aa r X i v : . [ a s t r o - ph ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 8 November 2018 (MN L A TEX style file v2.2)
Abundances of lithium, sodium, and potassium in Vega
Y. Takeda ⋆ † National Astronomical Observatory of Japan,2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Accepted 2008 May 12. Received 2008 May 8; in original form 2007 December 14
ABSTRACT
Vega’s photospheric abundances of Li, Na, and K were determined by using con-siderably weak lines measured on the very high-S/N spectrum, while the non-LTEcorrection and the gravity-darkening correction were adequately taken into account.It was confirmed that these alkali elements are mildly underabundant ([Li/H] ≃ − . ≃ − .
3, and [K/H] ≃ − .
2) compared to the solar system values, as generallyseen also in other metals. Since the tendency of Li being more deficient than Na andK is qualitatively similar to what is seen in typical interstellar cloud, the process ofinterstellar gas accretion may be related with the abundance anomaly of Vega, assuspected in the case of λ Boo stars.
Key words: stars: abundances – stars: atmospheres – stars: early-type – stars: indi-vidual: Vega.
While Vega (= α Lyr = HR 7001 = HD 172167 = HIP 91262;A0V) plays an important role as the fundamental photomet-ric standard, its photospheric chemical composition is knownto be sort of anomalous. Namely, most elements are mildlyunderabundant (by ∼ − . T c ) such as C, N, O, and S. Sincethis is the tendency more manifestly shown by a group ofA–F main-sequence stars known as “ λ Boo-type stars” (see,e.g., Paunzen 2004 and the references therein), Vega’s mod-erate abundance peculiarities have occasionally been arguedin connection with this λ Boo phenomenon (e.g., Baschek& Slettebak 1998; St¨urenburg 1993; Holweger & Rentzsch-Holm 1995; Iliji´c et al. 1998).Regarding the origin of unusual surface compositions of λ Boo stars, various models have been propounded so far,such as the diffusion/mass-loss model, accretion/diffusionmodel, or binary model (see the references quoted in Paun-zen 2004). Among these, the interpretation that the anomalywas built-up by the accretion of interstellar gas (where re-fractory metals of high T c are depleted because of beingcondensed into dust while volatile species of low T c hardlysuffer this process), such as the interaction model betweena star and the diffuse interstellar cloud proposed by Kamp ⋆ E-mail: [email protected] † Based on observations carried out at Okayama AstrophysicalObservatory (Okayama, Japan). & Paunzen (2002), appears to be particularly promising,in view of Paunzen et al.’s (2002) recent finding that the[Na/H] values of λ Boo stars (showing a large diversity) areclosely correlated with the [Na/H]
ISM values (sodium abun-dance of interstellar matter) in the surrounding environment(cf. Fig. 7 therein).Then, according to the suspected connection betweenVega and λ Boo stars, it is natural to examine the photo-spheric Na abundance of Vega. Besides, the abundances ofLi and K may also be worth particular attention in a simi-lar analogy, since these three alkali elements play significantroles in discussing the physical state of the cool interstellargas through their interstellar absorption (resonance) lines ofNa i /D ), Li i i W λ measurements were tried only twoof these) according to our literature survey: Hunger (1955;Na i W λ data byStrom, Gingerich & Strom (1966), Qiu et al. (2001; Na i W λ data by Saffe& Levato (2004). Unfortunately, none of these appear to besufficiently credible as viewed from the present knowledge,because of neglecting the non-LTE effect for the strong res-onance D /D lines which were invoked in all these studies(see below in this section).The situation is even worse for Li and K, for which anydetermination of their abundances in Vega has never been c (cid:13) Y. Takeda reported. Gerbaldi, Fraggiana & Castelli (1995) once triedto measure the W λ (Li i i i , Na i , and K i lines quickly fade out as T eff becomes higher owing to theenhanced ionization of these alkali atoms (only one valenceelectron being weakly bound), one has to contend with thedifficult task of measuring very weak lines (e.g., W λ valueeven as small as ∼ O or O generally exist to cause severe blending withstellar lines, which often prevent us from reliable measure-ments of equivalent widths. (see, e.g., Sect. 6.2 of Qiu et al.2001, who remarked the large uncertainty caused by this ef-fect in their measurements of Na i D lines).