A STUDY OF VEGA: A RAPIDLY ROTATING POLE-ON STAR

Vega was observed with the Dominion Astrophysical Observatory 1.22 m telescope using the coudé spectrograph with the long camera, the 800 l mm⁻¹ grating in the second order and the IS96B image slicer. For these observations, the detectors were initially a 1872 pixel bare Reticon with 15μ pixels and more recently the 4K SITe with 15μ pixels. A central stop in the beam removed light in the same manner as does the secondary mirror of the telescope. The exposures were flat fielded with exposures of an incandescent lamp, which was viewed through a filter placed in the coudé mirror train.

The portion of the spectrum of Vega discussed here extends from λλ4487 to 4553 in wavelength steps of 0.035 Å with differing S/N values for the continuum regions after the co-addition of many spectra. In the original paper cited above, a mean scattered light correction of 4% (J. M. Fletcher 1990, private communication) was applied to the co-added spectrum, but by comparing our observed hydrogen line profiles with those of Peterson (1969), we now favor a value of 3.5% for the Reticon. For the CCD spectra, we have written software that maps the behavior of scattered light along both sides of the spectrum making the correction straightforward—though not without its difficulties. Digitized instrumental profiles of the spectrograph plus detector were constructed by co-adding intensity-weighted lines from the comparison spectra. The resultant FWHM for the instrumental profile was 0.072 Å. Details of the spectroscopic data used in this analysis are given in Table 1.

Because typical exposures for these data were long (20–60 minutes), cosmic rays became a real limitation in extracting high S/N data. On average in one CCD image, we record a few hundred "hits" over a 10 minute exposure. When using the Reticon we had to accept the limitations imposed by cosmic rays, editing out the worst and most obviously affected data. With the CCD we scanned each image with the software package CCDSPEC. Broad details of this software, which is at the heart of all our spectroscopic reductions, have been reported elsewhere (Gulliver & Hill 2002). CCDSPEC can operate interactively or automatically to provide mean biases, lamps, and darks with which to correct the stellar data. Spectra are automatically scanned for cosmic rays; in extreme cases up to a few thousand have been removed for a success rate of from 95% to 100%, with the affected data being replaced by interpolation within the surrounding data. Any missed cosmic ray pixels are removed manually. Aside from the cosmic rays, we have only achieved this high S/N because of the time spent in gathering adequate numbers of biases, lamps, and darks for each night's observing. The comparison star, o Peg, used to correct the laboratory gf values was treated similarly and had a mean S/N = 840.

The continuous energy calibration of Vega has recently been revised and is a combination of ultraviolet fluxes from International Ultraviolet Observer and Hubble Space Telescope (HST; Bohlin & Gilliland 2004), and more recently (Bohlin 2007). The 2004 UV results are on the Web site ftp.stsci.edu/instruments/cdbs/cdbs2/calspec.html under the name alpha_lyr_stis_004.fits including various instrumental profiles. We convolve these profiles with our theoretical intensity spectra prior to any analysis.

Profiles of Hγ, Hβ, and Hα were taken from Peterson (1969). These profiles continue to be the best available observations. More modern instruments suffer from insufficient knowledge of the instrumental characteristics. Unfortunately, these profiles only extend 40 Å from each line center, and in main-sequence A stars, the wings of these lines have yet to reach continuum levels. We had to deal with this observational limitation as we modeled the data by including the height and slope of the continuum as unknowns during the fitting process.

We modeled Vega with the program STELLAR which is in turn based on an eclipsing binary modeling code, LIGHT2 by Hill (1979), itself a combination of the formulation of Collins (1963, 1964) and of Rucinscki (1969). STELLAR is written in Fortran for implementation on the OpenVMS system and is generally run on an AlphaServer DS25 and requires a theoretical database which provides specific intensities as a function of wavelength, emergent angle μ, T_eff, log g, and ξ_T. In its present form, the program occupies about 1 GB of memory but this is strongly dependent on the wavelength coverage and the attendant array sizes. Iterations to a solution are displayed on the screen and, depending upon the number of free variables and fitting points, solutions can be achieved in a few minutes for the line spectrum or over hours for the continuum data.

