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    Re: The Moon and Parallax
    From: David Iwancio
    Date: 2019 Oct 19, 02:46 -0700

    Simply using the tabular SD on the day pages will cost you accuracy because it doesn't take into consideration the issue of "augmentation."

    The semidiameter is calculated for an observer that is at the center of the earth, which is about 3400 nmi away from most of us.  As the moon gets higher over your horizon, you're also moving closer to the moon, and its SD for you is going to be sightly greater for you than it would be from the center of the earth.  The difference is small enough to ignore in the Air Almanac with its precision of 1', but it's noticeable at the NA's precision of 0.1'.  

    To see this play out in the NA table, consider what happens when the altitude of the center of the moon is 90°.  The refraction and parallax corrections are then both 0.0' and the only correction left should be SD.  But if you enter the tables with a LL altitude of (90°-SD) you'll always end up with a corrected altitude a few tenths of an arcminute shy of 90°00.0'.  That difference is the moon's augmentation at 90° altitude.

    Tables of the moon's augmentation were included in old navigational books.  The modern altitude correction tables get around this by reframing it all in terms of parallax.

    Consider the P-in-A critical tables in the Air Almanac that you mentioned earlier.  If the altitude of the center of the moon happens to be on or near one of the critical points in the table, then the parallax correction for the lower limb will be higher than for the upper limb, i.e. the lower limb will be even lower than it "should" be, requiring a greater correction.  In a mathematical sense, this is what augmentation can be seen as:  the parallax correction is different for different points on the disc of the moon, causing the moon's disc to spread out more as it gets higher. 

    The table in the Nautical Almanac works by correcting the altitude for parallax (and refraction) specifially at the limb.  Once the limb's altitude is corrected, adding in the tabulated ("corrected") SD afterward is appropriate to get the corrected altitude of the center of the moon.

    Using the table with the upper limb altitude will give you the same precision as lower-limb sights (and you'll note that 30.0' + L - U is always twice the moon's tabulated SD).  The mechanic of subracting a nice, round 30.0' for UL sights is just a convenient way of keeping all other corrections additive.

    The final trick to the table is shown at the bottom of page 280:  in an attempt to take into account the true shape of the earth, 0.1' is subtracted from the HP before computing the parallax in altitude.  For those that have access, Bennett's paper "The Effect of Neglecting the Shape of the Earth on the Accuracy of Sight Reduction"  in the Journal of Navigation mentions correspondance with Yallop that confirms this.  (Bennett was apparently not impressed.)

       
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