Figure 17.2. The principle behind speckle imaging and the autocorrelation procedure. Courtesy of Dr. H. A. McAlister. Reproduced by permission from Sky Publishing Corporation.

If one assumes a point-source object, the corresponding image intensity, after going through the atmosphere and the optical system, is the convolution of the object source intensity with the Fourier transform of the atmospherically disturbed telescope pupil (the intensity distribution seen in "collimation" mode -most often seen by the amateur astronomer aligning a Newtonian telescope). The convolution process alters the original, ideal object source by a combination of atmospheric distortions and telescope aberrations.

The physical manifestation of this procedure can be seen from Figure 17.2. The top right image shows a short exposure, highly magnified star image taken through a narrowband filter. The envelope of individual speckle images is the seeing disk, in this case about 1'' in diameter. Inside the seeing disk can be seen the individual speckles but more importantly pairs of

Figure 17.2. The principle behind speckle imaging and the autocorrelation procedure. Courtesy of Dr. H. A. McAlister. Reproduced by permission from Sky Publishing Corporation.

speckles which have the same orientation and separation (such as that indicated by the arrows) can be made out. Each pair of speckles represents the components of the binary star imaged at full resolution of the telescope (in this case about 0.03''. The separation of the pair of stars is about 0.27'' and the position angle 293°. The position angle and separation could be measured directly from this frame but the power of the speckle method is that it uses many pairs of speckles to increase the reliability of the measurement.

The top left image shows five typical pairs of speckles. These move randomly inside the seeing disk but the relative separation of the speckles in each pair always remains the same. The essence of the Fourier transform is that it assesses the frequency of spatial separations of the speckles - each speckle from every other speckle. It can be seen from Figure 17.2 that there is a wide range of separations between speckles of different pairs but underlying this the most often occurring separation is that between the speckles representing the binary. However, because we are considering the separation of each speckle from every other then there are just as many pairs of speckles at PA 113° as there are at 293° so there is an ambiguity. Resolving this ambiguity is treated later in the chapter.

Calculation of the Fourier transform of the above-described image intensity produces a picture frequently referred to as the power spectrum of the image. The power spectrum represents the distribution of image "power" among the available "spatial frequencies". As an example, consider a stream of (one-dimensional) audio data. If this data represents two pure musical tones, the combined audio waveform will be the mixture of two sinusoids with different periods and (most likely) different amplitudes. If one were to take the Fourier transform of a finite section of this mixed waveform, the resulting diagram would reveal the constituent waveforms (and their relative amplitudes). Returning to the two-dimensional realm of astronomical imaging, the frequencies encountered are spatial rather than temporal. In the case of imaging a binary star system, the most likely spatial frequency to occur in the individual speckle snapshots is the separation between the two stars. In this case, the combined power spectrum of many snapshots will show bands of light and dark. The crest-to-crest distance is mapped to the separation of the two stars, while the axis perpendicular to the bands represents the position angle.

The reader interested in learning more about the Fourier transform and convolution is strongly encouraged to seek out the book by Bracewell.7 Alternately, most undergraduate level textbooks on optics will have sections describing these two concepts as well - see, for example, Klein and Furtak.8

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