As the first step in the modeling process, a grid of ATLAS9 models (Kurucz 1993) was calculated. This grid now extends from T_eff = 7500 K to 37,000 K in steps of 500 and 1000 K, from log g = 2.0 to 5.0 dex in steps of 0.25 dex, at metallicities of [M/H] = −2.0, −1.5, −1.0, −0.5, −0.3, −0.2, −0.1, 0.0, 0.1, 0.2, 0.3, 0.5, and 1.0 dex, and at ξ_T = 0, 1, and 2 km s⁻¹. We tried to converge all ATLAS9 models to ≲ 1% in the flux error and flux derivative at all 64 depth points. These models used the opacity distribution functions (ODFs) of Kurucz (1993). Although these ODFs have been supplanted by the ODFs of Castelli & Kurucz (2004), we have not yet implemented these into our extensive model grid. Castelli & Kurucz (2004) and R. L. Kurucz (2009, private communication) indicate that no significant differences would be expected at the T_eff of interest for Vega.

As a given stellar application is addressed, input files of synthetic specific intensity line spectra are calculated using the SYNTHE program (Kurucz & Avrett 1981) for the 17 values of μ = 1.0–0.01 normally used in ATLAS9 and for each T_eff, log g, and ξ_T point in the above grid. Memory addressing limitations required that the arrays not exceed (4500, 17, 30, 13, 4) for the five parameter groups (λ, μ, T_eff, log g, ξ_T) of observed spectral data and for the hydrogen line spectra. The wavelength intervals for investigation of Vega are restricted to λλ4510–4550 sampled every 0.01 Å for the metallic line spectra, and 200 Å intervals centered on Hγ, Hβ, and Hα sampled every 0.1 Å for the hydrogen line profiles. Continuum flux files were generated using SYNTHE for fitting to Vega's observed continuous flux distribution. This analysis is restricted to a (maximum) array of (100,000, 17, 6, 5, 1). Now that multi-gigabyte memory plus 64-bit memory addressing are available, these constraints will be relaxed.

The line parameters used were those provided with SYNTHE supplemented by astrophysical gf values determined from o Peg for the λλ4519–4535 interval as well as rest wavelength corrections. The high-resolution spectral region we are analyzing is shown in Figure 1. Anticipating our later results we have included a fit to the observed data. Departures of the fit from the observed spectrum are principally due to laboratory constants that need to be refined and/or determined, as discussed in the next section.

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There are outstanding issues in our knowledge of stellar spectra related to unidentified lines and oscillator strengths that are still uncertain and can bedevil the process of determining T_eff, log g, and stellar abundance [M/H]. We can see the effects of this in the residual fit over regions, λλ4524–4527 and λλ4532.6–4533.4, shown in Figure 1, where the synthetic spectrum provides a poor fit to the observations. Similarly, synthetic spectra of stars exhibiting sharp lines, when fitted by SYNTHE, reveal discrepancies in line strengths and positions that can only be attributed to imperfectly known oscillator strengths and rest wavelengths (see Figure 2).

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We can do nothing about the deficiencies in the laboratory data for very weak features. So we did our best to remove these wavelength regions from our solutions limiting the overall fitting to λλ(4519–4524.5; 4527.6–4530.3; 4531–4535). But since any model parameters derived from the line spectrum are critically dependent upon the gf values (often even the best available laboratory values do not produce satisfactory fits to our spectra) we have made empirical changes to the values of some gf values. In addition, the rest wavelengths in the database are not known with such precision that a generated spectrum will "exactly" match the observations in both position and line strength. This fact means that the two lines that are the main subject of the inclination study (Fe i 4528 Å and Ti ii 4529 Å) must be analyzed independently; the shifts are about 0.004 Å (∼ 0.2 km s⁻¹), but such shifts cannot be ignored. Smaller shifts are seen in the strong lines which we have used to provide us with information on rotational velocity and microturbulence.