— Third, the resonance lines in the longer wavelength re-gion are generally known to subject to considerable non-LTE corrections especially for the case of Na i lack of sufficient knowledge on non-LTE correctionsmay have hampered investigating their abundances.Given this situation, we decided to spectroscopically es-tablish Vega’s photospheric abundances of Li, Na, and Kscarcely investigated so far, while overcoming the problemsmentioned above, in order to see how they are comparedwith the compositions of other elements in Vega as well aswith those of interstellar gas. This is the purpose of thisstudy.This attempt was originally motivated by our recentwork (Takeda, Kawanomoto & Ohishi 2007) of publishinga digital atlas of Vega’s high-resolution ( R ∼ ∼ Few available at present may be only two studies on neutralsodium lines: Takeda & Takada-Hidai’s (1994) non-LTE analysisof various Na i lines in α CMa (in addition to Procyon and A–Fsupergiants), and Andrievsky et al.’s (2002) non-LTE study ofNa i D and D lines for late-B to early-F type dwarfs of λ Boocandidates. rotating Vega (Takeda, Kawanomoto & Ohishi 2008). There-fore, this would make a timely subject to address.
The basic observational data of Vega used for this studyare the high-S/N ( ∼ R ∼ which may be consulted for more de-tails.The following 11 lines were selected as the targetlines used for abundance determinations: Na i i i i i task telluric ) which are alsoincluded in Takeda et al. (2007) along with the airmass in-formation. Though this removal process did not always workvery successfully (e.g., for the case of very strong saturatedlines such as the O in the K i O lines in the Na i i ∼ The measurements of the equivalent widths (withrespect to the local continuum level specified by eye-inspection) were done by the Gaussian-fitting technique, andthe resulting W λ values are given in Table 1. The relevantspectra at each of the wavelength regions are shown in Figs.1 (Na i i i i i i While the spectrum atlas is presented as the elec-tronic tables of this paper, the same material is avail-able also at the anonymous FTP site of the AstronomicalData Center of National Astronomical Observatory of Japan: h ftp://dbc.nao.ac.jp/DBC/ADACnew/J/other/PASJ/59.245/ i . IRAF is distributed by the National Optical Astronomy Obser-vatories, which is operated by the Association of Universities forResearch in Astronomy, Inc. under cooperative agreement withthe National Science Foundation, USA. Though this results in a blurring of ∼
10 km s − correspondingto the width of 9-pixel boxcar function (1 pixel ∼ ∼ − .c (cid:13) , 000–000 bundances of lithium, sodium, and potassium in Vega N o r m a li ze d i n t e n s it y Vega/RegulusVegaRegulus
Na Nasimulation
Figure 1.
Spectra of 5885–5900 ˚A region comprising Na i v e sin i = 22 km s − ), Vega/Regulus (telluric lines removed), rawVega, and raw Regulus are arranged from top to bottom with ap-propriately chosen offsets. The scale of the ordinate correspondsto the second-top spectrum, for which equivalent-widths of theselines were measured by the Gaussian fitting method (as overplot-ted in solid lines on this spectrum). N o r m a li ze d i n t e n s it y Na NaVega/Regulus VegaRegulussimulation
Figure 2.
Spectra of 8180–8200 ˚A region comprising Na i Regarding the fiducial atmospheric model of Vega, Kurucz’s(1993) ATLAS9 model with the parameters of T eff = 9550 K(effective temperature), log g (cm s − ) = 3.95 (surface grav-ity), [X/H] = − . v t = 2 km s − (mi-croturbulence) was adopted for this study as in Takeda etal. (2007), which appears to be the best as far as withinthe framework of classical plane-parallel ATLAS9 modelsaccording to Castelli & Kurucz (1994).As the basic strategy, the LTE abundance ( A LTE ) wasfirst derived from the measured W λ by using the WIDTH9 Na Na
Wavelength (Å) N o r m a li ze d i n t e n s it y Vegasimulation
Figure 3.
