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4. Phase correction and resolving power


It has been shown that it is possible to predict the residual uncorrected phase variance after various calibration schemes have been applied. But what impact does this phase variance really have? Perhaps the most direct approach is to consider the response to a point-like object. What upper limit can be set on the size of the object? The problem is best addressed in the visibility domain, illustrated schematically in Fig. 8 (click here). Figure 8 (click here)a shows how the phase power distribution changes as the length of the baseline is increased. The phase power on short timescales is independent of the baseline length. When the peak of the distribution is on a timescale slower than tex2html_wrap_inline1736, the decorrelation in an integration becomes independent of baseline length (Fig. 8 (click here)b); when it is slower than tex2html_wrap_inline1738 the phase error is also independent of baseline length. The timescale for the peak ranges from 5 to 15 times tex2html_wrap_inline1740, for wind w blowing along and perpendicular to a projected baseline length tex2html_wrap_inline1744, respectively (Paper I). For tex2html_wrap_inline1746 s and w=5 m stex2html_wrap_inline1750 the residual phase power becomes constant for baselines exceeding 3.7 km if the wind blows along the baseline, and 1 km if the wind blows perpendicular. These can be reduced through decreasing tex2html_wrap_inline1752.

The scatter in the decorrelation also increases with baseline. The lower amplitudes measured on longer baselines can be corrected by dividing by the corresponding amplitude measured on the calibrator, but the uncertainty in the amplitude is increased, as shown in Fig. 8 (click here)c, and there is no fundamental gain in signal-to-noise. If radiometry is used to correct for the on-source fluctuations only, then the amplitude measured in each integration is restored to its correct value (Fig. 8 (click here)b) but the uncorrected phase errors reduce the amplitude obtained when a number of complex visibilities are averaged together (Fig. 8 (click here)d).

Figure 9: Model prediction for the overall coherence as a function of baseline length, for a frequency of 230 GHz under reasonable conditions (see main text for details). The crosses are the expected values for phase referencing only, and circles also include a radiometric correction for on-source fluctuations. The curves are Gaussians, and represent the visiblity curves that would be obtained under perfect conditions from circular Gaussian sources on the sky with FWHM of 0.6'' and 0.3''. These are a measure of the effective seeing. A full radiometric correction restores perfect seeing

Figure 9 (click here) shows a specific calculated example of the average coherence obtained over many integrations as a function of baseline length. This example is for a 5 m stex2html_wrap_inline1758 wind blowing a layer of turbulence 1 km thick in a direction perpendicular to the baseline. The strength of the turbulence corresponds to reasonable conditions in the Owens Valley (tex2html_wrap_inline1760 rms phase at 230 GHz measured on a 100 m baseline over a 5 minute interval) and the observing frequency is 230 GHz. The integration time tex2html_wrap_inline1762 is 300 s and the cycle time tex2html_wrap_inline1764 is 1500 s, the same as the values used for Fig. 4 and Fig. 7 (click here).

The total decorrelation with no radiometric correction, indicated by the crosses in Fig. 9 (click here), is the result of phase fluctuations on the target with tex2html_wrap_inline1766 and those on the calibrator with tex2html_wrap_inline1768 that are introduced by phase referencing (see Fig. 4). The solid Gaussian curve in Fig. 9 (click here) is the visibility curve that would be obtained under perfect conditions for a circular Gaussian source on the sky that has a Full-Width-to-Half-Maximum of 0.6''. This figure is therefore a measure of the effective "seeing" - the amount of smearing of the sky brightness distribution due to the atmosphere - when there is no radiometric correction.

The circles show the behavior to be expected if radiometry is used to correct the fluctuations during each on-source period only, with the reduced coherence resulting from just the aliased calibrator component of phase fluctuations, as shown in Fig. 7 (click here). The dashed curve represents the visibility obtained for a Gaussian source on the sky with FWHM of 0.3'' (note that a wide curve in this visibility plot corresponds to a narrow distribution on the sky).

The effective seeing is clearly improved by this limited radiometric correction, but obtaining the highest possible spatial resolution on faint objects requires the full correction (dotted line in Fig. 9 (click here)), which in turn requires very precise calibration of the radiometers.

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