Independent astrophysical gf values were determined for 31 lines in the λ λ4519–4540 region using the narrow-lined (Vsin i  = 6.6 km s⁻¹), A1 IV slightly metal-rich star, o Peg, assuming T_eff = 9600 K, log g = 3.60, ξ_T = 1.3 km s⁻¹, and the appropriate abundances (Adelman 1988). The empirical adjustment of these gf values was done iteratively in a time-consuming process. The rest wavelengths of four lines were also adjusted by up to 0.015 Å. After these changes were made, a subsequent analysis of o Peg using STELLAR indicated that the T_eff should be decreased to 9550 ± 10 K and log g increased to 3.75 ± 0.01 with ξ_T unchanged (S. J. Adelman et al. 2010, in preparation). New individualized abundances are also available in S. J. Adelman et al. (2010, in preparation). For the highly temperature sensitive Ti ii 4529 Å line, the T_eff is most important. Using STELLAR for o Peg, a new gf value was determined for this line by having STELLAR interpolate among a set of synthetic spectra, each for a different gf value; the log gf changed from −0.822 to −0.908 dex.

Because of the uncertainty that then existed in the value of ξ_T for o Peg, the gf values of the three strong lines, Fe ii 4520 Å, Fe ii 4522 Å, and Ti ii 4534 Å, were adjusted to fit the observed Vega profiles using the best spherical model of Vega (Castelli & Kurucz 1994) then available; T_eff = 9550 K, log g 3.95 with ξ_T = 1.1 km s⁻¹. These adjustments do not significantly influence the non-spherical model that we develop since the adjustments address primarily the overall strengths of the lines and not their shapes. The adjustments for the tabulated values of the Fe ii lines were small, [−0.05] and [−0.03] dex, respectively, while that for the Ti ii line was relatively large at [−0.34] dex. It should be noted that for these adjustments in the gf values, and that of Ti ii 4529 Å above, the assumed underabundance of both Ti and Fe was [−0.5] dex, as chosen from the set of [M/H] abundances of Kurucz. Fortuitously, this choice was within the errors of the results of Adelman & Gulliver (1990), who found that Ti and Fe were underabundant by [−0.46] and [−0.56] dex, respectively, and of Hill & Landstreet (1993) who found [−0.46] and [−0.51] dex.

For the analysis of the continuous flux, separate input files of continuum specific intensity generated by SYNTHE at the standard 17 values of μ = 1.0–0.01 and 93,000 wavelengths between 120 and 1050 nm for the grid of models are used. The continuous flux and the line spectra are convolved with the appropriate instrumental profile, which can range from an analytical function to a digitized observed profile.

To create the modeling program STELLAR a number of elements had to be integrated: a physical model, a database of line profiles and continuum files, an integration or quadrature scheme, optimizing software, and graphical output. The physical model is based on the formulation given by Collins (1963, 1964) with the model and quadrature software adapted from the program LIGHT2 (Hill 1979). The program control involves keywords adapted initially from LIGHT2 and later from the software imbedded in the spectroscopic reduction software known under the name REDUCE, see, for example, Hill & Fisher (1986) and many other papers in the same publication series.

The physical model generates the run of temperature and gravity over the surface of a star with these values defined initially at the pole and changing as a function of rotational velocity and a gravity brightening exponent given as 0.25 or 0.08 for radiative or convective atmospheres, respectively. The rotating star is then viewed at some inclination angle. The integrations are performed using Gauss–Legendre quadrature with 64 times 64 integration points in the θ and ϕ axes, respectively. The choice of quadrature algorithm results in exceptional integration accuracy as previously shown in LIGHT2 (Hill 1979). Thus for each surface integration point T_eff, log g, and μ are calculated and by successive parabolic interpolations in the database within T_eff, log g, μ, and ξ_T we find intensity as a function of wavelength. By summing these weighted intensities, we generate a theoretical spectrum that can be compared with observation. The optimizing follows the method of Marquardt given in Bevington et al. (1967) as outlined in LIGHT2 by Hill (1979). We can choose to solve for any variables selected from T_eff, log g, ξ_T, Vsin i, i, V_eq, radial velocity (RV), continuum height, and slope.

