A&A 408, 707-713 (2003)
DOI: 10.1051/0004-6361:20030966
A. Reiners
Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
Received 6 May 2003 / Accepted 20 June 2003
Abstract
Mechanisms influencing absorption line profiles of fast
rotating stars can be sorted into two groups; (i) intrinsic
variations sensitive to temperature and pressure, and (ii) global
effects common to all spectral lines. I present a detailed study on
the latter effects focusing on gravity darkening and inclination for
various rotational velocities and spectral types. It is shown that
the line shapes of rapidly and rigidly rotating stars mainly depend
on the equatorial velocity ,
not on the projected
rotational velocity
which determines the lines' widths.
The influence of gravity darkening and spectral type on the line
profiles is shown. The results demonstrate the possibility of
determining the inclination angle i of single fast rotators, and
they show that constraints on gravity darkening can be drawn for
stellar samples. While significant line profile deformation occurs
in stars rotating as fast as
km s^{-1}, for
slower rotators profile distortions are marginal. In these cases
spectral signatures induced by, e.g., differential rotation are not
affected by gravity darkening and the methods applicable to slow
rotators can be applied to these faster rotators, too.
Key words: line: profiles - stars: rotation
The information contained in line profiles of rotating stars has been studied since the recognition of spectral Doppler broadening itself. The multitude of different mechanisms influencing line profiles presents a confusing picture for the possibility of determining them. Approximations exist for rotational line broadening, (linear) limb darkening, differential rotation, turbulence, rotational flattening and gravity darkening, etc. While distinguishing these interacting mechanisms is a delicate issue, in the case of slow rotators ( km s^{-1}) rotational flattening and gravity darkening can be neglected. Utilizing the Fourier transform technique Gray (1973,1976) also showed that turbulent velocities to some extent can be distinguished from rotation.
In fast rotators the situation becomes more complex, since centrifugal forces distort the spherical shape of the stellar surface and the line profiles depend on the inclination the star is observed under. Beyond the additional degree of freedom this would not be a major problem if temperature (and pressure) were not connected to gravity. With different regions on the stellar surface having significantly different temperatures and gas pressures, line profiles no longer can be approximated by convolution between a single intrinsic line profile and the rotational broadening function. For this reason, the effects of fast rotation on line profiles often were studied for specific absorption lines (e.g., Hardorp & Strittmatter 1968; Stoeckley 1968); Collins & Truax (1995) come to the conclusion that the detection of secondary effects like limb darkening and differential rotation is improbable in line profiles of fast rotators.
Under what circumstances the approximation of various broadening mechanisms by convolutions breaks down, only depends on the sensitivity of the spectral line on temperature and gas pressure. Spectral quality in terms of signal-to-noise as well as in wavelength coverage has improved and makes possible analyses of weak absorption features. Observers are not restricted to the extremely temperature and pressure sensitive lines of the lightest elements H and He. Fast rotation is the rule in stars of spectral types as late as F5, and in the spectra of fast rotating A- and F-stars a large number of heavy ion lines exist where the approximation of rotational broadening by a convolution becomes valid again. Especially the deconvolution of an "overall'' broadening function inherent in all lines by using the methods of Least Squares Deconvolution (LSD, e.g., Cameron 2000) can provide reliable broadening profiles independent of line specific intrinsic mechanisms.
In this paper the effects of fast rotation are examined in detail focusing on geometric distortions and gravity darkening on stars observed under different inclination angles. These effects underly all spectral lines independent of their intrinsic shape and temperature or pressure dependencies. The modelled spectra assume no intrinsic temperature or pressure dependence; whether this can be applied to specific spectral lines or groups of lines has to be checked individually.
In Reiners & Schmitt (2002a) the possibility of detecting differential rotation in line profiles was demonstrated. This paper also answers the question of whether the technique used there is applicable to fast rotators, too. It shows what can be learned about gravity darkening and inclination angle from the shape of stellar absorption line profiles.
Absorption line profiles observed in fast rotating stars are affected by three mechanisms inherent in all lines: (a) the projected geometry of the star, (b) the flux distribution due to temperature variations (known as gravity darkening) on the rotating stellar surface, and (c) the rotation law. These three line-independent mechanisms will be discussed in the following.
Case (a) is well understood, fast rotation diminishes the gravitational potential at the equator and alters the star's sphericity. As a consequence, lines of constant projected rotational velocities no longer lie on chords but are bent (cp., e.g., Collins & Truax 1995). Although an analytical description of the surface velocity distribution is not then available, calculation by integrating over the stellar surface is unproblematic.