Spectra of 6151–6164 ˚A region comprising Na i Na Na
Wavelength (Å) N o r m a li ze d i n t e n s it y Vega/RegulusVegaRegulussimulation
Figure 4.
Spectra of 5680–5690 ˚A region comprising Na i program, , to which the non-LTE correction (∆ NLTE ) andthe gravity-darkening correction (∆ GD ) were then appliedto obtain the final abundance ( A NLTE+GD ).In this step-by-step way, one can quantitatively judgethe importance/unimportance of each effect by comparing This is a companion program to the ATLAS9 model atmo-sphere program written by R. L. Kurucz (Kurucz 1993), thoughit has been considerably modified by Y. T. (e.g., for treating mul-tiple component lines or including non-LTE effects in line forma-tion Complete simulations (where the non-LTE effect and thegravity-darkening effect were simultaneously taken into accountin a rigorous manner) show that the finally derived abundancesin this semi-classical approach are fairly reliable. This consistencycheck is separately described in Appendix B.c (cid:13) , 000–000
Y. Takeda
Vega/RegulusVegaRegulusK K N o r m a li ze d i n t e n s it y simulation Figure 5.
Spectra of 7660–7705 ˚A region comprising K i Li Wavelength (Å) N o r m a li ze d i n t e n s it y simulationVega (original)Vega (smoothed) Figure 6.
Spectra of 6700–6716 ˚A region comprising Li i two abundance corrections, and such factorized informationmay be useful in application to other different situations(e.g., the case of slow rotator where only ∆ LTE is relevant;cf. Appendix A). Also, the use of “relative correction” (in-stead of directly approaching the absolute abundance solu-tion) has a practical merit of circumventing the numerical-precision problem involved in the spectrum computation onthe gravity-darkened model (especially seen for the case ofthe extremely weak Li i NLTE werecarried out in the same manner as described in Takeda etal. (2003; Na i ), Takeda & Kawanomoto (2005; Li i ), andTakeda et al. (2002; K i ). Besides, all the relevant atomic data ( gf values, damping constants, etc.) for abundance de-terminations were so adopted as to be consistent with thesethree studies.The gravity-darkening corrections (∆ GD ) were derivedbased on the equivalent-width intensification factor com-puted by the program CALSPEC under the assumption ofLTE ( W /W , where W and W correspond to the gravity-darkened rapid rotator model and the classical rigid-rotationmodel, respectively; and the abundance was adjusted tomake W consistent with W obs λ ), the details about whichare explained in Takeda et al. (2008). Practically, the rela-tion ∆ GD = − log( W /W ) was assumed for the very weaklines ( W λ <
15 m˚A; i.e., Na i i i i GD val-ues for the remaining four lines (Na i i W obs λ and theperturbed W obs λ / ( W /W ).The results of the abundances and the abundancecorrections are summarized in Table 1, where the finallyadopted (average) values of h A NLTE+GD i and the abun-dances relative to the standard solar system compositions([X/H] ≡ h A XNLTE+GD i − A ⊙ ) are also presented. As canbe seen from this table, the non-LTE corrections are alwaysnegative (corresponding to the non-LTE line-intensification)with appreciably W λ -dependent extents (ranging from 0.1–0.7 dex). Similarly, the gravity-darkening corrections arealso negative with extents of 0.1–0.3 dex (especially im-portant for the weakest Li i T -lowering.Regarding errors in the resulting abundances, those dueto uncertainties in the atmospheric parameters (e.g., in T eff or log g ) may not be very significant in the present compar-atively well-established case (an abundance change is only ∼ . T eff by ∼
200 K or inlog g by 0.2; cf. Table 5 in Takeda & Takada-Hidai 1994 forthe case of α CMa). Rather, a more important source oferror would be the ambiguities in the measurement of W λ .While the photometric accuracy of W λ estimated by ap-plying Cayrel’s (1988) formula (FWHM of w ∼ . δx ∼ . ǫ ∼ / ∼ . ∼ − . Presumably, errors in W λ would amount to severaltens of % in the case of very faint lines (e.g., Na i i It may be worth noting that these moderate ∆ GD values cor-respond to Takeda et al.’s (2008) model No. 4 ( v e = 175 km s − )which they concluded to be the best. If higher-rotating modelswith larger gravity-darkening were to be used, the correctionswould naturally become more enhanced. For example, if we adoptTakeda et al.’s (2008) model No. 8 with v e = 275 km s − , whichis near to the solution suggested from interferometric observa-tions (Peterson et al. 2006, Aufdenberg et al. 2006), the extentsof ∆ GD turn out to increase significantly by a factor of ∼ − . − . − . − . − . − . − . − . − . − .