Obviously, depending on our needs, some of these variables are held constant. For example, the continuous flux analysis was limited to determining T_eff and log g, the values of which are assumed in the analysis of the high resolution data. In addition to the models, we incorporated results of parallax measurements (van Leeuwen 2007, p. 350) and interferometry (Ciardi et al. 2001; Aufdenberg et al. 2006) to aid us in completely specifying Vega in terms of mass, radius, temperature, and luminosity.

We wanted to create a program that could run with as little intervention as possible, a goal that has largely been achieved. The wavelength scale is defined by the observational data, with any extra binning to create a common scale between the observations and theoretical database taking place automatically. One can simply generate a line spectrum (either high-resolution or lower resolution hydrogen profiles) or a continuum energy distribution by specifying T_eff, log g, ξ_T, Vsin i, RV, and i. Although a solution is possible for any or all of these variables depending on the situation, finding the first solution can be difficult. In our original analysis, circa 1993, we believed that the line profile was dependent on i and we laboriously searched through the entire range of inclinations until we discovered that the flat-bottomed profiles we see in Figure 1 could only be produced for i < 10°. The flat-bottomed profiles analyzed here are matched by many other lines in the Vega spectrum. In Figure 3, we display the spectrum from λ λ4175–90 to demonstrate this.

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STELLAR can be run in a variety of modes using a series of keywords, for example, to plot the data, generate a theoretical spectrum, or solve an observed spectrum for various parameters. The results can be displayed in graph and text format as the program runs to culmination and the results are generated to disk and/or screen. To run the program one specifies which of these modes is desired: the data files (FITS for the high-resolution data and ASCII files for the continuum and hydrogen lines), the theoretical continuum and line files, the parameter values, the unknowns to be solved for, and the desired output (graphical and/or printed). Keywords are generally combined in a command file containing the commonly used input data, parameter values, and specific data files to minimize the number of keystrokes needed to begin execution. The remainder of the input is reserved for specifying unknowns to be determined and controlling run-time plotting or printing options.

In our previous solutions, problems have been identified involving erroneously derived masses and fractional breakup velocities (D. M. Peterson 2003, private communication; Aufdenberg et al. 2006). After one of us (G.H.) got the point, we corrected the code, but that did not alleviate the discrepancy between solutions. Aufdenberg et al. (2006) derived a model with an i ∼ 47 and a breakup fraction of about 0.90, in sharp contrast to our second model where i ∼ 8° and the breakup fraction is 0.47 (Hill et al. 2004). Part of the difference here was our incorrect formulation of the breakup fraction and another was the inadequacy of the model, a problem that also bedeviled our present analysis for months. Thus in tackling the solution again with more data and a corrected code, we had hoped to be able to confirm Aufdenberg's results which were based on high spatial resolution data and was almost totally independent of spectroscopy. But attempting to model the system with i < 5° revealed an immediate problem, one noted by Aufdenberg himself (J. P. Aufdenberg 2006, private communication). At low inclinations, the contribution to an observed profile by stronger, cooler lines arising from the equatorial regions produces marked reversals in the core of the weak lines—the very reason for flat-bottomed lines (see Figure 4)—a problem that Aufdenberg hoped might be resolved by invoking some new physics in the model such as differential rotation.

