How to measure the limiting magnitudes

Limiting Magnitudes

The limiting magnitudes of the images are measured in three ways: Number count histogram 5-sigma point source detection Adding fake objects The first method is quite simple, indeed crude. The magnitudes of the objects are sorted into a histogram. The peak value of the histogram, where the number counts start to turn over, is a rough measure of the limiting magnitude of the image. The second method is also simple. The estimated magnitude error of each source is plotted against its magnitude. In this case, the SExtractor MAG_AUTOor Kron-style magnitude is plotted. At the faint magnitudes typical of MegaCam images, the sky noise dominates the magnitude error. This means that extended objects (which have more sky in their larger Kron apertures) will be noisier for a given magnitude than compact sources. Turning this around, this means that, for a given fixed magnitude error, a point source will be fainter than an extended source. A 5-sigma detection corresponds to a S/N of 5 or, equivalently, a magnitude error of 0.198 magnitudes. Thus, to find the 5-sigma point source detection limit, one finds the faintest source whose magnitude error is 0.198 magnitudes or less. It must be a point source, therefore, its magnitude is the 5-sigma point source detection limit. A more refined approach would be to isolate the point sources, by using the half-light radius for example. In practice, the quick and dirty method gives answers that are correct to within 0.3 magnitudes or so, which is accurate enough for many purposes. This is the value that is given in the image headers for the MAGLIM keyword. This is also the value that is used in theimage search page.

The figure at the right illustrates these methods. The top panel shows the number count histogram. The number counts peak at 26 in magnitude as shown by the vertical redline. The bottom panel shows magnitude error plotted against magnitude. The horizontal red line lies at 0.198 magnitudes. The verticalred line intersects the horizontal line at the locus of the faintest object with a magnitude error less than 0.198 magnitudes. The magnitude limit by this method is 26.6 magnitudes. Similar plots to the figure at right are provided for each image in the MegaPipe stacks on their respective webpages.

The final way the limiting magnitudes of the images are tested is by adding fake galaxies to the images and then trying to recover them using the same parameters used to generate the real image catalogues. The fake galaxies used were taken from the images themselves, rather than adding completely artificial galaxies. A set of 40 bright, isolated galaxies are selected out of the field and assembled into a master list. Postage stamps of these galaxies are cut out of the field. The galaxies are faded in both surface brightness and magnitude through a combination of scaling the pixel values and resampling the images.

To test the recovery rate at a given magnitude and surface brightness, galaxy postage stamps are selected from the master list, faded as described above to the magnitude and surface brightness in question, and then added to the image at random locations. SExtractor is then run on new image. The fraction of fake galaxies found gives the recovery rate at that magnitude and surface brightness, An illustration of adding the galaxies is shown at the right. The same galaxy has been added multiple times to the image. The galaxy has been faded to various magnitudes and surface brightnesses. The red boxes contain the galaxy. The boxes are labeled by mag/surface brightness. Note the galaxy ati=23, μi=25 accidentally ended up near a bright galaxy and is only partially visible. Normally of course, the galaxies are not placed in such a regular grid.

To test the false-positive rate, the original image is multiplied by -1; the noise peaks became noise troughs and vice-versa. SExtractor is run, using the same detection criteria. Since there are no real negative galaxies, all the objects thus detected are spurious.

The magnitude/surface brightness plot at the right shows the results of such simulations. The black points are real objects. The bottom edge of the black points is the locus of point-like objects. Thegreen points show the false-positive detections. The rednumbers show the percent of artificial galaxies that were recovered at that magnitude/surface brightness. The blue contour lines show the 70% and 50% completeness levels.

Deriving a single limiting magnitude from such a plot is somewhat difficult. The cleaner cut in the false positives seems to be in surface brightness. Extended objects become harder to detect at brighter magnitudes whereas stellar objects are detectable a magnitude or so fainter. Note that this plot is of limited usefulness in crowded fields. In this case, an object may be missed even if it is relatively bright because it lies on top of another object. However, the objects in crowded fields are almost always stellar. This suggests the use of the DAOphot package rather than using the SExtractor catalogs provided as part of MegaPipe. Similar graphs to the above plot are provided for each group on their respective pages. From: http://www4.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/megapipe/docs/photometry.html