The Pi of the Sky project observes optical flashes of astronomical origin and other light sources variable on short timescales, down to tens of seconds. We search mainly for optical emissions of Gamma Ray Bursts, but also for variable stars, blazars, etc. Precise photometry with a very large field of view (20?×20?) requires a careful study and modelling of a point spread function (PSF), as presented in this paper

Piotrowski, L. W.

Zarnecki, A. F.

Acta Polytechnica

English

L. W. Piotrowski

A. F. Zarnecki

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Acta Polytechnica Vol. 51 No. 6/2011Analysis and Simulation of the Pi of the Sky Detector ResponseL. W. Piotrowski, A. F. ŻarneckiAbstractThe Pi of the Sky project observes optical flashes of astronomical origin and other light sources variable on shorttimescales, down to tens of seconds. We search mainly for optical emissions of Gamma Ray Bursts, but also for variablestars, blazars, etc. Precise photometry with a very large field of view (20◦ × 20◦) requires a careful study and modellingof a point spread function (PSF), as presented in this paper.Keywords: Gamma-ray Bursts (GRB), point spread function (PSF), wide-field camera.1 IntroductionThe Pi of the Sky experiment [2] is designed for con-tinuous monitoring of a large fraction of the sky withhigh (in astronomical terms) time resolution (∼ 10 s).A real time analysis of the data stream, based ona multi-level triggering system, allows discoveries ofGRB optical counterparts independent from satel-lite experiments. Additionally, the self triggering ca-pabilities allow the detection of other rapidly vary-ing sources, such as nova and flare stars or even asyet unclassified phenomena. This approach resultedin autonomous detection of the “naked-eye” burstGRB080319B at its very beginning [3].To meet the requirement for monitoring a largefraction of the sky, the Pi of the Sky apparatus makesuse of cameras with a very wide field of view — about20◦×20◦ each. However, for stars positioned far fromthe optical axis, this causes significant deformationsof images, much bigger than in other astronomicalexperiments (as in the case of GRB080319B, see Fi-gure 1, left).Therefore both profile and aperture astrometricand photometric algorithms, which assume standardPSF shape, introduce large uncertainties. To im-prove the brightness and coordinate measurements,an elaboration of PSF in the Pi of the Sky systemis required. This requires high-precision profiles ofthe point-sources in different parts of the frame, anda mathematical model that will describe them prop-erly.2 Laboratory measurementsand modellingThe derivation of a PSF requires a star profile withsub-pixel resolution, which can be obtained from asuperposition of single star images. However, the su-perposition of real images introduces big uncertain-ties due to the fact that the exact shape and thusthe centre of the star image is unknown. Addition-ally, stars are blurred due to mount vibrations whenfollowing the sky movement, etc. These factors showthat ample data needs to be obtained in laboratorymeasurements.The apparatus for the laboratory measurementsconsisted of a LED diode (colour or white) placedbehind a pinhole 0.1 mm in diameter and placed at adistance of 22 m from a CCD camera1. The pixel sizeFig. 