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We began by running extensive grids for all the data (continuous flux, hydrogen line, and high resolution spectra) in parameter space searching for some combination of T_eff, log g, and Vsin i that might produce a minimum at low i. In all cases, we saw minima between 6° and 7° but follow-up solutions were never adequate; much better fits were achieved at inclinations greater than 8°. But at these inclinations the best fits did not do justice to the extremely high quality of our data. The code was then altered to include differential rotation modeled after the Sun. It was further altered to include various ad hoc variations in the gravity darkening exponent between 0.25 and 0.08, including a linear variation between these two values at the pole and equator, respectively. Over many months running extensive grids and trial solutions, it was hoped that some combination of parameters might produce flat-bottomed profiles for the weak lines and excellent fits to the three strong lines. Neither differential rotation nor a variable gravity darkening exponent resulted in a satisfactory outcome and both were abandoned as a potential solution to line center reversals, leaving us with relatively high inclinations and fits that were not good enough, as well as inconsistent between the data sets. Even with an inadequate model the fact that three independent sets of data all indicated a solution near 6°–7° was encouraging, though it seemed an impossible task to find a fulfilling solution that satisfied the data at i < 6°.

The thought process giving rise to the solution of the problem was blindingly simple: Vega has high rotational velocity and consequent large temperature gradients from pole to equator; meridional circulation or some form of convection may be present; macroturbulence (ζ) should be investigated. Indeed, STELLAR already included a macroturbulence algorithm (courtesy of J. B. Rice 1997, private communication, after Gray 1992). Once Vega was modeled to include this, the problems of fitting accuracy disappeared and more stable solutions were immediately achieved; the fits to the strong lines improved (rms σ < 0.001) as well as those of the weak lines (σ < 0.0004). The poorer fit to the strong lines results from fitting the spectrum between λλ4519 and 4535 (excluding very weak and unknown lines between λλ4524.5 and 4527, 4530.4 and 4532) when there are obvious small velocity shifts between the lines; see the residuals characteristic of a wavelength error near the line cores in Figure 1 earlier. Macroturbulent (ζ) velocities of about 7 km s⁻¹ were needed to achieve the fits, a value we thought might well be plausible for a star rotating at about 90% of breakup velocity and where there had to be extreme meridional flow. In deriving the solutions shown below, we have made the ad hoc assumption that the tangential and radial components of ζ are the same; solutions erroneously made where they were different produced significantly poorer fits. The effect of not including macroturbulence is shown in Figure 5, which can be compared to the fit in Figure 1 that includes macroturbulence. The reader may gauge why we had so much trouble coming to a satisfactory solution with the earlier model.

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Recently, Yoon et al. (2008) have also reported independently an estimate for macroturbulence of 10 km s⁻¹ based upon the need to introduce additional broadening to lines such as Mg i 4703 Å. The same line was investigated using our Vega model with the Mg abundance of S. J. Adelman et al. (2010, in preparation) and T_eff, log g, i, and ξ_T values fixed, allowing Vsin i and macroturbulence (ζ) to vary. Using STELLAR, the log gf of Mg i 4703 Å was allowed to vary, producing a best fit to the line strength for log gf = −0.570, as compared to the tabulated value of −0.666. The excellent agreement between the observed and model profiles is shown in Figure 6. The Vsin i and ζ values were 19.9 ± 0.2 and 7.3 ± 0.2 km s⁻¹, respectively. The value of ζ agrees with those in Table 3 while Vsin i is slightly lower.

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The ability to fit the continuous flux was the last function to be incorporated into STELLAR. Naturally, we tried to see if these data would yield a value of i also. Prior to the 2004 HST calibration, we used the earlier one by Bohlin et al. (2001). In this, we were disappointed although ironically the earlier calibration of continuous flux (Bohlin et al. 1990) had successfully yielded a satisfactory estimate of i. The observed continuous data we used were contained in the fits file alpha_lyr_stis_001 (Bohlin et al. 2001) converted to H_ν but left as vacuum wavelengths for comparison with the ATLAS9 output. We tried to match the resolution of the observational data to the output from ATLAS9 which used "boxcar" smoothing that we reproduced in STELLAR. As we experimented, we found the continuous flux solution was quite sensitive to the smoothing so we made every attempt to reproduce ATLAS9 output as realistically as possible but these efforts did not produce a solution for i. In the end, we abandoned the search for an inclination from these data.

