A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Dec 4, 09:19 -0800
Sean, you wrote:
"I usually use the formula found on the Wikipedia page titled "Longitude by Chronometer" (which is another name for a time sight)"
Not really. Even that Wikipedia article (which was almost certainly written, and exclusively edited by, Andres Ruiz) draws a distinction. The phrase "longitude by chronometer" was co-opted c.1900, primarily in British navigation, to apply to an observation for local time ("Hour Angle" in modern terms). The presence of the chronometer is irrelevant and does not properly define what's going on. Of course, the phrase "time sight" isn't much better intrinsically, but it is the standard terminology in use today.
You quoted the standard equation:
"cos(LHA) = (sin(Ho) - sin(Dec.) · sin(Lat.)) / (cos(Dec.) · cos(Lat.))"
This is no more and no less than the usual great circle equation for altitude but turned around and solved for "LHA". I usually now write this "hour angle" simply as "HA" to avoid worrying navigators who have learned previously that LHA is angle always measured west from the meridian. Also, it's worth noting that the parentheses defining the numerator and denominator here make things more difficult for many navigation students and you can easily eliminate them:
cos(HA) = sin(alt) / cos(Dec) / cos(Lat) - tan(Dec) · tan(Lat),
and one can make the equation more symmetrical by swapping ZD for altitude:
cos(HA) = cos(ZD) / cos(Dec) / cos(Lat) - tan(Dec) · tan(Lat).
This last step has no computational advantage (in fact a slight disadvantage), but it's helpful pedagogically.
"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."
Since you've already modernized much of the process, there's no need to convert back to local time and deal with the equation of time. The equation of time is implicit in GHA. You don't even need to know it exists! So once you have calculated the hour angle (HA), just add it onto the Sun's GHA. In the afternoon you add; in the morning you subtract. For any other celestial body, you add the HA onto that body's GHA. If the body is descending (and therefore it's after meridian passage), you add. If the body is rising, you subtract. (BTW, excellent use of the UTF "about equal" character here! :) ).
Need a line of position? Just run the calculation for some Lat and then again for Lat+0.1°. You get two slightly different longitudes. Plot the points on common graph paper. This is faster and easier than most variants of the intercept method and requires fewer tables and forms and all that. It does require a caculator (solar, of course). This is exactly what I teach in my Modern Celestial Navigation classes.
Clockwork Mapping / ReedNavigation.com / HistoricalAtlas.com
Conanicut Island USA