Dear Andrew,
I think your 9-year old step son needs
a lesson on the theory of limits but
meantime he needs an answer to his
question:
> Is there a place on the surface of the
> earth where the sun will rise at midnight
> on the longest day of the year and then
> set at the next midnight and then the
> days get shorter again?
Consider how the following ratio changes as
we approach the summer solstice:
Hours of darkness
-----------------
Hours of daylight
If you are just a few degrees south of the
Arctic Circle this ratio steadily decreases
to a minimum, at the solstice, and then
increases again as the day gets shorter.
The ratio at the minimum depends just how
far south of the Arctic Circle but at the
Arctic Circle itself the value is Zero at
the solstice so the sun, in some sense,
rises at midnight and sets at midnight.
There are several caveats. First, this is
a purely geometric view that assumes a
point sun and no diffraction and other
tricky little inconveniences!
Secondly, you have to worry about being a
little north of the Arctic Circle where
there is no sunrise or sunset at the
summer solstice but...
If you study the time-of-sunrise at such
a place you find that it gets earlier and
earlier until one day it rises at (or very
close to) midnight and STAYS above the
horizon until some days or weeks later it
sets at (or very close to) midnight.
This is not the NEXT midnight though. It
is only on the Arctic Circle itself that
it will be the NEXT midnight which is the
condition that your step son requires!
One side effect of having the sun round the
clock is that Italian Hours cease to have
any useful meaning :-(
Your job now is to keep up your step son's
interest in three-dimensional geometry and
that could be a rewarding challenge!
Frank King
Cambridge, U.K.
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