We were eager to try the new calibrations (Bohlin & Gilliland 2004) and subsequently the latest (Bohlin 2007) which became available as we were finishing the paper. Solving these calibrations was more demanding in that the instrumental profiles were gluttonous in computing time but also complex to integrate with our theoretical data. As a start, we ran a grid of solutions covering polar temperatures (9500–10,300 K), log g (3.5–4.5 dex), and inclinations (5°–10°) with a Vsin i of 21.0 km s⁻¹. The task took days to run but indicated a strong minimum near i∼ 55. Values of polar T_eff, polar log g, and i near 10,000 K, 4.0 (dex), and 55, respectively, were used as the starting values for both sets of continuous flux data and solutions were readily found with polar T_eff of 10,000 K, polar log g of 4.04, and i of 61 for the 2004 data and 10,000 K, 4.04, and 53 for the 2007 data (see Table 3). We perturbed the solutions by using starting values 100 K higher and lower but the solutions were stable. A notable feature of the fit seen in Figure 7 is the relatively poor agreement between the models and the hydrogen lines, prompting us to exclude them from the fit.

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Because the value of Vsin i affects the fractional breakup rate by varying the distension of the star this solution is dependent on the spectroscopy; we turn to this now knowing that the whole solution must be iterated a number of times.

5.3.1. High Dispersion Data

At inclinations between 5° and 6° we are faced with a distorted and distended star whose shape is strongly dependent on the equatorial velocity, values which alter the polar T_eff and log g as i changes. The equatorial velocity also depends on the value of rotational velocity thus linking the continuum analysis to the spectroscopy. But the spectra lack sufficient lines to give reliable temperatures and surface gravities; this is the root cause of the original criticism by D. M. Peterson (2003, private communication) of our low value of log g (3.75) and hence the derived mass. What we can get from the spectra then are Vsin i, ξ_T, ζ, RV, and i as well as small corrections to the original continuum placement made during the spectrum rectification process. The spectrum used comprises three strong lines whose velocities are within 0.2 km s⁻¹ of each other and two weak lines differing by ∼ 0.2 km s⁻¹. We decided that the strong lines taken as a group should be used to find Vsin i, ζ, and ξ_T as well as the continuum correction; the latter amounts to 0.14%. Analyzed independently, the two weak lines would give Vsin i, ζ, and i. It was hoped that the parameters Vsin i and ζ would be consistent between the weak and the strong lines, thus giving some assurance that we had gotten things right. Anticipating later results, we found that adopting ξ_T of 0 km s⁻¹ in the weak lines produced a reduction of 0.5 in the σ² results and a fit to the weak lines noticeably better than when ξ_T was held to that found in the strong lines. But nothing is ever simple—and this analysis was never simple; the weak line at 4529 Å shows low level variability and only three of the five summed spectra available were judged to be compatible. This left five spectra combining three lines each for the strong line analysis and eight lines for the weak line analysis. In Figure 8, variability among the five spectra at the level of a few thousandths of the continuum is most obvious in the weakest lines.

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The solution proceeded iteratively by determining Vsin i, ξ_T, and ζ from the spectra as a function of i assuming approximate polar values for T_eff and log g. The results for Vsin i were included in the continuous flux analysis to give better values of T_eff and log g as a function of i. These polar values were then used to analyze the spectra yielding Vsin i as a function of i and so on. Finally, we had a table of Vsin i, polar T_eff, and polar log g as a function of i. Then, the weak lines were each analyzed for Vsin i, ζ, and i, fixing ξ_T to 0 km s⁻¹. These results happily converged to those from the strong lines and yielded an i of 57 ± 01 (95% confidence interval: 56, 58). Given the dependence of Vsin i on i, we solved the continuous flux data again for i and found a value of i = 53 ± 02 (confidence interval: 51, 55). We did not adopt an average from the spectra and continuous flux data because of the uncertain results stemming from the 2004 and 2007 continuum calibrations. The results are given in Table 2; the errors quoted there and throughout the paper are in terms of a 95% confidence interval.

