====== Array radio telescopes ======

Array Radio Telescopes (ARTs) are powerful instruments that combine multiple antennas to act together as a single, enormous [[telescope]].  
By measuring and correlating the signals received at different antennas, they achieve far higher angular resolution and sensitivity than any single dish could provide.

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An array radio telescope collects radio waves from astronomical sources—galaxies, pulsars, or the cosmic background—and converts them into electrical voltages.  
Each antenna has two perpendicular dipoles that measure the X and Y components of the incoming electric field. These voltages are stored as complex numbers to capture both **amplitude** and **phase**, which together contain the full information of the electromagnetic wave.

The voltage outputs from two antennas are then **cross-correlated** to form a visibility \(V_{pq}\).  
For an array of \(N\) antennas, there are \(N(N-1)/2\) unique antenna pairs or **baselines**, each sampling one Fourier component of the sky brightness distribution.

The fundamental measurement equation of an array radio telescope—the **Radio Interferometric Measurement Equation (RIME)**—is:

$$
\mathbf{V}_{pq}(u,v) = \iint \mathbf{B}_p\, \mathbf{E}(l,m)\, e^{-2\pi i(ul + vm)}\, \mathbf{B}_q^{\mathrm{H}} \, dl\, dm
$$

Here:
  * \(u,v\) are the projected baseline coordinates (in wavelengths) on Earth’s surface  
  * \(l,m\) are direction cosines on the sky  
  * \(\mathbf{E}(l,m)\) is the brightness matrix of the source  
  * \(\mathbf{B}\) is the primary beam or directional sensitivity of each antenna  

The visibility \(V_{pq}\) is the Fourier transform of the sky brightness distribution, meaning that an interferometer does not form a direct image but samples the Fourier plane of the sky.  
When the visibilities are calibrated and inverse-Fourier transformed, they reconstruct the radio image.

===== Resolution and Sensitivity =====

The resolution of a telescope describes how closely two points on the sky can be distinguished.  
For a single dish of diameter \(D\) observing at wavelength \(\lambda\),

$$
\alpha = 1.22 \frac{\lambda}{D}
$$

where \(\alpha\) is the angular radius of the central Airy disk.  
In an array, this resolution depends instead on the longest baseline \(b_{\max}\):

$$
\alpha \approx \frac{\lambda}{b_{\max}}
$$

Thus, the greater the separation between antennas, the finer the resolution.  
For example, the LOFAR array in Europe achieves sub-arcsecond precision using baselines up to 1000 km.

The sensitivity of a radio telescope, measured in janskys (Jy), represents the faintest flux detectable above noise.  
\(1~\text{Jy} = 10^{-26}~\text{W m}^{-2}\text{ Hz}^{-1}\).  
Modern instruments like MeerKAT can detect signals of a few microjanskys.

===== uv-Plane and Synthesis Imaging =====

Each antenna pair defines a baseline vector \((u,v)\), corresponding to one point in the Fourier domain.  
As the Earth rotates, the projected baselines trace elliptical tracks—filling the uv-plane.  
The pattern of sampled points is the uv-coverage, which determines the quality of the reconstructed image.  
The inverse Fourier transform of the uv-coverage defines the synthesized beam (the array’s point-spread function), while the field of view is set by the primary beam of a single antenna.

For example, at 21 cm wavelength:  
  * A 45 m GMRT antenna gives a field of view of about 16 arcminutes  
  * A 25 km baseline gives a synthesized beam of about 1.7 arcseconds  

Hence, the array simultaneously achieves a wide field of view and high angular resolution.

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