Determination of Vega’s rotational velocity based on the Fourier analysis of spectral line profiles

ABSTRACT

While it is known that the sharp-line star Vega (v_esin i ∼ 20 km s⁻¹) is actually a rapid rotator seen nearly pole-on with low |$i\, ({\lt}10^{\circ })$|⁠, no consensus has yet been accomplished regarding its intrinsic rotational velocity (v_e), for which rather different values have been reported so far. Methodologically, detailed analysis of spectral line profiles is useful for this purpose, since they reflect more or less the v_e-dependent gravitational darkening effect. However, direct comparison of observed and theoretically simulated line profiles is not necessarily effective in practice, where the solution is sensitively affected by various conditions and the scope for combining many lines is lacking. In this study, determination of Vega’s v_e was attempted based on an alternative approach making use of the first zero (q₁) of the Fourier transform of each line profile, which depends on K (temperature sensitivity parameter differing from line to line) and v_e. It turned out that v_e and v_esin i could be separately established by comparing the observed |$q_{1}^{\rm obs}$| and calculated |$q_{1}^{\rm cal}$| values for a number of lines of different K. Actually, independent analysis applied to two line sets (49 Fe i lines and 41 Fe ii lines) yielded results reasonably consistent with each other. The final parameters of Vega’s rotation were concluded as v_esin i = 21.6(±0.3) km s⁻¹, v_e = 195(±15) km s⁻¹, and i = 6.4(±0.5)^○.

1 INTRODUCTION

The spectrum of Vega (=α Lyr = HR 7001 = HD 172167 = HIP 91262; spectral type A0 V) shows a sharp-line nature indicating a small projected rotational velocity (v_esin i ∼ 20 km s⁻¹, where v_e is the equatorial rotation velocity and i is the angle of rotational axis relative to the line of sight), which is rather unusual among A-type main-sequence stars (many of them showing v_esin i typically around ∼100–300 km s⁻¹). It is nowadays known, however, that this star is actually a rapid rotator with large v_e like other A stars and the apparent smallness of v_esin i is simply ascribed to low i (i.e. this star happens to be seen nearly pole-on).

Its intrinsic rotational velocity can be observationally determined by detecting the gravity darkening effect, because it becomes more exaggerated as v_e increases. The mainstream approach used for this purpose is to analyse the shape of spectral lines, because lines of a specific group (e.g. weak Fe i lines) show a characteristic feature (i.e. flat-bottomed profile), which is caused by the lowered temperature near to the gravity-darkened limb (see e.g. fig. 1 in Takeda, Kawanomoto & Ohishi 2008a). Alternatively, in order to establish v_e, the extent of gravity darkening can be estimated from the brightness distribution on the stellar disc by direct high-resolution interferometric observations.

Beginning from 1990s and especially in the period around 2010, quite a few determinations of Vega’s v_e based on these two methods have been tried by various investigators as summarised in Table 1. However, the resulting literature values of v_e considerably differ from each other as seen from this table. Although the large discrepancy amounting to |${\gtrsim}100$| km s⁻¹ (from ∼160 to ∼270 km s⁻¹) seen in early 2000s has been mitigated up to the present, they are still diversified between ∼170 and ∼230 km s⁻¹ (which are the published results since 2008).

Meanwhile, the discovery of magnetic field in Vega by spectropolarimetry (Lignières et al. 2009) provided a new means to measure v_e, because such a Zeeman signature would show cyclic variation due to rotation; that is, the rotational period (P) may be directly evaluated by applying a period analysis to time-series data of spectropolarimetric observations, from which v_e is derived as v_e = 2πR_e/P by using an appropriately assigned R_e (equatorial radius). Following this policy, Vega’s rotation period was determined within several years after 2010, as shown in Table 1. Although this method is expected to establish P precisely, these published data are not necessarily in good agreement but somewhat discrepant by ∼±10 per cent (i.e. ∼±20 km s⁻¹ around v_e ∼ 200 km s⁻¹). Therefore, even such an independent technique (which is essentially different from the other in the sense that any modelling of gravity-darkened star is not required) has not yet significantly improved the situation regarding the ambiguity in v_e.

