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Register a new userDetermination of Vegas rotational velocity based on the Fourier analysis of spectral line profiles
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Authors
Yoichi Takeda
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While it is known that the sharp-line star Vega (vsini ~ 20km/s) is actually a rapid rotator seen nearly pole-on with low i (< 10 deg), 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 Vegas v_e was attempted based on an alternative approach making use of the first zero (q_1) of the Fourier transform of each line profile, which depends upon K (temperature sensitivity parameter differing from line to line) and v_e. It turned out that v_e and vsini could be separately established by comparing the observed q_1^obs and calculated q_1^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 Vegas rotation were concluded as vsini = 21.6 (+/- 0.3) km/s, v_e = 195 (+/- 15) km/s, and i = 6.4 (+/- 0.5) deg.
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It is known that stellar differential rotation can be detected by analyzing the Fourier transform of spectral line profiles, since the ratio of the 1st- and 2nd-zero frequencies is a useful indicator. This approach essentially relies on the conventional formulation that the observed flux profile is expressible as a convolution of the rotational broadning function and the intrinsic profile, which implicitly assumes that the local intensity profile does not change over disk. Although this postulation is unrealistic in the strict sense, how the result is affected by this approximation is still unclear. In order to examine this problem, profiles of several lines (showing different center-limb variations) were simulated using a model atmosphere corresponding to a mid-F dwarf by integrating intensity profiles for various combinations of vsini (rot. velocity), alpha (diff. degree), and i (inc. angle), and their Fourier transforms were computed to check whether zeros are detected at the predicted positions or not. For this comparison purpose, a large grid of standard rotational broadening functions and their transforms/zeros were also calculated. It turned out that the situation criticaly depends on vsini: In case of vsini>~20km/s where rotational broadening is predominant over other line broadening velocities (typically several km/s), the 1st/2nd zeros of the transform are confirmed almost at the expected positions. In contrast, deviations begin to appear as vsini is lowered, and the zero features of the transform are totally different from the expectation at vsini as low as ~10km/s, which means that the classical formulation is no more valid. Accordingly, while the zero-frequency approach is safely applicable to studying differential rotation in the former broader-line case, it would be difficult to practice for the latter sharp-line case.
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With an aim of getting information on the equatorial rotation velocity (v_e) of Sirius A separated from the inclination effect (sin i), a detailed profile analysis based on the Fourier transform technique was carried out for a large number of spectral lines, while explicitly taking into account the line-by-line differences in the centre-limb behaviours and the gravity darkening effect (which depend on the physical properties of each line) based on model calculations. The simulations showed that how the 1st-zero frequencies (q_1) of Fourier transform amplitudes depends on v_e is essentially determined by the temperature-sensitivity parameter (K) differing from line to line, and that Fe I lines (especially those of very weak ones) are more sensitive to v_e than Fe II lines. The following conclusions were drawn by comparing the theoretical and observed q_1 values for many Fe I and Fe II lines: (1) The projected rotational velocity (vsini) for Sirius A is fairly well established at 16.3 (+/-0.1) km/s by requiring that both Fe I and Fe II lines yield consistent results. (2) Although precise separation of v_e and i is difficult, v_e is concluded to be in the range of 16 < v_e < 30-40 km/s, which corresponds to 25 < i(deg) < 90. Accordingly, Sirius A is an intrinsically slow rotator for an A-type star, being consistent with its surface chemical peculiarity.
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