Imaging of very distant comets: current and future limits.

We describe observations and reduction techniques used to image sky fields around three extremely distant comets (Shoemaker 1984 XV at heliocentric distance r=17 AU, Bowell 1982 I at 23 AU and Schuster 1975 II at 31 AU) during a pilot programme (3 half-nights) at the 3.5-metre ESO New Technology Telescope. Very deep upper brightness limits could be set (V>27mag), consistent with their nuclei now being quiescent and implying unexpectedly small upper limits for their radii, 3.4km, 5.4km and 10.3km, respectively. We address in some detail the important problem of determining objectively the "limiting magnitude" actually achieved during CCD imaging of a moving object. We also discuss some of the specific problems related to observations of very faint, moving objects with the currently available instrumentation, and we make predictions about observational possibilities with the new generation of giant telescopes.