Accordingly, it is desirable to redetermine v_e of Vega with higher reliability than before, in order to clarify which of the recent results (between ‘low-scale’ value of ∼170–180 km s⁻¹ and ‘high-scale’ value of ∼220–230 km s⁻¹) is more justifiable.

Here, it may be worthwhile to mention the weak point of line profile analysis, which was once employed by the author’s group (Takeda, Kawanomoto & Ohishi 2008b; hereinafter referred to as Paper I) to evaluate Vega’s v_e. According to our experience, to derive v_e by searching for the best fit (minimising χ²) between the observed and modelled line profiles for a selected line feature (e.g. well-behaved weak Fe i line showing a flat-bottomed profile) is not so hard. However, there is no way to estimate how much uncertainty is involved in such a specific solution. Actually, since χ² residual is a rather broad function of v_e and quite vulnerable to a slight imperfection (e.g. improper placement of continuum level, existence of weak line blending, irregular noise in observed data, etc.), because extremely subtle difference of profile shape is concerned (typically of the order of ∼10⁻³ in unit of the continuum; cf. figs 4 and 5 in Paper I), an erroneous v_e solution is easily brought about (or even no solution is found). Therefore, it was decided in Paper I to analyse the profiles of a large number of lines (87 lines of neutral species and 109 lines of once-ionized species) with a hope of hitting as many correct solutions as possible. Nevertheless, from a critical point of view, the result obtained in Paper I was not very satisfactory due to the following reasons: (i) The final solution (v_e = 175 km s⁻¹) was simply selected from nine models (where v_e was varied from 100 to 300 km s⁻¹ with an increment of 25 km s⁻¹) as the one corresponding to the highest frequency of χ² minimum for the case of neutral lines, so an ambiguity of ∼20 km s⁻¹ due to the coarseness of model grid is inevitable from the start. (ii) While lines of neutral species yielded a Gaussian-like frequency histogram centred around 175 km s⁻¹ (cf. fig. 6a in Paper I), those of ionized species (many of them have ‘non-flat-bottom’ profiles) show a near-flat distribution (cf. fig. 6b in Paper I); this means that the latter set of ionized lines were almost useless because they made no contribution to the determination of v_e.

Consequently, the conventional line-profile matching in the wavelength domain applied in Paper I was not necessarily suitable for such a very delicate problem. In order to make a further step towards improving the precision, a more efficient approach has to be invoked, in which many lines of different properties can be effectively combined to increase the reliability of v_e solution while providing a reasonable procedure for error estimation.

Recently, in an attempt to estimate the intrinsic rotational velocity of Sirius A, Takeda (2020; hereinafter referred to as Paper II) made use of the first zero frequency (q₁) in the Fourier transform of the line profile. It then revealed that this quantity can be used for measuring the gravity darkening effect because it sensitively responds to a slight variation of the line profile; actually, q₁ was found to vary almost monotonically with v_e (inducing a gravity darkening). While how q₁ reflects a change of v_e naturally differs from line to line depending on its property, it was found to be the sensitivity of line strength (W) to temperature (T), which is represented by the parameter K (≡log W/log T), that essentially controls the v_e dependence of q₁. Therefore, since information of v_e may be extracted from the comparison of the observed |$q_{1}^{\rm obs}$| with a corresponding set of |$q_{1}^{\rm cal}\,(K,v_{\rm e})$| calculated for this line on the models of different v_e, the best solution of v_e (along with its probable error) can be established by combining many lines of different K. This technique turned out successful, and in Paper II it was concluded that Sirius A is an intrinsically slow rotator (⁠|$16 \le v_{\rm e} \lesssim$| 30–40 km s⁻¹).

