The shape of a scatter-broadened image - I. Numerical simulations and physical principles

We investigate the appearance of the scatter-broadened image of a point source in the presence of strong scattering. We present numerical simulations wiih a one-dimensional scattering screen and interpret the results using a physical model of the optics of scattering. For strong scattering and bandwidths less than the scintillation dccorrelation bandwidth, it is known that there are two characteristic time-scales for flux scintillation, namely the slow refractive time-scale, tref and the fast diffractive time-scale, tdiff. Associated with these arc length-scales, rref and rdiff. We show that there are related time-scales (and length-scales) in the imaging problem as well and we demonstrate that the shape of the image depends critically on the relative magnitude of the integration time, tint, compared to these time-scales. We consider first the case when the telescope has an infinitesimally narrow bandwidth and filled UV coverage of characteristic radius σt. We assume | _ t r diff|⁠, since the scatter-broadened image is not resolved otherwise. In the regime tref ≪ tint, such a telescope will measure the ‘ensemble-average’ image, provided we assume ergodicity. There is a well-known result relating the shape of the ensemble-average image to the phase structure function in the scattering medium. The next important regime is | (_ t/r_ diff) t_ diff| ≪ tint ≪ | (r_ diff/_ t) t_ ref|⁠. In this regime, the telescope measures what we call the ‘average image’. The appearance of the average image depends on the spectrum of density fluctuations in the scattering medium. For ‘shallow’ power-law spectra the image is relatively smooth, but with characteristic low-level substructure, whereas for ‘steep’ spectra the image is strongly fragmented with a fractal morphology. The regime corresponding to the average image does not appear to have been explored before. Finally, the regime, |t_ int t_ diff|⁠, leads to the ‘snapshot image’. This image displays deeply modulated structure down to the resolution limit of the telescope. When the telescope is displaced, the structure decorrelates for shifts greater than σt. In addition to the filled UV coverage case, we also consider the measurement of visibilities using a two-element interferometer. This mode of observation differs in important ways from the previous case; in particular, some of the time-scales that demarcate the different regimes are not the same. We also consider the effect of a finite bandwidth and a finite source angular size. Both the bandwidth and the source size need to be less than very stringent limits if snapshot images are to be observed; much more relaxed criteria apply for average images. In the case of interstellar scintillation, tref ∼ days to years and tdiff ∼ minutes. Since normal VLBI observations extend over a fraction of a day, images obtained by this technique correspond to the average image regime. Some of the ideas explored in this paper are therefore relevant for the interpretation of such observations.