NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Lars Bergman
Date: 2010 Feb 27, 15:21 -0800
Here is a mathematical disproof of Zevering's GHA-Dec Updating Technique (GD-UT) to transfer an earlier observation to the time of a later one.
At the time of the first observation, the following equation is true:
sin(ALT1)=sin(LAT1)*sin(DEC1)+cos(LAT1)*cos(DEC1)*cos(GHA1+LONG1) (1)
Similarily, at the position of the second observation, the following equation is true:
sin(ALT2)=sin(LAT2)*sin(DEC2)+cos(LAT2)*cos(DEC2)*cos(GHA2+LONG2) (2)
The run between the two positions is defined by
dLAT=LAT2-LAT1 and dLONG=LONG2-LONG1 (3)
Putting (3) into (1) yields
sin(ALT1)=sin(LAT2-dLAT)*sin(DEC1)+cos(LAT2-dLAT)*cos(DEC1)*cos(GHA1+LONG2-dLONG) (4)
So far, there are no difficulties, (2) and (4) must be satisfied by the two observations and the run in between. Now, Zevering's method fulfills (2) and the first observation is utilized by moving the GP of the first body "by the run" and solving at the second position:
sin(ALT1)=sin(LAT2)*sin(DEC1+dLAT)+cos(LAT2)*cos(DEC1+dLAT)*cos(GHA1+dLONG+LONG2) (4')
If the GD-UT method is correct, then (4') must equal (4), i.e.
0=sin(LAT2-dLAT)*sin(DEC1)-sin(LAT2)*sin(DEC1+dLAT)+cos(LAT2-dLAT)*cos(DEC1)*cos(GHA1+LONG2-dLONG)-cos(LAT2)*cos(DEC1+dLAT)*cos(GHA1+dLONG+LONG2)
The obvious solution to this last equation is found by inspection:
dLAT=dLONG=0, i.e. the observer is stationary.
For a moving observer, generally dLAT and dLONG have non-zero values and in the general case the last equation is not fulfilled. Thus, the GD-UT method of Zevering must be incorrect. Q.E.D.
Lars, 59N 18E
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