag{ 4 }$

The disk itself is assumed to be in pure Keplerian rotation.

Our disk models assume a solar composition and are computed with a grid of 36 radial (R) and 30 vertical (Z) points. The spacing of the points is non-uniform to ensure that the areas of the disk that are close to the central star and/or close to the equatorial plane (where changes are occurring more rapidly) are sampled with greater frequency. The radial extent of the calculation is 50 R_*, and vertically the disk is truncated when the condition ρ(Z)/ρ(Z = 0) = 10⁻⁴ is met.

While the full radial extent of the disk was formerly employed to compute line profiles, for this study we employed a disk size (R_disk) of 50 R_* in the cases where n ⩽ 3.5, and adopted R_disk = 30 R_* for n > 3.5. By analyzing the change in the equivalent width of the model Hα profiles as a function of the assumed disk size, it was determined that low n-value models can have significant contributions to the total flux in Hα originating at large radial distances. However, for n > 3.5, models created at any of our initial densities showed all of the flux in Hα to be contained within ∼20 R_*. Thus, truncating these disks at 30 R_* safely ensured we were capturing all of the emission, but represented a substantial savings in computational time.

4. COMPARISON WITH PREVIOUS RESULTS

Our procedure for creating theoretical Hα profiles is a two stage process: first, Bedisk computes a disk model (the density structure of the disk, the temperature solution, and all of the atomic level populations) for a user-defined set of input values, including the parameters of the central star and the other parameters discussed in Section 3. Additional considerations, such as the number of computational gridpoints and the actual extent of the disk (both in R and Z), as well as the chemical composition of the disk, must also be set in advance by the user. A key aspect of the disk models produced by Bedisk is that the temperature is iterated at each computational gridpoint to enforce radiative equilibrium, producing a set of self-consistent physical conditions for the disk. These disk models are then used as input to a secondary code which calculates the expected Hα emission profile.

In our previous work, Bedisk was used to compute disk models of representative B0, B2, B5, and B8 type Be stars. To fully encompass the range of values found in the literature, we created a grid of ρ₀ values of spanning from 5 × 10⁻¹³, 1 × 10⁻¹², ..., 5 × 10⁻¹⁰ g cm⁻³ and n values of 1.5, 2.0, ... , 4.5, and all combinations of ρ₀ and n values were computed for each of the aforementioned spectral types. From there, line profiles at three different inclinations (20°, 45°, and 70°) were computed using an auxiliary line shape code.

As briefly alluded to earlier, Bedisk computes a spectrum at i = 0°, which is accomplished by solving the transfer equation along the Z-grid at each radial distance R. The function of the former line shape code was to divide the disk into sections, and compute the radial velocity for each section. Then, the i = 0° spectrum computed by Bedisk was Doppler shifted by the appropriate amount for a given i value and added back into the final spectrum. Finally, all sections of the disk were summed. This treatment should provide a fairly accurate representation, especially at low inclination angles where the effects of self-absorption are at a minimum. It was fully expected that this treatment would not be sufficient to describe high inclination systems, and therefore we did not attempt to model the shell spectra presented in Paper I.

In this work, we first retain the same disk models that were initially created for Paper I, but use a new auxiliary code—Beray—to compute the Hα line profiles and compare them with our previous work. Beray solves the transfer equation along a series of rays (approximately 10⁵) through the star+disk system, so in general, it should provide an improvement in the realism of our models. We then create new disk models with spectral types chosen to match our eight shell star targets, and use Beray to create line profiles at high inclination angles. Through an iterative fitting procedure, the new models are adjusted to match the observations. As was done in Paper I, we convolve our synthetic profiles with a Gaussian of FWHM of 0.656 Å to bring the resolving power of the computed profiles down to 10,000 to match the resolving power of our observations. To maintain as much consistency as possible between this work and the previous work, the new disk models were all created under the same assumptions used in Paper I, i.e., the same choice of computational grid and domain, and the same chemical composition for the disk were adopted.

Overall, there is very good agreement between the methods in terms of the trends that emerge by varying n, ρ₀, and i in a systematic fashion, as was done in Paper I. For example, for a given initial density, ρ₀, the general manner in which the equivalent width varies as a function of the power law index n is consistent between the two regimes.

The two sets of profiles also show good agreement between the actual profile shapes that are predicted, i.e., singly versus doubly peaked. This is largely to be expected, as both sets are computed from the same initial disk model. Therefore, if, for example, there existed a case in Paper I where a large, cool region of the disk close to the star was suppressing emission and giving rise to a double-peaked profile, then we fully expected that this same result would be duplicated by the new line shape calculation and indeed this outcome was found. The fact that we find such good agreement between predicted shapes is reassuring, as it indicates that our simpler approximation in Paper I still produced realistic results and that improving the realism with our new procedure has not fundamentally altered any basic results we obtained in that work.