Motivated by this achievement, the author decided to apply this method to analysing the spectral line profiles of Vega, in order to revisit the task of determining its v_e as done in Paper I, hoping that a result of higher accuracy would be obtained, so that the diversified literature values may be verified. The purpose of this article is to report the outcome of this reinvestigation.

2 OBSERVATIONAL DATA

2.1 Selection of lines and their profiles

Regarding the basic observational material of Vega, the high-dispersion spectra of high signal-to-noise ratio (S/N ∼2000) and high spectral resolving power (R ∼ 100 000) were used as in Paper I, which were obtained at Okayama Astrophysical Observatory by using the HIDES spectrograph attached to the 188 cm reflector and published by Takeda, Kawanomoto & Ohishi (2007).

The selection of lines to be used for the analysis was done by following almost the same procedure as adopted in Paper II (cf. section 2.2 therein), where it was decided to employ only lines of neutral and ionized Fe in order to maintain consistency with Paper II. As a result, a total of 90 lines (49 Fe i and 41 Fe ii lines) were eventually sorted out,¹ which are listed in Table 2. The observed profiles of these lines are displayed in Fig. 1, and their original data are available in ‘obsprofs.dat’ of the supplementary material.

The equivalent widths (W^obs) of these 90 lines were measured by the Gaussian fitting method, which are in the range of 1 mÅ |$\lesssim W^{\rm obs} \lesssim 40$| mÅ. As the ‘standard’ plane-parallel model atmosphere for Vega, Kurucz’s (1993) ATLAS9 model with T_eff = 9630 K, log g = 3.94, v_t = 2 km s⁻¹ (microturbulence), and [X/H] = −0.5 (metallicity) was adopted in this study as in Paper I, which well reproduces the spectral energy distribution. By using this model along with the atomic data taken from Kurucz & Bell (1995)’s compilation, the abundance (A^std; called as ‘standard abundance’) was derived from W^obs for each line.

2.2 Zero frequencies of Fourier transforms

Then, the Fourier transform d(σ) of the line depth profile D_λ (≡1 − F_λ/F_cont) was calculated for each line as done in Paper II (cf. section 2.3 therein), and the first and second zero frequencies (σ₁ and σ₂, in unit of wavelength^-1) were measured from the cuspy features of |d(σ)|, which were further converted to wavelength-independent quantities (q₁ and q₂, in unit of velocity⁻¹) for convenience by the relation q ≡ σ(λ/c) (c: velocity of light). The resulting q₁ and q₂ are plotted against the line parameters in Fig. 2, from which the following arguments can be made:

• These zero frequencies show an appreciable line-dependent scatter; especially, those of a fraction of Fe ii lines are remarkably higher in comparison with the theoretical values expected from the classical rotational broadening function (cf. Fig. 2a)

• This implies that the conventional Fourier analysis of spectral line profiles, which assumes that the observed profile is expressed by a convolution of the rotational broadening function and thus the zero frequency of the rotational broadening function (dependent on v_esin i) should be simply inherited in the observed transform equally for any line, is no more applicable for precise v_esin i determination in this case.

• The cause for this scatter in q₁ as well as q₂ is that they tend to systematically increase with K as shown in Fig. 2(b). This is because the line profile characteristics are determined by this T-sensitivity parameter; that is, a line of small/negative K (e.g. weak Fe i line of low excitation) shows a boxy U-shape, while that of large/positive K (e.g. weak Fe ii line of high excitation) has a sharp V-shape. Such a difference in the line profile (even if very subtle) is reflected by the position of zero frequency, which is actually verified by theoretical calculations based on the gravity-darkened rotating star model (cf. Section 3.3).

• These q₁ and q₂ also show some systematic trends with respect to χ_low (Fig. 2c) and W^obs (Fig. 2d); however, they can be reasonably explained by the dependence of K on χ_low and W^obs, as discussed in appendix A2 of Paper II. Accordingly, it is the difference in K that causes the line-by-line different characteristics in the profile (and the zero positions).

