F v v

where 206 265 is the number of arcseconds in 1 radian and 1000 is the conversion factor between millimeters and microns.

For a 1-m telescope of // = 7.5, the focal length (f) of the primary would be 7500 mm. If we were to use a Loral CCD with 15-micron pixels as an imager, the above expression would yield an image scale on the CCD of 0.41 arcsec/pixel. This image scale is usually quite a good value for direct imaging applications for which the seeing is near 1 or so arcseconds.

There are times, however, when the above expression for the plate scale of a CCD may not provide an accurate value. This could occur if there are additional optics within the instrument that change the final f-ratio in some unknown manner. Under these conditions, or simply as an exercise to check the above calculation, one can determine the CCD plate scale observationally. Using a few CCD images of close optical double stars with known separations (e.g., the Washington Double Star Catalog - http://ad.usno.navy.mil/wds/), measurement of the center positions of the two stars and application of a bit of plane geometry will allow an accurate determination of the pixel-to-pixel spacing, and hence the CCD plate scale. This same procedure also allows one to measure the rotation of the CCD with respect to the cardinal directions using known binary star position angles.

4.2 Flat fielding

To CCD experts, the term "flat field" can cause shivers to run up and down their spine. For the novice, it is just another term to add to the lexicon of CCD jargon. If you are in the latter category, don't be put off by these statements but you might want to take a minute and enjoy your thought of "How can a flat field be such a big deal?" In principle, obtaining flat field images and flat fielding a CCD image are conceptually easy to understand, but in practice the reality that CCDs are not perfect imaging devices sets in.

The idea of a flat field image is simple. Within the CCD, each pixel has a slightly different gain or QE value when compared with its neighbors. In order to flatten the relative response for each pixel to the incoming radiation, a flat field image is obtained and used to perform this calibration. Ideally, a flat field image would consist of uniform illumination of every pixel by a light source of identical spectral response to that of your object frames. That is, you want the flat field image to be spectrally and spatially flat. Sounds easy, doesn't it? Once a flat field image is obtained, one then simply divides each object frame by it and voilà: instant removal of pixel-to-pixel variations.

Before talking about the details of the flat fielding process and why it is not so easy, let us look at the various methods devised to obtain flat field exposures with a CCD. All of these methods involve a light source that is brighter than any astronomical image one would observe. This light source provides a CCD calibration image of high signal-to-noise ratio. For imaging applications, one very common procedure used to obtain a flat field image is to illuminate the inside of the telescope dome (or a screen mounted on the inside of the dome) with a light source, point the telescope at the bright spot on the dome, and take a number of relatively short exposures so as not to saturate the CCD. Since the pixels within the array have different responses to different colors of light, flat field images need to be obtained through each filter that is to be used for your object observations. As with bias frames discussed in the last chapter, five to ten or more flats exposed in each filter should be obtained and averaged together to form a final or master flat field, which can then be used for calibration of the CCD. Other methods of obtaining a CCD flat field image include taking CCD exposures of the dawn or dusk sky or obtaining spatially offset images of the dark night sky; these can then be median filtered to remove any stars that may be present (Tyson, 1990; Gilliland, 1992; Massey & Jacoby, 1992; Tobin, 1993).

To allow the best possible flat field images to be obtained, many observatories have mounted a flat field screen on the inside of each dome and painted this screen with special paints (Massey & Jacoby, 1992) that help to reflect all incident wavelengths of light as uniformly as possible. In addition, most instrument user manuals distributed by observatories discuss the various methods of obtaining flat field exposures that seem to work best for their CCD systems. Illumination of dome flat field screens has been done by many methods, from a normal 35-mm slide projector, to special "hot filament" quartz lamps, to various combinations of lamps of different color temperature and intensity mounted like headlamps on the front of the telescope itself. Flat fields obtained by observation of an illuminated dome or dome screen are referred to as dome flats, while observations of the twilight or night sky are called sky flats.

A new generation of wide-field imagers and fast focal length telescopes presents some problems for the normal "dome" screen approach to flat fielding. To achieve large-scale, uniform flat fields Zhou et al. (2004) have developed a method by which an isotropic diffuser is placed in front of the telescope and illuminated by reflected light from the dome screen. They claim to obtain flat fields with a measurement of the detector inhomogeneities as good as supersky flats over a 1° field of view. Shi and Wang (2004) discuss flat fielding for a wide field multi-fiber spectroscopic telescope. They use a combination of fiber lamp flat fields and offset sky flats to calibrate the pixel-to-pixel variations.

CCD imaging and photometric applications use dome or sky flats as a means of calibrating out pixel-to-pixel variations. For spectroscopic applications, flat fields are obtained via illumination of the spectrograph slit with a quartz or other high intensity projector lamp housed in an integrating sphere (Wagner, 1992). The output light from the sphere attempts to illuminate the slit, and thus the grating of the spectrograph, in a similar manner to that of the astronomical object of interest. This type of flat field image is called a projector flat. While the main role of a flat field image is to remove pixel-to-pixel variations within the CCD, these calibration images will also compensate for any image vignetting and for time-varying dust accumulation, which may occur on the dewar window and/or filters within the optical path.

