Radio Interferometry and Aperture Synthesis

• Phil gave a detailed picture of homodyne interferometry • Have to combine the light beams physically for interference • Imposes many stringent conditions on the instrument

• Heterodyne receivers retain phase information about the photons in the IF signal • Interference can be achieved using just the electronic signals • It can even be achieved by recording the signals and combining them later in some different place

• We ended the discussion of radio telescopes by pointing out that it was impossible to achieve high resolution images by making bigger telescopes • The ease and flexibility of interferometry solve this problem

Here is a radio interferometer: There are two telescopes separated a distance B. Their outputs are combined in the correlator. If the signal voltages are VS1 and VS2 and the noise voltages are VR1 and VR2, then the correlator output is The simplification is possible because the noise terms average to zero. If we expand the extreme right hand expression and simplify, we find

=

RC is defined as the cosine response. As θ varies, the correlated signal has a sinusoidal behavior on the sky, i.e., the response is in the form of fringes. However, the response is also subject to the diffraction-limited beam of the individual telescopes, which defines the “primary beam” of the interferometer. If we define a “beam” as half the fringe period, its diameter is

The situation we have just described is shown in (a). If we add more telescopes, their outputs can be combined to yield N(N-1)/2 baselines. Thus, the field patterns (b) and (d) yield rapidly improving images (c) and (e). Remember that these sharper images are just in the direction along the interferometer; perpendicular to this direction, the image is just the primary beam of the telescopes. That is, the image is long and thin.

To make a better image we need to rotate the interferometer. We think this in terms of the uv plane, defined as a coordinate system perpendicular to the direction toward the source. As the interferometer is rotated (a), it yields a corresponding a set of baselines extending in all directions around the origin of the UV plane (b). By combining the results from all these baselines, we can synthesize a high-resolution beam. But how are we going to rotate a row of heavy, large radio telescopes?

We let the earth do the work. Of course, this means that the rotation is in only one specific pattern, so typically the uv plane coverage will not be in the circles we would like, but will be in an ellipse, and the ellipse will be interrupted where the earth blocks the view of the source.

The uv plane coverage for a linear interferometer is far from ideal. Well, at the celestial pole, it is the circle we would like, but at the equator it collapses to a line and close to the equator it is not much better. As a result, the synthesized beam for a source near the equator is just the width of the primary beam in one direction.

To fix this issue, the JVLA antennas are placed in a “Y” configuration (left). One observation gives the baselines indicated in the middle, and a full track on a source at the equator gives the ensemble of baselines to the right. The imaging of the interferometer can be modified by changing the spacing of its elements; note the different spacings along the arms of the Y. The Y is a popular pattern, but others have their uses, and ALMA can move the antennas to give a variety of patterns depending on the science goals.

The Y shape of the VLA gives good coverage at all declinations (except far south).

Recent upgrades make interferometers even more powerful 1.) conveying the IF signals by optical fibers (rather than, say, waveguides) greatly increases the bandwidth 2.) Receivers, particularly at high frequencies, have been improved, including features such as sideband separation 3.) and of course, ALMA

The ALMA antennas are starting to arrive. Eventually there will be > 60 of them at 5000 m elevation (very low water vapor), each with extremely high

performance mm- and submm-wave receivers (up to ten sets, most likely initially from 90 to 850 GHz. At 1mm (300 GHz) the resolution will be up to 0.02”.

The future (?): interferometric arrays of many small telescopes Current-generation interferometers have relatively few, large antennae. For example, the VLA has 27 antennae, each 25 meters in diameter. The result is that the primary beam, over which the VLA images, is relatively small (l/D for the antenna size, about 0.5 degree at 21 cm). With cheaper correlators–the electronics that link the antenna signals –it is possible to have more, smaller antennae. The Allen Telescope Array (top) is to have 350 6-meter telescopes, so 4 times the field of the VLA with about the same collecting area. The Square Kilometer Array (not funded) is planned to have thousands of antennae. The issue is that the number of correlators goes as N(N-1), where N is the number of antennae, so the SKA would be prohibitively expensive with today’s technology.

And the ultimate resolution through combination of signals from telescopes around the world in Very Large Baseline Interferometry (VLBI). Of course, even greater

resolution would be possible with an antenna in orbit or on the moon.

However, we have to remember the fundamental limitations of interferometers: missing low spatial frequencies and image artifacts. To start with low spatial frequencies, for example the best JVLA images of M33 are missing 85% of the total flux from the galaxy (around 1 GHz). Here is the VLA image at 6 cm (~5 GHz):

Here, to the left, is a filled dish image with the Effelsberg 100m telescope (~ 2 GHz). The image to the right is at 160 microns. It does not match the VLA image at all but corresponds closely to the Effelsberg one. The far IR appears to arise from the same regions as the low surface brightness extended emission that is invisible to the VLA.

Image artifacts arise because the uv plane is not covered in full. The gaps in coverage produce structure in the image of a point source that departs significantly from the ideal Airy pattern:

This image is a “snapshot” obtained in a single observation without the benefit of extended tracking and earth rotation. The sidelobes reach 20% of the peak response. Full track observations improve on this PSF, but there are still significant artifacts. These untreated images are called “dirty”

The standard way to remove the artifacts is with “CLEAN”

1.) assume that the image can be approximated by a field of point sources;

2.) locate the position of the brightest point in the dirty map;

3.) subtract a scaled version of the dirty beam from this position; the subtraction

should account for only a modest fraction of the brightness at this point;

4.) Record the position and subtracted intensity in a “CLEAN component” file;

5.) Find the brightest position in the dirty map left from the subtraction;

6.) repeats steps 3.) – 5.) until no subtraction is possible without making part of the

dirty map negative.

This doesn’t work so well for extended sources. For them, approaches based on the maximum entropy method (MEM) are used.

The above discussion assumes that the phases of the signals are completely unmodified by the atmosphere. In general, this is not correct, particularly at high frequencies. The closure phase can be used to mitigate this issue. From the diagram, and if the error phases introduced by the atmosphere are ε1, ε2, and ε3, and the phase differences from a celestial source are, e.g., Φ12 , then the observed phases, e.g. Φ12 are:

(9.24)

And in the sum of these three observed phases, the error terms cancel! By modeling the closure phases, it is possible to deduce the source structure independent of the atmospheric effects.

Self-calibration is another useful approach. If there is a point source in the primary beam that is bright enough to be detected at reasonable signal to noise in a coherence time (the time when the atmospheric effects can be taken to be constant), then corrections can be bootstrapped from the imaging of this source. Another class of problem occurs when a bright source lies outside of the official primary beam, but it is SO bright that its artifacts contaminate the observations even if they are observed at low efficiency. These artifacts are not dealt with using CLEAN because there is no point source within the image to associate them with. To deal with this issue, observations often include a quick excursion to the bright source periodically to obtain an image of it under the conditions that prevail at that time, allowing its artifacts to be removed accurately. Nonetheless, because of the constantly changing atmospheric effects, reduction of deep radio interferometric images is very challenging.