next up previous
Next: VLBI Up: Fundamentals of interferometry Previous: Aperture Synthesis

Connected element Interferometer







The visibility function $ V ( u , v ) $ needs to be determined for many points

in the $uv$-plane as possible. If many telescopes are available, the visibility

can be measured simultaneously for many baselines.




Connected element interferometers are essentially of two kinds :



1) linked directly by means of cables or waveguides $\Rightarrow \, \, \vert \mbox{\boldmath$B$} \vert \, < \, 40$ km .



2) radio linked $\Rightarrow \, \, \vert \mbox{\boldmath$B$} \vert \, < \, 300$ km .





Even if few telescopes are available, better $uv$ coverage can be obtained

making use of the rotation of the earth ("supersynthesis"). An east-west

interferometer of baseline length $ \vert \mbox{\boldmath$B$} \vert$ traces out an ellipse in the

$uv$-plane :



\begin{displaymath}
u \, = \, \frac{\vert \mbox{\boldmath$B$} \vert}{\lambda} \; \cos (t)
\end{displaymath}



\begin{displaymath}
v \, = \, \frac{\vert \mbox{\boldmath$B$} \vert}{\lambda} \; \sin (\delta_{0}) \; \sin (t)
\end{displaymath}



where $t$ and $\delta_{0}$ are respectively the hour angle and the declination

of the field center, and $\lambda $ is the observed wavelenght.





If several telescopes are aligned on an east-west line (Westerbork Synthesis

Telescope (NL)) several concentric ellipses are traced out simultaneously.

For $\delta_{0} \, = \, 0$ the ellipses degenerate into straight lines. Therefore north-south

interferometer arms are needed to have good $uv$ coverage also for equatorial

sources.



Due to the earth rotation, the phase difference $\phi$ between the signals at

two telescopes separated by a baseline $\mbox{\boldmath$B$}$, is rapidly changing in time:



\begin{displaymath}
\phi \, = \, 2 \pi \, \nu \; \frac{\mbox{\boldmath$B$} \cdot \mbox{\boldmath$s_{0}$}}{c}
\end{displaymath}



\begin{displaymath}
f \, \equiv \, \frac{1}{2 \pi} \, \frac{d \phi}{d t} \, = \...
... \frac{ \vert \mbox{\boldmath$B$} \vert}{\lambda} \; \mbox{Hz}
\end{displaymath}



where $f$ is called the fringe frequency and $\mbox{\boldmath$\omega$}$ is the earth angular

velocity vector.




For istance, if $ \vert \mbox{\boldmath$B$} \vert \, \approx \, 10$ km and $\lambda \, = \, 1$ cm , $f \, = \, 73$ Hz .




In order to be able to calculate the cross correlation of the signals, with

typical integration times of few seconds, this effect has to be compensated for.



The signal of one of the two telescopes is opportunely delayed so that, at the

correlator input, the phase difference $\phi ( \mbox{\boldmath$s_{0}$} ) $ between the signals propagating

in the direction $\mbox{\boldmath$s_{0}$}$ , is zero. This operation ("tracking delay") needs an

accurate geometric and atmospheric model at the correlator.


next up previous
Next: VLBI Up: Fundamentals of interferometry Previous: Aperture Synthesis
Luca Moscadelli 2004-04-20