Well, so far so good. So what is the big deal about flat field exposures? The problems associated with flat field images and why they are a topic discussed in hushed tones in back rooms may still not be obvious to the reader. There are two major concerns. One is that uniform illumination of every CCD pixel (spatially flat) to one part in a thousand is often needed but in practice is very hard to achieve. Second, QE variations within the CCD pixels are wavelength dependent. This wavelength dependence means that your flat field image should have the exact wavelength distribution over the band-pass of interest (spectrally flat) as that of each and every object frame you wish to calibrate. Quartz lamps and twilight skies are not very similar at all in color temperature (i.e., spectral shape) to that of a dark nighttime sky filled with stars and galaxies.1 Sky flats obtained of the dark nighttime sky would seem to be our savior here, but these types of flat fields require long exposures to get the needed signal-to-noise ratio and multiple exposures with spatial offsets to allow digital filtering (e.g., median) to be applied in order to remove the stars. In addition, the time needed to obtain (nighttime) sky flats is likely not available to the observer who generally receives only a limited stay at the telescope. Thus, whereas very good calibration data lead to very good final results, the fact is that current policies of telescope scheduling often mean that we must somehow compromise the time used for calibration images with that used to collect the astronomical data of interest. Modern telescopes often observe in queue mode or service mode thereby removing the "at the telescope" interaction of the observer whose data are being collected from the data collection process itself. Often the calibration frames desired are not what is obtained.

1 One good sky region for twilight flats has been determined to be an area 13 ° east of zenith just after sunset (Chromey & Hasselbacher, 1996).

Flat fielding satellite CCD imagers (such as those on HST) and space misions (such as Cassini) are often hard to achieve in practice. Laboratory flat fields taken prior to launch are often used as defaults for science observations taken in orbit. Defocused or scanned observations of the bright Earth or Moon are often used for these. Dithered observations of a star field can be used as well in a slightly different way. Multiple observations of the same (assumed constant) stars as they fall on different pixels are used to determine the relative changes in brightness and thus map out low frequency variations in the CCD. An example of such a program is discussed in Mack et al. (2002).

Within the above detailed constraints on a flat field image, it is probably the case that obtaining a perfect, color-corrected flat field is an impossibility. But all is not lost. Many observational projects do not require total perfection of a flat field over all wavelengths or over the entire two-dimensional array. Stellar photometry resulting in differential measurements or on-band/off-band photometry searching for particular emission lines are examples for which one only needs to have good flat field information over small spatial scales on the CCD. However, a project with end results of absolute photometric calibration over large spatial extents (e.g., mapping of the flux distribution within the spiral arms of an extended galaxy) does indeed place stringent limits on flat fielding requirements. For such demanding observational programs, some observers have found that near-perfect flats can be obtained through the use of a combination of dome and sky flats. This procedure combines the better color match and low-spatial frequency information from the dark night sky with the higher signal-to-noise, high spatial frequency information of a dome flat. Experimentation to find the best method of flat fielding for a particular telescope, CCD, and filter combination, as well as for the scientific goals of a specific project, is highly recommended.

A summary of the current best wisdom on flat fields depends on who you talk to and what you are trying to accomplish with your observations. The following advice is one person's view.

What does the term "a good flat field" mean? An answer to that question is: a good flat field allows a measurement to be transformed from its instrumental values into numeric results in a standard system that results in an answer that agrees with other measurements made by other observers. For example, if two observers image the same star, they both observe with a CCD using a V filter, and they each end up with the final result of V = 14.325 magnitudes in the Johnson system then, assuming this is an accurate result, one may take this as an indication of the fact that each observer used correct data reduction and analysis procedures (including their flat fielding) for the observations.

The above is one way to answer the question, but it still relies on the fact that observers need to obtain good flat fields. Without them, near perfect agreement of final results is unlikely. While the ideal flat field would uniformly illuminate the CCD such that every pixel would receive equal amounts of light in each color of interest, this perfect image is generally not produced with dome screens, the twilight sky, or projector lamps within spectrographs. This is because good flat field images are all about color terms. That is, the twilight sky is not the same color as the nighttime sky, neither of which are the same color as a dome flat. If you are observing red objects, you need to worry more about matching the red color in your flats; for blue objects you worry about the blue nature of your flats. Issues to consider include the fact that if the Moon is present, the sky is bluer then when the Moon is absent, dome flats are generally reddish due to their illumination by a quartz lamp of relatively low filament temperature, and so on. Thus, just as in photometric color transformations, the color terms in flat fields are all important. One needs to have a flat field that is good, as described above, plus one that also matches the colors of interest to the observations at hand.

Proper techniques for using flat fields as calibration images will be discussed in Section 4.5. Modern CCDs generally have pixels that are very uniform, especially the new generation of thick, front-side devices. Modern thinning processes result in more even thickness across a CCD reaching tolerances of 1-2 microns in some cases. Thus, at some level flat fielding appears to be less critical today but the advances resulting in lower overall noise performance provide a circular argument placing more emphasis on high quality flats. Appendix A offers further reading on this subject and the material presented in Djorgovski (1984), Gudehus (1990), Tyson (1990), and Sterken (1995) is of particular interest concerning flat fielding techniques.

4.3 Calculation of read noise and gain

We have talked about bias frames and flat field images in the text above and now wish to discuss the way in which these two types of calibration data may be used to determine the read noise and gain for a CCD.

Noted above, when we discussed bias frames, was the fact that a histogram of such an image (see Figure 3.8) should produce a Gaussian distribution with a width related to the read noise and the gain of the detector. Furthermore, a similar relation exists for the histogram of a typical flat field image (see Figure 4.1). The mean level in the flat field shown in Figure 4.1

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