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The Point Spread Function in Lucky Imaging and Variations in Seeing on Short Timescales A&A 480, 589–597 (2008) Astronomy DOI: 10.1051/0004-6361:20079214 & c ESO 2008 Astrophysics The point spread function in Lucky Imaging and variations in seeing on short timescales J. E. Baldwin1,P.J.Warner1, and C. D. Mackay2 1 Cavendish Laboratory, Madingley Rd, Cambridge CB3 0HE, UK e-mail: [jeb;pjw]@mrao.cam.ac.uk 2 Institute of Astronomy, Madingley Rd, Cambridge CB3 0HE, UK e-mail: [email protected] Received 7 December 2007 / Accepted 22 December 2007 ABSTRACT Aims. We investigate the properties of astronomical images made by combining the best images selected from a sequence of short- exposure frames (Lucky Imaging); we assess the match between modelling and observation and discover what variations in seeing occur on very short timescales. Methods. Numerical simulations of a random phase-changing screen passing across a telescope aperture with ideal optics are used to determine the expected point spread function and isoplanatic properties for a range of seeing conditions for comparison with observations. Results. All the model images comprise a diffraction-limited core with Strehl ratios from 0.05–0.5 and an underlying broad disk. The isoplanatic patch sizes are large and coherence times long. The observations are a close match to the models in most respects. Large variations in seeing occur on temporal scales as short as 0.2 s and spatial scales as small as 1 m. Key words. atmospheric effects – techniques: high angular resolution 1. Introduction listed above were a consequence of selecting the moments of favourable fluctuations in a period of constant mean seeing or We have used the term Lucky Imaging in several earlier papers whether in practice the selection was for periods of better aver- (Baldwin et al. 2001; Tubbs et al. 2002; Mackay et al. 2003;Law age seeing. For instance, the large isoplanatic patch size might et al. 2006) to describe the technique of improving angular reso- have occurred only during periods when the mean seeing was ex- lution in astronomical images by combining images with the best ceptionally good. This distinction may seem of little importance Strehl ratios selected from a sequence of short-exposure frames to an astronomer whose sole interest is in obtaining images with using recentring on the brightest speckle. This follows the orig- better resolution; but it is essential to an understanding of the ap- inal usage by Fried (1978) applied to the selection of the single plication of the technique to telescopes of differing sizes and to best image from a sequence. Law et al. (2006) presented obser- the basic question of whether the behaviour of seeing conforms vations made with the 2.5 m Nordic Optical Telescope (NOT) at to that expected from the models in general use. A second unan- wavelengths close to 800 nm giving a quantitative assessment of swered question was by how much the small errors in figuring the improved resolution obtained in practice. The main conclu- of the NOT mirrors reduced the performance of Lucky Imaging sions were: relative to that potentially achievable with ideal optics. In this 1. The full width to half maximum (FWHM) of images was paper we explore these questions using simulations of a random improved, relative to conventional imaging, by a factor of phase screen moving past the telescope aperture. In Sect. 2 we about 2 if all of the data were recentred and by a factor up describe the model and in subsequent sections we attempt to an- to 4 if the best 1% of data were selected for recentring. swer the following questions: 2. These factors of improvement in the FWHM applied over a wide range of seeing conditions. 3. The radius of the area of isoplanatism was unexpectedly 1. What are the expected properties of Lucky images made with large, ∼20 arcsec for a reduction in the Strehl ratio by a fac- ideal telescope optics under conditions of constant mean see- tor of 2 relative to the value at the centre of the field. ing? 4. Stars as faint as m =+16 could be used as reference objects 2. Are the conclusions from the NOT observations consistent for the selection of frames when using an L3CCD camera with a constant value of the seeing during one sequence of with very low readout-noise. frames? 