37, and − .
82, for Na i i i (cid:13) , 000–000 bundances of lithium, sodium, and potassium in Vega to incompletely removed telluric lines (e.g., Na i i ∼ The abundances of sodium could be derived from 8 linesof different strengths (2 resonance lines of D +D and6 subordinate lines originating from the excited level of χ low = 2 .
21 eV). The three kinds of abundances ( A LTE , A NLTE , A NLTE+GD ; cf. Table 1) for each of the 8 lines areplotted against W λ in Fig. 7. We can read from this fig-ure that the serious inconsistency seen in the LTE abun-dances is satisfactorily removed by the non-LTE corrections,which are significantly W λ -dependent from − . i − . i W λ <
10 m˚A). This maysuggest that the applied non-LTE corrections are fairly re-liable and the adopted microturbulence of v t = 2 km s − is adequate (which affects the abundances of stronger Na i A NLTE+GD values of 8 lines, is 6.01 with the standard de-viation of σ = 0 .
10, indicating a mildly subsolar composi-tion ([Na/H] = − .
3) compared to the solar abundance of A ⊙ = 6 . +D lines are available for the Na abundanceof Vega. Hunger (1955) tentatively derived A ∼ . A ∼ . W λ (5890/5896)= 195/115 m˚A, though commenting that these abundancesare very uncertain. Thereafter, based on Hunger’s W λ data,Strom et al.’s (1966) model atmosphere analysis concluded A ∼ . ± .
4) ( T eff = 9500 K). Then, after a long blank,Qiu et al. (2001) obtained A = 6 .
45 from their measured W λ (5896) value of 94 m˚A, though they also remarked thatthis result is quite uncertain because of blending with wa-ter vapor lines. Soon after, Saffe & Levato (2004) derived A = 6 .
37 using Qiu et al.’s W λ data. Accordingly, all theseprevious work reported supersolar or near-solar Na abun-dances for Vega, contradicting the conclusion of this study.It is evident that they failed to obtain the correct abundancefrom the Na i According to Table 1, while potassium is only mildly subso-lar ([K/H] ≃ − .
2) to an extent similar to sodium ( ∼ − . ≃ − . ∼ . l (mÅ) A ( N a ) LTENLTENLTE+GD
Figure 7.