1: Left: The second exposure of GRB 080319B taken with the Pi of the Sky apparatus, during which the opticalemission was strongest. The GRB was recorded close to the corner of the frame, thus its shape is very deformed; Right:The setup for laboratory measurements1The standard Pi of the Sky camera consists of custom designed electronics with readout noise of ∼ 16e− at 2 Mpixels/s,commercial ∼ 2 000 × 2 000 pixel CCD and Canon EF lenses with f = 85 mm and f/d = 1.2.57Acta Polytechnica Vol. 51 No. 6/2011of the CCD sensor was 15× 15 μm (Figure 1, right).This setup gives a geometrical spot size of the diodeon the CCD sensor of less than 0.1 pixel — fulfillingthe requirement of a point source.2.1 Intra-pixel measurementsThe way in which the CCD sensor is designed causesa single-pixel sensitivity to light to be spatially non-uniform [4]. This is mainly due to the electrodesplaced across the pixels and channel stops separat-ing the columns of the sensor. This setup with anadditional diaphragm 20 mm in radius on the lenses(reducing the PSF size) enabled measurements of twofunctions describing the non-uniformity: the pixel re-sponse function (PRF) and the pixel sensitivity func-tion.Fig. 2: Left: pixel response function; Right: pixel sensi-tivity function for a red diodePRF describes a single pixel signal value vs. theposition of the spot relative to the pixel edge. A spotonly partially contained in the pixel causes a sud-den drop in PRF close to the pixel border (Figure 2,left). However, the function is non-zero for a spotfully outside the pixel. This may be caused by PSFsize, by diffraction of the spot, or by a charge diffu-sion between the pixels. In the last case, the PRFcontributes to the whole shape of PSF.The pixel sensitivity function is defined in a sim-ilar way to PRF, but an overall CCD signal value istaken instead of the single pixel signal. Changes inpixel sensitivity are the main factor responsible forthe signal changes caused by movement of the imageacross the CCD. With knowledge of the function andthe position of the centre of the source in the pixelone can compensate for this effect, performing a moreprecise measurement of the brightness.2.2 PSF measurements andmodellingA high resolution profile for selected coordinates inthe frame was obtained using multiple images of adiode. Each exposure was taken for a specific po-sition of the centre of the diode and the full set ofimages covered 10 × 10 points inside a single pixel.All the images were superimposed, taking into ac-count the coordinates of each image. Sample profilesobtained for 0, 800 and 1 400 pixels from the framecentre along the diagonal are shown in Figure 3.The spread of the image of the point source, PSF,is caused by deformations of the wavefront of the lightpassing through the optical system [1]. In the Piof the Sky case, the PSF should be described by aRayleigh-Sommerfeld diffraction formula with aber-rations, within a Kirchhoff approximation. In thecase of the wide-field lenses used in our project, itis not possible to use any of the paraxial approxima-tions that lead to an FFT, thus the formula has tobe calculated directly. Therefore, even though thisapproach should finally lead to satisfactory results, afaster model had to be developed. We use a set ofmodified Zernike polynomials to describe the imagedirectly. For each measured profile, PSF modellingis obtained by fitting polynomial coefficients to thedata (Figure 4). The general model is obtained byinterpolating the coefficients determined for selectedcoordinates on the frame.Fig. 4: PSFs for 800 pixels from the frame centre. Left:measured; Right: polynomial modelFig. 