Table 2. A Summary of Spectroscopic Results for Pole-on Model

Vsin i		Strong Lines
km s−1	ξT	ζ	σ2	File Name
	km s−1	km s−1
21.0 ± 0.1	1.11 ± 0.01	6.5 ± 0.2	1.7e-6	RVEGASC4520S.fts
21.1 ± 0.1	0.92 ± 0.01	7.7 ± 0.2	2.0e-6	RVEGA458803.fts
21.0 ± 0.1	1.01 ± 0.01	7.2 ± 0.2	1.7e-6	RVEGA458804.fts
21.1 ± 0.1	1.04 ± 0.01	6.5 ± 0.2	1.9e-6	RVEGA458805.fts
21.1 ± 0.1	1.06 ± 0.01	7.3 ± 0.2	1.5e-6	RVEGA458806.fts
Mean Values
21.1 ± 0.1a	1.03 ± 0.06a	7.0 ± 0.5a
		Weak Lines
Vsin i	i°	ζ	σ2	File Name
		Fe ii 4528 Å
20.2 ± 0.1	5.55 ± 0.02	7.6 ± 0.5	3.0e-7	RVEGASC4520S.fts
20.4 ± 0.1	5.72 ± 0.04	8.2 ± 0.3	1.5e-7	RVEGA458803.fts
20.4 ± 0.1	5.45 ± 0.04	9.7 ± 0.3	1.3e-7	RVEGA458804.fts
20.7 ± 0.1	5.70 ± 0.04	7.2 ± 0.3	1.9e-7	RVEGA458805.fts
20.4 ± 0.1	5.54 ± 0.03	8.4 ± 0.4	1.8e-7	RVEGA458806.fts
		Ti ii 4529 Å
20.7 ± 0.2	5.68 ± 0.07	6.7 ± 0.6	2.5e-7	RVEGASC4520S.fts
21.0 ± 0.1	5.74 ± 0.06	8.2 ± 0.4	1.5e-7	RVEGA458804.fts
21.3 ± 0.1	5.80 ± 0.05	7.2 ± 0.4	1.2e-7	RVEGA458806.fts
Mean values
20.6 ± 0.2a	5.7 ± 0.1a	7.9 ± 0.6

Note. ^a95% confidence interval.

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In the preliminary stage of considering the result of Aufdenberg et al. (2006), we noticed that the grid gave much better fits for the hydrogen lines at i ∼ 6°–7° than at higher angles, and this provided an additional spur to keep trying to improve a model that was inadequate. Knowing that the profiles give poor values of T_eff, we fixed the parameters to those resulting from the above analysis and solved for log g. The results are given in Table 3 and yield a mean polar gravity of 4.01 ± 0.03 dex. The fits to the data are quite superb (see Figure 9) in that the fit is almost indistinguishable from the data but unfortunately the values of log g are not in great accord. As with the inclination, we have not included log g from the hydrogen lines in the adopted mean log g.

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The various results are presented in Table 3 along with our derived values. T_eff and log g come from the continuous energy flux solution and the other parameters from the line analysis. The value of ξ_T = 1.03 km s⁻¹ can be compared to that of 0.6 km s⁻¹ from Adelman & Gulliver (1990) using classical techniques or 1.0 km s⁻¹ of Hill & Landstreet (1993) from a spectrum fitting approach. The adopted parameters are presented in Table 4. The mean log g value of 3.95 is equal to the value normally used to model Vega, log g = 3.95 (Castelli & Kurucz 1994), and the mean T_eff = 9560 ± 30 K is nearly equal to the direct determination of T_eff = 9555 ± 111 K by Ciardi et al. (2001).

