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    Re: Lunar trouble, need help
    From: George Huxtable
    Date: 2008 Jul 5, 23:46 +0100

    Kent Nordstom seems keen to understand the remaining differences between his 
    lunar calculation and my own, and I agree that its well worthwhile. Let's go 
    on, to the bitter-end!
    First, let's go back to the matter of correcting the sextant reading of 
    observed lunar distance, for semidiameters, arriving at the value d (which 
    will later be "cleared" to provide the true distance D). I thought we had 
    resolved that between us, but it seems to be worrying Kent, still.
    He now writes-
    "For finding the apparent distance (what I believe George defines as d) my 
    model does the following:
    Obs. distance + corr. for index error +/- SD for the moon +/-SD for the sun 
    (if used) +/- corr. for augmentation of the moon +/- refraction correction 
    for the moon +/- refraction correction for the sun (if used)."
    Yes and no. I agree with all that except for those last two terms. This is 
    not the place for refraction correction to come in.
    "The correction for augmentation is based on the arguments moon's 
    semi-diameter and apparent altitude. Is this what George means... "but 
    neglected to do so in correcting the lunar distance for semidiameters (when 
    it's crucial)."??? Is the "augmentation factor "someting else?"
    No, nothing else, that's exactly what I was referring to. As far as I could 
    tell, Kent took his Moon semidiameter from Henning Umland, but then omitted 
    to adjust it for augmentation, when correcting to obtain d. We are in 
    agreement about Moon SD (as it would have been measured from the Earth's 
    centre) as 15m 25s, but that valued should then be "augmented". I had done 
    that by calculating a factor to multply by, which was 1.0147, as explained 
    in [5615], but augmentation could just as well have been taken from a table 
    in Norie's, which instead gives it as an amount to be added, of about 13 
    arc-seconds. It appeared that Kent hadn't applied that correction to SD when 
    obtaining d from observed lunar distance, which seemed to explain the 
    difference between us rather well. If I've misunderstood, I hope Kent will 
    put me right.
    Next, back to the first part of his mailing, about a discrepancy in our 
    values for Moon parallax. That seems to be simply explained by a slip (easy 
    to make) of a degree in the altitude.
    But let's touch, now on what Kent's recent message states-
    "What my model does is as per below. As can be seen the calculation includes
    two corrections for earth flatness (maybe a better English term is
    - 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."
    I'm not familiar with that correction term, 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 
    He continues-
    "Next is a small correction to the moon's HP with the arguments latitude and
    HP. This gives a "HP" of 56m 36s - correction 0,78s = 56m 35,22s."
    This seems to correspond with the table in Norie's headed "reduction of the 
    Moon's horizontal parallax", which I mentioned in my last post-
    "Now, if we're bothered to, we can make the correction for the reduction in 
    the Moon's HP on account of the spheroidal shape of the Earth, described by 
    Kent as "Earth flattening. At such a low latitude of 15�, this amounts to 
    only .01' (taken from a table in a modern Norie's), so we end up with a 
    corrected HP of 56.60' [or 56m 36 sec]".
    That value is within a second of Kent's result; which isn't surprising 
    really, because the only difference between our procedures was that I had 
    ignored a correction that turned out to be perfectly negligible.
    I asked Kent for some other quantities he used in his calculation, and he 
    came up with-
    "My input data were:
    - moon's refraction -29,82s
    - sun's refraction -1m 21s
    - sun parallax 7,3s"
    For two of those, the Sun parallax and Moon refraction, he and I agree 
    For Sun refraction, however, we diverge a bit, as I had made it 1m 30s. That 
    was a bit sloppy, as I simply hadn't bothered to make the correction for 
    atmospheric temperature and pressure. That was a mistake, especially as 
    Jeremy had reported such a high temperature, of 98�F, which implies that the 
    correction is well worth making. I had simply taken the refraction 
    "straight" from the table in the Almanac, to be 1.5 arc-minutes. If I'd made 
    the necessary correction (which was on the borderline between 0.1 and 0.2), 
    it would have reduced the Sun refraction to 1.3 or 1.4 (1m 18 sec or 1m 24 
    sec), very much in line with Kent's value.
    For working a lunar to highest sccuracy, the steps in refraction, of 0.1 
    arc-minutes, in the table in the Almanac, are rather coarse. At the 
    high(ish) altitudes we are looking at, refractions are very predictable, and 
    known to better precision than that table implies. Norie's has a table which 
    gives refraction, and its corrections for temperature and pressure, to an 
    extra decimal place, and that table predicts a Sun refraction of 1.37 
    minutes, or 1m 22 s, almost identical with Kent's figure.
    So in the end, by a process of give-and-take, I think that between us, Kent 
    and I have eliminated all our differences, down to a very few seconds. If he 
    thinks that any remain, no doubt he will say so.
    For me, it's been a rather informative exercise. I hope Kent feels the same.
    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.
    Navigation List archive: www.fer3.com/arc
    To post, email NavList@fer3.com
    To , email NavList-@fer3.com

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