The atomic line data and the values of W^obs, K, q₁, and q₂ for 90 lines are presented in Table 2. Besides, more complete data (including A^std and the main lobe height as well as the first sidelobe height) are summarised in ‘obsparms.dat’ of the supplementary material.

3 MODELLING OF LINE PROFILES

3.1 Adopted model parameters

Regarding the simulation of theoretical line profiles of a gravity-darkened rotating star, this study follows the same assumptions and procedures (including the adopted set of parameters for Vega) as described in Paper I, where the stellar mass (M), rotational velocity at the equator (v_e), inclination angle of rotation axis (i), polar radius (R_p), and polar effective temperature (T_eff,p) are the fundamental parameters to be specified.

The mass was fixed at M = 2.3 M_⊙. 10 v_e values were chosen as 22, 100, 125, 150, ⋅⋅⋅ 275, and 300 km s⁻¹ (numbered as models 0, 1, 2, 3, ⋅⋅⋅, 8, and 9), and the corresponding i values were derived from the assumption of v_esin i = 22 km s⁻¹ (which is a reasonable value for Vega). Based on the requirement of spectral energy distribution, R_p and T_eff,p can be expressed as second-order polynomials in terms of v_e (cf. equations 1 and 2 in Paper I). The model parameters for each of the 10 models are summarised in Table 3, which is the same as table 1 in Paper I. Note that model 0 (v_e = 22 km s⁻¹ and i = 90^○) is a special model different from others, in the sense that it is a spherically symmetric rigid model where the gravity effect (darkening and distortion) is intentionally suppressed. This model 0 is almost equivalent to the ‘standard model’ mentioned in Section 2.1.

3.2 Simulation of line profiles

The emergent line flux profile was simulated with the program CALSPEC (cf. section 4.1 in Paper I) by integrating the intensity profile at each point on the visible disc, which was generated by using the local model atmosphere corresponding to T_eff(Θ), g(Θ), v_t = 2 km s⁻¹, and [X/H] = −0.5 (where Θ is the co-latitude).

Here, a point to notice is how to assign the elemental abundance (A). If A^std (standard abundance derived from the classical plane-parallel model) is simply used, the equivalent width of the calculated line profile (W^cal) turns out generally stronger than W^obs because of the gravity darkening effect,² and this discrepancy progressively increases towards higher v_e (as can be recognized in figs 4 and 5 in Paper I). In Paper I, this problem was circumvented by renormalising the calculated profile (cf. equation 7 therein) so as to force W^cal = W^obs, although its validity was not necessarily clear.

Fortunately, this equality does not have to be strictly realised in the present case of Fourier analysis, because it is the ‘characteristics’ of the line shape that are essential. Accordingly, the following procedure was adopted in this study:

• First, the provisional equivalent width |$W_{*}^{j}$| (j = 0, 1, 2, ⋅⋅⋅, 9) was calculated with CALSPEC for each model by using A^std.

• Then, the corresponding abundance |$A_{*}^{j}$| was derived from |$W_{*}^{j}$| with the help of Kurucz (1993)’s WIDTH9 program by using the standard plane-parallel model (cf. Section 2.1).

• The abundance difference defined as |$\Delta A^{j} \equiv A^{\rm std} - A_{*}^{j}$| (which is mostly negative) is used as abundance correction to be applied to A^std; that is, the abundance actually adopted in CALSPEC for calculating the profile corresponding to model j is A^std + ΔA^j.

It should be remarked that this procedure is based on two assumptions that (i) the classical curve of growth (log W versus A relation) for the plane-parallel model is applicable even for the gravity-darkened case, and (ii) the absolute change of log W in response to perturbation by ±ΔA around A^std in this curve of growth is almost the same (i.e. locally linear). Despite these rough approximations, the discrepancy between W^cal and W^obs seen for the case of simply using A^std is considerably reduced by application of this correction (ΔA), as shown in Fig. 3.

