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    Re: Lunar trouble, need help
    From: George Huxtable
    Date: 2008 Jul 8, 00:37 +0100

    Kent wrote, about the refraction corrections-
    
    "It might be easiest to explain by
    starting with what has to be done when reducing altitudes. In Jeremy's case
    the UL of the moon was measured. The refraction has to be done for the
    geocentre of the moon and not on the altitide for the UL."
    
    He and I disagree here. In practice, in the case we are considering, the 
    difference in refraction between the Moon's centre and upper limb is pretty 
    infinitesimal: no more than 0.4 arc-seconds. Presumably, Kent is keen on 
    such details because he wants to be completely rigorous, and get the 
    principles right. And it seems to me that he hasn't got the principle right 
    here. I will try to convince him.
    
    The light-ray, coming down through the atmosphere, that is being observed 
    with the sextant, is the light that is from the Moon's upper limb. And it's 
    the bend in that ray that constitutes the refraction. Refraction is the 
    difference between the angle at which the light arrives, and the angle at 
    which it would arrive if the Earth had no atmosphere. So when we correct for 
    refraction the observed sextant angle (above the true horizon) of the Moon's 
    upper limb, that then gives us the true elevation of that limb, above the 
    true horizon, as would be seen by that observer if the Earth had no 
    atmosphere. Then the Moon's semidiameter allows for the difference between 
    that true altitude and the Moon's centre; the true difference between them, 
    refraction no longer playing a part.
    
    Of course, if working with tables or formula that work with zenith distance 
    rather than altitude, you have to use the complement (to 90�) of the 
    altitude, but the reasoning doesn't change.
    
    He continues [with my comments in square brackets]-
    
    "Consequently in my way to do the reduction I start with refraction on the 
    UL ..."  [all right so far...  ] "and then I calculate the difference in 
    refraction for the UL - SD for the moon." [that's the bit that's 
    unnecessary, and indeed wrong].
    This is then
    a correction for achieveing a correct refraction value. In this case the
    angle between the distance and the vertical is 90d (of course). If George is
    already from the beginning calculating the refraction for the geocentre [no, 
    I'm not] that
    is fine but then there is, as I see it, a difficulty coming up later when
    treating the distance."
    
    ====================
    
    Kent continues, about another correction which could be made to the lunar 
    distance-
    
    "I all cases there is an angle between the distance line and the vertical
    used for measuring the altitude. Refraction compresses the body - more on
    the LL than on the UL - which means that there is a need to correct also
    the distance for refraction. The body is not circular but compressed in
    directions apart from th the compression in the vertical. The arguments for
    this correction is altitude and angle between the distance and the vertical.
    This correction should be done for the moon and the sun. As an example for
    altitude 8d 30s and angle of 45d the correction is about 5 arcsec's, while
    when the angle is 0d, as in altitude reduction, the correction is 10
    arcsec's. If the other body is the sun then a similar correction should be
    used. So it is not negligible to avoid thiscorrection when reducing the 
    distance."
    
    [I agree that in principle, this is a correction that might be considered. 
    In Raper, (1864) it's table 53, "correction of the lunar distance for the 
    compression of the vertical semidiameter". And I agree that if one were 
    forced into observing at very low altitudes, such as Kent quotes in his 
    examples, then it might indeed be worth making. But most navigators would do 
    the damnedest to avoid measuring lunars below 15� altitudes, in which case 
    this correction never exceeds 3". In the examples we are considering, it's 
    less than 1".]
    
    
    "If George takes a look in the English reference below and puts this into
    graph I believe George will agree with me."
    
    [It depends on the precision one is hoping to work to, which is limited by 
    the observing instrument, the observer's skills, and the environmental 
    conditions he has to work under. I would ignore such corrections, for 
    observations made in a marine environment. ]
    
    "I have three Swedish references and one English: Jeans, Navigation and
    Nautical Astronomy 1853, page 105, header: Refraction, oulines the need for
    this correction. Note: I have not searched actively for English 
    referencies."
    
    ========================
    
    About another correction, for the contribution to Moon's parallax due to the 
    Earth's oblateness.
    
    "George wrote: I'm not familiar with that correction term >which was cos
    aziumuth x (diff geographic and geocentric latitude>,
    and perhaps Kent will explain it, or refer to a text that does. But as far
    as I can estimate,
    its practical effect in our exercise is less than an arc-second, and I doubt
    if it can ever work out to be much more than that, so it seems well worth
    ignoring
    
    Firstly I conclude that George does not compensate for the earth oblateness.
    
    Unfortunatle I do not have any
    English reference for this way to correct for earth oblateness, but I have
    two Swedish. I guess that this way of
    calculating has been described in the German "Lehrbuch der Navigation"
    concerning "moondistanzen, strenger metode" from about 1860 (textbook
    navigation...rigirous method). But I am sure that George os able to do a
    search amongst English litterature.
    
    [The correction  I made is from Chauvenet, "Spherical and Practical 
    Astronomy" (1863 and many later reprints), table XIII "Correction of the 
    Moon's eq. parallax" (actually, a reduction)and in text Vol I, page 104 to 
    126. Kent should take a look at Chauvenet, who provides all the rigour he 
    could ever ask for! He also seems to get everything right.
    
    It's also in a table in a modern Norie's "Reduction of the Moon's horizontal 
    parallax", from which I took, in [5530] the value -0.0002�, to slightly 
    tweak the parallax of 0.4622.
    
    I agree with Kent that this correction can be worth applying at higher 
    latitudes, but al lat = 14, it was hardly worth bothering with.
    
    This corresponds to one of Kent's corrections to parallax due to oblateness, 
    but leaves his second oblateness term unaccounted for.
    
    In [5701], Kent describes another correction to parallax for oblateness, 
    which depends on Moon azimuth, and I haven't even considered that one, 
    described by Kent as follows-
    
    "- find the azimuth to the moon
    - find the difference between the geographic and geocentric latitude
    - multiply  this difference with cosine for the azimuth
    The azimuth is approx. 111d and the diff. between the latitudes is 5m 45s.
    The product is +2m 6,46s, which gives a "local altitide" of  60d 38m 57,73s
    + 2m 6,46s =  60d 41m 04,19s to be used for parallax calculation. Due to the
    earth oblateness the value is added to the true local altitude if the
    azimuth is greater than 90d (the moon is pointing away from the pole),
    otherwise the value is negative."
    
    By my rough estimate, that might shift parallax either way by up to 5 
    arc-sec, so when striving for high accuracy, it could be well worth making. 
    In the present case, it seems to increase parallax by 2". Do I have that 
    right?
    
    However, I haven't made any such correction, that depends on Moon's azimuth, 
    and haven't found a reference to one, yet, in text in English. Still 
    looking, though.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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