5. The results 3) and 4) implied that the method could be used 3. What is the best measure of seeing on short timescales and ff over about 20% of the sky. what variations in seeing on di erent time-scales can be es- tablished from the NOT data? It was noted on several occasions that the mean seeing appeared to change by a large amount on a timescale of ∼1sbutwe were not able to distinguish and quantify the changes because We comment briefly on the implications of the rapid changes of the intrinsic random fluctuations expected in the turbulent at- which are found in the NOT data for other techniques of high- mospheric screen. It was therefore not clear whether the results resolution imaging and for the model of seeing in common use. Article published by EDP Sciences 590 J. E. Baldwin et al.: The point spread function in Lucky Imaging and variations in seeing on short timescales 2. Simulations of seeing Numerical simulations of stellar images observed through a ran- dom phase screen were adopted for this discussion. They pro- vide images which can be analysed in exactly the same way as the observational data and they allow estimation of observational parameters which are difficult to extract using the analytic meth- ods employed in many investigations of seeing. The simulations were made to match the circumstances of the observations made with the 2.5 m Nordic Optical Telescope, mostly with a 20% bandwidth at wavelengths near 800 nm, un- der seeing conditions ranging from 0.4−1.2 arcsec. Their appli- cation is much wider since the properties of the images are deter- minedbytheratio,D/r0, of the telescope aperture D to the Fried seeing parameter r0. For instance, they would apply equally to a 4 m aperture at 1.3 µm and an 8 m aperture at 2.2 µm under similar seeing conditions. It is likely that a similarly wide band- width will be necessary at most wavelengths to allow the use of faint reference stars. We therefore present the results as a func- tion of D/r0 in Sect. 3 and convert them to the NOT parame- ters for comparison with the observations in Sect. 4. The range of values of D/r0 in these simulations and observations is ap- proximately 6−18. This range was already identified by earlier authors (Hecquet & Coupinot 1984) as being one in which sig- nificant improvements in angular resolution could be achieved by selection and recentring of images. A 2-dimensional random phase screen was generated within a 2048 × 2048 pixel array, based on the code developed by Buscher (1988). For each screen the amplitude a(k) for each cosine and sine component of spatial frequency k were chosen as independent Gaussian random variables with zero mean and 2 ∝ −11/3 variance a (k) k . The diameter, D, of the telescope aper- / = . ture was chosen to be 56 pixels. Then, in the comparison with the Fig. 1. a) One instantaneous image; D r0 8 3, Strehl ratio 0.16. b) Average of best 10% of images; D/r = 8.3, Strehl ratio 0.19. NOT observations, the largest scale of turbulence in the simula- 0 tion is ∼100 m and the smallest ∼0.09 m. The consequent trun- cations of the Kolmogorov spectrum have no significant effect images on the sky is proportional to λ, averaging of the images on the results. The largest scales provide only tip-tilt variation also required linear interpolation of each image to a standard an- of phase across the 2.5 m aperture and are therefore removed gular scale. The second feature was to move the aperture across in the shift-and-add process. 100 m is, in any case, as large as the screen from the lower left to the upper right corner to create the outer scale of turbulence believed to exist for most sites. The a sequence of 664 successive instantaneous images correspond- power missing at the shortest scales is too small to affect the ing to a “frozen” turbulent screen carried by the wind crossing results except for seeing poorer than we have used in these sim- the telescope aperture. Comparisons could then be made with ulations. We have not explored the effects of alternative slopes in temporal and isoplanatic aspects of the observations. The spa- the power spectrum of turbulence but draw attention in Sects. 3.1 tial interval between successive images corresponded to shifts and 4 to observational aspects of short-exposure images which of 0.07575D, or 0.187 m for the NOT comparison, which is ad- reveal it. equate sampling for all aspects of the analysis. Images of stars were generated by taking a 512 × 512 sub- This procedure was repeated for a number, typically 8, of array centred on the telescope aperture with unit amplitude different phase screens for each value of D/r0. The residual sta- and the corresponding phases within the aperture and zero am- tistical fluctuations in the results presented here due to the finite plitude outside. The square of the Fourier transform of this number of frames are small but may account for some of the 2-dimensional array gives one instantaneous image. Several differences between the curves in Figs. 6 and 7. checks were made on the overall normalisation of the process, Figure 1a shows a typical short-exposure image for D/r0 = first by averaging the images from a large number of phase 8.3.
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