A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Sean C
Date: 2019 Dec 3, 22:06 -0800
When doing a time sight, I usually use the formula found on the Wikipedia page titled "Longitude by Chronometer" (which is another name for a time sight):
cos(LHA) = (sin(Ho) - sin(Dec.) · sin(Lat.)) / (cos(Dec.) · cos(Lat.))
The LHA can then be converted to local apparent time and the equation of time added or subtracted to give local mean time. This can be compared to the UT (≈GMT) of the sight to give you your longitude. Or, you can simply compare the LHA of the sun to the GHA at the UT time of the sight.
Incidentally, I discovered some time ago that this exact same formula can be used to calculate the times of sunrise, set and all of the twilights as well.
There are also tables available to accomplish the same task, such as Martelli's. Hope that helps!