|
Telescope separations and orientations on the ground are represented by the (p-q) plane and the plane of the sky by the (u-v) plane. The two co-ordinates p and q are two perpendicular components of the vector (called baseline vector) from one dish to the other in a telescope pair (see diagram). A multiplication of the voltages from the two telescopes gives a quantity called the complex degree of coherence, m(p,q). The function m(p,q), the complex degree of coherence, is a measure of the relation (or coherence) between the amplitudes and phases at the two telescopes. It is a function of p and q. The Van Cittert-Zernike Theorem: This theorem forms the crucial link in the problem of radio mapping. We shall first state it formally and then explore its interpretation. The van CittertZernike Theorem states that the complex degree of coherence m(p,q) is just the normalised Fourier Transform of the Intensity function I(u,v) of the source.
Mathematically,
m(p,q) = òò I(u,v) eik(pu+qv) dudv òò I(u,v)dudv
I(u,v) µ òò m(p,q) e-ik(pu+qv) dpdq.
This means that to find the radio image, i.e. to determine I(u,v) we must measure m(p,q) for a large no. of different telescope-pair separations and orientations (baselines).The right hand side involves an integration over the (p,q) plane (the ground), so the points on the (p,q) plane (the telescopes) must be spread widely over the plane. The relationship between the observed m(p,q) and the desired radio image, represented by I(u,v) is shown below:
Fourier Transform The process of getting the radio image from the values of m(p,q) using the van CittertZernike Theorem can be shown.
Inverse Fourier Transform To implement our technique, we need to build a large number of telescope pairs with different baseline vectors (see diagram below), out of a finite number of radio telescopes (dishes) at our disposal. An efficient method is to place the radio telescopes in an array configuration over the ground.
Copyright © 2022 ICICI Centre for Mathematical Sciences |