3.3 Fourier transform and the trend of first zero

By using such corrected abundances, the theoretical line profiles were simulated for each of the 10 models and their Fourier transforms were computed, from which |$q_{1}^{j}$| and |$q_{2}^{j}$| (j = 0, 1, 2, ⋅⋅⋅, 9) were measured. These |$q_{1}^{j}$| and |$q_{2}^{j}$| values along with the adopted abundance corrections (ΔA^j) for all 90 lines are given in ‘calparms.dat’ of the supplementary material.

As demonstrative examples, the simulated profiles of Fe i 5133.681 (K = −13.76) and Fe ii 4951.584 (K = +2.92) lines and their Fourier transform amplitudes, which were calculated for models 0, 1, 3, 5, 7, and 9, are illustrated in Fig. 4, where the observed data are also overplotted for comparison. It can be seen from Fig. 4 that the behaviours of zero frequency for these two lines of different K are just the opposite in the sense that q₁ of Fe i 5133.681/Fe ii 4951.584 moves towards lower/higher direction as the gravity darkening effect is enhanced with an increase in v_e.

From now on, our discussion focuses only on the first zero frequency (q₁), which is less affected by measurement errors or noises in comparison to q₂. In order to elucidate the trend of q₁ as a function of K and v_e, the q₁ values are plotted against K in Figs 5(a)–(f) (each corresponding to models 0, 1, 3, 5, 7, and 9, respectively). Besides, the linear regression lines (determined from the q₁ versus K plots for Fe i and Fe ii lines separately) are also shown in each panel, and these regression lines for all models are depicted together in Fig. 5(g). An inspection of Fig. 5 reveals the following characteristics:

• q₁ generally increases with an increase in K, which was already mentioned in Section 2.2 in reference to Fig. 2(b). The q₁ values for Fe i lines are generally smaller than those of Fe ii lines because of the difference in K.

• The slope of the q₁ versus K plots is a systematic function of v_e; i.e. it becomes progressively steeper with an increase in v_e (Fig. 5g). This is a useful property for estimating v_e from the observed q₁–K relation.

• The sensitivity of q₁ to a change in v_e also depends on K (cf. Fig. 5h). While |$\langle {\rm d}q_{1}/{\rm d}v_{\rm e}\rangle \lesssim 0$| holds for most lines of |$K \lesssim 0$| (all Fe i lines and many Fe ii lines), a group of high-excitation Fe ii lines (χ_low ∼ 10 eV; such as Fe ii 4951.584 in Fig. 4) with positive K indicates 〈dq₁/dv_e〉 > 0.

4 RESULT AND DISCUSSION

4.1 Rotational velocity of Vega

Now that the observational data of zero frequencies (⁠|$q_{1}^{\rm obs}$|⁠) as well as the corresponding theoretically calculated values (⁠|$q_{1}^{{\rm cal},j}$| for j = 0, 1, ⋅⋅⋅, 9) to be compared are all set for 90 lines, we can address the main task of investigating Vega’s rotational velocity, while following the same procedure as adopted in Paper II (cf. section 4.3 therein).

The observed |$q_{1}^{\rm obs}$| values are plotted against K in Fig. 6. As seen from this figure, these |$q_{1}^{\rm obs}$| data show an increasing tendency with K and those for Fe i and Fe ii lines are distributed in separate two groups, which is quite similar to the theoretical predictions mentioned in Section 3.3 (cf. Figs 5a–f). Therefore, there is a good hope of successfully establishing v_e by comparing |$q_{1}^{\rm obs}$| and |$q_{1}^{{\rm cal}}$| for many lines altogether.

The behaviours of the resulting σ (3D surface and contour plots) are displayed in Fig. 7 (left- and right-hand panels are for Fe i and Fe ii, respectively). The trace line connecting (x*, v_e) is also overplotted by the dashed line, where x* corresponds to the minimum of σ trough for each v_e (in Table 4 are given the actual data of x* and the corresponding σ*). Besides, the run of σ with v_e across the tracing is depicted in Fig. 8(a), and the tracings for both species are drawn together in Fig. 8(b).

