4 Examples

The tables below show the fields of view resulting from various combinations of the parameters in eqs.(2) & (3). Remember, all these numbers are ultimately based on the formulae from Wrobel (1995), so all correspond to a field of view that has no more than a $10\%$ decrease in the response to a point source. If your needs are different, you'll have to adjust the tabulated fields of view accordingly (e.g., see Fig.13-1 in Bridle & Schwab (1989) for a graph of the behavior of the peak response loss due to bandwidth smearing). Here, we consider:

$BW_{\rm SB}$
= 16MHz, 8MHz, 2MHz -- the largest possible subband bandwidth for sensitivity in faint-source surveys; other more commonly-used bandwidths
$N_\nu$
= 2048, 512, 16 -- Max. $N_\nu$ for 8 & 16 stations; ``normal" continuum to show other extreme
$B$
= 2000km, 8000km -- EVN baselines in Europe; Ef to Sh or VLA
$t_{\rm int}$
= 1s, 0.25s -- the previous & the current minimum $t_{\rm int}$.
$D$
= 25m, 100m -- a VLBA dish; Ef

Ttable 2 lists fields of view in arcseconds (via eq.2). The two columns following the configuration description are for $B=2000$km and 8000km, respectively. This helps with determining whether a single correlation pass could map a large enough solid angle on the sky to include all your sources of interest in a given pointing, or whether you'd need multiple correlation passes to get to all your sources (requests for multiple passes should in principle be cleared through the EVN Program Committee during the proposal-review process).


Table 2: Fields of View in arcseconds for some representative configurations of observation/correlation parameters.
Bandwidth smearing      
$BW_{\rm SB}$ $N_\nu$ $B=2000$km $B=8000$km
16MHz 2048 $3168.\!''00$ $792.\!''00$
16MHz 512 $792.\!''00$ $198.\!''00$
16MHz 16 $24.\!''75$ $6.\!''19$
8MHz 2048 $6336.\!''00$ $1584.\!''00$
8MHz 512 $1584.\!''00$ $396.\!''00$
8MHz 16 $49.\!''50$ $12.\!''38$
2MHz 2048 $25344.\!''00$ $6336.\!''00$
2MHz 512 $6336.\!''00$ $1584.\!''00$
2MHz 16 $198.\!''00$ $49.\!''50$
     
Time smearing      
$\lambda$ $t_{\rm int}$ $B=2000$km $B=8000$km
18.0cm 1.00s $166.\!''50$ $41.\!''62$
18.0cm 0.25s $666.\!''00$ $166.\!''50$
6.0cm 1.00s $55.\!''50$ $13.\!''88$
6.0cm 0.25s $222.\!''00$ $55.\!''50$
1.3cm 1.00s $12.\!''02$ $3.\!''01$
1.3cm 0.25s $48.\!''10$ $12.\!''02$



Table 3: Fields of View in terms of single-dish beams for some representative configurations of observation/correlation parameters.
Bandwidth smearing
$\lambda$ $BW_{\rm SB}$ $N_\nu$ $B$ $D=25$m $D=100$m
18.0cm 16MHz 2048 2000km 2.13 8.53
18.0cm 16MHz 512 2000km 0.53 2.13
18.0cm 16MHz 16 2000km 0.02 0.07
18.0cm 8MHz 2048 2000km 4.27 17.07
18.0cm 8MHz 512 2000km 1.07 4.27
18.0cm 8MHz 16 2000km 0.03 0.13
18.0cm 2MHz 2048 2000km 17.07 68.27
18.0cm 2MHz 512 2000km 4.27 17.07
18.0cm 2MHz 16 2000km 0.13 0.53
18.0cm 16MHz 2048 8000km 0.53 2.13
18.0cm 16MHz 512 8000km 0.13 0.53
18.0cm 16MHz 16 8000km 0.00 0.02
18.0cm 8MHz 2048 8000km 1.07 4.27
18.0cm 8MHz 512 8000km 0.27 1.07
18.0cm 8MHz 16 8000km 0.01 0.03
18.0cm 2MHz 2048 8000km 4.27 17.07
18.0cm 2MHz 512 8000km 1.07 4.27
18.0cm 2MHz 16 8000km 0.03 0.13
         