Na abundances derived from each of the 8 lines plot-ted against the equivalent widths, based on the results in Table1. Squares, triangles, and circles correspond to the LTE abun-dances, the non-LTE abundances, and the final abundances (in-cluding both the non-LTE and gravity-darkening corrections), re-spectively. The horizontal dotted line indicates the average of thefinal abundances (6.01). in interstellar gas and its accretion? Since the quantitativeamount of such a depletion is different from case to case de-pending on the physical condition (e.g., see the variety of[Na/H]
ISM in Fig. 7 of Paunzen et al. 2002), it is not muchmeaningful to discuss the “absolute” values of [X/H] here.Rather, we should pay attention to their “relative” behav-iors with each other or in comparison to those of variousother elements.The [X/H]
Vega values (those for Li, Na, and K are fromthis study, while those for other elements were taken fromvarious literature) are plotted against the condensation tem-perature ( T c ) in Fig. 8 (filled circles), where the [X/H] ISM results of typical interstellar gas in the direction of ζ Oph(taken from Table 5 of Savage & Sembach 1996; see alsoFig. 4 therein) are also shown by open circles. The followingcharacteristics can be recognized from this figure:— (1) The [X/H]
Vega values, falling on a rather narrow rangebetween ∼ − ∼
0, tend to decrease with T c , a qualita-tively similar trend to that seen in [X/H] ISM .— (2) The run of [X/H]
Vega with T c appears to be slightlydiscontinuous at T c ∼ Vega ∼ − . − . T c . Vega ∼ − . − . T c & ISM with T c alsoshows a discontinuity around this critical T c ( ∼ | d [X/H] ISM /dT c | becomes evi-dently steeper on the T c & Vega ( − .
6) be-ing more deficient than [Na/H]
Vega ( − .
3) and [K/H]
Vega ( − .
2) is qualitatively similar to just what is seen in ISM([Li/H]
ISM < [Na/H] ISM ≃ [K/H] ISM ).All these observational facts may suggest the existenceof some kind of connection between Vega’s photosphericabundances and those of interstellar cloud, which naturallyimplies that an accretion/contamination of interstellar gas islikely to be responsible (at least partly) for the abundance c (cid:13) , 000–000 Y. Takeda c (K) [ X / H ] ArKr C N O Tl TePb S SeSnZnB ClGeNaGa KCu AsPMn Li SiMg VCr FeCo Ni TiCa
Na K Li
Figure 8. [X/H] values (relative abundances in comparison withthe solar-system values of Grevesse & Noels 1993) of various ele-ments plotted against the condensation temperature ( T c ). Opencircles indicate the compositions of interstellar gas in the directionof ζ Oph (cool diffuse clouds), which were taken from Savage &Sembach (1996; cf. their Table 5 and Fig. 4). The [Na/H], [K/H],and [Li/H] for Vega derived in this study are shown by largerfilled circles. Besides, Vega’s [X/H] values for other elements areplotted by smaller filled circles for a reference, which were takenfrom from various sources: Przybilla & Butler (2001) (for C, N,and O; cf. their Table 7); Takada-Hidai & Takeda (1996) (for S);Adelman & Gulliver (1990) (for Mg, Ca, Ti, Cr, Mn, Fe, andNi); Qiu et al. (2001) (for Si and V). [Note. Since these litera-ture [X/H] values of Vega were derived in a conventional mannerbased on classical atmospheric model, further corrections for thegravity-darkening effect (∆ GD ) are to be expected. According toTakeda et al. (2008), however, the extents of (negative) ∆ GD (becoming appreciable only for very weak lines and for specificelements/stages) are only ∼ . i lines hadbetter be reduced by ∼ . . . peculiarity of Vega, such as being proposed for explainingthe λ Boo phenomenon. This is the conclusion of this study.
ACKNOWLEDGMENTS
The author thanks S. Kawanomoto and N. Ohishi for theirhelp in the observations of Vega, based on the data fromwhich this study is based.