An inspection of Fig. 8 yielded satisfactory results, because three kinds of (x, v_e) solutions turned out consistent with each other: (191, 21.8) from the minimum of σ_I (Fig. 8a), (194, 21.5) from the minimum of σ_II (Fig. 8a), and (201, 21.5) from the intersection of two trace lines (Fig. 8b).

The uncertainties involved in x* were estimated as ∼0.14 km s⁻¹ (Fe i) and ∼0.36 km s⁻¹ (Fe ii),³ which are indicated by dashed lines in Fig. 8(b). From this figure, errors in v_e and x were roughly evaluated (from the size of the parallelogram area embraced by four dashed lines around the intersection) as ∼±15 km s⁻¹ and ∼±0.3 km s⁻¹, respectively.

Consequently, by averaging these three solutions, Vega’s equatorial and projected rotational velocities were concluded as v_e = 195(±15) km s⁻¹ and v_esin i = 21.6(±0.3) km s⁻¹, which further result in i = 6.4^○(±0.5^○). Among the 10 models adopted in this study (cf. Table 3), model 5 (v_e = 200 km s⁻¹) is the most preferable model; this can be actually confirmed in Fig. 6, where the linear-regression lines defined in Fig. 5(b)–(f) are overplotted (after the v_e-dependent difference between x* and 22 has been corrected).

4.2 Comparison with previous results

As mentioned in Section 1, although the considerably large differences of Vega’s v_e amounting to |${\gtrsim}100$| km s⁻¹ seen in the literature of early time were reduced in the more recent results (most of them were published within several years around 2010), they are still diversified ranging from ∼170 to ∼230 km s⁻¹. Interestingly, the v_e value (∼200 km s⁻¹) derived in this study is almost in-between this dispersion. It may be worth briefly reviewing these literature v_e values (published since Paper I; cf. Table 1) in comparison with the consequence of this investigation.

• Line profile method:

  Paper I’s result (175 km s⁻¹) based on the conventional profile fitting has been revised upwards by ∼+20 km s⁻¹ in this reinvestigation by applying the Fourier transform method to line profiles. While Yoon et al. (2010)’s 236 km s⁻¹ is somewhat too large, Hill, Gulliver & Adelman (2010)’s 211 km s⁻¹ is in tolerable agreement as compared with the present result.

• Interferometry method:

  Monnier et al. (2012)’s conclusion of v_e = 197 km ⁻¹ (derived from v_esin i and i given in their table 2 as Model 3) based on optical interferometry is in good agreement with this study. Actually, fig. 2 of Monnier et al. (2012) shows that their Model 3 matches well with model 5 (v_e = 200 km s⁻¹) of Paper I.

• Magnetic modulation method:

  Vega’s rotational period (P) was directly determined by way of detecting the magnetic modulation based on time-sequence data of spectropolarimetric observations: 0.732 d (Petit et al. 2010), 0.678 d (Alina et al. 2012), 0.623 d (Butkovskaya 2014), and 0.678 d (Böhm et al. 2015). Among these four, it is the P-value of 0.678 d derived by both Alina et al. and Böhm et al. that is most consistent with the v_e result (195 km s⁻¹) of this investigation, which corresponds to P = 2πR_e/v_e = 0.685 d (where R_e = 2.784 R_⊙ for model 5 is adopted).

4.3 Advantage of Fourier analysis

Finally, some comments may be in order regarding the superiority of exploiting the zero frequency (q₁) measured from the Fourier transform of line profiles in comparison with the ordinary profile fitting approach in the wavelength domain.

