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Astron. Astrophys. 355, 743-750 (2000)

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5. Results

5.1. Choice of the fitting window

To obtain approximate values of all of the parameters, including the asymmetry factor, B, we make a preliminary fit to each of the pairs [FORMULA]=0 - [FORMULA]=2 and [FORMULA]=1 - [FORMULA]=3. These values are then used as input parameters in simulations, aimed at quantifying the interaction between the components during the fitting procedure.

In the first simulation, we apply the Nigam fitting on only one of the two components of a pair, in order to measure the influence of the neighbouring mode. This shows that for the pair [FORMULA]=0 - [FORMULA]=2, the influence of one mode on the other is such that fitting separately leads to a significant error on the asymmetry parameter (up to a factor of 2 at 2700 [FORMULA]). For the pair [FORMULA]=1 - [FORMULA]=3, fitting [FORMULA]=3 alone leads to large errors (the sign of B may even be reversed). However, due to the relatively low amplitude of the [FORMULA]=3 mode, the situation is more favourable for [FORMULA]=1, which can reasonably be fitted alone up to n=22, with an error inferior to 10 [FORMULA] on B and insensitive to the choice of the [FORMULA]=3 B parameter.

With confidence in the possibility of fitting the [FORMULA]=1 alone, we then examine the influence of the fitting window width on the determination of B. We might anticipate two effects, limiting the window width in opposing senses: the Taylor expansion used to derive the Nigam expression is valid close to the resonance and may therefore require a small window width, whilst the effects of stochastic excitation noise are minimised by increasing the amount of data through the use of a wider window. Fits using GOLF data show that for the [FORMULA]=1 fit alone, the variations of measured B with the window width are consistent with the stochastic noise effect alone. This gives us a reasonable estimate of B for [FORMULA]=1, which we can use to examine the validity of the [FORMULA]=1 - [FORMULA]=3 pair fit. We examine fits on this pair for different window widths. We find that a window width value exists for which B is compatible with the value when fitting [FORMULA]=1 alone. As seen in Fig. 4, the residuals (shown as a function of frequency) are not systematic, but are consistent with the stochastic effect. Also shown in the figure is the effect of a larger window, for which random errors are reduced, as expected, but the determinations are systematically too small.

[FIGURE] Fig. 4. Determination of the asymmetry parameter on the data for [FORMULA]=1: effect of the fitting window. Diamonds refer to the global fit of the pair [FORMULA]=1 - [FORMULA]=3 and stars for [FORMULA]=1 alone. The dashed line refers to the optimized window and the solid line to a larger window.

For [FORMULA]=3, we must also verify that the window is appropriate, i.e. that it does not extend too far to high frequencies. Using the reliable estimate of the [FORMULA]=1 asymmetry already obtained, we check the influence of the high-frequency side of the window for [FORMULA]=1 alone. No bias was found on B when this limit was chosen identical to that of [FORMULA]=3 in a global [FORMULA]=1 - [FORMULA]=3 fit, giving confidence in the choice of window and the resulting [FORMULA]=3 asymmetry parameter. The window width chosen for the [FORMULA]=1 - [FORMULA]=3 pair, optimized for the [FORMULA]=1 alone, varies from 27 [FORMULA] for n=14 up to 38 [FORMULA] for n=25.

For the pair [FORMULA]=0 - [FORMULA]=2, individual fitting being impossible, we extrapolate the results of [FORMULA]=1, to derive an initial window width. To gain in statistics, we increased slightly the width of the extrapolated window, checking at each step that B was not biased and that the [FORMULA]=4 wing does not enter the window.

5.2. Asymmetry parameter

Fig. 5 shows the variation of the B parameter with frequency for the pairs [FORMULA]=0 - [FORMULA]=2 and [FORMULA]=1 - [FORMULA]=3, with the optimized windows. These are found to be consistent with the results of Toutain et al. and do not show a clear difference between the two pairs. To study an eventual B dependence on degree, we performed also the same fitting but leaving free the two values of B inside a pair. As already indicated, the value of B for [FORMULA]=1 is not modified. The value of B for [FORMULA]=3 is not useful, displaying very large error bars, even when it is fitted alone with the fixed wing of [FORMULA]=1 included in the formula.

[FIGURE] Fig. 5. The asymmetric parameter as a function of frequency assuming that it is the same for the two components of each pair.

For modes [FORMULA]=0 and [FORMULA]=2, Fig. 6 suggests that the asymmetry of the mode [FORMULA]=0 is on average lower than for [FORMULA]=2. Moreover, Fig. 6 exhibits a mirror effect between the [FORMULA]=0 and [FORMULA]=2 curves which is explained as following: first, starting with different values of B for [FORMULA]=0 and [FORMULA]=2, Monte-Carlo simulations show that the fitting procedure is capable of reproducing the two values on average without bias, and when forcing them to be equal in the fitting, the value obtained is on average the mean value of the two. Second, the length of the time series is such that the variation of B with radial order is very smooth when only one value is taken for the pair [FORMULA]=0 - [FORMULA]=2. These two properties of convergence and smoothness explain why the effects of stochastic excitation on the curves obtained when separate asymmetry parameters are allowed, lead to anticorrelated variations. As far as a difference of B in the pair [FORMULA]=0 - [FORMULA]=2 is concerned, we checked if the overlap of the two modes could bias in opposite ways the estimation of their B parameter producing an apparent difference. Monte Carlo simulations show that there is an equal probability to get either the [FORMULA]=0 B greater or the [FORMULA]=2 B greater, when the two parameters are equal in the simulated profile. This permits to estimate the probability that the lower asymmetry for the [FORMULA]=0 mode is real. It is found to be 93%, which is marginally significant due to the low number of radial orders in the statistics.

[FIGURE] Fig. 6. The asymmetric parameter as a function of frequency, allowing a variation between [FORMULA]=0 and [FORMULA]=2. The dashed line corresponds to the curve of Fig. 5.

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© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000