# 3. Rectification

The degree of misalignment of the spectral and spatial axes from the detector axes is a substantial problem for good wavelength calibration. It also makes difficult any determination of the dispersion and resolution of the spectrograph. However, one can correct for this misalignment using lamp exposures. Such a correction is called {\it rectification}. I give here an example of rectification of one of the lamps obtained on 15 July 2008. The spectrum in question is the low-dispersion Hg spectrum 025.

The slope of the 5480 Angstrom line in this spectrum is m = -0.126. If one takes row 1 (y = 1) as the starting point, one can compute the required shift for any row as follows:

One then applies the resulting dx shifts to the spectrum to produce a rectified spectrum. A potential problem with this is that the resulting dx values are generally not integers. Application of non-integer shifts to images increases the noise in the image. (In the language of information mechanics, such shifts increase the informational entropy of the data). I thus produced two rectified spectra. For the first, I rounded all dx values to the nearest integer. For the second, I allowed non-integer shifts. In Figure 39, I show a portion of spectrum 025 around the 5769,5790 Angstrom doublet. On the left are the original data. In the middle are the integer-shifted data. On the right are the real-shifted data.

• Figure 39: Unrectified (left), integer rectified (middle) and real rectified (right) portion of Hg spectrum 025 from 15 July 2008 in the region of the 5769,5790 Angstrom doublet.

The real-shifted case clearly looks better than the integer-shifted case. But, as noted above, the integer-shifted case will provide more reliable spectrophotometry. As there can be situations in which spectral resolution matter more than photometric accuracy, it is worth studying the issue of spectral resolution with a bit more care. For each of the three cases shown above, I extracted a set of 1-D spectra from the 2-D spectral images. The 1-D spectra were generated by summing a number, n, of rows, where n ranged from 1 up to 250 (the entire spectral image). I then plotted the resulting 1-D spectra for various choices of n. As the data were obtained without the focal reducing lens, the spatial scale along the slit is 0.25" per pixel. Assuming seeing of ~2", the most realistic case in my test extractions is that with a 9-pixel summation. In Figure 40, I show the region of the spectrum around the 5769,5790 Angstrom doublet in the unrectified, integer-rectified and real-rectified cases. I note that there is no substantial disagreement in the resolution of the spectra.

• Figure 40: Unrectified (solid), integer rectified (dotted) and real rectified (dashed) 1-D spectra of the 5769,5790 Angstrom doublet with a 9-pixel summation (equivalent to 2.25" along the slit).

One might conclude from Fig. 40 that rectification is unneeded, as the unrectified spectrum does not appear to have significantly degraded resolution compared to the rectified spectra. Figure 41 shows a similar plot for an extraction aperture of 28 pixels, or about 7". This is an entirely reasonable aperture size for spectra of extended objects. In this example the unrectified spectrum clearly has substantially worse resolution than the rectified spectra.

• Figure 41: Unrectified (solid), integer rectified (dotted) and real rectified (dashed) 1-D spectra of the 5769,5790 Angstrom doublet with a 28-pixel summation (equivalent to 7" along the slit).

Analysis of lamp-spectra obtained through the wide slit (at both low and high dispersion) shows that such data also require rectification for extraction apertures larger than ~10 pixels.

Updated: 2010 July 19 [pbe]