The distinct merit of using q₁ is that it can discern very subtle differences in the profile shape. Fig. 4 provides a good demonstrative example. While the profile of Fe i 5133.681 undergoes a comparatively easy-to-detect change with an increase in v_e (Fig. 4a), that of Fe ii 4951.584 is apparently inert (Fig. 4b), which means that getting information on v_e from the profile of the latter is more difficult (this is the reason why Fe ii lines could not be used for determining v_e in Paper I). However, the situation is different in the Fourier space, where the shift of q₁ (reflecting the change of line profile) is sufficiently detectable with almost the same order of magnitude for both cases (cf. Figs 4e and f). Accordingly, Fe i as well as Fe ii lines are equally usable for v_e determination if q₁ is invoked, as done in this study.

Besides, q₁ is precisely measurable and easy to handle as a single parameter, which is a definite advantage from a practical point of view. Actually, q₁ data of many lines can be so combined as to improve the precision of v_e (while statistically estimating its error) as done in this paper. Such a treatment would be difficult in the conventional approach of fitting the observed and theoretical profiles.

4.4 Line profile classification using q₁ and K

Another distinct merit of q₁ is that it provides us with a prospect for quantitative classification of spectral line shapes founded on a physically clear basis. Since the discovery around ∼1990 that a number of spectral lines in Vega (e.g. weak lines of neutral species) show unusual profiles of square form, there has been a tendency to pay attention to this specific line group (e.g. compilation of flat-bottomed lines in Vega by Monier et al. 2017). However, the actual situation of Vega’s spectral lines in general is not so simple as to be dichotomized into two categories of normal and peculiar profiles; as a matter of fact, the individual profiles of most lines should more or less have anomalies of different degree. Unfortunately, the detection of such details has been hardly possible so far, because the judgement of profile peculiarity was done by simple eye inspection due to the lack of effective scheme for describing/measuring the delicate characteristics of line profiles.

The first zero frequency (q₁) is just what is needed in this context, which is not only sensitive to a slight difference of line shape but also easily measurable in the Fourier space. Moreover, thanks to its close relationship with K, the behaviour of q₁ (representing the line shape characteristics) can be reasonably explained in terms of the underlying physical mechanism. We now have a unified understanding as to why different spectral lines exhibit diversified profiles in Vega, as summarised below.

• It is the parameter K (temperature sensitivity) that essentially determines the observed line shape. The contribution to the important shoulder part of the profile away from the line centre (⁠|$|\Delta \lambda | \lesssim \lambda v_{\rm e}\sin i / c$|⁠) is mainly made by the light coming from near to the gravity-darkened limb of lowered T. Accordingly, lines of K < 0, K ∼ 0, and K > 0 show boxy (U-shaped), normally round (like classical rotational broadening), and rather peaked (V-shaped) profiles, each of which results in appreciably different q₁ values. For example, in Fig. 4, these three groups correspond to those of lower q₁ (∼0.025 km⁻¹s), medium q₁ (∼0.03 km⁻¹s), and higher q₁ (∼0.035 km⁻¹s), respectively.

• The peculiarity degree of the line shape (i.e. departure from the classical rotationally broadened profile) is described by K, because |$(q_{1} - q_{1}^{\rm classical}) \propto K$| (⁠|$q_{1}^{\rm classical} \simeq 0.03$| km⁻¹s) and the gradient (>0) of this relation progressively increases with v_e, as manifested in Fig. 5. As such, the profile of any line in Vega can be reasonably predicted if K and v_e are specified.

• As explained in appendix A of Paper II, the value of K for each spectral line depends on χ_low (lower excitation potential) and W (equivalent width). It is important to note that the line strength affects K in the sense that |K| tends to decrease with an increase in W (i.e. as the line becomes more saturated), which means that chemical abundances are implicitly involved. In the present case of A-type stars, K values for Fe i lines are determined mainly by W, while those for Fe ii lines are determined primarily by χ_low (cf. fig. A1 in Paper II), which are also indicated from Figs 2(c) and (d).