Attempts to fit Vega to a spherical model have been less than satisfactory in that they produce IR fluxes that are too low by 3%–6% than those observed (see papers by Ciardi et al. 2001; Castelli & Kurucz 1994, and references therein). We have made similar fits to the data as described above but using a spherical model with i = 90°. The spectroscopic results are given in Table 5 and summarized in Table 6. Although we include the spherical results here for comparison with the work of others, they patently do not match the profiles of the two weak lines—see Figure 10.

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The results of our analysis are shown in Table 7. In addition, we have gathered together the astrometric data along with those from this present paper. The mass comes from M = gR²/G, the breakup velocity V_crit = (gM/1.5R_p)^1/2, and the astrometric T_eff is given by T_eff = 2341(F_bol/Θ²)^0.5. In comparing our results with that of Aufdenberg et al. (2006), we find a difference in inclination of 08. Moreover if we replace our value (his Equation (14)) of Vsin i with that which he adopted, we end up trying to reconcile his inclination of 44 with our 565. While the difference is small, the effects on the predicted spectrum are large. At 565 we have no central reversal, but at i ∼ 44 it is large, and despite many attempts we found it impossible to satisfactorily fit the observations. However, fretting over 1° should be a happy circumstance. When further developments to STELLAR are concluded, we will revisit the star to examine the other weak lines that are susceptible to inclination and see what that analysis may yield.

It is unfortunate that the premiere photometric standard upon which our spectrophotometry is anchored should be the object it is—a rapidly rotating star seen almost pole-on and surrounded by a ring of dust—but this circumstance need not be an impediment to its use as a standard. It simply means that the fitting model should be more complex than a spherical one and in any case is far more testing of the models than a simple spherical comparison. The model flux calibration is available at http://www.brandonu.ca/physics/gulliver/atlases.html. A comparison of the relative IR flux from 0.3 to 10 μm of the rotating model compared to that of the spherical model is shown in Figure 11, including the observed IR fluxes of Cohen et al. (1992). The actual observed and calculated IR fluxes in units of W cm⁻² μm⁻¹ at the specific wavelengths are presented in Table 8 including fluxes for the 9400 K, 3.90 spherical model of Cohen et al. (1992). The results indicate that the observed fluxes exceed the model fluxes by 1%–2% and that the rotating model flux exceeds the spherical flux but only by about 0.2%. Clearly, a nearly pole-on rapidly rotating model cannot account for the excess observed flux at these wavelengths.

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Experience in modeling Vega has shown that there is considerable "overhead" before we can even embark on synthesizing spectra with STELLAR. The database of atmospheres and spectral synthesis has to be created for ranges of temperature, log g, [Fe/H], and ξ_T suitable for the task in hand. If abundances or a detailed spectroscopic analysis similar to the present paper are required, a separate investigation as to the reliability of the gf values for lines within the wavelength interval is also necessary. Obviously, we want to remove these limitations as well as those imposed by memory restrictions in our computer.

Thus, the development of STELLAR is continuing along several fronts. The need to maximize the wavelength range we can handle has increased the memory requirements which we have hitherto mitigated by selecting only a limited set of synthetic spectra to cover the expected range of the various parameters. We also need to be able to freely select the wavelength ranges we want to analyze without the restrictive creation of a database. The ultimate goal will be to incorporate atmosphere and synthesis codes within STELLAR that do not require the generation of a time-consuming, wavelength-limited database based on scaled solar abundances which themselves are a limitation. In fact, the synthesis code SYNSPEC (Hubeny & Lanz 1997) is already in a version of STELLAR as a subroutine running with both Kurucz and TLUSTY (Hubeny 1988; Hubeny & Lanz 1995) atmospheres. Incorporating both codes within STELLAR also facilitates the calculation of revised astrophysical gf values for a number of "normal" stars with well-determined parameters and abundances, thus eliminating the tortuous path required for us even to begin a detailed analysis of a star such as Vega. Ultimately these developments will give us the freedom to calculate theoretical spectra freely for any T_eff, log g, and abundance.