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Figure 19 shows a typical bias- and dark-corrected lamp-flat. The dispersion direction is roughly parallel to the x-axis. Note that the SBIG has a negative dispersion relation. That is, the wavelength decreases with increasing pixel number. There is a huge dynamic range between the well-exposed region of the chip (where the green light falls) and the red and blue ends. There is also a great deal of vertical structure caused by the non-uniform slit illumination from the LED. We need to process such frames to extract information on the pixel-to-pixel sensitivity variations. As a first step, I define a "sweet-spot" to use for statistical studies. That "sweet-spot" is the well-exposed region of the detector: a 380X350 pixel region enclosing columns 371-750 and rows 141-490. I then determined the average count-rate in that region for each lamp-flat exposure. Figure 20 shows plots of two sequences of lamp-flats, both obtained at the same detector temperature (T ~ -25 degrees C), but one obtained at an ambient temperature of about -5 degrees C, and the other at an ambient temperature of about -15 degrees C.
I include this figure as a caution against the assumption that the LED can be used as any sort of an absolute flux calibrator. The LED is powered by a 9-V battery. The drastic fall-off seen in the right panel of Fig. 20 is probably due to the inability of the battery to maintain constant output when it is too cold. The low-temperature limit for standard 9-V performance is about 0 degrees F (J. Pierce, private communication). Note that, although the LED output drops substantially when it operates for more than a few minutes at low ambient temperature, this does not imply any change in the detector behavior. Thus one can still use the full series of lamp-flats shown in the right panel of Fig. 20, if one properly accounts for the variation in the signal level over the course of the series.
The full field of the detector is not exposed to the sky when using the spectrograph. This is what leads to some of the vertical structure seen in Fig. 19. The lower part of the chip provides no useful information. The signal there is due to scattered light. Thus it is best to crop the frames at this stage of the reduction. The instrument manual reports a useful field of 765 pixels in the spatial direction. Thus the frames discussed in the remainder of Section 4.1 have been cropped accordingly. As will be discussed in Section 4.2, the actual region exposed to the sky is smaller than this, but for the moment, the usable field is taken to be rows 1-765, columns 1-1530. Figure 21 shows the cropped version of the image presented in Fig. 19.
The next step is to combine the flats from each sequence to improve their individual signal-to-noise ratio. To account for the problem show in Fig. 20, one must normalize the individual frames by their mean values. This results in a set of median-combined flats, with average pixel values of 1. Figure 22 shows an example of such a combined flat. The image show a great deal of large-scale structure, all of which has to do with the systematics of the scattering and dispersion of the light from the internal LED. Thus this large-scale structure is a useless annoyance. The utility of lamp-flat frames is in their (hopefully) high signal-to-noise information on the pixel-to-pixel sensitivity variations. In order to extract that information from an image such as is shown in Fig. 22, it is necessary to filter out all the large-scale structure. In signal-processing terms, we need to pass the image through a high-pass filter. Such a filter can be constructed from the lamp-flats by applying a spatial median filter to them.
To apply a spatial median filter, one must pick a spatial scale to sample on (a "kernel" in signal-processing terms). Some experimentation led me to adopt an 11 by 11 pixel kernel for the spectrograph data. This value should be regarded as provisional until the filtered flats are applied to actual astronomical data.
The median-filtered image was created using a task in the IRAF ("Imaging Reduction and Analysis Facility") software package. IRAF is the standard astronomical image processing software. Documentation on this package is available on-line. At each pixel in the input image, the median value of the pixels within the kernel (an 11 by 11 pixel square, in this case) is determined, and written to that same pixel location in the output image. Thus the output image will contain information on spatial structures larger than the kernel size only. One then takes this output image, and divides the initial image by it. Thus, the final image has the large spatial-scale structure removed, and retains the small spatial scale structure. This is exactly what is required to correct for pixel-to-pixel sensitivity variations. Figure 23 shows all three steps in this process. The top panel is a stacked lamp-flat. The middle panel shows the result of applying the spatial median filter to that image. The lower panel shows the final result of dividing the original image by the filtered image.
It is common to obtain lamp-flats at either the beginning or the end of an observing night. In fact, at many observatories, one obtains lamp-flats during the day. The lack of reliable detector temperature control with our system means that we cannot assume this will work at Andreas Observatory. In order to investigate this, we must see if there is any significant temperature dependence in the pixel-to-pixel sensitivity images produced as outlined above. There are two different sorts of tests we need to make. The first is a comparison of median-filtered lamp-flats taken at different temperatures on the same night. The second is a comparison of median-filtered lamp-flats taken at the same temperature on different nights. A final test is to compare the median-filtered lamp-flats obtained at similar temperatures, but through the narrow and wide slits. Figure 24 shows an example of the first such test.
The key conclusion to be drawn from Fig. 24 is that there is not a strong temperature variation in the pixel-to-pixel sensitivity of the detector. The noise varies as a function of exposure, but there are no obvious systematics in the bottom panel except for some narrow features along the dispersion direction. Future testing will be required to determine how important these features are. For the moment, however, we can conclude that variations in detector temperature of ~10 degrees C between the lamp-flats and actual object spectra will not adversely affect our ability to properly flat-field our spectra.
Figure 25 shows a ratio of two median-filtered lamp-flats taken at the same detector temperature (T ~ -15 degrees C), but on different nights. While there is general agreement between the two, a number of the systematic patterns on the lamp-flats differ enough that there is obvious structure on the ratio map. This is an indication that it is not safe to assume that one can use lamp-flats from a previous night. Always obtain a set of lamp-flats on the same night as any data you plan to actually reduce.
Finally, for the sake of completeness, Figure 26 shows a ratio of two median-filtered lamp-flats taken at the same detector temperature (T ~ -20 degrees C), but through different slit-plates. The range of systematic differences between the two lamp-flats is even larger than for the previous experiment. Therefore, under no circumstances should an observer use lamp-flats obtained with a different slit than used for the science observations.
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