The upper panel of Fig. 2 (click here) shows the distribution of the selected
lines at different pixel-scales (different symbols) versus relative line
strength (abscissa). The lower panel displays the information alternatively as
the fraction of rejected lines at a given pixel-scale and line intensity. One
should keep in mind that the lines useful for calibration purposes are in the
longward intensity tail, roughly at 
.
The fraction of rejected lines depends strongly on the pixel-scale. At lower
pixel-scales (higher resolutions) than considered in this paper, enough
totally unblended lines can be selected. At higher pixel-scales (lower
resolutions) than considered here, almost all Th lines are blended.
The absolute number of useful lines does not increase significantly
towards lower line intensities. This is true over almost the whole pixel-scale
interval under consideration. Hence, the use of more (and thus also weaker)
lines without appropriate selection for blends, in an attempt to average out
most of the blending influences, inevitably introduces more strongly blended
lines with lower S/N! The conclusions presented here remain valid for more
restricted wavelength intervals within our considered range.
Figure 2: Number of selected lines in bins of relative line strength, for
different pixel-scales (labels refer to 
/pixel-scale).
The full line is the input distribution (laboratory resolution).
The lower panel gives the fraction of rejected lines in the same
parameter space
From here on, we restrict our attention to the subsample of useful lines.
Figure 3 (click here) shows the distribution of the selected lines over
wavelength, together with the input distribution. At the higher pixel-scales,
the absolute number of useful lines in the red becomes higher than in the
blue, while the original input data show a line density ratio of roughly 3
in the other sense. The crowding of lines in the blue Th spectrum starts
to provide more useful calibrators than in the red only at
pixel-scales 
.
Figure 3: Distribution in wavelength (
-bins) of the original sample
and of the samples selected at the different pixel-scales. Symbols as in
Fig. 2 (click here)
The distribution of the displacements of the selected calibration lines is shown in Table 3 (click here) for the different pixel-scales. The strong concentration towards low values indicates that the number of useful lines does not increase significantly by including lines with displacements larger than a few 10-2 pix.
 [pix] | 100000 | 50000 | 43700 | 33333 | 25000 |
 | 2069 | 1209 | 1051 | 771 | 523 |
| 0.005-0.010 | 105 | 136 | 121 | 110 | 127 |
| 0.010-0.020 | 145 | 162 | 195 | 175 | 152 |
| 0.020-0.030 | 89 | 105 | 111 | 129 | 109 |
| 0.030-0.040 | 72 | 84 | 76 | 78 | 62 |
| 0.040-0.050 | 44 | 40 | 40 | 41 | 40 |
from the position of the principal component inside the range given in the
first column for different pixel-scales (column headers refer to
/pixel-scale)
| s [pix] | 100000 | 50000 | 43700 | 33333 | 25000 |
 | 2252 | 1448 | 1310 | 1019 | 740 |
| 0.001-0.003 | 163 | 166 | 149 | 168 | 146 |
| 0.003-0.005 | 66 | 65 | 80 | 64 | 65 |
| 0.005-0.010 | 35 | 34 | 34 | 30 | 41 |
| 0.010-0.015 | 8 | 19 | 18 | 17 | 19 |
/pixel-scale). The parameter s given in the first
column, is the rms of the blend wavelengths calculated for
20 different discretisations in steps of 0.05pix
The stability of the predicted blend wavelength with respect to sub-pixel
location is in the large majority of the selected cases excellent with respect
to the required accuracy (Table 4 (click here)).
The correlation between displacement and discretisation stability
s is shown
in Fig. 4 (click here). Also these data suggest that a selection on the
stability indicator that is stricter than the one used in the calculations
(
) is appropriate. We suggest to use
depending on the pixel-scale.
Figure 4: Line displacement against discretisation stability s
(see also
Tables 3 (click here) and 4 (click here)) for the highest pixel-scale
.
Only lines with 
and 
are shown.
At lower pixel-scales, the data occupy the same area, but with an increasing
density towards small displacements and high stability