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    Re: Precomputed lunar distances
    From: Frank Reed
    Date: 2005 Apr 16, 01:50 EDT

    C Roberts asked:
    "In postings on this list ,  altitudes and azimuths were required for
    accurate calculations of distances  between bodies. How can
    precomputed lunar distances be done as has been done  for centuries??
    It would seem that almanac GHA and DEC are the only data  available.
    This same calculation would be used for inter stellar distances  for
    checking sextant calibrations. It has been said that the values  in
    Bauer's book are wrong. In the lunar distance calculations this  value
    which is used to proportion for GMT is calculated. Can  precomputed
    values be used to avoid this ?"
    It's really just the same  as the situation in ordinary line of position
    celestial navigation when we deal  with a computed altitude versus an observed
    altitude. In fact, the best place to  begin with lunars or star-to-star distances
    is to consider a case where the  entire measurement amounts to a difference
    in altitude.
    Imagine two  stars that happen to be right on the celestial equator
    (Dec=0d00.0') and happen  to differ in SHA by exactly 30 degrees. Their separation
    could be tabulated in  advance as 30 degrees if desired. This is a fixed value
    valid for any observer  on Earth though even for stars it would change slowly ove
    r time. If you take out  your sextant and measure the apparent distance
    between these stars, you won't  get that value unless you correct for the effects of
    refraction (and in the case  of the Moon and other nearby objects, parallax,
    too). If you observe these two  stars from a point on the Earth's equator, you
    will always find them on the  prime vertical (due east/due west). Let's
    imagine a specific case...
    I am  on the equator and I see one of the two stars I've described above
    around 30  degrees high in the east and the other star at nearly 60 degrees
    altitude  directly above it. I measure the distance between them with my sextant and
    I  measure each star's altitude, too. I look up the measured altitudes in the
    star  refraction table in the almanac, and I find that the lower star has
    been lifted  by refraction by 1.7 minutes of arc, the upper by 0.6 minutes of
    arc. That means  that the apparent distance between them will be shorter than the
    predicted 30  degrees by 1.1 minutes. So if my measurement was 29d 29.0
    (after correcting for  index error) this would imply that my measurement was off by
    only 0.1 minutes of  arc --perfect by any practical standard. Now suppose I
    wait two hours. The stars  climb until the higher one is at the zenith. This
    time the lower star is lifted  by 0.6 minutes and the higher star is unaffected
    by refraction (it can't get any  higher!), so the measured distance should be
    shorter than the predicted 30 by  only 0.6 minutes of arc. This is the process
    of "clearing" a star-star distance  or a lunar distance. When the stars (or
    Moon and other body, for a lunar) are  not directly above each other, we have
    to calculate a geometric factor that  tells us what fraction of the altitude
    correction acts along the arc between the  objects, but that's the only
    additional complexity. The precomputed lunar  distances in the old almanacs were
    compared with measured distances *after*  going through this process of clearing
    for the effects of refraction and  parallax.

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