Pixel sensitivity; flat fielding

Due to imperfections in the manufacturing process the sensitivity of the pixels will vary slightly (usually by a few percent) across the grid. This effect is essentially random, and is not a function of, for example, position on the grid. The relative sensitivities of the pixels can be calibrated by imaging an evenly illuminated source, such as the twilight sky, and examining the variation in values recorded. Once this calibration is known, astronomical images of the sky can be corrected to the values they would have had if all the pixels had been uniformly sensitive. This correction is known as flat fielding and images of evenly illuminated sources, such as the twilight sky, are known as flat fields. The pixel-to-pixel sensitivity variations change with wavelength, so the flat fields should always be acquired using the same filter as the observations of the target objects. The flat fielding procedure also corrects for several other effects:

• small sharp dark features with the same percentage absorption on all flat fields. These come from dust particles on the CCD chip,

• vague ring or torus-shaped features. These come from dust on the filters, which are out of focus as seen from the chip. They are the same on all exposures with the same filter, but obviously differ from filter to filter, and can differ from time to time,

• vignetting, the dimming of objects observed towards the edge of the telescope field of view. Vignetting is caused by various out-of-focus obstructions in the light path, such as the support for the secondary mirror.

Two types of flat fields are usually used: dome flats and sky flats. Brief details are as follows.

Dome flats

are images of the inside of the telescope dome, illuminated by a bright continuum source free of emission lines. The interior surface of the dome is usually a smooth, diffuse reflector and is completely out of focus for the telescope optics. Consequently the image recorded is completely featureless. Dome flats are convenient because they can be taken in unlimited numbers during the day, rather than at night or during twilight when time is short. However, they have two disadvantages:

• light reflected from the dome is incident on the telescope at a slightly different angle to light from the sky. This difference does not affect the pixel-to-pixel sensitivity variations but can affect the vignetting and the shape of the images caused by dust particles,

• the colour (that is the wavelength distribution) of the lamp is not the same as that of the night sky. This effect is more important for observations made through a broad band filter than a narrow band one and can lead to fringing (see below).

Sky flats

are images of the sky taken during twilight when it is relatively bright. The sky should be much brighter than any stars which happen to be in the field of view, but not bright enough to saturate the chip. The optimum time to acquire the flat field depends on the filter. A narrow filter, a filter corresponding to a wavelength for which the chip is insensitive, or to a wavelength range where the Sun emits little light (such as the $U$ band), can be taken nearer to sunrise or sunset than a broadband filter at the peak of the chip's sensitivity. In an optimally exposed flat field the photon noise (see below) is negligible but the image is not saturated. However, it can sometimes be difficult to judge the exposure time correctly, particularly for frames acquired close to sunrise or sunset. Also, in such frames the interior of the dome is illuminated by sunlight and this light reaches the chip by internal reflections in the telescope. Thus sky flats show some of the vignetting and dust effects seen in dome flats. De-focussing the telescope to make any star images present less prominent is usually not viable because it may change the vignetting function.

An alternative to taking flat fields during twilight is to take then during the night. This approach is particularly common for infrared observations because at these wavelengths the sky is relatively bright.

It is possible to combine different sorts of flat fields to obtain the advantages of each. For example, you could use dome flats to correct pixel-to-pixel sensitivity variations and twilight flats to correct large-scale effects such as vignetting.

In outline, you use the flat fields to correct the target exposures as follows. Choose several correctly exposed flat fields, de-bias them and combine them into a single `master' flat field. The de-biassed images of the target objects are simply divided by this master flat field. You should always calibrate target images using flat fields obtained through the same filter (that is, in the same colour) and on the same night. Flat fields acquired with a 16-bit camera should ideally have a mean pixel count which averages around 20,000 in order to allow high accuracy to be obtained.