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Frank,

You asked:

about how long does that Port Chalmers time ball take to fall? Is it on the order of two seconds, like the video Rafael linked from Greenwich? Or is it slower?

Unfortunately this was not a controlled experiment and I only got to observe it once. However my impression was that it was a bit slower than the Greenwich version. At Greenwich there appears to be something close to an initial free fall followed by some braking. The Port Chalmers time ball seemed to descend at a pretty uniform rate but, again, I wasn't really paying particular attention to that aspect of it.

You also asked Rafael:

 "staff was too busy measuring the Sun’s meridian transit to drop balls." Was that really the issue? ...Why 13:00 and not 14:00 or 09:00?

To me it seems plausible that the astronomers were too busy around noon. If they were setting  some master time keeper using the Sun's meridian passage then that observation could occur at either side of local mean noon so you can't drop the time ball at 12:00. Once you've set the time keeper you'd want to publish the result via the time ball as soon as possible to minimize the effect of drift in the mechanical time keeper making 13:00 a logical choice. This interpretation, of course, rests on the assumption that the meridian transits of the Sun were used rather than those of stars.

Robin Stuart