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The
coverage
Using a Cartesian coordinate system
with
towards the pole,
towards the meridian, and
towards East, the
conversion matrix to
is
![$\displaystyle \left( \begin{array}{c} u \ v \ w \end{array} \right) = \frac...
...) \end{array} \right) \left( \begin{array}{c} X \ Y \ Z \end{array} \right)$](img414.png) |
(2.46) |
where
are the hour angle and declination of the phase tracking
center.
Eliminating
from Eq.2.46 gives the equation of an ellipse:
![$\displaystyle u^2 + \left( \frac{v-(Z/\lambda) \cos(\delta)}{\sin(\delta)} \right)^2 = \frac{X^2+Y^2}{\lambda^2}$](img417.png) |
(2.47) |
The
coverage is an ensemble of such ellipses. The choice of antenna
configurations is made to cover the
plane as much as possible.
Baseline measurement
Assume there is a small baseline error,
(
). The phase error is
Hence, if we observe
sources, we have for each source
![$\displaystyle \phi_k = \phi_0 + \cos(\delta_k) \cos(h_k) \Delta X - \cos(\delta_k) \sin(h_k) \Delta Y + \sin(\delta_k) \Delta Z$](img422.png) |
(2.50) |
i.e. a linear system in (
), with
equations
and 4 unknown (including the arbitrary phase
). This can be used
to determine the baselines from phases measured on a set of sources with
known positions
.
From the shape of Eq.2.49, one can see that the
determination of
requires large variations in
, preferably at declination
, while that of
requires large variations in
. However,
in Eq.2.50 is multi-valued (the
ambiguity...). Retaining the function in the [
] interval
only, the system to solve is in fact
![$\displaystyle mod( \phi_0 + \cos(\delta_k) \cos(h_k) \Delta X - \cos(\delta_k) \sin(h_k) \Delta Y + \sin(\delta_k) \Delta Z - \phi_k + \pi, 2 \pi) -\pi = 0$](img431.png) |
(2.51) |
which is a linear system of equations only if
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
phase steps) around the most likely values for
.
Next: 3. Millimetre Very Long
Up: 2. Millimetre Interferometers
Previous: 2.5 Fourier Transform and
  Contents
Anne Dutrey