A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2015 Nov 18, 20:42 -0800
When you do what Gary described, you'll discover a big simplification. Given the assumption that the true altitude is zero, the equation simplifies as follows:
0 = sin(Dec)·sin(Lat) + cos(Dec)·cos(Lat)·cos(t),
cos(t) = -tan(Dec)·tan(Lat),
where t is the time of starrise or starset measured from the time of the star's transit, and of course Dec and Lat are the declination and latitude. This assumes a few things: you can work out the time of transit separately, you're not worried about a minute or two here and there, and (related to that) you're willing to ignore refraction right at the horizon which slightly delays setting times and advances rising times. Since you can't see stars right near the horizon in almost all cases, these are reasonable assumptions.
Here's a couple of examples:
I. Star's dec is 30° N, observer's latitude is 40° N. Mean time of star's transit is 10:30pm. By the calculation above, t is 119.0°. We divide that by 15 to get hours: 7.93h which is 7h 56m. Add to or subtract that from the transit time to get set and rise times: 6:26am and 2:34pm.
II. Star's dec is 35° S, latitude is 40° N. Star transits at 5 minutes past midnight: 00:05. By the calculation, t is 54.0°. Convert to hours and minutes: 3.6h or 3h 36m. Therefore the star sets at 03:41am, and it rises at 8:29pm.
Note that the times are nearly correct for any longitude so long as we recognize that this is LMT or Local Mean Time. In other words, if Capella rises at 10:30pm in Boston, it also rises at nearly 10:30pm in Seattle.