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• Outline
• Glossary
• 7. Observing Systems
  • Previous: 7.4 Digital Correlators
  • Next: 7.6 Polarization and Antenna Feeds

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Import standard modules:

Import section specific modules:

TODO:

• why is it important to understand the primary beam
• a direction-dependent effect
• present the E jones
• what is intrinsic and apparent flux
• example: sky field with and without beam attenuation
• why is it a plane wave? source is in the far field, define far-field distance, define far/near field
• consider a single dish
• optics intution of a primary beam: reflecting a plane wave to a focus using a simple parabolic dish
• simple aperture of a parabolic dish is a disc -> bessel beam pattern ; fourier relation
• voltage beam, power beam ; coordinate system
• primary lobe, sidelobes, FWHM/resolution, directivity, gain
• adding complexity: aperture blockage, secondary reflector
• aperture efficiency: surface accuracy, blockage, taper, spillover
• typical types of dishes: primary focus, cassegrain, offset gregorian (with examples)
• antenna mounts
• parallactic angle
• example: primary beam of kat-7 (beam pattern, freq dependence, spatial slice, freq slice)
• example: primary beam of meerkat (beam pattern, freq dependence, spatial slice, freq slice)
• extra: pointing accuracy, jitter, deformation
• the relation of a primary beam of a single dish to the PSF of an interferometric array

• how is the primary beam used in calibration and imaging?

• https://www.cv.nrao.edu/course/astr534/ReflectorAntennas.html

• https://www.cv.nrao.edu/course/astr534/RadioTelescopes.html
• white book chapter 3
• malloux antenna theory

7.5 The Primary Beam (E- and P-Jones)

The primary beam of an antenna (also known as the radiation pattern) is the directional dependence of the gain of the antenna.

The primary beam of the antenna is the most important direction-dependent propagation effect.

It has a multiplicative effect in the image plane, and a convolutional effect in the visibility plane, due to the Fourier Transform relationship between the image and visibility planes.

Figure 7.5.1: Schematic diagram of an antenna radiation pattern (Image taken from https://commons.wikimedia.org/wiki/File:Sidelobes_en.svg).

Example: Primary beam of the JVLA (Jansky Very Large Array)

Figure 7.5.2: Primary beam of the JVLA at 1.41 GHz.

For a JVLA antenna, the primary beam gain varies with direction at a given frequency (see Figures 7.5.1 and 7.5.2), and with frequency towards a given direction (see Figures 7.5.3 and 7.5.4; the beam pattern scales with frequency, becoming more compact at higher frequencies).

Figure 7.5.3: Gain across a horizontal cross section through the centre of the beam pattern shown in Figure 7.5.2. The peak at the centre corresponds to the mainlobe, with the first null and the first sidelobe on either side.

Figure 7.5.4: Variation of the beam pattern for a JVLA antenna over the frequency range 1.3 $-$ 1.6 GHz. As the frequency increases, the beam pattern becomes more compact.

Figure 7.5.5: Variation of the beam gain with frequency at the position of the source marked by a black dot in Figure 7.5.4.

A JVLA antenna, which has an alt-azimuth mount, rotates relative to the sky during the course of an observation.

Figure 7.5.6: Relative rotation of the primary beam pattern wrt the sky during the course of an observation. The blue and green sources experience different beam gains at different times, while the beam gain for the red source at the phase centre remains unchanged.

The rotation of the primary beam causes the beam gain in a given direction to vary with time.

Figure 7.5.7: Rotation of the beam pattern in Figure 7.5.2 during the course of an observation.

Figure 7.5.8: Variation of the beam gain as a function of the hour angle at the position of the source marked by a black dot in Figure 7.5.7.

7.7 Antenna mounts and parallactic angle

TODO:

• P-Jones
• fold into primary beam section
• main point: different polarization calibration issues depending on mount
• introduce transit arrays here?

7.7.1 Antenna mounts

Antenna mounts can be of two types:

1. Alt-azimuth mount
2. Equatorial mount

An antenna with an alt-azimuth mount tracks a source in the sky by rotating along two axes - altitude (vertical) and azimuth (vertical).

An antenna with an equatorial mount tracks a source by rotating about the polar axis (i.e., an axis which points towards the celestial pole).

Figure 7.7.1: Alt-azimuth and equatorial mounts illustrated for the case of an optical telescope (Image taken from http://www.oasi.org.uk/Telescopes/CFTOB/Mounts.gif).

For an antenna with an alt-azimuth mount, the antenna primary beam rotates relative to the sky during the course of an observation. For an equatorially-mounted antenna, the relative orientation of the antenna primary beam and the sky remains unchanged throughout the course of an observation.

Figure 7.7.2: Relative rotation of the beam and the sky during the course of an observation, for an antenna with an alt-azimuth mount.

Figure 7.7.3: Relative rotation of the beam and the sky during the course of an observation, for an antenna with an equatorial mount.

7.7.2 Parallactic angle

The parallactic angle is the spherical angle between two great circles on the celestial sphere - one passing through the source and the zenith, and the other passing through the source and the north celestial pole.

Figure 7.7.4: Parallactic angle

The parallactic angle is given by

$$ q = \arctan \bigg( \frac{\sin(HA)}{\tan(L_a)\cos(\delta)-\sin(\delta)\cos(HA)} \bigg) $$

where

$q$ is the parallactic angle,

$HA$ is the hour angle of the source,

$\delta$ is the declination of the source,

$L_a$ is the latitude of the observer.

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• Next: 7.6 Polarization and Antenna Feeds