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    Re: Precomputed lunar distances
    From: Bill B
    Date: 2005 Apr 17, 20:54 -0500

    > 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 ?"
    Frank responded:
    > 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.
    Darn it Frank, there's that "horizontal component" of refraction again
    everyone denies exists.   Refraction lifts AND squeezes two orbs together
    (yielding the cleavage we expect from "stars" and other heavenly bodies ;-)
    In two bodies not at the same azimuth, refraction will act a line passing
    through the body from the observer's terrestrial equator to his/her zenith
    for each body.  Yes? Two sides of a spherical triangle meeting at the
    zenith. Both start off almost perpendicular to the horizon and progressively
    arc in to the zenith.  So from the *observers* frame of reference,
    refraction is acting up and in, less up and more in the higher the bodies.
    A recall accounts of an experiment Galileo proposed but supposedly never did
    because he already knew the answer.  A galloping horseback rider would
    outstretch his arm and drop a ball.  From the frame of reference of the
    rider, the ball would hit the ground below his outstretched hand.  From the
    frame of reference of an observer perpendicular to the rider and his path,
    the ball would scribe an arc.  So other than "observed horizontal
    displacement," what do we call it the observed horizontal displacement?
    Corner cosine?
    Alex had provided me a formula for calculating refraction adjustment from
    calculated separation, which I believe he credited to you.  (I do not know
    its pedigree past that.)  I had calculated a separation adjustment of
    -0.9661 between Procyon and Cappella (40d 28'N, 86d 56') April 3, 8PM EST.
    Hc's were 54d 23.6' and 59d 47.0' and refraction from almanac -0.7' and
    -0.6'.  Bauer's instructions would have yielded -0.1' (-.7 -(-.6))
    April 8, 11:15PM EST I recalculated for the same two bodies, this time Hc's
    were 26d 22.3' and 24d 44.7' and refraction corrections of -1.9' and -2.1'
    (no SHA or declination change).  I was a bit surprised to see a correction
    of -.9208--LESS than when the bodies were higher.  Bauer's instructions,
    -0.2 (-2.1 - (-1.9)).
    I was somewhat surprised that with approx. half the altitudes, and almost
    triple the refraction, the correction was an RCH less.  When I thought
    through it, even with triple the displacement, apparently the angle it acted
    (vertically instead of horizontally) from the observers frame of reference
    more than compensated for the increased displacement.
    For those like myself who are new to the concept (and as a sanity check),
    the formula for calculating refraction correction from calculated true
    separation as it was given to me as follows:
    CORR = REFR1((sin(ALT2) - cos(TRUED)*sin(ALT1))/(cos(ALT1)*sin(TRUED)))
     + REFR2((sin(ALT1) - cos(TRUED)*sin(ALT2))/(cos(ALT2)*sin(TRUED))).
    TRUED is calculated true separation.
    ALT1 and 2 are the body's Hc's.
    REFR 1 and 2 are almanac (or calculated) refraction for body altitudes.
    In Alex's words: "This number is always negative.
    The dimensions should match, of course. I mean if TRUED is in degrees, REFR1
    and REFR2 should be in degrees too, and the resulting CORR will be in
    degrees. (I compute all angles in radians, and translate to degrees only in
    the end."
    So even if a "horizontal component" to refraction does not exist, I can see
    it and have a formula for calculating it.  Then again, with the whole
    "artist" thing going, maybe it's just the voices in my head telling me I see
    it .... 

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