The most significant difference between the two methods is that the emission as calculated by Beray is consistently lower than that calculated by our previous treatment. Most likely, the approximation in Paper I overestimates the amount of emergent radiation since it was assumed that all escape occurred perpendicular to the disk and no further processing of radiation, such as scattering or absorption along the line of sight, was accounted for.

5. RESULTS FOR INDIVIDUAL SPECTRA

The results of the modeling are presented below. Each observation, and the model that best reproduces the observational signature, is discussed in its own subsection. We aim to reproduce four main aspects of the observation: peak height, peak separation, absorption depth, and the width of the absorption core.

The general strategy for modeling is to (1) fix the fundamental spectral type for the central star; (2) create a series of high inclination models for all combinations of n and ρ₀ previously listed; (3) eliminate models that do not show the characteristic shell shape; and (4) keeping our n values fixed at 1.5, 2.0, 2.5 etc..., iteratively adjust the corresponding ρ₀ value to match the peak height, and the i value to match the absorption depth.

In order to fix the spectral type, we conducted a thorough literature search for all targets; a discussion and our final spectral type determination of each star is given in its respective subsection.

From the systematic investigation of the theoretical profiles, it was found that there is typically a direct correlation between the absorption depth and the adopted inclination value: as expected, higher i values produce a deeper absorption core. Thus, for our targets with shallower absorption cores, we initially created a series of models at 70°, but for targets with deeper absorption cores, the initial models were created with 75° or 80°. Eliminating profiles that were clearly of the wrong shape, we were usually left with only one model at each of our seven n values that approximately replicated the observation. Adjusting the models from that point was a fairly straightforward task, as we found that at these high inclination values, the choice of ρ₀ mainly affected the peak height, i mainly affected the absorption depth, and the two were not strongly coupled. They could therefore be adjusted nearly independently until these aspects of the observation were matched.

Furthermore, in our models, it was the n value that primarily governed the peak separation, and therefore the width of the absorption core as well, with smaller n values corresponding to a smaller peak separation. Hence, once peak separation and absorption depth were fixed, we could eliminate all models created with n values that did not match the peak separation.

In all cases, a χ² goodness-of-fit test was employed to ensure that the retained fits were quantitatively better than the eliminated fits, in addition to being better visually. In a few cases, there were several models that were statistically indistinguishable and very similar visually, and these cases are discussed in more detail in their proper subsection.

We order the discussion of our targets by their spectral type, from earliest (B1V) to latest (B9V). We note that the best fits to the observations are obtained for the stars of earliest spectral types, and that the later-type stars usually show extended emission wings not reproduced by our model. One possible explanation for this is that we are overestimating the stellar photospheric absorption component. Determining the spectral type of Be stars is known to be a particularly complicated task: not only do the high rotational velocities broaden and distort the lines, but, as detailed in Steele et al. (1999), the disk emission can partially fill in lines of He i (as well as H i and several metallic ions), which often affects the main classification criteria. Moreover, a well developed shell spectrum containing many metallic absorption lines can completely veil the photospheric absorption spectrum, meaning that Be shell stars are even more difficult to correctly analyze than ordinary Be stars.

The effect of employing an incorrect spectral type for the central star becomes more important at later spectral types where the Hα photospheric absorption is much larger. If the adopted model spectral type is cooler than the true spectral type of the star, the absorption component of the Hα line will be overestimated since Hα absorption is quite weak for early-type B stars but increases steadily toward the later types, reaching a maximum at A0. Thus, the emission component produced by the disk will be insufficient to completely fill in the absorption component, and a discrepancy between the model and the observation will result.

Another factor possibly contributing to the discrepancy in the wings is that in our models we include only coherent electron scattering, both in the opacity and the emissivity. While non-coherent electron scattering is often neglected in the calculation of line profiles, Auer & Mihalas (1968) showed that its effect in emission lines is to decrease the intensity slightly and cause the development of extensive emission wings. Poeckert & Marlborough (1979) explicitly calculated this effect for Be stars, and were able to show that scattered envelope radiation can broaden the Hα emission line considerably. However, they only considered ϕ Per, a B1III star with very strong Hα emission. In our models, the Hα emission is much weaker, and so the effect is probably much smaller. Furthermore, because we see the most discrepancy in the wings at later spectral types, where we do not expect the electron density to be as high, this effect is probably secondary to the effect of employing a spectral type that is later than the true spectral type of the star. Finally, our models also do not include pressure broadening caused by the Stark effect, and while we do not anticipate this to be a large effect in our stars, it could be important in the case of very dense disks.