(b) Since the presence of gravity darkening was initially described by
von Zeipel (1924), many publications were written on the correct gravity
dependence of surface temperature. In general, gravity darkening is
described in terms of a parameter
used in the equation
(c) Stellar rotation laws are poorly known. Our picture comes from
observations of the Sun, where the angular velocity can be
approximated as
Stellar line profiles are subject to a number of different mechanisms affecting their shape. Besides the mentioned effects of gravity darkening and rotation law, limb darkening, additional (turbulent) velocity fields and stellar surface structure like spots can significantly contribute to the line profiles' shape. Utilizing the Fourier transform turns out to be of great advantage in disentangling the different effects. When approximating line broadening by convolutions, their interaction can be studied most easily using Fourier transforms, since the computationally complicated convolutions become multiplications in Fourier domain (cf., e.g., Gray 1976).
Reiners & Schmitt (2002a) utilized the zeros of the Fourier transformed line
profiles to search for the subtle effects of differential rotation.
They showed that the ratio of the first two zeros q_{1} and q_{2} is a
direct indicator for solar-like - i.e., Equator faster than Pole -
differential rotation (
in Eq. (2)). The
value of the ratio
q_{2}/q_{1} is extremely sensitive to the
profile's shape, and is unaffected by the "standard'' turbulent
velocity fields like micro- and macroturbulence even for rotational
velocities as low as 10 km s^{-1}. Using a linear limb darkening
law with a parameter ,
in a rigidly rotating star the value
of
q_{2}/q_{1} can be approximated as
Since the ratio q_{2}/q_{1} turns out to be an easily accessible parameter characteristic for the line's shape, I continue to use it for studying the influence of gravity darkening in rapidly rotating stars. For a more detailed discussion of the correlations between differential rotation , the inclination angle i and q_{2}/q_{1} see Reiners & Schmitt (2002a,2003).
The approach of this paper is to study the effects of fast rotation universal for all absorption lines. Line specific dependencies on gas pressure, ionization stages, etc. are not taken into account. This approach is especially suited for studies incorporating different lines of similar (preferably heavy) ions, e.g., with LSD methods. For heavy ions pressure broadening is not expected to play a major role compared to the dominant rotational broadening. For light elements like H and He, this is not true and great care has to be take when applying the methods to them. Although the Lorentzian profiles induced by pressure broadening should not add additional zeros to the line profile Fourier transforms (cp. Heinzel 1978), errors from approximating line broadening effects by convolutions may become significant in these cases.
The calculations were done using a modified version of the package developed and described by Townsend (1997). Surface integration is carried out over 25 500 visible surface elements scaling the flux with temperature according to a Planck law. A Gaussian profile was used as the input function, since the large rotational velocities make the shape of the input function unimportant. For all cases the same input function was used regardless of gravity and temperature. To measure the zeros in Fourier transformed line profiles the accuracy must be higher for the faster rotators; 32 768 points were used in the transform algorithm for values of km s^{-1}, 65 536 points for values of between 100 and 200 km s^{-1}, and 131 072 points for km s^{-1}. Thus the sampling error in q_{2}/q_{1} does not exceed a value of 0.01 for all calculations.
Broadening profiles are calculated and the values of q_{2}/q_{1}determined for four stellar models with polar effective temperature and mass given in columns two and three in Table 1. A grid is computed in three values of (0.0, 0.08 and 0.25), five values of inclination angle i(10, 30, 50, 70 and 90), and values of between 50 and 300 km s^{-1} in steps of 25 km s^{-1} with the equatorial velocity km s^{-1}. Altogether more than 450 models are computed. The dependence of the ratio q_{2}/q_{1} on different parameter choices is analyzed and studied in the following.
Table 1: Parameters of the calculated models, a and b are parameters used in Eq. (4) with values of gravity darkening according to Claret (1998). Note that for all models profiles with and 0.25 were analyzed.
Figure 1: Derived values of q_{2}/q_{1} for a rigidly rotating F0-type star with . Different symbols indicate inclination angles as indicated in the figure. The value of q_{2}/q_{1} is determined by independent of the values of i and . The slope is well approximated by the second order polynomial shown as a solid line. | |
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To show the dependence of q_{2}/q_{1} on , in Fig. 1 q_{2}/q_{1} is plotted vs. for the F0 model with a gravity darkening parameter of . The region in q_{2}/q_{1} typical for a spherical surface and a linear limb darkening law with is indicated with dotted lines (cp. Eq. (3)). Values for stars seen under various inclination angles are distinguished by different symbols, all values are in good agreement with the marked polynomial indicating a smooth decline of q_{2}/q_{1} for stars rotating faster than km s^{-1}.