6.0cm 16MHz 2048 2000km 6.40 25.60
6.0cm 16MHz 512 2000km 1.60 6.40
6.0cm 16MHz 16 2000km 0.05 0.20
6.0cm 8MHz 2048 2000km 12.80 51.20
6.0cm 8MHz 512 2000km 3.20 12.80
6.0cm 8MHz 16 2000km 0.10 0.40
6.0cm 2MHz 2048 2000km 51.20 204.80
6.0cm 2MHz 512 2000km 12.80 51.20
6.0cm 2MHz 16 2000km 0.40 1.60
6.0cm 16MHz 2048 8000km 1.60 6.40
6.0cm 16MHz 512 8000km 0.40 1.60
6.0cm 16MHz 16 8000km 0.01 0.05
6.0cm 8MHz 2048 8000km 3.20 12.80
6.0cm 8MHz 512 8000km 0.80 3.20
6.0cm 8MHz 16 8000km 0.02 0.10
6.0cm 2MHz 2048 8000km 12.80 51.20
6.0cm 2MHz 512 8000km 3.20 12.80
6.0cm 2MHz 16 8000km 0.10 0.40
         
1.3cm 16MHz 2048 2000km 29.54 118.15
1.3cm 16MHz 512 2000km 7.38 29.54
1.3cm 16MHz 16 2000km 0.23 0.92
1.3cm 8MHz 2048 2000km 59.08 236.31
1.3cm 8MHz 512 2000km 14.77 59.08
1.3cm 8MHz 16 2000km 0.46 1.85
1.3cm 2MHz 2048 2000km 236.31 945.23
1.3cm 2MHz 512 2000km 59.08 236.31
1.3cm 2MHz 16 2000km 1.85 7.38
1.3cm 16MHz 2048 8000km 7.38 29.54
1.3cm 16MHz 512 8000km 1.85 7.38
1.3cm 16MHz 16 8000km 0.06 0.23
1.3cm 8MHz 2048 8000km 14.77 59.08
1.3cm 8MHz 512 8000km 3.69 14.77
1.3cm 8MHz 16 8000km 0.12 0.46
1.3cm 2MHz 2048 8000km 59.08 236.31
1.3cm 2MHz 512 8000km 14.77 59.08
1.3cm 2MHz 16 8000km 0.46 1.85
Time smearing
$t_{\rm int}$ $B$ $D=25$m $D=100$m
1.00s 2000km 0.11 0.45
1.00s 8000km 0.03 0.11
0.25s 2000km 0.45 1.80
0.25s 8000km 0.11 0.45


Table 3 lists fields of view as a fraction of a single-dish beam (via eq.3). The two columns following the configuration description are for $D=25$m and 100m, respectively. This helps to determine whether you can recover, with a single correlation pass, all the sources for which the individual antennas in your array have themselves been able to observe during the pointing. On the other hand, if a given configuration provides a field of view considerably greater than the single-dish beam of the smallest telescope in your array, then there's clearly not much point in using such a large $N_\nu$ and/or small $t_{\rm int}$. If the bandwidth smearing is not a meaningful constraint, you could reduce $N_\nu$, possibly ``trading" this for more subbands to get higher sensitivity. If the time smearing is not a meaningful constraint, reducing $t_{\rm int}$ would in turn reduce the final data size. However, it turns out for our current situation that time smearing will most likely be your limiting factor in trying to map full single-dish beams, as we'll illustrate presently.

The tie between the bandwidth- and time-smearing portions of tables 2 & 3 is the set of valid correlator configurations derivable from eq.(4) -- see also table 1. The maximum $N_\nu$ limit for 8(16) stations is 2048(512) for $t_{\rm int}\ge 0.25$s. These $N_\nu$ assume only one subband and one polarization. Thus not every configuration listed in these tables will necessarily be available for your specific experiment (e.g., any $N_\nu=2048$ configuration would be ruled out for $N_{\rm sta}\ge 9$.

We can see from delving into table 3 that the time smearing is currently the limiting factor more often than is bandwidth smearing. Only for shorter-baseline arrays at the minimum $t_{\rm int}$ does the time-smearing field of view approach a full single-dish beam, whereas there are numerous configurations that provide a bandwidth-smearing field of view greater than this. The ratio of the two field of views can be easily computed from eq.(2), with $\lambda$ in cm, $BW_{\rm SB}$ in MHz, and $t_{\rm int}$ in seconds:

\begin{displaymath}
\frac{\mathrm{Bandwidth} FoV}{\mathrm{Time} FoV} \quad \simeq \quad
2.67 \frac{N_\nu t_{\rm int}}{BW_{\rm SB}\lambda}.
\end{displaymath} (7)

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