WHICH DISTRIBUTION OF VISIBILITIES?

This section summarizes the results of Paper II .

The distribution of samples in Fourier plane that should be taken as target when optimizing an array for imaging is determined by:

• the "work" that can be demanded to the image reconstruction process (i.e. the deconvolution algorithm performance and the prior information available),
• the level of sidelobes tolerated in the clean map,
• the resolution wanted in the clean map.

The work that can be demanded to the imaging process can be characterized by 2 parameters $\alpha$ and $\beta$ representing respectively the work that can be demanded to the interpolation and to the extrapolation processes:

 

$$\alpha=\frac{\delta u_M}{\delta u_o}\, ,\,\,\,\mbox{and}\,\,\,\beta=\frac{\Delta u_e}{\Delta u_s}\, ,$$ (1)

where

• $\delta u_M$ is the sampling accuracy (exact definition in paper II) required to allow estimation of the visibility function within the sampled uv-region.
• $\delta u_o$ is the Nyquist interval: $\delta u_o=1/\Delta\theta_{P\!B}$ , where $\Delta\theta_{P\!B}$ is the angular size of the primary beam.
• $\Delta u_e$ is the radius of the sampled uv-disc.
• $\Delta u_s$ is the radius up to which the visibility function can be reasonably extrapolated.

When no prior information on the source are available$\alpha=1$ . When mosaicing is possible $\alpha=2$ .

Let us define the clean weighting function as the Fourier transform of the wanted clean beam. The level of sidelobes in the clean map is directly related to the truncation level of the clean weighting function. Let $\gamma_c$ be this truncation level and $\Gamma_c$ its value in dB. At a given wavelength specifying a resolution is equivalent to specifying a size for the array. The largest projected baseline of the array $B$ is related to the resolution in the clean map $\delta\theta_c$ by:

 

$$\delta\theta_c=\frac{2\sqrt{b\ln 2}}{\pi}\,\frac{{\Gamma_c}^......frac{\lambda}{B} \;\;\; \mbox{ with } b=\frac{\ln \!10}{10}\,.$$ (2)

Then, for an array of $n_a$ antennas of diameter $D$ and for a maximum duration of observation in hour $\Delta t_{max}$ , two characteristic baseline lengths are introduced:

 

$$B_{o}=\frac{ n_a(n_a-1)\,\alpha \,D\,\Delta t_{max}}{48\,\ch......c}\left[\exp\! \left(\frac{b\Gamma_c}{\beta^2}\right)-1\right]$$ (3)

and,

 

$$B_{max} = \frac{ n_a(n_a-1)\,\alpha \,D\,\Delta t_{max}}{48\times 1.44} \, .$$ (4)

The first baseline length corresponds to the largest array size allowing single configuration observations without loss of sensitivity and the second one to the largest array size allowing single configuration observations. For larger sizes image reconstruction is in principle not possible.

For example if we take $\alpha=\beta=1$ (i.e. no particular deconvolution algorithm and no prior information available), $\Gamma=10$ dB and $\Delta t_{max}=8$ h then we get : $B_o=1.8$ km and $B_{max}=4.9$ km for 60 antennas of 12 m. For $\alpha=2$ we get $B_{max}=9.8$ . It is therefore justified for ALMA to build an extended configuration of 10 km in diameter.

Then three cases have to be considered:

1. $B<B_o$ : the array can be optimized for a distribution of samples equal to the Fourier transform of the wanted clean beam. If the wanted clean beam is Gaussian then the FWHM, $\delta_c$ , of the target distribution is given by:
 

$$\delta_c=\frac{\beta}{a \sqrt{2b\Gamma_c}}\frac{B}{\lambda} \;\;\;\;\;\mbox{where}\,\, a=\frac{1}{2\sqrt{2\ln2}}$$ (5)

The duration of the observation required is then given by:
 

$$\Delta t=\frac{48\,\chi\, B}{n_a(n_a-1)\,\alpha \,D} \, .$$ (6)

2. $B_o<B<B_{max}$ : the duration of observation is maximum. To allow the uv-disc to be well sampled everywhere and to minimize the loss of sensitivity the array should be optimized for a wider Gaussian distribution of samples (for a higher truncation level). The FWHM, $\delta_s$ , of the target distribution of samples is solution of:
 

$$\frac{2 a^2 \lambda^2 \delta_s^2}{B^2}\left[\exp\!\left(\fra......mbda^2 \delta_s^2}\right)-1\right]-{\frac{B}{1.44\,B_{max}}}=0$$ (7)

3. $B>B_{max}$ : the array should be optimized for multiconfiguration observations.

 

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