Aperture photometry

Aperture photometry is a technique that makes no assumption about the actual shape of the source PSF but simply collects and sums up the observed counts within a specified aperture centered on the source. The aperture used may be circular (usually the case for point sources), square, or any shape deemed useful. Aperture photometry is a simple technique, both computationally and conceptually, but this same simplicity may lead to errors if applied in an unsuitable manner or when profile fitting is more appropriate (e.g., severe blending).

The basic application of aperture photometry starts with an estimate of the center of the PSF and then inscribes a circular software aperture of radius r about that center. The radius r may simply be taken as three times the full-width at half-maximum (r = 3-FWHM): the radius of a PSF that would contain 100% of the flux from the object (Figure 5.6) (Merline & Howell, 1995). Summing the counts collected by the CCD for all the pixels within the area A = ^r2, and removing the estimated background sky contribution within A, one finally arrives at an estimated value for /. We see again that partial pixels (a circular software aperture placed on a rectangular grid) are an issue, even for this simple technique. Using a square or rectangular aperture alleviates the need for involving partial pixels but may not provide the best estimate of the source flux. Noncircular apertures do not provide a good match to the 2-D areal footprint of a point source, thereby increasing the value of npix that must be used and decreasing the overall S/N of the measurement. Remember, however, that for bright sources, npix is essentially of no concern (see Section 4.4).

It has been shown (Howell, 1989; Howell, 1992) that there is a well-behaved relation between the radius of the aperture of extraction of a point source and the resultant S/N obtained for such a measurement. An optimum

o —■—1—■—'—1—'—■—1—■—1—■—1—1—1—■—

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 RADIUS (units of FWHM)

o —■—1—■—'—1—'—■—1—■—1—■—1—1—1—■—

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 RADIUS (units of FWHM)

Fig. 5.6. For any reasonable PSF approximation, the figure above shows the run of the total encircled signal with radius of the PSF in FWHM units. Note that within a radius of 3-FWHM essentially 100% of the signal is included.

radius aperture, that is, one that provides the optimum or best S/N for the measurement, can be determined for any given PSF and generally has a radius of near 1 ■ FWHM. This optimum radius is a weak function of the source brightness, becoming smaller in size for fainter sources. Figure 5.7 illustrates this idea for three point sources of different brightness.

Fig. 5.7. The S/N obtained for the measurement of a point source is not constant as a function of radius. There is an optimum radius at which the S/N will be a maximum. The top panel shows this effect for three point sources that differ in brightness by 0.3 (middle curve) and 2.0 (bottom curve) magnitudes compared with the top curve (filled squares). The bottom panel presents the same three stars as a function of their photometric precision. The image scale is 0.4 arcsec/pixel and the seeing (FWHM) was near 1.2 arcsec. From Howell (1989).

Aperture Radius (pixels)

Fig. 5.7. The S/N obtained for the measurement of a point source is not constant as a function of radius. There is an optimum radius at which the S/N will be a maximum. The top panel shows this effect for three point sources that differ in brightness by 0.3 (middle curve) and 2.0 (bottom curve) magnitudes compared with the top curve (filled squares). The bottom panel presents the same three stars as a function of their photometric precision. The image scale is 0.4 arcsec/pixel and the seeing (FWHM) was near 1.2 arcsec. From Howell (1989).

To understand the idea of an optimum radius and why such a radius should exist, one simply has to examine the S/N equation given in Section 4.4 in some detail. To obtain a higher S/N for a given measurement, more signal needs to be collected. To collect more signal, one can use a larger aperture radius, up to the maximum of 3 ■ FWHM. However, the larger r is, the more pixels that get included within the source aperture and the larger the value of npix. As npix increases, so does the contribution to the error term from noise sources other than the source itself. Thus, a balance between inclusion of more signal (larger r) and minimizing npix in the source aperture (smaller r) leads to an optimum extraction radius for a given source.

