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    Re: Moon corrections: how computation order affects the result
    From: David Iwancio
    Date: 2020 Sep 1, 00:56 -0700

    Fernandez:

    Short version: 

    If you correct for semidiameter (SD) before you correct for parallax in altitude (P-in-A), you need to determine the augmented SD.  If you instead correct for P-in-A before SD, you can skip having to correct for augmentation.

    Long version:

    Both augmentation and parallax in altitude are caused by a shift of the observer from the geocenter to the surface of the earth.  Augmentation is caused by a change in the observer's x coördinate, and parallax by a change in the observer's y coördinate.  However, our x and y coördinates are related to each other due to the fact that we're all bound to the circular surface of the earth; when one coördinate is at maximum, the other is at zero.

    Further, because the earth-moon distance is much bigger than the radius of the earth, so that all the angles involved are ~1° or less, we can use small angle approximations for sines and cosines, simplifying what could be some ugly math.  Specifically:

    • Once you remember to convert to radians, the difference between 61.5' and sin(61.5') is less than 0.000 001
    • Similarly, the difference between cos(61.5') and {1 - [(61.5')^2] / 2} is less than 0.000 000 01

    By limiting our movement to the surface of the earth (plus or minus a few hundred kilometers) and being as far away from the moon as we are, the augmentation of its semidiameter can be treated as "differential parallax."  As the moon gets further from the horizon, sin(Ha) gets smaller, but the difference between sin(Ha) and sin(Ha+30') gets bigger; both the upper and lower limbs get closer to Hc, but further away from each other.

    If you scan across the moon correction tables on page xxxv of the Nautical Almanac, you can see that, though the corrections in the top part of the table get smaller, the difference between one correction and the correction three lines below it starts to creep up.

    To the precisions we care about, this differential parallax is numerically the same as augmentation.

    The upshot is that, when you apply a parallax correction first, you are correcting for parallax at the limb to get you to the geocentric position of that limb.  After that it's appropriate to use the geocentric SD to shift from the geocentric limb to the geocentric center.  Correcting for augmentation before parallax is for when you want to correct for parallax from the topocentric center of the moon's disc to the geocentric center.

    Side note:  if you're looking to compare the corrections you get from other sources with those you'd get from the Nautical Almanac tables themselves, add 0.1' to the HP before entering the table to undo the simplified "OB correction" described on page 280 the Almanac.  And in the strictest sense the -30' correction for upper limb measurements should be applied after the first correction from the upper half of the tables.

       
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