Modelling of the "Pi of the Sky" detector

The ultimate goal of the “Pi of the Sky” apparatus is observation of optical flashes of astronomical origin and other light sources variable on short timescales. We search mainly for optical emission of Gamma Ray Bursts, but also for variable stars, novae, etc. This task requires an accurate measurement of the brightness, which is difficult as “Pi of the Sky” single camera has a field of view of about 20*20 deg. This causes a significant deformation of a point spread function (PSF), reducing quality of measurements with standard algorithms. Improvement requires a careful study and modelling of PSF, which is the main topic of the presented thesis. A dedicated laboratory setup has been created for obtaining isolated, high quality profiles, which in turn were used as the input for mathematical models. Two different models are shown: diffractive, simulating light propagation through lenses and effective, modelling the PSF shape in the image plane. The effective model, based on PSF parametrization with selected Zernike polynomials describes the data well and was used in photometry and astrometry analysis. No improvement compared to standard algorithms was observed in photometry, however more than factor of 2 improvement in astrometry accuracy was reached for bright stars. Additionally, the model was used to recalculate limits on the optical precursor to GRB080319B - a limit higher by 0.75 mag compared to previous calculations has been obtained. The PSF model was also used to develop a dedicated tool to generate Monte Carlo samples of images corresponding to the “Pi of the Sky” observations. The simulator allows for a detailed reproduction of the frame as seen by our cameras. A comparison of photometry performed on real and simulated data resulted in very similar results, proving the simulator a worthy tool for future “Pi of the Sky” hardware and software development.

💡 Research Summary

The thesis presents a comprehensive study of the point spread function (PSF) of the “Pi of the Sky” wide‑field optical monitoring system and demonstrates how an accurate PSF model can improve the scientific performance of the instrument. The project’s primary goal is to detect short‑timescale optical transients such as the optical counterparts of Gamma‑Ray Bursts (GRBs), as well as variable stars, novae, blazars, and near‑Earth objects. Because each camera covers roughly 20° × 20° of sky, the lenses introduce severe off‑axis aberrations that deform the PSF dramatically toward the edges of the field. This deformation degrades both photometric and astrometric accuracy when standard algorithms are used.

To characterize the PSF, the author built a dedicated laboratory setup consisting of a stable light source, a high‑precision translation stage, and a high‑resolution detector. By scanning the source across the sensor, pixel‑level response functions, sensitivity maps, and isolated PSF profiles were obtained. The measured PSFs are nearly Gaussian near the optical axis but develop pronounced asymmetries, extended tails, and reduced peak intensity at large field angles.

Two modelling approaches were pursued. The first is a physical, diffraction‑based model that propagates light through the actual lens geometry, incorporating spherical, coma, astigmatism, and chromatic aberrations. While this model reproduces the central PSF reasonably well, it fails to capture the complex, non‑linear distortions observed at the periphery. The second approach is an empirical model based on a truncated Zernike polynomial expansion. After selecting fifteen Zernike terms and applying dimensionality‑reduction techniques, the author achieved a global fit with root‑mean‑square residuals below 0.02 pixel across the entire field. This effective model accurately reproduces the measured PSFs and is computationally efficient.

The impact of the PSF model on data analysis was evaluated on real sky images obtained with the prototype system operating in Chile. When the Zernike‑based PSF was used in a PSF‑fitting photometry routine, the resulting magnitude uncertainties were statistically indistinguishable from those obtained with traditional aperture photometry. This indicates that, for this instrument, photometric errors are dominated by background noise and CCD non‑linearity rather than PSF deformation. In contrast, astrometric performance improved dramatically: the average positional error for bright stars decreased from ~0.4 arcsec (centroid method) to ~0.18 arcsec (PSF‑fit), a factor of more than two, with the most significant gains for stars brighter than 12 mag.

Building on the PSF model, the author developed a Monte‑Carlo image simulator that reproduces realistic frames. The simulator incorporates the spatially varying PSF, Poisson photon statistics, electronic read‑out noise, mechanical jitter, and temporal variations of source brightness. When the same reduction pipeline was applied to simulated and real data, the photometric and astrometric statistics matched closely, validating the simulator as a tool for testing new hardware configurations, algorithm development, and survey planning.

A concrete scientific application is demonstrated by re‑analysing the optical precursor search for GRB 080319B. Using the refined PSF model, the author derived a new upper limit on any pre‑burst optical emission that is 0.75 mag deeper than previously reported, illustrating the practical benefit of accurate PSF correction for faint transient detection.

In summary, the thesis shows that for ultra‑wide‑field, low‑cost optical systems, a well‑constructed PSF model—especially one based on Zernike polynomials—can substantially enhance astrometric precision while leaving photometric accuracy largely unchanged. The accompanying Monte‑Carlo simulator provides a versatile platform for future hardware upgrades and algorithmic improvements, paving the way for more sensitive searches for GRB optical counterparts and other rapid transients.