Astron. Astrophys. 324, 177-184 (1997)

Appendix A: precision of LQ-fits to gaussian profiles

In Yoshizawa et al. (1985) we can find the generalized solution of the precision of a least square fit to a given photonic profile, taking into account the poisson noise of the distribution. In the case of a simple gaussian profile of maximum flux , total intensity I and full width at half maximum f the uncertainty on the determination of the full width and on the centroid find the simple expressions, in the case of a large amount of photons I:

These results translate the fact that the uncertainties scale with the spatial width of the profile, but vary inversely to the square root of the total flux under the profile, i.e. the integrated signal to noise value.

Supposing now that an emission line of integrated intensity is rising on top of the continuum of intensity . After a hypothetically perfect subtraction of the continuum the residual profile will be the one of the line, of integrated flux , but the noise of the profile will be determined by the noise due to the photons of the line and the photons of the continuum. Therefore, the precision of the least square fit to the residual profile should now be expressed as the following:

If several columns m in the spatial direction are averaged before the fitting, the total flux to be used is obviously , i.e. the flux of a single column multiplied by the number of columns.

As an example, let us estimate the different uncertainties of the least square fit in the case of the long-slit spectrum of AB Aur. The continuum right- and leftwards of the line peaks at 3560 ADU (analog/digital units), while the line exceeds the continuum of about 520 ADU. In spectral direction the FWHM of the line is close to 7 pixels (uncorrected for the instrumental resolution), in spatial direction 4.1 pixels. The CCDs conversion factor is close to 2 electrons/ADU. We average one column of continuum left- and one rightwards the emission line and then expand the result in the spectral direction before subtracting it finally from the total spectrum. For the precisions of the continuum fit equations A1 and A2 apply, with I multiplied by 2 (m = 2 here). The obtained uncertainties are therefore: pix, and pix. These theoretical results are based on the rough hypothesis that the spatial profile is a perfect gaussian. The line profile is collapsed in the spectral direction before performing a LQ fit. The total flux in the line is therefore photoelectrons. The flux in the continuum relevant for the noise determination is the flux in one column divided by the square root of the number of averaged columns (m = 2 here), multiplied by the spectral FWHM in pixel of the line: photoelectrons. Therefore: pix, pix. These pixel uncertainties are translated in angular uncertainties by the multiplication with 0.41"/pix. Results for the other stars are listed in Table 5.

Table 5. Theoretical uncertainties on the determination of the FWHM and the centroid of the forbidden emission lines by least square fit to a gaussian profile built by photons. Examples calculated for different stars of the sample following appendix A. Columns: (1) Star and position angle. (2) Maximum signal in ADU of the continuum and the line. (3) FWHM of the continuum in spatial direction and FWHM of the line in spectral direction (uncorrected for resolution, in pixel). (4) Uncertainty on the spatial centroid of the continuum (left) and uncertainty on the spatial centroid of the line with respect to the continuum (quadratic addition of the uncertainties), (right). The spatial centroid uncertainties are translated in angular uncertainties (arcseconds).(5) Uncertainty on the spatial FWHM of the continuum (left) and the line (right), expressed in percentage of the seeing.

We see that all these uncertainties are dominated by the measured errors indicated in cols. 3, 6 and 9 of Table 4, most probably due to the unprecise continuum subtraction, the presence of telluric lines, etc.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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