Aperture Synthesis Lab

What is modelled

Each pair of antennas measures one Fourier component of the sky brightness, the visibility, at a spatial frequency (u, v) given by the baseline between them in units of wavelengths. As the Earth turns, the baseline seen from the source turns too, so one pair traces an arc and the uv plane fills in. The inverse Fourier sum of the sampled visibilities is the dirty image, which is the true sky convolved with the dirty beam.

Equations

• Number of baselines N (N − 1) / 2, each also sampled at (−u, −v).
• Projected baseline u = (X sin H + Y cos H) / λ, v = (−X sinδ cos H + Y sinδ sin H + Z cosδ) / λ.
• Dirty beam B(l, m) = (1/Nuv) Σ cos[2π(u l + v m)], dirty image I = Σ Re[V e2πi(ul+vm)] / Nuv.
• Resolution about λ / Bmax; a filled line of baselines gives FWHM 0.6034 λ / Bmax.

Checks run in the page

• Two antennas: the first null of the fringe pattern sits at λ/(4b).
• Filled uv coverage: the beam width equals 0.6034 λ/B.
• Baseline and uv sample counting, and the highest sidelobe.
• The east-west ellipse and the invariance of the baseline length under Earth rotation.

Honest simplifications

The array is at 34 degrees north, the antennas are identical point receivers, there is no noise, no bandwidth smearing, no w-term correction and no deconvolution (CLEAN), and the sources are two points with fixed flux. It is a teaching sketch of the Fourier relation, not a calibration pipeline.

• Press Snapshot, east-west line, then Earth rotation, 6 hours: the line of points becomes ellipses and the beam rounds out.
• Press Unresolved pair and compare with Y array snapshot.
• Raise the wavelength and see the beam widen in proportion.