TRIPPy: TRAILED IMAGE PHOTOMETRY IN PYTHON

We test the precision of this process using two archival images from the Mosaic2 camera on the Blanco telescope of asteroid 105217 (Elliot et al. 2005) and four archival images from MegaCam on the Canada–France–Hawaii Telescope that serendipitously imaged the bright asteroid (2006) Polonskaya (Gwyn et al. 2012). Two images of Polonskaya were available in each of the r' and g' filters, with exposure times of 480 and 240 s, respectively. During the observations, the seeing was ∼11 (nearly 6 pixels), and Polonskaya was moving at 30''–31'' hr⁻¹. During the 240 s images of 105217, which were acquired in the VR filter, the asteroid had a rate of motion of ∼587 hr⁻¹and the seeing was ∼13 (2.6 pixels). The images of Polonskaya resulted in an S/N ∼ 800 while those of 105217 resulted in an S/N ∼ 350 when measured in a pill aperture with r = 1.4 FWHM. The long trails and extremely high S/N present both a strong challenge and excellent test case for our TSF generation procedure. Three of the six images are shown in Figure 4, and a pill aperture using Polonskaya's rate of motion can be found in Figure 2.

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For all six images, the PSF and the TSF were generated using the above procedure with a super-sampling factor, ssf = 10. On-sky rates of motion for each image were provided by the JPL horizons ephemeris service.⁹ Point sources were visually confirmed to be isolated stars, and only those with S/N > 200 (measured in a circular aperture) were used. We note, however, that very little difference in performance was found when considering reference star S/N thresholds in the 100–300 range. That is, the worst residual value did not change by more than a few percent.

Source centroids and S/N estimates were measured with SExtractor (Bertin & Arnouts 1996). We tested possible improvement with measuring centroids during the moffat profile fitting (step 2). No noticeable improvement was found, and the centroids provided by SExtractor were deemed as sufficient for our purposes.

The quality of the PSF and TSF generated with ssf = 10 are presented in Figures 4 and 5. The stellar residuals of those stars used to generate the PSF are particularly good, with no consistent residual structure between different sources, and peak residuals <1% of the peak pixel amplitudes of each source.¹⁰ This performance slightly surpasses that of the daophot–iraf package, which produces peak residuals of ∼2% using identical PSF reference stars, albeit with slightly worse chi-squared. Our residuals increase by a factor of ∼2 if 5× super sampling is used, and by ∼5 if 3× is used.

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For the asteroids, the residuals do not fare as well; the pixel residuals are <5%–10% the peak. Furthermore, each image shows a structure that shares an axis with the trailing direction. These structures dominate the residuals at all super-sampling factors between 3 and 30, and are simply structures in the Polonskaya image that are not accounted for by our TSF.

The most likely cause of the residual structure is the inherent assumption by our routine of perfectly stable conditions during an exposure. That is, for our method to work perfectly, there can be zero variation in the PSF during the entire exposure, a condition that is impossible to meet under even the best observing circumstances. We provide basic tests of this scenario, by planting artificial trailed sources in our images containing Polonskaya, and then subtracting a source with the same flux, but slightly adjusted FWHM. These experiments revealed that ∼8% of peak residuals are found with FWHM variations as little as 5%. With this result in mind, residual structures in trailed image subtractions are an inevitable consequence of the fact that point sources do not preserve a record of on-image seeing variations during an exposure. It should be noted, however, that trailed subtraction residuals will be lower for less trailed sources; the 240 s exposures of Polonskaya have a peak residual that is nearly half the 8% residual of the 480 s exposures, and the longer trailed image 105217 has higher peak residual still at 10% of peak pixel value.

Despite the modest subtraction residuals, accurate photometry for all trailed sources is still afforded by the TSF. This is because accurate aperture corrections can be measured directly from the TSF. We demonstrate this with the extremely high S/N images of Polonskaya and 105217. Specifically, we measured the aperture corrections as a function of r in the standard way, using the TSF, and then directly from the image of the asteroids themselves. The difference is presented in Figure 6 for one of the 480 s, r' images of Polonskaya and for one image of 105217. For completeness, we also include an estimate of the aperture correction when the lookup table is not utilized. As can be seen, for nearly all r, the difference between the aperture corrections measured from the TSF, and that measured directly from the asteroid's image is less than 10 mmag. In an r = 1.4 FWHM pill aperture, the S/N = 889 corresponds to a source photon noise uncertainty of ∼1 mmag. Clearly, even for highly elongated sources, accurate aperture corrections can be measured from the TSF alone. It should be noted that with a pill radii of r < 2.5, for the extremely high S/N images of Polonskaya, the aperture correction uncertainty is the dominant source of error, and a pill aperture with larger r should be used. For images with S/N < 200 (or still in the flux limited regime), however, the uncertainty is dominated by source photon noise, and apertures of smaller r will provide the lowest final uncertainty. The value of r that provides maximal S/N depends on the trail length, but we have found in practice that for background images with trail lengths <2 FWHM, 1.2 < r < 1.6 typically provides maximal S/N.