The parameter determining the value of q_{2}/q_{1} in a given stellar model obviously is , independent of the inclination angle (and thus ). This means, that under the assumption of rigid rotation, and with assumed values of , in fast rotators one can in principle obtain the value of from the value of q_{2}/q_{1}, and thus, with the known value of , the inclination angle i. I will return to this point in Sect. 4.3.
Figure 2: Derived values of q_{2}/q_{1} vs. equatorial velocity for four different stellar models (B0, A0, F0 and G0). For each model different values of the gravity darkening parameter (0.0, 0.08 and 0.25) are indicated using different symbols. The dependence of q_{2}/q_{1} on differs for the different cases but is monotonic in and . | |
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In the four panels of Fig. 2 the derived values of q_{2}/q_{1} are plotted vs. for the models of spectral types B0, A0, F0 and G0 as specified in Table 1. In each panel calculations for three different values of are shown, the different choices of (0.0, 0.08 and 0.25) are plotted using different symbols. The slope of q_{2}/q_{1} is well described by a second order polynomial for all cases. The ratio q_{2}/q_{1} is diminished in rapid rotators with larger deviations from the standard value of q_{2}/q_{1} = 1.75 the larger the value of . In the A0, F0 and G0 models, q_{2}/q_{1} becomes less than 1.72 for very fast rotators depending on the value of , i.e., deviations from standard line profiles are significant and in principle observable in these stars.
The behavior of q_{2}/q_{1} with different values of the gravity darkening parameter can be studied in the four panels of Fig. 2. In the B0 model no value of q_{2}/q_{1} < 1.72 appears, but the influence of on the ratio q_{2}/q_{1}grows with later spectral type. In all cases q_{2}/q_{1} depends monotonically on the rotational velocity with diminished q_{2}/q_{1} for larger rotational velocities. The result that the line profiles depend more strongly on the value of for stars of later spectral types, reflects the fact that the relative temperature contrast on the stellar surface is larger for cooler stars, since is comparable for similar values of . Furthermore, for all models no significant variation of q_{2}/q_{1} appears for , i.e., without gravity darkening the geometrical deformation does not suffice to significantly change the shape of the line profiles.
In a larger sample of stars of similar spectral types these results could make it feasible to determine the value of gravity darkening in terms of the parameter when plotting q_{2}/q_{1} vs. . While for many stars in such a plot the values should fall below an upper envelope expected from the q_{2}/q_{1}- relation (the real values of are larger than the measured values of ), a clear threshold should mark the upper envelope indicating the correct value of . Using a larger sample of fast rotators of similar spectral types the ratio q_{2}/q_{1} could thus be used to determine the values of from measurements of single stars.
Figure 3: Stars with measured values of and q_{2}/q_{1} are seen under an inclination angle i. In the four panels (B0, A0, F0 and G0-type stars) the approximated dependence of i on the ratio q_{2}/q_{1} is shown for values of and 300 km s^{-1} (solid lines, top to bottom as indicated in the upper left panel). The dashed line marks the lowest ratio of q_{2}/q_{1} consistent with slow rotation and extreme limb darkening, no values of q_{2}/q_{1} > 1.75 are expected. | |
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The ratio
q_{2}/q_{1} monotonically depends on equatorial velocity
and is independent of the inclination angle i. Assuming
a value of
for a fast rotator of known spectral class, the
equatorial velocity
can thus be determined from a
measurement of
q_{2}/q_{1}. The functions
shown in Fig. 2 are approximated by the second order
polynomial
In each panel of Fig. 3 the values of i are plotted vs. the ratio q_{2}/q_{1} for four different values of . As in Fig. 2 each panel represents a stellar model from Table 1. Functions for values of and 300 km s^{-1} are plotted as examples; for calculating intermediate values of , Eq. (4) can be used with values of a and b as given in columns five and six of Table 1. For example, a value of q_{2}/q_{1} = 1.70 measured in a profile of an A0-type star with km s^{-1} can indicate that a rigidly rotating star is observed under an inclination of ( km s^{-1}).
Solar-like differential rotation with the Equator faster than the Pole diminishes the value of q_{2}/q_{1} as shown in Reiners & Schmitt (2002a); a measurement yielding q_{2}/q_{1} < 1.72 can be due to a solar-like rotation law with the Equator faster than the Pole. Whether very fast rotators are subject to strong differential rotation will not be discussed here. In Fig. 2 q_{2}/q_{1} appears to be larger than 1.72 for all stars with km s^{-1}and values of gravity darkening according to Claret (1998) (cp. Table 1). This means that a measurement of differential rotation by determining the ratio q_{2}/q_{1} is not affected by gravity darkening if km s^{-1}. Interpreting gravity darkening as the reason for a measured value of q_{2}/q_{1} < 1.72, the rotational velocity ( km s^{-1}) and inclination angle of the star can be determined. Consistency of with breakup velocity can be checked. For , differential rotation and very fast rotation remain indistinguishable without further information. An example is given in Sect. 6.