5.1. 60 Cygni

60 Cyg (HR 8053, HD 200310) has long been known as an emission star. It is classified as B1V according to Jaschek et al. (1980), and, similarly, as B1Ve according to Slettebak (1982). 60 Cyg exhibits pronounced long-term spectral variations, transitioning from normal B to Be phases (Koubský et al. 2000). Wisniewski et al. (2010), who monitored the target between 1992 August and 2006 December, found the Hα EW reached a maximum in December of 1998, and then steadily declined until Hα was in pure absorption in early May of 2001. Their Figure 1 shows a slow, steady increase in Hα EW after the minimum in 2001 May until the end of their monitoring in 2006 December, with the emission being in a generally low state throughout this entire period.

60 Cyg is a binary and Wisniewski et al. (2010) estimate that its companion truncates the disk at a tidal radius of 120–135 R_☉ based on the range of masses for the primary and secondary given in Koubský et al. (2000). This truncation range corresponds to 19–21 R_* given our adopted model (see below), and since we find the Hα emission to be contained within ∼20 stellar radii in the vast majority of our models, we do not anticipate that this disk truncation will have a large effect on our models. Furthermore, while it has been hypothesized that binary companions can trigger mass loss events from the central star (e.g., δ Sco; Miroshnichenko et al. 2001), Wisniewski et al. (2010) find this is probably not the case in 60 Cyg because the time for disk dissipation corresponds to almost six complete orbits of the binary. Thus, in the particular case of 60 Cyg, the binary companion does not seem to have a strong effect on the primary.

We adopt a B1V model for this star with stellar mass M = 13.21 M_☉, radius R = 6.42 R_☉, effective temperature T_eff = 25, 400 K, and gravity log g = 3.9. (See Table 1 for a complete list of all model spectral types and their associated parameters used in this work.)

Table 1. Adopted Stellar Parameters

Spectral Type	Stellar Radius	Stellar Mass	T_eff	log g
Spectral Type	(R_☉)	(M_☉)	(K)	log g
B0V	10.00	17.00	25,000	4.0
B1V	6.42	13.21	25,400	3.9
B2V	5.33	9.11	20,800	3.9
B3V	4.80	7.60	18,800	4.0
B5V	3.90	5.90	15,200	4.0
B6IV	5.42	5.58	13,800	3.8
B6V	3.56	5.17	13,800	4.0
B7V	3.28	4.45	12,400	4.1
B8V	3.00	3.80	11,400	4.1
B9V	2.70	3.29	10,600	4.1
A0V	2.40	2.90	10,000	4.1

Notes. The parameters assumed for the B0V, B2V, B5V, and B8V stars are identical to those assumed in Paper I. As noted in that paper, the B0V parameters are the ones employed in the modeling of γ Cas in previous works by some of the authors. All other values are interpolated from Cox (2000). We note that the masses and radii given here are generally slightly larger than those given by Harmanec (1988) for the same spectral types.

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The spectrum presented and modeled here (Figure 1) was obtained on JD 2453640.5. It shows a clear shell signature with a very slight asymmetry in peak height: the red peak is marginally higher than the blue peak. This probably represents a very slight asymmetry in the density distribution of the circumstellar material. As our model assumes an axisymmetric geometry, this small asymmetry cannot be matched, and we elect to choose a model with a peak height that falls in between the two peak heights in the observation to represent an average disk density. We adopt a similar strategy in the other seven spectra, all of which display some small degree of asymmetry in the peak height.

Interestingly, we found the observation of 60 Cyg could be represented equally well by a model with n = 3.5, ρ₀ = 1.8 × 10⁻¹¹ g cm⁻³, i = 65° or n = 4.0, ρ₀ = 3.3 × 10⁻¹¹ g cm⁻³, i = 64°. Both models fit the peak height and separation, as well as the absorption core depth and width, extremely well. Performing a χ² goodness-of-fit test, it was determined that the two models were statistically indistinguishable. It is not surprising that there is degeneracy in the choice of n and ρ₀: both parameters effectively control the same thing, i.e., the density structure of the disk. What is of interest is that both best-fitting models had nearly identical values of i. This trend was found repeatedly throughout the modeling process, for all of the spectra: in all cases, even where the underlying spectral type was changed or many combinations of n and ρ₀ could adequately reproduce the peak height and separation, there was only ever a very small range of i (usually a maximum span of 4°) that produced models with the correct absorption depth.

