NavList:

Martha Noyes, you wrote:
"At what degree of altitude is a star (magnitude 2.5 -ish) first visible? At sea level at 21° N 158° W."

First you have to decide what the faintest star is that you can see near the zenith --the zenith limiting magnitude. For most people on a dark clear night, this would be magnitude 6.0 roughly. Then you need a table of "extinction". The definitive extinction tables usually cited are the "International Comet Quarterly" or ICQ extinction tables. In table "Ia" on that site, you will find average extinctions for sea level (in the left column) versus zenith distance. There is already some exctinction at the zenith, but you could ignore that given the uncertainty of the actual zenith limiting magnitude. Then you need to find the extinction that would reduce a star of magnitude 2.5 to the equivalent of a star with that limiting magnitude. If there is extinction of 3.5 magnitudes, that would pull that star from 2.5 down to 6.0 --the limiting magnitude. From the table, you can see that this happens at a zenith distance of 86° or 4° altitude. According to ICQ extinction tables "Ib" and "Ic", seasonal variations could shift that altitude up or down by half a degree. So somewhere between 3.5° and 4.5° altitude, a star of magnitude 2.5 (without extinction) will become barely visible to the unaided eye.

-FER