3: Measured PSFs 0, 800 and 1 400 pixels from the centre of the frame, along the diagonal58Acta Polytechnica Vol. 51 No. 6/2011Fig. 5: Left: Comparison of the position determination accuracy from polynomial model profile astrometry and fromASAS aperture astrometry, for stars brighter than 9m; Right: The signal resulting from the PSF model fit at the “naked-eye” burst position prior to the explosion (bars) and the extracted precursor limits (lines) (data from one camera)Fig. 6: Left: Polynomial photometry results for simulated and real sky images; Right: Relative error of the photometryperformed on the simulated data, as a function of the star brightness (points) and the estimated contribution fromdifferent sources of variability (lines)3 Model applicationsThe polynomial model that is obtained can be usedfor multiple purposes. The most straightforward useis for photometry and astrometry of stars, GRBsand other objects of interest. As for the photome-try, when testing on selected data samples the modeldid not prove better in determining the brightnessof stars than the ASAS aperture photometry used inour experiment. This is probably due to the real fluc-tuations of the star signal, caused by miscellaneousfactors in the experiment that dominate the measure-ment. However, determining the position of the starson the frame is up to a factor of 2.5 more precise inthe polynomial PSF model (Figure 5, left).Another application of the model is in a dedicatedsearch for a signal in specific coordinates, such as asearch for an optical precursor to the “naked-eye”GRB080319B [5]. Pi of the Sky started observingthe position of the burst more than 19 minutes be-fore the trigger, thus providing data well suited forthis task. No signal exceeding 3σ, coinciding on twocameras, was found when fitting the modelled PSFto the frames preceding the burst, and the limitingmagnitudo was improved from 11.5m (resulting fromstandard photometry) to 12.25m (Figure 5, right).Additionally, a feature-rich simulator of the Piof the Sky frame was created, utilizing the modelleddetector response. The photometry results for realand simulated data are in very good agreement (Fi-gure 6, left). This can be taken as proof of goodsimulation quality. To obtain photometric uncertain-ties similar to the real uncertainties, fluctuations suchas photon statistics, gain fluctuations, readout noise,position fluctuations and star turbulences had to betaken into account. The simulation results show thatthe photometry quality for stars darker then 8.5m ismost probably limited by gain fluctuations, and notby readout noise, as had previously been anticipated(Figure 6, right).4 SummaryDue to the very large field of view, PSF in the Pi ofthe Sky detector is highly deformed and is not de-scribed by general models. On the basis of dedicatedlaboratory measurements, we have created an effec-tive model describing the dependence of PSF on theposition of the star in the frame. These results mayhelp in further understanding and developing the Piof the Sky detector and other future wide field ex-periments. So far we have successfully introduced59Acta Polytechnica Vol. 51 No. 6/2011the model into a photometric algorithm, a dedicatedsignal search and astrometric algorithms, the last ofwhich has given very promising results. Additionally,a detailed simulator of the Pi of the Sky frame hasbeen created.AcknowledgementThis work describes a research project supported bythe Polish Ministry of Science and Higher Educationin 2009–2011.References[1] Rerabek, M., Pata, P.: The space variant PSFfor deconvolution of wide-field astronomical im-ages, Proceedings of SPIE — 7015 — Adap-tive Optics Systems. Washington : SPIE, 2008,p. 70152G-1–70152G-12.[2] Siudek, M., et al.: Pi of the Sky telescopes inSpain and Chile, these proceedings.[3] Racusin, J. L., et al.: Broadband observations ofthe naked-eye big γ-ray burst GRB 080319B, Na-ture 455, 183–188, 2008.[4] Toyozumi, H., Ashley, M. C. B.: Intra-pixel sensi-tivity variation and charge transfer inefficiency —results of CCD scans, Publications of the Astro-nomical Society of Australia, 2005, 22, 257–266.[5] Paczynski, B.: Optical Flashes Preceding GRBs,astro-ph/0108522, 2001.Lech Wiktor PiotrowskiAleksander Filip ŻarneckiFaculty of PhysicsUniversity of WarsawHoża 69, 00-681 Warsaw60

