How can one determine what "point" on Earth is directly beneath a?

Answer 1

"...how quickly is it changing..."

depends on where you are. if i've thought this thru correctly, as the earth spins on its axis, every star appears to be directly over a particular line of latitude. the stars are so far away that i don't think this line of latitude changes over the course of a year; the parallax is negligible. the only thing that changes is what point on that line is directly under a star at any given time.

stars that appear to be directly over the poles (if there were such a thing; even the north star is 3/4 of a degree away from being directly above the north pole, and there is no south pole star visible to the naked eye) would not appear to move at all, while stars directly over the equator would appear to circumnavigate the earth every 24 hours. since the equator is almost 25,000 miles long, the position "under" a particular star at the equator would be moving along the surface of the earth at over 1000 miles per hour.
How long a star would stay at the Zenith is rather arbitrary - they're both points, and the Zenith's moving, so you could argue that it's only there for an instant. If you allow some leeway, then the length of time will depend on the latitude, as explained by barneca. That lack of precision in time will have an equivalent lack of precision in the position on the Earth - not too much of a problem if the answer to the riddle is a major city, but it's going to be a show stopper if people are expected to dig up buried treasure!

As to how you'd do it, you'd need a good star catalogue, a good atlas, a calculator and Jean Meeus' excellent book "Astronomical Formulae for Calculators".
i hope u got the idea!!!!!

Answer 2

Assume the earth is spherical, which I assume will be accurate enough for you. Find your lattitude for point a. If it's a north lattitude flip it to south or vice versa. Find your longitude of point a. Subrtract this from 180degrees. If it's west flip it to east or vice versa. This new lattitude and longitude is the point directly through the earth from point a.

Answer 3

beneath a point in the air above?
by a plumb line.

Answer 4

a what?

Answer 5

Complete sentences, please!