What is the Darkest Hour of the Night?

John Ladasky writes:

George Dishman wrote:

It will be darkest at local midnight in midwinter
when the Moon is new and Venus is below the horizon.
I wonder how often that combination occurs (if ever).

How can Venus *not* be below the horizon at local midnight? Venus
reaches a maximum elongation of about 48 degrees relative to the Sun.
If I've done the math correctly, Venus should always be below the
horizon by 3 hours and 12 minutes after sunset.

For locations near the equator, yes. Near the poles, however,
it is possible for the Sun to be down and Venus to be up at
local midnight.