Analysis and Simulation of the Pi of the Sky Detector Response

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 51 No. 6/2011Analysis and Simulation of the Pi of the Sky Detector ResponseL. W. Piotrowski, A. F. Z˙arneckiAbstractThe Pi of the Sky project observes optical ﬂashes of astronomical origin and other light sources variable on shorttimescales, down to tens of seconds. We search mainly for optical emissions of Gamma Ray Bursts, but also for variablestars, blazars, etc. Precise photometry with a very large ﬁeld of view (20◦ × 20◦) requires a careful study and modellingof a point spread function (PSF), as presented in this paper.Keywords: Gamma-ray Bursts (GRB), point spread function (PSF), wide-ﬁeld camera.1 IntroductionThe Pi of the Sky experiment [2] is designed for con-tinuous monitoring of a large fraction of the sky withhigh (in astronomical terms) time resolution (∼ 10 s).A real time analysis of the data stream, based ona multi-level triggering system, allows discoveries ofGRB optical counterparts independent from satel-lite experiments. Additionally, the self triggering ca-pabilities allow the detection of other rapidly vary-ing sources, such as nova and ﬂare stars or even asyet unclassiﬁed phenomena. This approach resultedin autonomous detection of the “naked-eye” burstGRB080319B at its very beginning [3].To meet the requirement for monitoring a largefraction of the sky, the Pi of the Sky apparatus makesuse of cameras with a very wide ﬁeld of view — about20◦×20◦ each. However, for stars positioned far fromthe optical axis, this causes signiﬁcant deformationsof images, much bigger than in other astronomicalexperiments (as in the case of GRB080319B, see Fi-gure 1, left).Therefore both proﬁle and aperture astrometricand photometric algorithms, which assume standardPSF shape, introduce large uncertainties. To im-prove the brightness and coordinate measurements,an elaboration of PSF in the Pi of the Sky systemis required. This requires high-precision proﬁles ofthe point-sources in diﬀerent parts of the frame, anda mathematical model that will describe them prop-erly.2 Laboratory measurementsand modellingThe derivation of a PSF requires a star proﬁle withsub-pixel resolution, which can be obtained from asuperposition of single star images. However, the su-perposition of real images introduces big uncertain-ties due to the fact that the exact shape and thusthe centre of the star image is unknown. Addition-ally, stars are blurred due to mount vibrations whenfollowing the sky movement, etc. These factors showthat ample data needs to be obtained in laboratorymeasurements.The apparatus for the laboratory measurementsconsisted of a LED diode (colour or white) placedbehind a pinhole 0.1 mm in diameter and placed at adistance of 22 m from a CCD camera1. The pixel sizeFig. 1: Left: The second exposure of GRB 080319B taken with the Pi of the Sky apparatus, during which the opticalemission was strongest. The GRB was recorded close to the corner of the frame, thus its shape is very deformed; Right:The setup for laboratory measurements1The standard Pi of the Sky camera consists of custom designed electronics with readout noise of ∼ 16e− at 2 Mpixels/s,commercial ∼ 2 000 × 2 000 pixel CCD and Canon EF lenses with f = 85 mm and f/d = 1.2.57Acta Polytechnica Vol. 51 No. 6/2011of the CCD sensor was 15× 15 μm (Figure 1, right).This setup gives a geometrical spot size of the diodeon the CCD sensor of less than 0.1 pixel — fulﬁllingthe requirement of a point source.2.1 Intra-pixel measurementsThe way in which the CCD sensor is designed causesa single-pixel sensitivity to light to be spatially non-uniform [4]. This is mainly due to the electrodesplaced across the pixels and channel stops separat-ing the columns of the sensor. This setup with anadditional diaphragm 20 mm in radius on the lenses(reducing the PSF size) enabled measurements of twofunctions describing the non-uniformity: the pixel re-sponse function (PRF) and the pixel sensitivity func-tion.Fig. 2: Left: pixel response function; Right: pixel sensi-tivity function for a red diodePRF describes a single pixel signal value vs. theposition of the spot relative to the pixel edge. A spotonly partially contained in the pixel causes a sud-den drop in PRF close to the pixel border (Figure 2,left). However, the function is non-zero for a spotfully outside the pixel. This may be caused by PSFsize, by diﬀraction of the spot, or by a charge diﬀu-sion between the pixels. In the last case, the PRFcontributes to the whole shape of PSF.The pixel sensitivity function is deﬁned in a sim-ilar way to PRF, but an overall CCD signal value istaken instead of the single pixel signal. Changes inpixel sensitivity are the main factor responsible forthe signal changes caused by movement of the imageacross the CCD. With knowledge of the function andthe position of the centre of the source in the pixelone can compensate for this eﬀect, performing a moreprecise measurement of the brightness.2.2 PSF measurements andmodellingA high resolution proﬁle for selected coordinates inthe frame was obtained using multiple images of adiode. Each exposure was taken for a speciﬁc po-sition of the centre of the diode and the full set ofimages covered 10 × 10 points inside a single pixel.All the images were superimposed, taking into ac-count the coordinates of each image. Sample proﬁlesobtained for 0, 800 and 1 400 pixels from the framecentre along the diagonal are