REFERENCES
Adelman S.J., Gulliver A.F., 1990, ApJ, 348, 712Andrievsky S.M., et al., 2002, A&A, 396, 641Aufdenberg J.P., et al. 2006, ApJ, 645, 664 (erratum: 651,617)Baschek B., Slettebak A., 1988, A&A, 207, 102Burkhart C., Coupry M.F., 1991, A&A, 249, 205Castelli F., Kurucz R.L., 1994, A&A, 281, 817Cayrel R. 1988, in Cayrel de Strobel G., Spite M., eds, The Impact of Very High S/N Spectroscopy on Stellar Physics,Proc. IAU Symp. 132, Kluwer, Dordrecht, p.345Coupry M.F., Burkhart C., 1992, A&AS, 95, 41Gerbaldi M., Faraggiana R., Castelli F., 1995, A&AS, 111,1Grevesse N., Noels A., 1993, in Prantzos N., Vangioni-FlamE., Cass´e M., eds, Origin and evolution of the elements,Cambridge University Press, Cambridge, p.15Gulliver A.F., Adelman S.J., & Friesen T.P. 2004, A&A,413, 285,Holweger H., Rentzsch-Holm I., 1995, A&A, 303, 819Hunger K., 1955, Zs. Ap., 36, 42Iliji´c S., Rosandi´c M., Dominis D., Planini´c M., PavlovskiK., 1998, Contrib. Astron. Obs. Skalnat´e Pleso, 27, 467Kamp I., Paunzen E., 2002, MNRAS, 335, L45Kurucz R.L., 1993, Kurucz CD-ROM, No. 13,Harvard-Smithsonian Center for Astrophysics,http://kurucz.harvard.edu/cdroms.htmlPaunzen E., 2004, in Zverko J., ˇZiˇzˇnovsk´y J., Adelman S.J.,Weiss W.W., eds, The A-Star Puzzle, Proc. IAU Symp.224, Cambridge University Press, Cambridge, p.443Paunzen E., Iliev I.Kh., Kamp I., Barzova I.S., 2002, MN-RAS, 336, 1030Peterson D.M., et al. 2006, Nature, 440, 896Przybilla N., Butler K., 2001, A&A, 379, 955Qiu H.M., Zhao G., Chen Y.Q., Li Z.W., 2001, ApJ, 548,953Saffe C., Levato H., 2004, A&A, 418, 1083Savage B.D., Sembach K.R., 1996, ARA&A, 34, 279Strom S.E., Gingerich O., Strom K.M., 1966, ApJ, 146, 880St¨urenburg S., 1993, A&A, 277, 139Takada-Hidai M., Takeda Y., 1996, PASJ, 48, 739Takeda Y., Kawanomoto S., 2005, PASJ, 57, 45Takeda Y., Takada-Hidai M., 1994, PASJ, 46, 395Takeda Y., Kawanomoto S., Ohishi N., 2007, PASJ, 59, 245Takeda Y., Kawanomoto S., Ohishi N., 2008, ApJ, 678, 446Takeda Y., Zhao G., Chen Y.-Q., Qiu H.-M., Takada-HidaiM., 2002, PASJ, 54, 275Takeda Y., Zhao G., Takada-Hidai M., Chen Y.-Q., SaitoY.-J., Zhang H.-W., 2003, ChJAA, 3, 316
APPENDIX A: LI ABUNDANCE IN O PEG
The enhanced deficiency of Li (compared to Na and K)was an important key result in deriving the conclusion ofthis study (cf. Sect. 4.2), since we interpreted it as a man-ifestation of ISM compositions. In this connection, it maybe worth mentioning another remarkable very sharp-linedA1 IV star, o Peg, for which we could also get informationof the Li abundance. As the only available measurementof the Li i W λ (6708) =1.3 m˚A for this star, which has atmospheric parameters ( T eff = 9650 K, log g = 3 .
6) quite similar to those of Vega exceptfor its near-normal metallicity ([Fe/H] ≃ +0 .
1; Burkhart &Coupry 1991).Now, their W λ ( o Peg) value twice as large as that ofVega (0.7 m˚A) would raise the Li abundance by +0.3 dex.Furthermore, since any flat-bottomed shape has never beenreported in the spectral lines of o Peg (as we can con- c (cid:13) , 000–000 bundances of lithium, sodium, and potassium in Vega Table 1.
