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# Field of View

- Contents

### Primary Beam

The ultimate factor limiting the field of view is the diffraction-limited response of the individual antennas. An approximate formula for the full width at half power in arcminutes is θ_{PB} = 42/ν_{GHz }for frequencies between 1 and 50 GHz (L- through Q-band). At P-band the approximate value is θ_{PB} = 50/ν_{GHz}. New precise measurements of the primary beam shape have been reported in EVLA Memo 195; these allow for the correction of the primary beam attenuation in wide-field images. Both AIPS and CASA (5.0 and later versions) have these new parameters incorporated.

With the wide-bandwidths of the VLA it is necessary to account for the variation of the primary beam with frequency in order to achieve high-dynamic range images. For this and other imaging details we refer to the Limitations on Imaging Performance section of the OSS.

Objects larger than approximately half this angle cannot be directly observed by the array. However, a technique known as mosaicking, in which many different pointings are taken, can be used to construct images of larger fields. Refer to References 1 and 2 in Documentation for details.

Guidelines for mosaicking with the VLA are given in the Guide to Observing with the VLA.

### Chromatic Aberration (Bandwidth Smearing)

The principles upon which synthesis imaging are based are strictly valid only for monochromatic radiation. When visibilities from a finite bandwidth are gridded as if monochromatic, aberrations in the image will result. These take the form of radial smearing which worsens with increased distance from the delay-tracking center. The peak response to a point source simultaneously declines in a way that keeps the integrated flux density constant. The net effect is a radial degradation in the resolution and sensitivity of the array.

These effects can be parameterized by the product of the fractional bandwidth (Δν/ν_{0}) with the source offset in synthesized beamwidths (θ_{0}/θ_{HPBW}). Table 3.5.1 shows the decrease in peak response and the increase in apparent radial width as a function of this parameter and should be used to determine how much spectral averaging can be tolerated when imaging a particular field.

(Δν/ν_{0})*(θ_{0}/θ_{HPBW}) | Peak | Width | ||
---|---|---|---|---|

0.0 | 1.00 | 1.00 | ||

0.50 | 0.95 | 1.05 | ||

0.75 | 0.90 | 1.11 | ||

1.0 | 0.80 | 1.25 | ||

2.0 | 0.50 | 2.00 |

Note: The reduction in peak response and increase in width of an object due to bandwidth smearing (chromatic aberration). Δν/ν0 is the fractional bandwidth; θ_{0}/θ_{HPBW} is the source offset from the phase tracking center in units of the synthesized beam.

### Time-Averaging Loss

The sampled coherence function (visibility) for objects not located at the phase-tracking center is slowly time-variable due to the motion of the source through the interferometer coherence pattern, so that averaging the samples in time will cause a loss of amplitude. Unlike the bandwidth loss effect described above, the losses due to time averaging cannot be simply parametrized, except for observations at δ = 90^{°}. In this case, the effects are identical to the bandwidth effect except they operate in the azimuthal, rather than the radial, direction. The functional dependence is the same as for chromatic aberration with Δν/ν_{0} replaced by ω_{e}Δt_{int}, where ω_{e} is the Earth's angular rotation rate, and Δt_{int} is the averaging interval.

For other declinations, the effects are more complicated and approximate methods of analysis must be employed. Chapter 13 of Reference 1 in Documentation considers the average reduction in image amplitude due to finite time averaging. The results are summarized in Table 3.5.2, showing the time averaging in seconds which results in 1%, 5% and 10% loss in the amplitude of a point source located at the first null of the primary beam. These results can be extended to objects at other distances from the phase tracking center by noting that the loss in amplitude scales with (θΔt_{int})^{2}, where θ is the distance from the phase center and Δt_{int} is the averaging time. We recommend that observers reduce the effect of time-average smearing by using integration times as short as 1 or 2 seconds (also see the section on Time Resolution and Data Rates) in the **A** and **B** configurations.

Amplitude loss | ||||||
---|---|---|---|---|---|---|

Configuration | 1.0% | 5.0% | 10.0% | |||

A |
2.1 | 4.8 | 6.7 | |||

B |
6.8 | 15.0 | 21.0 | |||

C |
21.0 | 48.0 | 67.0 | |||

D |
68.0 | 150.0 | 210.0 |

Note: The averaging time (in seconds) results in the listed amplitude losses for a point source at the antenna first null. Multiply the tabulated averaging times by 2.4 to get the amplitude loss at the half-power point of the primary beam. Divide the tabulated values by 4 if interested in the amplitude loss at the first null for the longest baselines.

**Note:** For both the chromatic aberration and the time-averaging loss, the issue is not a simple reduction in amplitude for sources far from the phase center, but a convolution to the extent that a point source far from the phase center will become resolved due to bandwidth and/or time smearing. Furthermore, the description given above for the bandwidth smearing is based on the assumption that the radiation is monochromatic to parameterize the smearing, and does not take into account the consequences of having wide-bandwidths as is the case for the VLA. Therefore, while proposing and planning for VLA observations, and depending on the objectives of the science and the location of the sources of interest within the field, including confusing sources which may be far outside the science field, the above noted guidelines need to be used to conservatively estimate the proper channel width and the correlator integration time in order to minimize the effects of the bandwidth smearing and the time smearing, respectively.

### Non-Coplanar Baselines

The procedures by which nearly all images are made in Fourier synthesis imaging are based on the assumption that all the coherence measurements are made in a plane. This is strictly true for E-W interferometers, but is false for the VLA with the single exception of snapshots. Analysis of the problem shows that the errors associated with the assumption of a planar array increase quadratically with angle from the phase-tracking center. Serious errors result if the product of the angular offset in radians times the angular offset in synthesized beams exceeds unity: θ > λB/D^{2}, where B is the baseline length, D is the antenna diameter, and λ is the wavelength, all in the same units. This effect is most noticeable at 90 cm and 20 cm in the larger configurations, but will be notable in wide-field, high fidelity imaging for other bands and configurations.

Solutions to the problem of imaging wide-field data taken with non-coplanar arrays are well known, and have been implemented in AIPS task *IMAGR* and CASA task *clean*. Refer to the package help files for these tasks, or consult with the NRAO Helpdesk for advice. More computationally efficient imaging with non-coplanar baselines is being investigated, such as the W-projection method available in CASA (see EVLA Memo 67 for more details).