We saw in Figure 5.7 that if extracted at or very near an aperture radius of 3 ■ FWHM, 100% of the light from a point source would be collected. However, to obtain the maximum S/N from your measurement, extraction at a smaller radius is warranted. If one extracts the source signal using an aperture that is smaller then the actual PSF radius itself, some of the source light that was collected by the CCD is not included in the summing process and is thus lost. This sounds like an incorrect methodology to use, but remember that inclusion of many pixels lying far from the source center also means inclusion of additional noise contributions to the aperture sum. Therefore, while one may wish to obtain the maximum S/N possible through the use of a smaller aperture (i.e., summation of less than the total collected source counts), for the final result it is often necessary to correct the answer obtained for this shortcoming.

In order to recover the "missing light," one can make use of the process of aperture corrections or growth curves as detailed by Howell (1989) and Stetson (1992). Growth curves do not make any demands on the underlying PSF except through the assumption that any bright stars used to define the aperture corrections are exact PSF replicas of any other (fainter) stars that are to be corrected. As we can see in Figure 5.8, growth curves for the brightest stars follow the same general shape, leading to minor or no necessary aperture corrections at a radius of 3 ■ FHWM. The fainter stars, however, begin to deviate from the canonical growth curve at small radii, resulting in 0.5 up to 1.5 magnitudes of needed aperture correction. In general, as a point source becomes fainter, the wings of the source will contain pixels that have an increasingly larger error contribution from the background, leading to greater deviations from a master growth curve at large r, and thus a larger aperture correction will be needed.

As we will see below, if differential photometric results are desired, the aperture corrections used to bring the extracted source signal back to 100% are not necessary. This is only strictly true if all point sources of interest

Fig. 5.8. Growth curves for five stars on a single CCD frame. The three brightest stars follow the same curve, which is very similar to the theoretical expectation as shown in Figure 5.6. The two faint stars start out in a similar manner, but eventually the background level is sufficient to overtake their PSF in the wings and they deviate strongly from the other three. Corrections, based on the bright stars, can be applied to these curves to obtain good estimates of their true brightnesses. The top panel presents growth curves as a function of normalized aperture sums while the bottom panel shows the curves as a function of magnitude differences within each successive aperture. The relative magnitudes of the point sources are given in the top panel and the image scale is the same as in Figure 5.6. From Howell (1989).

Aperture Radius (pixels)

Fig. 5.8. Growth curves for five stars on a single CCD frame. The three brightest stars follow the same curve, which is very similar to the theoretical expectation as shown in Figure 5.6. The two faint stars start out in a similar manner, but eventually the background level is sufficient to overtake their PSF in the wings and they deviate strongly from the other three. Corrections, based on the bright stars, can be applied to these curves to obtain good estimates of their true brightnesses. The top panel presents growth curves as a function of normalized aperture sums while the bottom panel shows the curves as a function of magnitude differences within each successive aperture. The relative magnitudes of the point sources are given in the top panel and the image scale is the same as in Figure 5.6. From Howell (1989).

(those to be used in the differential measures) are extracted with the same (optimum) aperture and have identical PSFs. It is likely that on a given CCD image all stars of interest will not be of exactly the same brightness and will therefore not all have exactly the same optimum aperture radius (see Figure 5.7). Thus, a compromise is usually needed in which the extraction radius used for all sources of interest is set to that of the optimum size for the faintest stars. This procedure allows the faintest sources to produce their best possible S/N result while decreasing the S/N for bright stars only slightly. Another method is to use two or three different apertures (each best for 1/2 or 1/3 of the magnitude range) with the final differential light curves separated in halves or thirds by aperture radius (i.e., magnitude).

Advances in the technique of differential photometry have led to an output precision of 1 milli-magnitude for the brightest stars almost routinely. Everett and Howell (2001) outline the procedure in detail, providing the technique and equations to use and discuss a few "tricks" that help not only achieve very high precision but provide good results even for the faintest stars. The use of local ensembles of stars and production of an ensemble for every frame (not an average frame) are the main ones. Ensemble differential photometry is the method that provides the highest precision photometry one can obtain. This method will be used for the NASA Kepler Discovery mission to search for terrestrial size extra-solar planets, the GAIA Mission, and numerous ground-based time-resolved photometric surveys even in fairly crowded fields (e.g., Howell et al, 2005; Tonry et al., 2005)

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