Atomic data, equivalent widths, and abundance results.Species RMT λ χ low log gf W λ A LTE ∆ NLTE ∆ GD A NLTE+GD [X/H](˚A) (eV) (m˚A)Na i − − i − − − i − − i − − i − − − i − − − i − − − i − − − − K i − − i − − − − Li i − − − − − − − − − Following the atomic data of spectral lines (species, multiplet No., wavelength, lower excitation po-tential, and logarithmic gf value) in columns 1–5, column 6 gives the measured equivalent width.The results of the abundance analysis are given in column 7 (LTE abundance; in the usual normal-ization of H=12.00), column 8 (non-LTE correction), column 9 (gravity-darkening correction), andcolumn 10 (non-LTE as well as gravity-darkening corrected abundance). The finally adopted averageabundance and the corresponding [X/H] value ( ≡ A VegaX − A ⊙ X ) are also given at the bottom of thesection (in italics; columns 10 and 11), where Grevesse & Noels’s (1993) values of 6.31 (Na), 3.31(Li), and 5.13 (K) were used as the standard solar-system abundances ( A ⊙ X ). Note that the analysisof W λ (Li i Li (neglecting the contributionof Li; cf. Takeda & Kawanomoto 2005), while all the remaining lines of Na and K were analyzed bythe single-line treatment as was done by Takeda et al. (2003; Na) and Takeda et al. (2002; K). firm from the high-quality spectrum atlas of this star pub-lished by Gulliver, Adelman, & Friesen 2004), the appli-cation of the gravity-darkening correction ( ∼ − . oPeg ( ≃ [Li/H] Vega + 0.6) ∼ .
0; an interesting result that the photosphericLi abundance of o Peg almost coincides with that of thesolar-system composition. This result, that [Li/H] (as wellas [Fe/H]) is deficient/normal in Vega/ o Peg, may suggestthat the mechanism responsible for producing the under-abundance of these elements in Vega is irrelevant for o Peg.Thus, according to our interpretation, o Peg would not havesuffered any pollution due to accretion of interstellar gasdepleted in volatile elements of higher T c .Yet, we had better keep in mind another possibilitythat the mechanism causing this Li deficiency in Vega mightdifferent from that of other metals. For example, this un-derabundance could be attributed to some process of en-velope mixing (e.g., meridional circulation or shear-inducedturbulence which are supposed to be more significant as astar rotates faster), because Li atoms are burned and de-stroyed when they are conveyed into the hot stellar interior ( T & . × K). Namely, since we know that Vega rotatesrapidly (as fast as v e ∼
200 km s − ) while o Peg does notso much (at least the gravity-darkening effect is not so sig-nificant as in Vega), the underabundance of Li in the atmo-sphere of Vega might stem from the rotation-induced mixing(which is not expected for slowly rotating o Peg). If this isthe case, however, the rough similarity in the extent of de-ficiency in Li as well as other metals has to be regarded asa mere coincidence, which makes us feel this possibility asrather unlikely.
APPENDIX B: CHECK FOR THE FINALABUNDANCES: COMPLETE SPECTRUMSYNTHESIS
Our basic strategy for deriving the abundances of Na, K,and Li in Sect. 3 was as follows.— (1) First, the A LTE was derived from W obs λ based on theclassical plane-parallel model atmosphere.— (2) Next, the non-LTE correction (∆ NLTE ) was derivedin the conventional way by using this classical model.— (3) Then, the gravity darkening correction (∆ GD ) wasevaluated from the line-intensification factor W LTE4 /W LTE0 (where the abundance was so adjusted as to satisfy W LTE4 ≃ c (cid:13) , 000–000 Y. Takeda W obs λ ), which was computed by applying the CALSPEC pro-gram (Takeda et al. 2008) with the assumption of LTE to thegravity-darkened model 4 and the rigid-rotation model 0.— (4) Finally, A NLTE+GD was obtained as A LTE + ∆