Figure 4: Values of as measured from the first zero of the Fourier transformed broadening profiles plotted vs. the real values given as input to the models. The four panels show cases of B0, A0, F0 and G0-type stars, the value of the gravity darkening parameters are chosen according to Claret (1998). In each panel series for four different inclination angles ( and ) are differentiated, the respective values are connected by solid lines. Dotted lines mark the identity between real and measured values. | |
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To what accuracy a measurement of is possible was discussed in a number of publications (e.g., Collins & Truax 1995; Stoeckley 1968). It is known that measured values of underestimate the real values especially for very fast rotators regardless of the data quality. To what extent a measured value is too low, depends on the gravity darkening law and on the stellar inclination. In the four panels of Fig. 4 the differences between real and measured values for the B0, A0, F0 and G0-type stars are shown using the most probable values of gravity darkening as done above. The measured values of have been derived using the first zeros of the Fourier transform. Values for different inclination angles are differentiated and shown with different symbols, those derived for identical values of i are connected by solid lines. Dashed lines mark the identity of real and measured values. As expected, the measured values of may underestimate the real ones but will never indicate too high a projected rotational velocity. This reflects the fact that the (equatorial) surface regions with the largest projected rotational velocities become cooler due to gravity darkening, and thus less flux is observed from them.
The extent, to which a measured value of is systematically underestimated, depends strongly and in a well-defined manner on the inclination angle. For constant inclination angles, the measured value of is a smooth function of the real one (cp. to Fig. 6 in Collins & Truax 1995). Underestimation of is stronger for smaller inclination angles in all models. The deviation naturally is higher for large values of (and thus also of ). Thus, measured values of in very rapid rotators can be corrected by determining the ratio and calculating the inclination angle i as shown in Sect. 4.3.
It should be noted that - in contrast to Fig. 12 in Collins & Truax (1995) - inclination has a strong effect on the zeros of Fourier transformed line profiles of rapid rotators; the case shown there only holds for zero projected rotational velocity.
As an observational example the ten stars for which differential rotation was claimed in Reiners & Schmitt (2003) are given in Table 2 with their determined values of and q_{2}/q_{1}. According to the present calculations, these values of q_{2}/q_{1} < 1.72 could also be interpreted as fast rotation seen under small inclination angles i. The required values of and i for this interpretation are calculated and given in columns five and six in Table 2. With one exception all values of are larger than break-up velocity and thus can be excluded (it is furthermore very unlikely to find inclination angles below 6 in ten out of the 32 objects used in that sample). Thus differential rotation remains the most probable explanation for the peculiar shapes of these profiles.
Table 2: Stars with measured values q_{2}/q_{1} < 1.72 from the sample of Reiners & Schmitt (2003); differential rotation is claimed for these objects. Interpreting the values of q_{2}/q_{1} as due to very fast rotation seen under small inclination angles, the required values of and i are given. Nine of the ten are larger than break-up velocity.
Line independent effects of fast stellar rotation on line profiles were presented. Neglecting line specific variations in shape and line depth, the effects of geometrical deformation and the role of the gravity darkening parameter have been outlined. Grid calculations in gravity darkening , inclination angle i, rotational velocity and spectral type with the parameter q_{2}/q_{1} classifying the profiles' shapes were carried out. The results summarized below can be applied especially to high quality data covering large wavelength regions where line-independent broadening profiles may be obtained using the methods of LSD. Applicability must be proved for each case individually, but especially for lines of heavier ions in stars of later spectral types, intrinsic variations of line profiles are sufficiently small and the results of this study can directly be applied. The results can be summarized as follows:
As for differential rotation, it turned out that its measurement is not affected by gravity darkening in stars rotating slower than km s^{-1}. For faster rotators, small values of q_{2}/q_{1} can principally be due to three reasons; (i) strong differential rotation, (ii) very fast rigid rotation (perhaps seen under a small inclination angle), and (iii) both. The constraints on a value of can help to clarify the situation, e.g., if a value of larger than breakup velocity is needed to rule out differential rotation.
Acknowledgements
A.R. acknowledges financial support from Deutsche Forschungsgemeinschaft DFG-SCHM 1032/10-1.