A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2022 Sep 23, 12:58 -0700
Homer Smith, you asked:
"I figured that if I moved to a more easterly longitude I could find a meridian where the 2 events occured simultaneously, but what west longitude would that be? My limited knowledge base only allowed me to determine that it would be 5 degrees east of me (20 minutes = 5 degrees). I have a friend that helped me with the finer points of the determination (i.e. taking EOT into consideration), but I am interested in how others would approach this problem."
What is sunset? This is not a trivial question. One definition, which is relevant to your question, is the historical/classical definition which sees it as the moment when the center of the true Sun (unrefracted) reaches the true horizon (the set of points exactly 90° away from your own zenith. That's the sense of sunset that was used to define the equinox as equal day and night. And at the exact instant of the equinox, when the Sun is on the celestial equator and directly above the Earth's equator, the Sun will be setting (in this classical sense) as seen from a longitude 90° away from the subSun point.
Where was the subSun point at the time of the equinox? The Declination is the latitude, and of course by the setup that is exactly 0° 00.0' at the moment of the equinox. The GHA of the Sun at that time is the longitude of the subSun point. We can look up the GHA of the Sun at 1:04 UT on 23 Sep 2022 in a variety of resources. It was 197°52' W (GHAs are West longitudes... this one is equiv. to 172°18' E but we don't need that). At any location 90° in longitude east from there, the Sun would be just setting in the classical sense and exactly due west (azimuth 270° 00'). So the longitude we seek is 107° 52' at any latitude on thee globe. You can verify this easily enough in a sky simulation like Stellarium. I would say that this is probably what you were looking for.
The modern definition of sunrise and sunset tells us the time when the center of the true Sun has a zenith distance of 90°50'. This definition makes some sense for ocean observers in small-ish boats. It assumes a mean sea level refraction of 34' and it assumes that the Sun's semidiameter is 16'. The sum of those two is 50' so the upper limb of the Sun would just be disappearing behing the sea horizon when the center of the Sun is 50' below the true horizon. This definition requires zero height of eye for the observer.
For inland observers, the modern definition of sunset is nothing more than an arbitrary, agreed-upon standard of comparison. It has almost no connection with reality. You mentioned that sunset at your location was at 19:24 MDT. That is not a physically meaningful time. Even for observers on inland lakes, the definition ignores height of eye (as already noted) and it also ignores the decreased refraction higher in the atmosphere. If you attempt to puzzle out this problem using tabulated values for sunset times, you'll run into major problems. When we see the Sun at the time of the modern definition of sunset and see that last flash of the upper limb, it is already well below the horizon and therefore its azimuth has shifted significantly from true west approximately by 50'·sin(latitude). This means that by the modern, standardized defintion of sunset and sunrise the Sun will not set due west if you observe your sunset at that exact time of the equinox.