Models created for 60 Cyg with an n value of 3.0 or lower displayed a peak separation that was too small as well as an absorption core depth that was systematically too narrow. The effect was noticeable, but not extreme. Similarly, employing an n value of 4.5 resulted in models whose absorption core widths were consistently too wide, although again, the effect was fairly subtle. Models created with n = 3.0 required a corresponding i value of 68° to match the absorption depth, and n = 4.5 models required an i value of 64° to match the absorption depth. It thus appears that the value of i needed to match the observation is fairly constant despite hugely different theoretical density distributions. This reinforces our finding that the value of i needed to match the observation is only marginally affected by the choice of the other two model parameters. It appears that, within the framework of our models, i does not suffer the same degeneracy and that we can specify a unique inclination angle (± a few degrees) that reproduces the obser-vation.

The viscous disk decretion (VDD) theory, which has been shown recently to describe observations of Be stars quite well, predicts an n value of 3.5 in the simplest case of isothermal, isolated disks (see, e.g., Carciofi 2011). It was also shown in Paper I that a power law index of n = 3.5 was strongly preferred by the Hα fits performed in that study. We therefore marginally prefer our model created with n = 3.5, and adopt n = 3.5, ρ₀ = 1.8 × 10⁻¹¹ g cm⁻³, and i = 65° as our choice of final, best-fit model. (See Table 2 for a summary of the best-fit model for each observation.) For completeness, it should be noted that VDD theory naturally predicts a departure from n = 3.5 with the inclusion of other factors such as a non-isothermal radial temperature structure, disk scale height, and viscous transport. Therefore, while n = 3.5 may be the preferred value, it cannot be assumed a priori, and deviations from this value are quite a normal occurrence.

Table 2. Best Fit Models

HR	HD	Name	Spectral Type	Reference	Model Parameters
HR	HD	Name	Spectral Type	Reference	n	ρ₀ (g cm⁻³)	i (°)
1180	23862	28 Tau, Pleione	B8V	1	2.5	6.2 × 10⁻¹²	76
2343	45542	ν Gem	B6V	2	2.0	1.6 × 10⁻¹²	77
5938	142926	4 Her	B9V	3	3.5	6.7 × 10⁻¹¹	67
6664	162732	88 Her	B7V	4	3.5	2.3 × 10⁻¹⁰	79
7040	173370	4 Aql	B8V	5	2.5	3.3 × 10⁻¹²	46
7836	195325	1 Del	B9V^a	...	3.5	3.0 × 10⁻¹¹	84
8053	200310	60 Cyg	B1V	4, 5	3.5	1.8 × 10⁻¹¹	65
8260	205637	Cap	B3V	6	3.0	1.7 × 10⁻¹¹	80

Note. ^aSee Section 5.8 for a discussion of the spectral type determination of 1 Del. References. (1) Abt & Levato (1978); (2) Cucchiaro et al. (1977); (3) Cowley (1972); (4) Jaschek et al. (1980); (5) Slettebak (1982); (6) Levenhagen & Leister (2006).

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5.2. Capricorni

epsilon Cap (HR 8260, HD 205637) is listed as a B3IIIe type star according to Slettebak (1982), but more recently has been typed as B3V (Levenhagen & Leister 2006). We adopt this latter spectral type with a corresponding M = 7.6 M_☉, R = 4.80 R_☉, T_eff = 18, 800 K and log g = 4.0.

epsilon Cap is a close binary with an orbital period of 129 days (Rivinius et al. 2006). It is also known to display both short-term and long-term variations in its spectroscopic and photometric measurements. Because we model one spectrum for each of our targets, our models only represent the disk system at the time of the observation.

The spectrum shown in Figure 2 was obtained on JD 2454011.5. The blue peak is very slightly higher than the red peak, but otherwise the spectrum is very symmetrical. Note, however, the presence of "shoulders" starting at the continuum and extending up to F/F_c ≈ 1.15; the effect is slightly more noticeable on the blue shoulder than the red shoulder. The source of this excess emission could be due to a slight density enhancement close to the star. Rivinius et al. (2001) demonstrated that early-type Be stars often show periodic mass ejection episodes which result in substantial differences in the profile wings. To test this hypothesis, we constructed models where the equatorial density structure of the disk was governed by two power laws (instead of the usual one): a smaller n value was adopted for the first five stellar radii, and a larger one was employed for the remainder of the disk. This simulates a ring-like region of enhanced density close to the central star, as one might reasonably expect to find following an episode of mass ejection. Our investigations confirmed that we could indeed produce excess emission in the wings if the density enhancement was carefully selected to be significant enough to make a noticeable signature in the synthetic spectrum, but not so dense as to become optically thick, and therefore cold (i.e., not radiate efficiently). Because modeling the excess emission in the wings introduces many new free parameters in the model that cannot be constrained, we elected to retain our initial strategy of matching the key features of the observation (again, peak height and separation, and absorption depth and width) with the equatorial density governed by a single power law.

