A&A 471, 925-927 (2007)
DOI: 10.1051/0004-6361:20066909
H.-G. Ludwig
GEPI, CIFIST, Observatoire de Paris-Meudon, 5 place Jules Janssen, 92195 Meudon Cedex, France
Received 10 December 2006 / Accepted 6 May 2007
Abstract
Context. Spectral synthesis calculations based on three-dimensional stellar atmosphere models are limited by the affordable angular resolution of the radiation field. This hampers an accurate treatment of rotational line broadening.
Aims. We aim to find a treatment of rotational broadening of a spherical star when the radiation field is only available at a modest number of limb-angles.
Methods. We apply a combination of analytical considerations of the line-broadening process and numerical tests.
Results. We obtain a method which is closely related to classical flux convolution and which performs noticeably better than a previously suggested procedure. It can be applied to rigid as well as differential rotation.
Key words: hydrodynamics - radiative transfer - stars: atmospheres - line: profiles - methods: numerical
Spectral synthesis calculations based on time-series of three-dimensional (3D) background structures from hydrodynamical model atmospheres are computationally demanding and pose limits on the affordable resolution for representing the angular dependence of the radiation field. In such calculations, the number of employed azimuthal directions typically ranges from four to eight, the number of inclined directions representing the center-to-limb variation being between three and five. Obviously, the resolution is not high, and in particular makes an accurate implementation of rotational line broadening somewhat difficult. To our knowledge, the only published method handling rotational broadening in the 3D case is that of Dravins & Nordlund (1990). Here, we describe an alternative procedure which provides higher accuracy at similar computational complexity. It is closely related to standard flux convolution (e.g., Gray 1992). Its development was motivated by the demand for an accurate description of the rotational broadening of spectral lines in the solar spectrum (Caffau et al. 2007).
For the moment, we neglect differential rotation, and treat the star as spherical and rigidly rotating. Local-box hydrodynamical model atmospheres provide a statistical realization of a small patch of the surface flow pattern. Formally, we want to obtain an estimate of the expectation value of a rotationally broadened, disk-integrated line profile. Rotational symmetry with respect to the stellar disk center implies that there is no azimuthal dependence of the radiation field. All we need to know for evaluating the disk-integrated rotationally broadened line profile are temporal and azimuthal averages of the emergent radiation intensity of the local model as a function of asterocentric angle . Hence, the problem is equivalent to the rotational line broadening problem in standard, plane-parallel model atmospheres.
Figure 1: Illustration of the apparent radial velocity distribution on a stellar disk for solid body rotation: vertical lines of constant radial velocity are labeled by their velocity in units of . They lie parallel to rotational pole - disk center direction. For further explanations see text. | |
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We start by describing the procedure of Dravins & Nordlund (1990,
Sect. 2.1) who model rotational broadening as a superposition of line
profiles at various
,
broadened by the velocity field
along circles
around the stellar disk center. We refer to
one of these circles as "-circle'' (see Fig. 1). The
broadening effect of the projected rotational velocity along a -circle
located at
can be expressed as a convolution according to
(3) |
The original implementation of Dravins & Nordlund did not use a formulation as convolution but a discrete integration over -circles using polar coordinates. While mathematically equivalent to Eq. (1), it somewhat obscures the critical role of the most extreme velocities on a -circle for the smoothness of the rotationally broadened spectrum.
