Apparent magnitude is the measure of the observed flux of a celestial object as seen from the Earth. The ancient Greek astronomers divided the stars visible to the naked eye into six magnitude groups. The first magnitude stars were the brightest, whilst the faintest stars visible to the naked eye were of 6th magnitude. When, in the 19th century, a quantitative system was developed it was made to agree closely with the old system. This led to the defining equationmag = constant − 2.5 log10 (flux) or flux = 10−0.4 ( mag − constant )
where mag is the apparent magnitude, flux is the observed flux, i.e. photons per second, and the constant defines the zero-point of the magnitude scale.
In practice the apparent magnitude of a object, X is measured by reference to a standard star of known apparent magnitudemagX = magSTD − 2.5 log10 ( fluxX / fluxSTD )
The magnitude of the standard star thus defines the magnitude of the star whose brightness we have measured; in other words, photometry is relative. Once you grasp this crucial point, photometry makes sense.
There are magnitude systems for different wavebands, i.e. the flux is measured in a restricted range of wavelengths, e.g. the Visual band (V-band) wavelength centre = 550, width = 100 nm, the Blue band (B-band) wavelength centre = 450 nm, width = 100 nm.
Chapter 8 (Photometry) of "The Handbook of Astronomical Image Processing" by Berry and Burnell is an excellent introduction.
Measuring Photometry from Images
A star is a point source, but due to the atmosphere, the light on a CCD is spread over a number of pixels, and extends to a considerably greater distance that it first appears. To extract the brightness of a star from the image, we must first sum the star's contribution over all pixels illuminated by it, and somehow estimate and subtract the contribution from the sky background.
The standard technique for this is called aperture photometry. To collect 99% of a stars flux, it is usually necessary to choose an aperture four to five times the size of the star image. The size of a star image is defined as a radius corresponding to the Half Width Half Maximum (HWHM) (alternatively, the diameter is the Full Width Half Maximum; FWHM).
Measuring the starlight requires you to make an aperture large enough to include all of the light from the star, but keep the aperture small enough that uncertainties in measuring the sky background are relatively small. The first criterion requires making the aperture as large as possible, the second criterion required making the aperture as small as possible. Thus, the first step in your photometry calculation is to determine the optimal aperture radius.
The tool for this purpose is to construct a radial profile of the star image and look at the curve of growth. This is simply a plot of the number of counts in the star image versus distance from the center of the star image. It is then easy to measure the FWHM from the profile, and to see where the star light merges into the background. The FWHM is usually between 1.5 and 2.5 pixels, and the star blends into the background between 5 and 10 pixels in radius.
The sky background is measured from a region near the star. The simplest way to do this is to place an annulus immediately surrounding the star image. The annulus usually starts immediately outside of the star aperture and ends twice the radius of the aperturee.
The first calculation is now to determine the sky background level (from the annulus). This average sky value is subtracted from all pixels inside the star aperture. The total flux in the star is then simply the sum of the pixel values in the star aperture. The annulus, however, covers enough of the image that it is likely to be contaminated by one or more background stars. This will raise the average pixel value in the annulus, thereby biasing the measured brightness of the star.
A more robust solution is to sort the pixels in the sky annulus, and exclude the brightest percentage of the pixels. Averaging the remaining pixels will ensure that the sky background is "star free".
Now that you understand the basics of aperture photometry, it is now necessary to actually derive the magnitudes for some of the objects in your data. Fortunately, there exists an image display and analysis tool for this purpose called GAIA. GAIA provides the usual facilities of image display tools, plus more astronomically useful ones such as aperture & optimal photometry, contouring, source detection, surface photometry, arbitrary region analysis, celestial coordinate readout, calibration and modification, grid overlays, blink comparison, defect patching and the ability to query on-line catalogues.
Once you have determined the optimal aperture for the telescope, CCD camera and night, use the same aperture for all measurements. The software package takes care of summing pixels inside the aperture:
Measuring Photometry from lots of images
Once you have mastered manual aperture photometry from a single image, the next step is to measure the variability from a series of images. Clearly, this could be achieved by loading each image into GAIA and creating the appropriate aperture. However, this is time consuming, and completely impractical, especially with the extremely long image sequences possible on some nights. Instead, we have developed a set of scripts which process images in a fully automatic manner.