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    Re: Advancing position circles: Huxtable vs. Zevering
    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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