epsilon — **Figure 2.** Cap. Symbols are as in Figure 1.
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Apart from the emission described above, our model matches the observation quite well: peak height and separation, and absorption depth and width, are all fit very well by a disk model with n = 3.0, ρ₀ = 1.7 × 10⁻¹¹ g cm⁻³, and i = 80°. This case seems to suffer less degeneracy than many of the other stars: a model created with n = 2.5 and ρ₀ = 6.2 × 10⁻¹² g cm⁻³ at i = 79° matched the peak height and absorption depth, but had an absorption core width that was clearly too narrow. Similarly, a model with n = 3.5, ρ₀ = 4.5 × 10⁻¹¹ g cm⁻³ and i = 81° could also match the peak height and absorption depth, but displayed an absorption core width that was obviously too wide. Thus, we adopt the model with n = 3.0, ρ₀ = 1.7 × 10⁻¹¹ g cm⁻³, and i = 80° as our best-fitting model. Although our n value departs slightly from the one predicted by VDD, it certainly is within the range of n normally determined for Be stars.

5.3. ν Geminorum

Our spectrum of ν Gem (HR 2342, HD 45542) was obtained JD = 2454452.5. Older catalogs (e.g., Figure 8 of Slettebak & Reynolds 1978) show the Hα line of ν Gem in emission but with little to no core. Therefore, ν Gem is one of the as yet unexplained Be stars that has transitioned from a normal Be phase to a shell Be phase; this transition is difficult to explain if the shell phenomenon arises from inclination angle alone.

Rivinius et al. (2006), who observed the target various times between 1994 and 2003, found ν Gem to be V/R variable. Our spectrum shows the red peak (which originates from the portion of the disk that is receding from the observer) is noticeably stronger than the violet peak (from the region moving toward the observer), which is commonly interpreted as a density difference in the regions responsible for forming each peak. Since the effect is not very large, we model the spectrum by selecting a density that results in an intermediate peak height; thus, our model represents an average disk density.

The spectral type determinations of ν Gem range from B5 to B7, and luminosity class determinations range from type III to V, with nearly every combination of the two having been suggested at some time.

We initially modeled ν Gem as a B6V star (Cucchiaro et al. 1977) with M = 5.17 M_☉, R = 3.56 R_☉, T_eff = 13, 800 K and log g = 4.0. Adopting an n value of 2.0, an initial density of ρ₀ = 1.6 × 10⁻¹² g cm⁻³, and i = 77°, our model successfully matched most aspects of the observation (see Figure 3). At this spectral type, the models created with n = 1.5 were markedly too narrow overall, and had a peak separation that was too small, while models created with n = 2.5 or greater were visibly too wide and had a peak separation that was too great. Thus, if the spectral type is indeed B6V, the only choice of n that can reproduce the observations is one with a value quite close to 2.0.

**Figure 3.** ν Gem. Symbols are as in Figure 1.
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While we cannot detect a definite trend of disk loss or gain from our archival observations, they do show nonetheless that the equivalent width of Hα emission of ν Gem is quite variable. Furthermore, the equivalent width did decrease substantially between the last two observations, from −0.98 Å on JD 2454158.5 to −0.78 Å at the time of our observation, which may indicate the beginning of a disk loss phase. Thus, it is not unreasonable that the best-fitting n value departs significantly from the one predicted by VDD.

We investigated the effects of modeling ν Gem with a slightly evolved central star (Slettebak 1982, for instance, classifies this star as B6IV) to see if this would affect the best-fitting n value. Interpolating values from Cox (2000) in a manner consistent with our previous work, our B6IV model corresponds to a stellar model of M = 5.58 M_☉, R = 5.42 R_☉, T_eff = 13, 800 K and log g = 3.8. We again found an n value of 2.0 was necessary to match the peak separation of the observation, but did not pursue the investigation beyond that given that there is so much uncertainty in the spectral typing. We thus retained the model with a spectral type of B6V, n = 2.0, ρ₀ = 1.6 × 10⁻¹² g cm⁻³, and i = 77° as our best-fitting model.

While the peak height, peak separation, absorption depth, and the width of the absorption core are all satisfactorily matched, there is a discrepancy in the wings. As discussed at the beginning of this section, this discrepancy is common for the late-type stars in our study, and could originate from the uncertainty in the underlying spectral type, as well as the scattering that is not included in the model.

An additional complication, however, is that Rivinius et al. (2006) found that ν Gem is a triple system, with the shell absorption features originating from the Be star, but possible contamination to the photospheric spectrum from the other stars in the system. This could affect our continuum normalization, as well as other aspects of the model, depending on the extent of the contamination.