Figure 2: Rotationally broadened line profiles (solid lines) for different total numbers of asterocentric angles employing the broadening procedure of Dravins & Nordlund in comparison to the exact profile (diamonds). For clarity, the broadened line profiles have been vertically offset. Wavelengths are given as Doppler velocities in units of the width of the Gaussian non-broadened profile (dash-dotted line). | |
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Figure 2 illustrates the result of a test of Dravins & Nordlund's procedure. We broadened an artificial Gaussian line profile with a rotational speed of three times the line's Doppler width. The rotational speed was chosen to be particularly critical. Effects of smaller rotational velocities become less pronounced due to the smoothing by the Gaussian line profile, at larger rotational speeds deviations are less conspicuous since less localized. For the test case we assumed that the relative line shape is independent of , and that the intensity in the continuum follows a linear limb-darkening law with a limb-darkening coefficient of 0.6. Hence, standard flux convolution can be applied to obtain the exact disk-integrated line profile. Figure 2 shows the convergence of the numerical approximation towards the exact result with an increasing total number of limb-angles . As evident in the figure, noticeable deviations between the exact and the numerical result are present up to and including .
The major reason for the "wiggly'' behavior of the broadened line profile
shown in Fig. 2 are the pronounced spikes in the integrand in
Eq. (1). One can reduce their impact by associating a given
not only with an infinitely thin -circle but with a
-ring (see Fig. 1) of finite extent. The
contribution
to the rotationally broadened flux profile stemming
from the stellar disk area subtended by
is
given by the convolution
Figure 3: As for Fig. 2 for our broadening procedure. | |
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Figure 3 illustrates the outcome of this procedure. A comparison with Fig. 2 shows a more rapid convergence towards the exact result. The improvement is related to the fact that the new method can at least handle exactly the simple case of a globally -independent intensity which is not the case for the method of Dravins and Nordlund. One could further refine it by introducing an analytical expression for the -dependence of the intensity in each -ring - perhaps motivated from a fit to the continuum intensity available at the discrete . However, we did not implement this since from the test it appeared that the accuracy at an affordable number of is already sufficient. There is a caveat to this statement: the accuracy of the methods also hinges on the level of the differential line-shift and -broadening as a function of which we did not test here. This should be checked on a case by case basis.
Up to this point we considered solid body rotation only. However, the previous discussion made clear that integration over -rings should also perform better in the case of differential rotation. The referee made the point that putting efforts into 3D model and spectral synthesis calculations warrants the inclusion of effects of differential rotation to maintain highest level of accuracy. In the following we give a brief demonstration that differential rotation on the level observed in the Sun can indeed be relevant for resulting line profiles. We implemented our method for a case of differential rotation.
We chose the possibly simplest parameterization of the solar
differential rotation pattern of the form
Figure 4 depicts an example comparing rotationally broadened
profiles assuming rigid as well as differential rotation. We arbitrarily
selected an Fe I line (at 6082 Å) of a 3D spectral synthesis
calculation for the Sun, and solar-like rotational parameters,
V=1.8 km s^{-1},
,
and for the case of differential
rotation
.
The plot shows that - as expected - our procedure of
integrating over -circles (here )
leaves no obvious imprint of
spikes in the resulting profiles. Moreover, noticeable differences are present
for fixed V and .
One might interpret the smaller degree of
broadening in the differentially rotating case as simply a result of the
the smaller disk-averaged rotation rate for
.
However,
from Eq. (6) we obtain for the root-mean-square radial
velocity
due to the rotation over the stellar disk
Figure 4: Comparison of rotational broadening assuming a rigidly (solid line) and differentially (diamonds) rotating Sun. Thirty times enlarged differences of the resulting profiles (in the sense differential-rigid) are plotted in the lower part of the panel. For further explanations see text. | |
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The approach suggested in this paper provides an accurate treatment of rotational line broadening of a spherical star employing intensities at a modest number of limb-angles only. This is of practical importance in the context of 3D spectral synthesis calculations where angular resolution is computationally expensive. This also holds for differential rotation but the rotational kernel functions must be evaluated numerically. Our approach is not restricted to the 3D case but could be equally well applied in the standard 1D case. The issue is of course less pertinent in 1D since high angular resolution is affordable making the broadening method uncritical.
Acknowledgements
We thank Ansgar Reiners for helpful discussions about differential rotation. The work was funded by EU grant MEXT-CT-2004-014265 (CIFIST).