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In Figure 6, we also present the results when a lookup table is not used. This facilitates a comparison with the trail-fitting method of Vereš et al. (2012) which utilizes a Gaussian profile without a lookup table, but otherwise, uses essentially the same technique to generate TSFs as we present here. Without the lookup table, the error in the aperture correction estimate is a factor of four worse. Therefore, for the most accurate photometry, the use of a lookup table is required. For sources with S/N ≥ 50, the lack of a lookup table will cause a bias in the flux measurement larger than the amplitude of the photon noise uncertainty.

Recently, Sonnett et al. (2013) presented a comparison of the relative performances of various photometry techniques in the scenario of moderate S/N observations (≳20), of a Kuiper Belt Object, moving at 28 hr⁻¹. At the expense of knowledge of the PSF and any aperture corrections, the telescope was tracked at a non-sidereal rate to produce a (nearly) untrailed image of the KBO, allowing circular apertures to be used, facilitating comparison of a wide range of photometry packages.

In a lowest rms sense, one of the highest performing techniques is the SExtractor MAG_ISOCOR photometry technique. This technique attempts to provide full flux measurements of a source by tracing an isophotal curve around the source and measuring photometry within that aperture. A correction is then applied to account for the flux excluded by that aperture based on a two-dimensional Gaussian profile fit to the source. In this way, source elongation can be at least roughly accounted for, with the rms uncertainty increasing by modest amounts—approximately 50% in the case discussed by Sonnett et al. (2013).

One large difficulty arises in the use of MAG_ISOCOR when tracing isophotal apertures around low S/N sources. While the S/N of this technique is generally as good or better than other techniques, a bias toward measuring a flux lower than the true value exists. This is presented in Figure 7 where we compare simulations of photometry reported by pill aperture photometry and that from MAG_ISOCOR. This comparison was made by planting 50 synthetically elongated Moffat profiles at various flux levels in an image with a background of 1000 ADU, and appropriate Gaussian noise. The Moffat profile extracted from the first r' image of asteroid Polonskaya was used. The rate, and one-third the rate of motion of the asteroid in that image, were considered. The pill aperture photometry was corrected for the aperture correction measured on frame to provide comparison with MAG_ISOCOR, which includes their own corrections. All measurements at each flux level are normalized by the true flux, and are presented in Figure 7, along with the mean measurement at each flux level.

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As can be seen, both produce similar S/Ns at all flux levels. Similiarly, at high S/N levels, both techniques reproduce the correct flux with no appreciable biases. At low S/N, however, on average, MAG_ISOCOR produces a mean measurement that is biased faintward while the pill aperture produces an unbiased measurement. For example, the MAG_ISOCORR bias rapidly degrades to ∼3% at S/N∼50. The reason pill photometry avoids this bias is simple; measurement of the PSF, and estimation of the TSF afforded by sidereally tracked images allows accurate aperture corrections to be calculated, at least to the precision demonstrated in Figure 6.

In addition to the pill aperture, we also introduce a new technique to estimate the background modal value in an image. A common technique used to estimate the mode is to take three times the median value minus two times the mean value of a sourceless region of an image, or small variations on this equation (see, for example, Bertin & Arnouts 1996). Our modal technique involves multiplying the pixel values within a region by some factor of x < 1 and rounding it to an integer to produce values y_i. The mode m of all y_i integers is then found. The background estimate is then the median value of all pixels with $\mathrm{int}(x\times {y}_{{\rm{i}}})=m$. Our tests demonstrate that for values of x ∼ 0.1, the background estimated by our modal technique is less sensitive to bright background sources than the more standard background estimation techniques.

A comparison of techniques is demonstrated in Figure 8 where we have cut out a small 41 × 41 pixel region specifically chosen to contain two bright sources, which occupy roughly half of the cutout. We present a histogram of pixel values, and background estimations using various techniques, including mean, median, Gaussian fit (which produces the same value as the mean), 3 × median − 2 × mean with initial 3σ away from the mean rejection, and our modal technique using x = 0.1 and x = 0.2. As can be seen, the modal technique produces visually the most satisfactory result.

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It should be noted that the modal technique does depend on the choice in x. Our experience suggests that x ≳ 0.3, while still outperforming other techniques, produces background estimates that are a few percent too low, while values of x ≲ 0.05 produce values that are too large unless the region is large, containing of the order of 10,000 or more pixels. We consider x = 0.1 to be the best overall choice.