5.4. 88 Herculis

The spectral type of 88 Her (HR 6664, HD 162732) is given simply as Bpshe on the SIMBAD database, and it is the only star in this study that was not included in Slettebak (1982). Jaschek et al. (1980) classify it as B7Vn, but both the type and luminosity class appear to be fairly uncertain. We initially adopted a model spectral class of B7V, which translates to stellar parameters of M = 4.45 M_☉, R = 3.28 R_☉, T_eff = 12, 400 K and log g = 4.1.

88 Her undergoes long-term spectroscopic and photometric variations. It was studied in some detail by Harmanec et al. (1972a, 1972b, 1974) and Doazan et al. (1982), who suggested that the star is probably a single-line spectroscopic binary. They also propose that the long term variations the star displays may have their origin in the binarity of the system.

Our spectrum of 88 Her was obtained JD 2453844.5. This target shows the strongest emission of all the observations, with a maximum peak height very nearly reaching F/F_c = 2.2. It also shows very deep absorption, with its core extending almost to F/F_c = 0.6. A model with n = 3.5, ρ₀ = 2.3 × 10⁻¹⁰ g cm⁻³, and i = 79° successfully reproduces the peak height and absorption core depth, but no combination of n and ρ₀ can recover the exceedingly large width of the observation (see Figure 4). This model also exhibits a lack of emission in the wings, which again points to an overestimation of the stellar photospheric absorption component. Especially in light of the other discrepancies in the fit, it seems highly plausible that the spectral type for 88 Her has not been accurately determined. Note also that the density of the best fit model (2.3 × 10⁻¹⁰ g cm⁻³) is quite high, so this is one case where Stark broadening may play an important role in the line profile shape.

**Figure 4.** 88 Her. Symbols are as in Figure 1.
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5.5. 28 Tauri = Pleione

28 Tau, or Pleione (HR 1180, HD 23862), acquired an opaque shell in 1972 that was still present at the time that Slettebak (1982) observed and attempted to classify it. His final determination was B8(V:)e-shell. It had previously been classified as B8nn by Cowley (1972) and as B8V shell by Abt & Levato (1978). We modeled our spectrum, obtained JD 2454452.5, assuming a star of B8V spectral type with M = 3.80 M_☉, R = 3.00 R_☉, T_eff = 11, 400 K and log g = 4.1.

Like many Be stars, Pleione has undergone several phase transitions from B to Be to Be shell. Pleione also exhibits a very strong cyclic V/R variability. As shown in Rivinius et al. (2006), this affects both the peak heights and the amount of absorption in the wings of the Balmer lines. Pleione is also a binary, with a 218 day period (Nemravová et al. 2010). Because this represents a relatively close binary, it is possible that dynamical interaction could influence the disk structure and its variability.

The observation displays an additional asymmetry not present in the other seven: the top approximately one-third of the blue peak appears to be skewed slightly to the left (see Figure 5). Naturally, our axisymmetric models cannot reproduce this asymmetry. Setting the asymmetry aside, a model with an n value of 2.5 best reproduces the width of the absorption core, and matches the placement of the red peak. The ρ₀ value needed at this n to achieve the peak height approximately halfway in between the two is 6.2 × 10⁻¹² g cm⁻³. The resultant spectrum created at i = 76° appears just slightly too shallow to match the observation, but increasing i to 77° produces a model that is distinctly too deep.

**Figure 5.** 28 Tau = Pleione. Symbols are as in Figure 1.
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To be thorough, models were created with n = 1.5, 2.0, 3.0, 3.5, 4.0, and 4.5 (and corresponding varying densities, adjusted to match the peak height), but values of n < 2.5 consistently produced models whose peak separation and absorption cores were too narrow, and higher n values made both features too wide. As in the case of 88 Her (above), no combination of n, ρ₀, and i that could successfully match the peak height and absorption depth could reproduce the overall (very large) width of the observation. We did find, however, that to match the absorption depth of the profile, all models created with 1.5 ⩽ n ⩽ 2.5 required an i value of 76°, and all higher n-value models required i = 77°. Thus we feel that the inclination angle of Pleione is fairly well determined.

Although our best-fitting model has an n = 2.5, which departs significantly from VDD, our archival observations show that Pleione was in a phase of evident disk loss at the time of our observation (see Jones et al. 2011). The n value of 3.5 predicted by VDD applies only to a steady-state disk, which clearly does not apply in this case. We retain the model with n = 2.5, ρ₀ = 6.2 × 10⁻¹² g cm⁻³, and i = 76° as our best fit.

