### 10 Measuring Instrumental Magnitudes

The first step to producing a set of calibrated magnitudes for your list of programme objects is to measure the instrumental magnitudes of the programme objects and standard stars recorded in your CCD frames. Instrumental magnitudes are usually measured relative to the sky background in each CCD frame. Prior to measuring the instrumental magnitudes the various instrumental effects should be removed from the CCD frames. Typically you will need to de-bias and flat-field the frames and remove the effects of bad pixels (possibly including whole bad rows and columns) and cosmic-ray events. The CCDPACK package (see SUN/139[19]) is available for this task. The cookbook SC/5: The 2-D CCD Data Reduction Cookbook[18] gives a set of recipes for removing instrumental effects and is a good introduction to the topic. The support staff at the observatory where your observations were made should be able to advise about any peculiarities associated with the CCD detector that you were using.

Once the CCD frames have been corrected for instrumental effects they are ready to be used to measure instrumental magnitudes. There are two classes of techniques for measuring instrumental magnitudes in widespread use: aperture photometry and point-spread-function fitting. There are numerous variations on each technique, but the essentials of the two methods are as follows.

Aperture photometry
The principle of aperture photometry is simple. For the star which is to be measured a circular region of the CCD frame (or ‘aperture’) is defined which entirely encloses the image of the star (that is, all the light from the star falls inside the aperture)7. The flux in all the pixels inside the aperture is added to give the total flux. A similar measurement is made of a region containing no stars to give the flux from the background sky. The two are then subtracted to yield the flux from the star. The same principle may be applied to extended objects such as galaxies or nebulæ, though here an elliptical aperture may be used. Packages for performing aperture photometry include PHOTOM (see SUN/45[22]) and GAIA (see SUN/214[20]).
This technique is used to measure images in crowded star fields, such as the central regions of a globular cluster. In such regions the images of individual stars overlap and it is impossible to position an aperture so that it simultaneously includes all the light from a given star and excludes all the light from its neighbours. Stars are, of course, unresolved by a conventional telescope and the star images recorded in a CCD frame simply trace out the point-spread function of the telescope. For a properly designed telescope the point-spread function will be independent of the position of the star in the focal plane, at least for positions close to the optical axis. The point-spread-function fitting technique makes the underlying assumption that all the star images have the same shape. Since CCD detectors are usually positioned on the optical axis and have a small field of view this assumption is usually valid. The light distribution in the CCD frame is modelled by assuming positions and brightnesses for the observed stars and knowing the point-spread function (it can be measured using isolated stars). The positions and brightnesses of the stars are iteratively varied until the observed light distribution in the CCD frame is reproduced. The actual mathematical details are not important here (and vary between different packages).

It is possible to perform accurate photometry of crowded regions using point-spread-function fitting. However, it is clearly important that all the images should have the same profile. Thus, the presence of extended objects with their own unique profiles will invalidate the technique.

The DAOPHOT (see SUN/42[23] and Stetson[68]) and STARMAN (see SUN/141[62]) packages are available for point-spread-function fitting. However, their use is beyond the scope of this cookbook and they are not considered further here.

In the case of aperture photometry the instrumental magnitude, ${m}_{inst}$, is defined as:

 ${m}_{inst}=A-2.5{log}_{10}\left[\frac{\left(\sum _{i=1}^{n}{C}_{i}\right)-n{C}_{sky}}{t}\right]$ (14)

where:

$A$
is an arbitrary constant which is often added to the instrumental magnitudes,
${C}_{i}$
is the count in the $i$th pixel inside the aperture,
${C}_{sky}$
is the average count in a background sky pixel,
$n$
is the number of pixels in the aperture,
$t$
is the integration time of the frame.

That is, the instrumental magnitude is computed from the sum of the pixels inside the aperture with the average sky background subtracted. For simplicity the number of pixels inside the aperture, $n$, has been presented as an integer number here. However, if necessary pixels which partly overlap the aperture can be properly accounted for. There are various ways of measuring the average8 sky background value. One is to use an aperture positioned on a neighbouring region of blank sky, another is to use an annulus surrounding the original aperture; this latter technique is perhaps preferable. Though the details of the point-spread-function fitting technique are very different the definition of the instrumental magnitude is the same.

It is often a good idea to set the arbitrary constant, $A$, to a silly value (say 30) so that the instrumental magnitudes have very different values from the corresponding calibrated magnitudes and hence the two are unlikely to be inadvertently confused.

You will need to determine instrumental magnitudes for all your programme objects and standard stars and in all the colours in which you made observations. Section 14 gives a recipe for measuring instrumental magnitudes using PHOTOM and Section 15 one using GAIA.

7The name of the technique comes from an earlier generation of astronomical instrumentation when photometry was carried out with a single-element photoelectric photometer. A circular aperture was placed in front of the photometer to limit its view of the sky. Using software to define a circular region in a CCD frame mimics the effect of a physical aperture limiting the field of view of a single-element photometer.

8Though it is simplest to think of determining the average sky background level, in practice a straightforward mean is often not a good measure of the sky background, because of contamination of the chosen region by faint stars. Instead techniques are often used which minimise the effects of outlying values in the sky background histogram, such as computing the median instead of the mean.