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Longitude by Mintaka
From: Frank Reed
Date: 2015 Mar 16, 22:36 -0700

We all know that you get latitude at noon from the Sun from a simple equation:
Lat = ZD + Dec.
Lat is the observer's latitude. Dec is the Sun's declination. And ZD is the "zenith distance" which, of course, is 90°-altitude after the usual corrections. All three of these terms can be positive or negative (north or south). The sign for the ZD is governed by which way you're facing (if your shadow points south, then the ZD is labeled "South" or counted a negative). This relationship holds on any day of the year, anywhere on Earth, and it can be used on any body making meridian passage so long as we can measure its altitude well enough.

There is a similar simple relationship for longitude:
Lon = ZD + GHA.
But this doesn't work very often. I usually teach this for the (as I call it) "e cubed" textbook sight: E3 = easy, equatorial, equinoctial. If the Sun's declination is near zero (meaning we're at either equinox), and the observer is very close to the equator, then it's easy. The Sun rises straight up from the eastern horizon, and the local hour angle is just the zenith distance. So if the corrected altitude is 30°, then we are 60° or four hours from meridian passage, and the local apparent time must be 0800 on the dot.

The star Mintaka (or δ Orionis), northermost in Orion's belt, and "upper right" when seen "riding high" in the northern hemisphere, is less than a third of a degree from the celestial equator. That's unique. Its declination is around 0°17' S this year. Although Mintaka isn't one of the standard 57 navigational stars, it is very close to Alnilam (middle star in the belt; one of the 57) so we can get the SHA of Mintaka on any date by adding 1° 03' to the SHA of Alnilam. Then, to some degree of accuracy, within some range of latitude near the equator, we'll be able to use the same simple equation for longitude:
Lon = ZD + GHA.
By my estimation, within 5° of the equator, as long as Mintaka is below 45° altitude, this simple equation will yield a longitude accurate to within 10' or since we're near the equator that's within 10 nautical miles. That's a fair trade for a longitude without sight reduction.

As with any star sights, we're limited by the ability to see Mintaka while the horizon is still visible, but Mintaka is only slightly fainter than Polaris and about the same brightness as Mizar in the Big Dipper. Most of us, with a little practice, have no trouble picking those out during twilight and catching Mintaka is no more difficult. For five months out of the year, an observer within that 10° band centered on the equator could get longitude without sight reduction tables in either morning or evening twilight using Mintaka. The only calculation required is for GHA Aries --simple enough if you have a standard Nautical Almanac.

Example: May 1, 2015, north of the Galapagos at latitude 4° 00' N, 92° 30' W exactly. In evening twilight, I measure the altitude of Mintaka at exactly 01:00:00 UT as follows:
Observed Alt: 32° 03.3'
Corrected Alt: 31° 56.1' (after IC, dip, refraction)
ZD: -58° 03.9' (subtracting alt from 90°, labeled negative since we're looking west)
GHA Alnilam: 149° 24.3'
GHA Mintaka: 150° 27.3' (adding 1° 03' to Alnilam GHA)
Lon = ZD + GHA: 92° 23.4' (error is 6.6 nautical miles)

That's the entire sight reduction process. No trig, no tables, no calculator. Unlike the symmetrically simple "Noon Sun" sight and other meridian sights for latitude, this "Longitude by Mintaka" sight only works reasonably well within the band of latitude from 5° N to 5° S, and if we insist on taking the sights in twilight when they're most likely to yield good results, we're limited to about five months out of the year. We can also throw in two days of Sun sights on the equinoxes with the same conditions on latitude and maximum altitude.

Frank Reed
Conanicut Island USA

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