5.6. 4 Aquilae

4 Aql (HR 7040, HD 173370) was classed as B8Ve by Slettebak (1982), and we adopted this spectral type, with M = 3.80 M_☉, R = 3.00 R_☉, T_eff = 11, 400 K and log g = 4.1, for our central star.

To the best of our knowledge, 4 Aql has not been monitored in any long term observing campaigns, and thus we cannot make any strong statements regarding its general variability. It does not appear to be a binary star.

The observation of 4 Aql, obtained on JD 2454255.5 and shown in Figure 6, differs slightly from the other seven presented here in that it displays very broad absorption wings that extend ≈ ± 30 Å outside of line center. The absorption in the wings actually extends below the absorption core at line center, which is quite shallow. Our archival data suggests that 4 Aql was in a phase of disk loss at the time of observation, which could explain the overall weak emission that this observation exhibits.

We were able to satisfactorily match the peak height and separation, and absorption depth and width of the observation, with models created with n = 2.5, 3.0, and 3.5 and correspondingly adjusted initial densities. However, the model with n = 2.5 did have a marginally lower χ² value and a slightly better fit visually, so we adopted n = 2.5, ρ₀ = 3.3 × 10⁻¹² g cm⁻³, i = 46° as our final, best fit to the observation. We note that this i value departs significantly from the near 90° values that are expected for shell stars, and that the two other models (above) that matched the observation had even lower i values of 43°. Despite our best efforts, no models with high (i.e., i ≈ 70°) inclination values could be matched to the observation, as the absorption depth of such models was consistently and markedly too strong.

5.7. 4 Herculis

4 Her (HR 5938, HD 142926) is a rather well-known Be and shell star that has been frequently observed. Like many Be stars, it undergoes transitions from normal B to Be phases, with the length of emission and non-emission phases varying from three to 20 yr (Koubsky et al. 1994). We model it as a B9V star (Cowley 1972), with M = 3.29 M_☉, R = 2.70 R_☉, T_eff = 10, 600 K and log g = 4.1, although we note that it was classified as B8p shell by Molnar (1972) and as B7IV e-shell by Slettebak (1982). 4 Her is a binary star with a nearly circular 46 day orbit (Koubsky et al. 1997). This represents another case in which a close binary may exhibit influence over the structure and dynamics of the circumstellar disk.

Our observation, obtained on JD 2454273.5, shows a rather shallow absorption core that only extends to F/F_c ≈ 0.85. Our archival data indicates that 4 Her was in a phase of disk loss at the time of our observation, with the equivalent width of its Hα emission declining from −2.73 Å on JD 2453510 (2005) to only −1.13 Å at the time our spectrum was collected (2007). Like some of the other spectra discussed above, several combinations of n and ρ₀ adequately reproduce the peak height, peak separation, and absorption width. In all cases, however, an i value between 65° and 68° was necessary to match the depth of the absorption.

Figure 7 shows the best fit obtained for 4 Her: n = 3.5, ρ₀ = 6.7 × 10⁻¹¹ g cm⁻³, i = 67°. However, as in the case of 4 Aql, above, there were two other models that were statistically indistinguishable from this fit. A model created with n = 2.0, ρ₀ = 1.8 × 10⁻¹² g cm⁻³, i = 68°, and another with n = 3.0, ρ₀ = 1.9 × 10⁻¹¹ g cm⁻³, i = 65°, also fit the observation well, although upon close visual inspection, the aforementioned n = 3.5 model matched the shape and width of the absorption core more closely, and so we adopted this model as our final best fit.

5.8. 1 Delphini

1 Del (HR 7836, HD 195325) is one of very few Be stars whose Hα emission profile has remained nearly constant for decades. Our observation, obtained on JD 2454453.5 and shown in Figure 8, shows a very deep and sharp absorption core flanked by weak emission shoulders, similar to most observations that can be found in spectral catalogs. 1 Del is an isolated star with no known binary companion.

**Figure 8.** 1 Del. Symbols are as in Figure 1.
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Spectral type determinations for 1 Del vary considerably: Slettebak (1982) classified it as B8-9:(e)-shell, while other authors had previously suggested its type was as late as A1 (see, e.g., Cowley et al. 1969). It was also recently classified A1:III shell by Abt & Morrell (1995).

The Hα line of 1 Del was previously modeled with a stellar mass, radius, effective temperature, and log g of 4 M_☉, 3.3 R_☉, 12, 600 K, and 4.0, respectively, by Marlborough & Cowley (1974) and subsequently by Millar & Marlborough (1999). These parameters are quite close to the ones we assume for a B7V model (see Table 1). Despite the previous authors' success at reproducing the observations, in light of other determinations that type the star as A1, we felt it prudent to adopt a spectral class in between the two extremes, and model 1 Del as a B9V star. We also preferentially retained a B type spectral class for the underlying star due to reports by other authors of the presence of He i lines in its spectrum, as stated in Slettebak (1982). The parameters for our B9V model are M = 3.29 M_☉, R = 2.70 R_☉, T_eff = 10, 600 K and log g = 4.1.

