The Shapes Of Cross Correlation Interferometers Daniel Birman Boaz Chen

Aperture Synthesis In astronomy carried out at radio frequencies it is possible to record both the amplitude and phase from stellar objects. This enables us to add up different observations after they have been recorded. This is used for high-resolution sub- millimeter and radio astronomy observations. The farther apart the observations are the better the resolutions can get by Rayleigh condition:

In order to produce a high quality image, a large number of different separations between different telescopes are required in order to get a good quality image. [n*(n-1) baselines are produced from n telescopes] Most aperture synthesis interferometers use the rotation of the Earth to increase the number of different baselines included in an observation.

The problem Positioning an array of optical sensors (Telescopes) in order to produce the best general propose interferometer using aperture synthesis While trying to maximize resolution signal/noise sampling accuracy

Analytic approach The best response is the one that provides the most complete sampling of the Fourier space of the image out to the limit of some best spatial resolution The sampling should be invariant to measuring different directions This implies that the response function should be circular symmetric confined in the boundaries of the best resolution

If we want to maximize the signal to noise ratio and the accuracy of the measurement the Fourier plane ought to be sampled uniformly. Signal to noise ratio decreases as we use nonuniform weighting (as result of uneven spread in the Fourier plane as well as using different weights for the measurements) Uniform sampling will provide images least susceptible to errors arising from the unmeasured Fourier components

Analytic approach Therefore the best positioning would be on a curve of constant width, (so that all the sampling would be done in a radius of that width) More uniform sampling would be achieved by reducing the symmetry of the sampling pattern

The Reuleaux triangle Is a symmetric curve with constant width (the length of the original triangle side) Has the lowest degree of rotational symmetry (3) from all possible shapes of constant width

So we understand we should position our sensors on the curve of the Reuleaux triangle However placing the sensors non-uniformly on that curve as well as small perturbation off that curve might further reduce symmetry and produce a more uniform sampling pattern

Numeric approach Calculating the best perturbation is a hard exponential (by the number of sensors) problem. As the cross correlation function is not invertible and even calculating it takes O(n 2 ) The problem is analogous to a continuous traveling salesman problem and might be treated with similar tools i.e.: Simulated annealing Genetic algorithms Neural/Elastic networks

The results of these algorithms are quite good as seen in the picture We must not forget however that most aperture synthesis interferometers use the rotation of the Earth to increase the number of different baselines included in an observation.

Different shapes under Earth’s rotation

According to this model the SMA observatory was constructed The spiral galaxy M51. The SMA observations reveal gas and dust in the spiral structure and regions of active star formation.

References Keto, Eric, 1997, "The Shapes of Cross-Correlation Interferometers", ApJ 475, p Optical Physics. Third Edition. By S. G. Lipson and H. Lipson and D. S. Tannhauser