NLTE +∆ GD .Actually, such a phased approach has a distinct meritof clarifying the importance/contribution of two different ef-fects (the non-LTE effect and the gravity-darkening effect).Besides, from a practical point of view, the necessary amountof calculations to arrive at the final abundance solution (suchthat reproducing the observed spectrum) can be consider-ably saved.However, there is some concern about whether such astep-by-step approach (treating the two effects separately)really yield sufficiently correct results, because both are ac-tually related with each other (e.g., how does the largelyvariable non-LTE corrections over the gravity-darkened stel-lar surface differing in T or g play roles? How reliable isthe gravity-darkening correction derived by neglecting thenon-LTE effect?). Therefore, it may be worth checking thevalidity of the finally derived abundances ( A NLTE+GD ) bycarrying out a complete spectrum synthesis including bothNLTE and GD effects simultaneously. For this purpose,the CALSPEC program was modified so as to allow inclu-sion of the non-LTE departure coefficients (correspondingto the different conditions at each of the points over thegravity-darkened stellar surface), and the NLTE+GD pro-files ( R NLTE4 ) were computed for each of the 9 lines (Na i i i A NLTE+GD values (6.03, 6.01, 5.82, 6.01, 6.13, 6.12, 5.97, 5.99, 4.82,5.01, and 2.74, respectively). Such obtained R NLTE4 profilesare depicted (in thick solid lines) in Fig. 9, where the rele-vant three kinds of line profiles [ R NLTE0 (thick dashed line), R LTE4 (thin solid line), and R LTE0 (thin dashed line)] are alsoshown for comparison.The resulting theoretical equivalent widths ( W NLTE4 )computed by integrating R NLTE4 ( λ ) are 129.7, 101.3, 18.4,43.2, 0.8, 1.5, 4.0, 7.8, 12.8, 10.3, and 0.4 m˚A for these 9lines, respectively. Comparing these values with the observedequivalent widths ( W obs λ ) given in Table 1, we can see a sat-isfactory agreement between W NLTE4 and W obs λ (typically towithin several percent except for Li i ), by which wemay conclude that the approach adopted in Sect. 3 is prac-tically validated. We see a rather large discrepancy for this Li line ( W NLTE4 =0.4 m˚A, while W obs λ = 0.7 m˚A). This must be due to the in-evitable numerical errors in R NLTE4 (Li i ◦ × ◦ in latitude and longitude), which becomes apprecia-ble especially for this case of Li i GD ≃ − log( W /W ) is not affected by this numer-ical error in W because it is canceled by taking the ratio of W .In this sense, our semi-classical approach of establishing the finalabundance (application of two corrections to the classical LTEabundance solution; cf. Sect. 3) is surely advantageous as far asthis Li i Na I 5890
NLTE+GD NLTE (rigid)LTE (rigid) LTE+GD
Na I 5896
Na I 8183
Na I 8195
Na I 6154
Na I 6161
Na I 5683
Na I 5688
K I 7665
K I 7699
Li I 6708
Wavelength (Å)
Figure B1.
Theoretical profiles of the Na, K, and Li lines com-puted by the CALSPEC program (developed for synthesizing theflux spectrum for a given rotationally-distorted gravity-darkenedstellar model; cf. Takeda et al. 2008), which has been modifiedto allow inclusion of the non-LTE effect. The A NLTE+GD valuesgiven in Table 1 were assumed as the abundances for each of thelines. Results for the gravity-darkened model and the classicalrigid-rotation model (model No. 4 and No. 0 in Takeda et al.2008) are discriminated by the line type (solid and dashed lines,respectively), and those for NLTE and LTE are by the line thick-ness (thick and thin lines, respectively). As a result, four profilesare shown for each of the 9 lines: R NLTE4 (thick solid line) R NLTE0 (thick dashed line), R LTE4 (thin solid line), and R LTE0 (thin dashedline). Shown in the ordinate is the normalized flux divided by thetheoretical (pure) continuum; therefore, the local continuum levelsometimes turns out to be slightly less than unity because of theextended wings of H lines. (Note that the theoretical equivalentwidths W NLTE4 corresponding to R NLTE4 profiles discussed in Ap-pendix B were calculated with respect to the local continuum, i.e.the maximum level in the neighborhood of the line profiles, irre-spective of the scales in the ordinate.) (a) Na i i i i i i i i i i i (cid:13)000