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2.6 Array Geometry & Baseline Measurements



The $ uv$ coverage


Using a Cartesian coordinate system $ (X,Y,Z)$ with $ Z$ towards the pole, $ X$ towards the meridian, and $ Y$ towards East, the conversion matrix to $ u,v,w$ is

$\displaystyle \left( \begin{array}{c} u \ v \ w \end{array} \right) = \frac... ...) \end{array} \right) \left( \begin{array}{c} X \ Y \ Z \end{array} \right)$ (2.46)

where $ h , \delta $ are the hour angle and declination of the phase tracking center.

Eliminating $ h$ from Eq.2.46 gives the equation of an ellipse:

$\displaystyle u^2 + \left( \frac{v-(Z/\lambda) \cos(\delta)}{\sin(\delta)} \right)^2 = \frac{X^2+Y^2}{\lambda^2}$ (2.47)

The $ uv$ coverage is an ensemble of such ellipses. The choice of antenna configurations is made to cover the $ uv$ plane as much as possible.



Baseline measurement


Assume there is a small baseline error, ( $ \Delta X, \Delta Y, \Delta Z$). The phase error is

$\displaystyle \Delta \phi$ $\displaystyle =$ $\displaystyle \frac{2 \pi}{\lambda} \Delta \ensuremath{\text{\boldmath$b$\unboldmath }}.\ensuremath{\text{\boldmath$s$\unboldmath }}_0$ (2.48)
  $\displaystyle =$ $\displaystyle \cos(\delta) \cos(h) \Delta X - \cos(\delta) \sin(h) \Delta Y + \sin(\delta) \Delta Z$ (2.49)

Hence, if we observe $ N$ sources, we have for each source

$\displaystyle \phi_k = \phi_0 + \cos(\delta_k) \cos(h_k) \Delta X - \cos(\delta_k) \sin(h_k) \Delta Y + \sin(\delta_k) \Delta Z$ (2.50)

i.e. a linear system in ( $ \Delta X, \Delta Y, \Delta Z$), with $ N$ equations and 4 unknown (including the arbitrary phase $ \phi_0$). This can be used to determine the baselines from phases measured on a set of sources with known positions $ h_k,\delta_k$.

From the shape of Eq.2.49, one can see that the determination of $ \Delta X, \Delta Y$ requires large variations in $ h$, preferably at declination $ \delta \sim 0$, while that of $ \Delta Z$ requires large variations in $ \delta $. However, $ \phi_k$ in Eq.2.50 is multi-valued (the $ 2 \pi$ ambiguity...). Retaining the function in the [$ -\pi,\pi$] interval only, the system to solve is in fact

$\displaystyle mod( \phi_0 + \cos(\delta_k) \cos(h_k) \Delta X - \cos(\delta_k) \sin(h_k) \Delta Y + \sin(\delta_k) \Delta Z - \phi_k + \pi, 2 \pi) -\pi = 0$ (2.51)

which is a linear system of equations only if $ \Delta X, \Delta Y, \Delta Z$ are small enough so that the shifted modulo function is the identity. Baseline determination usually proceeds through a ``brute force'' technique, by making a grid search (with $ \pi$ phase steps) around the most likely values for $ X,Y,Z$.
next up previous contents
Next: 3. Millimetre Very Long Up: 2. Millimetre Interferometers Previous: 2.5 Fourier Transform and   Contents
Anne Dutrey