shown in Figure 3.The spread of the image of the point source, PSF,is caused by deformations of the wavefront of the lightpassing through the optical system [1]. In the Piof the Sky case, the PSF should be described by aRayleigh-Sommerfeld diﬀraction formula with aber-rations, within a Kirchhoﬀ approximation. In thecase of the wide-ﬁeld lenses used in our project, itis not possible to use any of the paraxial approxima-tions that lead to an FFT, thus the formula has tobe calculated directly. Therefore, even though thisapproach should ﬁnally lead to satisfactory results, afaster model had to be developed. We use a set ofmodiﬁed Zernike polynomials to describe the imagedirectly. For each measured proﬁle, PSF modellingis obtained by ﬁtting polynomial coeﬃcients to thedata (Figure 4). The general model is obtained byinterpolating the coeﬃcients determined for selectedcoordinates on the frame.Fig. 4: PSFs for 800 pixels from the frame centre. Left:measured; Right: polynomial modelFig. 3: Measured PSFs 0, 800 and 1 400 pixels from the centre of the frame, along the diagonal58Acta Polytechnica Vol. 51 No. 6/2011Fig. 5: Left: Comparison of the position determination accuracy from polynomial model proﬁle astrometry and fromASAS aperture astrometry, for stars brighter than 9m; Right: The signal resulting from the PSF model ﬁt at the “naked-eye” burst position prior to the explosion (bars) and the extracted precursor limits (lines) (data from one camera)Fig. 6: Left: Polynomial photometry results for simulated and real sky images; Right: Relative error of the photometryperformed on the simulated data, as a function of the star brightness (points) and the estimated contribution fromdiﬀerent sources of variability (lines)3 Model applicationsThe polynomial model that is obtained can be usedfor multiple purposes. The most straightforward useis for photometry and astrometry of stars, GRBsand other objects of interest. As for the photome-try, when testing on selected data samples the modeldid not prove better in determining the brightnessof stars than the ASAS aperture photometry used inour experiment. This is probably due to the real ﬂuc-tuations of the star signal, caused by miscellaneousfactors in the experiment that dominate the measure-ment. However, determining the position of the starson the frame is up to a factor of 2.5 more precise inthe polynomial PSF model (Figure 5, left).Another application of the model is in a dedicatedsearch for a signal in speciﬁc coordinates, such as asearch for an optical precursor to the “naked-eye”GRB080319B [5]. Pi of the Sky started observingthe position of the burst more than 19 minutes be-fore the trigger, thus providing data well suited forthis task. No signal exceeding 3σ, coinciding on twocameras, was found when ﬁtting the modelled PSFto the frames preceding the burst, and the limitingmagnitudo was improved from 11.5m (resulting fromstandard photometry) to 12.25m (Figure 5, right).Additionally, a feature-rich simulator of the Piof the Sky frame was created, utilizing the modelleddetector response. The photometry results for realand simulated data are in very good agreement (Fi-gure 6, left). This can be taken as proof of goodsimulation quality. To obtain photometric uncertain-ties similar to the real uncertainties, ﬂuctuations suchas photon statistics, gain ﬂuctuations, readout noise,position ﬂuctuations and star turbulences had to betaken into account. The simulation results show thatthe photometry quality for stars darker then 8.5m ismost probably limited by gain ﬂuctuations, and notby readout noise, as had previously been anticipated(Figure 6, right).4 SummaryDue to the very large ﬁeld of view, PSF in the Pi ofthe Sky detector is highly deformed and is not de-scribed by general models. On the basis of dedicatedlaboratory measurements, we have created an eﬀec-tive model describing the dependence of PSF on theposition of the star in the frame. These results mayhelp in further understanding and developing the Piof the Sky detector and other future wide ﬁeld ex-periments. So far we have successfully introduced59Acta Polytechnica Vol. 51 No. 6/2011the model into a photometric algorithm, a dedicatedsignal search and astrometric algorithms, the last ofwhich has given very promising results. Additionally,a detailed simulator of the Pi of the Sky frame hasbeen created.AcknowledgementThis work describes a research project supported bythe Polish Ministry of Science and Higher Educationin 2009–2011.References[1] Rerabek, M., Pata, P.: The space variant PSFfor deconvolution of wide-ﬁeld astronomical im-ages, Proceedings of SPIE — 7015 — Adap-tive Optics Systems. Washington : SPIE, 2008,p. 70152G-1–70152G-12.[2] Siudek, M., et al.: Pi of the Sky telescopes inSpain and Chile, these proceedings.[3] Racusin, J. L., et al.: Broadband observations ofthe naked-eye big γ-ray burst GRB 080319B, Na-ture 455, 183–188, 2008.[4] Toyozumi, H., Ashley, M. C. B.: Intra-pixel sensi-tivity variation and charge transfer ineﬃciency —results of CCD scans, Publications of the Astro-nomical Society of Australia, 2005, 22, 257–266.[5] Paczynski, B.: Optical Flashes Preceding GRBs,astro-ph/0108522, 2001.Lech Wiktor PiotrowskiAleksander Filip Z˙arneckiFaculty of PhysicsUniversity of WarsawHoz˙a 69, 00-681 Warsaw60

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Analysis and Simulation of the Pi of the Sky Detector Response

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