Modeling 1 Del presented us with a unique quandary: models created with low n values (2.0 or 2.5) matched the peak separation of the observation, but had absorption cores that were distinctly too narrow. Models created with n = 3.5 exhibited a very good match to the width of the absorption core, but had a peak separation greater than the one observed. We investigated the effect of changing the spectral type by creating models using B8V and A0V (i.e., the neighboring) spectral types. These models displayed the same behavior; it appears that, in our models, there is no choice of n that can reproduce both the peak separation and the absorption core width. Interestingly, at all n values and for all the spectral types, an i value of either 83° or 84° was required in order to match the depth of the observation. Thus, while the spectral type, n and (corresponding) ρ₀ values all remain relatively uncertain, the inclination of 1 Del is confined to a very narrow range in our models.

Given that 1 Del is known for its stability, it is expected that it would be well-described by VDD, and the value of n should be close to 3.5. As discussed above, our model with n = 3.5 did not match the peak separation, but did reproduce the depth and width of the absorption core. It also had a slightly lower χ² value than the n = 2.0 model, which did match the peak separation but had an absorption core that was too narrow. For all of these reasons, we adopt a final best fit model of n = 3.5, ρ₀ = 3.0 × 10⁻¹¹ g cm⁻³ and i = 84°.

6. SUMMARY AND DISCUSSION

We have employed the use of a new code, Beray, to compute Hα emission profiles of Be stars. The new profiles, computed from the same disk models that were created for Paper I, were compared to our previous profiles, and in general there was good agreement between the trends that arise from systematically varying the input parameters. In addition, the shapes (i.e., singly versus doubly peaked) of the profiles computed in the two different regimes also largely agreed. Under the new treatment, however, the equivalent widths of the computed profiles, and therefore the estimated amount of flux, are consistently lower than in the previous work. Still, the main conclusions of Paper I stand: profile shapes are strongly dependent on the physical conditions within the disk, and the profile shape alone cannot uniquely define the inclination angle of the system.

We have also modeled shell spectra of eight well-known Be shell stars. In all cases, we can simultaneously match the peak height and separation and the absorption depth and width of the observations. However, in nearly all cases, it was possible to do so with more than one combination of initial density ρ₀ and power-law index n. This finding emphasizes the need to employ other kinds of observations to remove this degeneracy and better constrain the models.

For the shell stars of later spectral type (B6 and later), it was found that our models consistently underestimated the amount of flux in the wings of the line. Our initial assumption was that the derived spectral type for the targets was probably quite inaccurate, thereby leading to an overestimation of the photospheric profile in our models. Increasing the spectral type of the central star by two or even four subclasses improved the discrepancy found in the wings, but did not fully resolve the issue. Therefore, it is probably the combined effects of incorrect spectral typing and sources of scattering not included in our model (non-coherent electron scattering and Stark broadening, both of which broaden the spectral line and contribute to extended emission wings) that results in this discrepancy.

The values of the inclination angles determined by this modeling were of great interest. Shell stars are generally considered to be Be stars viewed edge-on, and the expectation was that most of our targets would be modeled at very close to 90 deg. However, our highest inclination angle value was 84°, which we obtained for 1 Del: a star whose central absorption is markedly stronger than any of the other seven targets. Six of the eight targets were best represented by models created between 65° and 80°, which is considerably lower than expected. Finally, the case of 4 Aql is quite puzzling: we consistently found the observation could only be fit by models created at a mid-inclination value of roughly 45°.

A remarkable result of our modeling was that the inclination angle required to fit the observation was relatively insensitive to any of the other variables in our model. Within the framework of our models, it is almost entirely the effect of the inclination angle that dictates the depth of the central absorption trough. Thus, despite using vastly different density structures (i.e., different combinations of n and ρ₀), we typically found that the i value stayed constant to within a few degrees.

We thank our anonymous referee for comments and suggestions that helped to improve this work. We also thank Alex Carciofi for his helpful discussions on this project. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We thank the Lowell Observatory for the telescope time used to obtain the Hα line spectra presented in this work, and C.T. acknowledges, with thanks, FRCE grant support from Central Michigan University. This research was supported in part by NSERC, the Natural Sciences and Engineering Research Council of Canada.

Facility: Hall - Lowell Observatory's 42in Hall Telescope