3. Method

Table 1 (click here) summarizes the parameter combinations used in the computations. The five moderately high resolutions considered here are described by their pixel-scale, defined as the pixel-size expressed in wavelength units. The lines are assumed to have Gaussian profiles. The critical assumption is the symmetry of the lines, the exact shape has less importance. Their width is representative for critically sampled Th-Ar lines obtained with several moderate dispersion spectrographs (e.g. ARC at Yerkes, BME at CTIO, CASPEC at ESO, ELODIE at OHP, ESPRESSO at SPM, UCLES at AAO, UES at La Palma, ...).

 

quantity specification(s) 100000, 50000, 43700, 33333, 25000

Line shape Gaussian Line width Fit function Fit interval Ar I intens. Ar II intens.

Table 1:   Characteristics of the degraded laboratory spectra and the fitting process. The Ar line intensities in the last two rows refer to the "standard" and to the "Ar rich" case (line positions in Å)

 

reference element, ion

Minnhagen¹ (1973) ArI

Norlén (1973) ArI & ArII

Palmer & Engleman (1983) ThI & ThII

Table 2:   Sources of laboratory data

¹ We scaled the sources for ArI to each other by where S is the author's intensity parameter in a log scale with base .

The relative intensities of the lines are taken from the laboratory sources mentioned in Table 2 (click here). The Ar intensities are scaled against the Th intensities in two different ways (Table 1 (click here)). The applied ratios correspond to the output of the Th hollow cathode lamps provided by S&J Juniper (CAT NO 4160QA) with an Ar gas fill pressure of and a quartz window, operated "standard" at , but the "Ar rich" spectra have been obtained presumably at a higher, but not exactly known, lamp current. The main purpose of analyzing the blending in these two different cases is to gain insight in how far this scaling might invalidate the selection of useful calibrators. The output of Th-Ar lamps used with other spectrographs differs, as far as we could check, much less from one of these cases than the factor which separates the here adopted Ar-to-Th ratios, although not all lie in between these two cases. It should be noted that the result for most blends does not depend on this scaling: it turns out that at least 80% of the useful lines (and much more at smaller pixel-scales) is not or not significantly affected by a large change in the Ar-to-Th intensity ratio. Moreover we indicate in Sect. 5 (click here) several ways to detect or remove lines that would have been inappropriately selected as useful because of bad assumptions in the ratios of the blending components.

Anyway, the use of preset intensities is a drawback in predicting the effective blend wavelengths; even Th-Th blends might be affected by the operating lamp current and aging effects. It cannot be the purpose of such computations to predict large corrections precisely for any lamp, and surely not for different lamps. In view of the need for general usefulness of the final calibrator lists, one is therefore obliged to include in these lists only blends with corrected positions deviating from the laboratory wavelength of the stronger component by less than twice the rms due to random noise. Hence, lines that were found to have a displacement due to blending have been eliminated in an early stage of our two-step calculation method.

In the first step, a grid of synthetic spectra consisting of two Gaussians separated by a distance , and with , is subjected to a least-squares Gaussian fitting routine to find out when displacements are to be expected. The subscripts b and 0 refer to the blending and the principal component. The fit function includes a (constant) background term a₀ for consistency with several software packages in use in astronomical spectroscopy. The principal Gaussian is placed exactly in the centre of the seven pixels long fit interval. The resulting grid of displacements is used to decide for each laboratory line under study whether it is with certainty intolerably blended, or whether a more detailed analysis is useful. An important reduction in computing time results: 40% and 80% of the lines are rejected based on this simple criterion, for pixel-scales of and respectively. It is possible that in the case of a line blended by several others, a more detailed computation would give a displacement slightly below 0.05pix, since blending lines on different sides of the principal component can cancel partly the adverse effect of each other. Tests indicate that this occurs rarely ( probability); and even then, the result is very sensitive to the sub-pixel location of the centre of the line and thus a discretisation-independent predicted displacement would remain inaccurate (see the discussion of the second step for an evaluation of discretisation-dependent effects).

  
Figure 1: Displacement of the line centre due to one single blending component at a distance d. The curves are labelled with their corresponding relative intensity

Figure 1 (click here) clarifies what kind of blends are marked as intolerable. A blending line with an intensity of only 10% of the intensity of the principal component can produce a displacement when it is situated at a distance of . Stronger blending lines produce shifts in the order of tenth(s) of a pixel over a wide range in separation.

If in the first-step estimation the predicted displacement , then the line is subjected to a more stringent test in a second step, or, if the line is found to be quasi-isolated, then its laboratory position is marked as unaltered. Quasi-isolation was defined by:
 
for all blending components. This corresponds to a (neglected) displacement . The intensity of the blending components was arbitrarily enhanced by a factor 2 over the intensity given in the laboratory source to account for the case of blending with several lines near to each other that have similar intensities. In this step the only requirement is to label no line errorneously as quasi-isolated, while the number of missed quasi-isolated lines should remain small relative to the total number of lines that need a further check. The fraction of quasi-isolated lines detected in this way varied from 40% at a pixel-scale of to only 5% at . Hence, after the elimination of the intolerably blended lines and the quasi-isolated ones in this first step, there remain only of the lines for further testing irrespective of the pixel-scale considered.

In the second step, the spectrum around the considered laboratory line is calculated for a full grid of sub-pixel locations of the line centre, as the Gaussian fit depends on discretisation because of model-mismatch (David & Verschueren 1995). The line centre was shifted in steps of 0.05 pix (until the same discretisation repeated after 20 steps) and the Gaussian fit was explicitly made for each case. The average position of the fitted Gaussian line centre over these twenty trials defines the effective wavelength of the blended laboratory line for a given pixel-scale. In the case of complicated blends, the result may be too sensitive to discretisation and the average would become meaningless. The same may occur when the wing of a strong blending line affects the intensity at the edge of the fit interval. Such cases are marked as unsuitable for calibration purposes, like the ones eliminated in the first computational step. The rejection criterion used is that , where s is the rms of the fitted line centre computed from the twenty discretisation trials.

The complications discussed here imply that the decision on usefulness depends for a minority of lines critically on the fit interval used. A seven pixel interval corresponds in our sampling case to 3.7FWHM. There is no reason to consider in a rich spectrum a larger interval, since in the absence of blending effects at the edges, the background is well sampled. A smaller interval is less vulnerable to wide blends, but more to overfitting of closer blends, to errors in background estimation and to systematics with sub-pixel location (in the case of our 4-parameter fit). However, the fit function used might be constrained better using a priori information: background subtraction is in principle easy and the FWHM of the PSF could be constrained to vary at most slowly over the whole frame. At present, we only state that improvements in this sense might be considered, but are not applied in the commonly used software packages. The computation with the seven pixel interval is safe in the sense that it rejects too many lines (for a given line width), rather than provide the user with a selection containing some potentially bad calibration lines.