• These behaviours of K in terms of the line parameters reasonably explain why different spectral lines of Vega reveal various characteristic shapes. For example: (1) Flat-bottom profiles (manifestation of K < 0) are seen in Fe i lines but not in Fe ii lines, because of the distinct difference in K between these two line groups; i.e. |$-20 \lesssim$| K(Fe i) |$\lesssim -10$| and |$-5 \lesssim K$|(Fe ii) |$\lesssim +5$|⁠. (2) The reason why typical flat-bottomed shape is observed mainly in weak Fe i lines (e.g. 4707.272, 4903.308 with W of several mÅ) but not clearly in moderate-strength Fe i lines (e.g. 4202.028, 4920.502 with W of a few tens mÅ) is that the (negative) K values of the former group are generally lower than those of the latter owing to the dependence on W. (3) Regarding Fe ii lines, some lines have clearly peaked V-shape (e.g. Fe ii 5004.195 with χ_low = 10.272 eV and K = +2.92) while others exhibit rather rounded profile (e.g. Fe ii 4993.358 with χ_low = 2.807 eV and K = −6.27), which is naturally attributed to the apparent distinction of K (the sign is inversed) due to the large difference in χ_low.

5 SUMMARY AND CONCLUSION

It is known that the sharp-line star Vega (v_esin i ∼ 20 km s⁻¹) is actually a rapid rotator seen nearly pole-on with low |$i\, ({\lt}10^{\circ })$|⁠. However, its intrinsic rotational velocity is still in dispute, for which rather diversified values have been published.

In the previous studies (including Paper I by the author’s group), analysis of spectral line profiles has been often invoked for this purpose, which contain information on v_e via the gravity darkening effect, However, it is not necessarily easy to reliably determine v_e by direct comparison of observed and theoretically simulated line profiles. Besides, this approach is not methodologically effective because it lacks the scope for combining many lines in establishing the solution.

Recently, the author applied in Paper II the Fourier analysis to the profiles of many Fe i and Fe ii lines of Sirius A and estimated its v_e by making use of the first zero (q₁) of the Fourier transform, which turned out successful. Therefore, the same approach was decided to adopt in this study to revisit the task of establishing v_e of Vega.

As to the observational data, the same high-dispersion spectra of Vega as adopted in Paper I were used. From the Fourier transforms computed from the profiles of selected 49 Fe i and 41 Fe ii lines, the corresponding zero frequencies were measured for the analysis. The K values (T-sensitivity parameter) of these Fe lines are in the range of |$-20 \lesssim K \lesssim -10$| (Fe i lines) and |$-5 \lesssim K \lesssim +5$| (Fe ii lines).

Regarding the gravity-darkened models of rotating Vega, the model grid (comprising 10 models) arranged in Paper I was adopted, which cover the v_e range of 100–300 km s⁻¹ while assuming v_esin i = 22 km s⁻¹ as fixed. The theoretical profiles of 90 lines were simulated for each model, from which Fourier zero frequencies were further evaluated.

An inspection of these |$q_{1}^{\rm cal}$| values for the simulated profiles revealed an increasing tendency with K and the slope of this trend becomes steeper towards larger v_e, which suggests that v_e is determinable by comparing |$q_{1}^{\rm cal}\,(K,v_{\rm e})$| with observed |$q_{1}^{\rm obs}$| for many lines of different K.

It turned out that v_e and v_esin i could be separately established by the requirement that the standard deviation of the residual between |$q_{1}^{\rm cal}$| and |$q_{1}^{\rm obs}$| be minimized (while taking into account the difference between the actual v_esin i and 22 km s⁻¹ assumed in the model profiles), and independent analysis applied to two sets of Fe i and Fe ii lines yielded solutions consistent with each other.

The final parameters of Vega’s rotation were concluded to be v_esin i = 21.6(±0.3) km s⁻¹, v_e = 195(±15) km s⁻¹, and i = 6.4(±0.5)^○.

SUPPORTING INFORMATION

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ACKNOWLEDGEMENTS

This research has made use of the SIMBAD data base, operated by CDS, Strasbourg, France.

DATA AVAILABILITY

The data underlying this article are available in the supplementary materials.

Footnotes

REFERENCES