A Generalized Point Spread Function

 

The first order of business toward discerning extended sources (galaxies) from point sources (stars) is to accurately characterize the point spread function (PSF). The unique shape of the PSF derives from a combination of factors: the optics, large 2" pixels (frame images), dithering pattern of the six samples that comprise the coadd, focus, sampling/convolution algorithm to generate the coadds, and atmospheric seeing (although seeing generally only changes the size of the PSF, not the shape). As such, the 2MASS PSF corresponding to frame-coadded images is not well fit with a gaussian function. Fortunately it is sufficiently characterized with a generalized exponential function (see below) out to a radius ~2´ FWHM, which is all that is required for star-galaxy discrimination.

The 2MASS PSF typically varies on time scales of ~minutes due to two effects: atmospheric "seeing" and variable telescope focus (thermal driven). The 2MASS telescopes are designed to be mostly free of afocal PSFs (under most conditions), but experience has shown that 2MASS images can be slightly out of focus during periods of rapid change in the air temperature – conditions that generally only occur during the hottest summer months. Out of focus images have the ill-desired property of possessing asymmetric PSFs (i.e., elongation). Fortunately, under most/typical observing conditions for the survey, the PSFs are symmetrically round (throughout the focal plane) and thus are in focus. That leaves the atmospheric seeing as the primary dynamic to the radial size of the PSF. Given the long exposure times per sample (1.3 sec) and the six sample coaddition (with optimal dithering to produce round PSFs), seeing changes result in a symmetric ‘puffing’ in/out of the resultant coadd PSF. We can represent the image PSF with the generalized radially symmetric exponential of the form:

Where f0 is the central surface brightness, r is the radius in arcsec, and a and b are free parameters. This versatile function not only describes the 2MASS PSF, but it also used to characterize the radial profiles of galaxies, from disk-dominated spirals (b close to unity) to ellipsoidal galaxies (b ~ 4, de Vaucouleurs law). The scale-length, a, and the modifier, b, can be combined, a ´ b, to form an attribute fit parameter. This new parameter is referred to as the "shape parameter". The "shape parameter," a ´ b, itself is a powerful discriminate: galaxies tend to have larger values of both a & b than compared to stars; thus, the multiplicative join of the exponential fitting parameters amplifies the difference between point sources and extended sources.

 

Our ability to track the seeing on short time scales depends on the density of stars. The more stars available to measure a statistically meaningful measurement of the "shape" the higher the frequency of seeing changes that can be tracked. The stars in question must be isolated sources free of contamination from other stars and fainter background stars. A reasonable shape value can be derived from a minimum of about 10 stars. Consequently, for low stellar density regions, say the north galactic pole, ~300 stars per deg2 brighter than 14th mag at K, the seeing is tracked on time scales of about 20 seconds, and for high density regions, >104 stars per deg2, is tracked on time scales of a few seconds of time. Experience has shown that the seeing can indeed significantly change on times scales as fast as a seconds of time (see below).

 

The mean "shape" is determined from an ensemble of isolated stars spatially clustered along the 2MASS in-scan direction. The sample population must be free of extended sources (galaxies) and double stars to be a meaningful measure of the PSF. We perform a robust separation of isolated stars from the larger population (of the spatially correlated sample) by employing an iterative selection method that is keyed by using an initial boot-strap from the lower quartile of the total population histogram. Since isolated stars will have an inherently smaller "shape" value than extended sources (or double stars), the lower quartile (25%) is dominated by isolated stars and conversely, the upper quartile by galaxies. Thus, the lower quartile serves as a good guess to the actual mean shape value. To zero in on the desired shape, we iteractively search the histogram now using hard limits set by the lower quartile: -3s to +2s of the quartile, where s is the scatter in the "shape" value (first iteration we use an apriori determined s). For each iteration thereafter, we set hard limits of +-2s. The final "shape" value corresponds to the median (50% quartile) of the isolated sample, and the s to the rms scatter or standard deviation of the population.

 

In this way we build a "stellar ridgeline" of shape values as a function of scan position (or simply, scan coordinate). Two very different examples are illustrated in Figures 1 and 2. In Figure 1 we show the resultant ridgeline for a scan passing through the Hercules cluster of galaxies. The plots show the median "shape" values (large triangles) along the scan. Extracted sources (including stars and galaxies) are denoted with small points. The corresponding FWHM of the PSF (fit with a gaussian) are also shown (in red) to give some idea of the scale in arcseconds. The stellar number density is not large (galactic latitude of Hercules is about 30° ), but there are still plenty of isolated stars easily separated from extended sources (galaxies are located above the mean "shape" ridge). The seeing is fairly stable for each band all throughout the 6° scan (Dtime ~ 6 minutes). The same cannot be said for the second case, Figure 2, which demonstrates both poor seeing conditions and very rapid changes in the seeing. Fortunately, the stellar density is rather high in this field, 4000 stars per deg2, and the rapid seeing diversions are, for the most part, sufficiently tracked. Scans for which the seeing is poorly tracked or the absolute value of the seeing is greater than 1.3" (PSF FWHM > 4") are considered low quality data and are generally rescheduled for re-observation. The stellar ridgelines are used in the extended source processing to separate point sources from ‘resolved’ sources.