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Optimal Photometry

Inner radius

There is an issue that OPPHOT addresses which was not discussed in the original paper. It concerns the inner radius of the box used to determine the sky flux. The flux in the sky box from the star is simply the integral of the profile over the sky box. Since the sky box excludes the inner part of the stellar profile, this should be a small fraction of the total flux. Using the notation of the original paper, this is TZ, where T is the total flux from the star, and we introduce Z, the integral of the normalised profile, P, over the sky box. (The integral of P over all area is one.) The average counts-per-pixel in the sky box is S + TZ/Ns, where S is the counts-per-pixel from the sky, and Ns is the number of pixels in the sky box. Thus the sky level is over-estimated by TZ/Ns counts per pixel. Since our weighting scheme is a linear superposition, we can fold this through the weight mask given in equations (9) and (10) of the paper by substituting D-S with TZ/Ns. Further, we know from Section 3.3 that the variances of each pixel are equal. Finally, Ns is determined in Section 3.1 to be 200 times the sum of P**2, which allows us to arrive at a final error of TZ/200.

What is interesting about this result is that provided Z is the same for all stars, this inaccuracy is simply a fraction of T, the total counts, and is therefore the same fraction for all stars. The way of ensuring this is the case is to always set the inner radius of the sky box, and hence of the integration of Z to be the same for all stars. In OPPHOT it is set at twice the FWHM of the profile. Integrating a two dimensional Gaussian from here to infinity gives an upper limit on Z, and hence an upper limit on the error of 0.026 percent. Of course the sky is not determined by simply taking an average of the counts in the sky box; the algorithm described in Section 3.1 should reject some of the pixels closest to the star. However, in practice this estimate is not far wrong, with tests on real data suggesting an error of 0.02 percent. Even this is a string upper limit to the final error which will result from relative photometry. Since the error is approximately the same percentage for all stars, it will be removed at soon as the photometry is placed on a relative scale.

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