A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2022 Jun 6, 09:47 -0700
Joe Wong, you wrote:
"if a navigator choose to shoot a relatively close body, say the moon, would the hs value vary when shooting at different altitudes(20,000ft, 30,000ft or shooting at near ground level like myself). If so, would the "hs"s deviate so greatly from each other that a correction be needed at corresponding flight levels?"
In principle, yes! In practice, maybe, but not for manual sextant observations, like with your Kollsman periscopic sextant.
The Moon is deflected downward by parallax on average by 57 minutes of arc when it is near the horizon. The exact value of the Moon's "HP" at the date and time tells us how much the 'horizon parallax' deflection would be. On average, that's 57' but it ranges from 53' to 61', depending on the Moon's actual distance from the Earth. That deflection is the result of shifting our observing location away from the line passing through the Earth's and Moon's centers. The deflection is proportional to the observer's distance from that line. So if the subMoon point, where the Moon is at the observer's zenith, happens to be on the equator, then the maximum distance from the line passing through the centers of the Earth and Moon would be close to the geographic poles of the Earth or anywhere on the great circle centered on the subMoon point. Standing at the north pole at sea level in this scenario, an observer would be just about 3950 miles off the geocentric line (3950 miles or 6357 km is the polar radius of the Earth). Now suppose you're in an aircraft flying at an altitude of 7 miles, near the tropopause. That's two percent of the radius of the Earth so the parallactic deflection in the Moon's position would increase by two percent, and two percent of 57 minutes of arc is just about a tenth of a minute of arc. That's below the practical concerns of aerial celestial navigation with bubble sextants, but you could imagine an automated system using an INS (inertial nav system) derived vertical instead of a bubble in an aircraft flying at still higher altitude, and for that you would need to apply a small increase in the Moon's parallax. It is effectively an increase in the radius of the Earth.
But wait. The Earth itself has different radii in different latitudes. The Earth's geoid or "effective sea level" is well-approximated by an oblate spheroid, "flattened at the poles", as they say. The difference from equatorial to polar latitudes is about one part in 300. Flying an aircraft near the poles puts observers closer to the Earth's center by some 13 miles even before we climb to cruising altitude. So why don't we modify the Moon's parallax in different latitudes? Actually, we can and do, under some circumstances. There's an "oblateness" correction formula available for "calculator" users in the back of the Nautical Almanac (see image). For observers near the geographic poles, it's nearly equivalent to -0.2'·cos(H), where H is the angular altitude of the Moon. This formula has been included in the standard Nautical Almanac for over thirty years. But for bubble sextant applications, and even for most marine sextant observations, this correction is negligible.
Just for fun... My "back of the envelope" modification to this polar version of the oblateness correction, for aircraft flying near the poles incorporating both oblateness and flight level, would be
where F is true flight level and F0 is 70,000 feet or 21 km.
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