A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2010 Apr 3, 09:15 -0700
It really is quite un-necessary to specify delta-T in these historical analyses. While there remains some moderate uncertainty in the value of delta-T in the late 18th century (and that will not change unless some vast store of accurate astronomical observations from that period is found in archives somewhere), the values are known with enough certainty for any navigational calculations. They are not inputs to the problem.
Next, the online lunar calculator on my web site does not calculate any position from the observed altitudes. You can do so yourself easily enough using the calculator by adjusting the "DR" lat and lon until the calculated errors in the altitudes go to zero, but that's a secondary issue, extremely useful for historical sights but not an essential part of the lunar clearing process. When you are given a lunar observation consisting of an altitude of the Moon, an altitude of the other body, and the measured distance between them, you can clear it (nearly) without knowing anything at all about the observer's position on the Earth. The correction to clear the lunar distance is simply a determination of the amount of refraction and parallax that acts along the lunar arc (the great circle from the Moon to the other body). This correction, to first/linear order, is just dh1*A+dh2*B where dh1 and dh2 are the standard altitude corrections for the Moon and the other body and A and B are factors between -1 and +1. They tell us the "percentage" of the standard altitude correction that is projected along the lunar arc and they are simple functions of the geometry of the objects in the sky. They are the "corner cosines" --the cosines of the angles between the lunar arc and the vertical at each body. For example, when the Moon and other body are aligned in a vertical circle (in the same azimuth or on opposite azimuths), the factors A and B are exactly +/-1. Now, as you know the standard altitude corrections, here called dh1 and dh2, do not depend on where you are on the Earth. They are functions of the apparent altitudes. Therefore, since A and B depend only on the relative geometry of the objects in the sky, the total correction to the lunar distance dh1*A+dh2*B also does not depend on your location on the Earth. Note that this is the "first order" analysis of the terms but the higher order terms do not change anything qualtitatively. The bottom line here is that you can clear a lunar given ONLY the observed altitudes. Here's an example: Moon LL alt 30d, star alt 45d, observed LD 60d. Assume standard temperature and pressure, height of eye = 12 feet, and IC=0. Note that we need one more piece of almanac data: the Moon's HP, which also uniquely fixes the Moon's SD. Given HP=56.0, this set clears to 59d 47.3'. If we want a Greenwich Time as our final output, then we also need predicted geocentric LDs for some known times before and after the observation (or equivalently one time near the observation time and the rate of change around that time). At no point in any of this do we need the observer's location on the Earth, but if we want the very smallest corrections, then we do require an approximate position on the Earth and the Moon's approximate position to calculate the correction for the oblateness of the Earth.
By the way, I agree with you that "oblateness" of the Earth can be called "flattening" and probably even "should be". But this hasn't been the case in the history of nautical astronomy in the English language. Next you'll try to tell me that the semi-diameter of the Sun should really be called the radius.
Crazy navigators with their crazy navigational terminology! :-) To avoid confusion, I have modified the text labels on the web pages of my online lunars tools to make sure the distinction is clear.
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