Coordinate Transformation

To compute the ($u,v,w$) co-ordinates of the antennas, the antenna locations must first be specified in a terrestrial co-ordinate system. The terrestrial coordinate system generally used to specify the position of the antennas is a right-handed Cartesian coordinate system as shown in Figure 10.2. The ($X,Y$) plane is parallel to the earth's equator with $X$ in the meridian plane and $Y$ towards east. $Z$ points towards the north celestial pole. In terms of the astronomical coordinate system ($HA,\delta$), $X=(0^h,0^o)$, $Y=(-6^h,0^o)$ and $Z=(\delta=90^o)$. If the components of $\bar D_\lambda$ are $(X_\lambda, Y_\lambda, Z_\lambda)$, then the components in the ($u,v,w$) system can be expressed as

$$\left[\matrix{ u\cr v\cr w\cr }\right]=\left[\matrix{ sin(H... ...sin(\delta)\cr }\right]\left[\matrix{ X\cr Y\cr Z\cr }\right].$$ (10.1.6)

As earth rotates, the $HA$ of the source changes continuously, generating different set of ($u,v,w$) co-ordinates for each antenna pair at each instant of time. The locus of projected antenna-spacing components $u$ and $v$ defines an ellipse with hour angle as the variable given by

$$u^2+\left({v - Z cos \delta_o \over {sin \delta_o}}\right)^2 = X^2 + Y^2,$$ (10.1.7)

where $(HA_o,\delta_o)$ defines the direction of phase center. In the $uv$-plane, this is an ellipse, referred to as the $uv$-track with $HA$ changing along the ellipse. The pattern generated by all the $uv$ points sampled by the entire array of antennas over the period of observation is referred to as the $uv$-coverage and as is clear from the above transformation matrix, is different for different $\delta$. Examples of $uv$-coverage for a few declinations for full synthesis with GMRT array are shown in Figure 10.4.

Figure 10.2: The (X,Y,Z) co-ordinate system used to specify antenna locations.

The $uv$ domain is the spatial frequency domain and $uv$-coverage represent the spatial frequencies sampled by the array. The shorter baselines ($uv$ points closer to the origin) provide the low resolution information about the source structure and are sensitive to the large scale structure of the source while the longer baselines provide the high resolution information. GMRT array configuration was designed to have roughly half the antennas in a compact ``Central Square'' to provide the shorter spacings information, which is crucial mapping extended source and large scale structures in the sky. The $uv$-coverage of the central square antennas is shown in Figure 10.5. Notice that there are no measurements for ($u=0,v=0$). $V(0,0)$ represents the total integrated flux received by the antennas and is absent in the visibility data. Effect of this on the image will be discussed later.

The astronomical coordinates depend on the line of intersection of the ecliptic and equatorial planes. The $uv$-coverage in turn depends on the position of the source in the astronomical coordinate system. Since the reference line of the this coordinate system changes because of the well known precession of the earth's rotation axis, the $uv$-coverage also becomes a function of the reference epoch for which the source position is specified. For the purpose of comparison and consistence in the literature, all source positions are specified in standard epochs (B1950 or J2000). Since each point in the $(u,v,w)$ plane measures a particular spatial frequency and this spatial frequency coverage differs from one epoch to another, it's necessary to precess the source coordinates to the current epoch (also called the ``date coordinates'') prior to observations and all processing of the visibility data for the purpose of mapping must be done with $(u,v,w)$ evaluated for the epoch of observations. Precessing the visibilities to the standard epoch prior to inverting the Eq. 10.2.10 will require specifying the real and imaginary parts of the visibility at ($u,v,w$) coordinates which are in fact not measured (since the $uv$-coverage changes with epoch) introducing errors in the mapping procedure.