# Making the Measurements

To aid further the determination of binary star astrom-etry, the Fourier transform can be used yet again. By calculating the transform of the power spectrum, these bands of light and dark can be converted into a sequence of three co-linear, circularly symmetric peaks. Binary star astronomers call this picture the autocorrelogram. The autocorrelogram consists of a large central peak and two smaller peaks, one on either side of the central peak, exactly 180° apart. Figure 17.3 shows a typical autocorrelogram.

These peaks are the result of a random process, which gives them a Gaussian profile. Therefore, centres are easy to determine. The distance between the centres of the central peak and one of the other peaks

### Figure 17.3. A

background subtracted autocorrelogram of the binary star K UMa. Adapted from McAlister et al.1 Printed by kind permission of the American Astronomical Society

### Figure 17.3. A

background subtracted autocorrelogram of the binary star K UMa. Adapted from McAlister et al.1 Printed by kind permission of the American Astronomical Society

gives the separation angle. The angular orientation of the line of three peaks gives the position angle, though with a 180° ambiguity.

This process of creating the power spectrum and the autocorrelogram can be done using analogue or digital techniques. Gezari et al.9 describe a typical analogue autocorrelogram generating process. They start with a standard 35 mm film camera attached to an image intensifier tube. The camera is equipped with a rapid film advance system to more efficiently use telescope time, though, in the strictest sense, it is optional. The camera records a sequence of short exposure, high magnification images. This film is then developed and negative reversed (so the film is now a positive image). Each individual frame is stepped through a laser-illuminated optical system (employing the classical aperture/image relationship of coherent optics) and individually exposed onto a separate emulsion to form the power spectrum. This process is done again to form the autocorrelogram. This is usually a slightly toxic process in that an index-matching fluid to the film has to be used to keep laser scattering from micro-scratches from ruining the power spectrum and auto-correlogram emulsions. For traditional 35 mm film, this fluid is usually a variant of the standard dry cleaning fluid, naphthalene.

This whole process can be done digitally if digitised frames of the individual speckle frames are available. While one can digitise the individual photographic frames, video frames from a bare CCD or intensified CCD are the more likely source. Taking the Fourier transform of each individual frame, co-adding them all to form the power spectrum, and taking the Fourier transform again produces the autocorrelogram.

In the digital realm, there is a shortcut, a way to go straight from individual speckle frames to the autocor-relogram. This is done by correlating every pixel in the individual speckle frame with every other. To see this, imagine an "autocorrelogram canvas" four times the size of an individual speckle frame. For each individual frame, a value of one is assigned to pixel values above a certain threshhold, and zero below. For each above-threshhold value in the individual frame, its pixel address is aligned with the centre of the canvas and the frame is added to the canvas (remember that the frame is now a collection of ones and zeros). Morphologically, this procedure produces a diagram much like that of generating the autocorrelogram through the use of

Fourier transforms. In fact, taking into account the multi-bit nature of the data, and adjusting the thresh-hold value, these numerical techniques can be shown to be the same.